[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.53,0:00:01.37,Default,,0000,0000,0000,,- [Voiceover] So we started
Dialogue: 0,0:00:01.37,0:00:02.51,Default,,0000,0000,0000,,with a square wave
Dialogue: 0,0:00:02.51,0:00:04.24,Default,,0000,0000,0000,,that had a period of two pi,
Dialogue: 0,0:00:04.24,0:00:05.52,Default,,0000,0000,0000,,then we said, hmm,
Dialogue: 0,0:00:05.52,0:00:07.15,Default,,0000,0000,0000,,can we represent it
Dialogue: 0,0:00:07.15,0:00:09.54,Default,,0000,0000,0000,,as an infinite series
Dialogue: 0,0:00:09.54,0:00:11.88,Default,,0000,0000,0000,,of weighted sines and cosines,
Dialogue: 0,0:00:11.88,0:00:14.49,Default,,0000,0000,0000,,and then working from that idea,
Dialogue: 0,0:00:14.49,0:00:15.32,Default,,0000,0000,0000,,we were actually able
Dialogue: 0,0:00:15.32,0:00:16.77,Default,,0000,0000,0000,,to find expressions
Dialogue: 0,0:00:16.77,0:00:17.91,Default,,0000,0000,0000,,for the coefficients,
Dialogue: 0,0:00:17.91,0:00:19.80,Default,,0000,0000,0000,,for a sub zero and a sub n
Dialogue: 0,0:00:19.80,0:00:21.09,Default,,0000,0000,0000,,when n does not equal zero,
Dialogue: 0,0:00:21.09,0:00:22.47,Default,,0000,0000,0000,,and the b sub ns.
Dialogue: 0,0:00:22.47,0:00:23.72,Default,,0000,0000,0000,,And evaluating it for this
Dialogue: 0,0:00:23.72,0:00:25.30,Default,,0000,0000,0000,,particular square wave,
Dialogue: 0,0:00:25.30,0:00:26.13,Default,,0000,0000,0000,,we were able to get
Dialogue: 0,0:00:26.13,0:00:27.34,Default,,0000,0000,0000,,that a sub n is going
Dialogue: 0,0:00:27.34,0:00:28.31,Default,,0000,0000,0000,,to be equal,
Dialogue: 0,0:00:28.31,0:00:29.47,Default,,0000,0000,0000,,or a sub zero is going
Dialogue: 0,0:00:29.47,0:00:30.93,Default,,0000,0000,0000,,to be 3/2,
Dialogue: 0,0:00:30.93,0:00:31.76,Default,,0000,0000,0000,,that a sub n is going
Dialogue: 0,0:00:31.76,0:00:32.73,Default,,0000,0000,0000,,to be equal to zero
Dialogue: 0,0:00:32.73,0:00:33.96,Default,,0000,0000,0000,,for any n
Dialogue: 0,0:00:33.96,0:00:35.66,Default,,0000,0000,0000,,other than zero,
Dialogue: 0,0:00:35.66,0:00:37.04,Default,,0000,0000,0000,,and that b sub n
Dialogue: 0,0:00:37.04,0:00:39.10,Default,,0000,0000,0000,,is going to be equal
Dialogue: 0,0:00:39.10,0:00:41.59,Default,,0000,0000,0000,,to zero if n is even
Dialogue: 0,0:00:41.59,0:00:43.11,Default,,0000,0000,0000,,and six over n pi
Dialogue: 0,0:00:43.11,0:00:44.11,Default,,0000,0000,0000,,if n is odd.
Dialogue: 0,0:00:45.34,0:00:46.31,Default,,0000,0000,0000,,So one way to think about it,
Dialogue: 0,0:00:46.31,0:00:49.80,Default,,0000,0000,0000,,you're gonna get your a sub zero,
Dialogue: 0,0:00:49.80,0:00:50.64,Default,,0000,0000,0000,,you're not gonna have any
Dialogue: 0,0:00:50.64,0:00:52.05,Default,,0000,0000,0000,,of the cosine terms,
Dialogue: 0,0:00:52.05,0:00:53.19,Default,,0000,0000,0000,,and you're only going to have
Dialogue: 0,0:00:53.19,0:00:54.77,Default,,0000,0000,0000,,the odd sine terms.
