Signal Extraction Using Compressed Sensing for Passive Radar with OFDM Signals Christian R

1 Signal Extraction Using Compressed Sensing for Passive Radar with OFDM Signals Christian R. Berger, Student Member, IEEE, Shengli Zhou, Member, IEEE, and Peter Willett, Fellow, IEEE

Abstract—Passive radar is a concept where possibly multiple 10 non-cooperative illuminators are used in a multi-static setup. A freely available signal, like radio or television, is decoded 8 and used to identify moving airborne targets based on their Doppler shift. New digital signals, like Digital Audio/Video 6 Broadcast (DAB/DVB), are excellent candidates for this scheme, 4 as they are widely available, can be easily decoded, and employ orthogonal frequency division multiplex (OFDM), a multicarrier 2 transmission scheme based on channel equalization in the frequency domain using the Fast Fourier Transform (FFT). After 0 successfully decoding the digital broadcast, the channel estimates can be used to estimate targets’ bi-static range and range-rate by y−axis [km] −2 separating different multi-path components by their delay and −4 Doppler shift. While previous schemes have simply projected receiver available measurements onto possible Doppler shifts, we employ −6 Compressed Sensing, a type of sparse estimation. This way we illuminators can enhance separation between targets, and by-pass additional −8 target signal processing necessary to determine the actual target within bistatic range −10 a “blotch” of signal energy smeared across different delays and −10 −5 0 5 10 Doppler frequencies. x−axis [km]

