Example of CTFT application
$ x(t) = e^{j2\pi f_0 t} $
What is the CTFT of the given function?
Possible approaches to this problem:
- Integration using the CTFT equation
- Guess and invert
- Use existing pairs to arrive at the solution
Let's look at the first possible approach:
$ \begin{align} X(f) &= \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt \\ &= \int_{-\infty}^{\infty} \! e^{j2\pi f_0 t} e^{-j 2\pi f t} dt \\ &= \int_{-\infty}^{\infty} \! e^{-j2\pi (f-f_0)t} dt \\ &= \text{Oops! Where do we go from here?} \end{align} $
Second possible approach:
Well, we know that $ e^{j2\pi f_0 t} $ is an impulse function in the Fourier domain. Can we guess that this would be a shifted delta?
$ \begin{align} X(f) &= \delta (t) e^{-j2\pi f_0 t} \\ &= \delta (t - f_0) \end{align} $
Maybe??
Third possible approach:
If $ CTFT[\delta (t)] = 1 $ or $ CTFT[1] = \delta (t) $
Then using reciprocity/duality,
$ \begin{align} e^{j2\pi f_0 t} &= \delta (t) \ \text{at frequency} \ f_0 \\ &= \delta (t-f_0) \end{align} $
From the CTFT, we can sail smoothly into the Continuous-Space Fourier Transform.
