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http://doi.org/10.1109/TAES.2018.2879534

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

1

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

Steven Mercier, Student Member, IEEE, Damien Roque, Member, IEEE and Stephanie Bidon, Member, IEEE

Abstract—This paper investigates a radar-communicationswaveform sharing scenario. Particularly, it addresses the self-interference phenomenon induced by independent single-pointscatterers throughout a low-complexity monostatic OFDM-basedradar receiver from a statistical viewpoint. Accordingly, an an-alytical expression of the post-processing signal-to-interference-plus-noise-ratio is derived and detection performance is quanti-fied in simulated scenarios for rectangular and non-rectangularpulses. Both metrics suggest that this phenomenon must befurther handled.

Index Terms—Detection, non-rectangular pulses, OFDM-based, radar-communications, self-interference, waveform shar-ing.

I. INTRODUCTION

THE idea of using a joint or shared waveform to simulta-neously sense the environment and transmit information

has been around for some time now [1]. This co-design ap-proach reviewed in [2], [3] has several merits such as favoringhardware integration and spectrum sharing. As such, it di-rectly addresses the well-known spectrum congestion problemwhich is mostly due to the never-ending hunger for spectralresources of radar and communications systems [4]. Accord-ingly, for the last two decades, several radar-communicationssystems involving direct-sequence spread spectrum (DSSS)techniques [5], linear frequency modulated (LFM) wave-forms [6], [7] and multicarrier signals, among others, havebeen investigated in various configurations. This paper focuseson the “monostatic broadcast channel topology” [3] alsoreferred to as RadCom [8], as sketched in Fig. 1.

Radar and wireless communication channels both implytime and frequency selectivity caused by motion-inducedDoppler effect and multipath propagation [9]. In this context,multicarrier modulations may be particularly robust by usingtime-frequency shifted pulses to carry information symbols[10]. Consequently, symbols reconstruction and channel es-timation (or radar sensing) tasks are made easier comparedto single-carrier waveforms (e.g., [11]). Among a wide va-riety of multicarrier schemes [12], Orthogonal Frequency-Division Multiplexing (OFDM) has been designed to accountfor channel’s frequency selectivity while enjoying a very low-complexity transmitter and receiver based on fast Fouriertransform (FFT) and rectangular pulse shaping [13]. A guard

This paper has been presented in part at the Asilomar Conference onSignals, Systems, and Computers, Pacific Grove, CA, November 6-9, 2016and October 29-November 1, 2017.

The authors are with ISAE-SUPAERO, Universite de Toulouse, France(firstname.lastname@isae-supaero.fr). The work of Steven Mercier is sup-ported by DGA/MRIS under grant 2017.60.0005 and Thales DMS.

interval may be added to cancel interference at the cost of aspectral efficiency loss [14]. As an example, Cyclic Prefixed(CP)-OFDM is an extensively used transmission techniquefound in many standards such as LTE (Long Term Evolution),WiMAX (Worldwide Interoperability for Microwave Access),and IEEE 802.11p a standard for vehicular communicationsystems [15], to name a few examples.

Multicarrier and especially OFDM waveforms have origi-nally been introduced in radar for the sole purpose of sens-ing [16], [17]. In such approach, symbols do not transmit in-formation per se; instead they are used as a tunable parameterto augment the radar performance. In [16], [17], the waveformis shaped to lower the range sidelobes and limit the spectralextent at the same time. In [18], optimal phase code sequencesare derived to optimize some figures of merits such as theso-called peak-to-mean envelope power ratio. In [19]–[21],symbols are derived adaptively to optimize the transmittingwaveform with respect to its environment.

Using OFDM in a RadCom scenario has been initially con-sidered in [1], [22]. It may be widely applicable, for instance,in radar networks [1], [23], [24], intelligent transportationsystem such as car-to-car networks [8], and commercial or mil-itary aviation including unmanned aerial vehicles. Although itshigh peak-to-average power ratio remains a substantial issuetoday [25], OFDM exhibits some interesting attributes from aradar perspective. For instance, by contrast with the traditionalLFM waveform [26, p. 208], it does not experience range-Doppler coupling [22]. Besides, it can offer a means to solveDoppler ambiguity [27].

So far, two radar receiver architectures have been thoughtof for the RadCom OFDM waveform. The first architectureis a conventional radar design based on correlating the re-ceived and transmitted signals to obtain the range profile [22].However, the resulting autocorrelation function depends thenon the transmitted data symbols and may thus have highsidelobes due to random effect. Alternatively, a so-calledsymbol-based architecture has been reported in [8], [28]–[31]. The received signal is passed through a conventionalmulticarrier linear receiver (i.e., a bank of correlators) [32]that outputs estimated data symbols containing phase shiftscharacterizing the radar channel (namely target’s range andvelocity). The range-Doppler map is then obtained by dividingthe estimated symbols with that transmitted and, finally, byapplying a bi-dimensional Fourier transform.

The symbol-based architecture has been originally describedfor the particular case of CP-OFDM and investigated in re-stricted scenarios as summarized in [33]. Particularly, target’srange and velocity are assumed small enough (in terms of

2

s(t)

r(t)

Radcom TX

Radar RX

{ck,m}

Radcom transceiver

Radar chan.

Com. chan.

