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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

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© 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 = 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 : = 0 Target absent in QTMS radars by exploiting quadrature entanglement (that H1 : 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 ρˆ from the radar’s voltage measurements, set a threshold, and declare a detection if ρˆ 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 √ √ ! ρ 2N −2 ln p p (p |ρ, N) = 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 √ be modeled as stationary, zero-mean Gaussian random pro-  p  p (p |ρ, 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 performance√ Given that these signals are Gaussian with known means, as quantified by the ROC curve depends only on ρ √2N. 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 = [I1 Q1 I2 Q2] . A full derivation of the remains constant. This implies the following tradeoff between covariance matrix is given in [13], so we omit the derivation ρ and N. Suppose we increase ρ by a factor of α. The radar here. We showed that the theoretical form of this matrix can would then require only N/α2 integrated samples in order to be written in block matrix form as achieve the same performance, since  σ 2 ρσ σ φ  6 112 1 2R( ) r √ QTMS = 2 (1) N ρσ1σ2R(φ) σ 12 (αρ) 2 = ρ 2N. (5) 2 α2 where σ 2 and σ 2 are the measured signal powers, φ is the 1 2 Because the number of integrated samples is proportional to phase shift between the signals, 1 is the 2×2 identity matrix, 2 the integration time, we may state the following: if the corre- R φ and ( ) is the reflection matrix lation coefficient ρ is improved by a factor of α, the integra- cos φ sin φ  tion time required to obtain the same detection performance R(φ) = . (2) sin φ − cos φ is reduced by a factor of α2.

VOLUME 8, 2020 185545 D. Luong et al.: Quantum Two-Mode Squeezing Radar and Noise Radar: Correlation Coefficient and Integration Time

FIGURE 2. ROC curves calculated using (3). From right to left, the dashed FIGURE 4. Comparison of experimentally obtained ROC curves for a = curves correspond to N 25 000, 50 000, 75 000, . . . , 225 000, respectively, QTMS radar and a noise radar. Solid lines: N = 50 000. Dashed line: ρ while is fixed at 0.01. The solid curves (from right to left) correspond to N = 50 000 × 9 = 450 000. Dotted lines are theoretical approximations. ρ = 0.01, 0.02, and 0.03, with N = 25 000.

Figure2 shows ROC curves for N = 25 000α for of 9. The figure shows that if the noise radar’s integration α ∈ {1,..., 9} when ρ = 0.01. The curves for α = 1, 4, and time were increased by a factor of 9, it would match the 9 coincide exactly with the ROC curves for ρ = 0.01, 0.02, performance of the QTMS radar as expected. We conclude and 0.03, N = 25 000. This illustrates the ρ-vs.-N tradeoff that the main result of this paper is experimentally supported. described above. The deviations in the experimental ROC curves from the ρ Figure3 shows what happens as the  1 approximation theoretically expected ones, particularly when pFA is low, breaks down. We plot ROC curves for ρ = 0.2α and N = can be attributed to the small quantity of experimental data 400/α2, where α ∈ {1,..., 4}. There is little difference available for this analysis. between the curves for α = 1 and α = 2, but as α continues The original claim in [12] was that the reduction in N to increase, the ROC curves become steeper. would be a factor of 8; we note that this was based on a different detector function.

V. CONCLUSION In this paper, we explored how the detection performance of a QTMS radar changes when the correlation ρ between the received and reference signals of the radar is changed. We found a simple rule which applies when ρ is small: increasing ρ by a factor of α is equivalent to increasing the integration time by α2. This gives us a simple way to understand the potential benefit of a quantum radar: it reduces the dwell time required to detect a target. Since the integration time is related to the amount of power received by the radar, it suggests that there is a similar rela- tionship between ρ and the transmit power. FIGURE 3. ROC curves for a QTMS radar with ρ = 0.2α and N = 400/α2, Now that the effect of changing ρ is understood, all that α = 1, 2, 3, 4. The ρ-vs.-N tradeoff described in the text holds only when ρ is small. remains in order to predict the performance of a QTMS radar is to understand how ρ varies with factors such as relative We may now apply our result to the QTMS radar exper- motion between the radar and target (Doppler), range, and iment described in [12]. Fig.4 shows experimental ROC radar cross section. We will also explore how the results in curves drawn from that experiment which confirm the this paper can be extended to traditional radars. These will be N-vs.-ρ tradeoff described above. It was found in [15] that the subjects of future work. the correlation coefficient for the QTMS radar prototype was approximately ρQTMS = 0.0127. A comparable noise radar REFERENCES (technically a slight variant of noise radar) was found to [1] S. Lloyd, ‘‘Enhanced sensitivity of photodetection via quantum illumina- have a correlation of ρN = 0.00419. The QTMS radar thus tion,’’ Science, vol. 321, no. 5895, pp. 1463–1465, Sep. 2008. improves ρ by a factor of approximately 3. According to the [2] S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, ‘‘Quantum illumination main result of the present paper, this means that the QTMS with Gaussian states,’’ Phys. Rev. Lett., vol. 101, no. 25, Dec. 2008, radar should reduce the needed integration time by a factor Art. no. 253601.

