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Receiver operating characteristic curves by false-alarm probability


[Pd,SNR] = rocpfa(Pfa)
[Pd,SNR] = rocpfa(Pfa,Name,Value)


[Pd,SNR] = rocpfa(Pfa) returns the single-pulse detection probabilities, Pd, and required SNR values, SNR, for the false-alarm probabilities in the row or column vector Pfa. By default, for each false-alarm probability, the detection probabilities are computed for 101 equally spaced SNR values between 0 and 20 dB. The ROC curve is constructed assuming a single pulse in coherent receiver with a nonfluctuating target.

[Pd,SNR] = rocpfa(Pfa,Name,Value) returns detection probabilities and SNR values with additional options specified by one or more Name,Value pair arguments.

rocpfa(...) plots the ROC curves.

Input Arguments


False-alarm probabilities in a row or column vector.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.


Maximum SNR to include in the ROC calculation.

Default: 20


Minimum SNR to include in the ROC calculation.

Default: 0


Number of SNR values to use when calculating the ROC curves. The actual values are equally spaced between MinSNR and MaxSNR.

Default: 101


Number of pulses to integrate when calculating the ROC curves. A value of 1 indicates no pulse integration.

Default: 1


This property specifies the type of received signal or, equivalently, the probability density functions (PDF) used to compute the ROC. Valid values are: 'Real', 'NonfluctuatingCoherent', 'NonfluctuatingNoncoherent', 'Swerling1', 'Swerling2', 'Swerling3', and 'Swerling4'. Values are not case sensitive.

The 'NonfluctuatingCoherent' signal type assumes that the noise in the received signal is a complex-valued, Gaussian random variable. This variable has independent zero-mean real and imaginary parts each with variance σ2/2 under the null hypothesis. In the case of a single pulse in a coherent receiver with complex white Gaussian noise, the probability of detection, PD, for a given false-alarm probability, PFA is:


where erfc and erfc-1 are the complementary error function and that function’s inverse, and χ is the SNR not expressed in decibels.

For details about the other supported signal types, see [1].

Default: 'NonfluctuatingCoherent'

Output Arguments


Detection probabilities corresponding to the false-alarm probabilities. For each false-alarm probability in Pfa, Pd contains one column of detection probabilities.


Signal-to-noise ratios in a column vector. By default, the SNR values are 101 equally spaced values between 0 and 20. To change the range of SNR values, use the optional MinSNR or MaxSNR input argument. To change the number of SNR values, use the optional NumPoints input argument.


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Plot ROC curves for false-alarm probabilities of 1e-8, 1e-6, and 1e-3, assuming no pulse integration.

Pfa = [1e-8 1e-6 1e-3];

Figure contains an axes object. The axes object with title Nonfluctuating Coherent Receiver Operating Characteristic (ROC) Curves contains 6 objects of type line, text.


[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005, pp 298–336.

Extended Capabilities

Version History

Introduced in R2011a