Study of the Target Self-Interference in a Low-Complexity OFDM-Based Radar Receiver

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(13) https://oatao.univ-toulouse.fr/21176. http://doi.org/10.1109/TAES.2018.2879534.

(14) . Mercier, Steven and Roque, Damien and Bidon, Stéphanie Study of the Target Self-Interference in a Low-Complexity. OFDM-Based Radar Receiver. (2018) IEEE Transactions on Aerospace and Electronic Systems. ISSN 0018-9251 .

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(24) 1. Study of the Target Self-Interference in a Low-Complexity OFDM-Based Radar Receiver Steven Mercier, Student Member, IEEE, Damien Roque, Member, IEEE and Stéphanie Bidon, Member, IEEE. Abstract—This paper investigates a radar-communications waveform sharing scenario. Particularly, it addresses the selfinterference phenomenon induced by independent single-point scatterers throughout a low-complexity monostatic OFDM-based radar receiver from a statistical viewpoint. Accordingly, an analytical expression of the post-processing signal-to-interferenceplus-noise-ratio is derived and detection performance is quantified in simulated scenarios for rectangular and non-rectangular pulses. Both metrics suggest that this phenomenon must be further handled. Index Terms—Detection, non-rectangular pulses, OFDMbased, radar-communications, self-interference, waveform sharing.. I. I NTRODUCTION. T. HE idea of using a joint or shared waveform to simultaneously sense the environment and transmit information has been around for some time now [1]. This co-design approach reviewed in [2], [3] has several merits such as favoring hardware integration and spectrum sharing. As such, it directly addresses the well-known spectrum congestion problem which is mostly due to the never-ending hunger for spectral resources of radar and communications systems [4]. Accordingly, for the last two decades, several radar-communications systems involving direct-sequence spread spectrum (DSSS) techniques [5], linear frequency modulated (LFM) waveforms [6], [7] and multicarrier signals, among others, have been investigated in various configurations. This paper focuses on the “monostatic broadcast channel topology” [3] also referred to as RadCom [8], as sketched in Fig. 1. Radar and wireless communication channels both imply time and frequency selectivity caused by motion-induced Doppler effect and multipath propagation [9]. In this context, multicarrier modulations may be particularly robust by using time-frequency shifted pulses to carry information symbols [10]. Consequently, symbols reconstruction and channel estimation (or radar sensing) tasks are made easier compared to single-carrier waveforms (e.g., [11]). Among a wide variety of multicarrier schemes [12], Orthogonal FrequencyDivision Multiplexing (OFDM) has been designed to account for channel’s frequency selectivity while enjoying a very lowcomplexity transmitter and receiver based on fast Fourier transform (FFT) and rectangular pulse shaping [13]. A guard This paper has been presented in part at the Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, November 6-9, 2016 and October 29-November 1, 2017. The authors are with ISAE-SUPAERO, Université de Toulouse, France (firstname.lastname@isae-supaero.fr). The work of Steven Mercier is supported by DGA/MRIS under grant 2017.60.0005 and Thales DMS.. interval may be added to cancel interference at the cost of a spectral efficiency loss [14]. As an example, Cyclic Prefixed (CP)-OFDM is an extensively used transmission technique found in many standards such as LTE (Long Term Evolution), WiMAX (Worldwide Interoperability for Microwave Access), and IEEE 802.11p a standard for vehicular communication systems [15], to name a few examples. Multicarrier and especially OFDM waveforms have originally been introduced in radar for the sole purpose of sensing [16], [17]. In such approach, symbols do not transmit information per se; instead they are used as a tunable parameter to augment the radar performance. In [16], [17], the waveform is shaped to lower the range sidelobes and limit the spectral extent at the same time. In [18], optimal phase code sequences are derived to optimize some figures of merits such as the so-called peak-to-mean envelope power ratio. In [19]–[21], symbols are derived adaptively to optimize the transmitting waveform with respect to its environment. Using OFDM in a RadCom scenario has been initially considered in [1], [22]. It may be widely applicable, for instance, in radar networks [1], [23], [24], intelligent transportation system such as car-to-car networks [8], and commercial or military aviation including unmanned aerial vehicles. Although its high peak-to-average power ratio remains a substantial issue today [25], OFDM exhibits some interesting attributes from a radar perspective. For instance, by contrast with the traditional LFM waveform [26, p. 208], it does not experience rangeDoppler coupling [22]. Besides, it can offer a means to solve Doppler ambiguity [27]. So far, two radar receiver architectures have been thought of for the RadCom OFDM waveform. The first architecture is a conventional radar design based on correlating the received and transmitted signals to obtain the range profile [22]. However, the resulting autocorrelation function depends then on the transmitted data symbols and may thus have high sidelobes due to random effect. Alternatively, a so-called symbol-based architecture has been reported in [8], [28]– [31]. The received signal is passed through a conventional multicarrier linear receiver (i.e., a bank of correlators) [32] that outputs estimated data symbols containing phase shifts characterizing the radar channel (namely target’s range and velocity). The range-Doppler map is then obtained by dividing the estimated symbols with that transmitted and, finally, by applying a bi-dimensional Fourier transform. The symbol-based architecture has been originally described for the particular case of CP-OFDM and investigated in restricted scenarios as summarized in [33]. Particularly, target’s range and velocity are assumed small enough (in terms of.

