Download "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]"

"videoThumbnail Нейроинтерфейсы #2 / Алексей Осадчий [MADE]

Углубленное программирование на С С++ Лекция 6

Углубленное программирование на С С++ Лекция 6

Channel: VK Team

Машинное обучение в продакшене #5 / Михаил Марюфич [MADE]

Машинное обучение в продакшене #5 / Михаил Марюфич [MADE]

Channel: VK Team

Product Management [Технострим]

Product Management [Технострим]

Channel: VK Team

7. Введение в Python. Курс "ВВЕДЕНИЕ В АНАЛИЗ ДАННЫХ" | Технострим

7. Введение в Python. Курс "ВВЕДЕНИЕ В АНАЛИЗ ДАННЫХ" | Технострим

Channel: VK Team

Продуктовые гипотезы | Курс Product management | Лекция 4

Продуктовые гипотезы | Курс Product management | Лекция 4

Channel: VK Team

Алгоритмы и структуры данных #4 / Григорий Шовкопляс [MADE]

Алгоритмы и структуры данных #4 / Григорий Шовкопляс [MADE]

Channel: VK Team

11. Введение в SQL. Курс "ВВЕДЕНИЕ В АНАЛИЗ ДАННЫХ" | Технострим

11. Введение в SQL. Курс "ВВЕДЕНИЕ В АНАЛИЗ ДАННЫХ" | Технострим

Channel: VK Team

Нейроинтерфейсы #5 / Алексей Осадчий [MADE]

Нейроинтерфейсы #5 / Алексей Осадчий [MADE]

Channel: VK Team

Advanced Scala #1 / Корольков Алексей [MADE]

Advanced Scala #1 / Корольков Алексей [MADE]

Channel: VK Team

Глубокое обучение в компьютерном зрении #8 / Иван Карпухин [MADE]

Глубокое обучение в компьютерном зрении #8 / Иван Карпухин [MADE]

Channel: VK Team

лекции по программированию

уроки программирования

как начать программировать

курсы программирования

мобильная разработка

анализ данных

технострим

linux

java

базы данных

golang

администрирование linux

алгоритмы и структуры данных

алгоритмы

тестирование по

нейронные сети

с++

hadoop

kotlin

android разработка

веб разработка

python

00:03:42

and well no more about

00:03:44

and accordingly this is the periodic

00:03:47

stimulation causes on the cortex

00:03:50

oscillations, that is, such a periodic

00:03:53

activity

00:03:55

with a frequency equal to the stimulus frequency at

00:03:58

whichever this one is located

00:04:00

moment in time in sight, here you go

00:04:04

because we are talking about one way or another

00:04:07

rhythms then without Fourier transform without

00:04:10

We cannot do without general frequency analysis

00:04:12

so now I’ll tell you a little

00:04:15

here about

00:04:16

frequency analysis harmonic analysis

00:04:19

but you probably know everything there and I don’t

00:04:21

I know someone at school who had it

00:04:23

Institute about the hordes of Fourier and that any

00:04:27

Let's assume the function is like this

00:04:29

periodic function was like this

00:04:33

be approximated by the red curve

00:04:35

here shows the approximation it can

00:04:38

be approximated by

00:04:39

set of harmonic functions

00:04:41

that is, a harmonic set

00:04:43

harmonic functions of sines and

00:04:45

cosines which

00:04:48

have frequencies that are multiples of frequencies

00:04:51

and the frequency

00:04:53

inversely proportional to base frequency

00:04:55

inversely proportional to no period

00:04:59

here, that is, 1 hour is our 2

00:05:02

pi over 2 frequency 2 pi over 2 multiply by

00:05:07

2 pin you and so on and

00:05:10

this frequency is essentially in we are here two and

00:05:13

therefore we call it the circular frequency but

00:05:15

you can do it without 2pi then it will be easy

00:05:17

something in hertz and

00:05:20

these are the coefficients a and b which

00:05:23

scales the sine contribution accordingly

00:05:27

1 from cosine 1 there sine 2 cosine 2

00:05:32

they are calculated using certain formulas

00:05:34

which I won’t explain to you where it comes from

00:05:36

that's all it takes, the only thing I'll say is that

00:05:38

if you look at this whole thing correctly then

00:05:41

it becomes quite easy to understand

00:05:44

these coefficients a and b are nothing more than

00:05:47

just expansion coefficient

00:05:49

representation of our signal

00:05:51

in a different basis in a certain one I'm afraid

00:05:55

this harmonic basis they policy

00:05:58

which we are all accustomed to being one and

00:06:00

then all zeros then 01 again all zeros

00:06:03

to where our actual reports are

00:06:06

the meaning of these signals

00:06:08

moment in time let's start drawing a picture

00:06:10

and more about that later, but just so you understand

00:06:13

and in essence this is an operation if

00:06:17

you look at her carefully then you

00:06:19

you will understand that they generally say scalar

00:06:21

product between signal x from t and

00:06:26

here are the basis functions and in

00:06:31

depending on how much this

00:06:33

signal what is a scalar product

00:06:36

this is the modulus of this signal multiplied by

00:06:37

module signal and multiplied by the angle between

00:06:40

these signals, respectively, than

00:06:43

the smaller the angle by the cosine of the angle, the smaller

00:06:46

the angle, the greater the cosine, the more our

00:06:48

our our basis function is more like

00:06:51

to this our basic signal signal

00:06:53

which we want to expand and the more

00:06:55

will be the contribution of this basis function to

00:06:57

description of this signal which will be expressed in

00:06:59

relatively large value

00:07:01

coefficient a well, the same thing about b

00:07:04

the only thing is that cosine and sine are like you

00:07:06

you know they are shifted relative to each other

00:07:08

friend on pi in half and

00:07:10

they are very effective that's the combination

00:07:13

namely cosine and sine with the same

00:07:16

frequencies but with different coefficients

00:07:19

very expensive allow you to fight with

00:07:23

shifts hit the signal, that is, when

00:07:25

I'll call you, the signal moves into phase

00:07:28

it doesn’t match like this

00:07:30

Odessa sinusoid.

00:07:33

and she is there somewhere, this one here yes.

00:07:36

the signal starts to grow then

00:07:38

correct selection of coefficients a and b

00:07:40

allows us to move the sum of the cosine and

00:07:43

sine provided that it is all the same

00:07:45

harmonic oscillation remains but with

00:07:47

the phase which is determined

00:07:50

from the base that is determined

00:07:53

I think there's a picture here

00:07:55

that is, this is our hesitation

00:08:00

video of sines gtb on cosines yes this is

00:08:03

in fact the roots are hidden in a square

00:08:06

multiply by sine x plus this is the phase

00:08:08

which depends on the attitude

00:08:10

coefficient fight coefficient otherwise

00:08:13

it is clear that this is also a harmonic

00:08:15

the destructor is also arranged or cosine

00:08:18

but shifted in phase by a certain

00:08:20

size now if you consider

00:08:23

what is essentially the calculation of these curia

00:08:26

coefficients but this is something other than

00:08:28

really just a transition to another

00:08:30

basis or if you present your

00:08:32

signal by some discrete reports on

00:08:35

gates in time

00:08:38

calculation of Fourier coefficients

00:08:40

transformation

00:08:41

or

00:08:43

series of Fourier series detailed being

00:08:46

let's talk about this essentially multiplying some

00:08:49

ballista of their sequences on yours

00:08:53

input signal that is, look here

00:08:55

0 0 coefficient but here sold itself

00:08:59

large marked there it was t0 it

00:09:01

actually averaging over everywhere

00:09:04

the coefficients are the same they are equal

00:09:05

one by 8 in this case here is one

00:09:09

the eighth is here and this is this

00:09:11

multiplying this guy by this one

00:09:12

vector will give you by and large

00:09:14

the average value of these X's up to

00:09:18

from 0 to 7

00:09:21

then you project this vector of yours onto

00:09:25

like this to a sine wave slow and on

00:09:27

sinusoid and look at what it is bigger for

00:09:29

similar to essentially counting a scalar

00:09:31

product further by sine wave

00:09:33

quickly then even faster then again

00:09:35

quickly, but it’s interesting that

00:09:38

the next sinusoid in frequency despite

00:09:41

to the fact that formally the index of our

00:09:43

components often grow until they no longer grow

00:09:45

and the inverse frequency decreases from frequency

00:09:48

last last sequence

00:09:51

coinciding with frequency not 1 here is the second

00:09:55

follower as you can see yes

00:09:57

the only thing you can reverse

00:09:59

note what's here

00:10:00

these are the imaginary components she is here

00:10:02

negative varnish and the year is here

00:10:04

positive sine this is unity

00:10:06

now you'll see why this is so

00:10:09

happens but this is so that we understand that

00:10:12

fingers up to what this is practically a transition

00:10:14

from one basis to another now well

00:10:19

How are these coefficients derived? Here's everything

00:10:21

there is an output that shows

00:10:23

which, but in reality it’s certainly not more convenient

00:10:25

use them cosines sines a

00:10:27

just use Euler's formula and

00:10:29

imagine the complex exponent here

00:10:33

such as according to Euler's formula as cosine

00:10:36

these are the frequencies 2 t na t where n is

00:10:40

index of our coefficient

00:10:43

about frequencies

00:10:46

plus carnisme titmouse jay on sinus 2 g

00:10:50

n a t t i

00:10:53

further if we imagine we will say that

00:10:56

are you okay I want to submit my signal

00:10:58

x from t as a certain sum, let it be infinite

00:11:02

but as a certain sum of cent coefficients

00:11:05

multiply by e powers like this

00:11:09

for this exhibitor's complexion

00:11:10

it is clear that these coefficients are now

00:11:14

case for example if x is from t

00:11:16

real these coefficients will be

00:11:18

comprehensive and

00:11:21

real part of these coefficients

00:11:23

will be responsible for the cosine will

00:11:25

scale cosines are frequencies and

00:11:27

and frequencies a

00:11:29

the imaginary part will be responsible for

00:11:31

Sine scaling here and there

00:11:34

if you just substitute

00:11:36

your x from t

00:11:42

that's the expression then you can it's you

00:11:45

you can do this quite easily

00:11:48

such passer actions and come to

00:11:50

because in fact NTP in a cent is

00:11:55

in essence this is what it is

00:11:57

expression

00:11:58

projected the projections of our vector

00:12:00

x from t based on the sequence

00:12:03

but it’s complex here, well, essentially it’s

00:12:05

separately imaginary part separately

00:12:06

the real part and it's enough

00:12:09

do as shown here only on

00:12:10

interval of one period

00:12:13

from 0 to

00:12:15

your lectures will remain in March, that’s all

00:12:17

follow but it's pretty standard

00:12:19

output when you use expression

00:12:22

what you want to get and

00:12:27

represent to the original and then

00:12:29

come to the rural expression for

00:12:31

coefficient

00:12:32

in essence this is a system of linear

00:12:36

equations And

00:12:38

as a result you get these

00:12:41

coefficient centers which are essentially

00:12:43

equal to the projection of your signal onto the cosine

00:12:48

and and by sine and yes and that is the imaginary part

00:12:51

center this projection onto the sinus begins

00:12:54

and projection a

00:12:56

really and soar projections on

00:12:59

cosine well, since they really

00:13:01

and gave a comprehensive excuse from a comprehensive

00:13:04

number e 2 component of a complex number

00:13:06

there is a modulus of number and

00:13:09

a complex number has

00:13:12

argument to the corner essentially here and there

00:13:16

if you have some omega purity km

00:13:18

Already watched it here with your sines

00:13:21

gtb cosine mega real part b

00:13:25

the real part was the module here

00:13:28

this tape is essentially the root and behind

00:13:30

square b square a phase is essentially

00:13:34

tangent inverse tangent arctangent b

00:13:37

divide by

00:13:40

content

00:13:43

but only

00:13:45

index then this is how it works

00:13:50

interpret it this way

00:13:52

we can draw

00:13:54

meaning a n because they are normal

00:13:57

real numbers and arrange them as

00:14:00

function from this here are the frequency numbers

00:14:02

components, well, for example, so and so

00:14:06

that is, that is, you can draw like this

00:14:09

pictures and

00:14:11

here we see that we have our signal

00:14:15

can be represented by a set of sines and

00:14:18

cosines and these sines and cosines

00:14:23

amplitudes

00:14:24

this is how it changes, that is, these are the root and

00:14:26

law plus b square is amplitude

00:14:28

range

00:14:29

called a

00:14:31

their phases are arctangent b divided by

00:14:35

so this is the zero phase

00:14:37

constant here, put on the phases here are different

00:14:40

depending on so that we can describe our

00:14:43

signal and here's what else you need and what for

00:14:46

you need to pay attention that because of this

00:14:49

that we have a periodic signal here

00:14:53

the number of these coefficients are these

00:14:54

the coefficients are so discrete yes

00:14:56

. we have a limited selection of these

00:14:59

coefficients because we actually

00:15:01

we take the signal in this dimensional

00:15:04

space and project it to

00:15:08

other basis vectors and we are completely

00:15:10

describe the signal in this way when

00:15:12

using a finite set of coefficients

00:15:14

it's just like going from one fighter to

00:15:17

there is nothing else like this here

00:15:20

there is nothing criminal here

00:15:21

complex

00:15:23

this becomes especially clear if you

00:15:26

look at our original signal

00:15:28

which is continuous and like this

00:15:30

dotted line phreak dotted line

00:15:31

indicated by a line and these are the reports

00:15:34

which we take

00:15:35

this is actually the meaning of

00:15:39

which we work when we deal

00:15:41

with data processing on a computer, yes

00:15:43

yes we do not process continuous

00:15:44

signal and we process them

00:15:46

versions of his time were discredited and

00:15:48

here we usually choose something else

00:15:52

scripting

00:15:53

he accordingly

00:15:56

characterizes the sampling frequency

00:15:58

which is the reciprocal of this interval

00:16:01

mine will often be designated FSS RF

00:16:04

water sampling it is measured in hertz well

00:16:07

for example, if we take the original

00:16:09

signal 2pi f 0 3 0 this is the current frequency f

00:16:13

0 to physical frequency yes that is it

00:16:15

hertz everything is fine you continuous time

00:16:18

we measure it in seconds, then further if we

00:16:21

let's take the countdown of this from a t in moments

00:16:25

time multiples of a large interval

00:16:28

sampling it was clear

00:16:31

insert we will get that from intel snt

00:16:34

essentially equals 2f 0 divided by f frequency

00:16:39

sampling on n yes now

00:16:42

look

00:16:43

it turns out that the digital system which

00:16:46

at the entrance to ourselves when we want to lead our

00:16:50

signal our system we're done

00:16:52

sampling and after this

00:16:54

sampling or during such

00:16:55

sampling system looks at

00:16:58

processes from the point of view and judge them

00:17:03

the speed of these processes these

00:17:04

processes in terms of frequency

00:17:07

sampling

00:17:08

that is, for example, if the frequency

00:17:11

sampling high then the same

00:17:14

process the same harmonic for example

00:17:17

will require many points to

00:17:21

so which month in one period

00:17:23

this harmonic and

00:17:25

then from the point of view of the system it is

00:17:28

quite a slow process because

00:17:30

points then a lot here she is many, many times

00:17:32

manages to open his eyes while the signal is here

00:17:35

so it changes smoothly and slowly

00:17:37

on the contrary, if we imagine that for example

00:17:40

we will pass these reports there and take each one

00:17:42

3 before, here you can already see what we are opening

00:17:46

our eyes were zero then they opened

00:17:48

We opened our eyes at most later

00:17:50

our eyes already have a negative signal

00:17:52

there we opened our eyes, we are already big

00:17:54

we have a lot of negativity in the hood

00:17:56

again positive then here already at

00:17:59

We are already alone at this sampling rate

00:18:01

and the same sinusoid which has the same and

00:18:04

the same physical frequency f0 it to the system

00:18:08

seems fast faster at least

00:18:11

at least than in the first case and this is it

00:18:14

frequency f 0 divided by fs is

00:18:18

dimensionless frequency is called but

00:18:20

in different ways from different books in different ways

00:18:23

sometimes they also multiply by 2pi

00:18:25

it's called normalized it's so it's

00:18:27

what do you call normalized

00:18:29

digital is pure and we can essentially

00:18:32

write down that our sequence of

00:18:35

1 we can you remove it because well

00:18:39

By and large, for the system it’s not you

00:18:41

we don't need you for processing

00:18:43

I need it here

00:18:45

i garden n equals sine omega n such

00:18:49

digital

00:18:50

by and large

00:18:52

that's

00:18:54

in this case, that is, I'm just now

00:18:56

I'll be back anodes, yes, that's if we're in the case

00:18:59

continuous signal see my dom and

00:19:02

continuous signal look please contact me

00:19:04

represented as the sum of our complex

00:19:07

exponent yes, that is, sines and

00:19:12

cosines of the favorite part and cosine

00:19:15

indeed, and these are the coefficients we

00:19:17

calculated as the integral from 0 to 3 x of t

00:19:21

but these basic functions, oh well

00:19:23

paired version and

00:19:25

here the integral is actually this one

00:19:27

replaced the procedure for us with this and there it is

00:19:30

definition of the scalar product of two

00:19:31

functions essentially because we have

00:19:35

time interval they sample and

00:19:37

no, practically you are with us endlessly

00:19:39

infinite between two points you are always

00:19:43

there will be others as close as you like

00:19:45

there will always be another one. means that you

00:19:48

so continuous that's why we

00:19:50

used the integral in the case when we have

00:19:53

for example sampling interval but one

00:19:55

chose well, let's assume to the origins

00:19:56

it doesn't matter for

00:19:59

period of infinity let's just

00:20:02

Let's assume that our period is equal to

00:20:04

our infinity signal then we can

00:20:08

like this one

00:20:10

here is an analogue to this one above

00:20:13

enter the coefficient

00:20:15

this is from omega this is just discrete

00:20:20

Fourier transform

00:20:22

this is a function of frequency because

00:20:24

depending on what frequency you have

00:20:26

our harmonics we will have different

00:20:28

the value of the scalar product is now

00:20:31

look at this pure scalar

00:20:34

product of two vectors one director

00:20:36

complex valued e to the power between

00:20:38

unnecessary g n that is, to the cosines of g n

00:20:40

plus the wife has sinuses and this day, well, we don’t

00:20:44

we have neither Odessa valid

00:20:45

signals and the son really and in the case

00:20:49

real data when we have

00:20:51

they are real data of course of course

00:20:53

they they are limited by hryvnia and so here

00:20:55

this amount goes there from 0 to there n

00:20:58

big for example where is it big

00:21:00

data quality what points of our data

00:21:04

this is what this thing is called

00:21:07

discrete Fourier transform to

00:21:09

you can see some books here

00:21:12

Here one to the root of 2 p or for example

00:21:16

unit per

00:21:19

Well, yes, here you can see the root of

00:21:21

two as a rule, this comes from

00:21:24

that

00:21:25

we just define it differently

00:21:27

transformation is important so that these two

00:21:30

pairs it is important that finding sata coefficients

00:21:33

mega further calculating restore our

00:21:36

signal back we will get our signal

00:21:39

there is no doubt about the original correctness

00:21:41

Ravan's headquarters to achieve this for us

00:21:43

you just need to take into account the module, but here’s the norm

00:21:46

these basis functions it is easy to show that

00:21:48

the norm of this disease function is equal to one

00:21:51

to the root of 2 and therefore it can be

00:21:52

take into account here they are with the root of two pi

00:21:55

and then all your functions will be normal 1

00:21:58

and in fact you will project your

00:22:01

son on a normalized basis and even then

00:22:04

you it will be just a coefficient if

00:22:06

if you don't do this, then it's the opposite for you

00:22:08

transformation in which you from your

00:22:10

signal from your förö conversion

00:22:12

discrete transform

00:22:15

this is called correctly in English

00:22:17

called descried time free transform

00:22:20

because we have discrete time

00:22:22

the frequency as you can see is continuous and therefore

00:22:25

for the reverse conversion we

00:22:26

we use the integral instead of the sum but on

00:22:29

in practice, of course we have three nights

00:22:30

I use discrete sums but like this

00:22:33

it will just be easier

00:22:36

it will be easier for the person to study

00:22:38

look at the thing so let's

00:22:40

let's get used to the fact that it's the opposite

00:22:42

Fourier transform it is given through

00:22:45

integral where you integrate

00:22:48

set of coefficients depending on purity

00:22:51

and essentially multiply each of these

00:22:55

coefficients on its basis function and

00:22:57

just like discrete Fourier series

00:23:00

continuous lady free you these guys have

00:23:03

you had a fixed period you them

00:23:05

sum to describe the signal

00:23:07

exactly the same here you are, these here

00:23:09

basis functions to a large extent

00:23:11

scale by s from Oleg and sum up

00:23:14

only you do it in integral

00:23:16

sense when you have a delta mega essentially

00:23:18

tends to zero and these are the organs

00:23:20

you can set the signal back

00:23:23

pair

00:23:24

discrete Fourier transform

00:23:27

time why is it called

00:23:28

discrete time because here I am

00:23:30

I repeat because there is only time

00:23:32

discretely this time is still simple

00:23:34

discrete non-Fourier rule like here

00:23:36

it is written but it is implied that then

00:23:38

there both time and frequency are discrete

00:23:41

specially heel thing decided to leave to

00:23:43

don't complicate the review there

00:23:46

circular convolutions and so on are essentially

00:23:48

doesn't change in the book you can always

00:23:50

look

00:23:51

here we will also talk about spectral

00:23:54

power density but we're probably getting there

00:23:56

let's go back let's look at some more now

00:23:59

since my example is about the fact that it’s a vector

00:24:02

data that you observe from not

00:24:04

there must be a sinusoid that

00:24:06

Anything could be this is our signal b and

00:24:11

you can imagine it as in

00:24:14

ordinary basis in which you have

00:24:15

along each direction

00:24:18

largely postponed

00:24:21

signal magnitude at the moment in time

00:24:24

Well, let's say there is 0 moment in time in the first

00:24:28

and in the second and so on here and there you have

00:24:31

such in this multidimensional n-dimensional

00:24:33

space where there are quite a lot of them

00:24:34

it turns out essentially the radius vector and .

