# npwgnthresh

Detection SNR threshold for signal in white Gaussian noise

## Syntax

## Description

calculates the Detection SNR Threshold in decibels
for detecting a deterministic signal in white Gaussian noise. The detection uses the
Neyman-Pearson (NP) decision rule to achieve a specified probability of false alarm,
`snrthresh`

= npwgnthresh(`pfa`

)`pfa`

. This function uses a square-law detector.

## Examples

### Compute Detection Threshold from Pfa

Calculate the detection threshold that achieves a probability of false alarm (pfa) of 0.01. Assume a single pulse with a `real`

detection type. Then, verify that this threshold produces a pfa of approximately 0.01. Do this by constructing 10000 real white gaussian noise (wgn) samples and computing the fraction of samples exceeding the threshold.

Compute the threshold from pfa. The detection threshold is expressed as a signal-to-noise ratio in db.

```
pfa = 0.01;
numpulses = 1;
snrthreshold = npwgnthresh(pfa,numpulses,"real")
```

snrthreshold = 7.3335

Compute fraction of simulated noise samples exceeding the threshold. The noise has unit power with 10000 samples.

noisepower = 1; Ntrial = 10000; noise = sqrt(noisepower)*randn(1,Ntrial);

Express the threshold in amplitude units.

threshold = sqrt(noisepower*db2pow(snrthreshold)); calculated_Pfa = sum(noise>threshold)/Ntrial

calculated_Pfa = 0.0107

### Detection Threshold Versus Number of Pulses

Plot the SNR detection threshold against the number of pulses, for real and complex noise. In each case, the SNR detection threshold is set for a probability of false alarm (pfa) of 0.001.

Compute detection threshold for 1 to 10 pulses of real and complex noise.

Npulses = 10; snrcoh = zeros(1,Npulses); snrreal = zeros(1,Npulses); Pfa = 1e-3; for num = 1:Npulses snrreal(num) = npwgnthresh(Pfa,num,"real"); snrcoh(num) = npwgnthresh(Pfa,num,"coherent"); end

Plot the detection thresholds against number of pulses.

plot(snrreal,"o-") hold on plot(snrcoh,".-") hold off legend("Real data with integration",... "Complex data with coherent integration",... Location="southeast") xlabel("Number of Pulses") ylabel("SNR Required for Detection") title("SNR Threshold for P_F_A = "+Pfa)

### Linear Detection Threshold Versus Number of Pulses

Plot the linear detection threshold against the number of pulses, for real and complex data. In each case, the threshold is set for a probability of false alarm of 0.001.

Compute detection threshold for 1 to 10 pulses of real and complex noise.

Npulses = 10; snrcoh = zeros(1,Npulses); snrreal = zeros(1,Npulses); Pfa = 1e-3; for num = 1:Npulses snrreal(num) = npwgnthresh(Pfa,num,"real","linear"); snrcoh(num) = npwgnthresh(Pfa,num,"coherent","linear"); end

Plot the detection thresholds against number of pulses.

plot(snrreal,"o-") hold on plot(snrcoh,".-") hold off legend("Real data with integration",... "Complex data with coherent integration",... Location="southeast") xlabel("Number of Pulses") ylabel("Detection Threshold") title("Linear Detection Threshold for P_F_A = "+Pfa)

## Input Arguments

`pfa`

— Probability of false alarm

scalar in the range `(0, 1)`

Probability of false alarm, specified as a scalar in the range
`(0, 1)`

.

**Data Types: **`double`

`numpulses`

— Number of pulses

`1`

(default) | positive integer

Number of pulses used in the integration, specified as a positive integer.

**Data Types: **`double`

`dettype`

— Type of pulse integration

`"noncoherent"`

(default) | `"coherent"`

| `"real"`

Specifies the type of pulse integration used in the NP decision rule,
specified as `"coherent"`

,
`"noncoherent"`

, or `"real"`

.
`"coherent"`

uses magnitude and phase information of
complex-valued samples. `"noncoherent"`

uses squared
magnitudes. `"real"`

uses real-valued samples.

**Data Types: **`char`

| `string`

`outscale`

— Scale of the output value

`"db"`

(default) | `"linear"`

Scale of the output value, specified as `"db"`

or
`"linear"`

. When `outscale`

is set
to `"linear"`

, the returned threshold represents
amplitude.

