0:00:00.528,0:00:01.369
- [Voiceover] So we started

0:00:01.369,0:00:02.506
with a square wave

0:00:02.506,0:00:04.239
that had a period of two pi,

0:00:04.239,0:00:05.524
then we said, hmm,

0:00:05.524,0:00:07.146
can we represent it

0:00:07.146,0:00:09.545
as an infinite series

0:00:09.545,0:00:11.877
of weighted sines and cosines,

0:00:11.877,0:00:14.487
and then working from that idea,

0:00:14.487,0:00:15.320
we were actually able

0:00:15.320,0:00:16.767
to find expressions

0:00:16.767,0:00:17.908
for the coefficients,

0:00:17.908,0:00:19.795
for a sub zero and a sub n

0:00:19.795,0:00:21.092
when n does not equal zero,

0:00:21.092,0:00:22.472
and the b sub ns.

0:00:22.472,0:00:23.723
And evaluating it for this

0:00:23.723,0:00:25.295
particular square wave,

0:00:25.295,0:00:26.128
we were able to get

0:00:26.128,0:00:27.341
that a sub n is going

0:00:27.341,0:00:28.307
to be equal,

0:00:28.307,0:00:29.467
or a sub zero is going

0:00:29.467,0:00:30.926
to be 3/2,

0:00:30.926,0:00:31.759
that a sub n is going

0:00:31.759,0:00:32.729
to be equal to zero

0:00:32.729,0:00:33.963
for any n

0:00:33.963,0:00:35.664
other than zero,

0:00:35.664,0:00:37.036
and that b sub n

0:00:37.036,0:00:39.105
is going to be equal

0:00:39.105,0:00:41.587
to zero if n is even

0:00:41.587,0:00:43.109
and six over n pi

0:00:43.109,0:00:44.109
if n is odd.

0:00:45.343,0:00:46.306
So one way to think about it,

0:00:46.306,0:00:49.804
you're gonna get your a sub zero,

0:00:49.804,0:00:50.637
you're not gonna have any

0:00:50.637,0:00:52.048
of the cosine terms,

0:00:52.048,0:00:53.189
and you're only going to have

0:00:53.189,0:00:54.772
the odd sine terms.

0:00:55.844,0:00:56.852
And if you think about it

0:00:56.852,0:00:57.885
just visually, if you look

0:00:57.885,0:00:59.039
at the square wave,

0:00:59.039,0:01:00.088
it makes sense

0:01:00.088,0:01:01.213
that you're gonna have

0:01:01.213,0:01:03.205
the sines and not the cosines

0:01:03.205,0:01:04.857
because a sine function

0:01:04.857,0:01:07.573
is gonna look something like this.

0:01:07.573,0:01:10.132
So a sine function is gonna look

0:01:10.132,0:01:11.696
something like this,

0:01:11.696,0:01:12.869
while a cosine function

0:01:12.869,0:01:14.619
looks something like,

0:01:17.052,0:01:19.360
let me make it a little bit neater,

0:01:19.360,0:01:20.731
a cosine function would look

0:01:20.731,0:01:22.398
something like that.

0:01:25.054,0:01:26.317
And so a cosine

0:01:26.317,0:01:28.226
and multiples of cosine

0:01:28.226,0:01:30.910
of two x, cosine of three x,

0:01:30.910,0:01:32.185
is gonna be out of phase,

0:01:32.185,0:01:33.501
while the sine of x,

0:01:33.501,0:01:35.071
or I should say cosine of ts

0:01:35.071,0:01:36.457
and the sines of ts,

0:01:36.457,0:01:38.082
sine two t, sine three t,

0:01:38.082,0:01:39.310
is gonna be more in phase

0:01:39.310,0:01:40.417
with the way this function

0:01:40.417,0:01:41.764
just happened to be.

0:01:41.764,0:01:43.875
So it made sense that our

0:01:43.875,0:01:45.758
a sub ns were all zero

0:01:45.758,0:01:48.066
for n not equaling zero.