Dialogue: 0,0:00:55.84,0:00:56.85,Default,,0000,0000,0000,,And if you think about it
Dialogue: 0,0:00:56.85,0:00:57.88,Default,,0000,0000,0000,,just visually, if you look
Dialogue: 0,0:00:57.88,0:00:59.04,Default,,0000,0000,0000,,at the square wave,
Dialogue: 0,0:00:59.04,0:01:00.09,Default,,0000,0000,0000,,it makes sense
Dialogue: 0,0:01:00.09,0:01:01.21,Default,,0000,0000,0000,,that you're gonna have
Dialogue: 0,0:01:01.21,0:01:03.20,Default,,0000,0000,0000,,the sines and not the cosines
Dialogue: 0,0:01:03.20,0:01:04.86,Default,,0000,0000,0000,,because a sine function
Dialogue: 0,0:01:04.86,0:01:07.57,Default,,0000,0000,0000,,is gonna look something like this.
Dialogue: 0,0:01:07.57,0:01:10.13,Default,,0000,0000,0000,,So a sine function is gonna look
Dialogue: 0,0:01:10.13,0:01:11.70,Default,,0000,0000,0000,,something like this,
Dialogue: 0,0:01:11.70,0:01:12.87,Default,,0000,0000,0000,,while a cosine function
Dialogue: 0,0:01:12.87,0:01:14.62,Default,,0000,0000,0000,,looks something like,
Dialogue: 0,0:01:17.05,0:01:19.36,Default,,0000,0000,0000,,let me make it a little bit neater,
Dialogue: 0,0:01:19.36,0:01:20.73,Default,,0000,0000,0000,,a cosine function would look
Dialogue: 0,0:01:20.73,0:01:22.40,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:01:25.05,0:01:26.32,Default,,0000,0000,0000,,And so a cosine
Dialogue: 0,0:01:26.32,0:01:28.23,Default,,0000,0000,0000,,and multiples of cosine
Dialogue: 0,0:01:28.23,0:01:30.91,Default,,0000,0000,0000,,of two x, cosine of three x,
Dialogue: 0,0:01:30.91,0:01:32.18,Default,,0000,0000,0000,,is gonna be out of phase,
Dialogue: 0,0:01:32.18,0:01:33.50,Default,,0000,0000,0000,,while the sine of x,
Dialogue: 0,0:01:33.50,0:01:35.07,Default,,0000,0000,0000,,or I should say cosine of ts
Dialogue: 0,0:01:35.07,0:01:36.46,Default,,0000,0000,0000,,and the sines of ts,
Dialogue: 0,0:01:36.46,0:01:38.08,Default,,0000,0000,0000,,sine two t, sine three t,
Dialogue: 0,0:01:38.08,0:01:39.31,Default,,0000,0000,0000,,is gonna be more in phase
Dialogue: 0,0:01:39.31,0:01:40.42,Default,,0000,0000,0000,,with the way this function
Dialogue: 0,0:01:40.42,0:01:41.76,Default,,0000,0000,0000,,just happened to be.
Dialogue: 0,0:01:41.76,0:01:43.88,Default,,0000,0000,0000,,So it made sense that our
Dialogue: 0,0:01:43.88,0:01:45.76,Default,,0000,0000,0000,,a sub ns were all zero
Dialogue: 0,0:01:45.76,0:01:48.07,Default,,0000,0000,0000,,for n not equaling zero.
