APPLICATION of compressed sensing (CS) in radar signal

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1 A Novel Joint Compressive Single Target Detection an Parameter Estimation in Raar without Signal Reconstruction Alireza Hariri, Massou Babaie-Zaeh Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran Abstract In this paper, a etector/estimator is propose for compresse sensing raars, which oes not nee to reconstruct the raar signal, an which works irectly from compressive measurements. More precisely, through irect processing of the measurements, an without the nee for reconstructing the original raar signal, the system performs target etection, an then estimates range, Doppler frequency shift, an raar cross section in the presence of a Gaussian clutter. It can be seen that for large compression ratios, the etection performance an estimation quality is comparable to a common raar system while having a much lower ata rate an with less computational loa. Inex Terms compresse sensing, etection-estimation, generalize likelihoo ratio test, receiver operating characteristic, traitional raar system, compression ratio. I. INTRODUCTION APPLICATION of compresse sensing CS in raar signal processing has recently attracte the attention of many researchers [], [], [3], [4], [5], [6] because raar signals usually have high banwiths at high carrier frequencies. Therefore, even when using banpass sampling theorem [7], in orer to have the signal not aliase, it is necessary to sample at a very high rate, leaing to huge amounts of ata. On the other han, it is well-known in the CS literature [8], [9], [0] that one can take many fewer measurements compare to Nyquist samples, without any loss of information. Raar signal is generally sparse in the elay-doppler omain [6] an CS theory may therefore be use to achieve more efficiency in storage space an computation time. On this basis, many works have been one in synthetic aperture raar SAR raw ata processing to get its image with less nee for onboar ata storing an processing [], [], [3]. Recently, CS has been employe in multiple input multiple output MIMO raars to achieve more efficient processing of jointly receive signals in orer to etect targets an estimate their parameters, such as position an velocity [4], [5], [6]. In all of the above mentione works, the esire signal use for etection an/or estimation is first reconstructe by using a general-purpose CS reconstruction algorithm [for example Orthogonal Matching Pursuit OMP [], Basis Pursuit De- Noising BPDN or Compressive Sampling Matching Pursuit CoSaMP []] or an algorithm evelope specially for the specific problem [5], [3], [4]. However, the nee for reconstructing the original signal for etection/estimation creates some ifficulties. When using compresse sensing, the ata rate is highly reuce, that is, the rate of measurements is much lower than for Nyquist samples. If, for processing the measurements, one nees to reconstruct the original signal, that is, reconstruct the Nyquist samples in some part of the system, then the ata rate in that part is again very high. Therefore, it is esirable for etection/estimation to be base on irectly processing the CS measurements, that is, by performing the processing within the CS omain. This has been the subject of some recent stuies [5] an [6]. Detection/Estimation using CS measurements without reconstructing the original signal has been also iscusse in fiels other than raar. For example, authors in [7] an [8] stuie blin communications in the CS omain. In [7], the carrier frequency of a blin phase shift keying PSK signal is estimate, which is a preliminary step for the moulation orer estimation an blin signal etection. In [8], the unknown signal is classifie between four moulations, namely Binary PSK BPSK, Quarature PSK QPSK, 8PSK an 6 Quarature Amplitue Moulation 6QAM. Detection, classification, estimation an filtering in the CS omain have been explore extensively in [5]. In fact, these were investigate for the most general form of the signal, an