In this paper, we consider a previously described low-complexity WCP-OFDM radar receiver and focus on the self-interference component induced by targets throughout this processing. Particularly, we assume and verify by simulation that this self-interference can be modeled in the range-Doppler map as an additive white Gaussian process independent from the internal noise. To that end, we propose an expression for the signal-to-interference-plus-noise ratio and verify by simulation that the expected performance of the well-known CA-CFAR detector are recovered and thus predictable

Bidon, Stéphanie

Mercier, Steven

Roque, Damien

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											 	   		 	  	     	 			 									 an author's https://oatao.univ-toulouse.fr/19820Mercier, Steven and Bidon, Stéphanie and Roque, Damien CA-CFAR Detection Based on an AWG Interference Modelin a Low-Complexity WCP-OFDM Receiver. (2017) In: 51th Annual Asilomar Conference on Signals, Systems, andComputers (ASILOMARSSC), 29 October 2017 - 1 November 2017 (Pacific Grove, United States).CA-CFAR Detection Based on an AWGInterference Model in a Low-ComplexityWCP-OFDM ReceiverTopics: E5 MIMO Radar, A3 OFDM/MulticarrierSteven Mercier, Ste´phanie Bidon and Damien RoqueInstitut Supe´rieur de l’Ae´ronautique et de l’Espace (ISAE-SUPAERO), Universite´ de Toulouse, 31055 Toulouse, FRANCETelephone: +33 5 61 33 83 56 Fax: +33 5 61 33 83 51Email: {steven.mercier,stephanie.bidon,damien.roque}@isae-supaero.frAbstract—In this paper, we consider a previously describedlow-complexity WCP-OFDM radar receiver and focus on theself-interference component induced by targets throughout thisprocessing. Particularly, we assume and verify by simulation thatthis self-interference can be modeled in the range-Doppler map asan additive white Gaussian process independent from the internalnoise. To that end, we propose an expression for the signal-to-interference-plus-noise ratio and verify by simulation that theexpected performance of the well-known CA-CFAR detector arerecovered and thus predictable.I. INTRODUCTIONIn modern applications such as vehicular technology, merg-ing communication and radar systems is tempting to saveresources (e.g., spectrum, energy, weight, volume). To thatend, several waveforms were proposed to jointly perform datatransmission and radar sensing [1]–[3]. Particularly, cyclic pre-fix orthogonal frequency-division multiplexing (CP-OFDM)multicarrier modulations have shown interesting capabilitiesto fulfill this dual function [2], [4]. More specifically in [4],the authors described a low-complexity radar receiver suitedfor such a waveform that consists of: i) linear estimation of thedata symbols; ii) symbols removal; iii) bi-dimensional discreteFourier transform (2D-DFT) to obtain the traditional range-Doppler map.Recently we extended this 3-stage radar processing to amore general class of multicarrier waveforms referred to asweighted cyclic prefix (WCP)-OFDM [5]. It allows non-rectangular short pulses to be used while preserving per-fect symbol reconstruction and low-complexity implementa-tion [6]. Orthogonal time-frequency localized (TFL) pulses [7]have thus revealed their benefit over CP-pulses in presence oflarge Doppler shifts at close range [5].Still, since the low-complexity receiver of [5] does not per-form matched filtering, interference terms appear throughoutthe 3-stage processing. Visually, this manifests as a loss onthe target peak that converts to a white-looking process in therange-Doppler map. Yet, this interference, or self-interferencefrom a target point of view, has been formally neglected so far.This paper focuses on deriving statistical properties of theseself-interference terms for