adaptive detection and diversity order in multistatic...

TABLE VISmall- and Medium-Scale Test Hub Positions

Total Displacement (cm)Data Set Polarization from Start

1/2 Vertical/Horizontal 0.03/4 Vertical/Horizontal 1.75/6 Vertical/Horizontal 2.57/8 Vertical/Horizontal 3.89/10 Vertical/Horizontal 5.011/12 Vertical/Horizontal 10.113/14 Vertical/Horizontal 15.215/16 Vertical/Horizontal 20.317/18 Vertical/Horizontal 25.421/22 Vertical/Horizontal 20.523/24 Vertical/Horizontal 50.825/26 Vertical/Horizontal 91.427/28 Vertical/Horizontal 111.729/30 Vertical/Horizontal 132.131/32 Vertical/Horizontal 172.733/34 Vertical/Horizontal 213.435/36 Vertical/Horizontal 254.037/38 Vertical/Horizontal 294.639/40 Vertical/Horizontal 335.8

RICHARD KETCHAMJEFF FROLIKDept. of Electrical and Computer EngineeringUniversity of Vermont33 Colchester Ave.Burlington, VT 05405E-mail: ([email protected])

JOHN COVELLGoodrich CorporationFuel and Utility SystemsVergennes, VT

REFERENCES

[1] Pottie, G., and Kaiser, W.Wireless integrated network sensors.Communications of the ACM, 43, 5 (May 2000), 51—58.

[2] Estrin, D., Girod, L., Pottie, G., and Srivastava, M.Instrumenting the world with wireless sensor networks.In Proceedings of the International Conference onAcoustics, Speech and Signal Processing (ICASSP 2001).

[3] MIT Technology Review10 emerging technologies that will change the world.MIT Technology Review, Feb. 2003.

[4] Furse, C., Chung, Y., Lo, C., and Pendayala, P.A critical comparison of reflectometry methods forlocation of wiring raults.Smart Structures and Systems, 2, 1 (2006), 25—46.

[5] Wireless Smoke DetectionOnline: http://www.securaplane.com/smoke2.html,Securaplane Technologies Inc. (21Nov06).

[6] St. John, E.Operational issues wireless overview.Presented at the World Airline Entertainment AssociationSingle Focus Workshop (WAEA SFW), Washington,D.C., Nov. 19—20, 2002.

[7] Securaplane Technologies, Inc.Press Release, “Securaplane Technologies AwardedBoeing 787 Contract for Wireless Emergency Lighting,”Dec. 21, 2004.Online: http://www.securaplane.com/releases/pr0095.html(21Nov06).

[8] Fitzhugh, C., and Frolik, J., Ketcham, R., Covell, J., andMeyer, T.2.4 GHz multipath environments in airframes.Presented at the IEEE Conference on Wireless andMicrowave Technology (WAMICON 2005), Clearwater,FL, Apr. 7—8, 2005.

[9] Frolik, J.A case for considering hyper-Rayleigh fading.IEEE Transactions on Wireless Communications, 6, 4 (Apr.2007).

[10] Durgin, G., Rappaport, T., and de Wolf, D.New analytical models and probability density functionsfor fading in wireless communications.IEEE Transactions on Communications, 50, 6 (June 2002).

[11] Chong, C-C., and Yong, S.A generic statistical-based UWB channel model forhigh-rise apartments.IEEE Transactions on Antennas and Propagation, 53, 8(Aug. 2005), 2389—99.

[12] Rappaport, T.Rayleigh and Ricean distributions.In Wireless Communication Systems: Principlesand Practice, (2nd ed.), Upper Saddle River, NJ:Prentice-Hall, 2002, sect. 5.6.

Adaptive Detection and Diversity Order inMultistatic Radar

We derive the generalized likelihood ratio test (GLRT) for

multistatic space-time radar where each sensor platform has

a coherent multi-channel array. A multistatic version of the

adaptive matched filter (AMF) detector is also developed and its

performance results compared with that of the GLRT. Results

show that the multistatic GLRT outperforms the multistatic AMF

and that the performance advantage increases with the number of

radars in the system. The concepts of geometry gain and diversity

gain for multistatic STAP are defined in terms of asymptotic

properties of the system’s probability of miss.

I. INTRODUCTION

Space-time adaptive processing (STAP) is anextension of adaptive antenna techniques that uses thesignals received on multiple elements of an array andmultiple pulses of a coherent processing interval (CPI)to form a space-time filter that maximizes systemsignal-to-noise ratio (SNR). The benefits of thisfiltering technique are enhanced detection performance

Manuscript received February 6, 2007; revised December 7, 2007;released for publication April 21, 2008.

