The Identification of Frequency Hopping Signal Using Compressive Sensing

Transcription

1 Communcatons and etwork, 2009, do: /cn Publshed Onlne August 2009 (htt:// The Identfcaton of Frequenc Hong Sgnal Usng Comressve Sensng Ja YUA 1, Pengwu TIA 2, Hong YU Deartment of Communcaton Engneerng, Insttute of Informaton Technolog, Zhengzhou, Henan, Chna Emal: 1 haa@126.com; 2 tw0802@163.com Abstract: Comressve sensng (CS) creates a new framework of sgnal reconstructon or aroxmaton from a smaller set of ncoherent roecton comared wth the tradtonal qust-rate samlng theor. Recentl, t has been shown that CS can solve some sgnal rocessng roblems gven ncoherent measurements wthout ever reconstructng the sgnals. Moreover, the number of measurements necessar for most comressve sgnal rocessng alcaton such as detecton, estmaton and classfcaton s lower than that necessar for sgnal reconstructon. Based on CS, ths aer resents a novel dentfcaton algorthm of frequenc hong (FH) sgnals. Gven the ho nterval, the FH sgnals can be dentfed and the hong frequences can be estmated wth a tn number of measurements. Smulaton results demonstrate that the method s effectve and effcent. Kewords: comressve sensng, frequenc hong sgnal, dentfcaton 1. Introducton Wth so man good advantages such as ant-am, antnterceton, hgh securt and so on, the technque of frequenc hong sread sectrum (FHSS) has been extensvel aled n man areas, esecall n mltar doman. The detecton and nterceton of FH sgnals can be addressed n several methods of whch wde band or channelzed recever, tme-frequenc dstrbuton, and cclostatonar rocessng are tcal ones [1-4]. For all the methods above, the extremel large requrement of measurements s one of the most serous dsadvantages, whch can be a bottleneck n the alcaton of dentfcaton of hgh seed wde band FH sgnals. Recentl, there have been some actve attemts on sgnal rocessng wth the advantage of CS for the sarse or comressve sgnals [5-8]. However, most of them are lmted wthn the area of statstcal nference tasks whch need the ror knowledge of the robablt denst dstrbuton of sgnals. Besdes, t s seldom to be studed on how to develo the otental of CS to make rocessng of FH sgnal whch s one of the most mortant sarse or comressve sgnals. Ths aer makes use of the sarst of FH sgnals on the local Fourer bass, and then resents a novel dentfcaton algorthm of FH sgnals wth the comressve measurements. Gven the ho nterval, the FH sgnals can be dentfed and the hong frequences can be estmated wthout reconstructng the sgnals. 2. Comressve Sensng Background 2.1 Reresentaton and Sarst of Sgnal qust-rate samlng s the classcal method to descrbe a sgnal wth ts bandlmtedness, whle CS ams to comletel descrbe a sgnal wth ts sarst or comressblt to reduce the requred number of measurements [9]. A sgnal x can be vewed as an 1 column vector n wth elements x[ n], n 1,2,...,. Let the matrx,, have columns whch form a bass of vectors n. And then, an sgnal x can be exressed as: x s or x s (1) 1 where s s the 1 column vector of weghtng coeffcents s x,. When we sa that x s K-sarse, we mean that t s well reconstructed or aroxmated b a lnear combnaton of ust K bass vectors from, wth K. That s, there are onl K of the s n (1) are nonzero and ( K ) are zero. 2.2 Incoherent Measurements Consder a generalzed lnear measurement rocess of a sgnal x whch s K-sarse. Let be an M measurement matrx, M where the rows of Corght 2009 ScRes

