Compressive Direction Finding Based on Amplitude Comparison
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1 Compressve Drecton Fndng Based on Ampltude Comparson Rumng Yang, Ypeng Lu, Qun Wan and Wanln Yang Department of Electronc Engneerng Unversty of Electronc Scence and Technology of Chna Chengdu, Chna { shan99, luypeng, wanqun, wlyang}@uestc.edu.cn Abstract Ths paper explots recent developments n compressve sensng (CS) to effcently perform the drecton fndng va ampltude comprarson. The new method s proposed based on unmodal characterstc of antenna pattern and sparse property of receved data. Unlke the conventonal methods based peak-searchng and symmetrc constrant, the sparse reconstructon algorthm requres less pulse and takes advantage of CS. Smulaton results valdate the performance of the proposed method s better than the conventonal methods. eywords - drecton fndng; ampltude comparson; beam scannng; sparse reconstructon; compressve sensng. I. INTRODUCTION Wth the development of radar technology and the complcaton of target background, more and more nformaton whch s not range but also angle need be known to target n order to track and orentate accurately. In most modern radar systems, the target drecton of arrval s estmated by the monopulse technque [], whch n prncple can work wth just a sngle pulse. Dfferent from the drecton-fndng methods of monopulse radar, there s another method that works as follows: The beam of radar antenna scans to fnd the user; then the user responses; fnally the radar measures the strength of the response sgnal, and fnds the user s locaton to the radar by the modulaton nformaton of the pattern. As the radar antenna pattern has obvous peak features, so the user poston relatve to the radar can be determned drectly usng the estmated peak locaton method. There are many ways to estmate the peak poston. An effcent algorthm for estmatng the peak poston of a sampled functon s the Hlbert Transform nterpolaton algorthm [2]. The algorthm s a computatonally effcent algorthm for the peak detecton and poston estmaton of a sgnal functon. It s based on a sgnal nterpolaton technque whch reles on the Hlbert Transform of the sampled sgnal. Besdes, another method such as the mult-resoluton method whch s able to overcome the samplng perod s nfluence on the peak poston estmaton accuracy, Fourer transform tme shft nvarant Methods and Snc functon nterpolaton method [3] can estmate the peak locaton too. Ths paper re-examnes the angle estmaton problem and uses recent results n sparse approxmaton [4] and Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna under grant , the Natonal Hgh Technology Research and Development Program of Chna (863 Program) under grant 28AA2Z36 and n part by Scence Foundaton of Mnstry of Educaton of Chna under grant 939.) compressve sensng to provde a fundamentally dfferent drecton fndng method. Frst we get a sparse representaton of the receved sgnal and then the user s locaton to radar s obtaned by the sparse soluton. Comparng wth the tradtonal unmodal characterstc and symmetry constrants based maxmum () methods, the proposed one requres fewer pulses, s wth the ablty of compressed samplng, and acheves a much smaller estmaton error than the tradtonal search method. Ths paper s organzed as follows. The Compressed sensng revew s descrbed n Secton II. In secton III we presented the measurements model. We ntroduce four drecton fndng methods n secton IV: the tradtonal maxmum method and symmetry constrants based maxmum method, the match pursut and bass pursut methods whch based on the compressve sensng. Secton V presents smulaton results that valdate the formulaton and demonstrate sgnfcant performance ncrease over tradtonal maxmum methods. Conclusons are presented n secton VI. II. CORESSED SENSING REVIEW Sparsty wdely exsts n wreless sgnals [5]. Consderng a sgnal x can be expanded n an orthogonal complete dctonary, wth the representaton as N NN N x Ψ b when most elements of the vector b are zeros, the sgnal x s sparse. And when the number of nonzero elements of b s S (S << L < N), the sgnal s sad to be S-sparse. Compressve Sensng (CS) provdes an alternatve to the well-known Shannon samplng theory. It s a framework performng nonadaptve measurement of the nformatve part of the sgnal drectly on condton that the sgnal s sparse. In CS, nstead of measure the sgnal drectly as Nyqust samplng, a random measurement matrx Φ s used to sample the sgnal. In matrx notaton, the obtaned random sample vector can be represented as M MN N y Φ x The measurement matrx should satsfy the restrcted sometry property (RIP) whch s a condton on matrces Φ whch