IEEE Log No. T-AES/44/4/930746.

Refereeing of this contribution was handled by M. Rangaswamy.

0018-9251/08/$25.00 c° 2008 IEEE

CORRESPONDENCE 1615

Authorized licensed use limited to: The University of Arizona. Downloaded on April 5, 2009 at 16:36 from IEEE Xplore. Restrictions apply.

and improved rejection of external interference[1—3].Even though STAP is a powerful technique in

clutter cancellation, certain radar/target geometriesare less favorable than others, and ground-basedinterference is still a significant factor in mostcircumstances. If the target velocity vector isapproximately perpendicular to the radar platform’sline of sight, the target will be hard to distinguishfrom background clutter. However, this can beovercome by adding additional viewing perspectivesthat realize more radial velocity from the target.Furthermore, we show below that the number of radarplatforms observing a fluctuating target controls theasymptotic slope of the system’s probability-of-miss(Pm) performance while the specific geometry controlsthe asymptotic shift.The system that we study here is a multistatic

radar system made up of widely separated airborneplatforms. Each platform is capable of acting as bothtransmitter and receiver. As a receiver, each platformpossesses a coherent, multi-channel array capableof collecting a space-time data set. Furthermore,we also assume that each receiver can collect andseparate the data due to multiple transmitters, whichrequires separable waveforms. For example, multiplewaveforms can be separated when transmitted onnonoverlapping frequency bands. The multistaticsystem consists of several unique propagationpaths. First, there are monostatic paths wherereceivers measure reflections due to their co-platformtransmitters. Second, there are bistatic paths wherereceivers measure reflections due to the waveformstransmitted from different platforms.Multistatic radar has the advantage of viewing

the same target from multiple perspectives. Asdescribed in [4], in the presence of ground clutter thisprovides two types of performance gains: diversitygain and geometry gain. Diversity gain is obtainedby observing multiple independent observations ofa fluctuating target. For example, the MIMO radarwork of [5]—[6] shows how these fluctuations can beexploited. The work in [5]—[6] considers spectrallywhite interference. For multistatic STAP, however,the effectiveness of these multiple observations iscontrolled by how well ground clutter and otherinterference can be rejected. This rejection, in turn,depends on the geometry of the radar platform and thetarget’s velocity vector. Fortunately, a moving targetthat has no radial velocity with respect to one radarwill likely have some radial velocity with respectto a second radar provided there is enough spatialdisplacement between the two viewing perspectives.Hence, geometry controls the diversity gain, and wehave called this factor the geometry gain.In this paper, we derive constant false-alarm

rate (CFAR) detectors for multistatic STAP andquantify their performance in terms of diversity and

geometry gain. The CFAR detectors are multistaticextensions of the monostatic CFAR tests of [7] and[8]. We compute Pm and show that in the presence ofa fluctuating target, Pm curves for both CFAR testspossess an asymptotic slope. This asymptotic behavioris analogous to the probability-of-outage behaviorused to define diversity order (DO) in communicationtheory. Therefore, we propose definitions of diversityand geometry gain in terms of the asymptoticbehavior.In the next section, we define our problem

statement and signal model. Section II also developsthe multistatic versions of the GLRT and AMFdetection methods and discusses their statisticalproperties. Section III presents the results from aseries of experiments using the detection methodsdeveloped in Section II. These experiments notonly compare the methods as a function of numberof radars employed, but also outline a way tocharacterize which geometries optimize performancefor a given multistatic configuration. Also discussed inSection III is the performance increase associated withincluding bistatic paths between radars as a meansof increasing diversity gain without increasing thenumber of radar platforms. Section IV concludes thepaper.