00:24:37

which largely corresponds

00:24:40

your temporary descendant who are you

00:24:41

you watch and it’s clear

00:24:44

looking at this thing from this point

00:24:46

point of view you can imagine this point

00:24:50

not as the sum of these basic

00:24:53

vectors, yes, that is, which are summed up

00:24:54

to it we are scaled and reached

00:24:57

which we scale the court with these

00:24:58

coefficients

00:24:59

and you can choose any other ballz

00:25:01

Well, in particular, the basis of vectors which

00:25:04

represent these harmonic

00:25:05

functions that is fireplace sines and cosines

00:25:07

in different numbers that's all there is

00:25:11

Fourier expansion

00:25:13

Here you go

00:25:15

if we return to this ours again

00:25:19

experiment yes yes now we already

00:25:21

maybe we can be a little more conscious

00:25:23

look at the frequency analysis here

00:25:26

we presented this situation

00:25:29

such an input signal was recorded for a person

00:25:32

him data from the primary visual cortex

00:25:35

And

00:25:38

what we did next, what we calculated next

00:25:43

amplitude spectrum

00:25:44

this is essentially xlt amount up to e in

00:25:49

degrees

00:25:50

2 pi and desampling for masha

00:25:53

remember we will reach our normalized frequency

00:25:57

m are our amplitude coefficients a

00:26:00

these are our angles this is our phase phase our

00:26:05

coefficient that tells us about

00:26:07

signal delay in relation to me

00:26:10

they say the countdown begins and if you

00:26:12

look at the distribution of these

00:26:15

amplitudes as amplitude relative to

00:26:18

cleanliness yes it depends

00:26:20

what phase was the input signal yes

00:26:24

here he is, fate means they have begun to look

00:26:27

for this

00:26:29

blinking thing she started blinking with

00:26:32

certain phases started here

00:26:33

blinking with such a phase started here

00:26:36

blink this phase is not broken here

00:26:38

such a phase, well, actually, just not

00:26:40

on the contrary, sorry, this is what an incentive

00:26:42

blue is what a stimulus and this is the answer yes

00:26:45

see here we actually started with

00:26:47

phases, well, which one do we need to determine 90

00:26:50

degrees, yes, that is, and here it’s over 90

00:26:53

degrees and the answer is still half a period

00:26:57

or add how much and your answer begins

00:27:00

almost from zero phase 3 quarter period

00:27:04

passed during this period the answer begins here

00:27:06

just like here is the phase

00:27:08

stimulation, you see it begins here

00:27:11

from here, that is, it's about half

00:27:13

date transfer for about half well

00:27:15

and it turns out that this frequency alone goes

00:27:17

Jeff's response here is also late, but

00:27:20

delayed in utero by half a day on the floor

00:27:22

the periods are therefore approximately equal to the phase

00:27:24

exactly so, but here it’s not, but here it’s appearance

00:27:26

we start with the negative phase, yes, that is

00:27:29

the signal already looks like here

00:27:32

and at the exit we have a detainee again

00:27:35

that is, still scrolling the signal in

00:27:38

time and here is some kind of phase

00:27:41

You can depict it all like this

00:27:43

graphics and

00:27:46

look see what's shown here

00:27:48

Well, it’s clear that these are the amplitudes, but

00:27:50

here we also have delayed amplitudes

00:27:53

amplitude is actually length

00:27:55

the distance from each of these points to

00:27:58

center and each one is . this is actually a

00:28:04

multiplied but in polar coordinates yes

00:28:06

that is, as it were, and this is the radius vector

00:28:09

which goes from the center to this point a

00:28:12

this is the actual angle from the origin

00:28:15

and now you can see that what if stimulation comes with

00:28:19

0 out of 0 the rest with 0 phase then we see

00:28:22

phase phase response around zero if

00:28:25

stimulation occurs around 90 degrees, which is

00:28:28

here we see the phases like this is equal to 90

00:28:30

then that is, you can display it like this

00:28:34

the brain's response to this stimulation

00:28:36

and thus it turns out that it is possible

00:28:39

take into account not only

00:28:41

at the date of the signal, that is, as it were

00:28:44

these here stimulation can be different

00:28:46

frequencies but also different different phases too

00:28:51

allow you to encode information if

00:28:54

say 1 part 1 part screen

00:28:57

illuminated at the same frequency

00:28:59

but from one phase and the other part of the screen is in

00:29:02

different frequency but different with the same

00:29:04

frequency of another phase then theoretical

00:29:06

you can distinguish between each other this too

00:29:09

used in

00:29:10

interface

00:29:13

well you see here when we talk about

00:29:16

spectrum but in general but purely

00:29:18

harmonic analysis of furies analysis of all

00:29:20

these things and frequencies cannot be bypassed

00:29:23

question related to filtering I just

00:29:26

I don’t know how familiar you are with the right

00:29:27

I will try to talk about filters very briefly

00:29:30

tell them, well, you probably already

00:29:33

encountered filters even if

00:29:35

think you haven't encountered it for sure

00:29:37

ever dealt with a signal with

00:29:40

some just follow the countdown experience and

00:29:41

anything could be there

00:29:44

set of temperature values or not there

00:29:47

I know there is a signal to sit down, are we sensor or well

00:29:50

anything can be sound

00:29:51

again but also if you saw on this

00:29:54

sound and signal such a beard

00:29:57

high frequencies fast changing you

00:29:59

wanted to get rid of her and you calculated

00:30:02

so-called moving averages

00:30:04

there you ran at your signal and

00:30:06

actually connected neighboring reports

00:30:09

connected then shifted by one

00:30:11

this very little window is yours again

00:30:14

connected, calculated value, and so on

00:30:17

and so they ran all over your

00:30:19

the signals were thus smoothed out

00:30:22

the result is essentially filtering

00:30:25

who are you in this situation?

00:30:28

thatcher wear filtering which

00:30:30

emphasized low frequencies and even killed

00:30:32

tried to absorb these here to suppress

00:30:35

these high-frequency noises, this noise

00:30:36

what do you really call it?

00:30:39

very bad filter can be used

00:30:40

much more correct filters that

00:30:43

will have certain

00:30:44

characteristics during operation of which

00:30:46

it will be predictable and it will not be

00:30:49

drag for example a ban

00:30:51

drag the result of such smoothing

00:30:55

for example higher frequency signals

00:30:57

why such a simple scheme

00:30:58

may happen here accordingly for

00:31:00

a whole big theory has been developed for this

00:31:03

digital filtering was first

00:31:05

theory is just analog filtering

00:31:07

then after it appeared

00:31:09

the ability to use computers for

00:31:12

signal processing and

00:31:14

processing is already practically

00:31:16

Total

00:31:17

carried out on a computer, that is,

00:31:19

you started with an ADC system that

00:31:22

transfers the analog signal to digital further

00:31:24

you process with 40 filters

00:31:26

and then analog is applied back

00:31:28

converter

00:31:29

if you need an analog signal

00:31:32

for example to send headphones to your

00:31:34

player here and here is the filter and

00:31:37

characterized by the so-called

00:31:40

amplitude-frequency response

00:31:42

in English it's called

00:31:44

April spawns or you can by you

00:31:48

hear these transfer functions

00:31:49

amplitude of the transfer function that is

00:31:52

This is what we're talking about here, let's assume

00:31:54

do you want to arrange such processing so that

00:31:57

signals starting from a certain frequency

00:31:59

your filter didn’t let through f-stop but

00:32:03

at the same time, so that it passes signals until

00:32:05

some frequency f pace and so on

00:32:08

you say that you are in bandwidth

00:32:11

want your filter to pass the signal

00:32:13

and may have some irregularities

00:32:16

here yes because this task

00:32:18

optimization you can actually think about

00:32:21

that you are trying to write here

00:32:22

spline that lives according to its own laws and

00:32:26

the more freedom you give him

00:32:28

here from him spline y those sorry those

00:32:31

steeper drop you can

00:32:34

provide here but generally speaking they went to

00:32:36

this transition strip from the last one

00:32:39

purity of the highest frequency in the band

00:32:41

transmission what is it called?

00:32:44

first frequency in band delay than

00:32:47

this transitional transitional piece he

00:32:50

the more solid the better the filter is not

00:32:53

you always need them like this but in general

00:32:55

speaking very often I want to

00:32:57

width transition transition transition

00:32:59

the stripes were quite low, the sky was small

00:33:02

here it comes

00:33:05

there is something we have just looked at

00:33:07

what is the transmittance here?

00:33:08

filter

00:33:10

units until sometimes but because

00:33:13

such engineering science with these filters

00:33:15

people love it very much

00:33:17

logarithms is all to ask, that is, this

00:33:19

the magnitude is now decibels is 10

00:33:22

logarithms, this is essentially hrtf and

00:33:26

regarding everything from f to wallet filtering

00:33:29

after filtering the signal power in

00:33:32

range before filter power cycle after

00:33:34

filtering, well, then it’s clear logo

00:33:37

the logarithm is zero then you have a band

00:33:39

transmission here and the logarithm

00:33:41

some small number that

00:33:43

by which components are multiplied

00:33:46

higher frequency devices read and delay

00:33:49

which components are weakened then this

00:33:53

so big from the face

00:33:56

negative logarithm number

00:33:59

there is a filter and low-frequency in which

00:34:02

us we pass filters down to yesterday

00:34:04

low frequency range

00:34:05

there are bandpass filters for precise transmission

00:34:08

You can also build a band filter using

00:34:10

the one who is delaying is simply blocking or

00:34:12

a notch filter is a filter after all

00:34:15

which also has one in the interval

00:34:18

here zero dress detention and again

00:34:22

one on the top

00:34:24

it is clear that what you require is narrower

00:34:27

the filter the more stringent the requirements

00:34:30

presented to the algorithm and especially not

00:34:34

calgary tmao quality by

00:34:35

possibilities, that is, it is more expensive with

00:34:37

computational points of view and such filters

00:34:39

may not behave very well from a point of view

00:34:43

vision

00:34:45

additional transients we

00:34:47

they will talk to you but before we

00:34:49

we can give it close so I owe you

00:34:51

lead such three, maybe even 2

00:34:54

basic basic digital

00:34:56

sequences

00:34:57

this is to follow this is just a one-off

00:35:00

impulses

00:35:01

it is written as our signal is equal to our

00:35:05

single impulse delta you she

00:35:07

indicated by a tinted unit

00:35:10

at which its height is equal to one

00:35:12

respectively

00:35:15

single jump it is zero for everything in

00:35:18

interval from minus infinity to 0 a

00:35:20

starting from zero not including 0 starting from

00:35:23

0 to pi plus infinity it's everywhere

00:35:25

fontanelles we can write it like this

00:35:28

x1 is delta 1 what what is this one

00:35:31

signal g entirely signal plus delta t n

00:35:36

shifted by one this is this one

00:35:38

here's the whole guy, here he is shifted

00:35:41

per unit to water throughout the entire interval

00:35:44

plus delta then -2 and so on is

00:35:47

it's actually critical that is

00:35:48

delta denotes non-specificity of the gurgle

00:35:51

wow, and delta means everything

00:35:54

this sequence that is, you just

00:35:56

take this sequence and shift it

00:35:58

because it is equal to zero everywhere except

00:36:01

one single point then you can

00:36:03

shift and thus compose here

00:36:05

this is unity you can do the same

00:36:07

create a more or less arbitrary signal

00:36:09

yes, just moving this del of yours

00:36:12

sequence to digital delta

00:36:14

so-called impulse and multiply it by

00:36:17

coefficients coefficients that are equal

00:36:19

signal values at these points, that is

00:36:23

let's write it like this

00:36:26

well here's an example for you

00:36:29

[music]

00:36:30

example

00:36:33

look here for an example of the design

00:36:37

filter here we see what's in the dress

00:36:40

detention we have suppression of almost 60

00:36:42

A decibel is 10 logarithms of the ratio with

00:36:46

the address can be recalculated as

00:36:49

there are some dresses

00:36:51

unevenness but still

00:36:53

coefficient is approximately equal to unity

00:36:56

because the logarithm is zero and so

00:36:59

this filter if you reverse it

00:37:02

Please note that this band filter is on

00:37:03

missing yes that is he misses

00:37:06

the signal is somewhere in the range of 10 hertz here

00:37:09

here from 9 to 12 herz so he misses

00:37:14

signal and delays the signal in all

00:37:16

other bands

00:37:18

throughout the main range it is

00:37:20

bandpass filter which we will often use

00:37:21

use to highlight something

00:37:23

rhythmic activity

00:37:27

filter

00:37:28

characterized well, for the first time

00:37:31

this one here

00:37:33

transfer function or frequency

00:37:35

amplitude-frequency response and

00:37:36

frequency phases but we don’t talk about substances

00:37:39

but also a filter from definitely

00:37:42

characterized by its so

00:37:45

called impulse responses

00:37:46

impulse response is response

00:37:49

of our filter to a single digital

00:37:52

impact that is, here it comes, it’s burning and

00:37:54

impulse and somewhere it’s at zero, yes here it is

00:37:58

here our filter is ringing

00:38:02

in jargon they call a ringing, this is the ringing

00:38:06

delayed until he his phase response see

00:38:10

that is, it also introduces some kind of delay

00:38:13

this filter, that is, us the signal itself was

00:38:15

more and it began to ring only there in

00:38:18

area there 0 9 seconds that is the maximum

00:38:22

he has this impulse

00:38:25

the characteristic is absolutely complete

00:38:28

full breasts are to the filter and on

00:38:30

actually a complete filter

00:38:31

characterizes the hour we will understand why

00:38:34

so look here

00:38:36

the thing is that the filters we have

00:38:39

the point is linear filters and therefore, well

00:38:42

here at the entrance if bed sheets are served

00:38:43

scene and which is simply delta 1 on

00:38:46

at the exit we see how as I already said

00:38:48

some last one with whom he is

00:38:49

impulse response and that's it

00:38:52

the time was the same, the delta fell out

00:38:54

they got to stall for time and the brown-haired man fell out

00:38:57

delta from m shifted by 5 counts you

00:38:59

get h&m shifted by 5 counts when

00:39:02

so-called properties of time

00:39:05

invariance, that is, like a filter

00:39:07

it doesn't change, now or in an hour

00:39:09

from 10 so that then the same

00:39:12

there is also the property of linearity and this is it

00:39:16

what if you are served at the entrance

00:39:17

Here's a sequence for example

00:39:20

just sorry we made a mistake today

00:39:23

should be delta delta tn or even

00:39:26

be x of n equals delta of n plus 2

00:39:29

delta t minus 5t delta

00:39:33

1 first at first at point pyramid a

00:39:36

the second at point n is equal to 5 up to 2nd method

00:39:40

Russian to the center 2 then at the exit you

00:39:42

get here you see linear

00:39:45

combination of 2d functions with different

00:39:46

coefficients and the output you get

00:39:49

linear combinations of impulse reports

00:39:51

impulse response characteristics

00:39:52

with the same coefficients 1st and 2nd floor with

00:39:56

taking into account these relevant

00:39:58

delays, these are the properties

00:40:01

time invariance

00:40:04

time connected to the property

00:40:06

linearity is exactly like this

00:40:09

combination of linearity or variation

00:40:10

in time gives you the opportunity to

00:40:13

effectively describe these

00:40:15

digital filters using this

00:40:17

called convolution operation you are sure

00:40:20

have you heard what a bundle is?

00:40:23

Have you ever trained a convolutional network?

00:40:25

or heard about them or read this

00:40:28

Yes only

00:40:29

It’s just clear that the packages

00:40:31

which in the context of filters is like this

00:40:33

standard mathematical operation

00:40:35

and convolutions which in neural networks are

00:40:38

exactly the same operation, well used

00:40:40

here is a fairly modern method

00:40:43

signal and data processing

00:40:45

but essentially mathematically it’s the same to someone

00:40:47

the best thing is, look, it means anyone

00:40:50

signal let's look at the input filter

00:40:54

we drop any signal as we have already seen

00:40:56

you can use the function yes

00:40:58

here is the delta sequence like this

00:41:00

then this is a single tribute shifted by

00:41:02

different points in time from

00:41:05

scaled with different

00:41:06

this can be written down as coefficients:

00:41:08

for example

00:41:09

you can imagine any

00:41:11

Shona sequence is arbitrary

00:41:13

the forms are like this

00:41:17

just take different coefficients and that's it

00:41:20

you will receive different ones

00:41:22

any just a countdown and who your data is

00:41:26

digitized your ccp gave you x0 and x1 x2

00:41:30

and so on, well, you use them for

00:41:32

so that your signals and this

00:41:35

the signal goes to a digital filter but

00:41:38

you know what to respond with

00:41:40

previous properties that from as hope

00:41:43

function with scaled and delayed

00:41:46

available with scaled and appropriate

00:41:49

way and delayed response to delta

00:41:53

function that was at point n

00:41:55

equal to zero had unit amplitude well

00:41:58

accordingly, at the exit then you

00:42:00

you can write down the output signal

00:42:02

this is essentially

00:42:05

x0 from n plus x0 x0 tinao brown-haired is not necessary

00:42:10

be up to + h 1 x 1 times h n

00:42:15

minus 1 plus x 2d two and here is two

00:42:19

yes what a deuce is this is the meaning of this

00:42:21

I will find your delta yes yes your signal

00:42:24

at point n is equal to there, that is, it

00:42:27

described using this one here

00:42:29

summer sequence shifted by 2

00:42:31

countdown yes well and here then at the exit

00:42:33

the filter we get is pulsed

00:42:35

characteristics moved by 2 counts from

00:42:37

scaled by the same factor with

00:42:40

which are scalable speakers only by

00:42:43

in the original constant the day off is here

00:42:46

further if you write this down then

00:42:48

instead of these here x0 with index 1 2 3

00:42:51

and so on but there should be mountains x0 too

00:42:53

ours to multiply yes that is if you

00:42:56

write down here and if you imagine that

00:42:59

their cattle and this is x in parentheses somehow I

00:43:01

just another sequence then you

00:43:04

you can write something like this:

00:43:06

the output to the output of the filter is essentially equal to

00:43:10

convolution of the input sequence with

00:43:14

impulse response of your

00:43:15

filter, this asterisk means

00:43:18

signal strongly signifies

00:43:20

convolution operation

00:43:23

Well, this is a package and about to be and

00:43:26

Complain it’s clear what can be changed

00:43:28

where to put he minus k where to put n

00:43:32

but here it is important that for calculating this

00:43:35

signal you, as it were, with this little window, yes

00:43:39

these h and n minus cats you you you

00:43:43

run along x to that local point

00:43:46

around this point n and here

00:43:48

it is important that dice summation is here and so far

00:43:51

two indexes k and one index n

00:43:55

that is, this is but this is the convolution operation

00:43:59

so look what it looks like here

00:44:02

this is the example about smoothing to the stirrup

00:44:07

reports so you run simply and

00:44:10

in fact, you consider each point

00:44:11

the average values of these reports later

00:44:14

avoid the window by one count and

00:44:16

again consider the average, this is it like this

00:44:19

animation

00:44:21

there and explain what any filters contribute

00:44:24

delay that works in real

00:44:27

time, that is, you don’t have the opportunity

00:44:29

first calibrate during the period and then

00:44:30

open back to compensate

00:44:32

this phase delay phase frequency

00:44:35

characteristics that were still

00:44:37

let's see here, but here you can see that here

00:44:41

our signal is like

00:44:42

this is the impulse response of filter a

00:44:45

this is the input sequence and this is

00:44:47

he is trying to suppress all these signals

00:44:50

and you filter this one like this

00:44:52

harmonic burst here and after

00:44:55

The filter characteristic is similar to that

00:44:58

the splash he's trying to make you

00:45:00

we filter at the output

00:45:02

highlight highlight yeah and on the way out

00:45:05

the result is a signal that is actually

00:45:08

has a significantly higher ratio

00:45:10

signal-to-noise if we compare this

00:45:14

the moment where is under the harmonic

00:45:16

splash

00:45:19

now look now you need it

00:45:22

connect us our

00:45:24

Temporary too, maybe there are questions

00:45:26

some

00:45:32

Can you basically ask a question?

00:45:34

something if it’s not clear there

00:45:40

you can write to chat

00:45:44

look here

00:45:46

Means

00:45:48

I'll be back now

00:45:59

I understand how to remove this thing, but well

00:46:03

let's see let's see

00:46:06

Next let's connect now these are ours

00:46:08

impulse responses of filters

00:46:09

filters and Fourier transform because

00:46:12

what is actually there to understand how

00:46:15

filters affect the signals we need

00:46:17

of course of course write down these filters and

00:46:19

the filtering process itself in frequency

00:46:22

region

00:46:23

Well, look at our code here

00:46:25

transformation

00:46:27

here we have a discrete transformation

00:46:31

discrete time free conversion

00:46:34

so it’s easy to show that it’s discrete

00:46:37

descried time free transform

00:46:39

sequence shifted by d

00:46:40

then we simply write down the readings according to

00:46:44

by definition, just this guy

00:46:46

put it here and get dreams

00:46:48

we insert this guy here

00:46:50

we get

00:46:52

CIS further we

00:46:55

we replace the variable because by

00:46:57

summation from between is infinite

00:46:58

infinity then we replace the minus here

00:47:03

d + d we write yes and then we have

00:47:06

free hanging ftp with knives gd

00:47:09

which we pull out for the amount of ships

00:47:13

for the sum sign and inside the sum we have

00:47:15

it turns out here Yandex and here the index

00:47:18

complexes exponentially the same and

00:47:20

considering that we have infinity, then this

00:47:22

yes yes of course this thing is not good for anything

00:47:25

influences and we actually say that in

00:47:27

transformation of the shifted

00:47:29

there are sequences like this one here

00:47:31

rotation factor so called but oil

00:47:34

lower megadeth driving factor

00:47:37

Fourier multiplied transform

00:47:39

the original sequence will be like this

00:47:42

this is very important it means that

00:47:45

by shifting the signal your sine wave is essentially

00:47:49

got a delay that is null

00:47:52

omega b divide by 2 n by 2 pi understand

00:47:54

yes, that is, in fact, in all of this

00:47:56

multiplication you just just naturally

00:47:59

this signal has moved, it’s clear that you

00:48:01

such a fan of these sine waves

00:48:02

avoid, well, that’s exactly what it’s about and

00:48:05

This is exactly what this thing is talking about

00:48:09

now look how we can understand how

00:48:12

us to describe the signal in the frequency domain

00:48:14

at the output of the digital filter, well

00:48:17

quite simply you take it again

00:48:19

represent your expressions for convolution

00:48:21

here you are, if I put this expression

00:48:24

yes here it is if you set it up for him

00:48:27

expression for washing here in this first

00:48:29

defining the transform form like this

00:48:32

this amount is quite easy

00:48:36

you can understand, see the transformation

00:48:40

summarized by n by reports over time and

00:48:45

not yet but accordingly here

00:48:50

as much as kataya acts simply as

00:48:52

coefficient in front of x which in turn

00:48:56

depends on n and does not depend on n

00:48:59

accordingly you can use

00:49:01

linearity transformation supreme fury

00:49:03

you can write it like this hkt and here

00:49:07

just this courier sign

00:49:09

transform apply to n minus to

00:49:11

because because it's a mess there

00:49:13

apply this simply scaling

00:49:15

understand the coefficient for everyone

00:49:17

fixed to it's just scaling

00:49:19

skoda coefficient which is on

00:49:21

which is multiplied like the last minus

00:49:23

here is our cat

00:49:26

dtf you are from X - cats, well, in accordance

00:49:30

with the previous result we can

00:49:32

replace with you d tft mediation

00:49:34

shifted on to the counts we can her

00:49:37

replace as

00:49:39

oil and 5 minutes I'm a mega size

00:49:42

countdown there was a hole here to here on

00:49:45

crown conversion of the original

00:49:48

sequences

00:49:49

xn you give so interestingly

00:49:52

it turns out this d tft Anapa n-yes

00:49:56

is done and it is actually to this dtf you

00:50:00

she has nothing to do with it either

00:50:02

actually a multiple multiplier here

00:50:04

and this in itself is also a fury

00:50:06

transformation but already pulsed

00:50:08

characteristics

00:50:09

hk you and thus and thus on

00:50:14

output Fourier transform convolution

00:50:17

fury acquired the output signal from

00:50:19

us is equal to the product of courier

00:50:23

converting the original signal to here

00:50:27

such a characteristic has already become obsolete

00:50:30

where h is actually called

00:50:34

transfer characteristic or

00:50:36

English frequency response of ours

00:50:38

filter and this is nothing more than just

00:50:40

our Fourier transform will follow

00:50:43

it's hard, look, let's do it now

00:50:46

let's go back here right here right here

00:50:48

for example, this filter is like this

00:50:51

us

00:50:52

this is this module this is this and this is ours

00:50:56

the answer is the degree of alive until it is

00:50:59

module our frequency characteristics

00:51:01

look he's focused narrowly

00:51:03

frequency range yes and then everywhere it is equal

00:51:06

zero and look at the pulse

00:51:09

This is essentially his characteristic

00:51:12

this is also a harmonic up to the frequency of which

00:51:15

central

00:51:17

coincides with the central frequency here in

00:51:20

this thing because that's exactly what

00:51:22

this is this fury transformation

00:51:25

this great characteristic

00:51:27

you see, yes, that is, because your

00:51:29

filter he

00:51:30

called it works so it highlights

00:51:33

certain frequencies are accordingly yours

00:51:36

impulse response it

00:51:38

has this characteristic shape

00:51:41

periodic and it is clear that if you are from

00:51:43

you will calculate the Fourier so that you take the floor

00:51:46

absolute value then you get this

00:51:48

such a thing that has a peak on a narrow

00:51:51

frequency range that corresponds

00:51:53

own what average period

00:51:56

this thing before the radio station

00:51:59

[music]

00:52:01

Well, here too, yes, that is, as it were

00:52:03

in order to emphasize the signal needed

00:52:06

frequency then we will take the peak here

00:52:08

in relation to the peak here we have it here

00:52:10

this relationship is much less than in

00:52:13

this after filtering because our

00:52:15

the filter had a pulse

00:52:16

characteristic

00:52:18

sharpened around a certain

00:52:21

date frequencies mysterious highlight

00:52:23

oscillations of a certain frequency a

00:52:27

so ok then look at this

00:52:31

Basically you have two methods description

00:52:34

digital filtering or this

00:52:36

time domain where the output signal

00:52:39

the signal at the filter output is convolution

00:52:42

input signal and pulse

00:52:44

characteristics

00:52:45

here it is here diagrams of the owl camera I have it

00:52:48

from here here it is

00:52:50

in addition you can in the frequency domain

00:52:53

if you take the Fourier transform from

00:52:56

x get hottel tour conversion from

00:52:59

h receive a fury interrupt error from y

00:53:02

get y and then in the frequency domain this

00:53:05

the thing is much simpler

00:53:07

look, the intuition here is that

00:53:11

actually decomposing the signal into a harmonic

00:53:15

your signal is the sum on your x from ep

00:53:17

alive years different sine and cosine and you

00:53:19

by skipping each sine you take into account

00:53:23

amplitude by which this yours is multiplied

00:53:25

the sine of this this and that is your norm

00:53:27

transfer characteristic is necessary

00:53:29

comprehensively and the phase for which it is yours

00:53:32

sine wave shifts

00:53:34

these are the two things you take into account

00:53:37

then you just drag out everything is worth it

00:53:39

collect your signal you need to go departments

00:53:42

inverse Fourier transform understand

00:53:44

Yes, what about the coefficients? Well, what about the coefficient?

00:53:46

decomposition of this signal is simple

00:53:49

decomposition coefficients of the original

00:53:51

signal with scaled according

00:53:54

with this module already

00:53:58

shifted to phase fiat mega which are

00:54:02

in fact, wearing an imaginary actual

00:54:05

parts of yours here are transfer parts

00:54:09

characteristics of your background

00:54:10

conversion from pulse

00:54:12

filter characteristics

00:54:14

Well, here the pulse is shown

00:54:17

characteristic assume here

00:54:19

shows the logarithm of

00:54:23

from module transfer functions

00:54:26

here it is equal to the constant 1000 this

00:54:29

low frequency

00:54:30

Korovin Konstantin and this interval

00:54:33

and then fades out quite sharply and

00:54:35

stops skipping and this is the phase

00:54:38

This is a direct clip in this situation

00:54:40

because we have this special filter

00:54:42

the filter we considered is like this

00:54:45

called finite impulse filters

00:54:47

characteristic of which is sleeve length

00:54:50

impulse response it is fixed and

00:54:52

equal to actually it is scary and

00:54:57

the delay here is always equal to half of

00:55:00

impulse response delay

00:55:01

the output signal is always equal to half

00:55:04

Impulse characteristics of Wagashi 515

00:55:06

I'll be back, look, they got it

00:55:07

characteristic is about to see filtering

00:55:11

between this peak this peak exactly

00:55:13

half of this one is impulse

00:55:15

characteristics

00:55:16

and since when we draw phase

00:55:20

phase characteristic there and this

00:55:22

delay

00:55:23

multiply by the purity because it is a sine wave

00:55:26

purity for example ten hertz per

00:55:29

interval there for example 100 milliseconds

00:55:32

one period will pass

00:55:34

if you delay if you describe

00:55:37

want to describe the same

00:55:38

interval of 100 milliseconds using

00:55:41

sine wave of the hour at which 20 hertz because

00:55:44

there will be two periods before, that is, the phase is already

00:55:46

there will be 24 n yes there was 254 and therefore y

00:55:51

us phase response phase phase spectrum

00:55:53

so-called he is from linear linear

00:55:56

so straight that

00:55:59

which garden is the corner of which is large

00:56:01

pants from the corner that matches like this

00:56:03

called group delay time

00:56:05

but for such filters, of course I am a cupcake

00:56:08

ready Tracy what k and x

00:56:12

to them this is this group time

00:56:15

the delay is equal to a constant it is all the time

00:56:17

constant and depends on the length

00:56:20

impulse characteristics

00:56:23

well look, it means

00:56:26

[music]

00:56:27

here were the filter with the end late

00:56:29

characteristics with you with them them

00:56:31

considered let's consider from usual

00:56:33

filters when he draws them, here he draws them

00:56:35

this is how the signal arrives at the input

00:56:38

xk you next is the delay module

00:56:41

on count 11 sometimes you will see z here

00:56:43

minus 1 index place t I won't

00:56:46

explain why this is very much for

00:56:47

outside the course here

00:56:50

here we moved one report further

00:56:53

multiplied by

00:56:55

destination filter coefficient after

00:56:57

I’ll move the crosses at this point again later

00:57:00

by one count multiplied by the second

00:57:02

meaning before but this just arose here

00:57:04

it's been a long time since it's purely convention, what's wrong with that?

00:57:06

orders here x n ok just a battle

00:57:08

nominally

00:57:10

on the world and text 0 of 0 but this detail and

00:57:14

that's it, but what's important is that you move it

00:57:15

input signal and multiply it by

00:57:17

coefficient and then sum up everything

00:57:19

here you get the output: filters

00:57:21

the ultimate holiday of interest in such

00:57:23

limited number of guys allowed

00:57:25

use this how you can people

00:57:27

figured out what filtration actually is

00:57:29

using endless pulse

00:57:31

vray characteristics are of course practically

00:57:33

not a very effective thing despite

00:57:35

that she has these

00:57:37

wonderful quality that she has

00:57:39

constant delay at all frequencies

00:57:41

the dates are all sine waves they all move on

00:57:43

the bottom is honestly one delta t but if

00:57:47

for example now here I am here you are

00:57:48

take a look at this example and

00:57:50

imagine that you have this one of yours

00:57:52

this signal is his original one

00:57:55

countdown and taken not so far from each other

00:57:58

friend and very, very, very often

00:58:00

very very often they and but we want to

00:58:03

average this over approximately the same

00:58:05

interval in time then we have ours

00:58:08

impulse response of our filter

00:58:09

should be very will be should be

00:58:12

very big

00:58:13

to long because quantity

00:58:16

readings that will be located here on this

00:58:18

the interval there is very large and

00:58:21

so according to the calculation here

00:58:23

this convolution is so long and functions up to

00:58:25

such a long sequence can

00:58:27

not be the most effective activity

00:58:29

especially in digital systems especially

00:58:31

here you can heal the device with dom.by by

00:58:33

three times plays a role that's why it exists

00:58:36

such a filter with infinite let

00:58:38

covering there's ticking who are doing what

00:58:40

they have the same block which of course

00:58:42

impulse responses but he usually

00:58:44

much shorter but further output signal

00:58:48

we are already sending the output signal

00:58:51

delay and send with odds

00:58:53

back to our filtering essentially and brother and

00:58:57

sum up the results

00:58:58

we get at the output we get no

00:59:01

only input signal we

00:59:04

delay and multiply but also take

00:59:07

previous output reports

00:59:09

you see here only minus 1 and n minus

00:59:12

there are 2 and minus q and so on is present

00:59:14

and and and the report is not present either

00:59:16

moment is what we don't think right now

00:59:18

we can use it, but here it is

00:59:20

the idea allows you to essentially

00:59:22

significantly save on quantity

00:59:24

multiplication and addition operations which

00:59:26

you need to calculate from like this

00:59:29

filter but the price for this is nonlinear basic

00:59:33

characteristics accordingly

00:59:34

inconsistent delay time distortion

00:59:37

output waveforms say if you

00:59:39

filter some heartfelt

00:59:40

you want to allocate an artifact to us?

00:59:42

recognized q then proceed to it

00:59:44

but it will be distorted, that is, as if

00:59:46

you will remove the film but at the exit for the fact that

00:59:49

different components different sines different

00:59:51

bone and immediately move differently

00:59:53

You will no longer have a filter at the output.

