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IEEE Transactions on Radar Systems

Noise-Type Radars With Gaussian Statistics: SNR and the Correlation Coefficient

David Luong, Ian W. K. Lam, Sreeraman Rajan

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Noise radars have the same mathematical description as a type of quantum radar known asquantum two-mode squeezing radar. Although their physical implementations are very different, this mathematical similarity allows us to analyze them collectively, for which reason we call themnoise-type radars. The target detection performance of noise-type radars depend crucially on a parameter called the “correlation coefficient”. In this paper, we show that the correlation coefficient reduces to the signal-to-noise ratio (SNR) in certain limiting cases when the SNR is appropriately defined. This allows us to translate between the correlation coefficient, which is the more fundamental parameter in the theory of noise-type radars, and the SNR, which is more familiar to engineers. To illustrate the ideas in this paper, we present experimental results from a laboratory-based noise radar.

DOI
10.1109/trs.2025.3575786

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•Jan 1, 2023•3 citations
Noise radars have the same mathematical description as a type of quantum radar known asquantum two-mode squeezing radar. Although their physical implementations are very different, this mathematical similarity allows us to analyze them collectively, for which reason we call themnoise-type radars. The target detection performance of noise-type radars depend crucially on a parameter called the “correlation coefficient”. In this paper, we show that the correlation coefficient reduces to the signal-to-noise ratio (SNR) in certain limiting cases when the SNR is appropriately defined. This allows us to translate between the correlation coefficient, which is the more fundamental parameter in the theory of noise-type radars, and the SNR, which is more familiar to engineers. To illustrate the ideas in this paper, we present experimental results from a laboratory-based noise radar.
Noise radars have the same mathematical description as a type of quantum radar known as quantum two-mode squeezing radar. Although their physical implementations are very different, this mathematical similarity allows us to analyze them collectively. We may consider the two types of radars as forming a single class of radars, called noise-type radars. The target detection performance of noise-type radars depends on two parameters: the number of integrated samples and a correlation coefficient. In this paper, we show that when the number of integrated samples is large and the correlation coefficient is low, the detection performance becomes a function of a single parameter: the number of integrated samples multiplied by the square of the correlation coefficient. We then explore the detection performance of noise-type radars in terms of this emergent parameter; in particular, we determine the probability of detection as a function of this parameter.
We derive a detector that optimizes the target detection performance of any single-input single-output noise radar satisfying the following properties: it transmits Gaussian noise, it retains an internal reference signal for matched filtering, all external noise is additive white Gaussian noise, and all signals are measured using heterodyne receivers. This class of radars, which we call noise-type radars, includes not only many types of standard noise radars, but also a type of quantum radar known as quantum two-mode squeezing radar. The detector, which we derive using the Neyman-Pearson lemma, is not practical because it requires foreknowledge of a target-dependent correlation coefficient that cannot be known beforehand. (It is, however, a natural standard of comparison for other detectors.) This motivates us to study the family of Neyman-Pearson-based detectors that result when the correlation coefficient is treated as a parameter. We derive the probability distribution of the Neyman-Pearson-based detectors when there is a mismatch between the pre-chosen parameter value and the true correlation coefficient. We then use this result to generate receiver operating characteristic curves. Finally, we apply our results to the case where the correlation coefficient is small. It turns out that the resulting detector is not only a good one, but that it has appeared previously in the quantum radar literature.
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.
Quantum radar is an emerging technique, currently at the experimental stage, that promises to disrupt the well-established field of conventional radar. Recent progress shows that it is possible to generate and use entangled pairs of microwave photons for detection purposes and to obtain an advantage over a classical radar. In this work, we study the currently experimentally feasible type of quantum correlation radar. To this end, we compare different radar architectures, quantum and classical, by analyzing their detection performances by means of the receiver operating characteristic (ROC), the minimum error probability as well as the Chernoff bound. The underlying system models are based on quantum mechanical formulations as well as conventional signal theory. Where it is appropriate and necessary to facilitate our analysis, we apply the central limit theorem to establish the Gaussianity of the observable quantities. A conceptual analogy between the quantum and classic points-of-view is drawn and supported by results showing the asymptotic behavior of the ROC curves of both physical descriptions depending on the power levels of signal and noise. We come to exact and comprehensible conclusions on the current state of quantum radar in comparison to classic radar with a detailed mapping of the radar operating regime (measurement time, signal-to-noise ratio, environmental noise level, target object size and distance) in which a quantum advantage is attainable.