II. SIGNAL MODEL AND DETECTION STATISTICS

A. Multistatic STAP Signal Model

Let a multistatic radar system be comprised ofmultiple airborne platforms at different locationssurrounding an area of interest. Each platformpossesses a multi-channel array; therefore, the systemmodel is defined by Q independent sets of space-timeobservations of the same area on the ground. In thecontext of this paper, independent observations canbe realized by geographical separation of the radarplatforms and from both monostatic and bistaticpropagation paths.Let the primary data corresponding to the

qth space-time data set be denoted by the vectorzq for 1· q·Q. Each zq is formed by stackingthe space-time data samples collected by a localmulti-antenna array over multiple pulses into acolumn vector where the length Nq is the numberof space-time measurements observed in a singlesnapshot of the qth data set. The qth primary datavector is defined as

H0 : zq = nq

H1 : zq = bqsq+nq:(1)

In (1), nq is the vector of noise and interferencecorrupting the qth data set, sq is the normalizedspace-time steering vector for the target hypothesisrelative to the specific propagation geometry of theqth data set, and bq is the complex amplitude of the

1616 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 44, NO. 4 OCTOBER 2008


target signature in the qth data set. H0 is the nullhypothesis, and H1 is the target-present hypothesis.The interference vector nq is a zero-mean complexGaussian process characterized by the covariancematrix Mq. The noise vectors are assumed to beindependent from data set to data set.Since the target’s absolute position and velocity

vector map into different relative angles and Dopplershifts for different data sets, each steering vectorsq is different even though they describe the sameabsolute target hypothesis. Therefore, the mappingof hypotheses from absolute parameters to relativeparameters must be very accurate, which impliesaccurate knowledge of the relative locations ofthe radar platforms. If the mapping is inaccurate,the target may still be detected, but estimates ofthe target’s absolute position and velocity may beinaccurate due to improper alignment of the relativehypotheses of each platform. In this paper, we do notaddress how the mapping of relative hypotheses takesplace, nor do we quantify the effects of inaccuratemapping on detection or parameter estimation.

B. Multistatic STAP Generalized Likelihood Ratio Test

Let Zq be a matrix of space-time measurementsthat comprise the qth data set. This matrix isdefined as Zq = [zq zq(1) zq(2) ¢ ¢ ¢zq(Kq)] wherezq is the primary data vector defined above, andzq(1), : : : ,zq(Kq) are Kq secondary data vectors. TheZqs combine to form the full matrix of observationsdue to all Q data sets, Z= [Z1 Z2 ¢ ¢ ¢ZQ]. Due to theindependence of the interference between data sets,the joint probability density function (pdf) of all datamatrices from all radar platforms reduces to a productof the individual pdfs from each of the data sets. Thejoint pdfs for the H0 and H1 hypotheses for Q datasets are

f(Z jH0) = f(Z1,Z2, : : : ,ZQ jH0) =QYq=1

fq(Zq jH0)

f(Z jH1) = f(Z1,Z2, : : : ,ZQ jH1) =QYq=1

fq(Zq jH1):(2)

Under the H0 hypothesis, the primary data vectorhas the same pdf as the secondary data. Since clutterreturns are assumed to be complex Gaussian, the nullhypothesis pdf for a single data vector and data setcan be represented as

fq(zq(k) jH0) =1

¼NqkMqkexp[¡zHq (k)M¡1q zq(k)]

(3)

where (¢)H denotes the conjugate transpose operationand k ¢ k denotes the determinant of a matrix. In (3),Mq is the interference covariance that applies to the

qth data set; therefore, Mq = Efzq(k)zHq (k)g where Ef¢gdenotes expected value. Equation (3) also applies toall zq(k) under the H1 hypothesis, but the pdf for theprimary data vector under H1 is

fq(zq jH1) =1

¼NqkMqkexp[¡(zq¡ bqsq)HM¡1q (zq¡ bqsq)]:

(4)If the space-time data snapshots within a given dataset are independent, the pdf for that data set is

fq(Zq jH0) = fq(zq,zq(1), : : : ,zq(Kq) jH0)

= fq(zq jH0)KqY

k=1

fq(zq(k) jH0)

fq(Zq jH1) = fq(zq,zq(1), : : : ,zq(Kq) jH1)

= fq(zq jH1)KqY

k=1

fq(zq(k) jH0):

(5)

Using the identity vHA¡1v= tr[A¡1V] where v is avector, A is a square matrix, V= vvH, and tr[¢] is thematrix trace operator, we can derive

fq(Zq jH0) =(

1

¼NqkMqkexp[¡tr(M¡1q TqjH0 )]

)Kq+1

(6)

and

fq(Zq jH1) =(

1

¼NqkMqkexp[¡tr(M¡1q TqjH1 )]

)Kq+1

(7)where

TqjH0 =1

Kq+1

Ãzqz

Hq +

KqXk=1

zq(k)zHq (k)

!(8)

TqjH1 =1

Kq+1

Ã(zq¡ bqsq)(zq¡ bqsq)H +

KqXk=1

zq(k)zHq (k)

!:

