Fourier Transform 101 — Part 2: Complex Fourier Series
Last time, I covered Real Fourier Series based on lectures from Dr. Wim van Drongelen.
In this article, we are going to extend this to “Complex Fourier Series”.
Introduction to Complex Fourier Series
Just a short recap from the last time. In Part 1, we went through “Real Fourier Series” where we wanted to approximate periodic signal f(t) with P(t) where P(t) was represented as follows:
So that when we figure out coefficients a’s and b’s, we get an approximated version of f(t) described as linear combinations of sines and cosines.
In the end, we got the equations for coefficients:
OK, that’s enough for revisiting Part 1. Now we are going to extend this to “Complex Fourier Series”.
When we talk about complex numbers along with sines and cosines, do you remember the famous formula that contains all the above contents I just mentioned?
If you guessed it right, great! Yes, it’s the famous “Euler’s formula”!
Here, “j” is the imaginary number, “omega” is the frequency term, “t” is for time. If you visualize it, it’s something like this.
If you are not familiar with the formula, check the below link.
So by using this formula, we could actually extend the previous “Real Fourier Series” to “Complex Fourier Series”.
The P(t) in the “Complex Fourier Series” becomes this.
With coefficients term becoming this.
Note that now instead of calculating a’s and b’s separately as we did in the “Real Fourier Series”, we only have to deal with one coefficient term, c’s.
Although the equations changed its figure, the actual calculation behind the scene hasn’t changed. To show this, let’s start from these “Complex Fourier Series” and show that these are indeed same as “Real Fourier Series”. One tips is that instead of trying to follow by using your eyes, try to bring a sheet of paper and write down the equations to follow the process. This would definitely help you understand the process rather than just looking at it. It’s a golden rule for understanding!
First, we separate P(t)’s summation range into 3 ranges.
Note that in the second equation, we just flipped the signs so that the summation range in Sigma is the same as the other Sigma.
Now for the coefficient “c”, we simply use the euler’s formula to convert it using a’s and b’s derived in Part 1.
Same can be done for minus version of “c”.
Now, let’s consider a specific case (n = 0). Using the c’s above, we can say the following.
And the last part is to use what we have to convert P(t) from “Complex Fourier Transform” form to “Real Fourier Transform” form.
If you remember the P(t) I showed you in Part 1, it’s exactly the same! But the imaginary part is now gone!
So in “Complex Fourier Transform”, we write P(t) like this.
And the coefficients a’s and b’s from last time,
becomes a single coefficient term “c” as below.
Summary
Complex Fourier Series is almost the same as Real Fourier Series, just rewriting sines and cosines using euler’s number. The benefit is that now it could consider imaginary numbers as well as deal with a single coefficient term “c” rather than dealing with two coefficient terms.
I hope this helps! See you next time!
Next time, we are actually stepping into the Fourier Transform itself based on the Complex Fourier Series so stay tuned!