Fig. 1. In passive radar, illuminators of opportunity are used in a multi-static I.INTRODUCTION setup; weak target signatures can be extracted from the dominating “direct In passive radar, one or several illuminators of opportunity blast” radio/television signal based on their Doppler frequency, rendering bi- are used in a multi-static setting to detect and track airborne static range and range-rate information. targets. The general multi-static setup, see Fig. 1, is modified in the sense that we still measure bi-static range and range- rate, but the illuminators (senders) are not only dislocated from Since terrestrial radio/television signals were dominantly the receiver, but are also non-cooperative. of analog modulation type, signal extraction of precise tar- Although the concept of passive radar has been known get information was extremely difficult. With the advent of for a long time (and has been used extensively by the mili- widespread digital radio/television broadcasting, i.e., Digital tary/intelligence communities), not much was published in the Audio/Video Broadcast (DAB/DVB), a new generation of sig- open literature until the last decade [1]. Now, technological ad- nal processing tools can be utilized to extract target informa- vances have made this topic interesting to a broader audience tion. This comes with the following advantages: i) both DAB and DVB employ orthogonal frequency division multiplexing [2], [3], [4], [5], mainly due to the widespread availability of 1 high-performance signal processing equipment (see detailed (OFDM) as their modulation scheme ; ii) the digital signal discussion in [2]). can be easily decoded and then treated as a known signal for The concept of passive radar has appeal in many ways, e.g., target signal extraction; iii) DAB and DVB are both wideband multi-static benefits: it brings new electronic countermeasures signals, giving ample frequency diversity; and iv) terrestrial challenges and the receiver operates in an entirely passive fash- broadcasting now operates in a single frequency network ion. Furthermore, since the illuminators seem unsuspecting, (SFN), i.e., all illuminators transmit in the same frequency the surveillance is in a sense covert and operation in the ra- band and their signals are combined at the receiver using the dio/television VHF/UHF frequency bands needs no frequency special OFDM structure – this results in each target being allocation, gives frequency diversity and has the potential to illuminated several times during each scan, while operating counter stealth efforts and to detect low-flying targets beyond just in a single frequency band. the horizon [6], [7]. While the previously mentioned advantages are encour- aging, they also lead to new implementation challenges. In Manuscript received February 25, 2008. This work was supported by ONR OFDM, usually all arriving paths are assumed to have the grant N00014-07-1-0429. C. Berger, S. Zhou and P. Willett are with the Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Way 1OFDM is a multi-carrier scheme that uses the efficient Fast Fourier U-2157, Storrs, Connecticut 06269, USA (email: [email protected]; Transform (FFT) for frequency domain channel equalization; this greatly [email protected]; [email protected]). simplifies resolving multi-path arrivals (like the target reflections) 2 same Doppler frequency, making estimation of multipath and the corresponding passband signal is arrivals a one-dimensional problem – now with additional varying Doppler frequencies, a two-dimensional plane has to x˜(t)= Re ej2πfct s[m]ej2πm∆ftq(t) (3a) be estimated. Also, the SFN generates one bi-static range and ( m∈S ) range-rate measurements per illuminator, per target at each X scan; this offers more target information, but it introduces = Re s[m]ej2πfmtq(t) , (3b) an additional association problem since we are not able to (m∈S ) X differentiate between the various illuminators. The association where s[m] is the transmitted symbol on the mth subcarrier. problem is not addressed in