Page 154 - Fundamentals of Radar Signal Processing

(2.102)

Equation (2.102) shows that the range is approximately a quadratic function of

time for the crossing target scenario of Fig. 2.25. Using this truncated series in
Eq. (2.98) gives

(2.103)

All of the terms are the same as in the constant-velocity case of Eq. (2.99)
except for the middle exponential. Recall that instantaneous frequency is
proportional to the time derivative of phase. The quadratic phase function
therefore represents a Doppler frequency shift that varies linearly with time due
to the changing radar-target geometry:

(2.104)

As the target aircraft approaches from the left in Fig. 2.25 (t < 0) the

instantaneous Doppler shift is positive. When the aircraft is abreast of the radar
(t = 0) the Doppler shift is zero because the radial component of velocity is
zero. Finally, as the aircraft passes by the radar (t > 0) the Doppler shift
becomes negative, as would be expected for a receding target. This quadratic
range case is important in synthetic aperture radar and will be revisited in Chap.
8.
The exponential term exp(– j4πR(t)/λ) in Eq. (2.98) is called the phase
history of the received signal. This terminology is applied both to the complex

exponential and to just its phase function (–4πR(t)/λ). The phase history encodes
the variation of the range between the target and radar during the data collection
time. For the constant-velocity example [Eq. (2.99)], the phase history is a
linear function of time corresponding to a constant frequency sinusoid, i.e., a
constant Doppler shift. For the crossing target example of Eq. (2.103), it is a

quadratic function of time, producing a Doppler shift sinusoid having a
frequency that varies linearly with time. Other radar-target motions will
produce other functional forms for the phase history.
More generally, the term phase history can refer to the variation of phase
(or the corresponding complex exponential) in any dimension of the radar data.
Two other common uses are to describe the fast-time phase function of a
frequency- or phase-modulated waveform or the spatial phase variation across

the face of an array antenna at a fixed time. As will be seen, the phase history is
central to radar signal processing. The design of many important radar signal