00:59:56

cardiac

00:59:58

artifact answer which

01:00:02

the peak that you are used to, it will be strong

01:00:05

distorted form, or not so much

01:00:07

your distorted steel is burning which but

01:00:08

sharpened can recognize this form

01:00:10

be wrong but now look but both of these

01:00:14

I think it's just that you will be theirs

01:00:15

use also they are there magician lobby and in

01:00:18

in python they are actually created these

01:00:19

coefficients a and b using one

01:00:21

teams but also coefficients a and b

01:00:24

naturally affect the characteristics

01:00:26

your filter, these filters are called c

01:00:28

infinite impulse response

01:00:30

because look technically she is

01:00:32

endlessly you any way out even very

01:00:34

small

01:00:36

submit as input but also the coefficient with these

01:00:37

zero here and then she's here all the time

01:00:40

this is the diet she keeps ringing up to

01:00:42

infinity is big, it’s clear that when

01:00:44

analysis there when calculating these

01:00:47

coefficients there and so on you

01:00:49

truncate this

01:00:50

characterization and continue to do some

01:00:53

calculations but

01:00:55

conceptually this is a filter with infinite

01:00:57

pulse

01:00:59

look what another ambush is

01:01:03

with these filters and so practical

01:01:05

almost a hack lifehack when you

01:01:10

you have a signal let's assume

01:01:12

let's look at we have a little blue one

01:01:13

gray signal is wrong we have red

01:01:15

signal

01:01:16

red signal and red signal it's here

01:01:20

such a smooth, more or less shaped one

01:01:22

we observe a noisy signal from this

01:01:24

here's a harmonica from above, they're [ __ ] like this

01:01:26

blue signal and we want to apply

01:01:28

filtering so that these harmonics

01:01:30

to filter it we make our

01:01:34

filter we find the coefficients and

01:01:36

let's use this piece of ours

01:01:38

data and we are interested to see further

01:01:40

how accurate is our original signal

01:01:44

red is reproduced by this

01:01:46

green filtered and we count

01:01:48

absolute error between red and

01:01:50

green signal for each point and

01:01:52

we draw and we see that we are at the edges

01:01:57

we see a significant increase in value

01:02:00

then mistakes happen here

01:02:03

why because when your filter

01:02:05

performs this process of filtering you

01:02:08

actually impulse response

01:02:09

see yes she should now like this

01:02:12

go to this filter until you see

01:02:15

so she gradually moves in, moves in and

01:02:18

not all reports are impulse yet

01:02:20

characteristics

01:02:21

have a corresponding response

01:02:25

signal to the corresponding comrades

01:02:28

multiply by 1000 multiply this is alien and

01:02:30

what to multiply shadows to actually

01:02:32

you multiply by 0 but this is not filtering

01:02:35

like this kind of poison or is it filtration

01:02:37

signal which has a step here and here

01:02:40

this is the ringing response of the filter to this one

01:02:42

a step actually because if here

01:02:43

this is 0 to 10 0 some kind of core this

01:02:47

and there is this transition process

01:02:49

which you see here is green

01:02:50

the signal demonstrates and generates such

01:02:53

way of deviation from the red signal to

01:02:56

coupon from grown plus what is from the original

01:03:00

that's why when you work

01:03:02

signals with some short epochs

01:03:05

That

01:03:06

always remember this edge effect and

01:03:09

or take a little more from the data

01:03:11

signals left and right where is it on the left

01:03:14

and on the right how much depends on the effective

01:03:16

length impulse characteristics of a given

01:03:18

look how the robot fades out

01:03:20

what is 100 milliseconds 200 milliseconds

01:03:22

300, well, these are 200 300

01:03:25

milliseconds flankers so called

01:03:27

left and right count from the file together

01:03:29

process, filter and then them

01:03:32

trim after filtering and then

01:03:35

you will have a clean signal without

01:03:37

any artifact

01:03:40

Well, now let's see an example of how

01:03:42

it looks like it's not an area, let's say

01:03:46

signal original blue noisy it is

01:03:49

red signal

01:03:52

added by this harmonica

01:03:54

a certain frequency which is

01:03:56

certain will be acquired and the signal enters and

01:03:58

this is the signal, you see that it’s ours

01:04:01

edge of the conversion of this signal and

01:04:03

absolute value ad-free

01:04:04

transformation she is drawn in blue

01:04:07

graph but here the red is superimposed

01:04:09

on blue therefore low frequencies it

01:04:12

actually everything matches the red

01:04:13

signal but on this one high

01:04:15

frequency you have this artifact this

01:04:18

interference you want to make a filter that

01:04:21

will absorb this interference, suppress it, that is, it

01:04:24

for example the situation if you have 50 hertz

01:04:26

forces your data here you are

01:04:29

make a filter using this command

01:04:31

to whom you give digital normalized

01:04:35

frequencies Nikita you all in detail

01:04:36

he will tell you, he will show you, but what is important is that you are here

01:04:38

give it yours will acquire frequency

01:04:40

characteristic and this amplitude

01:04:43

characteristic has has has minus

01:04:46

but small means small

01:04:48

missing people here somewhere here, that is

01:04:50

as if the interval will be 0 25 tons

01:04:53

2725 in this here you are when

01:04:57

apply this this this to your blue

01:05:01

the signal here is up to the frequency domain too

01:05:03

just multiplication remember as I lived mega

01:05:06

multiplied by x e alive this one this one

01:05:09

the blue guy is multiplied by this one

01:05:11

coefficient on a small one which is strong

01:05:13

less than one then the logarithm here

01:05:14

shown and here's the one like this

01:05:16

relaxes to see him as green

01:05:18

signals this will be a fury

01:05:20

filtered signal conversion

01:05:22

essentially this green one

01:05:25

it's still a sine wave if you

01:05:28

look carefully at the boats

01:05:30

completely depressed but of course she is

01:05:32

much weaker than in the original signal a

01:05:34

how depressed he is depends on

01:05:37

order of the filter, well, open the design

01:05:40

reste which you developed which

01:05:43

you specified in the requirement for generation

01:05:45

these coefficients b and this filter c

01:05:48

just super with endless pulse

01:05:50

shake kai and his phases you see she doesn't

01:05:53

It’s linear for some people

01:05:55

I mean it's terrible and

01:05:58

if you count the group time

01:06:00

delays which is essentially a derivative of this

01:06:02

curve here you will get quite

01:06:04

uneven and delay time at different

01:06:06

frequencies

01:06:07

not good, but at least saving money

01:06:11

in terms of computing resources and

01:06:14

in general, well, this is what it was like

01:06:16

educational program

01:06:17

perfect I hope I didn't bore you but

01:06:21

I thought it was necessary

01:06:23

since the course is still called

01:06:25

signal processing in interfaces

01:06:27

brain-computer and so absolutely well

01:06:30

perfect explanation on your fingers

01:06:32

Of course you need filtering

01:06:33

the collection will not be listened to with paint

01:06:35

look at the books, probably relevant

01:06:37

position right signal processing

01:06:39

oppenheimer best man for example here but or

01:06:41

see how this is the territory with the basic

01:06:44

before it just seems to me without

01:06:47

Well, just use these

01:06:49

here are the operations for generating coefficients

01:06:52

filter before computing background

01:06:53

transformation so on it seems to me

01:06:55

they just had no idea what they were talking about

01:06:58

we're talking, well let's go back to

01:07:01

his

01:07:03

interface

01:07:05

and let's look at a paradigm like this

01:07:07

model

01:07:09

well look how it is

01:07:12

usually implemented as

01:07:13

demonstrated

01:07:17

parts matter is the you have

01:07:20

the keyboard on the screen is like this

01:07:23

small numbers on it and the keys and

01:07:25

each key blinks at its own frequency

01:07:28

these frequencies are signed and indicate

01:07:31

the states still have their own phase, each one here

01:07:33

key corresponds to unique purity

01:07:36

well, there’s some kind of fan there

01:07:43

man sits and looks at this

01:07:47

keyboard and see how

01:07:49

interesting volume so biological

01:07:50

I’ll tell you something neurobiological

01:07:52

you see the eye is very interestingly designed

01:07:55

the left half of the field is pla right and along the floor

01:07:58

field of vision and each eye

01:08:01

left side head right side eye

01:08:05

sensitive to the left sex field a

01:08:07

the left side of the eye is sensitive well

01:08:13

and is sensitive to the right sex field and

01:08:17

then these are the nerves through through

01:08:21

optic

01:08:22

I'm super called come part 2 well

01:08:28

cross the oceans go through so

01:08:31

called the geniculate nucleus

01:08:33

greenhouse nucleus that's what it is

01:08:36

crowned but let it look like that

01:08:38

the knee looks like now there is a picture

01:08:40

you can not me, well, it doesn’t matter, but through this

01:08:44

nucleus and already goes to the primary visual

01:08:46

cortex here and in this visual cortex to this

01:08:50

visual cortex not far from it

01:08:52

are our ha electrodes signals

01:08:56

which we register

01:08:58

these signals travel through

01:09:00

some well there through through the server

01:09:03

spinal fluid through

01:09:05

bone there and so on. you give birth but

01:09:09

I told the first lecture and generates

01:09:11

the mind of the sals who would be right here

01:09:13

guys register this is what we are

01:09:15

register this is our column vector

01:09:17

Victor I will try almost everywhere

01:09:19

designate

01:09:20

in bold letters in small letters and

01:09:25

always all victor we have victor columns

01:09:29

if we need to put the vector on its side

01:09:31

to multiply by 4 x transposed

01:09:33

that is, as if here we attribute

01:09:35

letter d

01:09:37

it means but it’s such a setting, well

01:09:41

and it is assumed that depending on different

01:09:43

keys

01:09:44

depending on the frequency it is

01:09:47

the keys blink in your visual cortex

01:09:50

will arise

01:09:51

signals of different frequencies

01:09:54

dominant yes there in the skin of analysis by me

01:09:56

you can look for example yes and you can and

01:10:00

so you can

01:10:03

understand which key he is looking at

01:10:05

man because each of the keys

01:10:07

frequency is uniquely linked

01:10:11

that means it’s also interesting that well

01:10:16

firstly the brain is a nonlinear system yes but

01:10:18

everyone knows that

01:10:20

because from the steam there was a linear number

01:10:23

but we weren’t and this is a nonlinear system

01:10:29

when she doesn't come to the entrance

01:10:32

for example, even harmonic influence

01:10:34

then at the output it generates

01:10:37

linear or click the beach resort

01:10:40

after passing our sinusoid turns into

01:10:43

multiples of sinusoids, for example sines

01:10:45

burnt turns into a certain set of harmonics

01:10:47

there sine green sine 2 mega per m7

01:10:50

arrow megan aries and so on but how

01:10:52

guitar string

01:10:54

pulled and she was early in different fashions

01:10:57

fluctuates there is 1 fashion

01:11:02

here is his answer to look at the entrance if

01:11:06

our stimulation comes with a certain

01:11:08

frequency

01:11:09

these frequencies are here and then at the output

01:11:12

for example ten hertz then at the output we

01:11:14

we see the same ten hertz

01:11:17

20 hertz and even another 30 hertz and probably

01:11:21

you can even see 40 hertz here

01:11:25

but the higher the stimulation frequency, the

01:11:27

very high frequency harmonics

01:11:30

more they weaken but still you

01:11:33

you see

01:11:34

that the input is supplied with one frequency per

01:11:39

at the exit at the exit you have

01:11:41

several frequencies and these like this

01:11:45

their so-called harmonics are also at least

01:11:47

of course you will want to take it into account when analyzing

01:11:49

I'm talking about so that I understand it's just that

01:11:51

what we have is in this x from and

01:11:54

now a little more like this

01:11:56

the abstract model is that

01:11:59

that we have this one here, our primary

01:12:01

the visual cortex is how the signal arrives

01:12:03

and the primary visual cortex it can be

01:12:06

describe as a collection of neural populations

01:12:09

who receive these signals and

01:12:12

they don't receive a signal and

01:12:17

which population actually has

01:12:20

attitude towards the signal that comes from

01:12:22

eye yes, that is, well, let’s say which one

01:12:24

is right next to these

01:12:26

in sections

01:12:27

where do the nerves come oksana

01:12:30

the number of cores and some zones they

01:12:34

have populations that generally range from

01:12:36

I don’t even get a signal there and even in

01:12:38

society is not connected, maybe on the contrary the brain

01:12:39

this is how it works so that these signals

01:12:42

responded to the activity of these populations

01:12:44

was responsible for the signalmen and vice versa

01:12:47

wants to free the rest of the population

01:12:49

so that they occupy themselves with other tasks in

01:12:51

this is the time and all that needs to be processed

01:12:53

flashing lights are needed there and some kind of side

01:12:57

vision there is something else, well, in general

01:12:59

I mean that

01:13:01

the brain he can, well, we can describe it

01:13:05

this activity as a set of population

01:13:07

which is relevant to the task and

01:13:09

population set and which does not have

01:13:12

relationship of tasks and that's how we are with you

01:13:14

they said there must have been a lecture that us x

01:13:16

from t this

01:13:18

media from our relevant signals

01:13:21

tasks with signals that do not have

01:13:23

relation to the problem with coefficients

01:13:25

which are from essentially topography

01:13:29

our tomography sources

01:13:32

this is essentially

01:13:33

coefficients with which for example

01:13:36

the topography of this population is a vector

01:13:38

length equal to the number of our electrodes

01:13:42

which describe the coefficients with

01:13:45

with which a single activity is on

01:13:47

this population will be displayed on the electrode

01:13:50

suppose if this electron were

01:13:52

sensitive only to this population

01:13:54

other remaining electrons were not not

01:13:57

sensitive to this population then here

01:13:58

there would be something like 1000 but it’s not true because

01:14:01

that all electrodes are somehow

01:14:04

sensitive to all populations only

01:14:05

with different coefficients, yes, that’s understandable

01:14:08

what are these odds for how much?

01:14:09

electromagnetic stories remember there

01:14:11

beautiful blue profit pros and cons

01:14:14

coefficients can be like

01:14:15

positive and negative

01:14:16

this is the model it’s not like for us

01:14:20

will be directly useful, but understand that

01:14:23

we want these sources that we see

01:14:26

see they are essentially well projected into

01:14:29

in quotation marks there is I'll take projected onto

01:14:32

our sensors using these

01:14:33

linear linear that's linear

01:14:36

operations are simply essentially scaling

01:14:38

of our basis vector by 40 of ours

01:14:40

topography, it can be assumed that

01:14:44

to highlight the activity of this one

01:14:47

we need to use the source too

01:14:50

some kind of line for surgery, for example

01:14:52

some kind of multiplication of this x from t

01:14:54

on the left to the vector that will be suppressed here

01:14:58

for example, waiting for everyone else and you

01:15:01

besides this source we need, yes

01:15:03

and this is done with the help of

01:15:06

so-called spatial operation

01:15:08

filtering which is what we take

01:15:12

our input vector

01:15:14

this is our signal from our electrodes and

01:15:16

We multiply each of these signals by

01:15:19

coefficient coefficient w1 w2 and so on

01:15:22

then the vector of coefficients is clear that

01:15:24

its length is equal to the number of sensors here and there

01:15:27

so, well, consider it a package or something

01:15:29

that is, to the rapture but one point yes then

01:15:32

there are calculations here

01:15:34

this and this spatial time

01:15:37

signal up to with these coefficients and

01:15:39

describes it this way:

01:15:42

it's just a vector of sand placed on

01:15:43

tank multiplied by a column vector and x from

01:15:46

this is the scalar product of the output we

01:15:48

we get a scalar that's why here with

01:15:50

low-fat and thin, and generally speaking

01:15:53

it needs to be italic so

01:15:57

why is it called again

01:15:58

filtration remember during filtration times

01:16:00

chose temporary reports yes that is we

01:16:03

read our signal which lives in

01:16:05

time we took these reports from the claps

01:16:08

Valevskaya in cents is exactly the same

01:16:10

here we read our signal and

01:16:14

only now we are already taking a countdown of it

01:16:17

time and in space and on different

01:16:19

electrodes because the signal lives with us

01:16:21

spatially the scalp of his time here

01:16:24

Well, this is where we take these signals

01:16:25

reports and with what coefficients they are

01:16:27

Skokova and here we have it

01:16:30

space filtered let's

01:16:33

Let's see a little

01:16:34

by [ __ ] but do let's say if

01:16:37

write down this amount, yes there it is

01:16:39

Ghanaian member then we get that Rick Sad and

01:16:42

even one on s1 for the same two on i

01:16:44

brother I and so on and here is a mistake

01:16:46

again, the cheese should be low-fat

01:16:49

you see, yes, because it would just

01:16:50

scalars like here and not fat and

01:16:54

a little

01:16:56

x from t is this and this vector is like

01:16:59

received we hope to receive an assessment

01:17:03

signal from 1 at&t, that is, this guy

01:17:06

from 1 from t which corresponds

01:17:10

activity of 1 population of neurons

01:17:13

we hope to find a vector this way

01:17:16

w transposed w which we

01:17:19

transpose, multiply by the same 1 and

01:17:21

here we get the gain factor 1

01:17:23

so this thing will give us

01:17:26

signal from 1 from and we will try to find such

01:17:29

w so that the rest of the signalmen, that is,

01:17:32

topography of other sources

01:17:34

so that they lie in space

01:17:37

orthogonal eyelids as much as possible

01:17:39

as much as possible

01:17:40

this vector w w then the scalar

01:17:45

the product will be close to zero and then

01:17:47

at the exit from this whole

01:17:50

the amounts will only remain underlined and

01:17:52

signal from 1

01:17:55

Well, now the question is, how do we search?

01:17:59

this vector w

01:18:00

it is clear that if we had a good

01:18:02

the model is clear that if we knew for sure

01:18:05

where does this population of neurons live?

01:18:08

to which, well, this is yes, some kind of yes

01:18:10

which answers

01:18:12

Well, that's why influence with the help

01:18:16

these blinking buttons then probably

01:18:19

It would be possible to somehow solve the reverse

01:18:21

task to build a vector that listens

01:18:24

specific area of the cerebral cortex

01:18:26

here it is

01:18:30

in general like this

01:18:33

solve this problem but we don’t have that

01:18:37

but we have an understanding

01:18:40

tasks and we have an understanding of how and

01:18:44

what does it react to?

01:18:45

the brain reacts to periodic

01:18:49

influence through generation

01:18:52

the same periodic impact in

01:18:55

visual cortex and

01:18:57

look let's try to search

01:19:01

this vector of coefficients is here I already

01:19:02

designated it as but one and two why

01:19:06

because we will have two vectors

01:19:08

we will have a vector and a vector a and a vector b

01:19:11

and look how khidr we remember

01:19:15

you saw here that the response is

01:19:18

even at about ten hertz it consists

01:19:21

of several harmonies, that is, a harmonic

01:19:23

gum hertz then harmonic 20 hertz

01:19:26

multiples and remember I told you about

01:19:30

sine and cosine is a combination of sine and

01:19:32

cosine, it allows you to move the same

01:19:35

harmonic is an arbitrary frequency

01:19:37

phase well, let's put it on

01:19:40

such reference sequences such

01:19:43

vector support matrix

01:19:44

sequences where let's say the first

01:19:48

the frequency will be equal, well, that’s stimulating

01:19:51

frequencies that are actually

01:19:53

then there will be harmonics and these will be and

01:19:56

sines and cosines this is this sine

01:19:58

and the cosine of zero starts going up and

01:20:01

these guys should have a lot of ethics

01:20:04

should be cosines but it's not green

01:20:06

phase no matter past different coefficients

01:20:09

could fall down could simulate

01:20:11

different content and using it we can

01:20:13

moving means

01:20:16

and now these are cosine

01:20:18

here's the sequence

01:20:21

let's try this set for

01:20:25

cosine in after cosines last

01:20:26

ten days have actually moved

01:20:28

phase is scary here

01:20:30

let's do this, let's do it

01:20:34

Let's try to look for such coefficients with

01:20:37

with which we multiply each of these

01:20:39

here is each of these sequences

01:20:42

let's add it up so that what we get is

01:20:45

our output was as similar as possible to

01:20:49

some space above filtered

01:20:53

input signal because we know that

01:20:57

Our signal is somewhere on the sensors

01:21:00

we don't know which electrode anymore

01:21:02

sensitive and do not know typography we

01:21:04

we don't know anything, we model nothing here

01:21:06

here, but we hope that I’m doing something like this

01:21:09

processing looking for well 10w vector

01:21:12

we can also denote such a vector w

01:21:16

highlight the signals we are interested in

01:21:17

suppressing those not of interest is important correctly

01:21:20

it is important to set the task correctly

01:21:22

set generate target

01:21:24

functionality and exactly with the help here

01:21:27

these reference signals and so on

01:21:30

called canonical correlation

01:21:31

analysis we we will look for this one here

01:21:36

filter because our idea is

01:21:38

is that we want to find the best filter

01:21:40

here who will process our data for us

01:21:43

input and the best filter that we need

01:21:47

modifies our reference signals to

01:21:50

for each specific command, here it is

01:21:52

for frequency for example 10 gear 20 hertz

01:21:54

thirty hertz 40 that is, point . 10

01:21:58

20 10 20 30 40

01:22:00

50 these are the five harmonics

01:22:04

that means this team 1 and 2 are these teams

01:22:07

harmonics will be suppose 11 22 33

01:22:11

4455

01:22:13

but for each team we have here

01:22:15

there is such a sequence

01:22:17

set of support followers and we will

01:22:19

try as hard as possible

01:22:22

increase to that is, the correlation between

01:22:24

processed spatially filtered and

01:22:27

g signal and this reference

01:22:30

signal and so the correlation that we

01:22:34

will give the command dates for which we will see

01:22:36

the maximum of this coefficient and

01:22:40

and will be the team which will be the number

01:22:44

the keys will fit better

01:22:45

the number of the keys the person is looking at

01:22:47

this is how we can find the one

01:22:49

frequency with which the keys blink

01:22:51

which is watched by the person who

01:22:53

excites his primary visual cortex

01:22:55

that is, I don’t know, you probably can

01:22:58

be familiar with there is such there are such

01:23:01

here is my method give me a moment time

01:23:04

working is called it is used

01:23:06

speech processing when in well old old

01:23:09

systems when you have a template

01:23:11

some spoken word and you are this

01:23:14

teamplay, trying to put it into words

01:23:17

which you want to recognize

01:23:18

sound sequence omit

01:23:21

super recognize and understand that all people

01:23:23

speak differently at different speeds

01:23:24

there and so on and here you can do this

01:23:27

change the time scale so that

01:23:30

so that you can increase or decrease there

01:23:33

non-viscous quadratic increase

01:23:35

correlation yes it doesn’t matter here and here you are

01:23:39

best correlation coefficient value

01:23:41

you take it as

01:23:43

what you will compare for different

01:23:47

teamplay tov for different to different teamplays

01:23:49

there are words to recognize this word well

01:23:51

so here too, that is, like that, we are

01:23:52

we find a processing that

01:23:55

the signal will be emphasized to us as much as possible and

01:24:00

reference signal and bring it as close as possible

01:24:03

understand each other that's the point

01:24:07

then we calculate practically the cosine

01:24:09

between the sequence of dreams foamy well

01:24:12

that there is a correlation

01:24:14

so we're trying to maximize it

01:24:17

see here

01:24:20

this is essentially

01:24:22

that means so

01:24:24

x means if if simply means y

01:24:28

you

01:24:29

you have with 1 this is a transposed to

01:24:33

x1 is a scalar up depending on and like

01:24:36

time its meaning here you are it

01:24:40

scalar

01:24:41

multiply by another scalar that exists

01:24:45

would be transposed y and but since

01:24:47

for I wrote down the scalar from Odessa and one of

01:24:50

them I transpose for me just like that

01:24:52

it’s more convenient and that’s what it allowed me to endure

01:24:56

b and a beyond this amount and here

01:25:00

this amount is nothing more than, well, some

01:25:04

estimation of the correlation coefficient between

01:25:07

various channels of extender x and him

01:25:12

such Vicks and each of these here

01:25:15

look at the sequences

01:25:17

now I mean, that is, in essence, here I am

01:25:21

I have sequences in

01:25:22

brain activity here I am actually

01:25:26

I think there is a correlation between this guy and this guy

01:25:31

this guy then this guy with this

01:25:33

this guy this guy with this one

01:25:36

guy and so on and get one in

01:25:38

matrix size number of HD channels per

01:25:41

the number of these are my support ones

01:25:43

sequences look like this

01:25:46

this is this, this is autocorrelation, this is this

01:25:49

when I count with myself xxx

01:25:52

transposed dishes are considered here

01:25:54

this vector multiplied by this need

01:25:56

release who is chro city but even red

01:26:00

with red and then where is red green

01:26:03

red and orange we serve like this and

01:26:06

so on, here I get such a matrix

01:26:08

sigma x x which xx are indices

01:26:11

which are essentially in the matrix

01:26:13

autocorrelation autocorrelation

01:26:15

matrix, well, its assessment is not really

01:26:17

open it too 10 must determine on

01:26:20

t by the number of time reports and this

01:26:22

not you approximation provided that I

01:26:25

this is the expectation that

01:26:27

open matrix definition

01:26:29

present I replace it with

01:26:31

amount over time, yes, just like that

01:26:36

that means look, that means now there’s something else

01:26:39

there is ours x here is ours y this one

01:26:43

matrix here we have such functionality but

01:26:46

is the correlation coefficient between s and p

01:26:51

here you already see the fat ones because it's

01:26:54

Victor, that is, x is a matrix a

01:26:58

transposed vector multiplied by

01:27:00

the matrix is a recumbent vector because

01:27:02

which of the transposed n is a vector

01:27:06

vertical game transposed to b

01:27:08

and then you count with

01:27:11

transpose multiplied by p

01:27:12

scalar product but so that it is

01:27:15

correlation coefficient you must also

01:27:17

divide by the norm c and the norm p well

01:27:21

actually this is the denominator

01:27:22

this norm is done with and this is the norm a

01:27:27

this norm

01:27:29

correct because it would be transposed

01:27:32

y this is n times n n n

01:27:36

transposed and games planned

01:27:39

b is PPC is the root of p

01:27:42

transposed p is equal to the root of

01:27:46

modulus vector apparatus equal to modulus

01:27:49

vector not this not

01:27:52

said but here is the convention to pray

01:27:54

looks like this but just to

01:27:56

understood a little bit so simple at the end

01:27:58

already before the lecture I added because

01:28:00

that you didn’t show me because it’s important

01:28:02

you said there are collectibles

01:28:06

covariance matrix correlation

01:28:08

matrix it does not subtract the averages but in

01:28:12

in our business we usually believe that we

01:28:14

data with zero mean and therefore on

01:28:16

what is projection what is covariance but

01:28:17

for the sake of rigor, here is the correlation

01:28:20

matrix valuable matrices were stolen, she doesn’t subtract

01:28:22

this is the average it's just because

01:28:25

the variables are there

01:28:27

this is the first first 2 to about 1

01:28:31

channel second channel to they are among themselves

01:28:33

relate to relate and this

01:28:35

the ratio may even be up to and

01:28:37

average for example they have u2 not 0 average

01:28:39

Well, that means they are already among themselves

01:28:41

correlates with native positive

01:28:43

the average of the other negative means

01:28:45

here they are with negative coefficients

01:28:46

correlate with each other where if

01:28:48

both positive means positive

01:28:50

refers to a

01:28:52

the covariance matrix shows you

01:28:55

how two sequences and cry 2

01:28:57

vectors and so on as far as cubes are concerned

01:29:00

raipo no matter how much chi sum mi together

01:29:02

change with respect to each of their

01:29:05

average and that's why here this is this

01:29:07

average you read that is x 1 minus

01:29:09

average x1 exam average that's it

01:29:12

is considered to be the expectation of this product

01:29:13

in our business we are this expectation we

01:29:18

we replace it with time averaging

01:29:21

now we won’t discuss what kind there are

01:29:23

issues related to bodies with

01:29:26

organic sorry about organic

01:29:28

these processes to justify that

01:29:30

you can replace averages with averages according to

01:29:32

mid-time ensemble

01:29:34

relatively far beyond this

01:29:35

course theoretical questions like this but

01:29:39

for practice just you understand well

01:29:41

agree that you are pregnant

01:29:42

count sequences by how many

01:29:44

they are related, so what do we look at

01:29:47

calculation of one another here and here and this

01:29:50

says if it's auto covariance

01:29:52

the matrix is very noticeable

01:29:54

wonderful properties what if you

01:29:56

consider for example from a 2 by 2 matrix and

01:29:58

her own victor she matches

01:30:00

This is what our scatter plot looks like

01:30:03

our values, that is, the choice of these vectors

01:30:05

x1 and x2 and draw each point to 3

01:30:09

x one point in time 1 x 203 me

01:30:12

one meter. and then this and so on then

01:30:15

for example if you get a cloud like this

01:30:16

then calculating the correlation matrix y

01:30:18

Whose ration did he take?