this paper, but can be handled The channel impulse response for a multipath radio channel within a tracking algorithm, e.g., see [5], [8]. can be described by In this paper, we want to focus on the challenge of extracting target information from the OFDM signal. To differentiate the c(τ,t)= Apδ (τ − τp(t)) , (4) p weak target signatures, it is necessary to match the received X signal to tentative range/range-rate combinations, correspond- where Ap is the path amplitude and τp(t) is the time-varying ing to target detections. This is challenging, as to have a path delay caused by a moving sender, receiver or target. In sufficient resolution in both range and range-rate, a very this setup we assume both the senders and the receiver to be large number of combinations, many of which are highly stationary, so all direct-blast arrivals have zero Doppler. We correlated, have to be evaluated. This can be posed as a sparse assume that for the duration of a number of OFDM symbols estimation problem, as the number of tentative range/range- the Doppler-causing speeds can be assumed to be constant, rate combinations is large compared to the number of actual τ (t) ≈ τ − a t, (5) target signatures and we can arrive at a linear formulation. p p p Efficient sparse estimation approaches are being investigated where ap =r ˙p/c and r˙p is the range-rate of the pth path and c under the name Compressed Sensing (CS) [9], [10], [11], [12], the speed of light. We sort the paths by their Doppler causing where a signal is disected into a discrete representation using speeds, where if ap1 = ap2 , then p1,p2 ∈ Pk, leading to a a large dictionary of non-orthogonal signals. Using this sparse modified formulation, estimation approach, we can extract the signal information c(τ,t)= Apδ (τ − τp + akt) . (6) resulting in range/range-rate measurements, which can be used k p∈Pk in a tracking algorithm. X X When the passband signal in (3b) goes through the channel We define the following notation: vectors and matrices are described in (6), we receive: denoted as lower and upper case bold letters respectively, e.g., x, A; the conjugate complex, transpose and hermitian are x∗, xT and xH respectively. y˜(t)= Re s[m]ej2πfm(1+ak)t ( ∈S The paper has the following structure: In Section II we Xk mX introduce the OFDM signal model; in Section III we relate −j2πfmτp this to the target signatures and bring it into a linear problem × Apq (1 + ak)t − τp e +w ˜(t), (7) p∈P ) formulation in terms of unknown range/range-rate received Xk power; in Section IV we go over a sparse estimation algorithm where w˜(t) is additive noise. Define τmax = maxp τp, which and its efficient implementation; in Section V we look at some is usually less than the CP length Tcp. Using the definition of numerical results and finally conclude in Section VI. q(t) in (1), we obtain

II. OFDM SIGNAL MODEL (k) j2πfm(1+ak )t In OFDM transmission, the signal is divided into blocks of y˜(t)= Re Hm s[m]e +w ˜(t) ( k m∈S ) Ns symbols, which are modulated onto the subcarriers. One X X OFDM symbol is of duration T , then the subcarrier spacing t ∈ Tc ≈ [−(Tcp − τmax),T ] , (8) is ∆f = 1/T and the bandwidth is B = Ns∆f. Let fc where we define the channel transfer functions denote the carrier frequency, and f = f + m∆f denote m c −j2πfτp the frequency for the mth subcarrier in the passband, where Ck(f)= Ape (9) p∈Pk m ∈ S = {−Ns/2,...,Ns/2 − 1}. To handle the multi- X path propagation in radio channels and avoid inter-symbol- and the frequency response on the mth subcarrier for each k interference (ISI), a cyclic prefix (CP) is used. Let Tcp denote as (k) the length of the CP, and define a rectangular window of length Hm = Ck(fm). (10) Tcp + T0 as Converting the passband signal y˜(t) to baseband, such that j2πf t 1 t ∈ [−T ,T ], y˜(t)= Re y(t)e c , we have: q(t)= cp 0 (1) 0 otherwise. (k) j2π(m∆f+akfm)t ( y(t)= Hm s [m]e + w(t) (11a) ∈S One symbol in baseband can be written as Xk mX j2πak fct (k) j2πm∆ft x(t)= s[m]ej2πm∆ftq(t) (2) ≈ e Hm s[m]e + w(t), (11b) ∈S ∈S k m mX X X 3