Com. RX

Fig. 1. Typical RadCom scenario involving a shared waveform to simultane-ously perform radar sensing and data transmission. The radar transceiver ismonostatic with a perfect knowledge of the emitted symbols; it has also therole of communications transmitter.

guard interval length and subcarrier spacing, respectively) topreserve the principle of bi-orthogonality (cf. approximationin (8)). Later, the same architecture has been generalized tomore general multicarrier signals. For example, FBMC (FilterBank Multicarrier) [34] and WCP-OFDM (Weighted CP) [35],[36] enable the use of non-rectangular pulse shapes as a newdegree of freedom to improve the system’s performance. Forarbitrary target’s range and velocity, inter-symbol interference(ISI) and/or inter-carrier interference (ICI) appear. Althoughthese phenomena have been extensively characterized overradio-communication channels and from a data recovery stand-point (e.g., [37], [38]), their impact on the symbol-basedradar performance has only been witnessed so far. Particularly,in [34], [35] a dynamic range degradation was observed in therange-Doppler map, attributable to an increased noise flooralong with a target peak loss. It is worth noticing that thistwofold phenomenon is not to be mistaken for the interferencecaused in multi-user scenario [39]. To distinguish both wedenote the former as a target self-interference phenomenon.

In this work, we therefore extend the primarily workof [36] by thoroughly studying this self-interference assuminga white noise background. Our goal is to provide tools topredict the performance loss endured then by the symbol-basedradar receiver with WCP-OFDM waveforms. To that end, weformally demonstrate the expression of the target and self-interference signatures in the range-Doppler map and furtherprovide a second-order statistical analysis of these terms. Weparticularly prove that the self-interference is a white signalthat independently adds up to the thermal noise. A closed-form expression of the SINR (Signal-to-[self]-Interference-plus-Noise-Ratio) is thereupon easily obtained in single andmulti-target scenarios. We further quantify performance lossin terms of probability of detection in a conventional radardetector.

The remaining of the paper is organized as follows. Sec-tion II recalls the symbol-based radar architecture. Section IIIderives the resulting signal model when a WCP-OFDM wave-form is used. Self-interference terms and loss on target peakare evidenced. Section IV gives a theoretical second-orderstatistical analysis of signal components. Section V providesa numerical study of CA-CFAR detection performance whileassuming a Gaussian self-interference-plus-noise term. Thelast Section includes some concluding remarks.

Notation: We use ·T to denote transpose, ·∗ conjugate and·H conjugate transpose. Matrices (resp. vectors) are repre-

sented by uppercase (resp. lowercase) italic bold letters. Iand 0 are the identity and null matrices, [A]m,n denotes theelement in the mth row and nth column of A. ⊗ denotesthe Kronecker product, δm,n the Kronecker symbol, E{·}the expectation, ‖·‖ the `2-norm. DA is the diagonal matrixdefined as [DA]m,n = [A]m,nδm,n. Da is a diagonal matrixwith elements of a on the main diagonal. N, Z and Rare 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. LOW-COMPLEXITY RADAR SYSTEM USINGMULTICARRIER WAVEFORMS

A. RadCom transmitter

The multicarrier RadCom transmitter sends a message cen-tered around a carrier frequency Fc over K subbands and Mblocks. Let {ck,m} be a complex data symbol sequence to betransmitted. The baseband output of such transmitter is

s(t) ,K−1∑k=0

M−1∑m=0

ck,mg(t−mT0)ej2πkF0t

K1/2. (1)

In short, each ck,m is shaped by a pulse g and placed atcoordinates (mT0, kF0) in the time-frequency plane. Here, T0

and F0 represent the pulse repetition interval (PRI) and theelementary subcarrier interval, respectively. The transmissionduration is T = MT0 and corresponds to the conventionalcoherent processing interval (CPI).

We further consider K � 1 such that at each time t, thetransmitted signal s occupies approximately a band B = KF0.Thus, by sampling (1) at critical rate 1/Ts = B and providingthat T0 = LTs with L ∈ N∗, we obtain

s[l] =

K−1∑k=0

M−1∑m=0

ck,mg[l −mL]ej2π

kK l

K1/2, l ∈ Z (2)

with the definition s[l] , s(lTs). L and 1/K denote thenormalized versions of the elementary symbol spacing intime and frequency, respectively. Consequently, the ratio L/Kaccounts for the time-frequency spacing between symbols.

B. Radar channel

The transmitted signal s is partly reflected towards the radarreceiver 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) suchthat there is no range migration during the CPI. Thus, theDoppler effect simply translates into a phase shift of the carrierfrequency. The baseband received radar signal is

r(t) = αs(t− τ0) exp(j2πFdt) + 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 Gaussianprocess with zero-mean and power σ2 in the bandwidthB. Since the so-called self-interference is the core of this

3

investigation, we consider the target as being perfectly locatedin a range gate. That way, straddling effects are analyticallydiscarded in the following derivations. Therefore, if assumingno range ambiguity as well as Fd � B, then τ0 reduces toτ0 = l0Ts with l0 ∈ IL and (3) can be sampled at 1/Ts too,yielding

r[l] = α exp(j2πfdl/L)s[l − l0] + n[l], l ∈ Z (4)

where fd , v/va is the normalized Doppler frequency of thetarget, involving the ambiguous velocity va = c/(2FcT0).

C. Symbol-based radar receiver

On receive, the signal is passed through a symbol-basedradar architecture as described in [29]. We recall here its 3main stages (i.e., (6)-(10)-(11)) while adopting a more generalformulation suited for multicarrier waveforms [35].

1) Symbol estimation: As a first step, a linear estimation ofthe transmitted symbols is performed by cross-correlating thereceived signal r with a time-frequency shifted pulse g usingthe lattice m′T0, k

′F0, (k′,m′) ∈ IK × IM :

ck′,m′ ,∫ +∞

−∞r(t)g∗(t−m′T0)

e−j2πk′F0t

K1/2dt (5)

or equivalently, in the discrete-time domain

ck′,m′ =

+∞∑l=−∞

r[l]g∗[l −m′L]e−j2π

k′K l

K1/2(linear receiver).

(6)This operation corresponds to the first stage of a conventionallinear multicarrier communication receiver [32]. By injecting(4) in (6) we thus obtain [35]

ck′,m′ = α

K−1∑k=0

M−1∑m=0

ck,me−j2π k

K l0ej2π[

fdL + k−k′

K

]m′L

×Ag,g(l0 + (m−m′)L, fd

L+k − k′

K

)+ nk′,m′

(7)

where

Ag,g(l, f) =1

K

+∞∑p=−∞

g∗[p]g[p− l]ej2πfp

nk′,m′ =

+∞∑l=−∞

n[l]g∗[l −m′L]e−j2π

k′K l

K1/2

denote the cross-ambiguity function and the noise term. Theassumption that underlies the rest of the processing of [29] isthat of a range-Doppler tolerant waveform [35], i.e.,

Ag,g

(l0 + (m−m′)L, fd

L+k − k′

K

)≈ δm,m′δk,k′ . (8)

Hence, (7) can be approximated by

ck′,m′ ≈ α ck′,m′e−j2πk′K l0ej2πfdm

′+ nk′,m′ (9)

which provides an estimate of the data symbols ck′,m′ up tothe target’s signature characterized by its range and velocity.

2) Radar channel estimation: In a second step, the radarreceiver isolates the target signature by removing the transmit-ted data symbols from (9) as follows

ck′,m′ =ck′,m′

ck′,m′(symbol removal). (10)

This low-complexity direct channel estimation approach isusually detrimental in OFDM communications scenarios, sothat more costly estimates are often preferred [40], [41].However, it has been shown to be sufficient for target de-tection [36].

3) Range-Doppler map computation: In a third and laststep, the radar receiver depicts the former channel estimatein the range-Doppler domain by computing a discrete Fouriertransform (DFT) and inverse DFT along the slow-time andslow-frequency domains, respectively. The operation is abu-sively summed up as

xk′,m′ = 2D-DFT{ck′,m′} (Fourier transform). (11)

Since (8) usually remains an approximation, (7)-(10)-(11) ar-bitrarily result in ISI and ICI, or self-interference from a targetdetection viewpoint. In the following sections, we quantify theimpact of this phenomenon on the radar performance for aspecific multicarrier waveform referred to as WCP-OFDM.

III. SIGNAL MODEL WITH A WCP-OFDM WAVEFORM

WCP-OFDM is a multicarrier waveform that generalizesconventional CP-OFDM to non-rectangular pulse-shapes whilekeeping a time-domain low-complexity implementation. It hasthe following characteristics [38]:• Short-length pulses:

g[l] = g[l] = 0 if l /∈ IL (12)

i.e., g and g are shorter than the PRI. As a consequence,we can define two finite length vectors such that [g]l =g[l] and [g]l = g[l] for l ∈ IL.

• Bi-orthogonality criterion:

Ag,g(pL, q/K) = δp,0δq,0 (13)

i.e., symbols’ perfect reconstruction when r = s. Ifg = g, the system is said orthogonal. Note that bi-orthogonality requires L/K ≥ 1 which limits the spectralefficiency 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 asL/K increases (Fig. 3).

These are detailed in Appendix A and depicted in Fig. 2. Notethat by convention, we have set ‖g‖2 = K while ‖g‖2 isimplicitly determined by (13).

A. RadCom transmitter

In case of a WCP-OFDM system, the transmitted signal (2)can be simply rewritten using (12) as an LM -length vector

s =[IM ⊗ (DgPFH

K)]d (14)

4

0 5 10 15 20 250

√K/L

1

√L/K

Time index k

Am

plitu

de(-

)

CP L/K = 12/8 TFL L/K = 12/8

(a) Impulse response with L/K = 12/8

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−40

−20

0

Normalized frequency f

Nor

mal

ized

ampl

itude

(dB

)

CP L/K = 12/8 TFL L/K = 12/8

(b) Frequency response with L/K = 12/8

0 5 10 15 20 250

√K/L

√L/K

Time index k

Am

plitu

de(-

)

CP L/K = 9/8 TFL L/K = 9/8

(c) Impulse response with L/K = 9/8

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−40

−20

0

Normalized frequency f

Nor

mal

ized

ampl

itude

(dB

)

CP L/K = 9/8 TFL L/K = 9/8

(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 the so-called cyclic extension matrix that expands a K-length vector by the end with its first L−K elements,namely [P ]l,k = δl,k + δl,k+K [38];

FK the unitary DFT matrix with [FK ]k,k′ =1/√K exp(−j2πkk′/K);

d the KM -length column vector with [d]k+mK ,dk,m = ck,me

j2π kKmL.

Concretely, the transmitted signal is formed by concatenatingM independent WCP-OFDM blocks. Each one is built from anIDFT on K complex symbols, followed by a cyclic-extensionand a pulse-shaping. Note that d represents precoded elemen-tary symbols introduced for computational convenience.

B. Radar channel

The received WCP-OFDM signal can be recast, accordingto (4), as

r = αZs + n (15)

where n ∼ CNLM (0, σ2ILM ) is the thermal noise vectorand Z is an LM -size matrix with nonzero elements onlyon the l0th subdiagonal, i.e., for l, l′ ∈ ILM , [Z]l,l′ =ej2πfdl/Lδl,l′+l0 . Z models the range-Doppler shifts of theradar channel. In absence of range ambiguity, the signalreceived during the emission of a block firstly comes fromthe previously emitted block and then from the current block

(tagged by p and c, respectively). Hence, Z can be decom-posed into two subdiagonal matrices corresponding to thesetwo contributions

Z = Z(p) + Z(c) (16)

where

Z(p) =(Ded(fd) ⊗Dfd(LL−l0L )T

)LLLM (17a)

Z(c) = Ded(fd) ⊗DfdLl0L (17b)

withed(f) the Doppler steering vector with frequency f and

[ed(f)]m = ej2πfm;Dfd the diagonal matrix with [Dfd ]l,l = ej2πfdl/L;LL the L× L lag matrix with [LL]l,l′ = δl,l′+1;

so that (15) becomes

r = α[Z(p) + Z(c)

]s + n. (18)

C. Symbol-based radar receiver

1) Symbol estimation: The (pre-coded) WCP-OFDM sym-bols estimated in the first stage of the symbol-based architec-ture can be expressed, according to (6), as

d =[IM ⊗ (FKP TDH

g )]r (linear receiver). (19)

One may notice the duality with the transmitted signal (14)since P T then shrinks L-length vectors by removing their lastL−K elements after having them summed with their firsts.

5

−L9/8 0 l0 L9/80

0.2

0.4

0.6

0.8

1

L12/8−L12/8

|Agg(l0, 0)|

|Agg(l0 − L, 0)|

l

Am

plitu

deCP L/K = 12/8 TFL L/K = 12/8

CP L/K = 9/8 TFL L/K = 9/8

(a) Cross-ambiguity function for f = 0: |Agg(l, 0)|. Markers representits L-spaced samples with an offset l0.

− 3K

− 2K

− 1K

0 fdL

1K

2K

3K

0

0.2

0.4

0.6

0.8

1 ∣∣∣Agg

(0, fd

L

)∣∣∣∣∣∣Agg

(0, fd

L− 1

K

)∣∣∣

f

Am

plitu

de

CP L/K = 12/8 TFL L/K = 12/8

CP L/K = 9/8 TFL L/K = 9/8

(b) Cross-ambiguity function for l = 0: |Agg(0, f)|. Markers representits 1/K-spaced samples with an offset fd/L.

Fig. 3. Cross-ambiguity function cuts of CP and TFL pulses.

Combining the expressions of (19), (18) and (14) yields thesignal at the output of the multicarrier receiver

d = α[T (p) + T (c)

]d + n (20)

where we set

T (p) = [IM ⊗ (FKP TDHg )]Z(p)[IM ⊗ (DgPFH

K)] (21a)

T (c) = [IM ⊗ (FKP TDHg )]Z(c)[IM ⊗ (DgPFH

K)] (21b)

and n = [IM ⊗ (FKP TDHg )]n.

Lemma 1: The transfer matrices T (p) and T (c), that connectthe estimated data-symbols to the transmitted ones, reduce to

T (p) =(Ded(fd) ⊗A

(l0−L)g,g Der(l0−L)

)LKKM (22a)

T (c) =(Ded(fd) ⊗A

(l0)g,g Der(l0)

)(22b)

wither(l) the range steering vector with delay l and [er(l)]k =

e−j2πlk/K ;A

(l)g,g the K ×K cross-ambiguity Toeplitz matrix at delay l

with entries [A(l)g,g]k,k′ = Ag,g(l, fd/L+ (k′− k)/K).

A(l)g,g is circulant since Ag,g(l, f) is 1-periodic in

frequency, by construction.Proof is given in Appendix B.

Using lemma 1, we remark that the diagonal matrixDT (c) = Ag,g(l0, fd/L) Ded(fd) ⊗ Der(l0) contributes tothe target’s peak albeit a loss factor illustrated in Fig. 3.Conversely, as evidenced in the next Section, both matricesT (c) − DT (c) and T (p) generate intrablock and interblockself-interference, respectively. Consequently, it is meaningfulto rearrange the estimated symbols into

d = α[DT (c) +

(T (c) −DT (c)

)+ T (p)

]d + n. (23)

Finally, the last two steps (10)-(11) of the symbol-basedarchitecture lead successively to:

2) Radar channel estimation:

d = D−1d d (symbol removal). (24)

3) Range-Doppler map computation:

x = (FM ⊗ FHK)d (Fourier transform). (25)

Our present development is summarized in the followinglemma and flowchart of Fig. 4.

Lemma 2: The WCP-OFDM range-Doppler measurementresulting from the symbol-based receiver consists of

x = x(t) + x(ic) + x(ip) + x(n) (26)

where the right-hand side terms represent the target’s peak,target’s self-interference terms induced by the current andprevious block, and noise signal, respectively, such that

x(t) = αAg,g(l0, fd/L) FMed(fd)⊗ FHKer(l0) (27a)

x(ic) = α(FM ⊗ FHK)D−1

d

[T (c) −DT (c)

]d (27b)

x(ip) = α(FM ⊗ FHK)D−1

d T (p)d (27c)

x(n) = (FM ⊗ FHK)D−1

d

[IM ⊗ (FKP TDH

g )]n.

(27d)

Fig. 5 illustrates a typical range-Doppler map derived fromour low-complexity WCP-OFDM radar system. As in [35],we observe a loss on the target peak along with an increasednoise floor owing to the self-interference terms.

IV. STATISTICAL ANALYSIS OF THE SIGNAL

Herein we provide a statistical study of (27), with particularattention paid to the self-interference component. We formallydemonstrate i) the whiteness of the latter; ii) the expression ofthe post-processing SINR proposed in [36].

A. Statistical assumption about the constellation

In what follows, data symbols {ck,m} are assumed in-dependent and uniformly distributed according to a chosenconstellation, with zero-mean and power σ2

c . We also introduceσ2c−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 constel-lation with respect to the origin. This is actually fulfilled byusual constellations, e.g., phase-shift keying (PSK), quadratureamplitude modulation (QAM), amplitude-phase-shift keying(APSK).

6

[IM ⊗ (DgPFHK)]

dRadarchan-nel

s

n

αZs[IM ⊗ (FKP TDH

g )]r

D−1d

d(FM ⊗ FH

K)d x

WCP-OFDM linear transmitter WCP-OFDM linear receiver Symbolremoval

Range-doppler mapcomputation

Fig. 4. Flowchart of the low-complexity WCP-OFDM radar transceiver.

−0.4 −0.2 0 0.2 0.40

20

40

60

Normalized Doppler frequency (-)

Ran

gega

te(-

)

Expected target peak

0

10

20

(dB

)

Fig. 5. Range-Doppler map |x|. TFL-pulses, QPSK symbols, K = 64,M = 32, L/K = 12/8, σ2 = 1, σ2

c = 1, expected SNRth of 40 dB, asdefined 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-noisepower: 3.9 dB (theoretical value: 4.1 dB).

B. First and second order moments

First and second order moments of the range-Doppler mapcontributions are computed from (27). The target statisticalmean is simply

E{x(t)} = E{α}Ag,g(l0, fd/L) FMed(fd)⊗ FHKer(l0).

In addition, as d and n are independent and zero-mean thenE{x(n)} = 0. Finally, by noticing that D−1

d is independentfrom [T (c) − DT (c) ]d and T (p)d and using once again thezero-mean assumption of the symbols we obtain E{x(ic)} =E{x(ip)} = 0.

Similarly, using (27a), the target’s power in the range-Doppler map is given by

σ2t = E{x(t)Hx(t)} = E{|α|2}KM

∣∣∣∣Ag,g (l0, fdL)∣∣∣∣2 . (28)

It is worth noticing that the conventional integration gain KMendures a loss equal to |Ag,g(l0, fd/L)|2 as exemplified inFig. 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). Onthe one hand, symbols and noise samples were assumedindependent and E{nnH} = σ2ILM . Besides, with the pulse-shapes considered in this study (i.e., CP and TFL pulses), wehave

P TDHg DgP = K−1 ‖g‖2 IK

so that the post-processing noise covariance matrix reduces to

E{x(n)x(n)H} = σ2nIKM (29)

withσ2n = σ2σ2

c−1K−1 ‖g‖2 (30)

where we have used E{D−1d D−1H

d } = σ2c−1IKM . Noise

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 summa-rized in the following proposition.

Proposition 1: If M � 1 self-interference is white

E{x(ic)x(ic)H} = σ2icIKM

E{x(ip)x(ip)H} 'M�1

σ2ipIKM

(31)

with

σ2ic = E{|α|2}σ2

cσ2c−1

K−1∑k=1

∣∣∣∣Ag,g (l0, fdL +k

K

)∣∣∣∣2σ2ip = E{|α|2}σ2

cσ2c−1

K−1∑k=0

∣∣∣∣Ag,g (l0 − L, fdL +k

K

)∣∣∣∣2 .(32)

Furthermore, self-interference and noise terms are orthogonal

E{x(ic)x(ip)H} = E{x(ic)x(n)H} = E{x(ip)x(n)H} = 0KM .(33)

Proof is given in Appendix C.

C. Post-processing SINR

1) Single target scenario: Detection performance of a radarsystem is conventionally driven, to a large extent, by thepost-processing signal-to-noise ratio (SNR), e.g., [43]. In ourWCP-OFDM RadCom scenario, we rather define an SINRperformance metrics due to the self-interference phenomenon.This figure of merits accounts both for target peak loss andincreased noise floor. Assuming that the target is perfectlylocated in a range-Doppler bin, the latter is defined as

SINR ,E{|x(t)|2}

E{|x(n) + x(ic) + x(ip)|2}

where x denotes the signal in the target cell. As a consequenceof proposition 1, the expression boils down to

SINR =σ2t

σ2n + σ2

i

(34)

where σ2i , σ2

ic+ σ2

ip(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

7

−40 −30 −20 −10 0 10 20−20

0

20

40

E{|α|2}increases

INR (dB)

SIN

R(d

B)

TFL CP (l0, fd/L) = (10, 1/(2K))

TFL CP (l0, fd/L) = (60, 1/(8K))

Fig. 6. Post-processing SINR as a function of the INR (curves parametrized byE{|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, σ2 = 1, σ2

c = 1, α ∼ CN (0,E{|α|2}).

a simple SNR as considered so far in CP-OFDM RadComscenario [30]

SNRth =E{|α|2}K2M

σ2σ2c−1‖g‖2

. (35)

To provide more insight into the effect caused by self-interference, we depict in Fig. 6 the SINR as a functionof the INR (Interference-to-Noise-Ratio) while varying thetarget power E{|α|2}. The latter metrics is formally definedas INR , σ2

i /σ2n. Jointly, an asymptotic study leads to

SINR −→E{|α|2}→0

KM |Ag,g(l0, fd/L)|2

σ2cσ

2c−1W

INR

SINR −→E{|α|2}→∞

KM |Ag,g (l0, fd/L)|2

σ2cσ

2c−1W

with

W =K−1∑k=1

∣∣∣∣Ag,g (l0, fdL +k

K

)∣∣∣∣2+

K−1∑k=0

∣∣∣∣Ag,g (l0 − L, fdL +k

K

)∣∣∣∣2 .(36)

We thus observe two distinct modes: i) a linear regime whereboth SINR and INR augment with the target power; ii) asaturation of the SINR while the INR keeps growing withthe target power. The SINR actually saturates as soon as theself-interference power σ2

i becomes significant with respect tothe noise floor σ2

n. The saturation value strongly depends onthe range-Doppler shift of the target in combination with thepulse-shapes chosen as hinted by (36) and Fig. 3. For instancein Fig. 6, TFL-pulses distinctly outperform CP-pulses at low-range and high-velocity.

