Aspects of Delay-Doppler Filtering in OFDM Passive Radar

Aspects of Delay-Doppler Filtering in OFDM Passive Radar Stephen Searle Daniel Gustainis, Brendan Hennessy, Robert Young University of South Australia Defence Science and Technology Group, Australia Abstract—Delay-Doppler filtering enables the isolation of radar Recently emerging studies have exploited the OFDM CP returns from a designated region of the ambiguity plane. Such for purposes of clutter modelling and cancellation. Noting returns can be cancelled from the surveillance signal, lowering that in the frequency domain a time shift would manifest the ambiguity floor and reducing demands on dynamic range. Modern passive radar systems frequently use OFDM-based illu- as a frequency-dependent scalar phasor on each carrier, the minators of opportunity, and recent studies have demonstrated ZDC is estimated by projecting the observed time-history of how the structure of OFDM signals can be exploited to perform Fourier coefficients at each carrier onto the recovered symbols delay-Doppler filtering in the frequency domain. In this study [6]. Further noting that in the presence of small Doppler several aspects of one such algorithm are investigated: the shift the scalar phasor would advance over time, the method ability to cancel returns when delay is not a whole number of samples; the ability to conserve computation by approximating a was adapted to successfully estimate clutter exhibiting small projection matrix; the effect of returns having delays exceeding movements [12]. These methods could broadly be described the OFDM guard interval; and the application of the method to as Doppler filters, selecting and cancelling all returns having extended targets. The consideration of these aspects demonstrates near-zero Doppler shift. In response to a need to isolate high- the efficacy of the method in situations common to passive radar. power returns of limited extent and nonzero Doppler shift, the method was further extended to operate on any arbitrary region of the delay-Doppler plane [13]. I. INTRODUCTION The purpose of this study is to revisit the method proposed Many modern communication signals employ Orthogonal in [13] and expound upon certain of its aspects. The ability Frequency Division Multiplexing (OFDM) modulation. Such to subsample in delay, not easily obtained with a discrete tap- signals are typically of large bandwidth and have favourable delay Wiener filter, can be achieved with the present method. autocorrelation properties. The prevalence and properties of An approximation is suggested which can reduce the required OFDM signals makes them ideal Illuminators Of Opportunity amount of computation with minimal cost to performance. The (IOOs) for Passive Bistatic Radar (PBR) [1]. Recent research long-delay case is investigated, in which the delay of a target in the field of passive radar has exploited many diverse OFDM- return exceeds the duration of the CP, thereby invalidating an modulated signals, including DAB [2], DVB-T [3], LTE [4], assumption of the present method. Finally we demonstrate the WiFi [5], and DRM [6]. ability of the method to isolate extended target returns. An ongoing problem in PBR is the presence of Direct Path Interference (DPI) and other Zero Doppler Clutter (ZDC) in II. REVIEW OF ECA+ METHOD the surveillance signal. These unwanted returns are typically The evolution of Extensive Cancellation Algorithms (ECA) of high power despite attempts to suppress them [1], [6]. culminating in the ECA+ method for cancellation of arbitrary Their presence increases dynamic range requirements on the regions of delay-Doppler in OFDM signals is described in [13] receiver, and they can mask the presence of weak targets in and its references. This section summarises the ECA+ method. the signal. Any DPI and ZDC in the digitised signal is often handled by cancellation [7]. A Wiener filter provides optimal A. Signal Model cancellation but is computationally unwieldy. In practice nu- An OFDM signal r(t) is formed by the concatenation of merical approximations to the Wiener filter or adaptive filters OFDM symbols r1(t); r2(t); r3(t) ::: are used [8], [9]. J Each sequential OFDM symbol is extended by prepending X r(t) = rj(t − (j − 1)Te) (1) it with redundant data, the Cyclic Prefix (CP). This is done j=1 in order to mitigate the effects of multipath, as it causes the mutual orthogonality of the OFDM subcarriers to be An OFDM symbol is comprised of K carriers, orthogonal over preserved when filtered with a time-delayed copy of the the duration Tu, each modulated by a datum ckj drawn from signal. This feature has proven to be beneficial for PBR, as a constellation. it allows the orthogonality of subcarriers to be exploited in K−1 2π X i fsk(t−Tg ) t 1 rj(t) = ckje Nu Π( − ) (2) ambiguity computations, resulting in delay-Doppler surfaces Te 2 with extremely low pedestals [10], [11]. k=0 The full symbol duration is Te = Tu + Tg where Tg is the The columns of V are the phase-histories over the J symbol 1 1 CP duration. Π(·) is the rect function with support (− 2 ; 2 ). periods due to Doppler shift for each of the M components in The useful portion (post-CP) of the jth symbol period begins the signal. Similarly the columns of D are the phase-responses at time uj, of the K carriers for the delays of each of the M components. u = (j − 1)T + T (3) j e g B. ECA+ Method and thus the nth useful sample of the jth symbol is Delay in a returned signal weights each OFDM subcarrier rj[n] = r(uj + nTs); n = 0; 1;:::Nu − 1 (4) by a phase which increases linearly with frequency, and Consider x(t), a copy of r(t) weighted by a, delayed by τ Doppler shift weights each OFDM symbol by a phase which seconds and Doppler shifted by ν Hz, increases with time. The ECA+ method [13] forms clutter subspaces in the frequency domain from phasors corresponding to x(t) = ar(t − τ)ei2πνt (5) delays and Doppler shifts of interest. The surveillance signal Note that with short delays (τ < Te), the pth symbol period is projected into these subspaces to estimate the clutter which of x(t) will contain elements of two consecutive transmitted is then cancelled. symbols, rp(t) and rp−1(t). The samples taken from x(t) in It is assumed that the modulation symbols C are known, or the useful portion of the pth symbol period are can be estimated perfectly from a reference signal. This data is removed from the surveillance signal in the frequency domain xp[n] = x(up + nTs) (6) to produce Z which is the effects of delay and Doppler shift J K−1 2π i2πν(up+nTs) X X i fsk((p−j)Te+nTs−τ) on the components: = ae ckje Nu j=1 k=0 Zkj = Skj=ckj (18) × Π( (p−j)Te+nTs−τ+Tg − 1 ) Te 2 (7) Denote Doppler frequencies of interest as η1; : : : ; ηL, dis- tinct from the frequencies ν ; : : : ; ν actually in the signal. The Discrete Fourier Transform (DFT) of this is 1 M The corresponding phase-history vectors can be generated via Nu−1 X −i 2π `n (16, setting ν = ηi), forming a Doppler subspace matrix Q. Xp[`] = xp[n]e Nu (8) From this a projection matrix PQ can be formed. The data- n=0 J K−1 free signal is filtered in the frequency domain by multiplication 2π X X i fsk((p−j)Te−τ+Nuupν) with PQ, preserving components which correlate with the = a ckje Nu ρ (9) Doppler vectors, j=1 k=0 i2πη`Te(j−1) where Qj` = e (19) H −1 H Nu−1 P = Q(Q Q) Q (20) 2π Q X (p−j)Te+nTs−τ+Tg 1 i ((k−`)n+T νn) ρ = Π( − )e Nu u Te 2 YQ = ZPQ (21) n=0 (10) Similarly, for a set of delays of interest, corresponding phase- In the event that τ < Tg there will be no Inter-Symbol history vectors can be generated via (15) to form a delay Interference (ISI); that is, only contributions from symbol j subspace matrix W. A projection matrix PW formed from will be present in the pth period’s DFT. In this case Π(·) this is used to filter the data-free signal, preserving components Tu evaluates to 1 and, providing that ν, we have ρ ≈ Nu Nu which correlate with the delay vectors. for k = ` and 0 otherwise. The DFT then simplifies to 2π −i fskd` −i 2π f kτ i 2π N u ν Wk` = e Nu (22) X [k] = N ac e Nu s e Nu u p (11) p u pk H −1 H PW = W(W W) W (23) Given a signal comprising M returns, each with parameters YW = PWZ (24) a~m, τm and νm, and τm < Tg 8m, then the Fourier bin history of the signal over time can be written One can multiply Z by both PW and PQ to project the signal S = C DAVT (12) into a specific region of delay and Doppler. where Y = PWZPQ (25) Skj = Xj[k] (13) The filtered signal can be reconstructed from Y, or the filtered Ckj = ckj (14) component can be cancelled from the signal by reconstruction 2π −i fskτm with Z − Y: Dkm = e Nu (15) i2πνm(p−1)Te Vjm = e (16) Sfiltered = Y C (26) A = diag(~a1;:::; a~M ) (17) Scancelled = (Z − Y) C (27) 120 40 120 40 100 30 100 30 20 20 80 80 10 10 60 60 0 0 40 40 -10 -10 Doppler (Hz) Doppler (Hz) 20 20 -20 -20 0 -30 0 -30 -20 -40 -20 -40 60 70 80 90 100 110 120 130 60 70 80 90 100 110 120 130 delay (samples) delay (samples) Fig.

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