Fourier transforms of base functions

As already mentioned, Fourier transform computations often are easily facilitated by using known Fourier transforms of basic functions. Here, FTs of some of the functions defined in Sec. 3.1 are given, which were said to be particularly useful.

First, we are interested in the Fourier transform of the impulse function. This is easily computed by

$$\mathcal{F}(\delta)(\omega) = \int \delta(x) e^{-j2\pi \omega x} \textrm{d}x = 1$$ (3.19)

which also follows directly from Eq. 3.5 with t=0. This means that the FT of $\delta(x)$ is the unity function, i.e.,

$$\delta(x) \rightleftharpoons 1$$ (3.20)

This means that the Fourier transform of an impulse signal is a flat spectrum, i.e., it has equal energy at all frequencies and, inversely, a flat signal only has energy at $\omega = 0$. If we shift the impulse to some position t, we also get as a direct consequence of Eq. 3.5

$$\delta(x - t) \rightleftharpoons e^{2\pi j \omega t}$$ (3.21)

Second, we want to compute the FT of the comb function and, quite interestingly, it turns out that this is another comb function but with reciprocal spacing:

$$\mathrm{comb_T}(x) \rightleftharpoons \mathrm{comb_{1/T}}(\omega)$$ (3.22)