NOVEL ITERATIVE TECHNIQUES FOR RADAR TARGET DISCRIMINATION

Transcription

1 NOVEL ITERATIVE TECHNIQUES FOR RADAR TARGET DISCRIMINATION Phaneendra R.Venkata, Nathan A. Goodman Department of Electrcal and Computer Engneerng, Unversty of Arzona, 30 E. Speedway Blvd, Tucson, Arzona 857 Keywords Multple target dscrmnaton, sequental hypotheses testng, target ID, adaptve transmsson radar, waveform desgn. Abstract In ths paper, transmsson waveforms for an M-target ID problem are compared and analyzed under a hypothess testng framework. Waveforms based on mutual nformaton between a target ensemble and the receved waveforms, under tme and energy constrants, have already been determned. They are appled to the M-target ID problem, and compared wth the soluton obtaned by maxmzng the average dstance between hypotheses. A sngle llumnaton s consdered frst wth regard to the bnary and the M-ary target ID problem. A new teratve transmsson scheme s proposed that calculates the probablty of each hypothess and updates the transmsson waveform at each step. Ths s coupled wth a sequental mult-hypotheses testng procedure and shown that for the same error rate, the proposed teratve scheme combned wth an nformaton-based waveform can reduce the number of teratons to reach a decson. I. INTRODUCTION There have been varous attempts n the past to determne the optmum transmsson waveforms for varous applcatons. One such applcaton s the M-target ID problem, where one needs to dscrmnate a target from other possble targets. In general, ths s a hypotheses testng problem n whch one needs to dentfy one of M ( ) as the true target. Transmt-receve optmzaton theory was shown as an applcaton to the target ID problem n [5]. The soluton s shown to be optmal n case of two targets, and an extenson to the M-target problem s mentoned, but not proved or analyzed. Ths soluton s determned by maxmzng the weghted average dstance between the output sgnals assocated wth the hypotheses. Another potental crteron for selectng a fnte set of transmtter sgnals for radar s shown n [6]. Ths crteron maxmzes the mnmum dvergence between hypothess pars beng tested at the recever. Furthermore, t s shown that the average dvergence s bounded below by twce the nformaton rate of the channel. It s clamed n [7] that the average dvergence s a reasonable crteron for optmalty of sgnal sets, and probng sgnals are desgned for M-ary dentfcaton of lnear channels usng ths crteron. Varous sgnal desgn crtera and condtons for sgnal optmalty for a communcaton system n a coherent Gaussan channel are derved n [4] and compared wth dfferent cases of constrants on average energy and message probabltes. In general, the regular smplex sets prove to be optmum at relatvely all SNR wth equal probabltes and ether equal energy or bounded average power of the sgnals. Informaton theory was appled to radar waveform desgn n [] where optmal detecton and estmaton waveforms have been determned. It s shown that puttng as much energy as possble nto the mode wth the largest egenvalue may not be the best way to obtan nformaton for dentfyng the target or estmatng ts parameters. Instead, the best estmaton waveforms maxmze the mutual nformaton between the target responses (ensemble) and the receved waveform. In ths paper, we look at applyng the waveform determned n [] to an M-target ID problem and compare wth the approach gven by [5] and other standard procedures usng a hypothess-testng framework. Frst, we consder a sngle llumnaton for bnary and the M-ary cases. The recever makes ts decson usng maxmum lkelhood. The dfference n performance s studed. It s shown that one method maxmzes the average dstance between the target echoes, whle the other s observed to be better at maxmzng the mnmum dstance between echoes. A new teratve scheme s proposed where the probabltes of the hypotheses are updated at each stage along wth updatng the transmt waveform. Ths scheme s coupled wth the standard sequental mult-hypotheses testng procedure and dfferent transmt waveforms are compared. In the sequental procedure, the fgure of mert s the number of teratons to reach a decson for a fxed probablty of msclassfcaton. It wll be shown that the number of teratons s consderably reduced by choosng a waveform