proper bouns lower, upper or both were introuce for the parameters etermine for the algorithm qualification in each process. The bouns were erive base on the measurement matrix properties efine there. Authors in [6] propose the iea for raar applications generally, but they followe it for Space Time Aaptive Processing STAP. They then showe that statistical testing in the CS omain compressive statistical testing coul perform at a level close to the traitional metho for a sufficient number of measurements. In this paper, etection/estimation in the CS omain is stuie for a simple raar application. More precisely, the aim is to etect whether any target exists an, if yes, to estimate its parameters, such as range, Doppler frequency shift an raar cross section RCS. It will be shown experimentally that an excellent etection-estimation performance can be obtaine without the nee for a lot of measurements as is require in [6]. Therefore, even though the ata rate of this new system is much lower than that of a common raar system, it is possible to achieve comparable qualities with less

2 computational loa. The paper is organize as follows. In Section II, the problem is efine an moele mathematically. Section III is eicate to solving problem. The performance of the propose etection algorithm is given in Section IV. In Section V, the signal to noise an clutter ratio SNCR at the input an output of the measurement obtainer system is calculate, in orer to restate the results of Section IV in terms of SCNR. Finally, the simulations an their results are escribe in Section VI. Notations: For any vector y, its transpose conjugate Hermitian is shown by y H. For any square matrix A, its eterminant is enote by A. I is the ientity matrix with appropriate imensions. For any complex scalar a, its complex conjugate an real part are represente by a an Ra} respectively. Finally, E } stans for the expectation operator. II. PROBLEM STATEMENT Consier a scenario in which there is a target having unknown istance an velocity relative to the raar platform in the presence of clutter. As in [9], it is assume that the clutter probability ensity function PDF is Gaussian an also that the raar pulse repetition frequency PRF is so high that the target s relative istance an velocity are nearly constant uring ifferent pulses. Furthermore, the target RCS is consiere as a ranom variable an it is assume that its value is constant throughout ifferent pulses calle Swerling Case in [0]. The receive signal is sparse in time an in elay-doppler omains [6], so instea of sampling the receive signal at the Nyquist rate, it provies many fewer measurements with properties mentione in the CS literatures [8]. The etection problem can be moele in the form of a hypothesis testing problem as H 0 : y Φc + n H : y τ, f + c + n, in which y M is the measurements vector, Φ M N is the real ranom Gaussian measurement matrix with inepenent ientically istribute i.i.. elements having zero mean an unit variance an M N. It shoul be note that the measurement matrix is assume to be incoherent with time or elay-doppler omain in which the receive signal is sparse. c N is the vector of clutter samples, n N moels the aitive white Gaussian noise AWGN with variance σn at the receiver, is the complex RCS of the target, an s r τ, f is the receive signal with elay τ an Doppler frequency shift f relative to the sent signal. As in [9], it is assume that the istribution of c is circular normal with zero mean an covariance matrix R c enote as CN 0, R c. As well, there is a pre-estimation of R c. Therefore, if the clutter to noise ratio CNR is fixe, σn will be known. As mentione in [0], circular normal istribution can be consiere for as CN 0, σ, where σ is its variance. This is because is forme by the many scatterers in the target range-doppler cell. Moreover, as in [], an [], it is assume that σ is known a priori. The goal is to etect whether or not any target exists, an if yes, to estimate its parameters, namely range or equivalently, elay, Doppler frequency shift an