Page 148 - Fundamentals of Radar Signal Processing

would have R(t) equal to a fixed R meters, while a constant-velocity target
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would have R(t) = R – vt meters. It makes no difference whether the radar, the
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target, or both are moving such that the range between the two is R(t), so it can
be assumed without loss of generality that the radar is stationary and the target is
moving, and that all measurements are made in the frame of reference of the
radar. Under these conditions the received signal can be shown to be (Cooper,

1980; Gray and Addison, 2003)

(2.86)

where k absorbs all radar range equation amplitude factors and h(t) is the
function that satisfies

(2.87)

The dot over h(t) in Eq. (2.86) denotes the time derivative. The minus sign

(180° phase shift) is required by the boundary conditions at a perfectly
conducting surface. The function h(t), which has units of seconds, is the time at
which a wave must have been launched in order to intercept the moving target at
time t and range R(t). For example, if R(t) is a constant R , then h(t) = t – R /c.
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For instantaneous velocities that are a small fraction of the speed of
light (virtually always the case as will be discussed shortly), the “quasi-
stationary” assumption is commonly made. This holds that the range change
during the short flight of any particular point in the waveform from the

transmitter to the target is negligible. Then R[h(t)] ≈ R(t) so that (Cooper, 1980)

(2.88)

The last step also uses the assumption . This result is exact when the
target is stationary, R(t) = R . Then h(t) = t–R /c exactly and
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exactly.
The case of a constant-velocity target is of special interest. Returning to the
exact result of Eqs. (2.86) and (2.87), let R(t) = R –vt and define β ≡ v/c. It is
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v
easy to show that