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Russian
00:00:00
why hello little humans welcome to part00:00:03
three of Laurier series yeah all right00:00:07
so we have finally made it to the point00:00:09
where we are ready to do an actual00:00:11
example oh look at that00:00:13
okay so in the previous part in part two00:00:18
we looked at how do we find the00:00:21
coefficients of our Fourier series00:00:23
pretty important because otherwise we00:00:26
just have a general formula that doesn't00:00:28
actually apply to any given situation so00:00:31
okay we talked a lot about sound but00:00:33
sound waves are a little tricky so let's00:00:35
start with something even more simple00:00:36
but still super useful and it crops up00:00:39
all the time in the real world00:00:42
especially if you work with electronics00:00:44
so let's say that we are given um I00:00:48
guess this can be a straight line we are00:00:51
given a periodic voltage pulse that00:00:56
looks like this where we have on and00:00:59
then it's off I will say that this is00:01:02
high for the sake of having a periodic00:01:06
function with a period of 2pi so I'm00:01:08
going to write that in the corner so our00:01:09
period is 2pi so then it 2pi it starts00:01:17
again and then it goes until 3 pi and00:01:22
then likewise we can also go back and we00:01:25
can say that at a negative PI we have00:01:29
our signal going like boom all the way00:01:33
to negative 2 pi and this is zero ok so00:01:40
then this vertical axis is what we will00:01:43
call a X and this point right here is 100:01:46
so when you are creating a Fourier00:01:49
series maybe if you're lucky you're00:01:52
given a picture like this because they00:01:54
say a picture is worth a thousand words00:01:57
or an R case a picture is worth a00:01:59
thousand equations or maybe at least one00:02:01
equation ok so you are most likely given00:02:05
the function in math speak or whatever00:02:10
and so in this case our that was a00:02:12
hilariously bad racket um in this case00:02:15
our function is represented like this00:02:18
let's try this again okay there we go so00:02:22
our function is zero for negative PI00:02:27
less than X less than 0 and 1 for 0 less00:02:35
than X less than PI and since we are00:02:38
told that it repeats over time you00:02:40
basically take this function and you00:02:42
infer these and onward and so in this00:02:45
case this is where you're like oh ok00:02:48
this function would be really really00:02:51
difficult to implement in a coding00:02:54
program and so something that would be a00:02:56
little bit easier would be to use the00:02:59
sine and cosine especially if you're00:03:01
working in something like are the00:03:02
statistical program or if you have00:03:04
Python you can import these math00:03:07
functions now I mean just use sine and00:03:09
cosine without having to define them00:03:11
whoo makes our lives easier ok so now00:03:14
that we are given this function if you00:03:16
are not given a picture I would always00:03:19
recommend drawing a picture so I like to00:03:23
talk about how understanding is critical00:03:26
in mathematics and a lot of this is00:03:28
coming from having done physics because00:03:31
to be honest I didn't memorize a ton of00:03:34
equations once you have done the00:03:36
equations enough times they get burned00:03:38
into your brain but when we're given00:03:40
most of the time on a tests were given00:03:42
the equations but that doesn't00:03:43
necessarily mean that it's helpful you00:03:46
have to understand what you're doing and00:03:47
I can't tell you the number of times00:03:50
where I sat into a picture and that was00:03:52
the thing that made my brain click so if00:03:55
your visual like me definitely draw a00:03:57
picture and even if you're not still00:03:59
draw a picture ok so let's find a of n00:04:02
first so yeah I'm not gonna rewrite the00:04:09
general equation because we're just00:04:11
gonna apply it here as I look at my00:04:12
notes down below me but if you want to I00:04:15
would recommend when you watch part 200:04:17
that you were actually write down those00:04:19
equations or at least have your notebook00:04:20