RCS. III. THE PROBLEM SOLUTION If the target exists, its range an Doppler frequency shift are not known a priori, so the usual likelihoo ratio test LRT cannot be compute an use for the etection. Instea, the generalize likelihoo ratio test GLRT shoul be use, in which the LRT is maximize to fin the optimum elay or equivalently, range an Doppler frequency shift, which are also estimations of the suppose target s parameters. The LRT value at the optimal point shoul then be compare with a proper threshol. So at the first step, the PDF of the measurements vector for the two hypotheses is compute. Because noise an clutter vectors both have circular normal istribution, conitione on H 0, y is a vector with istribution CN 0, ΦσnI + R c Φ T. If A is efine as the covariance matrix of y H 0, for the null hypothesis, we have [3] fy H 0 π M A exp yh A y}, in which A σ nφφ T + ΦR c Φ T. Similarly for the other hypothesis H, fy H,, τ, f CN τ, f, A 3 π M A exp y τ, f H A y τ, f }. 4 As was assume in Section II, the istribution of is f πσ exp } σ, an it is shown in A that fy H, τ, f is as fy H, τ, f π M A exp y H A y } aτ, f σ + exp bτ, f σ } aτ, f σ, + 5 where aτ, f s r τ, f H Φ T A τ, f an bτ, f y H A τ, f. The LRT can then be erive as Ly τ, f fy H, τ, f fy H 0 aτ, f σ + exp bτ, f σ } aτ, f σ. + In orer to fin the GLRT, Ly τ, f shoul be maximize over τ an f i.e. 6 GLRTy max τ,f Ly τ, f. 7 As escribe in B, the GLRT will be in the form GLRTy Ly ˆτ, f b ˆτ, f y H A ˆτ, f Th, 8

3 in which ˆτ, f arg max τ,f Ly τ, f, p 0 is the a priori probability of the target existence an σ + p0 Th ln a ˆτ, p f σ +. 0 σ IV. DETECTOR PERFORMANCE For evaluating the etector performance, the receiver operating characteristic ROC curve, i.e. the etection probability p versus the false alarm probability p f, shoul be calculate. By efining z y H A ˆτ, f, its PDF conitione on the null hypothesis is fz H 0 CN 0,. 0 So p f is calculate as p f p z > Th H 0 9 Th p xx H 0 x Th p x x H 0 x } exp Th, in which x z has a Rayleigh istribution with parameter / / conitione on H 0, enote as Ray / / [4]. On the other han, the PDF of z conitione on H an can be written as fz H, CN, a ˆτ, f. If the above PDF is average over, as shown in C, fz H will be as fz H CN 0, σ + a ˆτ, f. 3 Therefore p will be p p z > Th H Th p xx H x Th p x x H x Th } exp σ, 4 + a ˆτ, f where, similar to the H 0 hypothesis, p x x H Ray σ + a ˆτ, f / /. In orer to obtain the ROC, it can be seen from that Th a ˆτ, } f a ˆτ, f ln p f ln pf so p exp or ˆτ, f in other wors a +a ˆτ, f σ +a ˆτ, f σ p p f. 5 V. SNCR AT THE MEASUREMENTS OBTAINER SYSTEM INPUT AND OUTPUT The signal at the input of the measurements obtainer system is y in s r τ, f +c+n. The signal term is s r τ, f an the noise-clutter term is c + n. Therefore, the signal power is sr E τ, f } σ N p P w, where N p is the number of pulses sent an P w is the number of samples in a pulse. Here it is assume that the pulses are rectangular. On the other han, the noise-clutter power is E c + n }. Because the noise an clutter are statistically inepenent, it can be written as E c + n } } } N E c + E n λ i + Nσn, i 6 where λ i s are the eigenvalues of R c. Therefore, the input SNCR will be SNCR in σn p P w Nσn + N i λ. 7 i As erive in 8, the output statistic for GLRT is x y H A ˆτ, f HΦ sr ˆτ, f T A y HΦ s r ˆτ, f T A ˆτ, f HΦ +s r ˆτ, f T A Φc + n HΦ + sr ˆτ, f T A Φc + n. 8 Therefore, x can be expane as x + sr ˆτ, f HΦ T A Φc + n + c + n H Φ T A ˆτ, f + s r ˆτ, f HΦ T A Φc + n c + n H Φ T A ˆτ, f. 9 If the input SNCR is high, it can be assume that sr ˆτ, f c + n. Then, x can be approximate as x + s HΦ r ˆτ, f T A Φc + n + c + nh Φ T A ˆτ, f HΦ +s r ˆτ, f T A Φc + n c + nh Φ T A ˆτ, f + s HΦ r ˆτ, f T A Φc + n + c + nh Φ T A ˆτ, f / /

