Page 279 - Fundamentals of Radar Signal Processing

(4.108)

The phase term that is quadratic in δt is a complex constant. In synthetic
b
aperture imaging it is called the residual video phase (RVP). The middle

complex exponential contains a term that is linear in t and therefore represents a
constant-frequency complex sinusoid. By inspection, the sinusoid frequency is
F = –βδt /τ Hz. F is proportional to δt and thus to the range of the scatterer
b
b
b
b
relative to the CRP. The differential range can be obtained from the mixer output
frequency as

(4.109)

Heuristically, the scatterer produces a constant frequency tone at the output
of the stretch receiver because the receiver not only removes the carrier from

the LFM echo but also combines it in a mixer with a replica of the LFM with a
delay corresponding to the CRP. In the conventional real-signal receiver of Fig.
1.13, the mixer produces sum and difference “beat” frequencies. The sum
frequency is removed by a lowpass filter. (This LPF is not needed in the
complex representation and is therefore not shown in Fig. 4.31.) The difference
frequency is the difference between the instantaneous frequency of the LFM
echo and the LFM reference. Since both have the same sweep rate, this beat

frequency is a constant.
If there are several scatterers distributed at ranges R and delays δt , the
i
i
stretch receiver output is simply the superposition of several terms of the form
of Eq. (4.108)

(4.110)

Thus the output of the stretch receiver contains a different beat frequency tone

for each scatterer. The reason for the Fourier transform block in Fig. 4.31 is
now apparent. Spectral analysis of y(t) can identify the beat frequencies present
in the mixer output and therefore the ranges and amplitudes of the scatterers
present in the composite echo. Figure 4.32 illustrates the instantaneous
frequencies and timing of the signals involved for three scatterers, one in the
middle and one at each edge of the scene.