0:01:48.066,0:01:50.044
And so based on what we found

0:01:50.044,0:01:51.409
for our a sub zero,

0:01:51.409,0:01:52.242
and our a sub ns,

0:01:52.242,0:01:53.662
and our b sub ns,

0:01:53.662,0:01:54.758
we could expand out

0:01:54.758,0:01:56.445
this actual,

0:01:56.445,0:01:58.018
we did in the previous video,

0:01:58.018,0:01:59.212
what is this Fourier series

0:01:59.212,0:02:00.247
actually look like?

0:02:00.247,0:02:01.779
So 3/2 plus six over pi

0:02:01.779,0:02:04.255
sine of t plus six over three pi

0:02:04.255,0:02:05.127
sine of three t plus

0:02:05.127,0:02:05.960
six over five pi

0:02:05.960,0:02:07.111
sine of five t,

0:02:07.111,0:02:08.532
and so on and so forth.

0:02:08.532,0:02:09.365
And so a lot of you might

0:02:09.365,0:02:10.198
be curious what does this

0:02:10.198,0:02:11.831
actually look like.

0:02:11.831,0:02:13.011
And so I actually just,

0:02:13.011,0:02:14.055
you can type these things

0:02:14.055,0:02:14.964
into Google and it will

0:02:14.964,0:02:16.645
just graph it for you.

0:02:16.645,0:02:17.801
And so this right over here

0:02:17.801,0:02:19.058
is just the first two terms.

0:02:19.058,0:02:22.090
This is 3/2 plus six over pi

0:02:22.090,0:02:22.923
sine of t.

0:02:23.973,0:02:24.806
And notice it's starting

0:02:24.806,0:02:26.356
to look right because our

0:02:26.356,0:02:27.635
square wave looks something

0:02:27.635,0:02:28.802
like, it goes,

0:02:30.612,0:02:33.595
it looks something like this.

0:02:33.595,0:02:35.492
So it's gonna go like that

0:02:35.492,0:02:38.106
and then it's gonna go down

0:02:38.106,0:02:40.606
to zero and then it's gonna go

0:02:42.831,0:02:44.559
up, looks something like that.

0:02:44.559,0:02:45.392
It doesn't have the pis

0:02:45.392,0:02:46.455
and the two pis

0:02:46.455,0:02:47.302
marked off between these

0:02:47.302,0:02:48.135
because it's gonna look

0:02:48.135,0:02:49.731
something like that.

0:02:49.731,0:02:51.128
So even just the two terms,

0:02:51.128,0:02:52.539
it's kind of a decent approximation

0:02:52.539,0:02:53.912
for even two terms,

0:02:53.912,0:02:54.745
but then as soon as you get

0:02:54.745,0:02:55.578
to three terms,

0:02:55.578,0:02:57.861
if you add the six

0:02:57.861,0:02:59.364
over three pi sine of three t

0:02:59.364,0:03:01.085
to the first two terms.

0:03:01.085,0:03:01.918
So if you look at these

0:03:01.918,0:03:03.566
first three terms,

0:03:03.566,0:03:04.399
now it's looking a lot

0:03:04.399,0:03:05.664
more like a square wave.

0:03:05.664,0:03:08.458
And then if you add the next term,

0:03:08.458,0:03:09.291
well, it looks like even more

0:03:09.291,0:03:10.519
like a square wave,

0:03:10.519,0:03:12.058
and then if you add to that

0:03:12.058,0:03:14.220
what we already wrote down here,

0:03:14.220,0:03:15.646
if you were to add to that

0:03:15.646,0:03:16.988
six over seven pi times

0:03:16.988,0:03:18.156
sine of seven t,

0:03:18.156,0:03:19.044
it looks even more

0:03:19.044,0:03:20.382
like a square wave.

0:03:20.382,0:03:21.675
So this is pretty neat.

0:03:21.675,0:03:22.945
You can visually see

0:03:22.945,0:03:24.405
that we were actually able

0:03:24.405,0:03:25.238
to do it.

0:03:25.238,0:03:26.454
And it all kind of just

0:03:26.454,0:03:28.860
fell out from the mathematics.