for t ∈ Tc, where we used the narrowband approximated In radio applications like DAB/DVB-T, the maximum nor- akfm ≈ akfc and w(t) is the noise at baseband. malized Doppler ǫk is limited by the range-rate and thereby the As expected for CP-OFDM, we observe a cyclic convolution maximum speed. Assuming no targets are faster than 300 m/s, between the signal and the channel in the specified interval, a carrier frequency of one GHz and a symbol duration of where each subcarrier is only multiplied by the corresponding T ≈ 0.1 ms, the maximum normalized Doppler is, frequency response. The narrowband approximation is very 300 9 −4 (19) accurate in radio channels, since (maxm fm−minm fm)/fc = ǫk = akfcT =r ˙k/c · fcT < 8 · 10 ∗ 10 =0.1 −3 3 · 10 B/fc < 10 , as the bandwidth is at most in the MHz, while Γ I the carrier is on the order of GHz. This keeps the matrix (ǫk) close to the identity matrix Ns ; The receiver correlates the received waveform with each additionally the paths reflected off targets are attenuated by subcarrier: about 10 dB with respect to the paths associated with the line T 1 j2πn∆ft of sight or reflections off large stationary objects like hills or yn = e y(t) dt (12) buildings. Therefore, we approximate the marix Γ(ǫ ) ≈ I . T 0 k Ns Z This leads to the following modified signal model, Inserting (11b) and carrying out the integration, we obtain ′ j2π(i−1)ǫk T /T H (k) (k) yi = e diag(si)Fk ck + w. (20) yn = Hm s[m]ρn,m + wn (13) k k m∈S X X X Further we assume a certain subset of each OFDM symbol s where we define: i is known. This can be either pilot symbols or data symbols sin(π[m − n + ǫ ]) (k) k jπ(m−n+ǫk) fed back from the decoding process, since we can assume ρn,m = · e (14) π[m − n + ǫk] the broadcast signal is using sufficient error-correcting codes and ǫk = akfcT . Rewriting (13) in vector form, we obtain: to correctly decode the data. The elements of the vectors yi corresponding to known symbols are compensated, while other y = Γ(ǫk)diag(s)hk + w, (15) components are dropped. This can be represented via a selector k X matrix S, which is a diagonal matrix with [S]nn = 1 if [s]n (k) is known; where [y]n = yn, [Γ(ǫk)]nm = ρn,m, [s]m = s[m], (k) − ′ [hk]m = Hm and [w]n = wn. Therefore we observe j2π(i 1)ǫkT /T H y¯i = e SFk ck + w¯ , (21) a linear superposition of the different hk, with dictionaries k Γ(ǫ )diag(s) and the h contain the information about the X k k where w¯ is a noise vector of reduced size. targets we want to estimate. When observing a series of packets, the phase shifts due to the Doppler frequencies accumulate, c.f. (11b); so the ith B. Sparse Estimation Problem Formulation packet would have modified dictionaries as follows, To arrive at a systematic problem formulation, we will esti- − ′ mate tentative c at a sampled grid of combinations between y = ej2π(i 1)ak fcT Γ(ǫ )diag(s )h + w, (16) p i k i k τ and ǫ. Since we can assume the CP is of sufficient length, Xk ′ we get, where T = T + Tcp is the length of one OFDM symbol. τp ∈{0, ∆τ, 2∆τ,...,Tcp} (22) Using the definition of the frequency response, where ∆τ is usually chosen as a fraction of the sampling time (k) −j2π(m∆f+fc)τp H = Ape (17a) m 1/(αB) for α =1, 2,... and there are about αNcp = αBTcp p∈Pk X possible delays (Ncp is the length of the CP in samples). −j2πmτp/T = cpe , (17b) Furthermore, defining a maximum normalized Doppler ǫmax,

