Evaluation of Modulus-Constrained Matched Illumination Waveforms for Target Identification

Transcription

1 Evaluaton of Modulus-Constraned Matched Illumnaton Waveforms for Target Identfcaton Junhyeong Bae Department of Electrcal and Computer Engneerng Unversty of Arzona 3 E. Speedway Blvd, Tucson, Arzona 857 dolbt@emal.arzona.edu Naan A. Goodman Department of Electrcal and Computer Engneerng Unversty of Arzona 3 E. Speedway Blvd, Tucson, Arzona 857 goodman@ece.arzona.edu Abstract In pror wor, we have appled matched llumnaton strateges to target dentfcaton by a closed-loop radar system. In e closed-loop system, multple waveforms are transmtted n successon, but each s customzed based on e returns from pror transmssons. In s pror wor, however, e matched waveforms were not constraned to be constant modulus. Ths current paper evaluates e performance of closed-loop radar w constant-modulus matched llumnaton. We also compare e performance of non-constant-modulus llumnaton under a pea power constrant. Fnally, we use smple target models and assume unnown orentaton, raer an e determnstc or Gaussan target models used n earler wor. I. INTRODUCTION Whle many modern radars focus on mprovng performance by adaptve sgnal processng n e recever, cogntve radar [] focuses on mprovng performance not only by adaptve sgnal processng, but also rough an adaptve radar transmtter. One example of cogntve radar at we have presented s where e adaptve transmtter generates a new temporal waveform optmzed for target dentfcaton based on pror measurements and pror nowledge. Thus, e cogntve radar encompasses e transmtter, recever, and envronment nteractng togeer n a closed-loop system. Our earler results show mproved performance n resource-constraned (.e., fnte energy) and nterference-lmted envronments. A hgh-resoluton range profle (HRRP) s a radar target sgnature at s bacscatterng power recorded as a functon of delay when a wdeband sgnal s transmtted []. However, even a slght change n target aspect angle generates a dfferent HRRP, so many HRRPs must be stored n a template lbrary n order to accurately represent e target. Ths paper bulds on earler wor n e followng ways. Frst, n earler wor e targets have been modeled as eer determnstc or Gaussan w a nown PSD. In practce, targets are neer, so here we have mplemented very smple scatterng models at vary w angle. The target orentaton s unnown, but assumed to le wn a certan range. Second, we apply constant modulus constrants to e matched llumnaton waveform desgned by e mutual nformaton metrc. Thus, e goal of s paper s to mplement closed-loop, matched llumnaton w more realstc target models and waveform constrants. The outlne of s paper s as follows. In Secton II, we develop e problem statement and sgnal model. In Secton III, we present e target model. In Secton IV, we brefly show e waveform desgn technque. In Secton V, we descrbe how to generate e constant modulus waveform from a gven Fourer transform magntude. We also dscuss e pea-power (pea modulus) waveform normalzaton. In Secton VI, we revew e decson-mang procedure and probablty updates for closed-loop radar. In Secton VII, we present smulated results, and n Secton VIII, we mae conclusons. II. PROBLEM STATEMENT AND SIGNAL MODEL The target dscrmnaton performance of monostatc radar w customzed waveforms s evaluated n e presence of addtve whte Gaussan nose (AWGN). The problem formulaton s as follows. It s assumed at one of M targets s present. Target sgnatures are senstve to target aspect angle, so e set of target profles are dvded nto unformly spaced sectors. The number of sectors s assumed to be N h for each target. Target profles are en calculated at multple angles wn each sector, and e profles are averaged to obtan a mean template for at sector [3]. The target profles for each target are assumed to be nown exactly for a gven angle; erefore, e mean templates for e N h sectors for all M targets h (, =,..., N h,..., MNh are also perfectly nown. The exact target orentaton s not nown, but assumed to le wn just a few sectors. Ths //$6. IEEE 87

