Fourier Series: part 5
06-Mar-2024
Common Fourier Transform Pairs and how it looks like
In this article, let’s have some fun and see how the function and its Fourier Transform corresponds to.
PS: the content maybe updated over time for my personal notes
For the rests of the sections, we use Fourier transform definition that is normalized with oscillatory factor/unit
For Any applicable function
Cycling property
Fourier transform behaves in such a way that applying it twice is the same as flipping the argument. Applying it 4-th times will recover the original function.
Suppose we have a dual and .
Due to this cycle rules, a special function has to be an even function if the function doesn’t change after being transformed twice. This is because the flipped sign has no effect for this function (invariant).
Because of this, we also have a special relationship since the cycle is modulo 4. Performing inverse Fourier Transform is the same as doing Fourier Transform 3 times.
For even function (function that is the same under flipped parameters), its inverse Fourier Transform is the same as its Fourier Transform, because the transform cycles in modulo 2.
Dirac Delta function and constants
Just to recap from the previous articles. We have defined a “function” (in quotes, because it is not a true function) called Dirac Delta which is used as a dual of a constant values/function.
As you can see that is an even symmetric function. So it doesn’t matter if it’s inverse or forward transform, it is always a pair from a constant.
Because any function can be thought of as product between a function and constants, dirac delta function can be seen everywhere.
Harmonics
A harmonic or as we can say a natural periodic signals can be decomposed easily using delta function.
We can proof the duality using forward or inverse transform, but for me it is easier to proof it backward.
Single frequency
Suppose that a harmonic is defined to be a signal with singular frequency . That means, in the Frequency domain, it can be viewed as a singular jump in value using delta function.
This yields the following relationship: