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# albersheim

Required SNR using Albersheim’s equation

## Syntax

SNR = albersheim(prob_Detection,prob_FalseAlarm)
SNR = albersheim(prob_Detection,prob_FalseAlarm,N)

## Description

SNR = albersheim(prob_Detection,prob_FalseAlarm) returns the signal-to-noise ratio in decibels. This value indicates the ratio required to achieve the given probabilities of detection prob_Detection and false alarm prob_FalseAlarm for a single sample.

SNR = albersheim(prob_Detection,prob_FalseAlarm,N) determines the required SNR for the noncoherent integration of N samples.

## Examples

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Compute the required SNR of a single pulse to achieve a detection probability of 0.9 as a function of the false alarm probability.

Set the probability of detection to 0.9 and the probabilities of false alarm from .0001 to .01.

Pd=0.9;
Pfa=0.0001:0.0001:.01;

Loop the Albersheim equation over all Pfa's.

snr = zeros(1,length(Pfa));
for j=1:length(Pfa)
snr(j) = albersheim(Pd,Pfa(j));
end

Plot SNR versus Pfa.

semilogx(Pfa,snr,'k','linewidth',1)
grid
axis tight
xlabel('Probability of False Alarm')
ylabel('Required SNR (dB)')
title('Required SNR for P_D = 0.9 (N = 1)')

Compute the required SNR of 10 non-coherently integrated pulse to achieve a detection probability of 0.9 as a function of the false alarm probability.

Set the probability of detection to 0.9 and the probabilities of false alarm from .0001 to .01.

Pd=0.9;
Pfa=0.0001:0.0001:.01;
Npulses = 10;

Loop over the Albersheim equation over all Pfa's.

snr = zeros(1,length(Pfa));
for j=1:length(Pfa)
snr(j) = albersheim(Pd,Pfa(j),Npulses);
end

Plot SNR versus Pfa.

semilogx(Pfa,snr,'k','linewidth',1)
grid
axis tight
xlabel('Probability of False Alarm')
ylabel('Required SNR (dB)')
title('Required SNR for P_D = 0.9 (N = 10)')

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### Albersheim's Equation

Albersheim's equation uses a closed-form approximation to calculate the SNR. This SNR value is required to achieve the specified detection and false-alarm probabilities for a nonfluctuating target in independent and identically distributed Gaussian noise. The approximation is valid for a linear detector and is extensible to the noncoherent integration of N samples.

Let

$A=\mathrm{ln}\frac{0.62}{{P}_{FA}}$

and

$B=\mathrm{ln}\frac{{P}_{D}}{1-{P}_{D}}$

where ${P}_{FA}$ and ${P}_{D}$ are the false-alarm and detection probabilities.

Albersheim's equation for the required SNR in decibels is:

$\text{SNR}=-5{\mathrm{log}}_{10}N+\left[6.2+4.54/\sqrt{N+0.44}\right]{\mathrm{log}}_{10}\left(A+0.12AB+1.7B\right)$

where N is the number of noncoherently integrated samples.

## References

[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005, p. 329.

[2] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill, 2001, p. 49.