4 HΦ sr ˆτ, f T A Φc + n } + R HΦ sr ˆτ, f T A Φc + n } + R + R s HΦ r ˆτ, f T A Φc + n}. 0 If is consiere in the polar form as e jψ, x can be written as x + R e jψ s r ˆτ, f HΦ T A Φc + n }. In, the signal term is an the noise-clutter term is R e jψ s r ˆτ, f HΦ T A Φc + n }. Therefore, the signal power is σ. For the noise-clutter power calculation, s is efine as s s r ˆτ, f HΦ T A Φc + n so the noise-clutter term NCT can be restate as NCT R e jψ s r ˆτ, f HΦ T A Φc + n } Rse jψ } s r cos ψ + s i sin ψ, where s r an s i are the real an imaginary parts of s respectively. The noise-clutter power will be E s r cos ψ + s i sin ψ } E s r cos ψ + s i sin ψ +s r s i cos ψ sin ψ } E s r cos ψ + s i sin ψ }, because c an n are circular normal an inepenent of each other, s is circular normal an therefore its real an imaginary parts are statistically inepenent an both are zero-mean. Furthermore, s is inepenent of ψ, which has a uniform istribution in the interval [0, π, so NCP Es r + s i } E s } aˆτ, f, 3 where NCP stans for noise-clutter power. From 3, it can be conclue that SNCRout σ, 4 which can be written from 5 in terms of the etection an false alarm probability as ln pf SNCRout. 5 ln p VI. RESULTS AND SIMULATIONS In this section, five simulations are conucte to experimentally evaluate the performance of the propose etectionestimation approach. In all of these simulations, the target RCS, its istance an spee relative to the raar, CNR an also the number of samples are fixe as RCS 0.6, R 3900m, v 98m/s, CNR 7B an N 30. In the following, compression ratio CR means the fraction M N, an in the case, the measurement matrix is set to ientity. In other M N p Fig.. M N M/N M/N M/N M/N 0.05 M/N M/N M/N 0. M/N M/N 0.8 M/N ROC curves for ifferent compression ratios at SNCR in 0B means a traitional raar system. wors, CR correspons to the traitional raar system, which oes not use CS. Experiments, an 3. Performance Analysis In these experiments, the presente etector performance is stuie an compare with the traitional metho, by using 5. At first, SNCR in is fixe at 0B so by 7 σ is known. The ROC curves for a fixe SNCR in 0B an for ifferent CRs are plotte in Fig.. The chosen CRs except constitute a geometrical sequence with the first term an common ratio to cover the interval 0, in a goo manner. As can be seen, the lower the CR, the better the etector performance. This was expecte because, by increasing the number of measurements, more information from the incoming signal is obtaine. These curves show that the performance egraation is very small, even for CR the etection probability is above 0.99 for all false alarm probabilities. For comparison, the ROC curves for the CS metho with reconstruction at SNCR in 0B are plotte in Fig.. The OMP algorithm is use for CS reconstruction. As it is observe, the etection is very goo the etection probability is near 0.99 for all false alarm probabilities. Seconly, the etection probability has been shown versus CR for ifferent input SNCRs at fixe false alarm probability p f 0 4. Here, the input SNCR is varie to have a single target with ifferent powers. For each one, p is then compute, while CR varies as the first simulation. The result is shown in Fig. 3. It is observe that, up to SNCR in 5B, the obtaine etection probability at the largest CR is above 0.95 for p f 0 4. In the thir simulation, the ROC curve is erive for ifferent input SNCRs at fixe CR ROC curves for a fixe CR an for ifferent input SNCRs, or equivalently ifferent target powers, have been plotte in Fig.4. From this figure, it can be euce that, to have a goo etection performance p f

5 p.05.0 M/N M/N M/N 0.05 M/N 0.05 M/N 0.05 M/N 0. M/N 0. M/N 0.4 M/N 0.8 p p f SNCR in 5B SNCR 0.85 in 0B SNCR in 5B SNCR in 0B SNCR in 5B SNCR in 30B p f Fig.. ROC curves for ifferent compression ratios at SNCR in 0B with CS reconstruction. p SNCR in 5B SNCR in 0B 0.65 SNCR in 5B SNCR in 0B 0.6 SNCR in 5B SNCR in 30B Fig. 4. The ROC curves for ifferent input SNCRs at fixe compression ratio M N p p f SNCR in 5B SNCR in 0B SNCR in 5B SNCR in 0B SNCR in 5B SNCR in 30B Fig. 3. Detection probability versus compression ratio for ifferent input SNCRs at constant false alarm probability p f 0 4 M N means a traitional raar system. etection probability above 0.97 at CR 0.003, the input SNCR shoul be