The cross-section data presented in this section lead to the conclusion that it would not be appropriate simply to select a single value and expect it to have meaning in the computation of the radar equation without further qualification. Methods for dealing with the cross sections of complicated targets are discussed in the next section.
(it:nsity function for the cross section er is given by tl1e density function p( r;) = 1
exp ( - a )
a., a.,.
a20 (2.39a)where era, is tile average cross section over all target lluctuations.
Case 2. The probability-density function for the target cross section is also given by Eq. (2.39a), hut the fluctuations arc more rapid than in case J and arc taken to be independent from pulse to pulse instead of from scan to scan.
Case 3. In this case. the fluctuation is assumed to be independent from scan to scan as in case I. hut tl1c ptohahility-drnsity function is given by
4cr (
2<T)
11(cr) = 2 exp -
CT av O"av
(2.39b) / Case 4. The fluctuation is pulse to pulse according to Eq. (2.39b)
The probability-density function assumed in cases l and 2 applies to a complex target consisting of many independent scatterers of approximately equal echoing areas. Although, in theory, the number of independent scatterers must be essentially infinite, in practice the number may be as few as four or five. The probability-density function assumed in cases 3 and 4 is more indicative of targets that can be represented as one large rellector together with other small rellectors. In all the above cases, the value of cross section to be substituted in the radar equation is the average cross section
<Ta".
The signal-to-noise ratio needed to achieve a specified probability of detection without exceeding a specified false-alarm probability can be calculated for each model of target behavior. For purposes of comparison, the nonlluctuating cross section will he called case 5.A comparison of these five models for a false-alarm number "I= 108 is shown in Fig. 2.22 for 11 = 10 hits integrated. When the detection probability is large, all four cases in which the target cross section is not constant require greater signal-to-noise ratio than the constant cross section of case 5. For example, if the desired probability of detection were 0.95, a signal-to- noise ratio of 6.2 dB/pulse is necessary if the target cross section were constant (case 5), but if the target cross section fluctuated with a Rayleigh distribution and were scan-to-scan uncor- related (case I), the signal-to-noise ratio would have to be 16.8 dB/pulse. This increase in signal-to-noise corresponds to a reduction in range by a factor of 3.28. Therefore, if the characteristics of the target cross section are not properly taken into account, the actual performance of the radar might not measure up to the performance predicted as if the target cross section were constant. Figure 2.22 also indicates that for probabilities of detection greater than about O.JO. a greater signal-to-noise ratio is required when the fluctuations are uncorrelated scan to scan (cases I and 3) than when the fluctuations are uncorrelated pulse to pulse (cases 2 and 4). In fact. the larger the number of pulses integrated, the more likely it will be for the fluctuations to average out. and cases 2 and 4 will approach the nonfluctuating case.
Curves exist i i .. n for various values of hits per scan, 11, that give the signal-to-noise ratio per pulse as a function of Pd and 111 . The signal-to-noise ratio per pulse can be used in the form of the radar equation as given by Eq. (2.32). It is not necessary, however, to employ such an elaborate set of data since for most engineering purposes the curves of Figs. 2.23 and 2.24 may be used as corrections to the probability of detection (as found in Fig. 2. 7) and as the integration improvement factor (Fig. 2.8a) for substitution into the radar equation of Eq. (2.33).
0.99 0.98 0.95 0.90 ((' 0.80
._g C 0.70
u
~ OJ 0.60 .... 0.50 'O
0
ro 'O
0.40 0.30 0.20 0.10 0.05 0.02 0.01
-10
- 15
OJ C 0 L
-10 0
.2 0
L
<lJ
<I) 0 C 0 ' ...-- 5
0 C 0 ,
<I)
0 c::
:;:: 0
"O 0
"O
<{
-5 0 5 10 15 20
Signal-to-noise ratio per pulse, dB
25
Figure 2.22 Comparison of detec- tion probabilities for five different models of target fluctuation for
11 = 10 pulses integrated and 30 false-alarm number n1 108.