2 provdes a guarantee on the performance of Φ n CS. It can be stated as: s 2 s y Φy y for all S-sparse y. The restrcted sometry constant, s defned as the smallest constant for whch ths property holds for all s-sparse vectors y. There are three knds of frequently used measurement matrces: ) Non-Unform Subsamplng (NUS) or Random Subsamplng matrces whch are generated by choosng M separate rows unformly at random from the unt matrx I N ; 2) Matrces formed by samplng the..d. entres (Φ) j from a whte Gaussan dstrbuton; 3) Matrces formed by samplng the..d. entres (Φ) j from a symmetrc Bernoull dstrbuton and the elements are N wth probablty /2 each. When the RIP holds, a seres of recoverng algorthm can reconstruct the sparse sgnal [6]. One s greedy algorthm, such as matched pursut () [7], O [8]; another group s convex programmng, such as bass pursut (), LASSO and Dantzg Selector (DS) [9]. DS has almost the same performance as LASSO. Both of the convex programmng and greedy algorthm have advantages and dsadvantages when appled to dfferent problem scenaros. A very extensve lterature has been developed that covers varous modfcatons of both algorthms so to emphasze ther strengths and neutralze ther flaws. A short assessment of ther dfferences would be that convex programmng algorthm has a more reconstructon accuracy whle greedy algorthm has less computng complex. And n contrast to, LASSO has addtonal denosng performance advantage. III. MEASUREMENT MODEL As descrbed n secton I, the radar antenna pattern has obvous peak features, so the user poston relatve to the radar can be determned drectly usng the estmated peak locaton method. Assume that the antenna pattern s p( ). Wthout loss of generalty, as shown n Fg., let p( ) be a Snc functon and represented as: 2 p( ) snc b b s the half of the manbeam wdth. In the k-th moment, the strength of the receved sgnal can be represented as: xk sk vk sk p( k) ; k s the radar antenna scannng angle n the k-th moment; v k s the nose, whch s n 2 dstrbuton, s k, 2,,, s the number of receved sgnal strength n the measurement perod. In (2), s k has only the nformaton of the receved sgnal strength along wth the change of radar antenna s scannng angle. Thus v k s already normalzed by the maxmum receved sgnal strength. IV. DIRECTION FINDING A.Maxmum Method Maxmum method takes advantage of peak characterstc of the pattern p( ). The user s relatve azmuth to the radar s estmated by fndng the locaton correspondng to the maxmum element n the receved sgnal s strength sequences x, x 2,, x,.e. ˆ k x k ˆ ˆ arg max k k B. Symmetry Constrants Based Maxmum Method The symmetry constrants based maxmum () method uses both the unmodal characterstc and the symmetry. It estmates the user s azmuth relatve to the radar by fndng the best symmetrcal locaton correspondng to the maxmum element n the user s receved sgnal strength sequences,.e: ˆ SYM kˆ arg max k k ˆ N x x x m km km m N N 2 2 xkmxkm m m N s the mnmum element between k and k N mn k, k,.e. C. Match Pursut Searchng over an extremely large dctonary for the best match s computatonally unacceptable for practcal applcatons. Mallat and Zhang proposed a greedy soluton that s known from that tme as Matchng Pursut. Matchng pursut s a type of numercal technque whch nvolves fndng the best matchng projectons of multdmensonal data onto an over-complete dctonary. The basc dea s to
3 represent a sgnal from Hlbert space as a weghted sum of functons (called atoms). By takng an extremely redundant dctonary we can look n t for functons that best match a sgnal. Fndng a representaton most of the coeffcents n the sum are close to (sparse representaton) s desrable for sgnal codng and compresson. Here we dvde the beam wdth nto 2L+ sectons wth the same length,.e. L, L,..., k,..., L, k b k / L, k L, L,,,,, L, and construct the Redundant dctonary as : D Toepltz d, d (8) = d p( ) p( ) p( ) T (9) d p( ) p( ) p( ) () 2 2 p( k) snc b k, and the column vector D (,..., 2L ) of the redundant dctonary s named as atom. x=ds+v x [ x,..., x ] T, v [ v,..., v ] T. The user s azmuth to the radar can be determned by fndng the non-zero element of vector s, and the correspondng atom can be obtaned by fndng the maxmum correlaton of the atom and receved sgnal x,.e.: D arg max D, x opt D. Bass Pursut To encourage sparsty, The optmzaton s optmal but non-convex and known to be NP-hard. In practce, a multtude of effcent algorthms have been proposed, whch acheve hgh recovery rates. The -mnmzaton method s the most extensvely studed recovery technque. In ths approach, the non-convex norm s replaced by the convex norm. Ths approxmaton s known as Bass Pursut () whch s a prncple for decomposng a sgnal nto an "optmal" superposton of dctonary elements, optmal means havng the smallest norm