increase to 5B. This is not surprising, because the number of measurements has been reuce very much, so the input SNCR has to be increase a little. This is the key point: by increasing the input SNCR a little, with very few measurements, a very goo performance can be obtaine. For comparison, the ROC curves for the CS metho with reconstruction at CR are plotte in Fig. 5. It can be again seen that, to have an acceptable etection performance, the input SNCR shoul be increase. Experiment 4. Computational Cost In this experiment, the computational cost of the presente etector is stuie an compare with the traitional raar Fig. 5. The ROC curves for ifferent input SNCRs at fixe compression ratio M N with CS reconstruction. system, i.e. CR. The average CPU time of the etection process over 00 simulations is use as a rough measure of the complexity of the etection algorithm for ifferent CRs. Simulations are performe in MATLAB R03a environment, using Intel Core TM Duo P8800,.67GHz processor with 4GB of memory, an uner 64 bit Microsoft Winows 7 operating system. The results are shown in Fig. 6. It is seen that at CR 0.003, aroun % reuction in computational cost has been achieve. Contrary to previous compressive sensing raars [], [], [3], [4], [5], [6], this system oes not reconstruct the Nyquist samples. So, as emonstrate in Figs. an 4, having a ata rate of only 0.3% of a common raar system, the etection probability is above 0.99, with about a % reuction

6 mean calculation time in secons mean estimate elay an its stanar eviation SNCR in 5B 4 SNCR in 0B SNCR in 5B SNCR in 0B SNCR in 5B 0 SNCR in 30B Fig. 6. The mean etection time in secons versus compression ratio M N means a traitional raar system. mean etection time sec Compressin Ratio M/N Fig. 7. The mean etection time in secons versus compression ratio with CS reconstruction. The average CPU time is increase, however, the key point is that with CS reconstruction the ata rate is increase. in computational loa. In Fig. 7, the mean etection time is plotte when CS reconstruction is use. As it is seen, the average CPU time is increase, however, the key point is that with CS reconstruction the ata rate is increase. Experiments 5, 6, an 7. Parameters Estimation Accuracy In these three experiments, the estimation accuracy of the target parameters is stuie. To estimate the target elay an Doppler frequency shift, 6 an 7 are use. In these experiments, the simulation is run 00 times an the mean an stanar eviation of the estimate parameters compute. Input SNCR is varie from 5B to 30B. Figures 8 an 9 epict the estimate elay ˆτ an Doppler frequency shift f respectively for ifferent values of CR an input SNCR. It is interesting to note that even at SNCR in 5B for CRs less Fig. 8. Mean estimate elay ˆτ an its stanar eviation versus compression ratio for input SNCR varie from 5B to 30B the true elay value is τ 6. Note: M N estimate Doppler frequency an its st means a traitional raar system SNCR in 5B 5000 SNCR in 0B SNCR in 5B 4000 SNCR in 0B SNCR in 5B 3000 SNCR in 30B Fig. 9. Mean estimate Doppler frequency shift f an its stanar eviation versus compression ratio at for input SNCR varie from 5B to 30B the true Doppler frequency shift value is f Note: M N traitional raar system. means a than 0.05, the stanar eviation of the estimate elay is negligible, while using less than % of the samples of the traitional raar system. Similarly for the Doppler frequency shift estimation at CRs less than 0.003, the stanar eviation is very small. It shoul also be note that although there are some errors in the estimate parameters for CRs greater than 0.05, the etection performance is still goo for input SNCRs greater than 0B the etection probability is above For signal moeling in the presence of a target, it was assume that the signal complex RCS is a circular normal

