phased.LRTDetector
Description
The likelihood ratio test (LRT) detector performs binary signal detection in the presence of noise. The binary detector chooses between the null hypothesis H0 and the alternative hypothesis H1 based on data measurements. The null hypothesis denotes the absence of any signal while the alternative hypothesis denotes the presence of some signal.
Creation
Description
creates a
likelihood ratio test detector
= phased.LRTDetectordetector
System object™ with default properties.
creates a likelihood ratio test detector
= phased.LRTDetector(Name
= Value
)detector
System object with the specified property Name
set to the specified
Value
. You can specify additional name-value pair arguments in any
order as (Name1
= Value1
, …
,NameN
= ValueN
).
Properties
Unless otherwise indicated, properties are nontunable, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
release
function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
DataComplexity
— Data complexity
'Complex'
(default) | 'Real'
Data complexity, specified as 'Complex'
or 'Real'
.
Complexity defines the format used to report output data. When
DataComplexity
is 'Complex'
, data is
complex-valued. When DataComplexity
is 'Real'
,
data is real-valued.
Example: 'Real'
Data Types: char
| string
ProbabilityFalseAlarm
— Probability of false alarm
1e-1
(default) | nonnegative scalar
Probability of false alarm, specified as a positive scalar between 0 and 1, inclusive.
Example: 1e-6
Data Types: single
| double
ThresholdOutputPort
— Output detection statistics and threshold
false
(default) | true
Output detection statistics and detection threshold, specified as false
or true
. Set this property to true
to output the detection statistics in the stat
argument and the detection threshold in the th
argument. Set this property to false
to suppress the output of detection statistics and threshold.
Data Types: logical
OutputFormat
— Format of output data
'Detection result'
(default) | 'Detection index'
Format of output data, specified as 'Detection result'
or 'Detection index'
. Output data is returned in the Y
argument.
Example: 'Detection index'
Data Types: char
| string
Usage
Description
Input Arguments
X
— Input data
real-valued N-by-1 vector | complex-valued N-by-1 vector | real-valued N-by-M matrix | complex-valued N-by-M matrix
Input data, specified as a real-valued or complex-valued
N-by-1 vector, or an
N-by-M real-valued or complex-valued matrix.
N is the signal length and M is the number of
data channels. Each data channel has N samples to yield an
N-by-M matrix. Each row represents the
components of a length-M data vector. When M =
1, X
represents a single channel of data. When
M > 1, X
can represent N
samples from M data channels. The detector processes input data
along the columns of X
. The size of each row M
cannot change during the simulation.
The LRT detector assumes the same signal model in each column of
X
. Xknown
contains the
N-dimensional noise-free signal. ncov
defines the power of additive Gaussian noise added to each column. For this signal
model, the LRT detector determines whether or not to reject the null hypothesis
Xknown
= 0
. Because there is only one known
signal model, the LRT detector outputs one detection result for each column of
X
.
Data Types: single
| double
Complex Number Support: Yes
Xknown
— Known noise-free signal
real-valued N-by-1 vector | complex-valued N-by-1 vector
Known noise-free signal, specified as a real-valued or complex-valued N-by-1 vector.
Data Types: single
| double
Complex Number Support: Yes
ncov
— Noise power or covariance
positive scalar
Noise power or covariance, specified as a positive scalar.
Example: 2.0
Data Types: single
| double
Output Arguments
Y
— Detection results
1-by-M logical-valued vector | 1-by-L integer-valued vector
Detection results, returned as a 1-by-M logical-valued vector
or a 1-by-L integer-valued vector. The format of
Y
depends on the value of the OutputFormat
property. By default, the OutputFormat
property is set to
'Detection result'
.
When the
OutputFormat
property is set to'Detection result'
,Y
is a 1-by-M vector containing logical detection results, where M is the number of columns ofX
.Y
istrue
if there is a detection in the corresponding column ofX
. Otherwise,Y
isfalse
.When the
OutputFormat
property is set to'Detection index'
,Y
is a 1-by-L integer-valued vector containing detection indices, where L is the number of detections found over all M channels.
Data Types: single
| double
| logical
Complex Number Support: Yes
stat
— Detection statistics
1-by-M vector (default) | 1-by-L vector
Detection statistics, returned as a 1-by-M
vector or 1-by-L vector. M is the number of
columns of X
. The format of stat
depends on
the setting of the OutputFormat
property.
When
OutputFormat
is'Detection result'
,stat
has the same size asY
.When
OutputFormat
is'Detection index'
,stat
is a 1-by-L vector containing detection statistics for each corresponding detection in Y.
Dependencies
To enable this argument, set the ThresholdOutputPort
property to true
.
Data Types: single
| double
th
— Detection threshold
scalar
Detection threshold, returned as a scalar.
Dependencies
To enable this argument, set the ThresholdOutputPort
property to true
.