The likelihood ratio test (LRT) is

¤(Z) =f(Z jH1)f(Z jH0)

=QYq=1

fq(Zq jH1)fq(Zq jH0)

: (9)

To obtain the GLRT, the numerator and denominatorof (9) are independently maximized over theirrespective unknown variables, which in this caseincludes all bqs and Mqs. Fortunately, both thenumerator and denominator of (9) can be maximizedby maximizing each of the fq(Zq jHi) termsindividually. Moreover, maximizing an individualterm is the same problem solved in [7] to derive themonostatic STAP GLRT. Under the H0 hypothesis,maximization of the qth term is accomplished by

CORRESPONDENCE 1617


the covariance estimate M̂qjH0 = TqjH0 . Under the H1hypothesis, the maximization is accomplished by thecovariance estimate M̂qjH1 = TqjH1 . Substituting theseestimates into (6) and (7), respectively, the ratio in (9)becomes

¤G(Z) =QYq=1

24kTqjH0kKq+1kTqjH1k

Kq+1

35 : (10)Next, we define a matrix using only the secondarydata as

Sq =

KqX

k=1

zq(k)zHq (k) (11)

and using the same manipulations as in [7], we findthat the signal estimates that maximize the ratio in(10) are b̂q = s

Hq S

¡1q zq=s

Hq S

¡1q sq. With these estimates,

the multistatic GLRT is

¤G(Z) =QYq=1

0BBB@ 1+ zHq S¡1q zq1+ zHq S¡1q zq¡

jsHq S¡1q zqj2sHq S¡1q sq

1CCCAKq+1

=QYq=1

(lq)Kq+1: (12)

C. Multistatic Adaptive Matched Filter

The difference between the GLRT derivation andthe procedure for deriving the adaptive matched filter(AMF) test statistic is the starting assumption forthe interference covariance matrix. In the case of themultistatic GLRT, it is assumed that the noise statisticsare unknown and must be estimated. In the case ofthe multistatic AMF, the noise statistics are assumedknown, but then an estimate is substituted into thefinal form of the test statistic. This simplifies thederivation and provides a simpler test statistic.As in the GLRT, the derivation begins with the

LRT of (9). Since the Mqs are treated as knownquantities, however, terms in the LRT attributed to thesecondary data cancel each other in the numerator anddenominator. Thus, the LRT can be simplified to

¤(Z) = ¤(z1,z2, : : : ,zQ) =QYq=1

fq(zq jH1)fq(zq jH0)

: (13)

Substituting in the Gaussian pdfs, simplifying, andtaking the logarithm yields

° = ln(¤) =QXq=1

(2Refb¤qsHqM¡1q zqg¡ jbqj2sHqM¡1q sq):

(14)

Again, we can maximize this expression withrespect to the unknown signal amplitudes by

maximizing each component of the summation.Therefore, we need to maximize

°q = 2Refb¤qsHqM¡1q zqg¡ jbqj2sHqM¡1q sq (15)with respect to bq, which results in

b̂q =sHqM

¡1q zq

sHqM¡1q sq: (16)

The result of (16) represents the same maximumlikelihood estimate for the target amplitude that wederived for the GLRT derivation except that herethe covariance matrix is still assumed to be known.Substituting (16) into (14) for each observationplatform yields

°AMF =QXq=1

jsHqM¡1q zqj2sHqM¡1q sq

: (17)

Finally, the AMF approach of [8] calls for substitutingin the maximum likelihood estimate of theinterference covariance matrix computed from thesecondary data. Thus, the multistatic AMF is

°AMF(Z) =QXq=1

jsHq M̂¡1q zqj2sHq M̂¡1q sq

(18)

where

M̂q =1Kq

KqX

k=1

zq(k)zHq (k): (19)

D. Properties of the Test Statistics

In order to set the threshold using theNeyman-Pearson criterion, knowledge of theprobability distribution of the test statistic under thenull hypothesis is required. Taking the logarithm of(12), we have

ln[¤G(Z)] =QXq=1

ln[(lq)Kq+1] =

QXq=1

(Kq+1)ln(lq):

(20)

Each lq term in (12) is equivalent to the monostaticGLRT of [7] where the probability of false alarm (Pfa)for a single data set is shown to be

Pfa = Prflq ¸ l0g=Z 1l0

fq(lq)dlq = (l0)Nq¡1¡K

q :

(21)

Since Prflq > l0g= Prfln(lq)> ln(l0)g, we definexq = ln(lq) and x0 = ln(l0) and rewrite (21) as

Pfa = Prfxq ¸ x0g=Z 1x0

fq(xq)dxq = (exp(x0))Nq¡1¡K

q :

(22)



Fig. 1. Multistatic geometries used for simulation of three-platform systems.

