in which both velocities appear, instead of V 2 as derived 
in the aircraft case. Indeed, some use the term "radar 
velocity" to refer to and replace it by a single 
parameter, a practice which in the view of this author is 
rather mis-leading. It forces the conceptualization of an 
orbital SAR into the aircraft mould, and hides the 
essential differences between spacecraft and beam 
velocities. The aircraft or "flat Earth" case is simply a 
limiting form of the more general expression derived from 
the orbital geometry. 
SAR SYSTEM MODEL 
In order to support the arguments of this paper, a simple 
SAR system azimuth model is sufficient, as in Figure 3. 
Figure 3. Simple azimuth SAR model. 
The signal input s is either from a coherent point scatterer 
(such as a corner reflector), or from a distributed diffusely 
scattering scene, and is combined with additive Gaussian 
noise. The signals (but not the noise) are subject to gain 
through the antenna w with Doppler bandwidth B and 
beamwidth (3 (both measured as equivalent rectangle 
widths in the two-way amplitude domain), and (peak) 
transmitted power, and R 2 and atmospheric losses, the 
latter represented by the factor K. The pulse repetition 
frequency f a delivers a sequence of samples r, to the 
processor consisting of a weighted digital filter and 
magnitude squared detection. The impulse response of 
the radar (Harger 1970, Raney 1983) to signals from the 
scene is 
K 2 \p(n)\ 2 
where p(n) is the discrete convolution of the prefilter with 
the processor filter, 
p(n) - Y, w(n-i)h(i) ( 12 ) 
In general, the expected output of the processor (per 
pixel) in response to signal input is given by 
E[gU)l - K 2 E 
Y p(j-n)s(n) 2 
(13) 
- E P(j-m)p(j~n) R(m,n) 
m n 
where R(m,n) is the discrete autocorrelation function of 
the signal at the scene. The sums are over all samples 
that contribute to the signal at the output as evaluated 
below. In the following, it is assumed that the processor 
filter h is conjugate in phase to the input filter w. Both w 
and h are normalized such that max |. | =1. 
Point Scatterer 
For a discrete point scatterer of radar cross section a 
isolated at the (arbitrary) pixel position n=0, the 
correlation function at the scene is 
R p (m,n) 
a m - n - 0 
0 m * n 
(14) 
which leads (after substitution into Eq. 13) to the 
expression for the peak value at the output of the 
processor 
S„-oK 2 N^ (15) 
where 
b(0)l - 3 (16) 
is the effective number of pulses integrated by the 
radar/processor combination as limited by the time 
duration T of scatterer observation sampled at the pulse 
repetition frequency f a . 
Additive Noise 
Noise enters the system free of the Doppler weighting 
imposed by the antenna pre-filter. For an ensemble of 
noise signals {n(m)} at the input to the processor h(tn), 
the mean image noise level is 
£[*J - E E A(m) *(«)*„(«.») (17) 
For additive noise with mean power N 0 whose different 
samples are statistically independent (such as is the case 
for receiver noise), the noise correlation function may be 
written 
R n (m,n) 
N 0 m - n 
0 m * n 
(18) 
It follows that the expected noise level (for each look 
filter output) at the image is 
£[«.1 - W„EI'>(">I 2 - "o'«,. (19) 
which is proportional to the effective number of 
pulses integrated. In more generality, the image mean 
noise power is always proportional to the number of 
pulses integrated in azimuth since noise samples in the 
azimuth dimension are statistically independent. 
Distributed Scatterers 
Consider a region of spatially distributed statistically 
independent scatterers having mean reflectivity per unit 
area of ct 0 . Of these, we are interested only in one 
subset, namely a neighbourhood of adjacent cells each of 
length A meters on the surface, for which the 
corresponding radar cross section of the cell is o 0 A . 
(The range depth of the cell is of no consequence to this 
discussion, and is taken to be unity.) The cell length is 
711