p∈Pk X ǫk ∈ {−ǫmax, −ǫmax + ∆ǫ,...,ǫmax} , (23) −j2πfcτp we deﬁne the complex target returns cp = Ape , which also include the direct signal propagation. Putting this into a and the Doppler spacing such that 2ǫmax/(K − 1)=∆ǫ. With T vector format, we have, this, all matrices Fk = T F are equal, with the DFT matrix j2πm(n−1)/αNs [Fk]nm = e . This can be easily implemented H hk = Fk ck, (18) via the FFT operation by inserting additional zero elements

j2πmτn/T where needed, which we express via the zero insertion matrix with [Fk]nm = e , where τn is the nth element of Pk T. and Fk is of dimension |Pk|× Ns. Finally the signal model, using i = 1,...,I OFDM symbols, is given by: III. SIGNAL EXTRACTION FOR DAB/DVB-T y¯ SFH T · · · SFH T w¯ A. Narrowband Radio Model 1 c 1 y¯ ν SFH T · · · ν SFH T 1 w¯ We want to detect energy with non-zero Doppler frequen- 2 1 K . 2  .  =  . .  . +  .  −j2πfcτp . . .  .  . cies, i.e., estimate the complex amplitudes cp = Ape . . . .    I−1 I−1  cK   belonging to some Pk. Practically, we will later sample both y¯  ν SFH T · · · ν SFH T   w¯   I   1 K     I  τ and ǫ on a grid and estimate which grid points are non-zero.      (24) 4

′ i−1 j2π(i−1)ǫk T /T where we define νk = e to abbreviate nota- where ak is the kth column of A. Next bi is added to the set tion. This is a linear system of the general form Ax+w = y, Ii = Ii−1 ∪ bi, and the new estimate of the non-zero entries where we want to detect non-zero entries of the vector x. of x is: −1 xˆ = arg min |y − A x |2 = AH A AH y (28) i x Ii i Ii Ii Ii C. Solution via FFT based Projection i Current practice simply solves the problem by taking the AIi is the matrix consisting of all columns ak with k ∈ Ii, following approach: the solution to y = Ax+w is simply the accordingly xî has only i elements. The new residual vector projection of the measurements onto all possible solution, xˆ = has to be calculated every time as AH y (equivalent to the correlation), which can be efficiently r = r − A xˆ , (29) implemented via FFT operation. The dimensions of A are i 0 Ii i 2 (I · ρNs) × (K · αNcp), where ρ is the ratio of ones to zeros and the algorithm stops when |ri| <ε. on the diagonal of matrix S. Theoretically, if all data is known, S = I , only K = I tentative ǫ are chosen with the right Ns k B. Implementation spacing, using Ncp = Ns and α =1, A is invertible and even hermitian. In this case the projection is optimal and can be To minimize complexity, some operations can be combined, efficiently implemented. while other simplifications are based on the specific signal In practice signals have zero-tones, so even after data decod- model, i.e., structure of A in (24), where we can take advan- tage of efficient implementations via FFT. ing S 6= INs . Also to get a sufficient Doppler scale resolution, a large I is necessary, this increases complexity and conflicts We define the projection vector after the ith iteration, with the assumption that the channel and target are constant H H pi = A (y − yî)= Ay − A AIi xî (30) during this time-period. This leads to the target signature being H H additionally “smeared” over several range/range-rate bins, as and accordingly p0 = A y. Also the matrix A A can be the target moves during one measurement cycle. computed analytically based on (24), then either be stored in Instead, we would like to pose (24) as a sparse estimation memory or evaluated as needed. problem, where A is a “fat” matrix, i.e., (I·ρNs) < (K·αNcp), One iteration of the algorithms can be implemented in the and therefore there are many choices of x how to explain following steps: the observations y. One possibility, e.g., would be the least bi = arg max |[pi−1]k| (31) squares solution, finding an x that satifies the equations and k∈ /Ii−1 − has minimum energy. This leads to many small, noise-like H 1 xî = A AI (pi−1) (32) observations in x – instead we want to pose the problem as Ii i Ii H finding a solution x with the fewest possible non-zero entries. pi = p 0 − A AIi xî (33) This is the focus of Compressed Sensing (CS) [10], [11], |r |2 = |r |2 − xˆH (p ) (34) i 0 i 0 Ii [12], which has been studied recently in many fields of signal Comments: (31) takes K·αNcp operations; in (32) (p − ) are processing. i 1 Ii the elements of the projection vector with indices in Ii and the matrix inversion can be accomplised via Gaussian elimination IV. ORTHOGONAL MATCHING PURSUIT with complexity i2; updating the projection vector in (33) takes A. Overview of the Algorithm i · KαNcp operations and (34) was simplified using the fact Orthogonal Matching Pursuit (OMP) is a greedy algorithm that xî is the least-squares solution when limiting to Ii, taking to solve the sparse estimation problem [9]. Given the obser- i +1 operations. Therefore the ith iteration is of complexity 2 H vations (i + 1)K · αNcp + i + i +1, assuming A A is pre-computed y = Ax + w, (25) and stored. The highest complexity operation is therefore calculating with x the desired unknown, w the noise and accordingly a p = AH y, which potentially takes (I · ρN ) · (K · αN ) signal-to-noise ratio (SNR) of y 2 w 2 . We 0 s cp E |[ ]k| /E |[ ]k| operations. This can be reduced by using the special structure want to solve for the unknown x in the following sense, of A, by taking y in parts, performing FFT operations and then i−1 min card(x), s.t. |y − Ax|2 <ε (26) adding them weighted with the respective ν . This reduces x k the complexity to I · Ns log2(Ns)+ KαNcpI. where “card” describes the cardinality operator, i.e., counting the number of non-zero entries in its argument x, and ε is V. NUMERICAL EXAMPLES chosen depending on the SNR. The algorithm goes through the following steps, starting A. Setup with x = 0, r0 = y, the current fitting error of the We study a simple scenario of a single target moving on observations, I0 = ∅ the set of non-zero elements of x: At the a straight line trajectory, while being illuminated by three ith iteration, we choose a new index of a non-zero element of stations, see Fig. 2. The signal parameters are a bandwidth x, bi, as: of B = 4 MHz, using Ns = 1024 subcarriers, one quarter H 2 |ak ri−1| of which are deactivated as guard bands or zero tones, all bi = arg max H , (27) ∈ −1 other symbols are assumed to be known. The cyclic prefix k /Ii ak ak 5