(25) 2. Com. RX Com. chan. Radcom transceiver Radcom TX. s(t). {ck,m } Radar RX. Radar chan. r(t). Fig. 1. Typical RadCom scenario involving a shared waveform to simultaneously perform radar sensing and data transmission. The radar transceiver is monostatic with a perfect knowledge of the emitted symbols; it has also the role of communications transmitter.. sented by uppercase (resp. lowercase) italic bold letters. I and 0 are the identity and null matrices, [A]m,n denotes the element in the mth row and nth column of A. ⊗ denotes the Kronecker product, δm,n the Kronecker symbol, E{·} the expectation, k·k the `2 -norm. D A is the diagonal matrix defined as [D A ]m,n = [A]m,n δm,n . D a is a diagonal matrix with elements of a on the main diagonal. N, Z and R are the sets of positive integers, integers and real numbers, respectively. IN is the finite set of integers {0, . . . , N − 1}. Unless otherwise stated an index n is defined in the set IN . II. L OW- COMPLEXITY RADAR SYSTEM USING. guard interval length and subcarrier spacing, respectively) to preserve the principle of bi-orthogonality (cf. approximation in (8)). Later, the same architecture has been generalized to more general multicarrier signals. For example, FBMC (Filter Bank Multicarrier) [34] and WCP-OFDM (Weighted CP) [35], [36] enable the use of non-rectangular pulse shapes as a new degree of freedom to improve the system’s performance. For arbitrary target’s range and velocity, inter-symbol interference (ISI) and/or inter-carrier interference (ICI) appear. Although these phenomena have been extensively characterized over radio-communication channels and from a data recovery standpoint (e.g., [37], [38]), their impact on the symbol-based radar performance has only been witnessed so far. Particularly, in [34], [35] a dynamic range degradation was observed in the range-Doppler map, attributable to an increased noise floor along with a target peak loss. It is worth noticing that this twofold phenomenon is not to be mistaken for the interference caused in multi-user scenario [39]. To distinguish both we denote the former as a target self-interference phenomenon. In this work, we therefore extend the primarily work of [36] by thoroughly studying this self-interference assuming a white noise background. Our goal is to provide tools to predict the performance loss endured then by the symbol-based radar receiver with WCP-OFDM waveforms. To that end, we formally demonstrate the expression of the target and selfinterference signatures in the range-Doppler map and further provide a second-order statistical analysis of these terms. We particularly prove that the self-interference is a white signal that independently adds up to the thermal noise. A closedform expression of the SINR (Signal-to-[self]-Interferenceplus-Noise-Ratio) is thereupon easily obtained in single and multi-target scenarios. We further quantify performance loss in terms of probability of detection in a conventional radar detector. The remaining of the paper is organized as follows. Section II recalls the symbol-based radar architecture. Section III derives the resulting signal model when a WCP-OFDM waveform is used. Self-interference terms and loss on target peak are evidenced. Section IV gives a theoretical second-order statistical analysis of signal components. Section V provides a numerical study of CA-CFAR detection performance while assuming a Gaussian self-interference-plus-noise term. The last Section includes some concluding remarks. Notation: We use ·T to denote transpose, ·∗ conjugate and H · conjugate transpose. Matrices (resp. vectors) are repre-. MULTICARRIER WAVEFORMS. A. RadCom transmitter The multicarrier RadCom transmitter sends a message centered around a carrier frequency Fc over K subbands and M blocks. Let {ck,m } be a complex data symbol sequence to be transmitted. The baseband output of such transmitter is s(t) ,. K−1 −1 XM X. ck,m g(t − mT0 ). k=0 m=0. ej2πkF0 t . K 1/2. (1). In short, each ck,m is shaped by a pulse g and placed at coordinates (mT0 , kF0 ) in the time-frequency plane. Here, T0 and F0 represent the pulse repetition interval (PRI) and the elementary subcarrier interval, respectively. The transmission duration is T = M T0 and corresponds to the conventional coherent processing interval (CPI). We further consider K 1 such that at each time t, the transmitted signal s occupies approximately a band B = KF0 . Thus, by sampling (1) at critical rate 1/Ts = B and providing that T0 = LTs with L ∈ N∗ , we obtain s[l] =. −1 K−1 X XM k=0. k. ej2π K l ck,m g[l − mL] 1/2 , K m=0. l∈Z. (2). with the definition s[l] , s(lTs ). L and 1/K denote the normalized versions of the elementary symbol spacing in time and frequency, respectively. Consequently, the ratio L/K accounts for the time-frequency spacing between symbols. B. Radar channel The transmitted signal s is partly reflected towards the radar receiver by a single point target characterized by: • a complex amplitude α; • an initial round-trip delay τ0 ; • a constant radial velocity v. We assume vT c/(2B) (with c the speed of light) such that there is no range migration during the CPI. Thus, the Doppler effect simply translates into a phase shift of the carrier frequency. The baseband received radar signal is r(t) = αs(t − τ0 ) exp(j2πFd t) + n(t). (3). where Fd = 2vFc /c is the target’s Doppler frequency and n(t) is the thermal noise modeled as a white circular Gaussian process with zero-mean and power σ 2 in the bandwidth B. Since the so-called self-interference is the core of this.

(26) 3. investigation, we consider the target as being perfectly located in a range gate. That way, straddling effects are analytically discarded in the following derivations. Therefore, if assuming no range ambiguity as well as Fd B, then τ0 reduces to τ0 = l0 Ts with l0 ∈ IL and (3) can be sampled at 1/Ts too, yielding r[l] = α exp(j2πfd l/L)s[l − l0 ] + n[l],. l∈Z. (4). where fd , v/va is the normalized Doppler frequency of the target, involving the ambiguous velocity va = c/(2Fc T0 ). C. Symbol-based radar receiver On receive, the signal is passed through a symbol-based radar architecture as described in [29]. We recall here its 3 main stages (i.e., (6)-(10)-(11)) while adopting a more general formulation suited for multicarrier waveforms [35]. 1) Symbol estimation: As a first step, a linear estimation of the transmitted symbols is performed by cross-correlating the received signal r with a time-frequency shifted pulse gˇ using the lattice m0 T0 , k 0 F0 , (k 0 , m0 ) ∈ IK × IM : Z +∞ 0 e−j2πk F0 t dt (5) c˜k0 ,m0 , r(t)ˇ g ∗ (t − m0 T0 ) K 1/2 −∞. 2) Radar channel estimation: In a second step, the radar receiver isolates the target signature by removing the transmitted data symbols from (9) as follows c¯k0 ,m0 =. c˜k0 ,m0 ck0 ,m0. (symbol removal).. (10). This low-complexity direct channel estimation approach is usually detrimental in OFDM communications scenarios, so that more costly estimates are often preferred [40], [41]. However, it has been shown to be sufficient for target detection [36]. 3) Range-Doppler map computation: In a third and last step, the radar receiver depicts the former channel estimate in the range-Doppler domain by computing a discrete Fourier transform (DFT) and inverse DFT along the slow-time and slow-frequency domains, respectively. The operation is abusively summed up as xk0 ,m0 = 2D-DFT{¯ ck0 ,m0 }. (Fourier transform).. (11). Since (8) usually remains an approximation, (7)-(10)-(11) arbitrarily result in ISI and ICI, or self-interference from a target detection viewpoint. In the following sections, we quantify the impact of this phenomenon on the radar performance for a specific multicarrier waveform referred to as WCP-OFDM.. or equivalently, in the discrete-time domain c˜k0 ,m0. III. S IGNAL MODEL WITH A WCP-OFDM WAVEFORM. k0. +∞ X. e−j2π K l r[l]ˇ g [l − m L] = K 1/2 l=−∞ ∗. 0. (linear receiver).. (6) This operation corresponds to the first stage of a conventional linear multicarrier communication receiver [32]. By injecting (4) in (6) we thus obtain [35] c˜k0 ,m0 = α. K−1 −1 XM X. ck,m e. k l0 j2π −j2π K. e. h. fd L. + k−k K. 0. i. WCP-OFDM is a multicarrier waveform that generalizes conventional CP-OFDM to non-rectangular pulse-shapes while keeping a time-domain low-complexity implementation. It has the following characteristics [38]: • Short-length pulses: g[l] = gˇ[l] = 0. m0 L. k=0 m=0. . × Agˇ,g l0 + (m − m0 )L,. fd k−k + L K. 0. (7) •. +n ˜ k0 ,m0. if l ∈ / IL. (12). i.e., g and gˇ are shorter than the PRI. As a consequence, we can define two finite length vectors such that [g]l = g[l] and [ˇ g ]l = gˇ[l] for l ∈ IL . Bi-orthogonality criterion: Agˇ,g (pL, q/K) = δp,0 δq,0. (13). where +∞ 1 X ∗ gˇ [p]g[p − l]ej2πf p Agˇ,g (l, f ) = K p=−∞. n ˜ k0 ,m0. k0. +∞ X. e−j2π K l = n[l]ˇ g [l − m L] K 1/2 l=−∞ ∗. 0. denote the cross-ambiguity function and the noise term. The assumption that underlies the rest of the processing of [29] is that of a range-Doppler tolerant waveform [35], i.e., k − k0 fd 0 + ≈ δm,m0 δk,k0 . (8) Agˇ,g l0 + (m − m )L, L K Hence, (7) can be approximated by. A. RadCom transmitter. 0. 0 −j2π k K l0 j2πfd m. c˜k0 ,m0 ≈ α ck0 ,m0 e. e. i.e., symbols’ perfect reconstruction when r = s. If gˇ = g, the system is said orthogonal. Note that biorthogonality requires L/K ≥ 1 which limits the spectral efficiency of the system [42, Ch. 10]. In this work, two types of WCP-OFDM pulses are considered: - Rectangular pulses leading to traditional CP-OFDM. - Time-frequency localized (TFL)-pulses, improved as L/K increases (Fig. 3). These are detailed in Appendix A and depicted in Fig. 2. Note 2 2 that by convention, we have set kgk = K while kˇ g k is implicitly determined by (13).. +n ˜ k0 ,m0. (9). which provides an estimate of the data symbols ck0 ,m0 up to the target’s signature characterized by its range and velocity.. In case of a WCP-OFDM system, the transmitted signal (2) can be simply rewritten using (12) as an LM -length vector h i s = I M ⊗ (D g P F H (14) K) d.

(27) 4. CP L/K = 12/8. Normalized amplitude (dB). Amplitude (-). CP L/K = 12/8. TFL L/K = 12/8. p L/K 1 p K/L. 0. 0. 5. 10 15 Time index k. 20. 0. −20. −40 −0.3. 25. (a) Impulse response with L/K = 12/8 CP L/K = 9/8. Normalized amplitude (dB) 5. 10 15 Time index k. −0.1 0 0.1 Normalized frequency f. CP L/K = 9/8. TFL L/K = 9/8. Amplitude (-). 0. −0.2. 0.2. 0.3. (b) Frequency response with L/K = 12/8. p L/K p K/L. 0. TFL L/K = 12/8. 20. TFL L/K = 9/8. 0. −20. −40 −0.3. 25. (c) Impulse response with L/K = 9/8. −0.2. −0.1 0 0.1 Normalized frequency f. 0.2. 0.3. (d) Frequency response with L/K = 9/8. Fig. 2. Time and frequency responses of transmitter’s (thick line) and receiver’s (thin line) pulse shapes for K = 16.. with P. FK d. the so-called cyclic extension matrix that expands a Klength vector by the end with its first L − K elements, namely [P ]l,k = δl,k + δl,k+K [38]; the√ unitary DFT matrix with [F K ]k,k0 = 1/ K exp(−j2πkk 0 /K); the KM -length column vector with [d]k+mK , k dk,m = ck,m ej2π K mL .. Concretely, the transmitted signal is formed by concatenating M independent WCP-OFDM blocks. Each one is built from an IDFT on K complex symbols, followed by a cyclic-extension and a pulse-shaping. Note that d represents precoded elementary symbols introduced for computational convenience.. B. Radar channel The received WCP-OFDM signal can be recast, according to (4), as r = αZs + n. (15). where n ∼ CN LM (0, σ 2 I LM ) is the thermal noise vector and Z is an LM -size matrix with nonzero elements only on the l0 th subdiagonal, i.e., for l, l0 ∈ ILM , [Z]l,l0 = ej2πfd l/L δl,l0 +l0 . Z models the range-Doppler shifts of the radar channel. In absence of range ambiguity, the signal received during the emission of a block firstly comes from the previously emitted block and then from the current block. (tagged by p and c, respectively). Hence, Z can be decomposed into two subdiagonal matrices corresponding to these two contributions Z = Z (p) + Z (c). (16). where 0 T Z (p) = D ed (fd ) ⊗ D fd (LL−l ) LL LM L. (17a). Z (c) = D ed (fd ) ⊗ D fd LlL0. (17b). with ed (f ). the Doppler steering vector with frequency f and [ed (f )]m = ej2πf m ; D fd the diagonal matrix with [D fd ]l,l = ej2πfd l/L ; LL the L × L lag matrix with [LL ]l,l0 = δl,l0 +1 ; so that (15) becomes h i r = α Z (p) + Z (c) s + n. (18) C. Symbol-based radar receiver 1) Symbol estimation: The (pre-coded) WCP-OFDM symbols estimated in the first stage of the symbol-based architecture can be expressed, according to (6), as h i d˜ = I M ⊗ (F K P T D H ) r (linear receiver). (19) g ˇ One may notice the duality with the transmitted signal (14) since P T then shrinks L-length vectors by removing their last L − K elements after having them summed with their firsts..

(28) 5. CP L/K = 12/8 CP L/K = 9/8. Using lemma 1, we remark that the diagonal matrix D T (c) = Agˇ,g (l0 , fd /L) D ed (fd ) ⊗ D er (l0 ) contributes to the target’s peak albeit a loss factor illustrated in Fig. 3. Conversely, as evidenced in the next Section, both matrices T (c) − D T (c) and T (p) generate intrablock and interblock self-interference, respectively. Consequently, it is meaningful to rearrange the estimated symbols into i h ˜. (23) d˜ = α D T (c) + T (c) − D T (c) + T (p) d + n. TFL L/K = 12/8 TFL L/K = 9/8. 1 |Agˇg (l0 , 0)|. Amplitude. 0.8 |Agˇg (l0 − L, 0)|. 0.6 0.4 0.2. −L12/8. 0. Finally, the last two steps (10)-(11) of the symbol-based architecture lead successively to: 2) Radar channel estimation:. L12/8. −L9/8. 0. L9/8. l0. l. ˜ d¯ = D −1 d d (symbol removal).. (a) Cross-ambiguity function for f = 0: |Agˇg (l, 0)|. Markers represent its L-spaced samples with an offset l0 .. (24). 3) Range-Doppler map computation: CP L/K = 12/8 CP L/K = 9/8. TFL L/K = 12/8 TFL L/K = 9/8. ¯ x = (F M ⊗ F H K )d (Fourier transform).. 1

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(31) Agˇg 0,. Amplitude. 0.8 0.6.

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(34) Agˇg 0,. fd L. −. 1 K. fd L. (25). Our present development is summarized in the following lemma and flowchart of Fig. 4. Lemma 2: The WCP-OFDM range-Doppler measurement resulting from the symbol-based receiver consists of.

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(37).

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(40). 0.4. x = x(t) + x(ic ) + x(ip ) + x(n). 0.2. where the right-hand side terms represent the target’s peak, target’s self-interference terms induced by the current and previous block, and noise signal, respectively, such that. 0 3 −K. 2 −K. 1 −K. 0 fd L. 1 K. 2 K. 3 K. f (b) Cross-ambiguity function for l = 0: |Agˇg (0, f )|. Markers represent its 1/K-spaced samples with an offset fd /L. Fig. 3. Cross-ambiguity function cuts of CP and TFL pulses.. x(t) x(ic ) x(ip ) x(n). Combining the expressions of (19), (18) and (14) yields the signal at the output of the multicarrier receiver h i d˜ = α T (p) + T (c) d + n ˜ (20) where we set. (26). = αAgˇ,g (l0 , fd /L) F M ed (fd ) ⊗ F H K er (l0 ) (27a) h i H −1 (c) = α(F M ⊗ F K )D d T − D T (c) d (27b) −1 (p) d (27c) = α(F M ⊗ F H K )D d T h i −1 = (F M ⊗ F H I M ⊗ (F K P T D H K )D d g ˇ ) n. (27d). Fig. 5 illustrates a typical range-Doppler map derived from our low-complexity WCP-OFDM radar system. As in [35], we observe a loss on the target peak along with an increased noise floor owing to the self-interference terms.. (p) T (p) = [I M ⊗ (F K P T D H [I M ⊗ (D g P F H g ˇ )]Z K )] (21a) (c) T (c) = [I M ⊗ (F K P T D H [I M ⊗ (D g P F H g ˇ )]Z K )] (21b). and n ˜ = [I M ⊗ (F K P T D H g ˇ )]n. Lemma 1: The transfer matrices T (p) and T (c) , that connect the estimated data-symbols to the transmitted ones, reduce to (l −L) T (p) = D ed (fd ) ⊗ Agˇ,g0 D er (l0 −L) LK (22a) KM (l ) T (c) = D ed (fd ) ⊗ Agˇ,g0 D er (l0 ) (22b) with er (l) the range steering vector with delay l and [er (l)]k = e−j2πlk/K ; (l) Agˇ,g the K × K cross-ambiguity Toeplitz matrix at delay l (l) with entries [Agˇ,g ]k,k0 = Agˇ,g (l, fd /L + (k 0 − k)/K). (l) Agˇ,g is circulant since Agˇ,g (l, f ) is 1-periodic in frequency, by construction. Proof is given in Appendix B.. IV. S TATISTICAL ANALYSIS OF THE SIGNAL Herein we provide a statistical study of (27), with particular attention paid to the self-interference component. We formally demonstrate i) the whiteness of the latter; ii) the expression of the post-processing SINR proposed in [36]. A. Statistical assumption about the constellation In what follows, data symbols {ck,m } are assumed independent and uniformly distributed according to a chosen constellation, with zero-mean and power σc2 . We also introduce σc2−1 = E{1/|ck,m |2 }. The digital modulation is further expected to verify E{1/ck,m } = 0 and E{ck,m /c∗k,m } = 0. A sufficient condition for this is to have a symmetric constellation with respect to the origin. This is actually fulfilled by usual constellations, e.g., phase-shift keying (PSK), quadrature amplitude modulation (QAM), amplitude-phase-shift keying (APSK)..

(41) 6. n d. [I M ⊗ (D g P F H K )]. s. Radar channel. αZs. r. WCP-OFDM linear transmitter. ˜ d. [I M ⊗ (F K P T D H g ˇ )]. WCP-OFDM linear receiver. ¯ d. D −1 d. (F M ⊗ F H K). x. Rangedoppler map computation. Symbol removal. Fig. 4. Flowchart of the low-complexity WCP-OFDM radar transceiver.. with. 20 60. Expected target peak. 2. gk σn2 = σ 2 σc2−1 K −1 kˇ. (30). 40. 10 (dB). Range gate (-). H. 20 0. −1 where we have used E{D −1 } = σc2−1 I KM . Noise d Dd whiteness is thus preserved throughout the radar processing. Finally, the main results of this Section concern the second order moments of the self-interference terms and are summarized in the following proposition. Proposition 1: If M 1 self-interference is white H. 0. E{x(ic ) x(ic ) } = σi2c I KM −0.4. −0.2 0 0.2 0.4 Normalized Doppler frequency (-). M 1. Fig. 5. Range-Doppler map |x|. TFL-pulses, QPSK symbols, K = 64, M = 32, L/K = 12/8, σ 2 = 1, σc2 = 1, expected SNRth of 40 dB, as defined in (35). Target located at (l0 , fd ) = (45, 0.25). Measured target peak: 32.6 dB (theoretical value: 32.7 dB). Estimated self-interference-plus-noise power: 3.9 dB (theoretical value: 4.1 dB).. B. First and second order moments First and second order moments of the range-Doppler map contributions are computed from (27). The target statistical mean is simply E{x(t) } = E{α}Agˇ,g (l0 , fd /L) F M ed (fd ) ⊗ F H K er (l0 ). In addition, as d and n are independent and zero-mean then E{x(n) } = 0. Finally, by noticing that D −1 d is independent from [T (c) − D T (c) ]d and T (p) d and using once again the zero-mean assumption of the symbols we obtain E{x(ic ) } = E{x(ip ) } = 0. Similarly, using (27a), the target’s power in the rangeDoppler map is given by

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(44) fd

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(46) 2 (t)H (t) 2 σt = E{x x } = E{|α| }KM

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(48) Agˇ,g l0 , . (28) L

(49) It is worth noticing that the conventional integration gain KM 2 endures a loss equal to |Agˇ,g (l0 , fd /L)| as exemplified in Fig. 5. This loss term actually generalizes the results of [22] for CP-pulses to the more general framework of WCP-OFDM. Let us now focus on the post-processing noise (27d). On the one hand, symbols and noise samples were assumed independent and E{nnH } = σ 2 I LM . Besides, with the pulseshapes considered in this study (i.e., CP and TFL pulses), we have 2 −1 P T DH kˇ gk I K ˇP = K g ˇ Dg so that the post-processing noise covariance matrix reduces to H. E{x(n) x(n) } = σn2 I KM. (31). H. E{x(ip ) x(ip ) } ' σi2p I KM. (29). with

(50) K−1 X.

(51) 2

(52) k

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(56) Agˇ,g l0 , L + K

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(62) Agˇ,g l0 − L, fd + k

(63) . σi2p = E{|α|2 }σc2 σc2−1

(64) L K

(65) k=0 (32) Furthermore, self-interference and noise terms are orthogonal σi2c. 2. }σc2 σc2−1. H. H. H. E{x(ic ) x(ip ) } = E{x(ic ) x(n) } = E{x(ip ) x(n) } = 0KM . (33) Proof is given in Appendix C. C. Post-processing SINR 1) Single target scenario: Detection performance of a radar system is conventionally driven, to a large extent, by the post-processing signal-to-noise ratio (SNR), e.g., [43]. In our WCP-OFDM RadCom scenario, we rather define an SINR performance metrics due to the self-interference phenomenon. This figure of merits accounts both for target peak loss and increased noise floor. Assuming that the target is perfectly located in a range-Doppler bin, the latter is defined as SINR ,. E{|x(n). E{|x(t) |2 } + x(ic ) + x(ip ) |2 }. where x denotes the signal in the target cell. As a consequence of proposition 1, the expression boils down to SINR =. σn2. σt2 + σi2. (34). where σi2 , σi2c + σi2p (cf. Proposition 1). In a theoretical case where the target would be located at (l0 , fd ) = (0, 0), self-interference disappears so that the SINR (34) reduces to.

(66) 7. TFL TFL. CP CP. TABLE I TARGETS PARAMETERS. (l0 , fd /L) = (10, 1/(2K)) (l0 , fd /L) = (60, 1/(8K)). 40 h. 0. 1. 2. 3. 4. 5. 6. l0,h fd,h L. 20. 5. 8. 55. 30. 45. 60. 1 6K. 1 12K. 1 2K. 1 12K. 1 6K. 1 4K. 1 2K. SNRth,h (dB). 20. 35. 20. 20. 25. 15. 10. E{|α|2 } SINR (dB). increases 20. 0. L/K = 12/8. L/K = 9/8. −5. −30. −20. −10 INR (dB). 0. 10. 20. Fig. 6. Post-processing SINR as a function of the INR (curves parametrized by E{|α|2 }). Comparison between analytical (lines) and Monte-Carlo (markers) results for different target parameters and pulse-shapes. QPSK symbols, K = 64, M = 32, L/K = 12/8, σ 2 = 1, σc2 = 1, α ∼ CN (0, E{|α|2 }).. a simple SNR as considered so far in CP-OFDM RadCom scenario [30] E{|α|2 }K 2 M SNRth = (35) 2 . σ 2 σc2−1 kˇ gk To provide more insight into the effect caused by selfinterference, we depict in Fig. 6 the SINR as a function of the INR (Interference-to-Noise-Ratio) while varying the target power E{|α|2 }. The latter metrics is formally defined as INR , σi2 /σn2 . Jointly, an asymptotic study leads to SINR SINR with. −→ 2. E{|α| }→0. −→ 2. E{|α| }→∞. KM |Agˇ,g (l0 , fd /L)|2 INR σc2 σc2−1 W KM |Agˇ,g (l0 , fd /L)| σc2 σc2−1 W. −15. −20. −25. 1. 2. 3. 4 H. 5. 6. 7. Fig. 7. Overall interference power σi2 for a varying number of targets H, with different values of L/K. TFL-pulses, QPSK symbols, K = 64, M = 32, σc2 = 1, αh ∼ CN (0, E{|αh |2 }) as specified in Tab. I.. remains unchanged albeit the definition of the self-interference power reformulated as σi2 ,. 2. H−1 X. 2 σi,h. (37). h=0.

(67) K−1 X.

(68) 2

(69) fd k

(70)

(71)

(72) W =

(73) Agˇ,g l0 , L + K

(74) k=1

(75)

(76) 2 K−1 X

(77)

(78)

(79) Agˇ,g l0 − L, fd + k

(80) . +

(81) L K

(82). −10 10 log10 (σi2 ). −20 −40. (36). k=0. We thus observe two distinct modes: i) a linear regime where both SINR and INR augment with the target power; ii) a saturation of the SINR while the INR keeps growing with the target power. The SINR actually saturates as soon as the self-interference power σi2 becomes significant with respect to the noise floor σn2 . The saturation value strongly depends on the range-Doppler shift of the target in combination with the pulse-shapes chosen as hinted by (36) and Fig. 3. For instance in Fig. 6, TFL-pulses distinctly outperform CP-pulses at lowrange and high-velocity. 2) Multitarget scenario: The expression of the SINR (34) can be easily generalized to a multitarget scenario assuming statistically independent targets with zero-mean amplitudes. Using the linearity of our WCP-OFDM radar transceiver, it is straightforward to show that the functional form of (34). where H is the number of targets and ()h refers to the hth target. Obviously the more target in the radar scene, the stronger the self-interference power. However this growth partly depends, in an intricate way, on both the waveform’s and the targets’ parameters through Wh (36). This is exemplified in Fig. 7 for a specific target scene described in Table I. It particularly shows the increased resilience of the TFL pulse to self-interference for high L/K ratio. 3) SINR and detection performance trend: The SINR saturation effect observed in Section IV-C1 is unconventional. Indeed, even if the target power increases, its peak does not emerge more from the self-interference-plus-noise floor once the self-interference power dominates that of the thermal noise. This may be deleterious in terms of detection especially for a high-range and high-velocity target. This trend is aggravated in multi-target scenario since, as discussed in Section IV-C2, target self-interference powers add up. Specifically, a weak target can be easily hidden under the self-interference-plusnoise floor thus disabling the use of conventional powerthresholding detectors. Note finally that the severity of the self-interference-induced loss depends largely on the target scene itself and cannot thus be assessed beforehand in practical scenarios..

(83) 8. TABLE II P HYSICAL PARAMETERS . I NSPIRED FROM THE AUTOMOTIVE SCENARIO [29]. Fc = 24 GHz B = 93.1 MHz Ts = 10.7 ns δr = 1.61 m. V. D ETECTION PERFORMANCE WITH A CONVENTIONAL. ·10−2 Gaussian fit Laplace fit. Pdf of <{x(i) + x(n) }. Carrier frequency Bandwidth Sampling period Range resolution. 8. 6. 4. 2. DETECTOR 0. In this Section, we further quantify performance loss induced by the self-interference phenomenon in terms of detection probability (PD) in a conventional radar detector. As explained hereafter we focus only on a single target scenario. Fixed simulation parameters are summed up in Table II.. To select an appropriate detector in the range-Doppler map, we first investigate the distribution of the self-interferenceplus-noise component x(i) + x(n) defined in (26). On the one hand, thermal noise Gaussianity is preserved due to the linearity of the radar receiver, i.e., x(n) ∼ CN KM (0, σn2 I KM ). On the other hand, given (27b)-(27c), the self-interference signal x(i) has to our knowledge no obvious distribution. Additionally, the latter depends on the target amplitude pdf (probability density function). To pursue our study, we assume a Swerling I type target, i.e., α ∼ CN (0, E{|α|2 }). In this context, the histogram of a single bin x(i) +x(n) shows that the interference-plus-noise tends to be Laplacian at high INR 1 and Gaussian at low INR 1 (cf. Fig. 8). More generally at low INR, it seems reasonable to assume the Gaussianity on the whole vector, viz x(i) + x(n) ∼ CN KM 0, (σi2 + σn2 )I KM . (38) Otherwise, the pdf may have an intricate form (its study is out of the scope of this work). In the remainder of the paper, we thus restrict our analysis to low INR scenario by considering a single-target in the radar scene (thereby avoiding accumulation of self-interference powers (37)) with a reasonable power. B. CA-CFAR detection performance Under the Gaussian assumption (38), a conventional CellAveraging Constant-False-Alarm-Rate (CA-CFAR) detector can be applied [44]. The theoretical CA-CFAR PD is then given by " #−Ns −1/Ns Pfa −1 (39) Pd = 1 + 1 + SINR with Ns the number of training cells, Pfa the desired probability of false alarm. The PD is depicted in Fig. 9 for varying range l0 , Doppler fd and ratio L/K for both CP and TFL pulse-shapes. First, analytical and Monte-Carlo PD curves do match thereby advocating for assumption (38) at low INR. Secondly, we clearly observe significant PD loss when either the target range. −20. 0. 20. 40. <{x(i) + x(n) } (a) INR ' 22 dB 1. Pdf of <{x(i) + x(n) }. A. Self-interference-plus-noise distribution. −40. Gaussian fit Laplace fit. 0.8 0.6 0.4 0.2 0 −3. −2. −1. 0. 1. 2. 3. <{x(i) + x(n) } (b) INR ' −30 dB Fig. 8. Histograms of the real part of a self-interference-plus-noise bin for different INR values (30, 000 runs). The bin is chosen far enough from the target cell, to avoid sidelobe effects. QPSK symbols, TFL pulses, σ 2 = 1, L/K = 9/8, (l0 , fd /L) = (27, 1/(4K)) and: (a) K = 32, M = 16, SNRth = 50 dB; (b) K = 256, M = 64, SNRth = 20 dB.. or velocity increase. The degradation is actually mostly due to the loss on the target peak since at low INR noise floor is barely increased. Finally, owing to its orthogonality the TFL pulse-shape should be preferred to CP-pulses in low-range and high-velocity scenarios (e.g., emergency braking in automotive case). At higher ranges, increasing L/K is a way to achieve better detection performance, at the cost of a reduced spectral efficiency. VI. C ONCLUSION In this paper, a monostatic multicarrier radar has been considered to simultaneously perform data transmission and environment sensing. Its receiver is built on an existing lowcomplexity symbol-based linear architecture, that practically results in target peak loss and self-interference components in the range-Doppler map. We proposed an analytical study of this undesirable twofold phenomenon in the context of WCPOFDM, which is a generalization of conventional CP-OFDM to non-rectangular pulse-shapes. Specifically, we proved that the self-interference signal is white, enabling a closed-form expression of the post-processing SINR. This figure of merit.

