Page 239 - Fundamentals of Radar Signal Processing

per second. For typical pulse lengths, these are fairly large values. For example,

a 10 μs pulse would exhibit a Rayleigh resolution in Doppler of 100 kHz, or in
velocity at X band (10 GHz) of 1500 m/s. Many systems do not observe such
high Doppler shifts, so Doppler mismatch effects are insignificant and targets
cannot be resolved in Doppler on a single pulse. If finer Doppler resolution is
desired, a very long pulse may be needed. For example, velocity resolution of 1
m/s at X band requires a 15-ms pulse. The range resolution is then a very poor

2250 km. This conflict between good range resolution and good Doppler
resolution can be resolved using a pulse burst waveform, which will be
addressed in Sec. 4.5.

4.4 The Ambiguity Function

4.4.1 Definition and Properties of the Ambiguity Function
In the preceding sections, the matched filter response for the simple pulse
waveform has been analyzed to show its behavior both in time and in response

to Doppler mismatches. The ambiguity function (AF) is an analytical tool for
waveform design and analysis that succinctly characterizes the behavior of a
waveform paired with its matched filter. The AF is useful for examining
resolution, sidelobe behavior, and ambiguities in both range and Doppler for a
given waveform, as well as phenomena such as range-Doppler coupling
(introduced in Sec. 4.6.4).
Consider the output of a matched filter for a waveform x(t) when the input

is a Doppler-shifted response x(t)exp(j2πF t). Also assume that the filter has
D
unit gain (α = 1) and is designed to peak at T = 0; this merely means that the
M
time axis at the filter output is relative to the expected peak output time for the
range of the target. The filter output will be

(4.30)

which is defined as the complex ambiguity function Â(t, F ). An equivalent
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definition can be given in terms of the signal spectrum by applying basic Fourier
transform properties:

(4.31)

The ambiguity function is defined as the magnitude of Â(t, F ),
3
D