2) Multitarget scenario: The expression of the SINR (34)can be easily generalized to a multitarget scenario assumingstatistically independent targets with zero-mean amplitudes.Using the linearity of our WCP-OFDM radar transceiver, itis straightforward to show that the functional form of (34)

TABLE ITARGETS PARAMETERS

h 0 1 2 3 4 5 6

l0,h 20 5 8 55 30 45 60fd,hL

16K

112K

12K

112K

16K

14K

12K

SNRth,h (dB) 20 35 20 20 25 15 10

1 2 3 4 5 6 7−25

−20

−15

−10

−5

H10log

10(σ

2 i)

L/K = 12/8 L/K = 9/8

Fig. 7. Overall interference power σ2i for a varying number of targets H , with

different values of L/K. TFL-pulses, QPSK symbols, K = 64, M = 32,σ2c = 1, αh ∼ CN (0,E{|αh|2}) as specified in Tab. I.

remains unchanged albeit the definition of the self-interferencepower reformulated as

σ2i ,

H−1∑h=0

σ2i,h (37)

where H is the number of targets and ()h refers to thehth target. Obviously the more target in the radar scene,the stronger the self-interference power. However this growthpartly depends, in an intricate way, on both the waveform’s andthe targets’ parameters through Wh (36). This is exemplifiedin Fig. 7 for a specific target scene described in Table I. Itparticularly shows the increased resilience of the TFL pulseto self-interference for high L/K ratio.

3) SINR and detection performance trend: The SINR sat-uration effect observed in Section IV-C1 is unconventional.Indeed, even if the target power increases, its peak does notemerge more from the self-interference-plus-noise floor oncethe self-interference power dominates that of the thermal noise.This may be deleterious in terms of detection especially for ahigh-range and high-velocity target. This trend is aggravatedin multi-target scenario since, as discussed in Section IV-C2,target self-interference powers add up. Specifically, a weaktarget can be easily hidden under the self-interference-plus-noise floor thus disabling the use of conventional power-thresholding detectors. Note finally that the severity of theself-interference-induced loss depends largely on the targetscene itself and cannot thus be assessed beforehand in practicalscenarios.

8

TABLE IIPHYSICAL PARAMETERS. INSPIRED FROM THE AUTOMOTIVE

SCENARIO [29].

Carrier frequency Fc = 24 GHzBandwidth B = 93.1 MHz

Sampling period Ts = 10.7 nsRange resolution δr = 1.61 m

V. DETECTION PERFORMANCE WITH A CONVENTIONALDETECTOR

In this Section, we further quantify performance loss in-duced by the self-interference phenomenon in terms of de-tection probability (PD) in a conventional radar detector. Asexplained hereafter we focus only on a single target scenario.Fixed simulation parameters are summed up in Table II.

A. Self-interference-plus-noise distribution

To select an appropriate detector in the range-Doppler map,we first investigate the distribution of the self-interference-plus-noise component x(i) +x(n) defined in (26). On the onehand, thermal noise Gaussianity is preserved due to the lin-earity of the radar receiver, i.e., x(n) ∼ CNKM (0, σ2

nIKM ).On the other hand, given (27b)-(27c), the self-interferencesignal 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 assumea Swerling I type target, i.e., α ∼ CN (0,E{|α|2}). In thiscontext, the histogram of a single bin x(i)+x(n) shows that theinterference-plus-noise tends to be Laplacian at high INR� 1and Gaussian at low INR� 1 (cf. Fig. 8). More generally atlow INR, it seems reasonable to assume the Gaussianity onthe whole vector, viz

x(i) + x(n) ∼ CNKM

(0, (σ2

i + σ2n)IKM

). (38)

Otherwise, the pdf may have an intricate form (its study is outof the scope of this work). In the remainder of the paper, wethus restrict our analysis to low INR scenario by considering asingle-target in the radar scene (thereby avoiding accumulationof self-interference powers (37)) with a reasonable power.

B. CA-CFAR detection performance

Under the Gaussian assumption (38), a conventional Cell-Averaging Constant-False-Alarm-Rate (CA-CFAR) detectorcan be applied [44]. The theoretical CA-CFAR PD is thengiven by

Pd =

[1 +

P−1/Ns

fa − 1

1 + SINR

]−Ns

(39)

with Ns the number of training cells, Pfa the desired proba-bility of false alarm.