185546 VOLUME 8, 2020 D. Luong et al.: Quantum Two-Mode Squeezing Radar and Noise Radar: Correlation Coefficient and Integration Time

[3] C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, BHASHYAM BALAJI (Senior Member, IEEE) ‘‘Quantum-enhanced noise radar,’’ Appl. Phys. Lett., vol. 114, no. 11, received the B.Sc. degree (Hons.) in physics from Mar. 2019, Art. no. 112601. the St. Stephen’s College, University of Delhi, [4] D. Luong, A. Damini, B. Balaji, C. W. Sandbo Chang, A. M. Vadiraj, and New Delhi, India, in 1990, and the Ph.D. degree C. Wilson, ‘‘A quantum-enhanced radar prototype,’’ in Proc. IEEE Radar in theoretical particle physics from Boston Univer- Conf. (RadarConf), Apr. 2019, pp. 1–6. sity, Boston, MA, USA, in 1997. [5] M. Dawood and R. M. Narayanan, ‘‘Receiver operating characteristics for Since 1998, he has been a Defence Scientist the coherent UWB random noise radar,’’ IEEE Trans. Aerosp. Electron. with the Defence Research and Development Syst., vol. 37, no. 2, pp. 586–594, Apr. 2001. Canada, Ottawa, ON, Canada. His research inter- [6] T. Thayaparan and C. Wernik, ‘‘Noise radar technology basics,’’ Defence Res. Develop. Canada, Ottawa, ON, Canada, Tech. Rep. DRDC Ottawa ests include all aspects of radar sensor out- TM 2006-266, Dec. 2006. puts, including space-time adaptive processing, multitarget tracking, and [7] R. Narayanan, ‘‘Noise radar techniques and progress,’’ in Advanced Ultra- meta-level tracking, as well as multisource data fusion. His theoretical wideband Radar: Signals, Targets, and Applications, J. D. Taylor, Ed. Boca research interests also include the application of Feynman path integral Raton, FL, USA: CRC Press, 2016, pp. 323–361. and quantum field theory methods to the problems of nonlinear filtering [8] K. Savci, A. G. Stove, A. Y. Erdogan, G. Galati, K. A. Lukin, G. Pavan, and stochastic control. Most recently, his research interests have included and C. Wasserzier, ‘‘Trials of a noise-modulated radar demonstrator–first quantum sensing, in particular, quantum radar and quantum imaging. results in a marine environment,’’ in Proc. 20th Int. Radar Symp. (IRS), Dr. Balaji was a recipient of the IEEE Canada Outstanding Engineer Jun. 2019, pp. 1–9. Award, in 2018. He is a Fellow of the Institution of Engineering and [9] K. Kulpa, Signal Processing in Noise Waveform Radar. Norwood, MA, Technology. USA: Artech House, 2013. [10] C. Wasserzier, J. G. Worms, and D. W. O’Hagan, ‘‘How noise radar technology brings together active sensing and modern electronic warfare techniques in a combined sensor concept,’’ in Proc. Sensor Signal Process. for Defence Conf. (SSPD), May 2019, pp. 1–5. SREERAMAN RAJAN (Senior Member, IEEE) [11] D. Luong, S. Rajan, and B. Balaji, ‘‘Entanglement-based quantum radar: From myth to reality,’’ IEEE Aerosp. Electron. Syst. Mag., vol. 35, no. 4, received the B.E. degree in electronics and pp. 22–35, Apr. 2020. communications from Bharathiyar University, [12] D. Luong, C. W. S. Chang, A. M. Vadiraj, A. Damini, C. M. Wilson, Coimbatore, India, in 1987, the M.Sc. degree and B. Balaji, ‘‘Receiver operating characteristics for a prototype quantum in electrical engineering from Tulane University, two-mode squeezing radar,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 56, New Orleans, LA, in 1992, and the Ph.D. degree no. 3, pp. 2041–2060, Jun. 2020. in electrical engineering from the University of [13] D. Luong and B. Balaji, ‘‘Quantum two-mode squeezing radar and noise New Brunswick, Fredericton, NB, Canada, radar: Covariance matrices for signal processing,’’ IET Radar, Sonar Nav- in 2004. igat., vol. 14, no. 1, pp. 97–104, Jan. 2020. From 1986 to 1990, he was a Scientific Offi- [14] D. Luong, S. Rajan, and B. Balaji, ‘‘Estimating