Andy your math textbook okay so so this00:04:27
is the equation for a of n which are the00:04:31
cosine terms so again we're going from00:04:35
negative pi to pi f of X so I'm like I'm00:04:39
not gonna rewrite them I'm gonna rewrite00:04:40
them00:04:41
cosine of n X DX okay00:04:46
so basically now what we want to do is00:04:48
we want to break apart this integral00:04:51
into these two chunks so you can see00:04:53
that we have one piece of our integral00:04:56
is zero so from negative PI to zero and00:04:59
the other piece of our integral is 100:05:01
from 0 to PI ok and that looks like this00:05:05
I'm going to pull out the 1 over pi and00:05:08
so our first integral is going to go00:05:10
from negative PI to 0 and we look and00:05:14
see what the value of f of X is between00:05:16
this interval and it's zero well that's00:05:18
easy so 0 times cosine of X DX huh easy00:05:24
peasy and then the second one goes from00:05:27
0 to PI and this is 1 times cosine of X00:05:35
DX it's almost off the page there okay00:05:40
so cool all right so now this is00:05:44
actually pretty straightforward I'm00:05:45
gonna double check my notes make sure I00:05:46
don't mess anything up ok so this goes00:05:50
to 0 easy peasy00:05:52
and we are left with 1 over PI 0 to PI00:05:56
cosine of n X DX that's not so bad we00:06:01
can do this ok 1 over PI the integral of00:06:05
cosine is sine so we get but we have a00:06:09
factor of n in here so we have to divide00:06:11
by n so we're gonna multiply by 1 over n00:06:15
and then we get sine of n X from 0 to PI00:06:25
ok yay and now I'm running out of space00:06:30
actually I can write below here okay so00:06:32
now I have 1 over n I sine of n PI minus00:06:42
sine of 0 okay so let me think about the00:06:47
unit circle so sine of 0 is 0 and sine00:06:54
of any multiple of Pi is also 0 ha ha00:06:58
look at that um wait a second hold on00:07:03
hold on I think I messed that up00:07:07
oh ok I got ahead of myself here ok so00:07:12
um I'm gonna back up a little bit yeah00:07:19
ok so this is where we have to ask what00:07:25
are the possible values of n well and00:07:28
could be anything from 0 onward so if we00:07:34
have cosine of 0 that's actually going00:07:37
to turn our integral into something very00:07:40
different so for N equals 0 we have 100:07:46
over pi the integral of 0 to pi cosine00:07:51
of 0 is actually 1 so we get DX and then00:07:56
this is gonna be 1 over pi times this is00:08:01
just gonna be x from 0 to pi equals 100:08:05
over pi pi minus 0 so we're just gonna00:08:09
get pi and this equals 1 do that pretty00:08:12
cool right00:08:13
ok but now all of a sudden I'm gonna00:08:17
leave this last line and erase this and00:08:23
so or n not equal to 0 that's when we00:08:27
have to actually evaluate this integral00:08:29
so for n not equal 0 we have a N equals00:08:34
1 over PI 0 whoops 0 to PI I can do that00:08:41
um so the integral like you just did00:08:44
before is one over n sine of X from 0 to00:08:51
PI and so this is what goes to 0 so you00:08:54
get again 1 over pi 1 over n pi times 000:09:00
equals 0 ok so for N equals 0 or a 0 you00:09:06
get 1 cool okay so I'm going to write00:09:11
that up here so a 0 equals 1 tada00:09:17
that's pretty great right most of the00:09:20
coefficients in the cosine term went to00:09:23
0 which when you know the picture of00:09:26
cosine how it looks like that makes00:09:28
sense right because sine starts at 0 and00:09:33
wait is that right yeah sine starts at00:09:36
zero and goes like this so this function00:09:41
you could say has mostly mostly sign in00:09:45
it whereas cosine is gonna come down00:09:47
like this um and it makes sense to have00:09:50
the first term be 1 because you are00:09:52
starting at 1 ok so no we do sign a sine00:09:59
terms or the B terms00:10:01
yay so to be N equals 1 over pi negative00:10:08
PI to PI of f of X times sine and xdx00:10:17
and again we have to break this apart00:10:19
into the two pieces so we have 1 over pi00:10:22
times the integral from negative PI to 000:10:28
which is 0 times sine and X plus 0 to PI00:10:35
which is just gonna be 1 times sine oh00:10:39
my gosh DX and X DX okay this goes to 000:10:46