detection purposes.To that end, the paper is organized as follows. Section IIrecalls the existing WCP-OFDM radar model and gives anexpression of the induced interference. In Section III, weprovide an analytical expression of the signal-to-interference-plus-noise-ratio (SINR) in the range-Doppler map of thesystem that allows the performance of the so-called cell-averaging constant-false-alarm-rate (CA-CFAR) detector [8]to be predicted. Finally, Section IV concludes and outlinespotential prospects.II. WCP-OFDM RADAR MODELHerein we express the target self-interference terms appear-ing in the range-Doppler measurements of the WCP-OFDMreceiver presented in [5] (Fig. 1(a)-(b)).A. Linear WCP-OFDM transmitterLet {ck,m}(k,m)∈I be a sequence of complex symbols totransmit with I = {0, . . . ,K − 1} × {0, . . . ,M − 1} whereK and M denote the number of sub-carriers and blocks(i.e., radar sweeps), respectively. The baseband output of thetransmitter, sampled at critical rate 1/Ts, is [6]s[l] =K−1∑k=0M−1∑m=0ck,mg[l −mL]ej2pi kK lK1/2, l ∈ Z (1)where g[l] is the transmit pulse shape; 1/K and L represent el-ementary symbol spacing in frequency and time, respectively.A WCP-OFDM transmitter is characterized by: (i) L/K > 1,(ii) g[l] = 0 for l < 0 or l > L− 1. As a result the transmittercan be implemented by a blockwise processing based on theinverse discrete Fourier transform (IDFT):sm[l] ={∑K−1k=0 dk,mej2pikKlK1/2g[l] if l ∈ {0, . . . , L− 1}0 otherwisesuch thats[l] =M−1∑m=0sm[l −mL], l ∈ Zwhere we introduced the precoded symbols dk,m ,ck,mej2pi kKmL applying a rotation to each data symbol ck,m.WCP-OFDMtrans-mitterRadarchannelWCP-OFDMreceiverSymbolsremoval 2D-DFT| · |2CA-CFARdetector{ck,m}s[l] r[l]n[l]{d˜k,m} {d¯k,m} {xk,m}(a) WCP-OFDM linear system (b) Radar measurements (c) Radar detectorFig. 1. Flowchart of WCP-OFDM radar system.B. Single target: linear time-varying channel modelTo express the self-interference terms, we consider a singlepoint target perfectly located in a range gate l0, with a constantradial velocity v during the coherent processing interval (CPI).We further assume no range ambiguities and a slow movingtarget, namely l0 < L and v < va where va = c/(2fcLTs)represents the ambiguous velocity, with c the speed of lightand fc the carrier frequency. In a narrowband scenario, thereceived signal can be expressed asr[l] = αs[l − l0] exp(j2pifdl/L) + n[l] (2)where α and fd = v/va are the complex amplitude and thenormalized Doppler frequency of the target; n is the receivernoise. Range resolution is given by δr = cTs/2.C. Linear WCP-OFDM receiver and radar processingLet us now follow the 3-stage low-complexity processingof [4]:i) A linear estimation of the symbols is performed at theradar receiver, as in the first stage of a conventional linearmulticarrier receiver, namely, for (k′,m′) ∈ I ,c˜k′,m′ =+∞∑l=−∞r[l]gˇ[l −m′L]e−j2pi k′K lK1/2(3)where gˇ[l] denotes the receive pulse shape. In the WCP-OFDM framework, gˇ[l] also verifies gˇ[l] = 0 for l < 0 orl > L − 1. Thus, by truncating the received signal (2) to itsfirst ML samples, a DFT-blockwise operation can again beimplemented:d˜k′,m′ =+∞∑l=−∞rm′ [l]gˇ[l]e−j2pik′K lK1/2, (k′,m′) ∈ Iwithrm′ [l] ={r[l +m′L] if l ∈ {0, . . . , L− 1}0 otherwise.and c˜k′,m′ = d˜k′,m′e−j2pik′Km′L. Note that since precoding is alinear and invertible operation, the remaining of the processingcan be expressed as a function of the d˜k′,m′ .ii) Symbols removal is then achieved as symbols are knownat the radar receiverd¯k′,m′ = d˜k′,m′/dk′,m′ , (k′,m′) ∈ I.Using (2) and the finite support of g, gˇ in WCP-OFDM, thisgivesd¯k′,m′ = d¯(t)k′,m′ + d¯(i,m′)k′,m′ + d¯(i,m′−1)k′,m′ + n¯k′,m′ (4)where each term of the sum, detailed thereafter, representthe useful signal (4a), the self-interference induced from thecurrent (4b) and previous (4c) multicarrier blocks, and thenoise (4d). Interestingly enough appears in (4) the cross-ambiguity function of gˇ, gAgˇ,g(l, f) = K−1+∞∑p=−∞gˇ[p]g[p− l]ej2pifp (5)sampled at different points as exemplified in Fig. 2. Thechannel’s selectivity is in fact responsible for breaking thebiorthogonality of the WCP-OFDM transmission system, de-noted as Agˇ,g(pL, q/K) = δp,0δq,0 with δ the Kroneckersymbol. As a result, the system is no longer able to ensurea perfect reconstruction of the symbols following (3) and thetarget peak incurs a loss Agˇ,g(l0, fd/L) for the benefit ofboth interference terms, driven by (L,1/K)-spaced versionsof Agˇ,g(l0, fd/L).iii) If we neglect at first the interference contributions as in[4] and [5], the range-Doppler map of the scene is deducedfrom (4a) by computing an IDFT and a DFT along theslow-frequency and slow-time domains, respectively. This isabusively summarized as{xk′,m′}k′,m′ = 2D-DFT{d¯k′,m′}k′,m′ .However, given the actual expression of (4) and since DFT islinear, we get, similarly, for (k′,m′) ∈ I ,xk′,m′ = x(t)k′,m′ + x(i,m′)k′,m′ + x(i,m′−1)k′,m′ + x(n)k′,m′ .III. CA-CFAR DETECTION PERFORMANCEFrom (4) we are able to derive an analytical expressionof the SINR in the target cell of the WCP-OFDM radar’srange-Doppler map. This expression allows us to predict theperformance of a CA-CFAR detector when applied to theoutput of the system.A. SINR expressionWe consider the following conventional framework: thereceiver noise n is white and circular-symmetric Gaussian withzero-mean and power σ2; the data symbols {ck,m}(k,m)∈Iare independent and uniformly distributed according to ad¯(t)k′,m′ = αe−j2pi l0K k′ej2pifdm′Agˇ,g(l0, fd/L) (4a)d¯(i,m′)k′,m′ = αK−1∑k=0k 6=k′dk,m′dk′,m′e−j2pil0K kej2pifdm′Agˇ,g (l0, fd/L+ (k − k′)/K) (4b)d¯(i,m′−1)k′,m′ = αK−1∑k=0dk,m′−1dk′,m′e−j2pi(l0−L)K kej2pifdm′Agˇ,g (l0 − L, fd/L+ (k − k′)/K) (4c)n¯k′,m′ =1dk′,m′+∞∑l=−∞n[l]gˇ[l −m′L]e−j2pi k′K (l−m′L)K1/2(4d)chosen constellation, with zero-mean and power σ2c . Theconstellation is further assumed to verify E{1/ck,m} = 0and E{ck,m/c∗k,m} = 0, which is actually fulfilled by usualmodulations (e.g., quadrature phase-shift keying, or QPSK,and 16-quadrature amplitude modulation, or 16-QAM).In this context we show that the self-interference terms arezero-mean, uncorrelated to each other and to the thermal noisen. As a consequence, in the target cell of the range-Dopplermap, denoted x ∈ {xk′,m′}(k′,m′)∈I , the SINR defined asSINR , E{|x(t)|2}E{|x(n) + x(i,m′) + x(i,m′−1)|2}simply boils down toSINR =E{|x(t)|2}E{|x(n)|2}+ E{|x(i,m′)|2}+ E{|x(i,m′−1)|2}or in a more compact way,SINR =σ2tσ2n + σ2i(6)with σ2t = E{|x(t)|2}, σ2n = E{|x(n)|2} and σ2i =E{|x(i,m′)|2}+E{|x(i,m′−1)|2} the power of the useful term,noise and self-interference, in the target cell, respectively.These are found to beσ2t = E{|α|2}KM |Agˇ,g(l0, fd/L)|2σ2n = σ2σ2c−1K−1‖gˇ‖22σ2i 'M1E{|α|2}σ2cσ2c−1×[K−1∑k=0|Agˇ,g (l0, fd/L+ k/K)|2 − |Agˇ,g (l0, fd/L)|2+K−1∑k=0|Agˇ,g (l0 − L, fd/L+ k/K)|2]where σ2c−1 = E{1/|ck,m|2}. The Monte-Carlo simulationsrun in Fig. 3 for different L/K and varying (l0, fd/L)confirm these analytical results. In fact for (l0, fd/L) = (0, 0)the SINR (6) does reduce to the SNR as expected, whoseexpression is known to be [5]SNR =E{|α|2}K2Mσ2σ2c−1‖gˇ‖22. (7)−L9/8 0 l0 L9/800.20.40.60.81L12/8−L12/8|Agˇg(l0, 0)||Agˇg(l0 − L, 0)|lAmplitudeTFL L/K = 12/8 TFL L/K = 9/8(a) Cross-ambiguity function for f = 0: |Agˇg(l, 0)|. Markers representits L-spaced samples with an offset l0.− 3K− 2K− 1K0 fdL1K2K3K00.20.40.60.81 ∣∣∣Agˇg (0, fdL )∣∣∣∣∣∣Agˇg (0, fdL − 1K )∣∣∣fAmplitudeTFL L/K = 12/8 TFL L/K = 9/8(b) Cross-ambiguity function for l = 0: |Agˇg(0, f)|. Markers representits 1/K-spaced samples with an offset fd/L.Fig. 2. Cross-ambiguity function cuts of gˇ, g maximizing TFL with K =1024 sub-carriers. Target parameters are chosen as l0 = 300 and fd/L =1/(4K).Then as either l0 or fd/L increases, the SINR incurs a losswith respect to the SNR, as hinted by Fig. 2. This gap remainsconstant at low SNR values, and corresponds to the loss|Agˇ,g(l0, fd/L)|2 incurred by σ2t . But as the SNR is enhanced,self-interference contributions become more significant andSINR curves eventually have an horizontal asymptotic behav-ior (plateau effect). Finally, it is worth noticing that the SINRcurves can be improved by increasing L/K, at the cost of adecreased spectral efficiency though.0 10 20 30 40 50 60 70 80 90−100102030405060708090SNR (dB)SINR(dB)theo (l0,fd/L) = (0, 0.00/K)simutheo (l0,fd/L) = (256, 0.25/K)simutheo (l0,fd/L) = (512, 0.25/K)simu(a) TFL pulse shape with L/K = 9/80 10 20 30 40 50 60 70 80 90−100102030405060708090SNR (dB)SINR(dB)theo (l0,fd/L) = (0, 0.00/K)simutheo (l0,fd/L) = (256, 0.25/K)simutheo (l0,fd/L) = (512, 0.25/K)simu(b) TFL pulse shape with L/K = 12/8Fig. 3. SINR as a function of the theoretical SNR. Comparison betweenexpected SINR and Monte-Carlo results. QPSK symbols, K = 1024, M =256, varying L/K, TFL pulses with ‖g‖22 = ‖gˇ‖22 = K, σ2 = 1, σ2c = 1,α ∼ CN (0,E{|α|2}) with E{|α|2} defined as in (7). When varying L/K,l0 and fd chosen to keep target physically located at same range and Doppler.B. CA-CFAR detection performanceIn this section, we characterize the performance of theCA-CFAR detector at the output of the studied WCP-OFDMradar. Considering the framework described in Section III-A,we make the following assumptions within the range-Dopplermap:(i) the self-interference is white, as suggested by Fig. 4;(ii) the self-interference-plus-noise, or total interference, iscircular-Gaussian distributed, zero-mean with variance σ2i +σ2nin each cell;(iii) the cells are independent to each other.Under these assumptions, the total interference is independentand identically distributed in the range-Doppler map, thusfollowing an AWGI model. As a result, in our single target0 50 100 150 200 250050100150200250 −30−20−100(dB)Fig. 4. Covariance matrix of self-interference cells (zoom on the first 256elements). QPSK symbols, K = 64, M = 32, L/K = 12/8, TFL pulseswith ‖g‖22 = ‖gˇ‖22 = K, σ2 = 1, σ2c = 1, l0 = 12, fd = 0.25, α ∼CN (0,E{|α|2}) with E{|α|2} defined as in (7) with SNR = 35 dB.scenario, the environment is said homogeneous and the CA-CFAR detector is supposed to provide optimum detectionperformance at a fixed probability of false alarm (PFA) Pfa [8].The output of the square law rectifier applied to any cellx ∈ {xk′,m′}(k′,m′)∈I is denoted by z = |x|2, as depicted inFig. 1(c). For a white Gaussian interference, it is well knownthat the CA-CFAR detection test becomes [8]zH0≶H1βσˆ2 (8)withβ = Ns[P−1/Nsfa − 1]σˆ2 = N−1sNs−1∑ns=0znswhere β is the so called CA-CFAR constant for a given Pfa andσˆ2 is an estimate of the local noise power from Ns secondaryrange-Doppler cells. Now, restricting the study to Swerling Itargets (i.e., α ∼ CN (0,E{|α|2})), the theoretical probabilityof detection (PD) is furthermore shown to be [8]Pd =[1 +β/Ns1 + SINR]−Ns.In Fig. 5 the PD obtained by Monte-Carlo simulationswith the CA-CFAR test (8) is compared to that obtainedassuming an AWGI with the derived SINR (6), using TFL-pulses and a QPSK modulation. Both match perfectly therebysupporting the appropriateness of our AWGI model at leastfor the chosen numerical values. Besides, trends observed inFig. 