Dialogue: 0,0:01:48.07,0:01:50.04,Default,,0000,0000,0000,,And so based on what we found
Dialogue: 0,0:01:50.04,0:01:51.41,Default,,0000,0000,0000,,for our a sub zero,
Dialogue: 0,0:01:51.41,0:01:52.24,Default,,0000,0000,0000,,and our a sub ns,
Dialogue: 0,0:01:52.24,0:01:53.66,Default,,0000,0000,0000,,and our b sub ns,
Dialogue: 0,0:01:53.66,0:01:54.76,Default,,0000,0000,0000,,we could expand out
Dialogue: 0,0:01:54.76,0:01:56.44,Default,,0000,0000,0000,,this actual,
Dialogue: 0,0:01:56.44,0:01:58.02,Default,,0000,0000,0000,,we did in the previous video,
Dialogue: 0,0:01:58.02,0:01:59.21,Default,,0000,0000,0000,,what is this Fourier series
Dialogue: 0,0:01:59.21,0:02:00.25,Default,,0000,0000,0000,,actually look like?
Dialogue: 0,0:02:00.25,0:02:01.78,Default,,0000,0000,0000,,So 3/2 plus six over pi
Dialogue: 0,0:02:01.78,0:02:04.26,Default,,0000,0000,0000,,sine of t plus six over three pi
Dialogue: 0,0:02:04.26,0:02:05.13,Default,,0000,0000,0000,,sine of three t plus
Dialogue: 0,0:02:05.13,0:02:05.96,Default,,0000,0000,0000,,six over five pi
Dialogue: 0,0:02:05.96,0:02:07.11,Default,,0000,0000,0000,,sine of five t,
Dialogue: 0,0:02:07.11,0:02:08.53,Default,,0000,0000,0000,,and so on and so forth.
Dialogue: 0,0:02:08.53,0:02:09.36,Default,,0000,0000,0000,,And so a lot of you might
Dialogue: 0,0:02:09.36,0:02:10.20,Default,,0000,0000,0000,,be curious what does this
Dialogue: 0,0:02:10.20,0:02:11.83,Default,,0000,0000,0000,,actually look like.
Dialogue: 0,0:02:11.83,0:02:13.01,Default,,0000,0000,0000,,And so I actually just,
Dialogue: 0,0:02:13.01,0:02:14.06,Default,,0000,0000,0000,,you can type these things
Dialogue: 0,0:02:14.06,0:02:14.96,Default,,0000,0000,0000,,into Google and it will
Dialogue: 0,0:02:14.96,0:02:16.64,Default,,0000,0000,0000,,just graph it for you.
Dialogue: 0,0:02:16.64,0:02:17.80,Default,,0000,0000,0000,,And so this right over here
Dialogue: 0,0:02:17.80,0:02:19.06,Default,,0000,0000,0000,,is just the first two terms.
Dialogue: 0,0:02:19.06,0:02:22.09,Default,,0000,0000,0000,,This is 3/2 plus six over pi
Dialogue: 0,0:02:22.09,0:02:22.92,Default,,0000,0000,0000,,sine of t.
Dialogue: 0,0:02:23.97,0:02:24.81,Default,,0000,0000,0000,,And notice it's starting
Dialogue: 0,0:02:24.81,0:02:26.36,Default,,0000,0000,0000,,to look right because our
Dialogue: 0,0:02:26.36,0:02:27.64,Default,,0000,0000,0000,,square wave looks something
Dialogue: 0,0:02:27.64,0:02:28.80,Default,,0000,0000,0000,,like, it goes,
Dialogue: 0,0:02:30.61,0:02:33.60,Default,,0000,0000,0000,,it looks something like this.
Dialogue: 0,0:02:33.60,0:02:35.49,Default,,0000,0000,0000,,So it's gonna go like that
Dialogue: 0,0:02:35.49,0:02:38.11,Default,,0000,0000,0000,,and then it's gonna go down
Dialogue: 0,0:02:38.11,0:02:40.61,Default,,0000,0000,0000,,to zero and then it's gonna go
Dialogue: 0,0:02:42.83,0:02:44.56,Default,,0000,0000,0000,,up, looks something like that.
