Page 196 - Fundamentals of Radar Signal Processing

(3.12)

Y (ω) is a function of a continuous frequency variable, despite the fact that the
s
signal y [m] is discrete. Furthermore, it is periodic in ω with period 2π radians
s
per sample.
Consider the K-point discrete spectrum Y [k] formed by sampling Y (ω) at
s
s
K evenly spaced points along the interval [0, 2π),

(3.13)

Interpret Y [k] as a K-point DFT. To find the relation between it and the original
s
signal y [m], compute its inverse DFT:
s

(3.14)

The inner sum can be evaluated as

(3.15)

6
where δ [·] is the discrete-time unit impulse function. Substituting Eq. (3.15)
back in Eq. (3.14) gives

(3.16)

Although a finite length signal y [m] was assumed in the previous analysis, the
s
result also holds for infinite-length signals.
Equation (3.16) shows that, in the dual of time domain sampling behavior,
sampling the frequency spectrum replicates the signal in the time domain with a
period proportional to the frequency sampling rate. Specifically, if the slow-
time signal spectrum is computed at K frequency points, the time domain signal