Data Types: single
| double
Object Functions
To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named obj
, use
this syntax:
release(obj)
Examples
Likelihood Ratio Test on Complex Data
Perform likelihood ratio test detection on 10 signals in complex Gaussian noise. The input is a known signal vector of ones. The noise covariance is unity. Set the desired probability of false alarm to 0.1. Perform the detection on all samples of the input. Show that the probability of false alarm is close to the desired level.
rng default lrt = phased.LRTDetector('DataComplexity','Complex', ... 'ProbabilityFalseAlarm',0.1); N = 10; M = 1000; x = 1/sqrt(2)*(randn(N,M) + 1i*randn(N,M)); xknown = 100*ones(N,1); noisecov = 1; dresult = lrt(x,xknown,noisecov); Pfa = sum(dresult)/M
Pfa = 0.1060
Likelihood Ratio Test on Real Data
Perform likelihood ratio test detection on 10 signals in real Gaussian noise. The input known signal is a vector of ones. Set the noise covariance to 1 and set the desired probability of false alarm to 0.1. Perform the detection on all samples of the input. Show that the probability of false alarm is close to the desired level.
rng default lrt = phased.LRTDetector(DataComplexity = 'Real', ... ProbabilityFalseAlarm = 0.1,ThresholdOutputPort = true); N = 10; M = 1000; x = randn(N,M); xknown = 100*ones(N,1); noisecov = 1; [dresult,stat,th] = lrt(x,xknown,noisecov);
Display the proobability of false alarm, the first 10 values of stat, and the threshold th.
Pfa = sum(dresult)/M
Pfa = 0.1060
stat(1:10)
ans = 1×10
1.9742 2.2291 1.0325 -1.3958 0.6511 -0.3462 0.4053 -0.8368 0.9581 -0.7792
th
th = 1.2816
False Alarm Rate for Real-Valued Signal in White Gaussian Noise
This example shows how to empirically compute the probability of false alarm for real-valued data in white Gaussian noise.
Determine the required signal-to-noise ratio (SNR) in decibels for the NP detector (also known as the LRT detector) when the maximum tolerable false-alarm probability is .
% Probability of false alarm pfa = 1e-3; % SNR threshold in dB snrThdB = npwgnthresh(pfa,1,'real'); % SNR threshold snrThNP = sqrt(db2pow(snrThdB));
Assume the variance is 2 and generate 1 million samples of real Gaussian noise under the null hypothesis.
rng default % Noise variance variance = 2; % Number of independent samples M = 1e6; % Data under null hypothesis x = sqrt(variance) * randn(1,M);
Verify empirically that the detection threshold results in the desired false-alarm rate under the null hypothesis. This can be verified by determining the proportion of normalized samples that exceed the required SNR threshold.
% Empirical false-alarm rate
falseAlarmRateNP = sum(x/sqrt(variance) > snrThNP)/M
falseAlarmRateNP = 9.9500e-04
Alternatively, use phased.LRTDetector
to empirically calculate the detection result and false alarm rate. The LRT detector requires inputting the noise-free signal under the alternative hypothesis. As we test the null hypothesis, the amplitude of the noise-free signal under the alternative hypothesis is irrelevant to the testing and can be set to any real value. Set the noise-free signal under the alternative hypothesis to be 10.
% Configure the LRT detector lrt = phased.LRTDetector('DataComplexity','Real',... 'ProbabilityFalseAlarm',pfa,'ThresholdOutputPort',true); % Known noise-free signal under the alternative hypothesis xnoisefree = 10; % LRT detection [dresult,stat,snrThLRT] = lrt(x,xnoisefree,variance); % Empirical false-alarm rate falseAlarmRateLRT = sum(dresult)/M
falseAlarmRateLRT = 9.9500e-04
The false alarm rates estimated using the above two methods are consistent.
% Difference between SNR thresholds calculated by npwgnthresh and LRT detector
thmse = abs(snrThNP-snrThLRT)^2
thmse = 1.9722e-31
The SNR thresholds generated from npwgnthresh and phased.LRTDetector are also consistent.
Plot the first 10,000 samples of detection statistics. The red horizontal line shows the detection threshold of the LRT detector. The yellow circle shows the detected false alarm.
% Plot detection statistics x1 = stat(1:1e4); plot(x1) % Plot detection threshold line([1 length(x1)],[snrThLRT snrThLRT],'Color','red') hold on % Plot detection results dindex = find(dresult(1:1e4)); plot(dindex,stat(dindex),'o'); xlabel('Sample') ylabel('Value')
You can see that few sample values exceed the threshold. This result is expected because of the low false-alarm probability.
False Alarm Rate for Complex-Valued Signals in Complex White Gaussian Noise
This example shows how to empirically verify the probability of false alarm in a system that uses coherent detection of complex-valued signals. Coherent detection means that the system utilizes information about the phase of the complex-valued signals.
Determine the required SNR for the NP detector (the LRT detector) in a coherent detection scheme with one sample. Use a maximum tolerable false-alarm probability of .