2 that can extract more nformaton about a target. In addton, t s consstent wth the fact that the overall number of teratons wll reduce only f you transmt the best waveform at each stage. In Secton II, we formulate the problem and descrbe the sgnal model and parameters. Secton III deals wth the background of the M-target ID problem. The soluton for M= and the general form of the M-ary soluton assumed n [5] are descrbed. Also, the waveform that maxmzes the mutual nformaton between a Gaussan target ensemble and the receved waveform s llustrated. In Secton IV, we show results for the sngle-llumnaton case. We compare the error performance of dfferent waveforms startng wth bnary and subsequently multple hypotheses. Secton V descrbes sequental testng appled to multple hypotheses. An teratve scheme s ntroduced, descrbng the update of the probabltes and the transmt waveform at each step. Ths scheme s combned wth the sequental multhypotheses testng. Results show the relatve performance of dfferent transmsson technques and the reduced number of teratons obtaned by transmttng the waveform that maxmzes mutual nformaton. In Secton VI, we make our conclusons. II. PROBLEM FORMUALTION AND SIGNAL MODEL It s assumed that we have M targets characterzed by ther mpulse responses h (, =,,3,..., M.The mpulse responses are real, tme-lmted and chosen as sample functons of a Gaussan random process wth a specfed power spectral densty (PSD). The receved sgnals are assumed to be corrupted by addtve whte Gaussan nose (AWGN). Our am s to dentfy the true target from among all possble targets. In a basc sense, t can be vewed as a hypothess testng problem, where one needs to choose a hypothess based on the receved sgnal. The receved sgnal can be represented as r ( = s( h( + n( where r( represents the receved sgnal, s( the transmt sgnal, h( the mpulse response of the true target, and n( s AWGN. The dscrete-tme verson of the above equaton s used n our smulatons. All mpulse responses are normalzed to have unt energy such that where T s N n= h ( n) T s = s the samplng nterval and N s the number of samples. Our problem of nterest s as follows: gven a known set of target mpulse responses, fnd the waveforms s( that maxmze the probablty of correct classfcaton of the targets. III. WAVEFORM DESIGN The M-target ID problem has been consdered n the lterature n dfferent ways. The transmt-recever optmzaton theory n [5] was shown to be an applcaton to the target ID problem wth two targets. In the case of just two targets n AWGN, the problem transforms nto maxmzng the L norm dstance between the target echoes n sgnal space. Therefore the problem becomes max y( y ( dt = max s( s( y( where y( = y( y (. The soluton to ths turns out to be the egenfuncton assocated wth the largest egenvalue of the Fredholm ntegral equaton of the second knd, namely λ T max opt τ) = sopt ( τ ) K( τ, τ ) 0 () s ( dτ () ( τ K τ, τ ) = h ( t τ ) h( t ) dt (3) where h( = h ( h (, λ max s the maxmum egenvalue of the kernel K, ( s the optmum transmt sgnal to be s opt determned, and K s the kernel formed from the mpulse responses as shown n (3). Furthermore, t has been suggested that ths can be extended to an M-target ID problem by maxmzng the weghted average separaton between hypotheses. The soluton wll then have the same form as () wth the kernel beng N K( τ, τ ) = wm, n hm, n ( t τ) hm, n ( t τ ) dt (4) m, n N where hm, n ( = hm ( hn ( and = N!. ( N )!! The weghts to be assgned to the ndvdual pars of hypotheses are not descrbed n [5]. Moreover, there s no proof that (4) provdes the optmal soluton for an M-target ID problem. Ths partcular soluton wll be referred to as the egensoluton n the later parts of ths paper.