Next, if Fq(x0) is the cumulative distribution function(CDF) of the random variable xq, then Fq(x0) = 1¡Pfa.Using the right hand side of (22) for Pfa and takingthe derivative of the cumulative distribution function(CDF), the pdf of xq under H0 is

fq(xq) = (Kq+1¡Nq)exp(¡xq[Kq+1¡Nq]):(23)

Therefore,xq = ln(lq) (24)

is exponentially distributed with E[xq]=1=(Kq+1¡Nq),and (Kq+1)ln(lq) is exponentially distributed with amean of (Kq+1)E[xq].Based on the above, the logarithm of (12) results

in a test statistic that is the sum of independentexponential random variables. Since Kq and Nqmay be different for each platform, the mean ofeach term in the resulting statistic may be different.Hence, calculating the probability of false alarm forthe multistatic GLRT requires the distribution fora weighted sum of exponential variates, which ispresented in [9]—[11]. Furthermore, we can now usethese relationships to determine the CFAR propertiesof the multistatic GLRT. The pdf of a weighted sumof independent exponential variates depends only onthe weights (means) of the individual terms. Since theabove expression for the mean of a single term doesnot depend on any clutter statistics, it follows that thecomplete multistatic GLRT is a CFAR test. The pdf ofthe test output under H0 depends only on the snapshotlength and number of secondary snapshots for eachplatform.It is not unexpected that the multistatic GLRT is a

CFAR test since the independence assumptions reducethe test to a summation of terms that individuallyare equivalent to the CFAR test of [7]. Similarly,the multistatic AMF consists of a summation inwhich the individual terms are themselves CFAR.Again, in [8] the individual terms are proven to befunctions of Kq and Nq, but not functions of theunderlying interference statistics. The summation ofthese individual test statistics into a single statisticpreserves the CFAR property of the multistaticAMF. Unfortunately, a closed-form solution for theprobability of false alarm for the single-platformAMF was not presented in [8]. In order to calculate

the probability of false alarm for the single-platformcase, Robey [12] performs a numerical integrationover the pdf of a random loss factor ½, which is betadistributed. With this being the case, a closed-formexpression for the probability of false alarm of themultistatic AMF is currently unavailable, and in ourresults below we have set detection thresholds for thistest empirically.

III. RESULTS

In this section, the performance characteristicsof the two multistatic detection methods areevaluated and compared. Performance was evaluatednumerically using a multistatic STAP Monte Carlosimulation. First, we evaluate probability of detection(Pd) as a function of input SNR for tests using one,two, and three radars in a multistatic system. Initially,the target has fixed amplitude as observed by allradars in the system. The geometry used for thefirst scenario is geometry A, shown in Fig. 1. AsFig. 1 indicates, each of the three radars observesthe scene from a different aspect. Each radar’s lookdirection is perpendicular to its velocity vector, andthe hypothesized target is assumed to be in the centerof the beam for each radar. After performing thisexperiment for the nonfluctuating target model, thesame performance is reviewed for a fluctuating targetmodel.In the first set of tests, only monostatic data

collections are employed. In other words, each radarplatform transmits and receives its own waveform, butdoes not observe the reflections due to transmissionsfrom the other platforms. The data from each platformare still used to produce a joint test statistic. Insubsequent results, we consider the use of bistatic datato improve the DO of the system.For this simulation, each radar platform has the

same operating frequency, 450 MHz, and pulserepetition frequency (PRF), 600 Hz. (We assume thatsome type of signal coding enables each receiver toseparate out reflections due to its own transmitter.In practice, the simplest way to do this might be totransmit on different bands.) All radars are travelingat 60 m/s in the direction indicated in geometry A ofFig. 1. Each radar platform uses a 5-element antennaarray with half-wavelength spacing. The interference

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Fig. 2. Multistatic GLRT and AMF performance as functionof Q.

