# Documentation

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# shnidman

Required SNR using Shnidman’s equation

## Syntax

SNR = shnidman(Prob_Detect,Prob_FA)
SNR = shnidman(Prob_Detect,Prob_FA,N)
SNR = shnidman(Prob_Detect,Prob_FA,N, Swerling_Num)

## Description

SNR = shnidman(Prob_Detect,Prob_FA) returns the required signal-to-noise ratio in decibels for the specified detection and false-alarm probabilities using Shnidman's equation. The SNR is determined for a single pulse and a Swerling case number of 0, a nonfluctuating target.

SNR = shnidman(Prob_Detect,Prob_FA,N) returns the required SNR for a nonfluctuating target based on the noncoherent integration of N pulses.

SNR = shnidman(Prob_Detect,Prob_FA,N, Swerling_Num) returns the required SNR for the Swerling case number Swerling_Num.

## Examples

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Find and compare the required single-pulse SNR for Swerling cases I and III. The Swerling case I has no dominant scatterer while the Swerling case III has a dominant scatterer.

Specify the false-alarm and detection probabilities.

pfa = 1e-6:1e-5:.001;
Pd = 0.9;

Allocate arrays for plotting.

SNR_Sw1 = zeros(1,length(pfa));
SNR_Sw3 = zeros(1,length(pfa));

Loop over PFA's for both scatterer cases.

for j=1:length(pfa)

SNR_Sw1(j) = shnidman(Pd,pfa(j),1,1);
SNR_Sw3(j) = shnidman(Pd,pfa(j),1,3);
end

Plot the SNR vs PFA.

semilogx(pfa,SNR_Sw1,'k','linewidth',2)
hold on
semilogx(pfa,SNR_Sw3,'b','linewidth',2)
axis([1e-6 1e-3 5 25])
xlabel('False-Alarm Probability')
ylabel('SNR')
title('Required Single-Pulse SNR for Pd = 0.9')
legend('Swerling Case I','Swerling Case III',...
'Location','SouthWest')

The presence of a dominant scatterer reduces the required SNR for the specified detection and false-alarm probabilities.

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### Shnidman's Equation

Shnidman's equation is a series of equations that yield an estimate of the SNR required for a specified false-alarm and detection probability. Like Albersheim's equation, Shnidman's equation is applicable to a single pulse or the noncoherent integration of N pulses. Unlike Albersheim's equation, Shnidman's equation holds for square-law detectors and is applicable to fluctuating targets. An important parameter in Shnidman's equation is the Swerling case number.

### Swerling Case Number

The Swerling case numbers characterize the detection problem for fluctuating pulses in terms of:

• A decorrelation model for the received pulses

• The distribution of scatterers affecting the probability density function (PDF) of the target radar cross section (RCS).

The Swerling case numbers consider all combinations of two decorrelation models (scan-to-scan; pulse-to-pulse) and two RCS PDFs (based on the presence or absence of a dominant scatterer).

Swerling Case NumberDescription
0 (alternatively designated as 5)Nonfluctuating pulses.
IScan-to-scan decorrelation. Rayleigh/exponential PDF–A number of randomly distributed scatterers with no dominant scatterer.
IIPulse-to-pulse decorrelation. Rayleigh/exponential PDF– A number of randomly distributed scatterers with no dominant scatterer.
IIIScan-to-scan decorrelation. Chi-square PDF with 4 degrees of freedom. A number of scatterers with one dominant.
IVPulse-to-pulse decorrelation. Chi-square PDF with 4 degrees of freedom. A number of scatterers with one dominant.

## References

[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005, p. 337.