Page 153 - Fundamentals of Radar Signal Processing

portion will be discussed shortly.

Equation (2.98) adequately describes not only constant but also time-
varying Doppler frequency shifts. If the target moves relative to the radar at
constant velocity, R(t) = R – vt,
0

(2.99)

Equation (2.99) is identical to the second line of Eq. (2.97), with the exception
that the constant phase shift is –4πR /λ instead of –(1 + β )4πR /λ radians. This
0
0
v
difference in the constant phase shift does not affect the magnitude or Doppler
shift of the echo and can be ignored. Thus the analysis approach of Eq. (2.88) is
consistent with the earlier results in all important respects.
For a more interesting example of the use of Eq. (2.88), consider Fig. 2.25
again. Let the radar be located at (x, y) coordinates (x = 0, y = 0) with its
r
r
antenna aimed in the +y direction, and let the coordinates of the target aircraft
be (x = vt, y = R ). This means that the target aircraft is on the radar boresight
t
0
t
at a range R at time t = 0 and is crossing orthogonal to the radar line of sight at
0
a velocity v meters per second. The range between radar and aircraft is

(2.100)

While it is possible to work with Eq. (2.100) directly, it is common to expand
the square root in a power series:

(2.101)

In evaluating this expression, the range of t that must be considered may be
limited by any of several factors, such as the time the target is within the radar
main beam or the coherent processing interval duration over which pulses will
be collected for subsequent processing.

Assume that the distance traveled by the target within this time of interest is
much less than the nominal range R so that higher-order terms in (vt/R ) can be
0
0
neglected: