Fourier Transform 101 — Part 3: Fourier Transform
Previously, we covered the basic ideas behind Fourier Series starting from the “Real Fourier Series”.
Extending to the “Complex Fourier Series”.
In this story, we are finally going to step into the “Fourier Transform”!
Materials covered in this story
- Understanding the context of Fourier Transform respect to Complex Fourier Series
- Derivation of Fourier Transform
This will be a very important topic so try spending some time understanding this!
Again, main part of the derivations are from lectures taught by Dr. Wim van Drongelen and you could see his lecture online below. I will be adding more context and some information so that you could understand this topic in more depth.
Understanding the context of Fourier Transform respect to Complex Fourier Series
Before stepping into the actual meaning of the Fourier Transform (Continuous), let’s first try to understand the relationship between what we already know (Complex Fourier Series).
In Complex Fourier Series, when we wanted to convert the signal f(t) into a Frequency domain, we utilized this formula:
On the other hand, when coming back to the Time domain, we used this formula:
You will learn the equivalent formulas in Fourier Transform in a couple of minutes, but when we summarize this into a table, it’s something like this.
What I want to point out here is the underlying assumptions in each equation. For example, if you look at the top left formula which is the Complex Fourier Series version of Time -> Frequency domain conversion, you notice that it’s integrating over time “t” for a full period. We are dealing with “Continuous” function here. On the other hand, if you take a look at the right side where you have Frequency -> Time domain conversion, now you see a summation operation sigma. This is “Discrete” operation.
Similarly, if you now focus on “periodicity”, the only formula that has periodicity is the Complex Fourier Series version of Time -> Frequency domain conversion because if you look at the range it’s integrating over, it’s a full period. All the other equations are either integrating or summing from minus infinity to infinity.
To summarize this, the table looks like this.
Now I hope you have another perspective looking at different types of “Time to Frequency” conversions.
I will come back to this again with a complete list after covering “Discrete Fourier Transform”, but the take away is that if you account for
- “continuous” and “discrete” (2 possible options)
- “periodic” and “aperiodic” (2 possible options)
there’s 2 x 2 = 4 possible combinations. That’s the reason why there are 4 different types of “Fourier Transform” to deal with each case. Now you can answer why there are so many different types of “Fourier Transform”!
OK, that’s enough for understanding the context. Let’s get to the main part where we derive the formula for Continuous Fourier Transform.
Derivation of Fourier Transform
Check the table that we just discussed. We are trying to take a step from the “Complex Fourier Series” to the “Fourier Transform” in Time to Frequency domain. What’s the difference? Yes, that’s right. “Periodicity”. So in a way, you could say that if you extend the “Complex Fourier Series” from periodic to non-periodic, that’s the “(Continuous Time) Fourier Transform”.
Let’s try to visualize this. Previously, we were only dealing with a periodic signal. Now, to extend to non-periodic domain, we are going to just take a full period of the periodic signal f(t), and then going to stretch that period to infinity.
Do you get the idea? In other words, if there’s a single segment of signal from minus infinity to infinity, that signal is surely aperiodic because it’s not repeating itself over and over again.
But how can we express that in an equation?
Let’s recap the formula from “Complex Fourier Series”.
Now what we want to do is to take a full period just like what we saw as an image. (Don’t worry about the division of “T” as we are going to account for this in the later part of the equation.)
Now we define a new “c” by taking the limit to infinity. This is the part where we stretched the extracted segment of the signal.
Here, we could consider the multiplication of “n” times the “omega_0” as a new continuous variable called “omega” as if you remember that “n” goes from minus infinity to infinity and “omega_0” going to 0.
Now we have our formula for the “Fourier Transform”! This is a formula to convert the time domain signal into a frequency domain signal. Just as a reminder, this is dealing with a “continuous” function and is no longer periodic as indicated in the equation.
One last equation to derive for this story. Coming back to the equation from the “Complex Fourier Series”.
Using the “Fourier Transform” equation that we just derived we could change this equation to the following.
Now we insert this back to the “Complex Fourier Series” equation of Frequency -> Time domain equation below.
Below is a slight modification so that it leads to the final goal.
Now we set the limits.
Using the above limits, we get
And at last, we have this “Inverse Fourier Transform”! This is the equation to convert from Frequency domain to Time domain again. Note that this is also “Continuous” and “Aperiodic” as we discussed in the beginning of the story.
Summary
To summarize, the relationship between “Complex Fourier Series” and “Fourier Transform” is as follows:
Sometimes, the “Fourier Transform” is abbreviated as “FT” and the inverse as “IFT”.
And the formula for the “FT” is:
The “IFT”:
That’s it! We are going to cover the “Discrete Fourier Transform” next time!
I hope this helps! See you next time~