and I am left with 1 over PI and I keep00:10:50
running down its pi 1 over PI 0 to PI00:10:56
sign and xdx and again I have to break00:11:01
it into two pieces so let me just double00:11:06
check yeah okay so for N equals zero we00:11:12
have sine of zero is zero so our00:11:18
integral just equals zero so we have 100:11:20
over pi integrals where the PI sine of 000:11:25
DX on this goes to 0 and for n not equal00:11:32
to 0 I might be writing a little small00:11:39
okay let's move a little beat so00:11:51
actually I'll do this so be n not equal00:11:55
to 0 equals 1 over PI integral from 0 to00:12:00
PI of sine and X DX00:12:05
okay so this one again pretty00:12:08
straightforward you have a factor of 100:12:10
over PI the end comes out of u divided00:12:12
by N and you have a negative negative00:12:16
negative cosine and X and then that is00:12:21
all from 0 to PI ok and erase that00:12:28
because we don't need it anymore just00:12:31
remember in your heads that the first00:12:32
term and the sine coefficients went to 000:12:36
ok so now we have 1 over and PI and we00:12:40
need to evaluate this so I'm going to00:12:42
bring the negative out just to make my00:12:44
life a little bit easier and I have00:12:46
cosine of n PI minus cosine of 0 okay so00:12:54
let's hold off on this for a sec this00:12:56
goes to 1 and so now I have negative 100:13:00
over n PI cosine of n pi ok so let's00:13:06
talk about this and I'm gonna draw the00:13:08
unit circle to help us visually00:13:10
so the unit circle is a circle with the00:13:12
radius one where this is X and this is y00:13:17
and so cosine is adjacent over00:13:20
hypotenuse so that means cosine00:13:22
corresponds to the x-axis and sine00:13:25
corresponds to the y-axis00:13:26
so cosine of zero is here so hence00:13:30
cosine equals one uhm cosine of of Pi is00:13:36
over here so this is going to be00:13:37
negative one but then we go back around00:13:41
so for N equals 0 we get 1 for N equals00:13:44
1 we get negative 1 for N equals 2 we00:13:49
get a positive 1 for N equals 3 we're00:13:52
back over here and we get a negative 100:13:54
and you keep going around and around the00:13:56
circle it's a periodic function so you00:13:58
alternate between 1 and negative 1 the00:14:02
even numbers of n give you a positive 100:14:05
the negative numbers of n give you a00:14:09
negative 1 ok so that's a little wonky00:14:13
how the heck do we represent that it's00:14:15
not so bad actually00:14:16
so a really cool trick is to say okay00:14:20
well it's just negative 1 to the N and00:14:22
you can double check this I would00:14:23
challenge you to do this on your own00:14:25
where if you have n to the 0 you get 100:14:28
if you have n do the one you get00:14:30
negative 1 check and square or N equals00:14:33
2 you get negative 1 squared that's00:14:35
going to be a positive 1 if you have N00:14:37
equals 3 this is gonna give you a00:14:39
negative 1 just like you would expect so00:14:41
how it works so but we can't forget this00:14:45
other term here it's super important so00:14:47
we are gonna subtract a 1 okay so this00:14:50
is really interesting because this tells00:14:52
us that when we have a positive 1 here00:14:59
and remember positive or sorry even00:15:03
numbers of n even n this is going to be00:15:09
a positive 1 and this whole thing is00:15:12
going to go to 0 so B of N equals 0 but00:15:17
for odd n B of n00:15:21
equals running out of space again yeah00:15:27
we'll do this up here keep this so be00:15:33
even N equals zero but be odd and equals00:15:43
negative one over and pi times this is00:15:49
gonna give me a negative one negative00:15:51
one minus one is negative two so this is00:15:56
gonna give me a positive 2 over n pi00:16:01
boom and let me just double-check that00:16:04
this is right yes00:16:06
all right so now um we have a B on N00:16:14
equals 2 over n hi red right so now we00:16:23
can write the Fourier series the00:16:26
expanded the Fourier series that expands00:16:32
this function in terms of sines and00:16:35
cosines so now our function is not00:16:39
discontinuous yay now we can write it as00:16:45
a series of sines and cosines which00:16:47
means we can do a lot more cool things00:16:49
