Page 200 - Fundamentals of Radar Signal Processing

frequency exactly equals one of the DFT frequencies, that is, ω = 2πk /K for
0
D
some k (k = 5 and K = 20 in this example, corresponding to ω = π/2 rads per
0
0
D
sample). One DFT sample falls on the peak of the asinc function, while all of
the others fall on its zeroes, so that the DFT becomes an impulse function. This
could be viewed as an ideal measurement, since the discrete spectrum indicates
a single sinusoid at the correct frequency and nothing else; but it does not reveal
the mainlobe width or sidelobe structure of the underlying DTFT.

FIGURE 3.12 The 20-point DFT of a sampled pure complex sinusoid of 20
samples length, normalized frequency 0.25 cycles per sample, and amplitude 1.
The dotted line shows the underlying DTFT of the same data from Fig. 3.11.

More importantly, the good result of Fig. 3.12 depends critically on the
actual sinusoid frequency exactly matching one of the DFT sample frequencies.
If this is not the case, the DFT samples will fall somewhere on the asinc

function other than the peak and zeros. Figure 3.13 shows the result when the
example of Figs. 3.11 and 3.12 is modified by changing the normalized
frequency from 0.25 to 0.275 (equivalently, changing ω to 0.55π), exactly
D
halfway between two DFT sample frequencies. Now a pair of DFT samples
straddle the actual underlying peak of the asinc function, while the other samples
fall near the sidelobe peaks. Even though the underlying asinc function is
identical in shape in both cases, differing only by a half-bin shift on the

frequency axis, the effect on the apparent spectrum measured by the DFT is
dramatic: a broadened and attenuated mainlobe, and the appearance of
significant sidelobes where before there apparently were none.