covariance matrices Mq are calculated using arbitraryground reflectivity profiles in such a way that eachground profile is unique for each radar perspective.Clutter-to-noise ratios (CNRs) ranged from 47 to53 dB. No antenna alignment or ownship positionerrors were considered; therefore, target steeringvectors were assumed to be accurate. A small amountof internal clutter motion (ICM) has been added to theinterference covariance matrices to simulate surfacewind conditions.The first experiment looks at multistatic GLRT

and AMF methods where all platforms have 10 degof freedom and use 20 secondary data snapshotsto form a sample covariance matrix. This particularsetup is used so that results for a single platform willbe comparable with those of [8]. The target has thesame constant amplitude for all radar observationplatforms. As can be seen in Fig. 2, the performanceof the combined system improves as the number ofradar platforms increases, regardless of the methodemployed. Also, the multistatic GLRT performsslightly better than the multistatic AMF, and themargin of that performance advantage increases asthe number of platforms increases.Fig. 3 expands the scale on the single-radar and

three-radar cases of Fig. 2 to show the AMF/GLRTperformance crossover phenomenon discussed in [8].It is interesting to note that the performance crossoverseems to be unique to the single-radar case. It ispossible that the crossover point still exists for themultistatic system, but if so, it is located at a Pd thatincreases as the number of platforms is increased.Since the multistatic system consists of multiple

moving airborne platforms, a fading target modelbetter represents the radar cross section (RCS)fluctuations associated with multifaceted complextarget returns. Therefore, the next experiment usesthe same parameters as those used in Fig. 2, butemploys a Swerling I target fluctuation model to

Fig. 3. GLRT versus AMF performance at high probability ofdetection.

Fig. 4. GLRT and AMF performance for Swerling I targetmodel.

simulate target fades with respect to target orientation.The target fades are modeled as being independentfrom radar to radar, which is a good assumption ifthe radar platforms are widely separated. However,we should point out that the detection statistics ofSection II were derived under the assumption ofdeterministic but unknown target amplitudes, not theSwerling I model. Hence, we are applying a somewhatmismatched signal model, but this approach is notwithout precedent. The behavior of the monostaticGLRT and AMF under a Swerling I target modelwas evaluated in [8]. Moreover, one of the significantbenefits of multistatic radar is the diversity gainassociated with random target fluctuations, whichimplies a fluctuating target model.The performance curves in Fig. 4 become steeper

as more radar platforms are added to the system.This is due to the fact that multistatic observationsof a fluctuating target enhance the likelihood thatthe target will be observed from an aspect where



its RCS is high. In a nonfluctuating target model,multistatic observation provides little benefit at lowSNR because each platform observes low SNR. Witha fluctuation model, however, the probability that thetarget is faded for all radar platforms is low. Also notein Fig. 4 that the performance advantage of the GLRTover the AMF again becomes larger as the number ofplatforms increases, just as it did in the nonfluctuatingtarget model case.An alternative way of looking at diversity gain

comes from looking at the metrics used to characterizeMIMO communication channels (or other systemsusing diversity reception). In communication theory,probability of outage is defined as the probabilitythat SNR drops to a level at which communicationbecomes unreliable. The diversity order (DO) of acommunication system is defined as the asymptotic(as SNR approaches infinity) slope of probability ofoutage versus SNR [13] plotted on a log-log scale:

DO =¡ limSNR!1

log(Poutage)

log(SNR): (25)

In MIMO communication systems, this slope is equalto the number of independent channels employed bythe system.In comparison, probability of miss (Pm) is the

probability that a radar system fails to detect atarget that is present. This radar performance metricis directly analogous to probability of outage incommunication systems. Therefore, the asymptoticlog-log rate at which Pm decreases in a multistaticradar system should be equal to the number ofindependent target fluctuations observed by thesystem. The radar equivalent of DO is then

DO =¡ limSNR!1

log(Pm)log(SNR)

: (26)

Fig. 5 uses the same data that were used to createFig. 4, but displays log(Pm) as a function of SNRto determine if this asymptotic relationship existsin a multistatic STAP system. As Fig. 5 indicates,DO is directly linked to the number of independentobservations obtained by the multistatic system. AsSNR becomes large, the performance curves in Fig. 5become linear in the log-log scale. Moreover, for each10 dB of SNR increase within the linear region ofthese curves, Pm decreases by Q factors of 10 whereQ is the number of data sets containing independenttarget fluctuations.In the previous simulations, all observations were

collected using the monostatic scenario describedabove. That is, each platform only received reflectionsdue to its own transmitted waveform. Next, weconsider how detection can be improved by addingdata collected from the available bistatic propagationpaths. If two platforms observe the target fromdifferent aspects, it is reasonable that the bistatic

Fig. 5. Multistatic radar DO as function of number of platforms.