3 Gven a Gaussan target ensemble of random mpulse responses g( wth spectral varance σ G ( f ), the waveforms confned to the symmetrc tme-nterval [- T/,T/] that maxmze the mutual nformaton between the receved waveform and the ensemble n addtve Gaussan nose wth one sded power spectral densty P nn ( f ), s derved n []. The soluton has the magntude-squared spectrum gven by Pnn ( f ) T S( f ) = max 0, A σ G ( f ) (6) and A s found by solvng the equaton Pnn ( f ) T Es A df W G f = max 0, σ ( ) where E s s the energy of the transmt sgnal and P( H ) s the probablty of hypothess. An nterestng observaton s that (7) performs waterfllng [3] on the Pnn ( f ) T functon r ( =. Ths soluton wll be referred to as σ ( f ) G the waterfllng soluton n subsequent sectons of ths paper. Snce we have fnte number of known mpulse responses, we do not have a Gaussan target ensemble n our case. We estmate the spectral varance of the target ensemble usng n n G ( f ) = H ( f ) P( H ) = = σ H ( f ) P( H ) (8) and then apply (6) and (7). It was mentoned n [] that the waveform whose spectrum s descrbed by (6) s partcularly useful n dentfyng a target or extractng nformaton about a target, as opposed to optmum detecton where one needs to maxmze SNR by focusng energy nto the mode correspondng to the largest egenvalue of the target response. Subsequent sectons of ths paper explore the applcaton of the above waveforms wth respect to bnary and M-ary dentfcaton. IV. SINGLE ILLUMINATION In ths secton, we explore target classfcaton usng a sngle actve transmsson. The transmt sgnal nteracts wth one of the targets and the echo s corrupted by AWGN. In general, the targets are allowed to have unequal pror probablty. The recever s assumed to perform maxmum-lkelhood detecton. (7) A. Bnary case (M=) Fg. Error rate for two targets When we have only two targets, t s proved that the egensoluton gves the optmal performance by maxmzng SNR. Ths s acheved by puttng maxmum energy nto the mode correspondng to the largest egenvalue of the dfference of target responses h ( and h (. When the targets have unequal probabltes, the detecton threshold changes, whle the optmal waveforms reman the same as determned for the case of equal probabltes. The performance of ths waveform s compared wth a wdeband mpulse sgnal, a rectangular pulse and the waterfllng soluton. Probabltes of error are determned analytcally usng standard methods for detecton of sgnals n AWGN [8]. Fgure shows error rates n the case of two targets. The egensoluton performs the best among the waveforms consdered. Ths s obvous from the fact that t tres to separate the two target echoes as far as possble n sgnal space. The waterfllng soluton s also seen to perform better than the mpulse and the rectangular pulse because these latter two waveforms are not ntentonally matched to the target responses. The mpulse outperforms the rectangular pulse because t s at least matched to the PSD from whch the mpulse responses were generated. B. Multple targets (M>) In the case of three targets, there s no sngle dstance to be maxmzed snce there are three possble dstances between the hypotheses. For the M-ary case, pror probabltes ndcate whch hypotheses are most mportant to separate n the receve sgnal space. Therefore, pror probabltes affect both the transmt waveform and the detecton thresholds. For unequal pror probabltes, weghtng the kernels by the product of the probabltes of the two hypotheses s a reasonable approach. If one of the targets has low

4 probablty, then t s most mportant to separate the other hypotheses. For example, f the probabltes are P = 0.4, P = 0.55, and P 3 = 0.05, respectvely, then we need to worry less about the dstances from the thrd hypothess to the other two. The weghts for use n (4) would be w, = 0., w,3 = 0.0, and w,3 = Therefore, the weghts reflect the relatve mportance of the dfferent dstances. In [6, 7], t s shown that average dvergence s a reasonable crteron for optmalty, snce t s bounded below by twce the nformaton rate of the channel. The average dvergence between hypotheses s gven by J ( H ) = P Pj y y j (8) where, are the pror probabltes of hypotheses and j P and, y are the target echoes wth y ( = s( h (. y P j j Therefore, the soluton n (4)-(5) s equvalent to maxmzng average dvergence n (8) when weghts are chosen as a product of pror probabltes. Fg. Error rates for P =0.5, P =0.5, P 3 =0 C. Sngle-Illumnaton Results For our analyss, mpulse responses were randomly chosen as sample functons of a Gaussan random process wth flat power spectral densty. For every set of mpulse