2 THE IDETIFICATIO OF FREQUECY HOPPIG SIGAL USIG COMPRESSIVE SESIG 53 are ncoherent wth the columns of. The ncoherent measurements can be obtaned b comutng M nner roducts between x and the rows of as n x,. It can also be exressed as: x= s= s (2) where : s an M matrx. It s roved that dose not deend on the sgnal x and t can be constructed as a random matrx such as Gaussan matrx. And the CS theor shows that there s an over-measurng factor c 1 such that onl M : ck ncoherent measurements are requred to reconstruct x wth hgh robablt [9-11]. That s, onl ck ncoherent measurements nclude all of the salent nformaton n the K-sarse sgnal x, whch rovdes the theor suort on the sgnal rocessng onl gven the ncoherent measurements wthout reconstructng the sgnals. 2.3 Reconstructon Wth the salent nformaton ncluded n the ncoherent measurements, there have been several knds of reconstructon algorthms ncludng l 1 mnmzaton, greed algorthm, matchng ursut and so on [12-15]. Snce ths aer s concentrated on FH sgnal dentfcaton wthout sgnal reconstructon, we don t dscuss reconstructon algorthms n detal here. 3. Comressve Identfcaton for FH Sgnal Wth the good sarst of FH sgnals on the local Fourer bass, we now show that ncoherent measurements can be used to solve the dentfcaton roblem wthout ever reconstructng the sgnal. In ths rocess, t s able to save sgnfcantl on the number of measurements requred. 3.1 Comressve Identfcaton Problem Setu FH sgnals are sarse n a tme-frequenc reresentaton as short-tme Fourer transform, and the are alwas wdeband when there s no ror restrcton on the frequences of the local snusod [16]. Therefore, the measurements obtaned wth the tradtonal qust-rate samlng could be excessve and hard to meet wth the resent ablt of hardware nstrument. ow, consder a FH sgnal whch conssts of a sequence of wndowed snusods wth frequences dstrbuted between f 1 and f 2 Hz. The bandwdth of ths sgnal s B=f 2 -f 1 Hz, whch asks for samlng above the qust rate of 2(f 2 -f 1 ) Hz to avod alasng. However, the exresson of the sgnal at an sngle ho s extremel smle: t conssts of onl one snusod of whch bandwdth s extremel less than B [16]. Hence, CS could make dent- T1 T2 T3 T4 T 1 Fgure 1. Ho ntervals and observaton ntervals n the condton of 1-sarse fcaton of FH sgnals ossble wth a samlng rate that s extremel less than the qust rate. Let the observaton nterval equal to the ho nterval. If the start of the FH sgnal can be catured exactl, the sgnal can be observed snchronousl as dected n Fgure 1 and t has 1-sarse reresentaton on the local Fourer bass wthn each of ho nterval. Otherwse, as dected n Fgure 2, the sgnal wthn each of ho nterval wll have 2-sarse reresentaton snce onl two of the hong frequences aear n ever sngle observaton nterval. We observe x nstead of x and our goal s to dentf the FH sgnal and estmate ts hong frequences wth and ts connecton wth. T 2 T 3 T Sarse Comressve Identfcaton The amltudes of Fourer coeffcents of some FH sgnal wthn an observaton nterval have been shown n Fgure 3 whch dedcates that all the coeffcents are almost zero excet for onl one sngle large coeffcent. Fgure 2. Ho ntervals and observaton ntervals n the condton of 2-sarse Amltude Frequenc (MHz) Fgure 3. The amltudes of Fourer coeffcents of some FH sgnal wthn an observaton nterval n the condton of 1-sarse Corght 2009 ScRes

3 54 THE IDETIFICATIO OF FREQUECY HOPPIG SIGAL USIG COMPRESSIVE SESIG M Y s s 400 Fgure 4. Measurement rocess n the condton of 1-sarse The rocess of 1-sarse comressve measurement s dected n Fgure 4. We am to fnd the oston of nonzero s ndcatng the hong frequenc of a artcular nterval. Snce s obtaned b multlng the nonzero s b ts corresondng column vector, the hong frequenc can be estmated gven and. A drect method to estmate the oston of nonzero s s to search for the oston of whch can be decded b calculatng the angles between and each column vector of n the vector sace as onl the angle between and s zero n the deal condton. Snce s also a random Gaussan matrx f s chosen to be a random Gaussan matrx, the angle between and another column vector of s also zero wth extremel low robablt. Takng account of the effect of nose, we desgn the estmaton algorthm of hong frequenc as follows: 1) Obtan the ncoherent measurements wth. 2) Calculate the cosne of angles between and each column vector n the vector sace cos(,) H 2 2 where H denotes conugate transose. 