An impulse train can be defined as a sum of shifted impulses:
Here, is the period of the impulse train, in seconds--i.e., the spacing between successive impulses. The -periodic impulse train can also be defined as
where is the so-called shah symbol :
Note that the scaling by in (B.46) is necessary to maintain unit area under each impulse.
We will now show that
That is, the Fourier transform of the normalized impulse train is exactly the same impulse train in the frequency domain, where denotes time in seconds and denotes frequency in Hz. By the scaling theorem (§B.4),
so that the -periodic impulse-train defined in (B.46) transforms to
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
so that . We may interpret as a sampled rectangular pulse of width seconds (yielding samples).By linearity of the Fourier transform and the shift theorem (§B.5), we readily obtain the transform of to be
Using the closed form of a geometric series,
with , we can write this as
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
as . Finally, we expect that the limit for non-integer can be neglected since
whenever and is some integer, as implied by §B.13.
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 .]
Poisson Summation Formula