7 mean estimate RCS an its stanar eviation Fig. 0. Mean estimate RCS an its stanar eviation versus compressin ratio at SNCR in 0B the true RCS value is 0.6. Note: M N a traitional raar system. means ranom variable. In practice, however, it can be estimate using the measurements vector. In fact, if the input SNCR is high, from 8 the target RCS can be approximate as y H A ˆτ, f b ˆτ, f HΦ s r ˆτ, f T A ˆτ, f. 6 The RCS approximation accuracy is simulate similar to the elay an Doppler frequency shift at SNCR in 0B. Figure 0 shows the results. For CRs less than 0.05, the error in the mean RCS approximation an the stanar eviation is tolerable less than 5%, but for CRs greater than 0.05, there may be a large error in the mean RCS approximation. Therefore, for RCS approximation, a few more measurements shoul be obtaine. VII. CONCLUSION In this paper, it has been shown that a target can be etecte in the presence of a Gaussian clutter, an its important parameters, such as range, Doppler frequency shift an RCS, can be estimate using compressive measurements without reconstructing Nyquist samples. In fact, by using very high compression ratios like 0.003, the etection performance is proper an the estimation quality is comparable to traitional raar systems while having a much lower ata rate an with less computational loa. So it seems that the propose etection algorithm is very suitable especially for high pulse banwith raars. APPENDIX A PROOF OF 5 It is obvious that fy H, τ, f fy H,, τ, f f r i, 7 in which r an i are the real an imaginary parts of, respectively. By substituting the istribution fy H,, τ, f from 4 in 7 an using f πσ exp } σ we have fy H, τ, f π M A πσ exp y H A y} exp s r τ, f H Φ T A τ, f } exps r τ, f H Φ T A y } } expy H A τ, f } exp r i. If we efine aτ, f s r τ, f H Φ T A τ, f it is real value because a τ, f aτ, f an cτ, f s r τ, f H Φ T A y, it can be written fy H, τ, f π M A exp πσ σ exp y H A y} aτ, f + } σ exp cτ, f + cτ, f } r i π M A πσ exp y H A y} exp eτ, f cτ, f eτ, f cτ, f } r i eτ, f π M A πσ exp y H A y} cτ, f exp eτ, f eτ, f cτ, f } eτ, f r i π M A σeτ, f exp yh A y} cτ, f } exp. 8 eτ, f In 8, eτ, f aτ, f + σ an the last equality is obtaine by Gaussian PDF integration [5]. By efining bτ, f cτ, f s r τ, f H Φ T A y it can be shown that fy H, τ, f π M A σaτ, f + exp yh A y} σ exp bτ, f } σaτ,, 9 f + which is the same as 5.

8 APPENDIX B PROOF OF 8 Accoring to 6 the likelihoo ratio test is bτ, Ly τ, f aτ, f σ + exp f σ } aτ, f σ, + p 0 p 0 30 in which p 0 is the a priori probability of the H 0 hypothesis. As is obvious, aτ, f is not a function of observations measurements. So the new likelihoo ratio test can be reformulate as bτ, f σ } L y τ, f exp aτ, f σ + p 0 aτ, f σ + p 0 L y τ, f bτ, f σ aτ, f σ + p0 ln aτ, f σ + p 0 L 3 y τ, f bτ, f aτ, f σ + ln σ p0 p 0 aτ, f σ + L 4 y τ, f bτ, f s r τ, f H Φ T A y aτ, f σ + σ p0 ln aτ, f σ p If L 4 y τ, f is maximize over τ an f simultaneously, the GLRT will be GLRTy max τ,f L 4 y τ, f in which ˆτ, f arg max τ,f L 4 y τ, f. This is exactly 7. So the GLRT can be written as GLRTy L 4 ˆτ, f b ˆτ, f 3 is the same as 8. y H A ˆτ, f σ + Th σ p0 ln a ˆτ, f σ p APPENDIX C FINDING fz H IN SECTION IV The starting point is in Section IV, i.e. fz H, CN, a ˆτ, f. 33 By integrating 33 over the PDF of, fz H can be foun as fz H fz H, f r i z a ˆτ, f } π exp πσ } exp r i, 34 σ where r an i are as in A. The integration can be compute as } fz H π exp z πσ exp } expz } expz } r i + σ π πσ exp exp e ˆτ, f π exp πσ z e ˆτ, f exp z z e ˆτ, f exp z e ˆτ, f } z z e ˆτ, f } } r i z } e ˆτ, f } r i } z z e ˆτ, f exp e ˆτ, f π exp πσ exp e ˆτ, f } r i exp e ˆτ, f π πσ z e ˆτ, f } z z e ˆτ, f } r i πa ˆτ, z } f exp πσ πe e ˆτ, ˆτ, f f σ π σ + z } exp σ + CN 0, σ + a ˆτ, f. 35 In 35, eτ, f is the same as efine in A. REFERENCES [] V. M. Patel, G. R. Easley, D. M. Healy, an R. Chellappa, Compresse synthetic aperture raar, Selecte Topics in Signal Processing, IEEE Journal of, vol. 4, no., pp , 00. [] A. Buillon, A. Evangelista, an G. Schirinzi, Three-imensional SAR focusing from multipass signals using compressive sampling, Geoscience an Remote Sensing, IEEE Transactions on, vol. 49, no., pp , 0.

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