equal to the inter-pulse spacing at the Earth’s
surface, A - V B /f a .
The correlation function for the reflecting neighbourhood
in the scene is
R d (m,n)
o 0 A m - n
0 m * n
(20)
Notice the subtle difference between this correlation
function and that of the discrete isolated scatterer of Eq.
14. The distributed scatterers are modelled as if they
were replaced by a set of individual (point) reflectors
having uniformly distributed random phase, spaced from
each other by the surface inter-pulse distance. (The inter
pulse spacing is generally smaller than the "resolution",
sometimes by an appreciable margin.) Whereas the cells
are statistically independent at the surface, the finite
bandwidth of the radar and processor so that their
energies spread over neighbouring cells in the image, thus
reinforcing each other. This gives rise to a processing
gain of relevance to the calibration question. Use of a
well known and measurable quantity such as surface
sample spacing avoids the insecurity encountered if the
model were cast in terms of the azimuth resolution cell as
is usually done. Resolution is not a fundamental
parameter, but a consequence of optional and sometimes
imprecise processor operations.
The discrete scene correlation function of Eq. 20 may be
substituted into Eq. 13 to find
E[g d \ - o 0 A l/>(”)l 2 (21)
The sum over the impulse response may be re-written as
El 2 - Ip(°)I 2
Ei^wi 2
n
U><0)| 2
(22)
in which the square of the number of integrated pulses
appears as in the point scatterer case, and a sum over the
normalized image impulse response which we let be p p .
Thus, the processing gain G P may be defined as
G,-1.2Ep ,00 (23)
/1-1
which describes the increase in mean signal level in the
image over that predicted from the impulse response
peak. The sum is over those cells neighbouring the
reference cell whose image over-laps the image pixel
being observed. In all cases the summation limit M needs
to be sufficiently large to span the spread of the modified
impulse response.
The expected image power (per pixel) from a set of
distributed scatterers may be written
(g d ) - o 0 A K'Nfa (24)
which should be compared to the response from an
isolated point scatterer, Eq. 15. It may be shown that Eq.
24 is robust in the presence of focus errors and multi
looking.
In order to illustrate behaviour typical of G P , let the
antenna prefilter be Gaussian and of Doppler bandwidth
B sampled by the pulse repetition frequency f a , and let
the processor filter be matched. Then
G p - 1 + 2 exp
n-1
2 „2)
B z n
f
Ja
(25)
Eq. 24 is plotted in Figure 4. It shows for over-sampling
ratios in excess of 1.2, that the approximation f a /B is
satisfactory. For orbital situations in which the over-
sampling ratio may be rather close to unity, the exact
expression derived through the correlation function of the
filters in use may be preferred. (Note that this
formulation converges to the correct form for under
sampling, a condition not satisfied by the usual
approximation.)
Processing gain
Oversampling ratio
Figure 4. Processing Gain as a function of
the over-sampling ratio, and the
conventional approximation.
DISCUSSION
The results above may used to find the usual signal to
noise and clutter (distributed scene) to noise ratios. In
both cases, it is worthwhile to note the way in which
spacecraft and beam footprint velocities enter the
expressions. The discussion focuses on the issue of
calibration, for which comparisons of the clutter to the
signal from a point reference signal are required. For
compactness in the following, all expressions are derived
assuming that only the chosen signals and/or noise are
present. In practice, adequate signal to noise ratios are
needed, and subtraction of noise and/or clutter is
required. The points of this paper would not be served by
going into details at that level, and good practical
references are available (e.g., Gray et al. 1990).
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