01:30:21

counting your own Victor you will get

01:30:23

that the first eigenvector which

01:30:25

there is a corresponding maximum

01:30:26

own number he looks along

01:30:29

direction of maximum variability

01:30:31

your vector your two-dimensional from

01:30:33

this vector to here and correspondence 2 but

01:30:36

automatically it is orthogonal to the first and

01:30:38

where he looks there he looks in this

01:30:39

control situations the smallest

01:30:41

variability but there are only 2 so

01:30:44

smallest and maybe direction

01:30:46

the next largest then still rules

01:30:48

next promotion then here we go

01:30:51

let's now see how we can have

01:30:54

this crocodile is our functionality

01:30:57

don't forget we need to find a and b

01:30:59

which maximizes this guy

01:31:02

in the morning correlation coefficient

01:31:06

so let's change it first

01:31:09

basis see here it is marked

01:31:11

very simple actually, but

01:31:14

it’s clear that a and b are here

01:31:18

the only thing is that this is me as it is and

01:31:20

articles and therefore here is a little

01:31:21

violation nataisha yes that is, as it were

01:31:23

here is Victor they are all vector and this is

01:31:26

also a vector, that is, this is it, but they are

01:31:29

not fat and they aren't that, well, that's it

01:31:33

as it is as it is I can attach it

01:31:36

articles links will be therefore you

01:31:38

you can understand where these variables are found

01:31:41

Well, here is this vector a and b, here we have them

01:31:44

multiplied this essentially this

01:31:46

the works are clearly all ours on these

01:31:49

Here

01:31:51

s transposed to p yes we have it

01:31:54

numerator and

01:31:57

but we can't just do it ourselves

01:32:01

we need to optimize this thing

01:32:02

definitely need to watch dota

01:32:03

normalization because normalization is

01:32:06

allows us to take into account the similarity

01:32:10

forms and not like some kind

01:32:12

the signal in which direction is very large

01:32:14

the contour is small, well, that is understandable

01:32:16

when arnie rationing is not something

01:32:18

strive to program things like this

01:32:21

I won't fight this

01:32:23

tell intuition but the essence of it

01:32:24

is that you need to go to

01:32:25

space in which whichever direction

01:32:29

you're a vector and didn't look at 10

01:32:32

vector c

01:32:34

no matter what side I look at you

01:32:38

data will be projected onto it

01:32:41

so their power is still equal

01:32:43

one therefore can spin like

01:32:45

whatever you want, and here on the projection of the dispersion of this

01:32:49

project it will not change wallpaper but

01:32:52

this thing will then depend only on

01:32:54

how much

01:32:56

Well, it correlates with this covariance

01:32:59

matrices are basically a year and look

01:33:01

that this is just covariance

01:33:03

matrix between the plaintiff and y that is, this is also

01:33:06

since this is x by y transposed then

01:33:10

there are x on y laid out lying down yes and

01:33:14

in size it is naturally the number of lines

01:33:18

x by the number of lines and

01:33:23

so, let's formally let's do this

01:33:26

it would be better to formally create and

01:33:27

we'll replace the basis, we'll just take new ones

01:33:30

variables c this is this this is this

01:33:32

covariance data matrix of degree 1

01:33:35

2 is the matrix square root of

01:33:38

well, you are the same, those who know

01:33:41

such bleaching is beginning to be understood before

01:33:43

that and this practically does not repent of

01:33:46

separate space because the thing

01:33:48

can you come here we don't know about it yet

01:33:49

We’ll just stupidly substitute ours further

01:33:52

euro ruble a expressed through c here in

01:33:55

this is the expression you will get like this

01:33:57

crocodile what's good here because a

01:33:59

in fact that's what

01:34:01

denominator and

01:34:03

the denominator is gone all sorts of oyster data

01:34:07

covariance matrix paint

01:34:08

aviation discovery matrix

01:34:11

seamstress look further here

01:34:13

a little complicated, well, well, just complicated

01:34:16

plus a lot of letters, see if you have them

01:34:19

Skalia eats such porridge in the Black Forest

01:34:21

Codex Cauchy-Schwartz inequality

01:34:23

this is what she's talking about, well basically she's

01:34:26

says that if you followed

01:34:27

attributed to you have a set of numbers and to

01:34:30

last take you have a set

01:34:32

numbers, then the correlation coefficient of these

01:34:35

two are consistently less than one

01:34:36

played at least or equal to one

01:34:38

it can't be more than one, that's it

01:34:41

exactly what he's talking about because it's

01:34:43

this is the scalar product in on

01:34:46

w

01:34:49

normalized to the norm in and to the norm w

01:34:52

less than one to cover but with

01:34:56

provided that of course armani is equal to zero

01:34:57

Here

01:34:59

that's why this is this one

01:35:01

you say your acquaintance now let's

01:35:02

let's look at the effect here at this one

01:35:04

this guys this is a vector, this is true

01:35:07

just a matrix is a vector which

01:35:11

multiplied by this matrix

01:35:15

that means this is a certain vector v, here it is

01:35:20

this is a certain vector u we simply denote

01:35:24

because here's Kashirin's shorts apartment

01:35:26

wrote in general terms that this is ours

01:35:27

it is clear that you can write something

01:35:30

this thing is smaller than you

01:35:32

transport to square. but someone here

01:35:37

in transposed on in exactly with this

01:35:41

the formula needs powers 1 2 here it is the root

01:35:45

Hyde scalar yes and

01:35:46

he will be unbanned by d to the power of 1 2

01:35:50

so from here we conclude that

01:35:55

from here we conclude that our mouth is smaller

01:35:59

how

01:36:01

this is the value in parentheses

01:36:04

our denominator is simply the center

01:36:06

planned for 1 2 this is very good

01:36:08

because we are already with this guy

01:36:13

quite easy to fight see the banners

01:36:15

the numerator here is big big

01:36:17

crocodile but nothing wrong with that

01:36:18

crocodile no this is just this this is this

01:36:21

their transposed to x yes this is it

01:36:24

really because you just plan x on x

01:36:26

this is x then the game is planned like this

01:36:29

it's just y air transposed

01:36:32

supporting our work up to a sine wave

01:36:35

cosine cosines and back from it

01:36:37

calculated this is y by x transposed

01:36:40

this is actually transposed here

01:36:42

this is all the matrix and this is

01:36:45

Well, the amount you need is the same as here

01:36:47

raised to the power minus one second

01:36:49

so this is the inverse of the matrix root

01:36:53

then there's nothing here, nothing at all

01:36:56

scary it's all considered the same

01:36:57

operator from Billy Python here we go

01:37:01

next here we get this guy this

01:37:04

in fact, if we label this thing

01:37:06

for m we get that our functional is not

01:37:08

forget, we need to find such c a

01:37:10

as soon as you find c we can

01:37:13

recalculate a and b and we can

01:37:16

calculate our centers it turns out

01:37:19

what but write this down like this

01:37:22

our material acquires this completely

01:37:24

the human appearance is no longer at all

01:37:26

the crocodile isn't scary at all, that's what it means

01:37:29

it is clear that their functionality does not depend

01:37:32

from scale c you can c

01:37:35

scale to alpha

01:37:36

then you will still succeed, that is

01:37:39

gays from alpha c on functionality depends on

01:37:43

mas tinted prices some

01:37:44

the coefficient is also equal to the same

01:37:47

because and that's why we can do this thing

01:37:49

consider for the case when the number

01:37:52

the norm of c is equal to one, that is, c

01:37:54

transposed c is equal to the square of ravich

01:37:57

sits down sponsored prone and want

01:38:00

maximize it

01:38:02

this is the optimization problem

01:38:06

this is our limitation

01:38:09

only the center of decline per square is equal to everything

01:38:13

equals one, let's use it

01:38:15

Lagrange multiplier method

01:38:17

that is, we simply take similar

01:38:18

functional

01:38:19

multiply it by Lagrange and

01:38:23

Lagrange by multiply by our constrain

01:38:25

so that there is only constraint

01:38:27

on the right is zero, that is, it is planned

01:38:29

minus 1 equals 0, that’s all we need next

01:38:32

what you need to do is take the derivative with respect to c

01:38:36

from this thing but we plan

01:38:39

this is very simple, you can actually

01:38:41

in fact, just write it down, take 2 on

01:38:43

Case 2 and write simply according to c1 and 21

01:38:47

22 take the derivative, just

01:38:49

write it down and you have it and then collect it

01:38:51

reverse excellent expression vectors here

01:38:53

and you will succeed in exactly this

01:38:55

the mnemonic rule is that

01:38:56

quadratic purchase form square then

01:38:59

here 2 m nation that is a

01:39:01

transposed swarm well logical

01:39:02

because we are the same as x

01:39:04

square is 2 x correct that is x

01:39:06

multiplied by x is 2 multiplied by x

01:39:08

derivative is also here but also here

01:39:10

the same

01:39:12

only there is no matrix, okay, you can continue

01:39:15

see that the value c which us

01:39:18

I'm interested in what they satisfy

01:39:19

here are the ratios that c multiplied by here

01:39:23

this matrix m actually retains its

01:39:26

direction but it just turns out to be with

01:39:27

scaled by it with the coefficient lambda a

01:39:29

this is something else by definition as

01:39:33

eigenvector of matrix m a

01:39:36

since we are interested in the maximum

01:39:39

the meaning is this one and transposed

01:39:41

mc as we see she is connected from the times yes

01:39:44

this is because if we multiply by

01:39:46

transposed here we get that

01:39:48

It’s just that we’ve been married for a long time

01:39:50

transposed crc ends equal

01:39:51

only one

01:39:54

we should then get that ours is this one

01:39:58

here is the value of our optimized

01:40:00

functional it is equal

01:40:03

actually Thailand and that's why we need

01:40:05

choose c as the eigenvector of the matrix

01:40:10

m corresponding to the largest

01:40:13

own number

01:40:14

this

01:40:19

that means yes and now besides this already

01:40:24

learned how to do

01:40:25

now we do the following we remember

01:40:28

that our robe was originally like this

01:40:31

we moved here like a crocodile because of this

01:40:35

such is the porridge Schwartz and we tilt it moms

01:40:39

went to this guy and beyond

01:40:41

look at our vector c

01:40:47

which we found and it gives us

01:40:50

vector c multiplied by this

01:40:52

crocodile and

01:40:54

like we do here we need to select a vector

01:40:56

d which maximizes our coefficient

01:40:59

correlations how to choose it but vector

01:41:01

d obviously should about the direction

01:41:03

coincide with this vector a

01:41:05

since the husband and wider sector of us

01:41:07

unit norm then we

01:41:10

let's find vector d like vector c

01:41:13

multiplied by the finished guy and similar

01:41:15

we simply reinforce to a unit length

01:41:17

Well, that’s how it’s written here, and so on

01:41:20

we can recalculate a and accordingly

01:41:22

b and we will already have spatial ones

01:41:24

filters that we can apply

01:41:26

directly to our signal

01:41:29

data a b c j with our reference victor

01:41:32

Here

01:41:34

this is the thing that means here

01:41:39

[ __ ], who knows?

01:41:41

apartment demonstrators yes yes yes I understand

01:41:43

that the Americans are a shooter here

01:41:46

this is the algorithm

01:41:50

interesting thing, generally speaking the most

01:41:52

simple implementation of this thing a and b

01:41:53

we don't need to count everything we have

01:41:56

we need this we need to calculate

01:41:58

brother's matrix and calculate its first

01:42:02

the eigenvalue is the largest and this is

01:42:04

own number will be ours

01:42:06

the functional will be our correlation and on

01:42:08

based on that correlation we can

01:42:09

rank and we can choose a team

01:42:12

which which well frequency to which

01:42:15

the person with whom Kurt Square blinks

01:42:18

which person is watching

01:42:20

here is the processing pipeline

01:42:26

the city of the tour has been finalized, here it is different

01:42:29

the teams here are different accordingly

01:42:31

the frequencies here are all the same

01:42:33

of course they are actually different, that is

01:42:35

like something here is 8 hertz there it is 85

01:42:38

hertz is 9 and so on there be it

01:42:41

it will show you a specific example, that’s what it means

01:42:44

we take the matrix to that team and

01:42:47

take our input data take ours

01:42:51

input data and calculate the cross

01:42:54

correlation matrix

01:42:55

paint rational matrix x a and support

01:42:59

matrix corresponding to the card team

01:43:01

so we also take it, count the paint, take it ourselves

01:43:04

x and calculate the autocorrelation matrix and

01:43:07

we calculate the auto covariance matrix

01:43:10

supporting things here and here we have it

01:43:14

here, but here you will need this matrix y

01:43:17

from x y from and this is this and this is essentially

01:43:20

transposed matrix

01:43:22

so here are all three elements that

01:43:26

we need allow us to create a matrix

01:43:28

m

01:43:29

rolling which correspond to that

01:43:31

team we don't know what team we have

01:43:33

for encoded we calculate the most

01:43:36

large eigenvalue and remember

01:43:38

then we take it into the following matrix

01:43:42

we're doing exactly the same thing again

01:43:44

calculate the eigenvalue again

01:43:46

we remember it and do everything as

01:43:48

teams and

01:43:50

decode the command as a number

01:43:54

the largest proper number here in

01:43:57

this list

01:43:59

this is the first pipeline option

01:44:03

processing means how is this whole thing possible

01:44:06

improve, well, for example, look here

01:44:08

this is our response, these three harmonics

01:44:13

Here

01:44:14

this is for the first command, but let's say

01:44:17

freeze the fortieth team and Vanya doesn’t

01:44:19

shifted up to a week at a distance between

01:44:21

but they all get these responses there

01:44:25

range there but even there they start from 8

01:44:28

gears yes somewhere here yes well rough there before

01:44:32

50 hertz well let's just

01:44:35

let's build a filter that we will filter

01:44:38

everything except this frequency range

01:44:40

that is, I will leave it only in this range

01:44:42

so we will give all sorts of

01:44:44

unwanted oculomotor

01:44:45

artifacts artifacts movement there is a lot

01:44:48

all sorts of this is not evil spirits and

01:44:50

low-frequency mother we can't get rid of it

01:44:53

improve, well then the pipeline is like this

01:44:55

yes we just take the filter

01:44:57

the only one and even everything is the same

01:44:59

Most of all, everything is a penny. following

01:45:02

the improvement is that in general

01:45:04

speaking when we decipher

01:45:06

signals we are trying to understand some kind of command

01:45:09

1 2 tanks then we can filter

01:45:12

use the corresponding answer here

01:45:14

which corresponds to what is expected from and

01:45:16

team that is we can take we can

01:45:20

take for example

01:45:21

team number

01:45:23

information about what kind of team we have

01:45:26

goes and which one is the same as which team

01:45:29

we count Linda we take this one

01:45:31

we provide the information to our filter and

01:45:35

create a filter comb like this

01:45:38

and if the first team is such a comb

01:45:41

if 2 40 team is like this so we

01:45:45

filter out a bunch of crap

01:45:47

around and what if, for example, we don’t have this

01:45:50

This is where we will suppress the signal

01:45:52

accordingly according to

01:45:53

the correlation will be even less at the output

01:45:55

ptr practically according to the data we do not want

01:45:58

I get the wrong signal and we just need

01:46:00

because you need to emphasize the signal

01:46:02

frequencies

01:46:03

which are actually there if you

01:46:06

us if we have these these these these these

01:46:09

peak of our frequency characteristics

01:46:12

our filter if they are not included in

01:46:14

peak of our response, but on the contrary suppressed

01:46:17

well that's good it means that

01:46:19

send the answer here and accordingly Linda

01:46:21

will be a small risk native power

01:46:24

there will be much more, by the way, you

01:46:26

look carefully this is what you need

01:46:28

will need to implement homework

01:46:29

we formulate it correctly like this

01:46:33

There is also an improvement to this pipeline

01:46:37

called

01:46:39

extended

01:46:41

sessions

01:46:43

which allows

01:46:46

which allows well which does

01:46:49

such a rather complicated thing, so if

01:46:52

honestly I'll try

01:46:54

look, it means that you and I

01:46:59

considered please note this is a method

01:47:01

which is not used

01:47:03

some do not use training

01:47:07

sample in general and he has a huge

01:47:10

plus that is, we just know which ones

01:47:13

supporting placenta what harmonics as with

01:47:15

our keys blink at these frequencies and

01:47:17

we use them to

01:47:19

engage decoded we don't use

01:47:21

no specific properties at all

01:47:23

person how he responded to these

01:47:24

sequences here and this is the extension

01:47:27

methods that allow you to take into account all the same

01:47:31

then as a person

01:47:33

reacted to data on some

01:47:35

pull up is his specific response but

01:47:38

for this it is natural for us to

01:47:39

blink these keys and say which one

01:47:42

look look at this you have a feeling

01:47:44

blink remember then look at this

01:47:46

player for each key for the end that

01:47:49

you are essentially we have our own vector

01:47:51

matrix x data, well, of course

01:47:54

eat vectors on our support matrix

01:47:56

signals next we do what is this

01:47:58

our standard thing is just ours

01:48:01

native data with y from f to this is what we do

01:48:05

canonical correlation analysis and

01:48:06

think about this about up to 1 1 and then we

01:48:10

we're making it smarter we're talking about let's take it

01:48:12

x current already current which we are now

01:48:15

classify the work on the current one

01:48:18

we correlated you on the wing with this very

01:48:20

We do sessions with the answer from the training

01:48:23

sample, that is, we will find such

01:48:24

spatial fig filter w for x

01:48:27

which will underline the answer

01:48:30

the one we saw in the training

01:48:33

sampling at this frequency you understand, yes

01:48:36

we do the same for our x x

01:48:39

relation to y as y f yes that is, as if

01:48:43

here but not really if

01:48:45

honestly x if I often don’t like this

01:48:48

guys it must be something like here he is

01:48:50

you already see the players

01:48:51

in vain not if this hole was drawn either

01:48:54

that's it but it doesn't matter here and further x by

01:48:58

relation x teaching to from training

01:49:01

samples in relation to y f and this too

01:49:04

there will be our filter and these filters will continue

01:49:06

we use it in a cunning way for

01:49:08

largely correlation analysis

01:49:11

here is between the teaching and the world between

01:49:14

examples from the training set and

01:49:16

With current data, I don’t want to go into detail right now

01:49:19

go into quite a lot here, but we still

01:49:21

I need to consider another method with you

01:49:22

I'll give you an article, you read it and but if

01:49:27

would you like to implement this method in

01:49:30

in principle it gives quite serious

01:49:32

an improvement if you compare

01:49:33

standard session is exactly

01:49:36

depending on the length, data for the segment

01:49:38

data and this is a network extender but in

01:49:44

this article is now

01:49:45

[music]

01:49:47

said if you figured it out what I

01:49:49

told you you take it, then you won’t have it

01:49:52

nothing would be completely incomprehensible

01:49:53

calmly watch this thing implement

01:49:55

here it just calculates different coefficients

01:49:58

correlations are further considered to be maximum

01:50:01

they will not determine the maximum way

01:50:04

is combined into one certain number and into

01:50:08

depending on this number on its

01:50:10

values you draw a conclusion about what

01:50:14

the team is so big but the number is big

01:50:17

just for those two minutes please rest

01:50:20

I need to run for exercises, I forgot this

01:50:22

no chat and let's continue the more often people

01:50:24

literally minutes

01:51:36

so this was the first method before the first two

01:51:40

method

01:51:41

based on canonical to relational

01:51:43

analysis

01:51:45

the second method and the first method of these was

01:51:49

without training, yes, that is, not required

01:51:52

training set records are very

01:51:54

convenient because by and large

01:51:56

can be from any person and water

01:51:58

right away, maybe not very high

01:52:00

accuracy but he will immediately use

01:52:02

your interface and no one interferes with

01:52:05

while he uses apply

01:52:07

then either extended this version, that is

01:52:09

how to get it or like this

01:52:10

I'll tell you the place now

01:52:13

which with which you will teach

01:52:16

the patient will learn how to use the

01:52:18

this thing

01:52:19

look at this again the article is all that

01:52:22

same ubiquitous nakanishi who are many

01:52:24

what have you done to this area?

