5.9. The Discrete Fourier Transform and Fast Fourier Transform

Most signals in computing systems take on values only at discrete moments in time. Specifically, the signal is defined only at times nT, where n is an integer, and T, a real, is the inverse of the clock frequency. We denote such a signal as x[nT] in contrast to a continuous signal, which is denoted x(t).

5.9.1. The Impulse Train

We define an impulse train s_T(t) with parameter T to be a signal that consists of an infinite series of delta signals with time-separation T. An impulse train is denoted as

What is the Fourier transform of an impulse train? By the linearity property of the Fourier transform, the transform of the infinite sum in Equation 5.45 is the sum of the individual transforms. From Example 5.12, . Therefore, the Fourier transform of an impulse train is given by