2 represents pror nowledge about e approxmate target orentaton. The true target sgnature, g (, s drawn from a random angle wn e range of angles specfed by e pror nowledge. We want to determne not only whch target g ( belongs to, but also whch angular sector. The radar waveform s defned as s (, and e receved sgnal y ( due to s ( s defned as y ( = g( * s( + n( () where * represents e convoluton operator and n ( ndcates AWGN at has normalzed power σ =. We consder e probablty densty functon (pdf) of y ( under e hypoess. We treat h ( as determnstc and snce n ( s AWGN, e dstrbuton of y ( under e hypoess s Gaussan. Swtchng to a dscrete-tme notaton [4], e receved sgnal s y = Sg + n [4], and e pdf of y under e hypoess s p y = exp( ( y μ ) K ( y μ )) () ( H ) H N y, y, y, ( π ) K y, where K y, = σ n I s e covarance matrx of y under target hypoess and μ y, = Sh s e waveform-dependent mean of y under e hypoess. In [5] and oer technques, classfcaton s based on e estmated target HRRP, so to use s meod we would need to estmate e target HRRP from e measured sgnal y. If e transmtted waveform s a hgh-bandwd waveform w a good (.e., low-sdelobe) autocorrelaton functon, en all at needs to be done s to apply matched flterng. However, e matched llumnaton waveforms at we calculate are desgned to classfy targets, not necessarly to generate a qualty estmate of e target s range profle. In oer words, a good waveform for target dentfcaton does not necessarly produce a good mage of e target. The reason at typcal classfcaton technques use estmated HRRP s s at HRRP s are naturally ntutve for humans to understand and to buld algorms around, not because ey have nherent advantages n classfcaton. Thus, we do not estmate e HRRP here because e waveform we use often has a poor autocorrelaton functon, and e resultng estmated HRRP does not gve a good result. Instead of estmatng e HRRP, every tme we update e transmtted waveform, we update e sgnal matrx S n e calculaton of e mean μ. y, III. TARGET MODEL The smplest way to model a target s usng a pont target model, n whch case e target s sgnature s just an mpulse w e proper delay, phase and ampltude. However, s model does not accurately reflect what can be observed by a radar when resoluton s suffcent for resolvng many scatterng centers on e physcal target. A more realstc n model s to consder targets such as planes and shps to have many scatterng ponts, each modeled as a pont target. Then, a target mpulse response can be modeled accordng to N s h( = γ δ ( t τ ) (3) = where N s s e number of scatterng centers on e target, γ s e scatterng coeffcent of e scatterng center, δ ( s e Drac delta functon, and τ s e round trp tme for e transmtted waveform to travel from e radar to e target and bac. The mpulse functon n (3) has nfnte bandwd n e frequency doman and cannot be modeled drectly n a computer smulaton. However, any observed radar sgnal wll be bandlmted, so t s possble to represent e target w a e bandlmted transfer functon H (w) at equals N H ( w ) = = s jw e τ γ (4) wn e radar bandwd and zero elsewhere. We can model e transfer functon accordng to (4) over e radar band and convert bac to e tme doman va nverse Fourer Transform to model e target mpulse response. The tmedoman mpulse response s now tme-lmted and can be represented w a fnte number of samples. To mae smple targets, we defne our own arbtrary target outlnes. The targets have physcal sze at s generally accurate for targets of nterest, and we place scatters n ey postons along e target outlne. As we change aspect angle, e targets are rotated as well as e scatters. Thus, e relatve range between radar and e scatters changes, causng fluctuatons due to fadng, and hence changes n e target mpulse response. The target outlnes are admttedly low-fdelty at s pont; ongong wor s beng done to ncorporate target models calculated usng hgh-fdelty electromagnetc (EM) software. IV. WAVEFORM DESIGN The waveform desgn strategy at we use s based on mutual nformaton. The strategy was orgnally presented n [6] by Bell and s summarzed as follows. It s assumed at we have an ensemble of target mpulse responses. If we assume at e transmt waveform s tme-, energy-, and approxmately frequency- lmted, e transmt waveform at maxmzes e mutual nformaton between e ensemble of mpulse responses and e radar receved sgnal exsts, and e waveform has an energy spectrum accordng to σ nty max, A S( f ) = σ H ( f ) f Ts f > T s (5) //$6. IEEE 87