responses, we performed Monte-Carlo averagng over the nose. In addton, we averaged over 50 sets of mpulse responses to avod the performance beng affected by the choce of a partcular set of mpulse responses. The extreme case of probabltes, where one can expect to see these product weghts show a sgnfcant dfference n performance, s when one of the probabltes s zero or close to zero. Ths case s shown n Fgure. In Fgure 3, we show results for a case wth pror probabltes that are unequal, but not as drastc as n Fgure. One nterestng thng to note s that the performance of the mpulse waveform approaches the optmal solutons at hgh SNR. Ths may be attrbuted to the fact that snce the mpulse responses are chosen from a process wth a flat power spectrum, t s enough just to send out a wdeband sgnal matched to the ensembles PSD at hgh SNR. We are assured of not wastng energy n frequences where there wll not be a response, snce the mpulse responses are spread across the spectrum. Fgure 4 shows that, for three equprobable targets, the waterfllng soluton approaches the egensoluton performance for hgh SNR. In the case of four targets, however, there are many more dstances to be maxmzed. For an equprobable, 4-target stuaton as shown n Fgure 5, the waterfllng and the Fg.3 Error rates for P =0., P =0., P 3 =0.6 egensoluton perform almost equally well for low SNR, but for hgher SNR, the waterfllng soluton performs better. Ths llustrates the fact that as the number of hypotheses gets larger, just puttng energy nto the mode correspondng to the largest egenvalue of (4) s not suffcent. The mplcaton s that we need to spread energy nto other modes as well. The waterfllng soluton seems to perform better as the number of hypotheses gets larger. As shown n (6)-(7), the waterfllng soluton tends to put more energy nto frequences that have greater varance among the target frequency responses. The egensoluton though, tends to maxmze the average separaton between the hypotheses. Average separaton, however, mght be maxmzed by makng one dstance much larger than all others, resultng n many hypotheses that are not separated well at all. Snce the dstance n sgnal space s drectly related to the dstance n

5 Fg.4 Error rates for the three targets equprobable Fg6. Comparson of frequency response for egensoluton Fg.5 Error rates for 4 hypotheses frequency space (Parseval s Theorem), the egensoluton tends to put energy nto the sngle frequency where the functon N, j G ( f ) G ( f ) j s the maxmum, where G ( f ) denotes the Fourer transform of the mpulse th response of hypothess. Ths concept s llustrated n Fgure 6. The waterfllng soluton, however, spreads energy over multple peaks n the ensemble s spectral varance, as long as there s enough SNR. Comparng Fgures 6 and 7, we see the multple peaks n the spectral varance functon. These peaks ndcate frequences that are useful for target dscrmnaton. At low SNR, only the largest peak s strong enough to provde useful nformaton. At hgh SNR, Fg.7 Comparson of frequency responses for waterfllng soluton however, there are other frequences that can be exploted. Ths s acheved wth the waterfllng soluton. V. ITERATIVE ADAPTIVE TRANSMISSION Instead of makng a decson after only a sngle step, another approach s to dvde up the avalable energy nto steps and transmt multple waveforms. In that case, every tme a sgnal s receved, we learn somethng about the target. Therefore, based on all prevous receved waveforms, we are able to send out an mproved waveform durng the next transmsson. A. Bayesan Update of Probabltes We have developed an teratve scheme, whch starts out by assumng that all targets are equally probable. After each

6 transmsson and recepton, the probabltes of the targets are updated usng the Bayesan update rule p( yk H ) p( H yk ) p( H y ) (9) k = n j= p( y where k s the teraton number, k H ) p( H y th k ) are the hypotheses, s the receved sgnal at the k teraton, and p( H yk ) s the probablty of Hypothess after k teratons. The updated probabltes are used n determnng the new waveform to be sent. In the case of fxed waveforms lke mpulse and rectangular waveforms, no update s performed. In case of the egensoluton, the weghts, w,j, change wth the new probabltes. Wth the waterfllng soluton, the varance of the target ensemble s updated accordng to (8). Ths new varance s used to determne the waveform for the next transmsson. H yk Fg.8 Iteratons to reach a decson for a flat PSD B. Sequental Hypothess Testng Sequental hypothess testng s a standard procedure for testng between hypotheses by successve observatons. Sequental