Dialogue: 0,0:02:44.56,0:02:45.39,Default,,0000,0000,0000,,It doesn't have the pis
Dialogue: 0,0:02:45.39,0:02:46.46,Default,,0000,0000,0000,,and the two pis
Dialogue: 0,0:02:46.46,0:02:47.30,Default,,0000,0000,0000,,marked off between these
Dialogue: 0,0:02:47.30,0:02:48.14,Default,,0000,0000,0000,,because it's gonna look
Dialogue: 0,0:02:48.14,0:02:49.73,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:02:49.73,0:02:51.13,Default,,0000,0000,0000,,So even just the two terms,
Dialogue: 0,0:02:51.13,0:02:52.54,Default,,0000,0000,0000,,it's kind of a decent approximation
Dialogue: 0,0:02:52.54,0:02:53.91,Default,,0000,0000,0000,,for even two terms,
Dialogue: 0,0:02:53.91,0:02:54.74,Default,,0000,0000,0000,,but then as soon as you get
Dialogue: 0,0:02:54.74,0:02:55.58,Default,,0000,0000,0000,,to three terms,
Dialogue: 0,0:02:55.58,0:02:57.86,Default,,0000,0000,0000,,if you add the six
Dialogue: 0,0:02:57.86,0:02:59.36,Default,,0000,0000,0000,,over three pi sine of three t
Dialogue: 0,0:02:59.36,0:03:01.08,Default,,0000,0000,0000,,to the first two terms.
Dialogue: 0,0:03:01.08,0:03:01.92,Default,,0000,0000,0000,,So if you look at these
Dialogue: 0,0:03:01.92,0:03:03.57,Default,,0000,0000,0000,,first three terms,
Dialogue: 0,0:03:03.57,0:03:04.40,Default,,0000,0000,0000,,now it's looking a lot
Dialogue: 0,0:03:04.40,0:03:05.66,Default,,0000,0000,0000,,more like a square wave.
Dialogue: 0,0:03:05.66,0:03:08.46,Default,,0000,0000,0000,,And then if you add the next term,
Dialogue: 0,0:03:08.46,0:03:09.29,Default,,0000,0000,0000,,well, it looks like even more
Dialogue: 0,0:03:09.29,0:03:10.52,Default,,0000,0000,0000,,like a square wave,
Dialogue: 0,0:03:10.52,0:03:12.06,Default,,0000,0000,0000,,and then if you add to that
Dialogue: 0,0:03:12.06,0:03:14.22,Default,,0000,0000,0000,,what we already wrote down here,
Dialogue: 0,0:03:14.22,0:03:15.65,Default,,0000,0000,0000,,if you were to add to that
Dialogue: 0,0:03:15.65,0:03:16.99,Default,,0000,0000,0000,,six over seven pi times
Dialogue: 0,0:03:16.99,0:03:18.16,Default,,0000,0000,0000,,sine of seven t,
Dialogue: 0,0:03:18.16,0:03:19.04,Default,,0000,0000,0000,,it looks even more
Dialogue: 0,0:03:19.04,0:03:20.38,Default,,0000,0000,0000,,like a square wave.
Dialogue: 0,0:03:20.38,0:03:21.68,Default,,0000,0000,0000,,So this is pretty neat.
Dialogue: 0,0:03:21.68,0:03:22.94,Default,,0000,0000,0000,,You can visually see
Dialogue: 0,0:03:22.94,0:03:24.40,Default,,0000,0000,0000,,that we were actually able
Dialogue: 0,0:03:24.40,0:03:25.24,Default,,0000,0000,0000,,to do it.
Dialogue: 0,0:03:25.24,0:03:26.45,Default,,0000,0000,0000,,And it all kind of just
Dialogue: 0,0:03:26.45,0:03:28.86,Default,,0000,0000,0000,,fell out from the mathematics.