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 Required SNR for Probability of Detection

Open Live Script

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)')

Figure contains an axes object. The axes object with title R e q u i r e d blank S N R blank f o r blank P indexOf D baseline blank = blank 0 . 9 blank ( N blank = blank 1 ) contains an object of type line.

Compute Required SNR for Probability of Detection of 10 Pulses

Open Live Script

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)')

Figure contains an axes object. The axes object with title R e q u i r e d blank S N R blank f o r blank P indexOf D baseline blank = blank 0 . 9 blank ( N blank = blank 1 0 ) contains an object of type line.

More About

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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

$A = \ln \frac{0.62}{P_{F A}}$

and

$B = \ln \frac{P_{D}}{1 - P_{D}}$

where $P_{F A}$ and $P_{D}$ are the false-alarm and detection probabilities.

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

$SNR = - 5 \log_{10} N + [6.2 + 4.54 / \sqrt{N + 0.44}] \log_{10} (A + 0.12 A B + 1.7 B)$

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

Version History

Introduced in R2011a