Fourier Series introduction

so I have the graph of y is equal to f of T here our horizontal axis is in terms of time in terms of seconds and this type of function is often described as a square wave and we see that it is a periodic function that it completes one cycle every 2 pi seconds and so we could say its period is equal to 2 pi if we want to put the unit's we could say 2 pi 2 pi seconds per cycle we could write it like that we could also just write s 4 seconds and it's frequency is going to be one over that so we could write its frequency its frequency is equal to 1 over 2 pi cycles cycles per second cycles per second it can also be described as Hertz and 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 of T so can we take our F of 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 and as we'll see that's going to be based on the average value of the function over over over one period so a sub 0 and then let's start adding some periodic functions here and so let's take a sub 1 times cosine of T now why am i starting with cosine of T and I could also add a sine of T so plus plus B sub 1 plus B sub 1 of sine of T why am i starting with cosine of T and sine of 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 and these weights will tell us how much they involve it if a one is much larger than B 1 it says ok this is this is a lot more of cosine of T in it then it has of sine of T in it and 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 and so what we're going to do is we're going to add we're going to add sinusoids of frequencies that are multiples of these frequencies so let's add let's add a sub two so another weighting coefficient times cosine of 2t this has this has a frequency of 1 over 2 pi this has twice the frequency this has a frequency of 1 over pi and then a sub 3 times cosine of 3 T and 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 plus B 2 sine of 2t plus B 3 sine of 3t and you might be saying well ok this this seems like a fun little mathematical exercise but why do folks even do this well this was a first explored and that 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 and they were interesting to him in the study of differential equations because a lot of differential equations Arkansas l'v when in 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 it's very used it's heavily used at Electrical Engineering is you can view these coefficients as they are their weights on these cosines and sines but another way to think about it is they tell you how much of different frequencies that this function contains so for example if a1 is much bigger than a 2 then that tells you that that function contains a lot more of the 1 over 2 pi hertz frequency then the one over pi frequency or maybe a two or maybe a three is bigger than a one or a two and so you start you can start to say hey this this helps us think of a function not just in terms of the time domain which f of 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 give us 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 and then we're going to solve for these and we're going to see how good we can approximate our function f