01:52:28

here it's like this here again on crocodiles but

01:52:32

It seems like they can feed me already

01:52:34

seem simpler here

01:52:37

so I'm trying to understand how

01:52:39

like this thing

01:52:47

that means look

01:52:57

look at this method, then I hid this one

01:53:02

component analysis for your 50s

01:53:06

Cascades channel company was here

01:53:08

invented quite a long time ago in 2013

01:53:10

Japanese and

01:53:14

that's it man that's it in general

01:53:18

how did you work until the main struggle

01:53:20

here comes the search for spatial

01:53:22

How can I find a filter like that?

01:53:24

spatial filter which gives us

01:53:26

data will highlight what we need what

01:53:28

is relevant to the problem guy on this

01:53:30

case of the series, we were looking for such a filter

01:53:33

which will give us the output activity

01:53:35

as similar as possible to some linear

01:53:37

the combination of our harmonics is here

01:53:40

we are looking for space filter a little

01:53:41

differently here we look for it based on

01:53:44

considerations

01:53:45

that since stimulation is on everyone

01:53:48

each epoch on each repetition

01:53:49

the same then this is stimulation oao

01:53:54

space filter should be like this

01:53:57

that the data is filtered at the output

01:53:59

of this filter space from epoch to

01:54:02

era should be as similar as possible

01:54:04

between themselves

01:54:06

that is, look, it means

01:54:12

we have our data there are eras here they are

01:54:16

here the highlighted and

01:54:19

there we denote them this matrix we

01:54:22

denote by capital letter x with index k

01:54:25

Dick this number is a repetition essentially here

01:54:28

next we can do these things

01:54:31

cut yeah or we can inside

01:54:33

inside each matrix we can

01:54:35

let's count

01:54:38

such

01:54:41

yes we can count and we can

01:54:44

assume that we have some kind of filter

01:54:46

w

01:54:48

which we multiply by x cat and this one for

01:54:52

we multiply the filter by xk + 1 suppose

01:54:55

and these two sequences which

01:54:58

we denote y from and y a + 1 these two

01:55:01

followed each other

01:55:02

as similar as possible, that is, we have a correlation

01:55:04

let's think it will be a big number

01:55:07

close to close and will ration

01:55:09

mean close to one if not

01:55:10

well, it's just big, look here

01:55:13

correlation between spaces above

01:55:15

finntroll and us to that and and and eras

01:55:18

this is y cats y el todo mathematical expectations here

01:55:22

I put this work here

01:55:25

index t because there is a mathematical expectation

01:55:28

I mean this is a connection along time

01:55:29

along the interval of this epoch

01:55:32

just a quick second

01:55:37

this is because y is our space

01:55:40

filtered x and

01:55:42

y from and a y el d is space

01:55:46

filtered X-el you Troil yes this and

01:55:49

repetition of water as described below

01:55:51

since this is my scalar I am the same

01:55:54

the method is used like last time I did this

01:55:56

I'm breading because it doesn't matter

01:55:58

and no further opportunity arises

01:56:00

enter the mathematical expectation operation

01:56:03

inside

01:56:04

relative to the operation space

01:56:07

filtration and

01:56:09

I get that I actually have this w on

01:56:13

some mathematical expectation of this work

01:56:16

average between the fifth and alcantara is

01:56:20

essentially a cross correlation matrix

01:56:22

VK data atoms altom troit that is, we

01:56:25

take

01:56:26

returning to our volume we take

01:56:28

Kotov Troil matrix take matrix k +

01:56:32

1, for example, is equal to k + 1 before this

01:56:35

Troilus, turn it from 1, bread it and

01:56:38

count the product of this 5

01:56:42

first this one with the second this one with

01:56:46

third, and so on, and the surveillance scanner

01:56:48

follow each and with each

01:56:49

follow south here and come on + 1

01:56:53

in this situation we get the matrices

01:56:56

such sigma carel this matrix size

01:56:59

number of channels and number of channels which is well

01:57:03

here is a matrix that essentially

01:57:05

Now, if the set on the left has to

01:57:08

laid out and to the right on w

01:57:10

will give you some non-standardized value

01:57:12

correlation coefficient

01:57:15

but does not roll openly, is not standardized and

01:57:17

correlation

01:57:18

covariance will be the sector will be zero

01:57:21

the average is not normalized covariance

01:57:23

between

01:57:24

between that and Altay eras

01:57:29

Now we understand that Odessa painted that

01:57:32

We only replace sigma cards

01:57:35

time expectation for

01:57:38

sum over time averaged yes but

01:57:41

time connection but this is actually

01:57:43

covalently if you are these segments

01:57:45

call to search for data from raids using matrices

01:57:48

excel this covalent by the way multiply

01:57:50

on x spain

01:57:52

Well, then you will say further that well

01:57:56

ok and then you also need women to live

01:57:58

all eras are taken into account therefore all pairs and

01:58:01

I'll try, you want to be like yourself

01:58:02

it was between all the couples, one of them was yours

01:58:05

sum up the epoch for all pairs except

01:58:07

source of the summation except when k

01:58:09

equals and like this and like this and substitute this

01:58:12

you will get that this is a certain number

01:58:16

this is a scalar this is essentially a long time ago

01:58:18

transpose multiplied by this

01:58:20

double the amount from these elementary highs

01:58:22

covariance matrices cross cross epoch

01:58:27

one day it would be necessary if we designate

01:58:30

this matrix through c like here then y

01:58:34

we will have such compacts and expressions

01:58:36

further it is clear that this is about this cross

01:58:40

covariance yes this member which is for us

01:58:42

talks about similarity and now we need

01:58:45

ration something dunno that for the army

01:58:47

for general energy and for one night and

01:58:49

bath energy is just simple

01:58:51

mathematical expectations auto product this

01:58:55

win this vector y canta y cats then

01:58:58

is it actually

01:58:59

the expectation of the square this variance can be

01:59:03

was it possible to write network networks for accuracy?

01:59:04

square I decided not to write in this

01:59:06

variance is essentially an estimate of variance here

01:59:09

she signs the same way

01:59:11

so it turns out that now we have

01:59:14

you need sigma to to matrix yes. like this

01:59:17

simple we will denote it with the letter q well

01:59:19

this is the sum of all these matrices for all

01:59:22

epochs here and either just take the data

01:59:24

just everything at once

01:59:28

you can break it down into epochs, here you get it

01:59:31

kindness planned but further

01:59:33

the functionality is what we want it to be

01:59:35

correlation between epochs covariance

01:59:38

between eras compared to well

01:59:40

similarity of pores compared to the general

01:59:42

power

01:59:43

the satoshi-filtered image was

01:59:46

the maximum is equal to this at

01:59:48

maximum is the so-called

01:59:50

Rayleigh ratio which we also need

01:59:54

maximize find such w again

01:59:57

make a change of variables

01:59:59

now we substitute w here

02:00:02

this guy q minus theta spirit we have this

02:00:06

symmetric matrix that's because in

02:00:09

difference up to ten and so media 34 because

02:00:12

what do you see us symmetrical here

02:00:14

there is also

02:00:15

k more whined more k

02:00:19

but you can actually do the amount

02:00:21

only from and equal to + 1 and then just

02:00:24

add it transposed

02:00:26

Well, that means it’s with us

02:00:29

symmetric and the matrix is therefore q minus

02:00:31

-3 second it’s all the same

02:00:35

What

02:00:36

no matter what

02:00:38

q minus 1 2 it’s clear then

02:00:42

the denominator again from it all the data

02:00:44

leave only the numerator remains

02:00:46

the numerator turns out like this

02:00:48

conversion matrix here again too

02:00:51

same argument that it doesn’t depend

02:00:54

from scale to and

02:00:55

here it is easy to understand what it essentially is

02:00:59

cake derivatives formulated as

02:01:02

Lanka Quadrant will lovers task here and

02:01:05

it turns out that essentially our our w and attack

02:01:09

minus 1 2 multiplied by a own

02:01:12

vector of this matrix q minus 1 2

02:01:15

with q minus one second respectively

02:01:17

the maximum existing here and

02:01:21

then pipeline vodka water in this case

02:01:25

it turns out to be calm to pay attention that

02:01:27

here we have a presence and be given

02:01:29

training yes that is how we are

02:01:32

these things we mean that we are them

02:01:34

we get from the data to from mine

02:01:36

this is our training data and we have it

02:01:40

processed in a certain way

02:01:41

find a w that is maximally

02:01:43

gives us reproducible results from

02:01:45

era to era before had their own filtering

02:01:48

so that means we have training

02:01:50

sample but we filter with a different frequency

02:01:52

range

02:01:53

we do

02:01:56

component analysis we find this there

02:02:00

here w is our filter and we also take

02:02:03

we average these of our gang and

02:02:06

trading delta u among averaging them here

02:02:09

and now we take the averaged data and

02:02:12

filtered using this our

02:02:14

spatial filter a

02:02:17

and the other entrance we have is actually ours

02:02:22

original data our original data

02:02:24

passed through the same filter for

02:02:26

the kind of team we want now

02:02:28

she understands this line as we read it

02:02:34

again we filter this data with this

02:02:36

the same spatial filter and

02:02:40

two sequences competing between

02:02:42

we look at how much we have

02:02:44

sequence from this command from

02:02:46

this command matches the training set

02:02:49

with the fact that we intend from the brain for this

02:02:53

will already be in real testing data

02:02:55

the data that is now coming here

02:02:56

we get the coefficient and further them but also

02:02:59

will be able to combine certain

02:03:00

way here you can do it separately for

02:03:02

each for each harmonic until so not

02:03:05

for all the glory harmonica separate on

02:03:07

each harmonic and in a cunning way here

02:03:09

just certain coefficients

02:03:10

collect one coefficient in nickname and then

02:03:13

based on what for what

02:03:16

signal for such a signal you have this

02:03:19

signal

02:03:22

content and trading data team you have

02:03:25

we all maximize this correlation you

02:03:28

you can you can sew what to to to to to

02:03:32

k k k k k k maximum mouth

02:03:36

number and team number equals number

02:03:39

maximum number of thieves on this list

02:03:42

This

02:03:44

the first option is the simplest and there is more

02:03:47

there is also an extender

02:03:51

versions it lies in the fact that you

02:03:56

say that in fact you have these

02:03:59

here are the filters filters

02:04:02

they should probably be the same

02:04:04

depending on

02:04:08

gender and for all teams these are

02:04:12

there should be several filters

02:04:14

remove the filter for each of the commands and

02:04:17

then you get this multidimensional

02:04:20

matrix in essence and this multidimensional

02:04:22

you are making a matrix essentially of networks but already

02:04:25

in this matrix are filtered by this

02:04:26

here is a set of filters that you have here

02:04:28

no correlation coefficient just

02:04:31

I like it and here you actually have you

02:04:33

look too

02:04:34

the best possible combination of these

02:04:37

signals which one you need is the best

02:04:40

way combines your testing

02:04:42

data too to maximize

02:04:44

correlation coefficient we are with this too

02:04:46

understand, I don’t want details here either

02:04:48

go into we will build so you

02:04:49

read these articles listen again

02:04:51

caught up with the lecture there and we ask questions

02:04:55

we'll try to answer Nikita, take a walk

02:04:57

4 hope here is a Python script

02:05:02

which Nikita himself will show here

02:05:04

try to implement your homework

02:05:06

which you for example implement based on

02:05:08

simple tercel

02:05:11

this one is ensemble

02:05:14

How many versions am I currently on?

02:05:17

enough told just time already

02:05:18

quite a lot I want Nikita to see you

02:05:21

spent at least 45 minutes so that we

02:05:23

told me just went through the scripts

02:05:26

although the scripts there are all very good

02:05:28

notebooks are commented out but still

02:05:30

it will be very useful, so if

02:05:33

compare the performance of this there

02:05:35

this compared to extended and

02:05:38

networks whose r-squared but this

02:05:43

accuracy depending on sample length

02:05:46

data lengths, here we see what actually

02:05:48

In fact, if the data is already 500 milliseconds then

02:05:50

methods they go we stretch approximately

02:05:52

work the same but on short data

02:05:54

ensemble enter or see turns out

02:05:57

noticeably better than simple tensei nothing

02:06:01

extended economy of course we have this

02:06:06

no other than that, that's all from me

02:06:09

for today

02:06:11

listen and I understand that it was difficult

02:06:14

but don't give up they are all for real

02:06:18

in fact, everything is not very difficult now

02:06:19

you look at Nika right scripts

02:06:22

look at these matrices, draw them yourself

02:06:25

you'll feel them and things will work out, but

02:06:30

It’s difficult at first I agree that is

02:06:32

ok that's it

02:06:34

question if you have one let's

02:06:41

ok then Nikita just met who

02:06:47

yes, I'm stopping Cher now Nikita

02:06:50

solve your screen

02:07:03

I leave Nikita in good hands

02:07:05

please, I understand that there are many

02:07:07

questions write to discord I'm these two days

02:07:10

I'll be really busy but after two

02:07:13

days I will be able to answer your question

02:07:15

maybe even tomorrow is freedom night

02:07:17

let's do some homework too

02:07:21

Nikita and we won’t send you this one at all

02:07:23

week

02:07:24

[music]

02:07:26

long live let's move on to

02:07:29

practical part

02:07:30

our lecture today

02:07:34

if along the way show some

02:07:36

questions or something becomes interesting

02:07:39

to see more

02:07:41

you either if stid comments feel free

02:07:44

you can stop me

02:07:47

let's get started

02:07:49

the first thing we need to do is of course

02:07:52

download the necessary ones for work

02:07:54

library data

02:08:00

used it reached lectures this is how we

02:08:03

convenient to visualize everything

02:08:06

he first of all we need functions

02:08:09

to load the data we import a

02:08:12

then we will work with matrices

02:08:15

ideal extra matrix operations

02:08:18

for this we are from the skype library and

02:08:21

apartment im a library or millennial with

02:08:24

functions from linear algebra

02:08:26

further we will filter the data but

02:08:30

works with signals so we need

02:08:33

module

02:08:34

didactic piece and who is called Signu

02:08:37

also good for signal processing

02:08:40

we know the library lamp of course

02:08:42

will be needed to basically work with

02:08:44

multidimensional arrays with matrices

02:08:46

visualization we will use smart

02:08:48

kid matplotlib mod

02:08:52

payment is the fruit and accordingly is a

02:08:55

bible library he doesn't need it

02:08:59

the action took place, showed a little bit, we didn’t see it

02:09:03

we will use it today

02:09:05

Let's take the functions from there, also one for

02:09:07

visualization

02:09:08

and here I’m asking simply

02:09:11

parameters for drawing output so that they

02:09:13

there were more

02:09:18

let's now load our

02:09:21

First we'll download it, then I'll explain what it is

02:09:23

for the data where they came from and what from

02:09:26

introduces himself

02:09:28

well, accordingly we use

02:09:30

loaded functions to download

02:09:33

data format mother

02:09:36

its given they can be stored in different

02:09:40

completely formats for example

02:09:42

standardized as food fgds with them

02:09:46

you can easily download from them

02:09:48

accordingly, using the function and we do not

02:09:51

run and name but also data can

02:09:54

be in free

02:09:57

Not

02:09:59

standardized with toast not

02:10:01

standardized form saved in

02:10:04

formats such as text and imam given

02:10:06

case and we used

02:10:11

data is posted in the public domain and gardens

02:10:15

topaz dedicated

02:10:17

interface research there means ours

02:10:21

data

02:10:22

saved in mat-la format babes.com

02:10:24

dashcam and here we are loading this data

02:10:27

they are depicted as a dictionary and we

02:10:30

We can first see what kind

02:10:32

there are fields in this dictionary but how

02:10:35

accordingly the structure and years it

02:10:37

should contain not only the data itself

02:10:39

not only the time series themselves but also

02:10:41

accordingly we need

02:10:42

patient information and records, etc.

02:10:47

further but let's see what it is

02:10:49

fields in general

02:10:50

well, the first avenger is still a field

02:10:54

it's called our temporary ones

02:10:56

rows is what we will analyze

02:10:57

then there are some fields

02:11:00

this is the age of the patients for example

02:11:04

age of the patient with which the subjects

02:11:08

we are looking for the center of the subject with whom we

02:11:10

recorded the data and then this

02:11:13

structure contains information about the channel with

02:11:15

which we recorded them, including them

02:11:17

names of positions on the head and so on it

02:11:19

we will need everything and then

02:11:21

encoded frequencies we further because

02:11:24

what's happened

02:11:25

further still no one information coded

02:11:28

phase a and a very important parameter frequency

02:11:31

sampling

02:11:32

some more parameters and related

02:11:36

information regarding this entry

02:11:38

accordingly we must load

02:11:40

remember the variables and some of the most

02:11:44

important a

02:11:45

Further

02:11:48

firstly, this is of course the frequency

02:11:49

we can't discretize anything

02:11:51

the right thing to do if we don't know

02:11:52

with what frequency

02:11:56

sampling we recorded yes and

02:11:59

then these are the encoded frequencies and phases

02:12:02

exactly the frequencies we will try

02:12:04

decant from our homes so it's

02:12:07

the field is also very important

02:12:10

which you need to remember and

02:12:12

of course, acute data, that is, ours

02:12:14

We also upload time series reader

02:12:16

from this library

02:12:18

because of this layer

02:12:23

respectively

02:12:25

I will also leave our coded ones

02:12:29

frequencies

02:12:36

I also saw the policy coded

02:12:39

frequencies we can see a

02:12:41

activities what we will decorate there

02:12:43

this is all in hertz, that is, from 87 frequency 8

02:12:48

up to 15 and 6 dense no more frequency

02:12:53

encoded phase so it's the same

02:12:56

the main thing is to understand how our data is organized

02:12:59

here I have displayed the form of arrays which

02:13:03

contain a multidimensional array which

02:13:05

contains our raw data we see that

02:13:07

it has as many as 4 dimensions 64 750 by 4 by

02:13:13

40 well, at first glance it’s difficult

02:13:15

let's figure out what it is now

02:13:17

let's move on to the experiment itself

02:13:20

accordingly, they are directly dedicated

02:13:23

interface that is

02:13:26

but one of the practical real

02:13:30

practical applications most clearing

02:13:32

steppe interfaces is

02:13:35

typing text using the interface is

02:13:38

means that the subject is introduced

02:13:41

and a keyboard with symbols on the keyboard

02:13:45

computer from keyboard on monitor

02:13:48

which simulates a computer keyboard

02:13:50

and with the symbols a respectively these

02:13:53

the symbols are blinking

02:13:57

they blink at a certain frequency

02:13:59

each symbol blinks at its own frequency

02:14:02

which differs from all from the number of all

02:14:04

in other symbols it also blinks with

02:14:07

certain beginnings and initial phase

02:14:09

accordingly, the subject looks at

02:14:13

pre-selected

02:14:17

supervision not specified symbol a

02:14:20

accordingly, this organ has symbols

02:14:24

it stimulates the brain im trying

02:14:26

decode

02:14:28

the signal that is the result of this

02:14:32

stimulation as a whole is organized

02:14:35

experiment and

02:14:38

according to the presentation of stimuli

02:14:40

happened so-called as in all

02:14:43

experiments occur in this way

02:14:45

called edges are separate

02:14:48

fixed individual

02:14:51

blocks of our experiment and in each of

02:14:55

which are presented with one stimulus

02:14:58

The timing of this rule is strictly defined

02:15:02

that is, what happens at the beginning of Troilus

02:15:04

first for 0 5 seconds and for

02:15:07

the subject is illuminated and

02:15:09

stimulus for which the stimulus is a symbol

02:15:12

Which one should I watch this on?

