Fourier Transform 101 — Part 1: Real Fourier Series
In this series, I’m going to explain about Fourier Transform. Have you heard of the term? If not, that’s totally fine. This will be the introduction to the concept for you. If you have heard of it, but haven’t really grasped the concept yet, no need to worry. I will start from the very beginning from Real Fourier Series, moving on to Complex Fourier Series, then Continuous Fourier Transform (CFT), Discrete Fourier Transform (DFT), and at last, Fast Fourier Transform (FFT).
I went through a lot of materials before trying to write about Fourier Transform. I feel like the YouTube video series from Dr. Wim van Drongelen and his textbook: “Signal Processing for Neuroscientists” is the easiest to understand from the beginning. So I will be following the derivations from him and his lectures, summarizing and adding some more information so that it’s easier for you to understand.
YouTube videos are online and you could take a look at the first video lecture below. Try to follow Lecture 4, 5A, 5B, and 5C for the topics covered in this series.
Materials covered in this story
In this story, I’m going to cover the following:
- What is Fourier Transform in general
- What are the orthogonal functions
- Introduction to Real Fourier Series
I hope this will be useful for those who are trying to understand Fourier Transform.
Enough said, let’s get started!
Introduction to Fourier Transform
So, to begin this story, let’s first take some time understanding what Fourier Transform is, without using any equations.
“Waves” are everywhere these days. It’s just you don’t really see them or deal with them if you are living your normal daily life. 4G or LTE that connects your phone to the internet is a form of waves. When you speak, a form of waves are carried to your friends ears so that it vibrates the eardrum and your friends can understand what you are saying.
Fourier Transform is a very important and useful technique to understand these waves. Long story short, what Fourier Transform is doing is it tries to approximate the signal (wave) of your interest using different kinds of sine and cosine waves. I hope you remember sines and cosines, but here’s some visualization for recap.
For visual introduction of Fourier Transform, I strongly recommend you watch the video from 3Blue1Brown.
I hope you understand the basic concept of what Fourier Transform is by now. Let’s jump into the mathematics to fully understand the concept.
What are the orthogonal functions
Before start deriving the equations for Fourier Transform, let’s take a moment to understand about orthogonal functions. This is a very useful property that will be used soon in derivation of Real Fourier Series.
Just to give you an example, think of the following orthogonal function.
This function is orthogonal because if you think about both cases m=n and m not equal n, you see why this is an orthogonal function. For the term that disappears, take a look at the plot on the right. Basically, for any sine and cosine function integrating over a period, it becomes zero. This is true regardless of the frequencies.
To know more about it, check the video I posted at the top or check the link below.
Introduction to Real Fourier Series
OK, finally we can start deriving the “Real Fourier Series” (RFS). Let’s say there’s a signal f(t) (assuming it’s a periodic signal, meaning there’s a repeated pattern over time) that you want to decompose into bunch of sines and cosines. In this case, what you want to do is to approximate f(t) with P(t):
where P(t) is represented by something like below:
Here, bunch of a’s and b’s are coefficients that you want to get in the end so that P(t) is a certain combinations of sines and cosines that describe f(t) pretty well.
Now, if you want to approximate f(t) with P(t), meaning that you want f(t) to be the same as P(t) as much as possible, one way to do this is to minimize the mean squared error between the two functions.
Note that the range of integral from here on is an arbitrary period of time unless specified otherwise. Here, I’m just writing it as T, but you could think of it as a full period [t, t+T].
To minimize the loss, we usually take the derivatives respect to coefficients. In this case, a’s and b’s and by setting them to zero, we can get the best coefficients that best describe the function P(t) which is the approximated version of f(t) using sines and cosines.
By solving the above derivatives, we could get coefficients a’s and b’s. So let’s start by solving a’s first.
It’s ok to use the above equation, but we could clean the equation a bit.
Since we don’t really need constant 2 removing it. Also, there’s no a’s in f(t) so by taking the derivative respect to a’s, we could also remove this term.
Now we are ready to move on. Let’s start from n = 0 case.
Case 1: n = 0
So now we have a simplified version of the derivative respect to a’s equals to zero:
To solve this equation, we want to know what’s the following term represent:
If you look back to check what P(t) was, we can easily calculate this derivative and get this.
Now plugging this back to the derivative = 0 equation, we get:
We could solve this left side just by simply taking the integral on each term of P(t). Note that most of the term just goes to zero if you remember what we have talked about in the orthogonal functions. (This part simply all the terms except the first one goes to zero because integration of any cosine and sine functions over a period of time is zero if you check the figure in the orthogonal function section)
Therefore we get coefficient a_0
Case 2: n = 1
Now we do the same for n = 1. First calculate the partial derivative from P(t).
Plug it in the equation that we are trying to solve.
And by using the properties of orthogonality, we get:
Therefore, the coefficient a_1 is
Case 3 and beyond: General a’s and b’s
So if you keep doing that for all the a’s and b’s, you will realize the general equation for a’s and b’s.
By using these equations, you can easily get coefficients that can then be plugged into the original equation of P(t).
After that, P(t) will be your approximation of f(t) as a real fourier series! Congrats!
Note that there are assumptions that the original signal f(t) has to be in the real domain and periodic function. In the next story, we are going to extend to complex fourier series where you can now even apply this to complex (imaginary) domain.
Summary
- What is Fourier Transform in general
It’s a technique to decompose signals into sine and cosine waves.
- What are the orthogonal functions
It only has a value when m = n and otherwise, it’s zero.
- Introduction to Real Fourier Series
One of the early steps before understanding the Fourier Transform. Decomposition of a periodic function using sine and cosine with coefficients applied in the real domain.
I hope this helps! See you next time~