3) Select the column vector that maxmzes cos(,), and defne the oston of ths vector as estmaton of hong frequenc (3) f ˆ arg max cos(, ) (4) After several ntervals of observaton and estmaton of hong frequences, the tme-frequenc curve of the sgnal can be obtaned and the FH sgnal has been dentfed n the condton of 1-sarse Sarse Comressve Identfcaton Dfferent from the condton of 1-sarse, Fgure 5 shows that there are two large coeffcents wthn an observaton nterval as each observaton nterval covers arts of two ho ntervals n the condton of 2-sarse dected n Fgure 2. Amltude Fgure 5. The amltudes of Fourer coeffcents of some FH sgnal wthn an observaton nterval n the condton of 2-sarse The rocess of 2-sarse comressve measurement s shown s Fgure 3 whch dedcates that s a lnear combnaton of two column vectors 1 and 2 corresondng to the two nonzero coeffcents s 1 and s 2 ndcatng the two hong frequences wthn a artcular observaton nterval. And s also a lnear combnaton of another two column vectors of wth extremel low robablt, snce s a random Gaussan matrx. Therefore, the two hong frequences can be estmated b decdng the subsace comrsed of 1 and 2 n. The estmaton algorthm s as follows: 1) Obtan the ncoherent measurements wth. 2) Calculate the orthogonal roecton of P onto the subsace L comrsed of an two column vectors and, where Frequenc (MHz) L P P (5) L P s orthogonal roector exressed b: H 1 H PL V( V V) V (6) where V,. 3) Select the two column vectors that maxmze P onto the corresondng subsace, and defne the ostons of these two vectors as estmaton of the two hong frequences [ fˆ, fˆ ] arg max( P ) (7) 1 2, Takng account of the reetton of hong frequences wthn two consecutve observaton ntervals n the condton of 2-sarse, we can use the estmaton results Corght 2009 ScRes

4 THE IDETIFICATIO OF FREQUECY HOPPIG SIGAL USIG COMPRESSIVE SESIG 55 M Y 1 s 2 s 1 s 2 Fgure 6. Measurement rocess n the condton of 2-sarse of the former nterval n the latter one. Onl n the frst nterval, the algorthm s a knd of two-dmensonal search as two column vectors have to be selected meanwhle. And n the successve ntervals, t can be executed as a one-dmensonal search (twce) as one column vector can be confrmed n accordng to the oston nformaton of two selected vectors of the former nterval. Ths teratve rocessng can effectvel reduce the comutaton, but obvousl the error roagaton can also be ntroduced. To solve ths roblem, an udatng wndow s desgned to searate the whole observaton tme nto several segments of ntervals. And n the frst nterval of ever udatng wndow, the two-dmensonal search s executed all over agan. As the condton of 1-sarse, the tme-frequenc curve of the FH sgnal can also be obtaned after several observaton ntervals, and the sgnal can be dentfed. 4. Smulaton Results To demonstrate the feasblt and effectveness of the roosed algorthm, a wdeband FH sgnal submerged n addtve Gaussan whte nose (AWG) s consdered to make the smulaton exerments. Ths FH sgnal has ten hong frequences whch are dstrbuted unforml between 20MHz and 200MHz, and the ho nterval s 1ms,.e hos er second. The other man smulaton arameters are as follows: 2048-ont local Fourer bass s chosen to be, random Gaussan matrx s chosen to be, and the number of observaton ntervals s set to Each exerment s made n the condton of both 1-sarse and 2-sarse. Frst, the estmaton erformance of hong frequenc s evaluated b normalzed mean square error (MSE) through several ntervals of observaton, where MSE s exressed b 2 1 ˆ T f f 1 T MSE f (8) where f ˆ s the estmaton of hong frequenc that exressed b f n the th observaton nterval and T reresents the number of observaton ntervals whch s set to 2000 here. Fgure 7 and Fgure 8 show the erformance curves of 1-sarse