10 22 8 ground truth 20 start point 6 blind detections 4 18 zone

2 16 0 14

y−axis [km] −2 range [km] 12 −4 receiver −6 10 illuminators target −8 8 trajectory −10 −10 −5 0 5 10 6 −400 −300 −200 −100 0 100 200 300 x−axis [km] range−rate [m/s]

Fig. 2. Overview of the simulation scenario; the target moves on a straight Fig. 3. Simultion results; the ground-truth in terms of range and range-rate line trajectory over a course of 45 s, being illuminated by three different are the lines, the superposition of all detections over the time interval are the stations. red markers. is one quarter of the symbol duration Ncp = 256, leading 26 ′ to a total of T = 1.25 · Ns/B = 0.32 ms. The carrier blurring ground truth frequency is 1 GHz and we observe I = 12 OFDM symbols. 24 start point widened detections All stations are assumed to broadcast in the same frequency 22 blind zone band, as defined for the single frequency network (SFN). This leads to the single target leaving three signatures, which still 20 have to be associated to the correct direct signal. Although the 18 positions of the stations and the receiver are generally assumed 16 sinc to be known, it might still be necessary to estimate the arrival sidelobes of the direct signal. These issues are not addressed here. range [km] 14 The broadcast signal is modelled as multipath, where each 12 path strength Ap in (4) is Rayleigh distributed and also the targets are similarly Swerling II targets. The direct arrival of 10 each station has power inversely proportional to the distance 8 squared, and is trailed by multipath arrivals that attenuate with increasing delay. The signal to noise ratio between the sum of 6 −500 0 500 all direct arrivals and the additive noise is 10 dB, while the range−rate [m/s] target signatures are 10 dB weaker than a single direct arrival path of the signal, but about 20 dB belwo the sum of all Fig. 4. Simultion results using projection via FFT; the superposition of all multipath arrivals. detections over the time interval are shown, we see that for few packets, I = 12 For simulation purposes, we generate the signal based on , adjacent range/range-rate bins are highly correlated leading to “smearing”, i.e., multiple adjacent detections. (21), were we assume the narrowband model, but evaluate the Doppler frequencies in continuous values, while our estimator assumes evenly sampled values. The target is observed over a time-frame of 45 s; moving at a speed of 182 m/s it covers a parameter ε in (26) or by applying an additional thresholding distance of about 8.2 km during this time interval. Although after the algorithm. The results of our algorithm are in Fig. 3, where we superimpose all detections over the 45 s period (90 using I = 12 OFDM symbols we could theoretically generate measurements at a rate of 1/(IT ′) ≈ 260 Hz, we only generate scans). We find that the target can be well seperated from estimates at 2 Hz. For simulation purposes the target positions the signal and that the receiver noise plays little role as with and channel response are assumed constant during each of a good detection performance, we observe almost no false these updates (which last less than 4 ms). alarms for this Swerling II target. Around the zero range-rate region, the dominating direct signal leads to a blind zone, which extends into a few range-rate bins on both sides. This B. Results is because neighboring range-rate bins are correlated, so the We run the algorithm as described in Section IV; the much stronger direct signal “spills over”. detection and false alarm probabilities can be controlled via the As comparison we plot results based on projection of the 6 observations via FFT operation as described in Section III-C, REFERENCES see Fig. 4. This is in effect the same data the OMP algorithm [1] P. Howland, “Target tracking using television-based bistatic radar,” IEE operates on defined as p0. We see that directly applying thresh- Proc. Radar, Sonar & Navig., vol. 146, no. 3, pp. 166–174, Jun. 1999. olding to this data makes no sense, as it suffers from blurring [2] ——, “Editorial: Passive radar systems,” IEE Proc. Radar, Sonar & Navig., vol. 152, no. 3, pp. 105–106, Jun. 2005. and strong side-lobes of the direct signal. This could be im- [3] P. Howland, D. Maksimiuk, and G. Reitsma, “FM radio based bistatic proved by using more OFDM symbols (larger I), which would radar,” IEE Proc. Radar, Sonar & Navig., vol. 152, no. 3, pp. 107–115, narrow the blurring, as with more observations the range- Jun. 2005. [4] D. Poullin, “Passive detection using digital broadcasters (DAB, DVB) rate resolution becomes better. In any case, some additional with COFDM modulation,” IEE Proc. Radar, Sonar & Navig., vol. 152, processing would be necessary, e.g., peak picking to avoid no. 3, pp. 143–152, Jun. 2005. multiple adjacent detections. Intuitively speaking, the OMP [5] M. Tobias and A. Lanterman, “Probability hypothesis density-based multitarget tracking with bistatic range and doppler observations,” IEE algorithm includes this, since starting from the projection, the Proc. Radar, Sonar & Navig., vol. 152, no. 3, pp. 195–205, Jun. 2005. algorithm iteratively picks a peak and subtracts it from the [6] H. Kuschel, “VHF/UHF radar part 1: Characteristics,” Electronics & signal, removing also correlated energy on neighboring bins. Communication Engineering Journal, vol. 14, no. 2, pp. 61–72, Apr. 2002. [7] ——, “VHF/UHF radar part 2: Operational aspects and applications,” VI. CONCLUSION Electronics & Communication Engineering Journal, vol. 14, no. 2, pp. 101–111, Apr. 2002. We presented in detail the OFDM signal model as used in [8] M. Daun and W. Koch, “Multistatic target tracking for non-cooperative DAB/DVB for the purpose of target detection and tracking illuminating by DAB/DVB-T,” in Proc. IEEE OCEANS Conference, Aberdeen, Scotland, Jun. 2007. using passive radar. We suggested using CS to extract the [9] S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictio- target signals, as in our opinion this is a well suited approach naries,” IEEE Trans. Signal Processing, vol. 41, no. 12, pp. 3397–3415, to this problem. We described a possible CS implementation Dec. 1993. [10] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact using the greedy OMP algorithm and detailed its implemen- signal reconstruction from highly incomplete frequency information,” tation. Our numerical results show very good performanec in IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. target detection and seperation. This shows especially for small [11] D. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. range-rate measurements, as they are highly correlated with the [12] E. Candes and T. Tao, “Near-optimal signal recovery from random pro- direct blast signal. jections: Universal encoding strategies?” IEEE Trans. Inform. Theory, In our opinion this setup warrants further investigation, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. including different CS algorithms, a more realistic generation of ground-truth for numerical simulation, possibly real experimental data, investiagtion of faster targets that cannot be modelled using the narrowband assumption and/or possibly multiple closely spaced targets.