Fourier Analysis — Pt 2. Fourier Transform
Quick recap from my last blog about Fourier Series …
Last time we talked about the fundamental Fourier Series where we establish a linear combination of different sinusoids to represent a periodic function. And in the end, we also get the complex Fourier series. Since we are talking about Fourier Transform here, we can further replace 1/T with frequency f.
From Fourier Series to Fourier Transform
I left a question last time asking what we should do to extend the Fourier Theory beyond periodic functions and discrete frequency spectra when Fourier series alone cannot.
Well.. the answer, as you might have already figured out from the title, is Fourier transform!
Fourier transform is the one that can be viewed as an extension of the Fourier series to non-periodic functions and allows us to deal with non-periodic functions.
So how can we go from Fourier series to Fourier transform then?
We first need to set nf in the complex Fourier series to f_n so that we have:
We can then put f_n back to complex Fourier series.
Now the most important part in transition to Fourier transform from complex Fourier series is to push the time limit T to ∞. And since T→∞, we have Δf → 0.
Then we have:
A little trick here is to multiple Δf/Δf to it…
We then also define a new function F(f_n) as:
We can then replace U_n/Δf with F(f_n) in f(t):
We can see from here fairly clearly that this is actually the definition of integral so…
And voilà, there you have it:
F(f) is what we normally would call Fourier Transform(FT) of the function f(t). And f(t) is what we normally call inverse Fourier Transform(IFT).
FT transforms the function in time domain into frequency domain while IFT transform frequency domain back to time domain.
Some notations: if we have our original signal x(t)
In the case of a finite-duration, discrete-time signal, we use something called Discrete Fourier Transform(DFT). DFT is simply replace integral with summation.
A useful function: Dirac-Delta function (Impulse Function)
First we define a unit are impulse of width w as:
which looks like so.
This function has the property that its integral equals to 1:
And we can see from the plot that if we make w smaller, the pulse height will increase while still preserving the unit area and if we limit w to 0, it will eventually look like:
This is how we define Dirac-Delta function and it still has area of 1 which means the integral of δ(x) still equals to 1:
Some properties of Dirac-Delta function:
(1) δ(x-a) is a pulse at a
(2) bδ(x) has area of b at x=0 (the pulse has height b)
(3) δ(cx)=δ(x)/|c| for c ≠ 0
(4) f(x)δ(x-a) = f(a)δ(x-a)
So, why is Dirac-Delta Function useful w.r.t Fourier Transform?
Let’s consider a shifted delta function: δ(t-a), it has a very simple FT.
And by calculating IFT of that, we can get δ(t-a)
Now if we have a function f(t)=cos(2πAt)
There you can see clearly that in frequency domain cos(2πAt) is represented by two frequencies A and -A both with amplitude 0.5.
That’s all I want to include for Fourier Transform in this blog! I hope you enjoyed it.
Although we still can talk a lot more about Fourier Transform, do you know that Fourier Transform has close relationship with Convolution Operation? Well, if you are curious about that, stay tuned for Part 3!