(84) 9. CP L/K = 12/8 CP L/K = 9/8. TFL L/K = 12/8 TFL L/K = 9/8. 0.8. Pd. 0.6. 0.4. 0.2. 0. 0. 200. 400. 600. 800. A PPENDIX A WCP-OFDM PULSE - SHAPES EXPRESSIONS Analytical expressions of the considered pulse-shapes are as follows: • CP-pulses: (p K/L if l ∈ IL CP g [l] = 0 otherwise (p L/K if l ∈ IL \ IL−K gˇCP [l] = 0 otherwise. 1,000. l0. •. (a) PD for fd = 0 CP L/K = 12/8 CP L/K = 9/8. TFL L/K = 12/8 TFL L/K = 9/8. where θ[l] is given in [45] through numerical optimization.. Pd. 0.85. A PPENDIX B S YMBOLS TRANSFER MATRICES Herein we provide details of computation to obtain the expression (22) of the transfer matrices T (p) and T (c) in lemma 1. First we focus on the derivation of T (c) . Using (17b) and (21b), the matrix can rewritten as. 0.8. 0.75. 0.7. with \ the set difference symbol. The CP length is given by L − K and should be chosen greater than l0 to avoid ISI. TFL-pulses:  if l ∈ IL−K  cos θ[l]  1 if l ∈ IK \ IL−K g TFL [l] = gˇTFL [l] = sin θ[l] if l ∈ IL \ IK    0 otherwise. 0. 0.1. 0.2 0.3 fd = v/va. T (c) = D ed (fd ) ⊗ B. 0.4. with. (b) PD for l0 = 0 Fig. 9. Probability of detection of the CA-CFAR as a function of l0 or fd at low INR (maximum value verifies INR < −30 dB). Comparison between analytical (lines) (39) and Monte-Carlo (markers) results for varying pulse-shapes and L/K ratios. Pfa = 1E − 6, Ns = 100, QPSK symbols, K = 1024, M = 256, σ 2 = 1, σc2 = 1, α ∼ CN (0, E{|α|2 }) with E{|α|2 } defined by (35) to achieve SNRth = 20 dB with TFL-pulses. Note that the maximum velocity here achieves the limit of model (3); beyond, range migration can no longer be neglected.. h i h i l0 H B = F K P T DH D L D P F f g g ˇ K . d L. To simplify the expression of B, let denote f H k the kth column of the Fourier matrix F K , i.e., H F K = [f H 0 , . . . , f K−1 ].. Note that the definition of f k can be easily extended to k ∈ Z since, by periodicity, f k+K = f k . We have then F K P T DH g ˇ = ∗ T gˇ[0]f 0 , . . . , gˇ[K − 1]f TK−1 , gˇ[K]f T0 , . . . , gˇ[L − 1]f TL−K−1. is convenient to assess the induced performance degradation in the radar receiver. We further evaluated the detrimental effect in terms of detection probability using the well-known CACFAR while detouring its domain of applicability (i.e., low INR scenarios). Overall, TFL-pulses were shown to be more resilient to self-interference over CP-pulses in low-range and high-velocity scenarios. In any event, the study motivates the need to develop new detection schemes to deal with the self-interference phenomenon. Future work may include the design of dedicated mitigation techniques such as successive interference cancellation. Presence of clutter should be also addressed.. Dg P F H K = T T g[0]f 0 , . . . , g[K − 1]f TK−1 , g[K]f T0 , . . . , g[L − 1]f TL−K−1 and D fd LlL0 D g P F H K =          . 01×K .. . 01×K ej2πfd l0 /L g[0]f 0 .. .. ej2πfd (L−1)/L g[L − l0 − 1]f L−l0 −1.      .    .

(85) 10. Thus, we obtain. then. L−l 0 −1 X. B=. l=0 L−1 X. =. ∗. gˇ [l + l0 ]g[l]e ∗. gˇ [l]g[l − l0 ]e. j2πfd (l+l0 )/L. j2πfd l/L. h i (i ) (i )H d¯mc d¯mc. fH l+l0 f l. = i,j. k=0 k0 =0 k6=i k0 = 6 j. fd k−i × Agˇ,g l0 , + L K 0 k −j fd × A∗gˇ,g l0 , + . L K. l=0. where we have used in the last line the short-filter assumption (12). Using the definition of f l , we have i h 0 1 = e−j2πlk/K ej2π(l−l0 )k /K fH l f l−l0 K k,k0 0 0 1 = ej2πl(k −k)/K e−j2πl0 k /K K. Therefore: • For j 6= i, since the constellation was assumed symmetric to the origin in Section IV-A, we have h i (i ) (i )H c E d¯m d¯mc = 0. i,j. •. that yields. For j = i then h. [B]k,k0 =. e. −j2πl0 k /K L−1 X. K. gˇ∗ [l]g[l − l0 ]e. j2πl. . fd L. +. k0 −k. . (i ) (i ) d¯mc d¯mc. H. i. =. K−1 X K−1 X. i,i. K. k=0 k0 =0 k6=i k0 = 6 i. l=0 0. = e−j2πl0 k /K Agˇ,g. . k0 − k fd + l0 , L K. dk,m d∗k0 ,m −j2πl0 (k−k0 ) K e di,m d∗j,m. . fH l f l−l0. 0. K−1 X K−1 X. dk,m d∗k0 ,m −j2πl0 (k−k0 ) K e |di,m |2. . k−i fd × Agˇ,g l0 , + L K k0 − i fd ∗ + × Agˇ,g l0 , L K. .. (l ). We thus proved that B = Agˇ,g0 D er (l0 ) and thus (22b). The derivation of T (p) is irigorously similar, oncei noticed that h h K L ) = I ⊗ (D g P F H LM L I M ⊗ (D g P F H M K ) LKM . K. with E{dk,m d∗k0 ,m } = σc2 δk,k0 so that h i (ic ) ¯(ic )H ¯ E dm dm i,i. A PPENDIX C S ELF - INTERFERENCE SECOND - ORDER STATISTICS. =. (ic ) ¯(ic )H. H. d. H }(F M ⊗ F H K). E. =. σc2 σc2−1. |dk,m |2 |di,m |2.