02:15:14

occurs within 0 5 seconds a

02:15:16

Illuminated with red light

02:15:18

during this time the subjects have time

02:15:20

bring your eye to this symbol from

02:15:21

fixate on it next happens

02:15:24

stimulation itself, that is, but in

02:15:27

stimulus selection time does not occur

02:15:29

blinking symbols during and then

02:15:33

self-stimulation occurs

02:15:35

try it, you can fix your vision on

02:15:41

the necessary incentive and then everything is incentive

02:15:43

Each one starts blinking at the same time

02:15:45

I said with my purity and with my

02:15:47

phases in this organ occur during

02:15:49

two seconds

02:15:50

this is exactly the section of the rules

02:15:53

and we will be interested after this

02:15:56

coloring seconds there is a rest when there is no

02:16:00

stimulation not choice of stimulus

02:16:01

accordingly, we have everything on the keyboard

02:16:04

40 different stimuli

02:16:08

in one block in random order

02:16:12

each wall was presented one at a time

02:16:15

once, that is, well, the subject had to

02:16:18

skim for one block

02:16:20

for all incentive one for all

02:16:22

symbols once each, as a result we have

02:16:25

one block 40 repetitions 40

02:16:27

presentation of stimuli or 40 rules a

02:16:29

this is exactly the size we see ours

02:16:32

houses are in last place and in total

02:16:37

Total

02:16:39

4 blocks were recorded and that is, well

02:16:44

four identical blocks, let's say

02:16:46

composition for the order of stimulus presentation

02:16:48

A

02:16:50

each dress with a theme was presented

02:16:52

once and we see this number by 3

02:16:56

positions a of our array

02:17:00

next are the remaining parts

02:17:04

64 is the number of the channel used

02:17:07

records, well, where can it be written, let’s compare

02:17:10

sense of channels

02:17:11

depending on what we are researching

02:17:15

in principle, not so much happier

02:17:18

must have high channels a note

02:17:21

1 standard number is 64 and then 750

02:17:26

Well, you can already get to what it is

02:17:27

number of time reports in one

02:17:29

trailer that is where does 750 come from

02:17:34

I output the sampling rate

02:17:37

from our experiment it is 50 hertz and

02:17:42

which means that within 1 second we

02:17:45

write down a 250 countdown and just like me

02:17:48

said in one rule happens

02:17:51

total one rules total lasts

02:17:53

three seconds and that means to understand

02:17:56

how many reports do we have in one rule?

02:17:58

must multiply the number of seconds by

02:18:01

sampling rate

02:18:02

multiplying 250 by 3 we get just

02:18:05

our 750 in the end is with us again

02:18:09

four-dimensional array which is 1 by 1

02:18:12

axis there are channels on the second axis

02:18:15

temporary reports are located

02:18:18

on the 3rd axis the block number and on the middle axis

02:18:23

incentive number

02:18:29

stimulus and present random

02:18:32

okay, but then for convenience these

02:18:36

the data has already been exposed to anyone

02:18:38

processing they deciphered them for

02:18:42

convenience is back in

02:18:44

sequential order

02:18:46

that is, from 8 and 6 to 15 8 and to 28 a

02:18:57

further here I announce several

02:19:00

functions which

02:19:02

lead some fields to a convenient

02:19:05

mind

02:19:06

let's see what these fields are first

02:19:08

will be needed to visualize our

02:19:10

data on

02:19:12

surface of the head opposition channel but

02:19:16

we load from our ours

02:19:18

dictionary general information about channels

02:19:21

right here

02:19:23

fields here

02:19:26

indexes 1 and 2

02:19:31

information channels it costs it too

02:19:34

the array consists of three

02:19:39

fields on the first field is set to 1 from

02:19:44

the field stands accordingly to the coordinates

02:19:47

used channels we download these

02:19:49

coordinates

02:19:50

but these coordinates are not very convenient

02:19:53

form for us

02:19:55

one of these coordinates is

02:19:59

azimuth azimuth angle

02:20:02

relative to the axis

02:20:05

here is a thematic drawing of our

02:20:08

coordinates this is the z axis

02:20:12

it corresponds to the axis which

02:20:15

pass from the base of the skull through

02:20:19

the base of the skull through the back of the head is the x axis

02:20:24

the y axis corresponds to the axis that passes

02:20:29

converges into the skull behind and comes out

02:20:32

front

02:20:34

according to our position

02:20:38

where are the cards encoded?

02:20:39

coordinates, that is, not through x and y but from

02:20:42

using the azimuthal angle and

02:20:44

inclination angle

02:20:47

azimuth angle and a we want to translate

02:20:50

besides one of these corners

02:20:56

one hare one of these coordinates

02:20:59

presented in radians, others in

02:21:02

degrees this is inconvenient accordingly we

02:21:06

we transform and our coordinates from

02:21:09

the same coordinates from steric so

02:21:11

called that is, from angles fi, etc.

02:21:15

[music]

02:21:16

x and y coordinates and in the end we print

02:21:20

positions of our channels accordingly

02:21:22

the dimension of this waving massive bangs

02:21:25

64 number of channels for 22 is two

02:21:29

coordinates

02:21:31

why do we have two coordinates and three goals?

02:21:34

three-dimensional after all, but for convenience

02:21:36

visualization we seem to project onto

02:21:38

x y plane that is on

02:21:41

horizontal plane and in the end he

02:21:42

there are two coordinates left

02:21:44

here is the opera fart then this is x 2 basket

02:21:49

the second field is y

02:21:51

further

02:21:55

in addition also for visualization to us

02:21:59

we need to ban the channels too

02:22:01

we do not load from our general dictionary

02:22:04

date dict and

02:22:06

converted too normal view otherwise

02:22:08

there is simply a list and a list of strings with

02:22:12

channel names and

02:22:15

we can print these channel names but here

02:22:19

we see the list of our channels so far

02:22:22

Most likely you are familiar with these names, but

02:22:25

then we will see where they are in the pictures

02:22:27

each channel is located

02:22:29

electrodes

02:22:32

further now let's get started

02:22:35

capitalization of our data and first

02:22:38

turn we have to take a

02:22:40

some kind of well

02:22:44

for clarity we should take

02:22:47

one kind of troil, that is, one segment

02:22:49

life 750 milliseconds some one

02:22:52

channel which we most likely should

02:22:54

be

02:22:55

activity must arise as a result

02:22:58

stimulation light

02:23:01

well you should take it accordingly

02:23:03

some block and

02:23:04

well, some frequency for which we will

02:23:07

visualize our signal these 3

02:23:09

lines are exactly what we do and we choose

02:23:11

one channel one block and one frequency

02:23:15

one incentive

02:23:18

this function is sour, it can’t be removed

02:23:21

we need

02:23:23

in this line we return the index

02:23:26

which corresponds

02:23:28

selected frequency and the squeeze function

02:23:31

selects those we don’t need autumn single

02:23:34

actors who will remain after the function

02:23:38

arch you and

02:23:41

next we ask

02:23:44

further to output graphs to a variable

02:23:47

timer inch is an array

02:23:51

time reports

02:23:54

converted to milliseconds translated

02:23:57

just in seconds

02:23:59

Now

02:24:01

no no no there is an array of our reports

02:24:09

accordingly and we we want us

02:24:13

interested as the segment said

02:24:15

lined up with which stimulation occurs

02:24:17

his activity he begins through

02:24:20

zero five seconds in half a second

02:24:22

after the start the rule ends in

02:24:24

five seconds left rule we

02:24:28

we want to cut it out of our rule here

02:24:31

part of the video actually happens

02:24:32

stimulation I take to visualize it for

02:24:35

we create this ourselves

02:24:38

an array of frequencies with which we will

02:24:41

cut

02:24:42

cut out the segment of interest to our

02:24:44

Troy a

02:24:47

further a

02:24:49

then we remember the changes

02:24:52

the number of our discrete samples then

02:24:55

there is the length of this array, this is for us

02:24:56

will come in handy, well, let’s create an array and

02:24:59

time for visualization and already in

02:25:02

seconds for this we simply divide and we

02:25:05

we create an array of length

02:25:09

integer array of naturals

02:25:11

numbers 1 whose length is

02:25:15

the length of the analyzed segments

02:25:18

in reports and a divide by frequency

02:25:21

sampling

02:25:22

sampling division results

02:25:24

discrete reports in seconds further we from

02:25:29

cut out our raw data

02:25:31

a necessary piece of data as I said

02:25:34

we set the variable in advance than Yandex

02:25:37

Yandex and which talks about how what

02:25:40

channel we will use which block

02:25:42

what frequency will we use

02:25:43

we will use and using an array

02:25:46

secret range we cut accordingly

02:25:48

set temporary piece of ours

02:25:51

so then we just visualize and here

02:25:54

this piece is 2 seconds long

02:25:56

let's take a look at it

02:25:59

this is what the rules are like

02:26:04

any area we see what is here but

02:26:07

here on the x axis and these are milliseconds

02:26:10

interrupted the seconds accordingly, here it is

02:26:12

it takes 2 thousand milliseconds or two seconds

02:26:15

here and along the y axis is the signal amplitude

02:26:18

that is, well, somewhere around 20

02:26:22

we see signals that there seem to be some

02:26:25

some kind

02:26:28

something is always present in principle

02:26:31

according to this schedule we can't do anything

02:26:32

say that's why we move from

02:26:35

frequency analysis

02:26:37

so how to find out from which frequency

02:26:41

component consists of our signal well for

02:26:44

this

02:26:45

but as the simplest option, is it possible

02:26:48

just take it

02:26:49

discrete fourier transform in python

02:26:53

discrete punctuation is implemented with

02:26:55

use the lamp function to eat

02:26:58

from under the model

02:27:02

f-f-f-f-f-f that means briefcase

02:27:04

transform posted on and pharm is the same

02:27:07

the most discrete transformation Fred

02:27:09

but only but it gives the same result

02:27:11

but it's just faster

02:27:12

implementation of this transformation that

02:27:15

what do we get as a result of this

02:27:18

transformation we had a time series

02:27:21

this is this

02:27:24

I bought this netrebko map told

02:27:27

but translates the signal into the frequency domain

02:27:31

here they are, we get an array with

02:27:33

length but only now

02:27:36

each point encodes an amplitude and

02:27:41

phase of the number of clicks corresponding

02:27:43

frequencies and components that make up

02:27:45

our signal is us first

02:27:46

interested in and the amplitude is

02:27:50

amplitudes a

02:27:53

found for the component that inserts

02:27:55

signal from its pre-conversion

02:27:57

free she will come in

02:28:00

complex assigned set assigned

02:28:03

time complex significant array a

02:28:05

because

02:28:08

every every . she codes

02:28:10

simultaneously m this and frequency

02:28:12

component and its phase phase is encoded

02:28:15

in absolute terms

02:28:19

this is a kit

02:28:21

this is this complex number a

02:28:25

phase is encoded

02:28:27

arctangent of the relation between it and the part to

02:28:31

[music]

02:28:32

real part of our complex number

02:28:35

and we are mainly interested in how we searched

02:28:38

namely the absurd component

02:28:41

amplitudes of the composition of the oldest to

02:28:43

we actually make the component

02:28:46

transforming ri and immediately taking absolute

02:28:47

values

02:28:49

that is, we immediately find the amplitude

02:28:52

amplitude that component is just

02:28:56

normalizing coefficient and which

02:28:58

needed for

02:29:00

in order for the result to be where

02:29:03

rendering and small then physical size

02:29:05

further

02:29:07

for insulation we create accordingly

02:29:10

array of frequencies and

02:29:12

after that we visualize the spectrum here

02:29:16

in these lines

02:29:18

let's look at narita renderings

02:29:21

this

02:29:24

picture

02:29:25

we see that during transformation it

02:29:30

in general it is symmetrical relative to

02:29:35

central frequency which is equal to

02:29:37

half the sampling rate

02:29:44

Accordingly, it is visualized here

02:29:46

so that components from zero to one hundred and twenty

02:29:49

five hertz why 120 hertz because

02:29:51

your sampling rate is 250

02:29:54

by half it's 120 but on

02:29:58

in fact, we are not interested in the game and in

02:30:00

in the slip we are not interested in high frequencies

02:30:03

usually she hears frequencies there

02:30:05

range from zero to 30-40 hertz

02:30:09

usually therefore for we visualize

02:30:12

only part of our spectrum from 0 to 50

02:30:16

hertz

02:30:19

accordingly this

02:30:21

shows

02:30:25

the contribution of each component is yours

02:30:28

signal here we see that for example but in

02:30:31

Is it our right to make a greater contribution please?

02:30:33

introduce

02:30:36

component frequency corresponding

02:30:38

12

02:30:40

further

02:30:42

but really I'm just taking

02:30:45

transformation

02:30:46

usually they don't do that usually research

02:30:50

associated with her fighting tongues I

02:30:54

transformation a

02:30:56

quantity which is called spectral

02:30:58

power density is actually simple

02:31:00

square

02:31:02

square

02:31:04

the result is straight

02:31:06

hello and education results module

02:31:12

respectively

02:31:14

It can be done

02:31:16

erect get make transformation

02:31:19

get the spectral density from it

02:31:21

power can be used using the built-in

02:31:24

functions in the library skype signal I'm

02:31:29

why such a strange name really?

02:31:32

In fact, this function is not entirely simple

02:31:35

takes our whole times series and finds

02:31:38

in transformation it goes to square on

02:31:40

in fact, for a more accurate assessment

02:31:42

spectral components

02:31:44

it splits the signal into overlapping

02:31:48

windows of small length, let's say 1 second

02:31:52

there are two seconds and in these windows there are

02:31:56

queue finds

02:31:58

finds limits when transforming

02:32:01

before that I apologize before that she

02:32:04

each window multiplies by some

02:32:06

smoothing window to prevent

02:32:15

A

02:32:21

to make these windows they change

02:32:24

the shape of our signals is a little bit so

02:32:26

our assessment of the components becomes

02:32:28

more accurately

02:32:31

these windows could have different shapes and

02:32:33

there are a lot of different types and in this

02:32:36

case we use a simply

02:32:38

she clicks on the rectangular window

02:32:41

signals to a rectangular window are simple

02:32:44

which is equal to one everywhere

02:32:45

accordingly and immediately that I forgave you

02:32:47

conversion to

02:32:50

in each window each window is multiplied by

02:32:54

1

02:32:56

each segment of our data for which

02:32:58

splits this function multiplies by the window

02:33:01

changing its shape a little after

02:33:04

this is done when converting after

02:33:06

this is located

02:33:10

we go out to square

02:33:12

we find absolute values

02:33:15

spectral densities in this window and

02:33:17

we do this for each window to average the windows

02:33:19

As a result, we obtain the spectral estimate

02:33:21

slender power spectral density

02:33:23

on everything

02:33:27

on everything that interests us in time

02:33:29

segment

02:33:30

accordingly let's look at

02:33:32

the result of this this function

02:33:38

on the left is the result of this function

02:33:41

we took the rules from the channel with the name

02:33:44

patreon is located in the occipital region where

02:33:47

we have

02:33:48

example is located

02:33:50

part of the cortex responsible for processing

02:33:52

visual information we see that

02:33:55

gives functions up in principle too

02:33:58

result what and if just take them

02:34:00

I take the transformation from him

02:34:02

absolute values only we see that

02:34:04

attitude here is the mind of the watch component they

02:34:07

changed a little because it

02:34:09

just an absolute value and this is a square

02:34:11

absolute values, respectively, we

02:34:13

we just took a rectangular window, we can

02:34:15

change it

02:34:17

take for example

02:34:19

window x

02:34:21

so in our case the shape of the spectrum is a little

02:34:26

will change but not

02:34:31

now we'll see it

02:34:37

so we changed our window we see that

02:34:41

now the spectrum is slightly different

02:34:42

obtained using ft and a spectral

02:34:45

power density but the overall picture is not

02:34:47

changed and then in these lines of code

02:34:52

in these last lines there is water and we

02:34:55

We also take as an example the rules from

02:34:57

another channel

02:34:59

which is called f-1 f1 it is located

02:35:02

somewhere in the front of our head

02:35:05

located on the front of our

02:35:07

heads and he in general in general

02:35:12

5 corr and

02:35:15

she is far from she herself

02:35:19

process information so I drive

02:35:21

assume that in this channel we do not have

02:35:24

will

02:35:26

frequencies of components arising in

02:35:28

as a result of stimulation

02:35:30

periodic and

02:35:32

here we compare in cultural density

02:35:35

power

02:35:36

in these two channels we watch and in

02:35:40

channel 3 which corresponds

02:35:42

visual area we have clearly expressed

02:35:46

peaks which matches first

02:35:48

stimulation frequency remember we chose

02:35:51

12 what to watch we see a clear peak

02:35:53

on 12 hearts as a result

02:35:57

free conversions except

02:36:01

talked about the brain as a nonlinear system and in

02:36:03

there is no response to this stimulation

02:36:05

it’s also natural that harmonics are

02:36:08

frequencies, that is, frequencies are multiples

02:36:10

that is, the frequencies are multiples of and a fundamental

02:36:13

frequency

02:36:14

that is, 24 hertz 36 well, 48 is no longer there

02:36:21

or it's there from the filter

02:36:25

50 hertz is usually interference from the supply

02:36:29

networks are therefore close to 50 hertz here

02:36:32

normal signal is always filtered immediately

02:36:35

so maybe this component is a

02:36:37

at the same time I also filtered 50 hertz

02:36:39

Now let's look at another channel f1 and in

02:36:42

which we, as you assumed,

02:36:44

it seems like it shouldn’t be, but what is a component?

02:36:47

caused by

02:36:49

stimulation

02:36:52

valid upon conversion to us

02:36:55

shows that

02:36:56

our power figure shows that

02:37:00

we are not there nor is the fundamental frequency here

02:37:05

and harmonics

02:37:06

accordingly we see that

02:37:09

build a spectral spectrum accordingly

02:37:11

power densities for other channels we

02:37:13

we also see different pictures and we see that

02:37:15

in some channel we have a

02:37:18

the signal we are interested in

02:37:20

other channels it may have

02:37:23

slightly different shape therefore bear

02:37:25

more information about some

02:37:26

our channel is generally just noise, they are noise

02:37:30

which may also be present in

02:37:31

channels of interest to us, therefore these

02:37:34

channels can also be used for

02:37:35

removing this noise as reference

02:37:38

roughly speaking

02:37:39

this is exactly what motivates us

02:37:41

use for processing

02:37:43

signals including

02:37:48

with slippy spatial filtering

02:37:51

such a combination of channels in which

02:37:53

would allow us to highlight all the useful

02:37:55

information

02:37:58

get rid of all useful information

02:38:01

get rid of information as best as possible

02:38:04

which we are not interested in

02:38:06

component from signal sources which

02:38:09

had an employee and they were interested

02:38:11

us information

02:38:15

Well, besides, we can’t work right away

02:38:18

with such data with raw

02:38:21

since we saw our frequencies

02:38:24

stimulations extend from

02:38:27

8 hertz

02:38:29

from 8 gear

02:38:33

that at and 16 hertz and only 40 something

02:38:38

simulations accordingly interests us

02:38:40

also their harmonics, including her

02:38:45

we usually take

02:38:49

usually we take a3 of the first harmonic

02:38:52

respectively at and above the frequency which

02:38:54

us be and draw this is the maximum

02:38:56

basic purity of stimulation that is 16

02:38:57

Hertz multiplied by 3, that is, it turns out

02:39:00

48 hertz

02:39:03

up to 48

02:39:07

accordingly a

02:39:09

the rest of the components are out

02:39:12

frequency gap

02:39:15

8-16 hertz doesn’t interest us, we don’t even

02:39:19

filter

02:39:21

we do this with the help of peter pro

02:39:23

which Alekseevich and I’ll tell you how

02:39:26

it can be done but first we must

02:39:28

determine here set the lower frequency

02:39:31

cutoff so-called and higher frequency

02:39:33

cut

02:39:34

which indicates exactly what

02:39:36

range

02:39:37

we will pass the signal to Kandy too

02:39:42

in what range will we suppress our

02:39:44

frequency components accordingly here

02:39:47

we set below 6 659 above 50 hertz

02:39:52

why don't we take exactly 8 hertz and

02:39:55

let's say 48 hertz exactly because we

02:39:59

we take with some indentation from the extreme ones

02:40:01

the purity of us our interests us

02:40:03

interested because the filter has it

02:40:07

it has a frequency response

02:40:08

some transition area

02:40:13

some transition area bottom

02:40:16

transition region some widths in

02:40:18

which a

02:40:21

in which little happens

02:40:24

suppression too

02:40:25

full-time companies respectively

02:40:28

states that interest us and preferably

02:40:30

that’s not why we take the spectrum further

02:40:33

but Vinci also told you about

02:40:37

odds you can which we will

02:40:38

milling is what is in the bathrobe nutrition

02:40:40

they can be

02:40:41

easy to get

02:40:43

using one function

02:40:48

In total there are two types of filtration

02:40:51

first we will look at filters with

02:40:53

of course the impulse response a

02:40:56

filtration coefficient designs for

02:40:58

they are made to sleep on using

02:41:00

functions ferre bin i.e. window filter

02:41:04

but because actually what I’m leading because

02:41:06

what actually is this

02:41:08

multiplying my coefficients by some

02:41:10

small sliding window according to our

02:41:13

given this function from the signal library

02:41:17

wireless library in one of the signal modules

02:41:20

Skype libraries then naturally these

02:41:24

all the functions they have are good

02:41:25

documentation the second is very easy and

02:41:28

look at the website let's take a look

02:41:31

what parameters does this function take?