and 2-sarse resectvel. Fgure 7. MSE of estmaton wth SR n the condton of 1-sarse, where =2048 and M reresents the number of measurements used n ths exerment exerments Fgure 8. MSE of estmaton wth SR n the condton of 2-sarse, where =2048 and M also reresents the number of measurements used n ths exerment exerments. And the length of udatng wndow s set to 40 Some conclusons can be demonstrated from Fgure 7 and Fgure 8. Frst, the hong frequences can be effectvel estmated wth a tn number of measurements when SR s hgher than 8dB. Second, the erformance of estmaton degrades wth the decrease of M, esecall n low SR. And fnall, the erformance of 1- sarse s better than that of 2-sarse. ext, the estmated tme-frequenc curves of the FH sgnal of 1-sarse and 2-sarse are dected n Fgure 9 and Fgure 10 resectvel when M /16 and SR s 10dB. From the Fgure 9 and Fgure 10, t s shown that the estmated tme-frequenc curve s qute close to the real one and the FH sgnal can be effectvel dentfed, esecall n the condton of 1-sarse. Corght 2009 ScRes

5 56 THE IDETIFICATIO OF FREQUECY HOPPIG SIGAL USIG COMPRESSIVE SESIG Based on CS, ths aer rovdes a novel method for the dentfcaton of wdeband FH sgnal wth a tn number of ncoherent measurements, whch s an nsraton of real-tme wdeband sarse sgnal rocessng. Ths method can also be of great hel for the detecton and recognton of wdeband sgnal n the non-cooeratve communcaton. There are man oortuntes for future research. Identfcaton wthout the nformaton of ho nterval, the cket fence effect of Fourer transformaton on the erformance of dentfcaton, and the theoretcal bounds of M wth a gven SR would be dscussed n the future work. REFERECES [1] AYDI L, POLYDOROS A. Ho-tmng estmaton for FH sgnals usng a coarsel channelzed recever. IEEE Trans. Communcaton, Ar. 1996, 44(4): [2] ZHAG X, DU X, ZHU L. Tme frequenc analss of frequenc hong sgnals based on Gabor sectrum method. Journal of Data Acquston & Processng, Jun. 2007, 22(2): [3] HIPPESTIEL R, KHALIL, FARGUES M. The use of wavelets to dentf hoed sgnals. In 1997 Forteth Aslomar Conf. Sgnals, Sstem & Comuter, 1997, 1: [4] FA H, GUO Y, XU Y. A novel algorthm of blnd detecton of frequenc hong sgnal based on second-order cclostatonart. Proc Image and Sgnal Processng Congr., 2008, 5: [5] HAUPT J, OWAK R, YEH G. Comressve samlng for sgnal classfcaton. In 2006 Aslomar Conf. on Sgnals, Sstem & Comuter, Oct. 2006, Conclusons [6] HAUPT J, OWAK R. Comressve samlng for sgnal detecton. Conf. Rec IEEE Int. Conf. Acoustcs Seech and Sgnal Processng, 2007, 3: [7] DUARTE M F, DAVEPORT M A, WAKI M B. Multscale random roecton for comressve classfcaton. Conf. Rec IEEE Int. Conf. Image Processng, 2007, 6: [8] DUARTE M F, DAVEPORT M A, WAKI M B, BRAIUK R G. Sarse sgnal detecton from ncoherent roecton. Conf. Rec IEEE Int. Conf. Acoustcs Seech and Sgnal Processng, 2006, 3: [9] BRAIUK R. Comressed sensng. IEEE Sgnal Processng Magazne, Jul. 2007, 24(4): [10] DOOHO D. Comressed sensng. IEEE Trans. Inform. Theor, Ar. 2006, 52(4): [11] CADES E, ROMBERG J, TAO T. Robust uncertant rncles: Exact sgnal reconstructon from hghl ncomlete frequenc nformaton. IEEE Trans. Inform. Theor, Feb. 2006, 52(2): [12] DOOHO D, TAER J. Sarse nonnegatve solutons of underdetermned lnear equatons b lnear rogrammng. Proc. atonal Academ Scence, 2005, 102(27): [13] TTOPP J A. Greed s good: Algorthmc results for sarse aroxmaton. IEEE Trans. Inform. Theor, Oct. 2004, 50(10): [14] HAUPT J, OWAK R. Sgnal reconstructon from nos random roecton. IEEE Trans. Inform. Theor, Se. 2006, 52(9): [15] CHE S, DOOHO D, SAUDERS M. Atomc decomoston b bass ursut. SIAM J. Sc. Comut., 1998, 20: [16] LASKA J, KIROLOS S, MASSOUD Y, BARAIUK R. Random samlng for analog-to-nformaon converson of wdeband sgnals. IEEE Dallas/CAS Worksho on Desgn, Alcaton, Integraton and Software, Oct. 2006, Corght 2009 ScRes