Page 150 - Fundamentals of Radar Signal Processing

on reception; for a receding target it is lengthened by a factor of α . The
v
compression (expansion) of the pulse in time results in an expansion
(contraction) of the pulse bandwidth by the factor α due to the reciprocal
v
spreading property of Fourier transforms. Finally, the amplitude of the
waveform is scaled by the factor α (in addition to the range equation effects), a
v
consequence of conservation of energy when the time scale is altered.
It is virtually always the case in radar that the ratio |β | = |v/c| is very small.
v
For example, a car traveling at 60 mph (26.82 m/s) has a ratio |v/c| of 8.94 × 10 –
–6
8 ; an aircraft at Mach 1 (about 340.3 m/s at sea level) has |v/c| = 1.13 × 10 ; and
even a low-earth orbit (LEO) satellite with a velocity of 7800 m/s has |v/c| = 2.6
–5
× 10 . Expand each of the terms 1/(1 ± β ) and α = (1 + β )/(1 – β ) in a
v
v
v
v
binomial series and retain terms only to first order in β :
v

(2.95)

Equation (2.90) and the sinusoidal pulse special case of Eq. (2.92) then reduce
to

(2.96)

The echoed pulse length τ′ = τ/α ≈ (1 – 2β )τ. This small change in the pulse
v
envelope duration of 2α τ seconds is insignificant and can be ignored. The
v
amplitude factor of α ≈ (1 + 2β ) is certainly negligible compared to range
v
equation effects and can also be ignored. The change in delay from 2R /c to 2(1
0
+ β )R /c represents a percentage change of β in the delay and is also usually
v
0
v
insignificant, though for a system with fine range resolution at long enough
ranges the error could become a significant fraction of a range resolution cell.
However, β cannot be neglected in the phase term because the factor of
v
4πβ R /λ will frequently be a large fraction or even a multiple of π. With these
v
0
three approximations to the envelope term and amplitude, the Doppler shift
effects on the sinusoidal pulse of Eq. (2.96) reduce to