with it so the first term a knot is one00:16:56
so remember we have that one half so we00:16:58
have a one half times one in there and00:17:00
now we just wiped out all of the cosine00:17:04
terms so those all went to zero and the00:17:06
only terms we have left are for odd00:17:09
numbers of n so that would be a 1 which00:17:14
is going to be for N equals 1 so it's00:17:17
going to be 2 over pi times sine of X00:17:22
and the even ones go to 0 so we skip00:17:25
that and so now we have 2 over 3 PI00:17:29
first sine of 3x plus00:17:34
two over five pi for sine 5x plus etc00:17:40
pretty red right that's it this is our00:17:44
Fourier series expansion of this00:17:47
discontinuous function um and the last00:17:50
thing that I want to note is that the00:17:55
magnitude or how big each of these00:17:58
coefficients are determines how much00:18:03
they come into play so for example the00:18:06
reason why we can say dot dot dot we00:18:08
don't really care is because as these00:18:11
fractions get smaller and smaller the00:18:13
contribution of the subsequent sign00:18:15
terms also gets smaller and smaller so00:18:19
what this means practically for example00:18:22
going back to sound because it's00:18:23
something that's near and dear to my00:18:24
heart what this means is that a higher00:18:28
quality audio signal would have more of00:18:32
these sine terms to the point at which00:18:35
the frequency gets so high and that the00:18:38
human ear can't hear that it is kind of00:18:42
interesting though to note when when I00:18:45
listen to records for the first time as00:18:49
a child of the 90s our Jam was CDs and00:18:55
so I actually heard digital music CDs00:18:58
before I heard records which is actually00:19:01
analog so the sound wave literally gets00:19:04
imprinted on the record so it's a really00:19:08
close sound to what you would actually00:19:12
hear live and I remember hearing a00:19:15
record for the first time and I was like00:19:16
oh my gosh this sounds like it's alive00:19:18
and especially with early digital music00:19:22
they would truncate these Fourier series00:19:24
pretty pretty early on because space was00:19:27
an issue we didn't have a lot of memory00:19:29
to store all of these terms but as space00:19:33
has increased audio files can be much00:19:36
larger and so that means that we can00:19:38
include a lot more of these fourier00:19:40
terms so that's literally what it means00:19:44
to to have more and more fourier terms00:19:47
and again at a certain point the human00:19:50
ear can't hear so you don't actually00:19:51
have to include an infinite number of00:19:53
terms we don't have an infinite number00:19:55
of storage space but you would include00:19:58
enough up to a point at which the human00:20:00
ear can't make a difference and then00:20:02
maybe you get close to the way record00:20:03
sounds pretty cool right all right so00:20:06
please let me know if you have any00:20:08
questions about the procedure here or if00:20:12
you're curious how you might apply this00:20:14
to a different type of function a00:20:17
different type a periodic function like00:20:19
I mentioned we will get to Fourier00:20:22
transforms I have a couple of other00:20:24
videos I want to do in the mean time but00:20:27
eventually I'll come back to Fourier00:20:29
transforms and we will look at non00:20:32
periodic functions for that so again00:20:36
please let me know if there are any math00:20:38
topics that you're super curious about00:20:40
and I'm also taking math myths whoo so00:20:44
what are the myths that you've heard00:20:45
that have to do with mathematics and00:20:47
let's tackle them together thank you00:20:50
very much for watching and I'll see you00:20:51
next time bye00:20:53
[Music]Description:
Finally!! Time to wield our math muscles, bbies!! We apply our new knowledge of Fourier Series to expand a given (discontinuous) function so we can create a continuous and more easily usable function! Questions about this or other math topics? Leave a comment and we will tackle it together! Like these videos? Please support my work by contributing to patreon: https://www.patreon.com/jenfoxbot
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