npwgnthresh
The threshold of the signal-to-noise ratio (SNR) for detecting a signal in white Gaussian noise.
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Syntax
Function call
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snrthresh = npwgnthresh(pfa)— Calculates the threshold of the signal-to-noise ratio (SNR) in dB to detect a deterministic signal in white Gaussian noise. During detection, the Neiman—Pearson crucial rule is used to achieve a given probability of a false alarm.pfa. This function uses a quadratic detector.
Arguments
Entrance
# pfa — the probability of a false alarm
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scalar in the range (0, 1)
Details
The probability of a false alarm, given as a scalar in the range (0, 1).
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# numpulses — number of pulses
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1 (default) | a positive integer
Details
The number of pulses used in the integration, set as a positive integer.
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# dettype — type of pulse integration
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"noncoherent" (default) | "coherent" | "real"
Details
The type of impulse integration used in the Neiman—Pearson decision rule.
Set as:
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"coherent"uses information about the magnitude and phase of complex samples. -
"noncoherent"uses quadratic values. -
"real"uses real counts.
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# outscale — the scale of the output value
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"db" (default) | "linear"
Details
The scale of the output value, set as "db" or "linear".
If for an argument outscale the value is set "linear" the returned threshold is the amplitude.
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Output
# snrthresh — detection threshold
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scalar
Details
The detection threshold returned as a scalar.
The detection threshold is expressed in the signal-to-noise ratio (SNR) in dB or linear units, if for the argument outscale the value is set linear.
The relationship between the linear threshold and the threshold in dB expressed as:
Examples
Linear detection threshold depending on the number of pulses
Details
Let’s plot the dependence of the linear detection threshold on the number of pulses for real and complex data. In each case, the threshold is set with the probability of a false alarm. pfa=0.001.
Calculate the detection threshold for 1–10 pulses of real and complex noise.
import EngeePhased.Functions: npwgnthresh
Npulses = 10
Pfa = 1e-3
snrreal = npwgnthresh.(Pfa, 1:Npulses, "real", "linear")
snrcoh = npwgnthresh.(Pfa, 1:Npulses, "coherent", "linear")
plot([snrreal snrcoh], label = ["Real data with integration" "Complex data with coherent integration"], xlabel = "Number of Pulses", ylabel = "Detection Threshold", title = "Linear Detection Threshold for P_FA = $Pfa", marker = :circle, legendposition = :bottomright)
Additional Info
Signal-to-noise ratio (SNR) threshold at signal detection
Details
Function output npwgnthresh defines the detection threshold required to achieve a certain probability of false alarm pfa.
The threshold increases if pulse integration is used in the receiver. This threshold is not the SNR of a single signal that is used as a function input. rocsnr, rocpfa or albersheim.
For any fixed value of the probability of a false alarm (pfa) you can reduce the signal-to-noise ratio of a single sample required to achieve a certain detection probability if pulse integration is used in the receiver.
Detection of real values of a signal in white Gaussian noise
Details
This function is designed to detect a non-zero average value in a sequence of Gaussian random variables. The function assumes that the random variables are independent and identically distributed, with a zero mean value.
Linear detection threshold for the Neiman — Pearson detector is equal to:
This threshold can also be expressed as a signal-to-noise ratio in dB:
In these equations
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— the variance of the sequence of white Gaussian noise;
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— number of signals;
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— the reverse function of the additional error;
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— the possibility of a false alarm.
If there is a possibility of a false alarm (pfa) is greater than or equal to 1/2 the formula for the detection threshold, as a signal-to-noise ratio, is invalid because is less than or equal to zero for its argument values greater than or equal to one. In this case, use the linear output of the function called by the argument. outscale with the value "linear".
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Detection of complex signal values in white Gaussian noise (coherent samples)
Details
Functions npwgnthresh The following assumptions are made:
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The variance of a complex Gaussian random variable is divided equally between the real and imaginary parts.
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The real and imaginary parts are uncorrelated.
According to these assumptions, the threshold of linear detection for the Neiman—Pearson detector is:
and expressed as the signal-to-noise ratio in dB:
If there is a possibility of a false alarm (pfa) is greater than or equal to 1/2 the formula for the detection threshold, as a signal-to-noise ratio, is invalid because is less than or equal to zero for its argument values greater than or equal to one. In this case, use the linear output of the function called by the argument. outscale with the value "linear".
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Detection of incoherent signal values in white Gaussian noise
Details
For incoherent signals in white Gaussian noise, the detection of a non-zero mean leads to a quadratic law detector. For a detailed conclusion, see [2], pp. 324-329.
The linear detection threshold for an incoherent Neiman—Pearson detector is:
The threshold value, expressed as a signal-to-noise ratio in dB, is:
where — inverse lower incomplete gamma function, — the probability of a false alarm, and — the number of pulses.