albersheim

Required SNR using Albersheim’s equation

Syntax

SNR = albersheim(Pd,Pfa)

SNR = albersheim(Pd,Pfa,N)

Description

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

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

Examples

Compute Required SNR for Probability of Detection

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

Set the probability of detection to 0.9 and the probabilities of false alarm from 0.0001 to 0.01.

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

Loop the Albersheim equation over all false-alarm probabilities.

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

Plot the SNR as a function of false-alarm probability.

semilogx(Pfa,snr)

grid
axis tight
xlabel("Probability of False Alarm")
ylabel("Required SNR (dB)")
title("Required SNR for P_D = "+Pd+" (N = 1)")

Compute Required SNR for Probability of Detection of 10 Pulses

Compute the required SNR of 10 noncoherently 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 0.0001 to 0.01.

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

Loop over the Albersheim equation over all the false-alarm probabilities.

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

Plot the SNR as a function of the false-alarm probability.

semilogx(Pfa,snr)

grid
axis tight
xlabel("Probability of False Alarm")
ylabel("Required SNR (dB)")
title("Required SNR for P_D = "+Pd+" (N = 10)")

Input Arguments

Pd — Probability of detection

positive scalar

Probability of detection, specified as a positive scalar.

Data Types: single | double

Pfa — Probability of false alarm

positive scalar

Probability of false alarm, specified as a positive scalar.

Data Types: single | double

N — Number of pulses for noncoherent integration

1 (default) | positive scalar

Number of pulses for noncoherent integration, specified as a positive scalar.

Data Types: single | double

More About

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

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

and

$$B=\ln \frac{P_D}{1-P_D}$$

where P_FA and P_D are the false-alarm and detection probabilities, respectively

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

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

where N is the number of noncoherently integrated samples.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

References

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

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

Version History

Introduced in R2011a