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.
Quantum radars use phenomena from quantum physics, such as quantum entanglement, to enhance detection performance compared with classical radars.This thesis presents an end-to-end analysis of an entanglement-based radar called quantum two-mode squeezing radar (QTMS radar).This type of radar is of particular interest because a QTMS radar experiment has actually been performed, and because QTMS radars are closely related to a type of classical radar known as noise radar (NR).As part of the analysis, we develop a mathematical theory of QTMS radars and show that QTMS radars and NRs can be united under a single probabilistic model.We then show how their signals are to be processed to determine the presence or absence of a target, and undertake an extensive analysis of the target detection performance of QTMS radars and NRs.These theoretical results are verified using data drawn from a QTMS radar experiment and a laboratory NR.One important conclusion, supported by both theory and experiment, is that the receiver operating characteristic curve for a QTMS radar is better than that of an NR with the same signal powers.Throughout the thesis, we emphasize the need to establish common ground between quantum physics and radar engineering, and it is hoped that the unified theory of QTMS radar and NR presented here will play a role in this. List of TablesCommon sense imagines that when it sees a table it sees a table.This is a gross delusion.Bertrand Russell The ABC of Relativity 2.1 A taxonomy of quantum sensors 11 5.1 System parameters for Wilson's QTMS radar experiment 104 6.1 QTMS radar variables and parameters 127 6.2 Rice distribution parameters fitted to simulated values of ρ ˆ145 6.3 Parameter estimates for Wilson et al.'s QTMS radar prototype 151 6.4 TVD between theoretical and experimental PDFs for the QTMS radar 152 6.5 Transmit powers and parameter values for the experimental NR 153 6.6 TVD between theoretical and experimental PDFs for the NR 156 7.1 Parameter values for the ROC curve plots 196 8.1 Correlation coefficient vs. transmit power: fit parameters 210 8.2 Correlation coefficient vs. range: fit parameters 218 8.3 Noise radar system parameters used in Example 8.10 232 8.4 QTMS radar system parameters used in Example 8.11 233 B.1 Relationships between probability distributions 281 xvi List of Figures As the Chinese say, 1001 words is worth more than a picture.John McCarthy 3.1 Time evolution of coherent states 64 3.2 Vacuum noise in a beam splitter 68 4.1 Target detection using a detector and a threshold 77 4.2 Probabilities of detection and false alarm for the envelope detector 78 4.3 ROC curves for the envelope detector 83 4.4 ROC curves for the NP detector (sinusoidal radar) compared with those for the envelope detector 92 5.1 Illustration of the basic QI protocol 95 5.2 Block diagram of Wilson's QTMS radar setup 102 5.3 Horn antennas used in the Wilson experiment 102 5.4 Interior and exterior of the dilution refrigerator 105 5.5 Interior of the can containing the JPA 106 5.6 Simplified diagram of a JPA 108 5.7 JPA mounted on a printed circuit board 109 5.8 Block diagram of the benchmark NR for the Wilson experiment 110 6.1 Correlation number line 135 6.2 Separable, entangled, and forbidden values of ρ as a function of σ 1 135 6.3 PDF of σ ˆ1 141 xvii LIST OF FIGURES xviii 6.4 Exact and approximate PDFs of ρ ˆ143 6.5 TVD between the exact and approximate PDFs of ρ ˆ146 6.6 Concentration parameter vs. Nρ 2 148 6.7 Exact and approximate PDFs of φ ˆ149 6.8 TVD between the exact and approximate PDFs of φ ˆ149 6.9 Theoretical and experimental PDFs for the QTMS radar 151 6.10 Theoretical and experimental PDFs for the experimental NR with transmit power -18.91 dBm 154 6.11 Theoretical and experimental PDFs for the experimental NR with transmit power -9.380 dBm 154 6.12 Theoretical and experimental PDFs for the experimental NR with transmit power 1.955 dBm 155 7.1 PDF of the NP detector (target present) 169 7.2 PDF of the NP detector (target absent) 170 7.3 LIST OF FIGURES xix 8.3 Correlation coefficient vs. range 8.4 ROC curves for the MF detector for various ranges 8.5 Correlation coefficient vs. range for an experimental NR 8.6 ROC curves for all detectors when ρ → 0 and N → ∞ 8.7 ROC curve for the MF detector when ρ → 0 and N → ∞: increasing ρ vs. increasing N 8.8 Probability of detection vs. Nρ 2 when ρ → 0 and N → ∞ 8.9 ROC curves for D MF and ρ ˆwhen ρ is large and N is small 8.10 ROC curves for the Wilson experiment, showing the increase in N required for the benchmark NR to match the QTMS radar 8.11 Maximum range vs. ρ 0 and N when p d = 0.9 and p fa = 10 -6 D.1 Simplified block diagram of the NR setup D.2 Photograph of the NR setup D.3 NR aimed at a wall D.4 Corner reflector used for the experiment in Section 8.2to x 0 in a given situation, without asserting that x = x 0 elsewhere.An index of symbols and abbreviations used in this thesis, together with the page at which each symbol or abbreviation is first defined, is included at page 262.The reader is encouraged to make liberal use of this index.