3 where e ensemble s spectral functon s denoted as σ H ( f ) = E{ H( f ) E{ H ( f )} }, H ( f ) s e target transfer functon, and T s e nterval at whch e waveform and mpulse responses are sampled. The energy constrant A s calculated accordng to e equaton E = TS TS σ nty max, A df. (6) σ H ( f ) The waveform desgn technque above s for a Gaussan ensemble, but for a fnte number of hypoeses n a target dentfcaton scenaro, e ensemble of potental transfer functons s not Gaussan. However, e procedure s stll ntutvely correct, and we extend e spectral varance functon accordng to [7] H MN h MN h P H ( f ) = = σ ( f ) = P H ( f ) (7) where H ( f ) s e transfer functon assocated w e mean template. Ths technque generates e waveform by pourng energy nto e functon σ nty σ H( f) untl e energy constrant s met. Ths technque s called waterfllng [6][8]. V. NEW WAVEFORM CONSTRAINTS Bell appled energy, tme, and frequency constrants to generate e nformaton-based waveform n [6], but here we add two more constrants, whch are constant modulus [9][] and maxmum modulus normalzaton [][] constrants. The two constrants are desred because we do not want e temporal waveform to have hgh pea ampltude and to operate neffcently by operatng durng part of e waveform at less an pea power. The technque to generate a constant modulus sgnal from a gven Fourer transform s summarzed below and based on e technque n [9]. We defne D M as e group of functons { v ( } at have e same Fourer transform magntude F (w) over e frequency nterval Ψ. Then, an arbtrary functon x ( can be projected to a closest pont on D M by a magntude projecton operator P M. When e Fourer transform of x ( can be represented by jω( w) X ( w) = X ( w) e, e magntude projecton of an arbtrary functon x ( s defned as jω( w) F( w) e, w Ψ PM x( =. (8) X ( w), w Ψ' We also denote D A as e set of functons { v ( } at have constant ampltude B over e tme nterval T. Then, an arbtrary functon x( can be projected to a nearest pont on D A by an ampltude projecton operator P A, and e projecton procedure s j B e PA x( = x(, φ (, t T oerwse jφ( where x ( can be represented by a( e. The magntude and ampltude projecton are combned accordng to x + ( PA PM x ( () where x ( s e projecton terated functon. After a number of magntude and ampltude projectons, e functon x ( has constant modulus envelope whle approxmately mantanng e prescrbed Fourer transform magntude. = Anoer constrant s e maxmum modulus normalzaton presented n [][], whch s essentally a constrant on e nstantaneous pea power. When applyng s constrant, we force e sgnal envelope to have maxmum ampltude of E A = s () N w where E s e waveform energy and N w s e number of dscrete-tme samples representng e waveform. We can apply s normalzaton to any of e waveforms we study, but because e constrant s essentally a scalng factor, e shape of e waveform does not change. If we desgn a nonconstant modulus waveform w energy E s, but en apply e maxmum modulus constrant, e resultng waveform wll have less energy an orgnally ntended. VI. FIXED NUMBER OF ITERATIONS AND BAYES THEOREM To evaluate waveform performance, we perform experments where e radar maes a predetermned, fxed number of transmssons before mang a decson. After every observaton, we compute e lelhoods for each hypoess. These lelhoods are used to update e waveform, but a decson s only made after e predetermned number of observatons. The lelhood expresson of e hypoess after e llumnaton can be formed accordng to Λ ( y) p ( y ) p ( y (9) = p ) P () where p ( y ) s e pdf of e receved sgnal due to e transmtted waveform for e hypoess (defned above n ()), y s e sgnal receved on e transmsson, and P s e pror probablty of e hypoess before tang any measurements. After e gven number of observaton, e experment s termnated. The decson H s made n favor of e target and sector w e hghest lelhood. At each transmsson, we update e hypoess probabltes to modfy e waveform. Usng Bayes //$6. IEEE 873

4 Theorem, e hypoess probablty after e transmsson s p( H y Λ ) = (3) p( y ) where p( y ) s a scalng factor such at e sum of e p( H y ) s one. The waveform s modfed accordng to updated hypoess probabltes as descrbed n [7]. VII. RESULTS We apply e eory we have dscussed to a computer smulaton. We assume at we have two targets. The orentaton for each target s dvded nto unformly spaced sectors. The sectors are one degree wde, and e lbrary s defned at.-degree ntervals. The target HRRPs wn a sector are averaged to obtan a mean template for use n e probablty updates and waveform calculatons. Thus, we have a mean template for every one degree. We randomly choose a target and orentaton angle and use two adjacent templates for each of e two targets to form four hypoeses. We compare e performance of four dfferent waveforms n dentfyng ese hypoeses. The frst ree waveforms are nformaton-based waveforms [6], and e last s a wdeband waveform. The frst nformaton-based waveform s obtaned drectly from waterfllng n e frequency doman. The tme-doman waveform s found va nverse DFT and results n a non-constant modulus temporal shape. The second nformaton-based waveform s a constant-modulus verson [9] at approxmates e optmum mutual-nformaton-based spectrum. The rd nformatonbased waveform s a scaled verson of e non-constant modulus waveform, scaled to meet e maxmum modulus constrant [][]. We perform fve transmssons to mae a decson. We run, Monte Carlo trals to estmate e probablty of an ncorrect decson. Fgure shows e complex constellaton of temporal waveform before and after e constant modulus constrant s appled. Before e constant modulus constrant s appled (e unconstraned optmzed waveform), e temporal waveform has fluctuaton ampltudes and also hgh pea ampltudes. However, after e constant modulus constrant s appled (e optmzed waveform w e constant modulus constran, e temporal waveform has constant ampltude. Fgures and 3 show e correspondng Fourer transform magntudes. Fgure shows e Fourer transform magntude of e non-constant modulus optmzed waveform. Fgure 3 shows e Fourer transform magntude of e optmzed waveform w e constant modulus constrant. The constant modulus constrant spreads e waveform energy nto addtonal frequency bands, but e two hgh pea magntudes are mantaned. The two Fourer transform magntudes loo smlar. Imagnary Before constant modulus After constant modulus Real Magntude Fgure. Complex constellaton of unconstraned and constant modulus constraned waveform. Orgnal frequency spectra 5 5 frequency ndex Fgure. Fourer transform magntude of unconstraned waveform. Magntude Constant modulus frequency spectra 5 5 frequency ndex Fgure 3. Fourer transform magntude of constant modulus constraned waveform //$6. IEEE 874

