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Quantum Two-Mode Squeezing Radar and Noise Radar Correlation Coefficient and Integration Time CAN UNCLASSIFIED Quantum Two-Mode Squeezing Radar and Noise Radar Correlation Coefficient and Integration Time David Luong Carleton University Bhashyam Balaji DRDC – Ottawa Research Centre Sreeraman Rajan Carleton University Electronics Letters IEEE Access VOLUME 8, 2020 pp. 185544-185547 DOI: 10.1109/ACCESS.2020.3029473 Date of Publication from External Publisher: October 2020 Terms of Release: This document is approved for public release. Defence Research and Development Canada External Literature (P) DRDC-RDDC-2021-P172 June 2021 CAN UNCLASSIFIED CAN UNCLASSIFIED IMPORTANT INFORMATIVE STATEMENTS This document was reviewed for Controlled Goods by Defence Research and Development Canada using the Schedule to the Defence Production Act. Disclaimer: This document is not published by the Editorial Office of Defence Research and Development Canada, an agency of the Department of National Defence of Canada but is to be catalogued in the Canadian Defence Information System (CANDIS), the national repository for Defence S&T documents. Her Majesty the Queen in Right of Canada (Department of National Defence) makes no representations or warranties, expressed or implied, of any kind whatsoever, and assumes no liability for the accuracy, reliability, completeness, currency or usefulness of any information, product, process or material included in this document. Nothing in this document should be interpreted as an endorsement for the specific use of any tool, technique or process examined in it. Any reliance on, or use of, any information, product, process or material included in this document is at the sole risk of the person so using it or relying on it. Canada does not assume any liability in respect of any damages or losses arising out of or in connection with the use of, or reliance on, any information, product, process or material included in this document. Template in use: EO Publishing App for CR-EL Eng 2021-02-11.dotm © Carleton University, 2020 CAN UNCLASSIFIED Received September 21, 2020, accepted September 30, 2020, date of publication October 7, 2020, date of current version October 21, 2020. Digital Object Identifier 10.1109/ACCESS.2020.3029473 Quantum Two-Mode Squeezing Radar and Noise Radar: Correlation Coefficient and Integration Time DAVID LUONG 1, (Graduate Student Member, IEEE), BHASHYAM BALAJI2, (Senior Member, IEEE), AND SREERAMAN RAJAN 1, (Senior Member, IEEE) 1Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada 2Radar Sensing and Exploitation Section, Defence Research and Development Canada, Ottawa, ON K2K 2Y7, Canada Corresponding author: David Luong ([email protected]) ABSTRACT Quantum two-mode squeezing (QTMS) radars and noise radars perform matched filtering of the received signal using a reference signal stored within the radar. Their target detection performance depends on the correlation between the received and reference signals, and the improved detection perfor- mance of QTMS radars over noise radars is due to the fact that it has an increased correlation. In this paper, we present a novel way of understanding how detection performance depends on the correlation by relating the correlation to the integration time. Concretely, we prove that increasing the correlation by a given factor is equivalent to increasing the integration time by the square of that factor. We apply this result to a recent QTMS radar experiment where the increase in correlation was approximately a factor of 3. This was found to be equivalent to increasing the integration time by a factor of 9, thus confirming the validity of our result and approximation. INDEX TERMS Quantum radar, quantum two-mode squeezing radar, noise radar, correlation, integration time. I. INTRODUCTION to reduce the integration time required for target detection Quantum radars have garnered increasing attention because by a factor of 8 or more relative to a comparable noise they can potentially improve sensing performance com- radar [12]. pared to conventional radars [1], [2]. Recently, a practically- In [13], we showed that as far as detection performance realizable quantum radar design at microwave frequencies is concerned, QTMS radars and noise radars effectively was proposed, which was called quantum-enhanced noise lie on a continuum parameterized by a certain correlation radar in [3] and quantum two-mode squeezing radar (QTMS coefficient ρ. This coefficient characterizes the two signals radar) in [4]. (We adopt the latter name in this paper.) These associated with a given QTMS radar or noise radar: the papers also showed that, from the point of view of per- received signal and a reference signal retained within the formance prediction, QTMS radars are closely analogous radar. In effect, QTMS radars have a higher ρ than noise to noise radars of the type described in [5]–[8]. This is