The impulse signal (defined in §B.10) has a constant Fourier transform:
We will now show that
Thus, the -periodic impulse train transforms to a -periodic impulse train, in which each impulse contains area :
Proof: Let's set up a limiting construction by defining
Using the closed form of a geometric series,
where we have used the definition of given in Eq. (3.5) of §3.1. As we would expect from basic sampling theory, the Fourier transform of the sampled rectangular pulse is an aliased sinc function. Figure 3.2 illustrates one period for .
The proof can be completed by expressing the aliased sinc function as a sum of regular sinc functions, and using linearity of the Fourier transform to distribute over the sum, converting each sinc function into an impulse, in the limit, by §B.13:
by §B.13. Note that near , we have
as , as shown in §B.13. Similarly, near , we have
See, e.g., [23,79] for more about impulses and their application in Fourier analysis and linear systems theory.
Exercise: Using a similar limiting construction as before,
show that a direct inverse-Fourier transform calculation gives
and verify that the peaks occur every seconds and reach height . Also show that the peak widths, measured between zero crossings, are , so that the area under each peak is of order 1 in the limit as . [Hint: The shift theorem for inverse Fourier transforms is , and .]