(Adapted from Swerling. 31)
Figure 2.23 Additional signal-to-noise ratio required to achieve a particular probability of detection, when the target cross section fluctuates, as com- _5 '--...l.--..1..----1.._--1.._..___.___._-1--..1._..1..-_.i.___--1.._-1-.,....J pared with a non fluctuating target;
0.01 0.05 0.1 0.2 0.3 0.5 0.7 0.8 0.9 0.95 0.99 single hit, n = L (To be used in con- Probability of detection junction with Fig. 2.7 to find (S//\1)1.)
30 ro "O
0 25
u
.E
c
<lJE 20
<I.I
>
e
a.~
C 15
g
!:'.
:E O'
'=
" 10
<::
~
5
10
~Case 4 - 50% Pd
Coses 1 and 3 - 50, 90, 99% Pd Cose 5 - 99% Pd
Cose 5 - 50% Pd
n, =
108100 Number of pulses inlegroted, n
1000
Figure 2.24 Integration-improvement factor as a function of the number of pulses integrated for the five types of target tluctuation considered.
The procedure ror usirig the radar equation when the target is described by one of the Swerling models is as follows:
1. Find the signal-to-noise ratio from Fig. 2.7 corresponding to the desired value of detection probability Pd and false-alarm probability Pr •.
2. From Fig. 2.23 determine the correction factor for either cases 1 and 2 or cases 3 and 4 to be applied to the signal-to-noise ratio found from step 1 above. The resultant signal-to-noise ratio (S/ N)i is that which would apply if detection were based upon a single pulse.
3. If" pulses are integrated, the integration-improvement factor l;(n)
=
nE;(n) is found from Fig. 2.24. The parameters (S/ N)1 and 11£;(11) are substituted into the radar equation (2.33) along with <Ta,.The integration-improvement raclor in Fig. 2.24 is in some cases greater than 11, or in other words, the integration efficiency factor E1(n)
>
1. One is not getting something for nothing, for in those cases in which the integration-improvement factor is greater than n, the signal-to-noise ratio required for n1
is larger thari for a nonfluctuating target. The signal- to-noise per pulse will always be less than that of an ideal predetection integrator for reasonable values of Pd. It should also be noted that the data in Figs. 2.23 and 2.24 are essentially independent of the false-alarm number, at least over the range of 106 to 1010.The two probability-density functions of Eqs. (2.39a) and (2.39b) that describe the Swerling fluctuation models are special cases of the chi-square distribution of degree 2m. 3 8 The probability density function is
m
(mu)m-
t (mu)
p(o-) = ·
1 - exp - - ,(m - 1).
!Tav <Tav O'av (i>
0(2.40)
It is also called the gamma distribution. In statistics texts, 2m is the number of degrees of freedom, and is an integer. However, when applied to target cross-section models, 2m is not required to be an integer. Instead, m can be any positive, real number. When m
=
l, the chi-square distribution of Eq. (2.40) reduces to the exponential, or Rayleigh-power, distribu- tion ofEq. (2.39a) that applies to Swerling cases 1 and 2. Cases 3 and 4, described by Eq. (2.39b ), are equivalent to m=
2 in the chi-square distribution. The ratio of the variance to the average value of the cross section is equal tom-112 for the chi-square distribution. The larger the value of m, the more constrained will be the fluctuations. The limit of m equal to infinity correspondsto the nonfluctuating target.
The chi-square distribution is a mathematical model used to represent the statistics of thc fluctuating radar cross section. These distributions might not always fit the observed data, but they are fair approximations in many cases and are used nevertheless for convenience. The chi-square distribution is described by two parameters: the average cross section a..,, and the number of degrees of freedom 2m. Analysis39 of measurements on actual aircraft Hying straight, level courses shows that the cross-section fluctuations at a particular aspect are well fitted by the chi-square distribution with the parameter m ranging from 0.9 to approximately 2 and with aav varying approximately 15 dB from minimum to maximum. The parameters of the fitted distribution vary with aspect angle, type of aircraft, and frequency. The value of mis near unity for all aspects except at broadside; hence, the distribution is Rayleigh with a varying average value with the most variation at broadside aspect. It was also found that the averagc value has more effect on the calculation of the probability of detection than the value of m.
Although the Rayleigh model might provide a good approximation to the radar cross sections of aircraft in many cases,
it
is not always applicable. Exceptions occur at broadside, as mentioned, and for smaller aircraft.38 There are also examples where no chi-square distribu- tion can be made to fit the experimental data.The chi-square distribution has been used to approximate the statistics of other-than- aircrafl targets. Weinstock38•40 showed that this distribution can describe certain simple shapes, such as cylinders or cylinders with fins that are characteristic of some satellite objects.