% Probability of false alarm pfa = 1e-3; % SNR threshold in dB snrThdB = npwgnthresh(pfa,1,'coherent'); % SNR threshold snrThNP = sqrt(db2pow(snrThdB));
Test that this threshold empirically results in the correct false-alarm rate. The sufficient statistic in the complex-valued case is the real part of the received sample.
rng default % Noise variance variance = 5; % Number of independent samples M = 1e6; % Data under null hypothesis x = sqrt(variance/2)*(randn(1,M)+1j*randn(1,M)); % Empirical false-alarm rate falseAlarmRateNP = sum(real(x)/sqrt(variance)>snrThNP)/M
falseAlarmRateNP = 9.9500e-04
Alternatively, use phased.LRTDetector
to empirically calculate the detection result and false alarm rate. The LRT detector requires inputting the noise-free signal under the alternative hypothesis. As we test the null hypothesis, the complex amplitude of the noise-free signal under the alternative hypothesis is irrelevant to the testing and can be set to any real value. Set the noise-free signal under the alternative hypothesis to be amplitude 10 with random phase.
% Configure the LRT detector lrt = phased.LRTDetector('DataComplexity','Complex',... 'ProbabilityFalseAlarm',pfa,'ThresholdOutputPort',true); % Known noise-free signal under the alternative hypothesis xnoisefree = 10*exp(1j*2*pi*rand); % LRT detection [dresult,~,snrThLRT] = lrt(x,xnoisefree,variance); % Empirical false-alarm rate falseAlarmRateLRT = sum(dresult)/M
falseAlarmRateLRT = 9.6300e-04
The false alarm rates estimated using the above two methods are consistent.
% Difference between SNR thresholds calculated by npwgnthresh and LRT detector
thmse = abs(snrThNP-snrThLRT)^2
thmse = 0
The SNR thresholds generated from npwgnthresh
and phased.LRTDetector
are also consistent.
More About
Data Precision
This System object supports single and double precision for input data, properties, and arguments. If any input data is single precision, the non-logical outputs are also single precision. If any input data is double precision, the non-logical outputs are double precision. The precision of the non-logical output is independent of the precision of the properties and other arguments.
Signal Model
The LRT detector assumes a linear deterministic signal model data with known white noise
covariance. In each case, the LRT detector assumes the same known signal model for each
column of X
. The LRT detector assumes that the
N-dimensional data follows the known signal model
X = Xknown + noise
where Xknown
is the N-by-1 signal vector
specified by the Xknown
argument. noise
is an
N-by-1 white Gaussian noise vector specified only by the noise
covariance scalar ncov
.
The input data X
contains M independent data
samples along each row. The data is processed along each column of X
.
For radar applications, each column of the input data X
can be:
an N-dimensional fast-time signal for range processing.
an N-dimensional array signal for spatial processing,
an N-dimensional slow-time signal for Doppler processing.
a stacked N-dimensional space-time signal for range-angle/Doppler-angle processing.
The LRT detector processes data column-by-column. Each column is a separate data channel
containing identical signals Xknown
in random noise. The LRT detector
determines whether to reject the null hypothesis given by Xknown = 0.
Because there is only one known signal model, the LRT detector outputs one detection result
for each channel of X
.
Detection Statistics and Threshold
The detector employs the Neyman-Pearson detection criterion. If the H0 hypothesis best accounts for the data, the detector declares that a target is not present. If the H1 hypothesis best accounts for the data, the detector declares that a target is present. In either case, the decision may be erroneous leading to false alarms and false dismissals. The LRT detector requires full knowledge of the likelihood functions p(x,H0) and p(x,H1). The LRT detector creates a detection statistics based on the ratio of the likelihood functions under H1 and H0. The detector compares the detection statistics with the detection threshold – the detector decides H1 if the likelihood ratio p(x,H1)/p(x,H0) > γ. If the detection statistics is larger than the threshold, then it decides H1 to be true; otherwise, it decides H0 to be true. The LRT detector is the optimal detector in noise when the likelihood function of each hypothesis is known. LRT detector assumes the signal and noise model is known. The detector that maximizes the probability of detection for a given probability of false alarm is the LRT detector or Neyman-Pearson (NP) detector.
For a real data model with known white Gaussian noise σ², the detection statistic is
while for complex data the test statistic is
For a given probability of false alarm Pfa, the detection threshold η is determined by solving
for complex data and
for real data.
References
[1] Steven M. Kay, Fundamentals of Statistical Signal Processing, Detection Theory, Prentice-Hall PTR, 1993.
[2] Mark A. Richards, Fundamentals of Radar Signal Processing, Third edition, McGraw-Hill Education, 2022.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced in R2023b
See Also
npwgnthresh
| rocsnr
| rocpfa
| phased.GLRTDetector
| phased.CFARDetector
| phased.CFARDetector2D
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