correlation coefficients for cer with the Reactor Control Division, Bhabha Atomic Research Center quantum radar and noise radar: A simulation study,’’ in Proc. IEEE Global (BARC), Bombay, India, after undergoing an intense training in nuclear Conf. Signal Inf. Process. (GlobalSIP), Nov. 2019, pp. 1–5. science and engineering from its training school. At BARC, he developed [15] D. Luong, S. Rajan, and B. Balaji, ‘‘Quantum two-mode squeezing radar systems for control, safety, and regulation of nuclear research and power and noise radar: Correlation coefficients for target detection,’’ IEEE Sen- reactors. From 1997 to 1998, he carried out research under a grant from sors J., vol. 20, no. 10, pp. 5221–5228, May 2020. Siemens Corporate Research, Princeton, NJ. From 1999 to 2000, he was with JDS Uniphase, Ottawa, ON, Canada, where he worked on optical compo- nents and the development of signal processing algorithms for advanced fiber optic modules. From 2000 to 2003, he was with Ceyba Corporation, Ottawa, where he developed channel monitoring, dynamic equalization, and optical power control solutions for advanced ultra-long haul and long haul fiber optic communication systems. In 2004, he was with Biopeak Corporation, where he developed signal processing algorithms for non-invasive medical devices. From December 2004 to June 2015, he was a Defense Scientist with the Defence Research and Development Canada, Ottawa. He joined Carleton University as a Tier 2 Canada Research Chair (Sensors Systems) in its Department of Systems and Computer Engineering, in July 2015. He is the Director of the Ottawa Carleton Institute for Biomedical Engineering, since July 2020, and was its Associate Director, from July 2015 to June 2020. He was an Adjunct Professor with the School of Electrical Engineering DAVID LUONG (Graduate Student Member, and Computer Science, University of Ottawa, Ottawa, from July 2010 to IEEE) received the B.Sc. degree in mathematical June 2018, and an Adjunct Professor with the Department of Electrical physics from the University of Waterloo, Waterloo, and Computer Engineering, Royal Military College, Kingston, Ontario, ON, Canada, in 2013, and the M.Sc. degree in since July 2015. He is the holder of two patents and two disclosures of physics (quantum information) from the Institute invention. He is an author of 170 journal articles and conference papers. His for Quantum Computing, University of Waterloo, research interests include signal processing, biomedical signal processing, in 2015, where he explored the practical aspects communication, and pattern classification. of quantum repeaters. He is currently pursuing the Dr. Rajan is currently the Chair of the IEEE Ottawa EMBS and AESS Ph.D. degree in electrical and computer engineer- Chapters and has served IEEE Canada as its board member, from 2010 to ing with Carleton University, Ottawa, ON. 2018. He was awarded the IEEE MGA Achievement Award in 2012 and From 2017 to 2020, he was a Defense Scientist with the Radar Sensing and recognized for his IEEE contributions with Queen Elizabeth II Diamond Exploitation Section, Defence Research and Development Canada, Ottawa, Jubilee Medal in 2012. IEEE Canada recognized his outstanding service where he helped develop a prototype microwave quantum radar. His research through 2016 W.S. Read Outstanding Service Award. He has been involved interests include quantum radar and signal processing. In 2020, he was in organizing several successful IEEE conferences and has been a reviewer awarded a Vanier Canada Graduate Scholarship, the highest scholarship for several IEEE journals and conferences. awarded to Ph.D. students by the Canadian government.

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