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Sensor Signal Processing for Defence (SSPD)

An Approximate Likelihood Ratio Detector for QTMS Radar and Noise Radar

David Luong, Bhashyam Balaji, Sreeraman Rajan

Published Sep 1, 2021• 3 citations
Publisher

Abstract

The most important task of any radar is to detect targets. From this perspective, they are machines for distinguishing between two hypotheses: target absent and target present. The test statistic—or detector function—used by the radar is clearly of primary importance. In this paper, we explore the properties of a detector function for quantum two-mode squeezing radar and noise radar. This detector is derived from a second-order approximation of the likelihood ratio, and is attractive because it has a simple mathematical form. In order to analyze the performance of this detector function, we derive an expression for the receiver operating characteristic curve and verify it via simulations.

DOI
10.1109/sspd51364.2021.9541444

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The most important task of any radar is to detect targets. From this perspective, they are machines for distinguishing between two hypotheses: target absent and target present. The test statistic—or detector function—used by the radar is clearly of primary importance. In this paper, we explore the properties of a detector function for quantum two-mode squeezing radar and noise radar. This detector is derived from a second-order approximation of the likelihood ratio, and is attractive because it has a simple mathematical form. In order to analyze the performance of this detector function, we derive an expression for the receiver operating characteristic curve and verify it via simulations.
We derive a detector function for quantum two-mode squeezing (QTMS) radar and noise radar that is based on the use of a generalized likelihood ratio (GLR) test for distinguishing between the presence and absence of a target. In addition to an explicit expression for the GLR detector, we derive a detector function which approximates the GLR detector in the limit where the target is small, far away, or otherwise difficult to detect. When the number of integrated samples is large, we derive a theoretical expression for the receiver operating characteristic (ROC) curve of the radar when the GLR detector is used. When the number of samples is small, we use simulations to understand the ROC curve behavior of the detector. One interesting finding is that there exists a parameter regime in which a previously-studied detector outperforms the GLR detector, contrary to the intuition that LR-based tests are optimal or nearly so. This is because neither the Neyman–Pearson lemma, nor the Karlin–Rubin theorem which generalizes the lemma to composite hypotheses, hold in this particular problem. However, the GLR detector remains a good choice for target detection in certain regimes.
A detector function for a newly proposed quantum radar known as quantum two-mode squeezing (QTMS) radar is analyzed in this paper. The detector function proposed in this paper behaves like a normal distribution function when the number of radar samples integrated for the detection is over 500 samples. A closed form expression for the receiver operating characteristic (ROC) curve for QTMS radar is derived under the assumption that the radar integrates more than 500 samples to detect the presence or absence of a target. The detector function can also be applied to noise radar.
We analyze the potential application of a generalized likelihood ratio (GLR)-based detector function to quantum two-mode squeezing (QTMS) radars and standard noise radars. We give an expression for the likelihood ratio (LR) in terms of the maximum-likelihood estimate of the correlation coefficient between the received and reference signals of the radar. Interestingly, we found that a previously-studied detector function outperforms the GLR detector, though not in all parameter regimes. This runs counter to the intuition, based on the Neyman-Pearson lemma, that the LR test is optimal. We discuss why the lemma does not hold in this particular case and why the search for detector functions for QTMS radars and noise radars remains open. However, the GLR detector is a good choice when the correlation coefficient is high, the number of integrated samples is low, and appropriate computational resources are available.
Quantum two-mode squeezing (QTMS) radars and noise radars detect targets by correlating the received signal with an internally stored recording. A covariance matrix can be calculated between the two which, in theory, is a function of a single correlation coefficient. This coefficient can be used to decide whether a target is present or absent. We can estimate the correlation coefficient by minimizing the Frobenius norm between the sample covariance matrix and the theoretically expected form of the matrix. Using simulated data, we show that the estimates follow a Rice distribution whose parameters are simple functions of the underlying, “true” correlation coefficient as well as the number of integrated samples. We obtain an explicit expression for the receiver operating characteristic curve that results when the correlation coefficient is used for target detection. This is an important first step toward performance prediction for QTMS radars.
For target detection, quantum two-mode squeezing (QTMS) radars and noise radars require estimation of a covariance matrix. A scalar function of the covariance matrix is then used for deciding the presence or absence of a target. In this paper, estimation of the covariance matrix is carried out by minimizing the Frobenius norm between the sample covariance matrix and the theoretically expected form of the matrix. Two scalars, a normalized correlation coefficient and an unnormalized cross-correlation, are used for detecting the target. Their performances are compared and found to be the same. Probability distributions are then fit to the histograms of these estimated correlations. Using these fitted distributions, expressions are obtained for receiver operating characteristic (ROC) curves that predict the performance of these functions in the task of deciding whether a target is present or not. This work is a first step toward understanding and modeling target detection performance of a QTMS radar or noise radar when using correlation-related measures as detector functions.
We analyze the signal-to-noise ratio (SNR) metric in the context of quantum two-mode squeezing (QTMS) radar and find that there are actually two SNRs associated with a QTMS radar, one for the received signal and another for a signal retained inside the radar. Definitions for these SNRs are proposed which are simpler than those hitherto used in the quantum radar literature. We plot receiver operating characteristic (ROC) curves for varying values of these two SNRs. These plots show that the quality of the matched filtering performed by the radar, as quantified by the SNR of the retained signal, can have a strong impact on detection performance.