The PD is depicted in Fig. 9 for varying range l0, Dopplerfd and ratio L/K for both CP and TFL pulse-shapes. First,analytical and Monte-Carlo PD curves do match therebyadvocating for assumption (38) at low INR. Secondly, weclearly observe significant PD loss when either the target range

−40 −20 0 20 400

2

4

6

8·10−2

<{x(i) + x(n)}

Pdf

of<{x

(i)+x

(n)}

Gaussian fitLaplace fit

(a) INR ' 22 dB

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

<{x(i) + x(n)}

Pdf

of<{x

(i)+x

(n)}

Gaussian fitLaplace fit

(b) INR ' −30 dB

Fig. 8. Histograms of the real part of a self-interference-plus-noise bin fordifferent INR values (30, 000 runs). The bin is chosen far enough from thetarget cell, to avoid sidelobe effects. QPSK symbols, TFL pulses, σ2 = 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 dueto the loss on the target peak since at low INR noise floor isbarely increased. Finally, owing to its orthogonality the TFLpulse-shape should be preferred to CP-pulses in low-range andhigh-velocity scenarios (e.g., emergency braking in automotivecase). At higher ranges, increasing L/K is a way to achievebetter detection performance, at the cost of a reduced spectralefficiency.

VI. CONCLUSION

In this paper, a monostatic multicarrier radar has beenconsidered to simultaneously perform data transmission andenvironment sensing. Its receiver is built on an existing low-complexity symbol-based linear architecture, that practicallyresults in target peak loss and self-interference components inthe range-Doppler map. We proposed an analytical study ofthis undesirable twofold phenomenon in the context of WCP-OFDM, which is a generalization of conventional CP-OFDMto non-rectangular pulse-shapes. Specifically, we proved thatthe self-interference signal is white, enabling a closed-formexpression of the post-processing SINR. This figure of merit

9

0 200 400 600 800 1,0000

0.2

0.4

0.6

0.8

l0

Pd

CP L/K = 12/8 TFL L/K = 12/8

CP L/K = 9/8 TFL L/K = 9/8

(a) PD for fd = 0

0 0.1 0.2 0.3 0.40.7

0.75

0.8

0.85

fd = v/va

Pd

CP L/K = 12/8 TFL L/K = 12/8

CP L/K = 9/8 TFL L/K = 9/8

(b) PD for l0 = 0

Fig. 9. Probability of detection of the CA-CFAR as a function of l0 orfd at low INR (maximum value verifies INR < −30 dB). Comparisonbetween analytical (lines) (39) and Monte-Carlo (markers) results for varyingpulse-shapes and L/K ratios. Pfa = 1E − 6, Ns = 100, QPSK symbols,K = 1024, M = 256, σ2 = 1, σ2

c = 1, α ∼ CN (0,E{|α|2}) withE{|α|2} defined by (35) to achieve SNRth = 20 dB with TFL-pulses. Notethat the maximum velocity here achieves the limit of model (3); beyond, rangemigration can no longer be neglected.

is convenient to assess the induced performance degradation inthe radar receiver. We further evaluated the detrimental effectin terms of detection probability using the well-known CA-CFAR while detouring its domain of applicability (i.e., lowINR scenarios). Overall, TFL-pulses were shown to be moreresilient to self-interference over CP-pulses in low-range andhigh-velocity scenarios.

In any event, the study motivates the need to developnew detection schemes to deal with the self-interference phe-nomenon. Future work may include the design of dedicatedmitigation techniques such as successive interference cancel-lation. Presence of clutter should be also addressed.

APPENDIX AWCP-OFDM PULSE-SHAPES EXPRESSIONS

Analytical expressions of the considered pulse-shapes areas follows:• CP-pulses:

gCP[l] =

{√K/L if l ∈ IL

0 otherwise

gCP[l] =

{√L/K if l ∈ IL \ IL−K

0 otherwise

with \ the set difference symbol. The CP length is givenby L−K and should be chosen greater than l0 to avoidISI.

• TFL-pulses:

gTFL[l] = gTFL[l] =

cos θ[l] if l ∈ IL−K1 if l ∈ IK \ IL−Ksin θ[l] if l ∈ IL \ IK0 otherwise

where θ[l] is given in [45] through numerical optimiza-tion.

APPENDIX BSYMBOLS TRANSFER MATRICES

Herein we provide details of computation to obtain theexpression (22) of the transfer matrices T (p) and T (c) inlemma 1. First we focus on the derivation of T (c). Using (17b)and (21b), the matrix can rewritten as

T (c) = Ded(fd) ⊗B

withB =

[FKP TDH

g

]DfdL

l0L

[DgPFH

K

].

To simplify the expression of B, let denote fHk the kth columnof the Fourier matrix FK , i.e.,

FK = [fH0 , . . . ,fHK−1].

Note that the definition of fk can be easily extended to k ∈ Zsince, by periodicity, fk+K = fk. We have then

FKP TDHg =[

g[0]fT0 , . . . , g[K − 1]fTK−1, g[K]fT0 , . . . , g[L− 1]fTL−K−1

]∗DgPFH

K =[g[0]fT0 , . . . , g[K − 1]fTK−1, g[K]fT0 , . . . , g[L− 1]fTL−K−1

]Tand

DfdLl0LDgPFH

K =

01×K...

01×Kej2πfdl0/Lg[0]f0

...ej2πfd(L−1)/Lg[L− l0 − 1]fL−l0−1

.

10

Thus, we obtain

B =

L−l0−1∑l=0

g∗[l + l0]g[l]ej2πfd(l+l0)/LfHl+l0f l

=

L−1∑l=0

g∗[l]g[l − l0]ej2πfdl/LfHl f l−l0

where we have used in the last line the short-filter assump-tion (12). Using the definition of f l, we have[

fHl f l−l0

]k,k′

=1

Ke−j2πlk/Kej2π(l−l0)k′/K

=1

Kej2πl(k

′−k)/Ke−j2πl0k′/K

that yields

[B]k,k′ =e−j2πl0k

′/K

K

L−1∑l=0

g∗[l]g[l − l0]ej2πl

(fdL + k′−k

K

)

= e−j2πl0k′/KAg,g

(l0,

fdL

+k′ − kK

).