target fluctuation would be independent from thefluctuations observed over the two monostatic paths.Thus, the bistatic paths provide an opportunity toincrease the number of independent target fluctuationsobserved by the system without increasing the numberof radar platforms.In the next simulation, the multistatic system

exploits both monostatic data collection and bistaticdata collection geometries. We only consider asingle direction for each bistatic path since thetarget fluctuation in the reverse direction wouldnot be independent if it is due to a transmitteroperating on the same frequency band with differenttemporal coding. If there are N radar platforms, thisassumption results in Q =N(N +1)=2 independenttarget fluctuations. Alternatively, if we were toassume that each transmitter operates on a differentfrequency band such that the frequency dependenceof the target’s reflection coefficient results innonreciprocal target fluctuations, we could obtainQ =N2 independent fluctuations. For the purposesof this study, any resolution issues associated withwide bistatic angles are ignored. Four geometries areconsidered in Fig. 6. We consider the three-platformscenarios of geometry A and geometry B where onlymonostatic paths are considered. We also consider atwo-platform system consisting of radars 2 and 3 inboth geometry A and B but allow for the bistatic pathbetween the two. All four scenarios have three uniquepropagation paths; hence, all scenarios have a DO ofthree.Despite constant DO, the four collection

geometries have different performance levels that canbe understood by considering the geometry of each.The shifts in the curves are a result of differences ingeometry with respect to the target’s velocity vector.Both configurations from geometry A outperformtheir corresponding configuration from geometry Bbecause the third radar in geometry A is in betterposition to observe the target’s Doppler shift.

CORRESPONDENCE 1621


Fig. 6. Example of geometry gain for multistatic GLRT with DOof three but different physical configurations.

Fig. 7. DO comparison for correlated versus independent targetfluctuations.

It is clear that when colored interference suchas ground clutter is present, target-radar geometryplays a key role in determining the diversity benefitrealized by the system. We can exploit the fact thatthe Pm curves are asymptotically linear to quantifythis controlling factor, which we term geometry gain.If DO is the asymptotic slope of the Pm curves, wedefine geometry gain as the shift between curves inthe asymptotic, linear-slope region. Thus, we cansay, for example, that the two-platform system ingeometry A has a geometry gain of nearly 5 dBover the three-platform system in geometry B, orthat the three-platform geometry A configurationhas a geometry gain of less than 1 dB over thethree-platform geometry B configuration. Theserelative shifts in the performance curves reflect thefact that good geometries reach their asymptoticperformance and realize their diversity benefit at lowerSNR than poor geometries.Finally, we consider the case where multiple

radar platforms are available, but are located in a

configuration where the target reflections are notalways independent. This may occur, for example,when two of the radar platforms observe the targetfrom nearly the same aspect angle. The angle spanover which the target reflections are correlateddepends on target properties as well as whetherthe same frequency band is being used by the twoplatforms. For now, we force two of the targetreflection coefficients in the simulation to be equal.It is expected that perfect correlation between two datasets will reduce DO by one, which is exactly what isseen in Fig. 7. Defining a signal coefficient vector asb= [b1 b2 ¢ ¢ ¢bQ]T, DO will, in general, be directlyrelated to the rank of the signal coefficient covariancematrix, E[bbH].

IV. CONCLUSIONS

The analysis shows that the multistatic GLRT teststatistic outperforms the AMF method, especiallyas the number of platforms collecting independentobservations is increased. Furthermore, a closed-formexpression for the probability of false alarm isavailable for the multistatic GLRT but not for themultistatic AMF.We have used asymptotic Pm performance to

isolate and quantify the differences between geometrygain and DO. We have noted that the number ofindependent observations collected by a multistaticsystem can be increased by incorporating bistaticconfigurations between radar platforms, therebyincreasing the DO of the system without additionalplatforms. When only monostatic data collections areemployed by the multistatic system, the number ofindependent looks increases linearly with the numberof observing platforms. However, when each uniquebistatic path is also included in the test statistic, theDO of the system increases roughly as the squareof the number of platforms according to either Q =N(N +1)=2 or Q =N2, depending on assumptions inthe target fluctuation model.

DONALD P. BRUYÈRENATHAN A. GOODMANDept. of Electrical and Computer EngineeringUniversity of Arizona1230 E. Speedway Blvd.Tucson, AZ 85721E-mail: ([email protected])

REFERENCES

[1] Melvin, W. L.A STAP overview.IEEE AES Systems Magazine, 19, 1, pt. 2 (Jan. 2004),19—35.

