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Quantum Radar Signal Processing

David Luong

Published Jan 1, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.22215/etd/2023-15615

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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.
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.
We built and evaluated a prototype quantum radar, which we call a quantum two-mode squeezing (QTMS) radar, in the laboratory. It operates solely at microwave frequencies; there is no downconversion from optical frequencies. Because the signal generation process relies on quantum mechanical principles, the system is considered to contain a quantum-enhanced radar transmitter. This transmitter generates a pair of entangled microwave signals and transmits one of them through free space, where the signal is measured using a simple and rudimentary receiver. At the heart of the transmitter is a device called a Josephson parametric amplifier, which generates a pair of entangled signals called two-mode squeezed vacuums at 6.1445 and 7.5376 GHz. These are then sent through a chain of amplifiers. The 7.5376 GHz beam passes through 0.5 m of free space; the 6.1445 GHz signal is measured directly after amplification. The two measurement results are correlated in order to distinguish signal from noise. We compare our QTMS radar to a classical radar setup using conventional components, which we call a two-mode noise (TMN) radar, and find that there is significant gain when both systems broadcast signals at -82 dBm. This is shown via a comparison of receiver operating characteristic curves. In particular, we find that the quantum radar requires eight times fewer integrated samples compared to the TMN radar to achieve the same performance.
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.
Recently, the authors have built and evaluated a prototype quantum radar in the laboratory which operates at microwave frequencies. This radar, which they call a quantum two‐mode squeezing radar (QTMS radar), generates a pair of entangled microwave signals and transmits one of them through free space, using the other signal as a reference to perform matched filtering. The specific type of entanglement is called a two‐mode squeezed vacuum, a type of continuous‐variable entanglement between two frequencies. Motivated by the success of these experiments, they try to better understand the entangled QTMS radar signals in this study. They do so by comparing it to a simpler, more conventional radar system, which they call a two‐mode noise radar (TMN radar). They also show how both types of radars are related to standard noise radars as described in the literature. They find that the signals for QTMS radar signals and TMN radar signals have the same mathematical form and that they are related to noise radar by a simple mathematical transformation. This shows that QTMS radar signals can be emulated by a fictional, idealised TMN radar and that it is possible to apply results from the noise radar literature to QTMS radar.
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.
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.