(86)

(87) 2

(88)

(89)

(90) Agˇ,g l0 , fd + k − i

(91)

(92)

(93) L K.

(94) K−1 X k=0 k6=i.

(95) 2

(96)

(97)

(98) Agˇ,g l0 , fd + k − i

(99) .

(100)

(101) L K. We thus demonstrated that (ic ) ¯(ic )H. E{d¯. with ¯(ic ). d. (c) = αD −1 − D T (c) )d. d (T. } = σi2c I KM. H. Let us cut d into M sub-vectors of length K and denote (i ) d¯mc the mth sub-vector, so that  (i )  d¯0 c  (i ) .   d¯ c = α   ..  . (i ) d¯Mc−1. Given the block diagonal structure of T (c) − D T (c) and since the dk,m (or indifferently, the ck,m ) were assumed (i ) independent, the d¯mc are also independent from each other. (i ) (i )H As a result, E{d¯ c d¯ c } is at least block diagonal which (i ) (i )H means that the computation of E{d¯ c d¯ c } is reduced to H (i ) (i ) that of E{d¯mc d¯mc }. Since we have for i ∈ IK [d¯mc ]i = ej2πfd m. d. and therefore. ¯(ic ). (i ). . k=0 k6=i. In what follows, we show that x(ic ) defined in (27b) is white. Since the same reasoning holds for x(ip ) when M 1, we can conclude with (31). First, we have ¯ E{x(ic ) x(ic ) } = (F M ⊗ F H K )E{d. K−1 X. K−1 X k=0 k6=i. dk,m k − i −j2πl0 k fd K Agˇ,g l0 , + e di,m L K. E{x(ic ) x(ic ) } = σi2c I KM where σi2c. 2. = E{|α|. }σc2 σc2−1. K−1 X

(102) k=1.

(103) 2

(104)

(105)

(106) Agˇ,g l0 , fd + k

(107) .

(108) L K

(109) H. As mentioned, the derivation of E{x(ip ) x(ip ) } is quite H similar to that of E{x(ic ) x(ic ) }. If keeping the same nota(i ) tions, the main difference is that d¯0 p is not defined. This side effect can potentially be avoided by ignoring the first received OFDM symbol. Otherwise, we obtain H. E{x(ip ) x(ip ) } =  (M − 1)I K   −I K σi2p M −1   ..  . −I K. −I K .. .. ... ... .... . −I K. −I K .. . −I K (M − 1)I K.      .

(110) 11. where we denote σi2p = E{|α|2 }σc2 σc2−1. K−1 X

(111) k=0.

(112)

(113) Agˇ,g

(114). .

(115) 2 fd k

(116)

(117) l0 − L, + L K

(118). which consequently leads to H. E{x(ip ) x(ip ) } ' σi2p I KM . M 1. The same reasoning is applied to prove (33). R EFERENCES [1] van Genderen, P. and Nikookar, H., “Radar network communication,” in Proc. Commun., Bucharest, Romania, June 8-10 2006, pp. 313–316. [2] Han, L. and Wu, K., “Joint wireless communication and radar sensing systems - state of the art and future prospects,” IET Microwaves, Antennas Propagation, vol. 7, no. 11, pp. 876–885, August 2013. [3] Paul, B., Chiriyath, A. R., and Bliss, D. W., “Survey of RF communications and sensing convergence research,” IEEE Access, vol. 5, pp. 252–270, 2017. [4] Griffiths, H., Cohen, L., Watts, S., Mokole, E., Baker, C., Wicks, M., and Blunt, S., “Radar spectrum engineering and management: Technical and regulatory issues,” Proc. IEEE, vol. 103, no. 1, pp. 85–102, Jan 2015. [5] Shaojian, X., Bing, C., and Ping, Z., “Radar-communication integration based on DSSS techniques,” in 8th Int. Conf. on Signal Process., vol. 4, 2006. 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(119) 12. Steven Mercier (S’17) was born in Fontenay-auxRoses, France, in September 1994. He received the engineer and master degrees in signal processing from Grenoble INP in 2017 and is now preparing a Ph.D. degree on waveform sharing for radar and communications at ISAE-SUPAERO, University of Toulouse, France.. Damien Roque (M’15) was born in Chambery, France, in July 1986. He received the Ph.D. degree from the University of Grenoble in 2012 and the ´ Dipl.Ing. in telecommunications from the Ecole Nationale Supérieure des Télécommunications (ENST) de Bretagne in 2009. From 2009 to 2013, he worked at the lab Grenoble Images Paroles Signal Automatique (GIPSA-lab), Grenoble, France. Since 2013, he is an associate professor at the Institut Supérieur de l’Aéronautique et de l’Espace (ISAE-SUPAERO), University of Toulouse, France. His research interests are in the area of spectrally efficient waveforms, including faster-thanNyquist signaling and multicarrier modulations.. Stéphanie Bidon (M’08) received the engineer degree in aeronautics and the master degree in signal processing from ENSICA, Toulouse, in 2004 and 2005 respectively. She obtained the Ph.D. degree and the Habilitation a` Diriger des Recherches in signal processing from INP, Toulouse, in 2008 and 2015 respectively. She is currently with the Department of Electronics, Optronics and Signal at ISAESUPAERO, Toulouse, as a professor. Her research interests include digital signal processing particularly with application to radar systems..

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