3 are recovered: (i) the higher (l0, fd/L), the higher theSNR must be to achieve a desired probability of detection;(ii) PD losses are reduced with high L/K, i.e., at the costof a spectral efficiency loss. Yet, note that the plateau effect−5 0 5 10 15 20 25 30 35 40 45 50 55 6000.10.20.30.40.50.60.70.80.91SNR (dB)Pdtheo (l0, fd/L) = (0, 0.00/K)simutheo (l0, fd/L) = (256, 0.25/K)simutheo (l0, fd/L) = (512, 0.25/K)simu(a) TFL pulse shape with L/K = 9/8−5 0 5 10 15 20 25 30 35 40 45 50 55 6000.10.20.30.40.50.60.70.80.91SNR (dB)Pdtheo (l0, fd/L) = (0, 0.00/K)simutheo (l0, fd/L) = (256, 0.25/K)simutheo (l0, fd/L) = (512, 0.25/K)simu(b) TFL pulse shape with L/K = 12/8Fig. 5. CA-CFAR probability of detection. Comparison of Monte-Carlosimulations (simu) with that of the AWGI model (theo) with SINR (6).QPSK symbols, K = 1024, M = 256, varying L/K, TFL pulses with‖g‖22 = ‖gˇ‖22 = K, σ2 = 1, σ2c = 1, α ∼ CN (0,E{|α|2}) with E{|α|2}defined as in (7), Pfa = 1E−6, Ns = 100. When varying L/K, l0 and fdchosen to keep target physically located at same range and Doppler.observed in Fig. 3 does not affect the performance curves ofFig. 5. In fact with the chosen numerical values, Pd = 1 isachieved before SINR curves reach their floor values. Notethat similar performance curves have been observed for CP-pulses and with 16-QAM modulations (not shown here due tospace limitation). Finally, the desired Pfa is also well retrievedby simulation.IV. CONCLUSIONIn this paper, we derive an expression of the interferencegenerated by a target throughout a 3-stage WCP-OFDMradar processing previously described. This self-interferenceis assumed to be uncorrelated from the ambient noise, whichallows us to deduce a closed-form expression of the SINRin the target cell of the computed range-Doppler map. Then,by assuming an AWGI model, we show that the expectedperformance in detection of the so-called CA-CFAR are recov-ered in simulations. This performance worsens at high rangeand/or velocity but this can be mitigated by adjusting spectralefficiency parameters. Natural prospects are to study more indepth the reliability of our AWGI model and its limits.REFERENCES[1] X. Chen, X. Wang, S. Xu, and J. Zhang, “A novel radar waveform com-patible with communication,” in Int. Conf. on Computational Problem-Solving, ICCP, Oct 2011, pp. 177–181.[2] G. E. A. Franken, H. Nikookar, and P. V. Genderen, “Doppler tolerance ofOFDM-coded radar signals,” in Radar Conference, 2006. EuRAD 2006.3rd European, Sept 2006, pp. 108–111.[3] S. Koslowski, M. Braun, and F. K. Jondral, “Using filter bank multi-carrier signals for radar imaging,” in Position, Location and NavigationSymposium - PLANS 2014, 2014 IEEE/ION, May 2014, pp. 152–157.[4] C. Sturm, T. Zwick, and W. Wiesbeck, “An OFDM system concept forjoint radar and communications operations,” in Veh. Tech. Conf., VTCSpring, April 2009, pp. 1–5.[5] D. Roque and S. Bidon, “Using WCP-OFDM signals with time-frequencylocalized pulses for radar sensing,” in 50th Annual Asilomar Conf. onSignals, Syst., and Comput. (ASILOMARSSC). Pacific Grove, UnitedStates: IEEE, Nov. 2016.[6] D. Roque and C. Siclet, “Performances of weighted cyclic prefix OFDMwith low-complexity equalization,” IEEE Commun. Lett., vol. 17, no. 3,pp. 439–442, 2013.[7] D. Pinchon and P. Siohan, “Closed-form expressions of optimal short PRFMT prototype filters,” in Proc. IEEE Global Telecommunications Conf.GLOBECOM ’11, 2011.[8] H. Finn and R. Johnson, “Adaptive detection mode with threshold controlas a function of spatially sampled clutter-level estimates,” RCA Review,vol. 29, pp. 414––464, Sep. 1968.

CA-CFAR Detection Based on an AWG Interference Model in a Low-Complexity WCP-OFDM Receiver

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CA-CFAR Detection Based on an AWG Interference Model in a Low-Complexity WCP-OFDM Receiver

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