02:41:33

first of all this is the order of the filter sort

02:41:36

the filter shows this parameter which

02:41:40

which tells us how much

02:41:43

well our filter will be strict

02:41:46

filter

02:41:47

they are the feeling components that interest us

02:41:50

100 components that are not of interest to us than

02:41:52

the longer the length of the filters the better in our

02:41:54

filter the higher the order the filter the

02:41:56

better we have our filter however here

02:41:59

there are problems that the larger the order

02:42:02

our filters first topics

02:42:04

wider window here we are sliding

02:42:08

we should take the wider the window

02:42:10

should take more for filtration

02:42:12

the more temporary we come

02:42:15

the delay would be temporary

02:42:17

delay if we were analyzed by Henri

02:42:19

signals were analyzed and filtered in

02:42:22

real time that is if

02:42:23

for example we took a huge order

02:42:26

filtrate lengths

02:42:27

arguing

02:42:29

[music]

02:42:30

which corresponds to two seconds then

02:42:33

we would be delayed

02:42:34

h1 seconds so we have to remove such

02:42:39

filters [music]

02:42:41

order filters compromise well that is

02:42:43

who would give

02:42:45

satisfactory filtration but at

02:42:47

I wouldn't give it too high

02:42:49

delay further and on the second

02:42:52

second field

02:42:54

the second parameter it takes

02:42:58

lower

02:43:00

lower and higher frequency light lead

02:43:03

the boundaries of which we will create our

02:43:05

signal next parameter

02:43:08

sampling rates and last

02:43:11

the parameter doesn't give a damn, that's all

02:43:13

says that we have canvases

02:43:15

We will use strip passes

02:43:17

transmitting filter, that is, filters

02:43:18

which passes the range of 6,550 hertz

02:43:22

they suppress these frequencies and if you do

02:43:25

pose ru then on the contrary he will

02:43:27

leave all frequencies except clearing with

02:43:30

ranges 6 550 hertz

02:43:37

accordingly we accordingly

02:43:40

how can I not see it anymore

02:43:43

sampling frequency is not necessary

02:43:45

pass to this function if we don't

02:43:47

convey the implementation principle then we

02:43:49

must transmit

02:43:51

frequencies not in hertz in parameters

02:43:55

receiving low and high frequencies

02:43:58

cut a

02:44:00

she reinforced the frequencies of more worlds

02:44:03

relative to frequency nike relative

02:44:05

half sampling rate

02:44:06

so anyway

02:44:08

our

02:44:11

to create normal to create

02:44:16

we need a filter for frequency

02:44:17

sampling simply we either as

02:44:19

we transmit the parameter and simply emit frequencies

02:44:22

or we ourselves normalize the frequencies

02:44:24

half sampling something and not

02:44:27

indicate in this function what is there

02:44:30

association but

02:44:34

Well, we transmit normalized frequencies

02:44:37

Here's how to visualize the slice further

02:44:40

what does this filter do for us

02:44:43

using also built-in functions

02:44:47

fixed

02:44:48

from these ethics coefficients to the center

02:44:51

we transform the so-called gearbox

02:44:53

functions or frequency characteristics

02:44:55

which shows the response of our filter

02:44:57

to different frequency components

02:44:58

return age accordingly

02:45:01

purity and a complex number which

02:45:04

Also

02:45:05

cinema bar of complex numbers which

02:45:08

correspond

02:45:09

from q

02:45:11

which are worth information about the opening

02:45:14

our filter to the relevant

02:45:16

components as well as

02:45:20

delay carpenters on different that you

02:45:22

company we translate returns function

02:45:26

cyclic frequency which is usually

02:45:28

frequency is connected simply through

02:45:29

coefficient a2 pi and

02:45:31

we from our transfer function it

02:45:35

complex meaning as well as

02:45:38

there are already principles in the analysis and we

02:45:42

just take absolute values

02:45:45

cleaning transfer function so that they

02:45:50

but to visualize on

02:45:56

suppression of our signal to destroy

02:45:58

For

02:46:00

to visualize ok domestic

02:46:03

the company is passed by our filter which ones

02:46:04

suppress a

02:46:07

accordingly we

02:46:12

it's called what else said and

02:46:14

is called the frequency response

02:46:17

there is a dependence of the output amplitude

02:46:18

signal from frequency

02:46:22

we witches visualize this

02:46:25

characteristics

02:46:28

let's see how it is here

02:46:31

frequency response here along the x axis

02:46:33

and also the y axis stats are ratio

02:46:36

output signal to pure hubble up to

02:46:39

output signal at a given frequency

02:46:41

input signal at frequency we see that

02:46:44

this is how our filter was required

02:46:46

misses that is, the relation is off

02:46:47

the input signal is equal to zero frequency above

02:46:50

50 hertz and also low frequencies

02:46:53

relations between the output Kokhanov signal

02:46:56

the bandwidth required is equal to

02:46:58

unit, that is, the filter is completely without

02:47:00

distortion misses these frequencies

02:47:02

components also we see that this

02:47:04

filter

02:47:06

that we consider this filter also for

02:47:10

this filter characteristic which

02:47:12

tells us how clean

02:47:14

different components are retained in our

02:47:16

filtering

02:47:17

if we filter online

02:47:20

as you can see and all the components and

02:47:25

This is the advantage of a filter with finite

02:47:27

pulse on their characteristics

02:47:28

delayed okay the same amount

02:47:30

report on 25

02:47:33

respectively, another type of filtering

02:47:36

which in this case is not possible

02:47:37

uses this filter with infinite

02:47:40

impulse response that is

02:47:41

which uses feedback on

02:47:44

filtering it is also set using

02:47:47

functions it parameters yours similar here

02:47:50

manages to order the filters here

02:47:53

accordingly, the filter order is

02:47:56

He characterizes women as if he usually

02:48:01

much less than a teacher

02:48:03

Of course, with the impulse response we

02:48:05

also give the type and filter type but

02:48:09

cutoff frequency and filter type for a given

02:48:11

case we are interested in and this is what we call

02:48:14

there is also a bandpass filter

02:48:17

We are interested in skipping the third packet

02:48:20

it only transmits stylistics to the filter

02:48:25

straightaway

02:48:26

22 array of coefficients and

02:48:29

and according to their words from these

02:48:34

accents again we find the characteristic

02:48:36

frequency filters

02:48:38

they use it to build a frequency cross

02:48:42

and chest group delay loans we

02:48:45

we see that

02:48:48

we see

02:48:50

here similarly for this filter we

02:48:54

built

02:48:56

we are building

02:48:58

absolutely what is the characteristic and

02:49:00

group delay in this case I

02:49:02

I forgot to change it and here we have it

02:49:05

amplitude-frequency response

02:49:07

it has the size you need

02:49:11

here on the y axis we have logarithmic

02:49:14

the scale is easy to change the hour

02:49:31

here we are now again white oil

02:49:36

we just see in relative units

02:49:40

that in principle

02:49:44

2 types of filters but they are similar, that is, well

02:49:47

yes it is clear and they fulfill ours

02:49:49

strength and same function disadvantage filter with

02:49:51

of course endless characteristics in

02:49:53

the fact that it does not have the same delay

02:49:56

different feelings different different sentries

02:49:59

companies, that is, he is delaying some

02:50:01

stronger some less including in

02:50:03

the bandwidth we are interested in

02:50:05

that is, here we have you lowering

02:50:08

somewhere here it starts no here

02:50:10

ends we see that even after

02:50:12

transmission

02:50:13

auto parts for which we don’t want how

02:50:16

we don’t want to touch anything, we don’t want to change it

02:50:19

we have different ones like pro and detain

02:50:21

they have

02:50:23

different distortion this leads to distortion

02:50:25

waveforms

02:50:29

Next, let's apply these filters to

02:50:32

to our real data for this we

02:50:34

again select the piece we are interested in

02:50:37

data about people of interest in the area of interest

02:50:39

us in the premium range and use these

02:50:42

filters using

02:50:45

using the built-in train function

02:50:48

called

02:50:50

Now

02:51:02

here we filter in this block

02:51:05

is found by filtering using the function from

02:51:07

fairy is a filter what function parameter

02:51:11

these are the coefficients we need

02:51:14

accordingly, for filtering, well, a piece

02:51:16

data it can be multidimensional, that is

02:51:19

you can immediately filter which

02:51:22

we want to filter and indicate

02:51:24

only to and filter reminding about

02:51:26

corresponds to this second axis filters

02:51:29

model not n.a.

02:51:32

we are using the built in phi function here

02:51:33

this is offline filtering, the so-called

02:51:36

there is it like filtering first in direct

02:51:39

with responsibility then she doesn’t give a beaver

02:51:41

no phase distortion for any filter

02:51:43

and accordingly in the online mode of this

02:51:46

impossible because we don't know and

02:51:48

Sarah's signal is not entirely but there is on analog

02:51:51

these functions to find accordingly

02:51:54

it also returns 3 documentation

02:51:56

just a visual signal

02:51:58

filtered by coefficients

02:52:00

which we will build in advance with the help

02:52:02

anyone using some function

02:52:06

using any function or constructions

02:52:08

filter

02:52:10

further

02:52:11

let's visualize

02:52:14

our filtered signal

02:52:16

this is what's happening here we're just

02:52:19

build a signal in time in time

02:52:21

region

02:52:24

2 seconds and this is an unfiltered signal

02:52:28

from we see that it becomes a little more

02:52:30

smooth and a lot of pain ron filtered

02:52:34

high frequency components we

02:52:36

also embedded below each graph

02:52:38

corresponding to spectral density

02:52:39

power and we see that first here we have

02:52:41

there was some kind of us down here

02:52:45

several component hours here are above 50

02:52:47

hertz number of hairs and

02:52:48

high quality components but ours

02:52:50

filter filtered

02:52:53

no low frequencies and high frequencies

02:52:56

now all these signals are already

02:52:58

filtered

02:52:59

We

02:53:01

we can work

02:53:04

here it's no longer heavy but here I'm just

02:53:09

gave examples of different types of responses

02:53:12

filters and on the left this is her with

02:53:16

comments file and this is the response this

02:53:19

Of course, not everything matches the filter

02:53:21

characteristics column is a filter with

02:53:23

endless red characteristics and we

02:53:25

we see a and

02:53:27

matches a low pass filter filter

02:53:29

low pass and the band was filter

02:53:32

filters filters

02:53:34

the high pass filter

02:53:37

high frequency treble

02:53:39

rectangular pulse in red is

02:53:41

the input signal shown here is a

02:53:44

in black you are the result of filtering

02:53:47

respectively rectangular pulses

02:53:49

quite convenient function convenient 1

02:53:51

influences so that by signal filters

02:53:54

guess

02:53:56

what type of filter are we using here

02:53:58

for example, it allows hair to pass through

02:54:01

seems to miss a vibration

02:54:03

sinusoidal harmonic vibrations

02:54:04

only in

02:54:06

a certain frequency range a low pass

02:54:09

for example he swallowed they miss

02:54:11

bring something from it by going out to

02:54:13

the eye listens strongly

02:54:15

so let's finally move on

02:54:17

already by processing method and with slippy

02:54:23

signals

02:54:24

For this purpose we have implemented two

02:54:26

functions

02:54:28

meaning bags there they are seen the

02:54:31

script component analysis

02:54:37

Let's see what these functions are, what do I need?

02:54:39

urge she takes it next

02:54:41

parameter

02:54:46

our

02:54:47

our data is entirely to the sampling staff

02:54:51

encoded frequencies

02:54:53

the window in which we want to encode

02:54:56

A

02:54:57

also additional arguments for

02:54:59

visualization and here first of all we

02:55:02

as if from ours

02:55:05

from ours from our data

02:55:10

remember remember some

02:55:13

parameters including number of channels

02:55:15

number of blocks, for example there are 4

02:55:17

quantity and quantity code

02:55:20

classes there are 40 40 characters and then after

02:55:25

This we transform the time window into

02:55:28

seconds explosive window in reports

02:55:30

cut ours from our raw data

02:55:34

required time period in all

02:55:36

rules

02:55:41

after that we remember the variable in

02:55:44

a variable number of our reports

02:55:46

we prepare in advance the array which

02:55:48

will guard accuracy and

02:55:51

decoding frequency a respectively y

02:55:53

we are 40 kilos that is

02:55:56

n n we have 40 classes and four blocks in

02:56:01

experiment in this four by 40 array

02:56:04

and after that we continue

02:56:07

in a loop we implement it like this

02:56:12

let's implement the method tr count from if

02:56:15

I remind you that he is trying to highlight and

02:56:17

space filters that

02:56:19

which would highlight

02:56:22

we are interested in the correlations between

02:56:24

between different rules and perhaps between

02:56:27

different blocks

02:56:29

sources and

02:56:32

and we want to maximize

02:56:38

simple filtering

02:56:42

the dispersion of these us in ours

02:56:44

sources of interest in relation to not

02:56:47

to the source of interest and we implement

02:56:50

actually

02:56:51

here is the solution

02:56:54

this method is for this we are for everyone

02:56:57

class we create a loop that we resort to

02:56:59

we will show the class and for each block we

02:57:03

let's understand our 4 blocks

02:57:12

in this cycle he seems to

02:57:14

shows

02:57:16

what block do we have 4 blocks and one of them

02:57:20

even use for dough and three for

02:57:22

good for others to learn but in this

02:57:25

cycle

02:57:26

change is the highest variable of this cycle

02:57:29

she shows she matches poorly

02:57:32

which we use for the test

02:57:34

respectively

02:57:39

we create in advance

02:57:43

create a storage of covariances in advance

02:57:47

c this covariance

02:57:48

this is a storage facility

02:57:51

this is the paint itself variations

02:57:56

cross covariance between different

02:57:58

cube blocks it

02:58:01

just store the covariances in the same volume

02:58:06

same blocks, here they are according to the formula

02:58:11

accordingly we

02:58:13

on the drops we consider poking around for

02:58:16

each block sat in covariance and

02:58:18

we accumulate their variable q&a and also

02:58:21

we consider the covariances between different

02:58:23

blocks of cross-covariance between

02:58:25

different blocks

02:58:28

if these blocks well their index is different

02:58:34

after this we get the finished matrix

02:58:38

qs which can be placed in this one

02:58:41

formula and and

02:58:43

find accordingly

02:58:47

from this formula we need the filters

02:58:50

and here is the solution to this type of expression:

02:58:54

Always

02:58:56

produced through the so-called a

02:58:59

general

02:59:00

opposition, that is, finding one’s own

02:59:03

eigenvectors which is Diogenes

02:59:06

leads simultaneously two matrices a

02:59:08

qui with and accordingly, at the output we

02:59:11

we get it honestly

02:59:12

corresponding to our own

02:59:14

vector a

02:59:16

these eigenvalues and eigenvalues

02:59:18

ram and are not ordered and we them

02:59:20

orderly court using function a

02:59:22

orc variety from on the largest own

02:59:25

numbers smaller than it and

02:59:27

let's take the very first one

02:59:31

filter filter or well vector

02:59:35

corresponding to on and

02:59:39

on the largest eigenvalue which

02:59:43

says that this is our highest

02:59:45

filter and remember the previously created one

02:59:48

also an array and this is an array it's simple

02:59:50

repositories of the best filters for

02:59:53

each class for each stimulus we have

02:59:56

your filter just listen to the dimension

03:00:00

this array is

03:00:02

4a as channels for up to 40

03:00:07

incentives, that is, 64 by 40, then this

03:00:11

visualization function from visualization and

03:00:14

here we have a filter

03:00:18

as a referent signal and what we take

03:00:20

we average

03:00:22

we average the activity between

03:00:26

between training blocks between 3 training blocks

03:00:29

blocks and this suppresses

03:00:33

signals that are not of interest to us and

03:00:37

enhances the signals we are interested in

03:00:39

which are repeated from block to block

03:00:41

We also apply our optimal

03:00:43

filter it happens here and

03:00:46

Thus we get our reference

03:00:48

the signal is

03:00:50

it seems to show this on-air signal

03:01:04

reaction

03:01:08

signal corresponding

03:01:14

response to a certain frequency

03:01:17

body which signals that appear

03:01:20

as a result of stimulating a certain

03:01:22

frequency and purity

03:01:25

filtered spatial and

03:01:28

filtered space

03:01:31

this is the turn for signals, as it were, the best

03:01:33

our performance our reactions to

03:01:36

stimulation and

03:01:38

we have this ferrino signal

03:01:44

for each class

03:01:49

multiply by

03:01:56

on this track, that is, we are trying

03:01:59

decode we don't know we take we

03:02:01

goes through each rule and

03:02:03

trying to understand

03:02:06

what what no frequency what what

03:02:09

problems decoded we multiply it by

03:02:11

pre-calculated spatial

03:02:14

filters

03:02:15

pre-calculated reference signals

03:02:18

for for each frequency and

03:02:21

let's see how clean it is

03:02:24

better, that is, the output is this one

Description:

On this page you can download the media «Нейроинтерфейсы #2 / Алексей Осадчий [MADE]» highest quality possible WITHOUT any restrictions on the number of downloads or download speed.

Preparing download options

Popular

HD video

Only sound

All

* — If the video is playing in a new tab, go to it, then right-click on the video and select "Save video as..."

** — Link intended for online playback in specialized players

Questions about downloading video

How can I download "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]" video?

http://unidownloader.com/ website is the best way to download a video or a separate audio track if you want to do without installing programs and extensions.
The UDL Helper extension is a convenient button that is seamlessly integrated into YouTube, Instagram and OK.ru sites for fast content download.
UDL Client program (for Windows) is the most powerful solution that supports more than 900 websites, social networks and video hosting sites, as well as any video quality that is available in the source.
UDL Lite is a really convenient way to access a website from your mobile device. With its help, you can easily download videos directly to your smartphone.

Which format of "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]" video should I choose?

The best quality formats are FullHD (1080p), 2K (1440p), 4K (2160p) and 8K (4320p). The higher the resolution of your screen, the higher the video quality should be. However, there are other factors to consider: download speed, amount of free space, and device performance during playback.

Why does my computer freeze when loading a "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]" video?

The browser/computer should not freeze completely! If this happens, please report it with a link to the video. Sometimes videos cannot be downloaded directly in a suitable format, so we have added the ability to convert the file to the desired format. In some cases, this process may actively use computer resources.

How can I download "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]" video to my phone?

You can download a video to your smartphone using the website or the PWA application UDL Lite. It is also possible to send a download link via QR code using the UDL Helper extension.

How can I download an audio track (music) to MP3 "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]"?

The most convenient way is to use the UDL Client program, which supports converting video to MP3 format. In some cases, MP3 can also be downloaded through the UDL Helper extension.

How can I save a frame from a video "Нейроинтерфейсы #2 / Алексей Осадчий [MADE]"?

This feature is available in the UDL Helper extension. Make sure that "Show the video snapshot button" is checked in the settings. A camera icon should appear in the lower right corner of the player to the left of the "Settings" icon. When you click on it, the current frame from the video will be saved to your computer in JPEG format.

What's the price of all this stuff?

It costs nothing. Our services are absolutely free for all users. There are no PRO subscriptions, no restrictions on the number or maximum length of downloaded videos.