of coeffcents among all such decompostons. Here the sparse soluton of () can be obtaned by optmzaton method. It can be modeled as: mn e L L 2 s s.t. e, s e x-ds. It s obvously that () s a convex programmng and the soluton can be obtaned by some optmzaton software, such as the software cvx []. V. SIMULATION EXPERIMENT In ths secton we present smulaton results that demonstrate the performance of our method. For the remander of ths secton we suppose that n the radar s beam scans process, the number of the responded mpulse s 3; and the changes n beam scannng angle correspondng to the adjacent pulse nterval s. degrees. The radar pattern s 2 p( ) snc b, b 7.5. We dd ndependent experments count the error probablty of the angle fndng. Fg. 2 to Fg. 4 demonstrate the performance of the four methods ntroduced n secton IV, when the data extracton rates are one-half, one-quarter, and one-eghth respectvely. The results of our bass pursut algorthm are marked by pentacle and labeled. Whle the ones of match pursut, method, and the tradtonal maxmum method are marked by damond, rectangle and crcularty, respectvely. They are respectvely labeled by, and. The table gves the correspondng mean square error of drecton estmaton by dfferent methods. The smulaton results above show that the estmates based on sparse sgnal representaton are better than the tradtonal maxmum method and method. The proposed method can gve a more accurate result even the volume of data s relatvely few. Matchng pursut method can get the user s azmuth to the radar only by calculatng the correlaton between the atom and receved sgnal. It s smple and easy to mplement. However, due to nfluence of the correlaton between adjacent atoms n the redundant dctonary, the performance of the match pursut s nferor to standard bass pursut method. The bass pursut can optmze the optmal atom to get the soluton, and the performance s much superor to others. However, compared wth the matchng pursut method, the base bass pursut method s more complex. VI. CONCLUSION Our results demonstrate that the angle fndng can be sgnfcant mproved f we ncorporate the sparse nformaton processng method nto the radar antenna pattern modulated drecton fndng. The expermental results show that sparse sgnal representaton based estmaton s better than the tradtonal maxmum method and the method. Although the base bass pursut method s more computatonal complex n contrast to the matchng pursut method, the performance of the algorthm s much superor to the method for as much as the nfluence of the correlaton between adjacent atoms n the redundant dctonary. ACNOWLEDGMENT The authors thank anonymous revewers for ther valuable suggestons. REFERENCES [] Skolnk, M., Introducton to Radar Systems, New York: McGraw-Hll, 2.
4 [2] S. S. Abeysekera, An effcent Hlbert transform nterpolaton algorthm for peak poston estmaton, Proceedngs of the th IEEE Sgnal Processng Workshop on Statstcal Sgnal Processng, 6-8 Aug. 2, pp [3] Chu-Xong Dng, Jng Ba, Peak poston estmaton algorthms for cross-correlaton functon n elastography, Proceedngs of the 2th Annual Internatonal Conference of the IEEE Engneerng n Medcne and Bology Socety, 29 Oct.- Nov. 998, pp [4] A. C. Glbert, J. A. Tropp, Applcatons of sparse approxmatons n communcatons, n Proc. of IEEE Int. Symp. Inf. Theory, 25. [5] Candes, E.J.; Wakn, M.B., An ntroducton to compressve samplng, IEEE Sgnal Processng Magazne, Volume 25, Issue 2, pp. 2-3, March 28 [6] E. Candes. The restrcted sometry property and ts mplcaton for compressve sensng, C.R.Math. Acad. Sc. Pars, Sers I, 346:589, 28 [7] S. G. Mallat and Z. Zhang.: Matchng pursuts wth tme-frequency dctonares, IEEE Tran. on ASSP, Dec. 993, vol. 4, no. 2, pp [8] Tropp J,Glbert A. Sgnal recovery from random measurements va orthogonal matchng Pursut, Transactng on Informaton Theory, 27, 53(2): [9] Emmanuel Candes, and Terence Tao, The Dantzg selector: statstcal estmaton when p s much larger than n, Annals of Statstcs 27. Volume 35, Number 6, pp [] M. Grant and S. Boyd, CVX: Matlab Software for Dscplned Convex Programmng. Onlne accessable :
5 .9 Normalzed beam pattern Error probablty of the angle fndng Angle(degree) Fg. the normalzed beam pattern of radar antenna Fg. 3 Standard devaton of the drecton error (extracton rate: onequarter,snr=5db) Error probablty of the angle fndng Error probablty of the angle fndng Fg. 2 Standard devaton of the drecton error (extracton rate: one-half, SNR=5dB) Fg. 4 Standard devaton of the drecton error (extracton rate: oneeght,snr=5db) TABLE I. MEAN SQUARE ERROR OF DIRECTION ESTIMATION (DEGREE) Sample s extracton Maxmum method method rate One-half One-quarter One-eght
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