5 .. waveform w non constant modulus waveform w constant modulus waveform w maxmum modulus normalzaton Fxed number of 5 teraton.8 Ampltude tme ndex Fgure 4. The temporal structure of ree mutual nformaton based waveform. Error rate mpulse waterfllng w non constant modulus waterfllng w constant modulus waterfllng w maxmum modulus normalzaton Es(dB) Fgure 5. Error rates versus waveform energy n fxed number of teratons. Fgure 4 shows e temporal structure of all ree mutual-nformaton-based waveforms. The non-constant modulus waveform has ampltude at vares, but e constant modulus waveform has constant ampltude. The maxmum modulus normalzaton waveform loos le e non-constant modulus waveform, but e pea s matched to e constant modulus waveform. Fgure 5 shows error rates of e target dscrmnaton. We apply new constrants for constant modulus and maxmum modulus normalzaton waveforms, respectvely. The result shows at e unconstraned optmzed waveform performs best, but e performance loss s mnor for e sub-optmum waveform w e constant modulus constrant. The reason s at e Fourer transform magntude of e sub-optmum waveform w e constant modulus constrant has smlar Fourer transform magntude as e unconstraned optmzed waveform. The performance loss for maxmum modulus normalzaton becomes hgh as e waveform energy ncreases, because e waveform w maxmum normalzaton constrant does not produce as much energy n e allotted tme. VIII. CONCLUSIONS We have mplemented a closed-loop radar system w two new constraned adaptve waveforms for target dscrmnaton. The constrants are constant modulus [9] and maxmum modulus normalzaton [][]. The waveforms were mplemented n e radar system such at e temporal waveforms are modfed based on prevous measurements and pror nformaton. We also used more realstc target models an used n prevous wor. The result shows at e constant modulus waveform has only a small performance loss compared to e optmzed non-constant modulus waveform. The maxmum modulus normalzaton shows hgh error rate as e waveform energy ncreases because s waveform operates neffcently. ACKNOWLEDGMENT The auors acnowledge support from e Ar Force Offce of Scentfc Research (AFOSR) va grant #FA95578 and from e ONR va grant #N REFERENCES [] S. Hayn, Cogntve radar: a way of e future, IEEE Sg. Proc. Mag., vol. 3, no., pp. 3-4, Jan. 6. [] S. Hudson and D. Psalts, Correlaton flters for arcraft dentfcaton from radar range profles, IEEE Trans. on Aerospace and Electronc Systems, vol. 9, no. 3, July, 993. [3] D. A. Garren, M. K. Osborn, A. C. Odom, J. S. Goldsten, S. U. Plla, and J. R. Guerc, Optmal transmsson pulse shape for detecton and dentfcaton w uncertan target aspect, n Proc. IEEE RadarConf., Atlanta, GA, May 3,, pp [4] D. A. Garren, M. K. Osborn, A. C. Odom, J. S. Goldsten, S. U. Plla, and J. R. Guerc, Enhanced target detecton and dentfcaton va optmzed radar transmsson pulse shape, Proc. IEEE, vol. 48, no. 3, pp. 3 38, Jun.. [5] H. Lu, Z. Yang, K.He, and Z. Bao, Radar hgh range resoluton profles recognton based on wavelet pacet and subband fuson, IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng, Phladelpha, PA, USA, 8-3 March, 5. [6] M.R. Bell, Informaton eory and radar waveform desgn, IEEE Trans. Info. Theory, vol. 39, no. 5, pp , Sept [7] N.A. Goodman, P.R. Venata, and M.A. Nefeld, "Adaptve waveform desgn and sequental hypoess testng for target recognton w actve sensors," IEEE J. Selected Topcs n Sgnal Processng, vol., no., pp. 5-3, June, //$6. IEEE 875

6 [8] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. New Yor: Wley, 99. [9] S.U. Plla, K.Y. L, and H. Beyer, Constructon of constant envelope sgnals w gven Fourer Transform magntude, IEEE Radar Conference, Pasadena, Calforna, USA, May 4-8, 9. [] L.K. Pattton, On e satsfacton of modulus and ambguty functon constrants n radar waveform optmzaton for detecton, Ph.D. dssertaton, Wrght State Unversty, 9. [] L.K. Patton, B.D. Rglng, Modulus constrants n adaptve radar waveform desgn, IEEE Radar Conference, Rome, Italy, May 6-3, //$6. IEEE 876