[2] Ward, J.Space-time adaptive processing for airborne radar.MIT Lincoln Laboratory, Lexington, MA, TechnicalReport 1015, Dec. 1994.



[3] Brennan, L. E., and Reed, I. S.Theory of adaptive radar.IEEE Transactions on Aerospace and Electronics Systems,AES-9, 2 (Mar. 1973), 237—252.

[4] Goodman, N. A., and Bruyere, D.Optimum and decentralized detection for multistaticairborne radar.IEEE Transactions on Aerospace and Electronics Systems,43, 2 (Apr. 2007), 806—813.

[5] Fishler, E., et al.Spatial diversity in radars-models and detectionperformance.IEEE Transactions on Signal Processing, 54, 3 (Mar.2006), 823—838.

[6] Fishler, E.MIMO Radar: An Idea Whose Time Has Come.In Proceedings of the IEEE Radar Conference, Apr. 2004,71—78.

[7] Kelly, E. J.An adaptive detection algorithm.IEEE Transactions on Aerospace and Electronics Systems,AES-22, 1 (Mar. 1986), 115—127.

[8] Robey, F. C., Fuhrmann, D. R., Kelly, E. J., and Nitzberg, R.A CFAR adaptive matched filter detector.IEEE Transactions on Aerospace and Electronics Systems,28, 1 (Jan. 1992), 208—216.

[9] Ali, M. M., and Obaidullah, M.Distribution of linear combination of exponential variates.Communications in Statistics–Theory and Methods, 11, 13(1982), 1453—1463.

[10] Hahn, G. J., and Shapiro, S. S.Statistical Models in Engineering.New York: Wiley, 1994.

[11] Johnson, N. L., Kotz, S., and Balakrishnan, N.Continuous Univariate Distributions (2nd ed.).New York: Wiley, 1994.

[12] Robey, F. C.A covariance modeling approach to adaptivebeamforming and detection.MIT Lincoln Laboratory, Lexington, MA, TechnicalReport 918, July 1991.

[13] Varadarajan, B., and Barry, J. R.The rate-diversity trade-off for linear space-time codes.Proceedings 56th Vehicular Technology Conference, vol. 1,Sept. 2002, 67—71.

Pulse Compression for a Simple Pulse

A new pulse compression method for a simple pulse is

proposed. The proposed method has a filter with impulse response

h(t) whose Fourier transform H(f) is given by D(f)=S(f), where

D(f) is the Fourier transform of desired output waveform d(t)

and S(f) is the Fourier transform of input signal s(t). Frequency

characteristics D(f) are considered not to make H(f) divergent,

since frequency characteristics S(f) have zero points at the

multiplication number of the reciprocal number of the input

pulsewidth. The proposed method has an advantage because it

can be compressed to the arbitrary pulsewidth given by waveform

d(t) and can be designed for compressed output with small peak

sidelobe level.

I. INTRODUCTION

Pulse compression has been employed tosimultaneously satisfy the requirements ofmaximum detection range and range resolution [1].Generally, pulsewidth compressed by these methodsapproximately equals the reciprocal of the mainspectrum width where the spectrum energy of thetransmitting pulse is concentrated. Therefore, it isnecessary to spread the bandwidth of the transmittedpulse to obtain narrower range resolution. It isimportant to obtain narrower range resolutionwithout wider bandwidth of the transmitted pulsebecause this leads to the effective use of radio waves.Many investigators have studied pulse compressiontechniques [2]. R. H. Barker proposed codes with anautocorrelation function whose peak sidelobe levelis only 1 [3]. J. Lindner [4] and M. N. Cohen et al.[5] investigated optimum binary codes that have anautocorrelation function with peak sidelobe levelswhich are the smallest possible level for a givencode length in the relatively small compression ratioregion (optimum binary codes). Rao and Reddyinvestigated the binary codes of relatively long codelength that have an autocorrelation function withcomparatively small peak sidelobe levels (PSLs)[6]. G. Coxson and J. Russo extended the upperlimit of the length to 70 [7]. These methods arecompressed to pulse wide given approximately by thereciprocal of the main spectrum of the transmittedpulse which is modulated. L. K. Cuomo tried tocompress an input pulse into a narrower pulsewidth

Manuscript received October 2, 2006; revised May 10, 2007;released for publication July 17, 2008.

IEEE Log No. T-AES/44/4/930747.

Refereeing of this contribution was handled by V. C. Chen.

0018-9251/08/$25.00 c° 2008 IEEE

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