Fourier Series introduction

So I have the graph of ( y ) is equal to ( F(T) ). Here, our horizontal axis is in terms of time, in terms of seconds. This type of function is often described as a square wave, and we see that it is a periodic function that completes one cycle every ( 2\pi ) seconds.

So we could say its period is equal to ( 2\pi ). If we want to put the units, we could say ( 2\pi ) seconds per cycle. We could write it like that; we could also just write ( s ) for seconds. Its frequency is going to be one over that, so we could write its frequency. Its frequency is equal to ( \frac{1}{2\pi} ) cycles per second. It can also be described as Hertz.

What we're going to explore in this video is: can we take a periodic function like this and represent it as an infinite sum of sines and cosines of different periods or different frequencies? So to write that out a little bit more clearly: can we take our ( F(T) ) and write it as the sum of sines and cosines?

So can we write it? So it's going to be some, let's say, baseline constant that'll shift it up or down. As we'll see, that's going to be based on the average value of the function over one period, so ( a_0 ). And then let's start adding some periodic functions here.

So let's take ( a_1 \cos(T) ). Now, why am I starting with ( \cos(T) )? And I could also add ( b_1 \sin(T) ). Why am I starting with ( \cos(T) ) and ( \sin(T) )? Well, if our original function has a period of ( 2\pi ), and I just set up this one, so it does have a period of ( 2\pi ), well, it would make sense that it would involve some functions that have periods of ( 2\pi ).

These weights will tell us how much they involve it. If ( A_1 ) is much larger than ( B_1 ), well, that says, "Okay, this has a lot more of ( \cos(T) ) in it than it has of ( \sin(T) ) in it." That by itself isn't going to describe this function because we know what this would look like. This would look like a very clean sinusoid, not like a square wave.

So what we're going to do is we're going to add sinusoids of frequencies that are multiples of these frequencies. So let's add ( a_2 \cos(2T) ). This has a frequency of ( \frac{1}{2\pi} ); this has twice the frequency, this has a frequency of ( \frac{1}{\pi} ), and then ( a_3 \sin(3T) ).

I'm going to keep going on and on and on forever, and I'm going to do the same thing with the sines. So let's add ( b_2 \sin(2T) ) plus ( b_3 \sin(3T) ). You might be saying, "Well, okay, this seems like a fun little mathematical exercise, but why do folks even do this?"

Well, this was first explored, and they’re named series like this; infinite series where you represent something by essentially weighted sines and cosines. This was explored originally by Fourier, and they're called Fourier series. They were interesting to him in the study of differential equations because a lot of differential equations can be easy to solve when you involve sines and cosines but not as obvious to solve when you have more general functions like maybe a square wave here.

But if you could represent that square wave as sums of sines and cosines, then all of a sudden you might be able to find more general solutions to your differential equations. Another really interesting thing about this—and this is really the foundation of signal processing—is that it’s heavily used in electrical engineering.

You can view these coefficients as weights on these cosines and sines, but another way to think about it is it tells you how much of different frequencies this function contains. So, for example, if ( A_1 ) is much bigger than ( A_2 ), then that tells you that the function contains a lot more of the ( \frac{1}{2\pi} ) Hertz frequency than the ( \frac{1}{\pi} ) frequency. Or maybe ( A_2 ) or maybe ( A_3 ) is bigger than ( A_1 ) or ( A_2 ).

So you can start to say, "Hey, this helps us think of a function not just in terms of the time domain, which ( F(T) ) does, but it can start bringing us to saying, 'Well, how much do we have of each frequency?'" And as we'll see with Fourier series and eventually Fourier transforms, that's going to get us into the frequency domain where we can start doing some signal processing.

So we're going to explore all of that in future videos. In order to understand how we can actually find these coefficients, we're going to review a little bit of our trigonometry, especially integrating trig functions. Then we're going to solve for these, and we're going to see how good we can approximate our function ( F ).