**Data Types: **`char`

| `string`

## Output Arguments

`snrthresh`

— Detection threshold

scalar

Detection threshold, returned as a scalar. The detection threshold is
expressed in terms of the signal-to-noise ratio in decibels or in linear
units if `outscale`

is set to
`"linear"`

. The relationship between the linear threshold
and the threshold in dB is

$${T}_{dB}=20{\mathrm{log}}_{10}{T}_{lin}$$

## More About

### Detection SNR Threshold

The output of `npwgnthresh`

determines the
detection threshold required to achieve a particular Pfa.

The threshold increases when pulse integration is used in the receiver. This
threshold is not the single sample SNR that is used as an input to `rocsnr`

or as the output of `rocpfa`

, `albersheim`

, and `shnidman`

. For any fixed Pfa, you can decrease the single sample SNR
required to achieve a particular Pd when pulse integration is used in the receiver.
See Signal Detection in White Gaussian Noise and Source Localization Using Generalized Cross Correlation for examples of how to
use `npwgnthresh`

in a detection system.

### Detection in Real-Valued White Gaussian Noise

This function is designed for the detection of a nonzero mean in a sequence of
Gaussian random variables. The function assumes that the random variables are
independent and identically distributed, with zero mean. The linear detection
threshold *λ* for an NP detector is

$$\frac{\lambda}{\sigma}=\sqrt{2N}\text{\hspace{0.17em}}{\text{erfc}}^{-1}(2{P}_{fa})$$

This threshold can also be expressed as a signal-to-noise ratio in decibels

$$10{\mathrm{log}}_{10}\left(\frac{{\lambda}^{2}}{{\sigma}^{2}}\right)=10{\mathrm{log}}_{10}\left(2N{\left({\text{erfc}}^{-1}(2{P}_{fa})\right)}^{2}\right)$$

In these equations

σ

^{2}is the variance of the white Gaussian noise sequence*N*is the number of samples*erfc*is the inverse of the complementary error function^{—1}*P*is the probability of false alarm_{fa}

**Note**

For probabilities of false alarm greater than or equal to 1/2, the formula for
detection threshold as SNR is invalid because erfc^{-1} is less than or equal to zero for values of its argument
greater than or equal to one. In that case, use the linear output of the
function invoked by setting `outscale`

to`'linear'`

.

### Detection in Complex-Valued White Gaussian Noise (Coherent Samples)

The NP detector for complex-valued signals is similar to that discussed in Source Localization Using Generalized Cross Correlation. In addition, the function makes these assumptions:

The variance of the complex-valued Gaussian random variable is divided equally among the real and imaginary parts.

The real and imaginary parts are uncorrelated.

Under these assumptions, the linear detection threshold for an NP detector is

$$\frac{\lambda}{\sigma}=\sqrt{N}\text{\hspace{0.17em}}{\text{erfc}}^{-1}(2{P}_{fa})$$

and expressed as a signal-to-noise ratio in decibels is:

$$10{\mathrm{log}}_{10}\left(\frac{{\lambda}^{2}}{{\sigma}^{2}}\right)=10{\mathrm{log}}_{10}\left(N{\left({\text{erfc}}^{-1}(2{P}_{fa})\right)}^{2}\right)$$

**Note**

For probabilities of false alarm greater than or equal to 1/2, the formula for
detection threshold as SNR is invalid because erfc^{-1} is less than or equal to zero for when its argument is greater
than or equal to one. In that case, select linear output for the function by
setting `outscale`

to `'linear'`

.

### Detection of Noncoherent Samples in White Gaussian Noise

For noncoherent samples in white Gaussian noise, detection of a nonzero mean leads to a square-law detector. For a detailed derivation, see [2], pp. 324–329.

The linear detection threshold for the noncoherent NP detector is:

$$\frac{\lambda}{\sigma}=\sqrt{{P}^{-1}(N,1-{P}_{fa})}$$

The threshold expressed as a signal-to-noise ratio in decibels is:

$$10{\mathrm{log}}_{10}\left(\frac{{\lambda}^{2}}{{\sigma}^{2}}\right)=10{\mathrm{log}}_{10}{P}^{-1}(N,1-{P}_{fa})$$

where $${P}^{-1}(x,y)$$is the inverse of the lower incomplete gamma function,
*P _{fa}* is the probability of false
alarm, and

*N*is the number of pulses.

## References

[1] Kay, S. M. *Fundamentals of Statistical Signal Processing: Detection
Theory*. Upper Saddle River, NJ: Prentice Hall, 1998.

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

## 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**

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