The parameter m varies between 0.3 and 2, depending on aspect. These have sometimes been called Weinstock cases.
The chi-square distribution with
m =
1 (Swerling cases 1 and 2) is the Rayleigh, or exponential, distribution that results from a large number of independent scatterers, no one of which contributes more than a small fraction of the total backscatter energy. Although the chi-square distribution with other thanm
= 1 has been observed empirically to give a reason- able fit to the radar cross section distribution of many targets, there is no physical scallaing mechanism on which it is based. It has been said that the chi-square distribution with m=
2 (Swerling cases 3 and 4) is indicative of scattering from one large dominant scatterer togel her with a colJection of small independent scatterers. However, it is the Rice distribution that follows from such a model.67 The Rice probability density function isl+s [
ul (
p(a) = exp
-s -
- { l+
s) /0 2O'av O'av
!!:_
s(l + s)),
Uav
(2.4 I}
wheres is the ratio of the radar cross section of the single dominant scatterer to the total cross section of the small scatterers, and I 0 ( ) is the modified Bessel function of zero order. 38 This is a more correct description of the single dominant scatterer model than the chi-square with m = 2. However it has been shown that the chi-square with m
=
2 approximates the Rice when the dominant-scatterer power is equal to the total cross section of the other, small scatterers, and so long as the probability of detection is not large.41The log-normal distribution has also been considered for representing target echo nuctua- tions. It can be expressed as
p(a)
=
(J> 0
(2.42)where s4 standard deviation of In (a/am), and <Tm= median or a. The ratio orthe mean to the median value of a is exp (sJ/2). There is no theoretical model of target scattering that leads to the Jog-normal distribution, although it has been suggested that echoes from some satellite bodies, ships, cylinders, plates, and arrays can be approximated by a log-normal probability
<l istribution. 42·4 ]
Figure 2.25 is a comparison of the several distribution models for a false alarm number of l06 when all pulses during a scan are perfectly correlated but with pulses in successive scans independent (scan-to-scan fluctuation).
The fluctuation models considered in this section assume either complete correlation 'between pulses in any particular scan but with scan-to-scan independence (slow fluctuations), -.. _., or else complete independence from pulse to pulse (fast fluctuation). These represent two extreme cases.
In
some instances, there might be partial correlation of the pulses within a scan (moderate nuctuation ). Schwartz44 considered the effect of partial correlation for the case of two pulses per scan (11 2). The results for partial correlation fall between the two extremes of completely uncorrclatc<l and completely corrclate<l,as
might be expected. Thus to estimate0 99 098 0 95 ..
090 -
r;:
0 80g
O 70%
u 060"O
- 050
0
:;;::. 0 40 ·
15
o
30 ·0 .0 !:.' 0 20 a.
010 0 0 0 01
-10
- i -
0 5 15 20 25 30
Signal to noise roho per pulse dB
Figure 2.25 Comparison of detection probabilities ror Rice, log normal, chi-square with m
=
2 (Swerling case J) and nonfluctuating target models with"=
10 hits and false-alarm number n1=
106• Ratio of dominant-to-background equals unity (s = 1) for Rice distribution. Ratio or mean-to-median cross section for log-normal distribution = p.performance for partially correlated pulses, interpolation between the results for the correlated and uncorrelated conditions can be used as an approximation. A more general treatment of partially correlated fluctuations has been given by Swerling.45 His analysis applies to a large family of probability-density functions of the signal fluctuations and for very general correla- tion properties. Methods for the design of optimal receivers for the detection of moderately fluctuating signals have been considered.46
It is difficult to be precise about the statistical model to be applied to any particular target. Few, if any, real targets fit a mathematical model with any precision and in some cases it is not possible to approximate actual data with any mathematical model. Even if the statistical distribution of
a
target were known,it
might be difficult to relate this to an actual radar measurement sincea
target generally travels on some well-defined trajectory rather than present a statistically independent cross section. to the radar. Thus the various mathematical models cannot, in general, be expected to yield precise predictions of system performance.It has been suggested38•39 that if only one parameter