We thus proved that B = A(l0)g,g Der(l0) and thus (22b).

The derivation of T (p) is rigorously similar, once noticed thatLLML

[IM ⊗ (DgPFH

K)]

=[IM ⊗ (DgPFH

K)]LKKM .

APPENDIX CSELF-INTERFERENCE SECOND-ORDER STATISTICS

In what follows, we show that x(ic) defined in (27b) iswhite. Since the same reasoning holds for x(ip) when M � 1,we can conclude with (31). First, we have

E{x(ic)x(ic)H} = (FM ⊗FHK)E{d(ic)

d(ic)H}(FM ⊗FH

K)H

with

d(ic)

= αD−1d (T (c) −DT (c))d.

Let us cut d(ic) into M sub-vectors of length K and denoted

(ic)m the mth sub-vector, so that

d(ic)

= α

d

(ic)0...

d(ic)M−1

.Given the block diagonal structure of T (c) − DT (c) andsince the dk,m (or indifferently, the ck,m) were assumedindependent, the d

(ic)m are also independent from each other.

As a result, E{d(ic)d

(ic)H} is at least block diagonal whichmeans that the computation of E{d(ic)

d(ic)H} is reduced to

that of E{d(ic)m d

(ic)H

m }. Since we have for i ∈ IK

[d(ic)m ]i = ej2πfdm

K−1∑k=0k 6=i

dk,mdi,m

Ag,g

(l0,

fdL

+k − iK

)e−j2πl0

kK

then [d

(ic)m d

(ic)H

m

]i,j

=

K−1∑k=0k 6=i

K−1∑k′=0k′ 6=j

dk,md∗k′,m

di,md∗j,me−j2πl0

(k−k′)K

×Ag,g(l0,

fdL

+k − iK

)×A∗g,g

(l0,

fdL

+k′ − jK

).

Therefore:• For j 6= i, since the constellation was assumed

symmetric to the origin in Section IV-A, we have

E{[

d(ic)m d

(ic)H

m

]i,j

}= 0.

• For j = i then[d

(ic)m d

(ic)H

m

]i,i

=

K−1∑k=0k 6=i

K−1∑k′=0k′ 6=i

dk,md∗k′,m

|di,m|2e−j2πl0

(k−k′)K

×Ag,g(l0,

fdL

+k − iK

)×A∗g,g

(l0,

fdL

+k′ − iK

)with E{dk,md∗k′,m} = σ2

cδk,k′ so that

E{[

d(ic)m d

(ic)H

m

]i,i

}=

K−1∑k=0k 6=i

E{|dk,m|2

|di,m|2

} ∣∣∣∣Ag,g (l0, fdL +k − iK

)∣∣∣∣2

= σ2cσ

2c−1

K−1∑k=0k 6=i

∣∣∣∣Ag,g (l0, fdL +k − iK

)∣∣∣∣2 .We thus demonstrated that

E{d(ic)d

(ic)H} = σ2icIKM

and therefore

E{x(ic)x(ic)H} = σ2icIKM

where

σ2ic = E{|α|2}σ2

cσ2c−1

K−1∑k=1

∣∣∣∣Ag,g (l0, fdL +k

K

)∣∣∣∣2 .As mentioned, the derivation of E{x(ip)x(ip)H} is quite

similar to that of E{x(ic)x(ic)H}. If keeping the same nota-tions, the main difference is that d(ip)

0 is not defined. This sideeffect can potentially be avoided by ignoring the first receivedOFDM symbol. Otherwise, we obtain

E{x(ip)x(ip)H} =

σ2ipM

−1

(M − 1)IK −IK . . . −IK

−IK. . .

......

. . . −IK−IK . . . −IK (M − 1)IK

11

where we denote

σ2ip = E{|α|2}σ2

cσ2c−1

K−1∑k=0

∣∣∣∣Ag,g (l0 − L, fdL +k

K

)∣∣∣∣2which consequently leads to

E{x(ip)x(ip)H} 'M�1

σ2ipIKM .

The same reasoning is applied to prove (33).

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Steven Mercier (S’17) was born in Fontenay-aux-Roses, France, in September 1994. He received theengineer and master degrees in signal processingfrom Grenoble INP in 2017 and is now preparinga Ph.D. degree on waveform sharing for radar andcommunications at ISAE-SUPAERO, University ofToulouse, France.

Damien Roque (M’15) was born in Chambery,France, in July 1986. He received the Ph.D. degreefrom the University of Grenoble in 2012 and theDipl.Ing. in telecommunications from the Ecole Na-tionale Superieure des Telecommunications (ENST)de Bretagne in 2009. From 2009 to 2013, he workedat the lab Grenoble Images Paroles Signal Automa-tique (GIPSA-lab), Grenoble, France. Since 2013,he is an associate professor at the Institut Superieurde l’Aeronautique et de l’Espace (ISAE-SUPAERO),University of Toulouse, France. His research inter-

ests are in the area of spectrally efficient waveforms, including faster-than-Nyquist signaling and multicarrier modulations.

Stephanie Bidon (M’08) received the engineer de-gree in aeronautics and the master degree in signalprocessing from ENSICA, Toulouse, in 2004 and2005 respectively. She obtained the Ph.D. degreeand the Habilitation a Diriger des Recherches insignal processing from INP, Toulouse, in 2008 and2015 respectively. She is currently with the Depart-ment of Electronics, Optronics and Signal at ISAE-SUPAERO, Toulouse, as a professor. Her researchinterests include digital signal processing particu-larly with application to radar systems.