hypothess testng has the property that t mnmzes the number of teratons to reach a decson for a fxed probablty of msclassfcaton, compared to the fxed teraton method. The teratve procedure llustrated n the prevous secton can be coupled wth the sequental hypotheses testng procedure to observe the relatve performances wth respect to the number of teratons to reach a decson, for a fxed error rate. At each teraton the lkelhood ratos of all pars of hypotheses are computed, and f the lkelhood ratos for any one of the hypotheses aganst all others s more than a set threshold, that partcular hypothess s decded as the true one []. If no sngle hypothess s a suffcently clear choce, we contnue to take measurements by updatng the probabltes and transmsson waveforms. The threshold s fxed based on the error rate set for the probablty of msclassfcaton, whch s decded before the experment begns [9]. C. Sequental-Testng Smulaton Results Sequental mult-hypotheses testng was performed wth the probablty of msclassfcaton set to The mpulse responses were randomly chosen from a Gaussan random process wth a flat PSD for M = 4. The dfferent transmt schemes were mplemented assumng that the targets were equally lkely. The mpulse and the rectangular pulse were transmtted wthout any update. For both the egensoluton and the waterfllng soluton, the probabltes and waveforms were updated after each transmsson. Fg.9 Dstances between hypotheses for flat PSD The waterfllng soluton seems to perform the best as shown n Fgure 8. It reduced the number of teratons consderably compared to the other waveforms under consderaton. Assgnng equal weghts or product of probabltes for the egensoluton does not affect results n ths case as they seem to overlap. Snce the probablty of error n decdng on a hypothess depends manly on the dstance between the hypotheses, t s reasonable to nvestgate the mnmum and the average dstances between the hypotheses for dfferent transmsson waveforms. Fgure 9 s consstent wth the fact that the egensoluton maxmzes the average separaton between the hypotheses. Although the waterfllng soluton does not strctly maxmze the mnmum dstance, t does so more often than not as shown n Fgure 9. Ths makes the waterfllng soluton perform better on average than other waveforms Next, we consder the mpulse responses from a Gaussan random process wth a low-pass hammng PSD, for whch

7 Fg.0 Iteratons to reach a decson for a Hammng PSD Fg. Dstances between hypotheses for Hammng PSD the results are shown n Fgure 0. The waterfllng soluton performed the best, and the wdeband sgnal the worst. Ths s reflected n the mnmum dstances between the hypotheses n Fgure, where the waterfllng soluton maxmzes the mnmum dstance and the mpulse yelds the mnmum, often. Moreover, snce the mpulse responses contan low frequency content, transmttng wdeband sgnal wastes energy n unwanted frequences. VI. CONCLUSIONS We have analyzed varous potental transmsson waveforms for an M-target ID problem. The waveform that optmzes mutual nformaton between a target ensemble and the receved waveform was appled to ths problem. The M=3 and M=4 cases were smulated, and t was seen that the waterfllng soluton performs well as the number of hypotheses ncrease, especally at hgh SNR. An teratve procedure, whch calculates the probabltes of targets at each step and updates the transmtted waveform, was ntroduced. Ths teratve procedure was coupled wth sequental mult-hypotheses testng. It was shown that the waveform that optmzes mutual nformaton s the best for an M-target ID problem for M >. REFERENCES [] P. Armtage, Sequental analyss wth more than two alternatve hypotheses and ts relaton to dscrmnant functon analyss, J.Roy. Statst. Sot. Suppl., vol. 9, pp , 947. [] M.R.Bell, Informaton Theory and Radar Waveform Desgn, IEEE Trans. Inform. Theory, vol. 39, pp , Sept [3] T.M.Cover and J.A.Thomas, Elements of Informaton Theory. New York: Wley, 99. [4] B. Dunbrdge, Asymmetrc Sgnal Desgn for the Coherent Gaussan Channel, IEEE. Trans. Inform. Theory, vol. IT-3, No.3, pp , July 967. [5] J. C. Guerc and S.U. Plla, Adaptve Transmsson radar: The Next Wave, Natonal Aerospace and Electroncs Conference (NAECON) 000, Proceedngs of the IEEE 000. [6] T.L. Grettenberg, Sgnal selecton n communcaton and radar systems. IEEE. Trans. Inform. Theory, vol. IT-9, pp , Oct.963. [7] E.Mosca, Probng Sgnal Desgn for Lnear Channel dentfcaton. IEEE Trans. Inform. Theory, vol. IT-8, No.4, pp , July 97. [8] J.Proaks, Dgtal Communcatons. New York: McGraw Hll, 00. [9] A.Wald, Sequental Analyss. New York: Wley, 947.