radars. of great interest because noise radars are inherently low In this paper, we present a novel way of understanding probability-of-intercept radars and have excellent spectrum the effect of increasing ρ by relating it to the integration sharing capabilities [9]–[11]. Noise radars (and, by exten- time, which is a far more familiar quantity to radar engineers sion, QTMS radars) have noteworthy potential applications than ρ is. This relationship can be quantified as follows: in the biomedical realm, where interference with medical when ρ 1, increasing ρ by a factor of α is equiva- devices should particularly be avoided. Preliminary exper- lent to increasing the integration time by a factor of α2. imental results showed that QTMS radar has the potential To verify this relationship, we compare our theoretical result to results drawn from the experiment described in [12] The associate editor coordinating the review of this manuscript and and show that the the experimental data agrees with the approving it for publication was Wei Wang . theory. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. 185544 For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ VOLUME 8, 2020 D. Luong et al.: Quantum Two-Mode Squeezing Radar and Noise Radar: Correlation Coefficient and Integration Time II. BASIC PRINCIPLES OF QTMS AND NOISE RADARS (Noise radars are characterized by a similar matrix, but with a QTMS radars operate by generating a pair of correlated rotation matrix instead of a reflection matrix.) The variable ρ, Gaussian noise signals, one of which is transmitted toward which is the Pearson correlation coefficient between I1 and I2 a target and the other retained within the radar as a refer- when φ D 0, dictates the strength of the correlation between ence for matched filtering (Figure 1). Noise radars work the the received and recorded signals. This can be seen from the same way [5], the difference being that the signals generated fact that it controls the ``magnitude'' of the off-diagonal block by a QTMS radar are more highly correlated compared to of the matrix (1). noise radars. This is due to the presence of quantum noise It turns out that ρ is intimately related to the detection inherent in every electromagnetic signal, which degrades the performance of a QTMS radar. In terms of statistical detection correlation between the two signals of a noise radar. In effect, theory, the detection problem in a QTMS radar reduces to noise radars perform imperfect matched filtering, wherein the distinguishing between two hypotheses, H0 and H1, where reference signal is not exactly the same as the transmitted ρ signal. The effect of quantum noise is partially suppressed H0 V D 0 Target absent in QTMS radars by exploiting quadrature entanglement (that H1 V ρ > 0 Target present is, quantum entanglement between the voltages of the two ρ signals), resulting in better matched filtering. Therefore, we can treat as a detector function: calculate an estimate ρO from the radar's voltage measurements, set a threshold, and declare a detection if ρO lies above the threshold. We have shown in [15] that the receiver operating charac- teristic (ROC) curve for a QTMS radar, with ρ as the detector function as outlined above, is approximately given by p p ! ρ 2N −2 ln p p (p jρ; N) D Q ; FA : (3) D FA 1 1 − ρ2 1 − ρ2 where pD is the probability of detection, pFA is the probability of false alarm, N is the number of integrated voltage samples, and Q1 is the Marcum Q-function. This approximate formula FIGURE 1. Block diagram illustrating the basic idea of QTMS radar and noise radar. This figure first appeared in [12]. holds for N greater than approximately 100. Detailed explanations of the concepts summarized here can IV. INTEGRATION TIME AND THE CORRELATION be found in [12]–[14]. COEFFICIENT We are now in a position to prove the main result of this paper. ρ ρ2 III. TARGET DETECTION PERFORMANCE AS A FUNCTION When becomes very small, we may neglect . This is a OF THE CORRELATION COEFFICIENT valid approximation when the signal-to-noise ratio is very low We denote the in-phase and quadrature voltages received by (which may occur when signal powers are very low or the target is very far away). Under this approximation, we find a QTMS radar by I1 and Q1; the voltages of the reference signal are I and Q . In a QTMS radar, these variables can that 2 2 p be modeled as stationary, zero-mean Gaussian random pro- p p (p jρ; N) ≈ Q ρ 2N; −2 ln p : (4) cesses [13]. We implicitly assume that the radar is aimed at a D FA 1 FA stationary target with a large radar cross section. Thus, when ρ 1, it is clear that detection performancep Given that these signals are Gaussian with known means, as quantified by the ROC curve depends only on ρ p2N. they are fully characterized by the 4 × 4 covariance matrix Given a fixed pFA, pD will not change so long as ρ 2N T ; ; ; T E[xx ], where x D [I1 Q1 I2 Q2] .
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