Parameter Estimation Based Coherent

Multistatic ProcessingAmin Jaffer

Space and Airborne Systems Raytheon Corporation

El Segundo, USA ajaffer@raytheon.com

Bruce Evans

Space and Airborne Systems Raytheon Corporation

El Segundo, USA

Muralidhar Rangaswamy AFRL/SNHE

Hanscom AFB Massachusetts, USA

Abstract—This paper develops the coherent generalized likelihood ratio test (GLRT) and the MVDR statistic for multistatic radar systems, conditioned on estimation of certain parameters that render the system coherent. Analytical and computer simulation results are presented to show substantially enhanced detection and geolocation of moving targets in clutter.

I. INTRODUCTION

The need to improve air and ground moving target indication in difficult environments may necessitate the use of multiple spatially separated receive sensors deployed one or more transmitters. The use of multiple spatially separate receive sensors with a transmitter, comprising a multi-static radar system, can provide a significant improvement in target detection and geolocation performance, especially for slow-moving targets embedded in clutter.

The work reported here addresses the problem of combining and processing the spatial-temporal return data received at multiple airborne sensors, resulting from signal transmissions from an airborne transmitter, with the goal being to enhance the detection of moving targets in clutter and simultaneously improve target localization accuracy tracking. As is known the surface clutter returns observed in airborne bistatic radar systems exhibit a Doppler spread which tends to mask the detection of slow-moving targets in clutter. The spatial diversity afforded by a multi-static system allows, however, diverse “views” of the target and clutter returns to be obtained and the possibility of combining them so as to enhance target detection.

We consider here the conditional coherent case i.e. the situation where the target return is regarded as coherent across the multiple separate receivers subject to the estimation of certain parameters pertaining to the reception

of the signal in the different receivers. In particular, we assume that the target return signal received in any particular i’th receiver can be modeled as being related to that received in some reference receiver in terms of the (unknown) path length difference between the target and the i’th receiver and the target and the reference receiver. The phase difference of the target return signal in the receivers is proportional to this path length difference and its power is inversely proportional to the square of the same path length difference. It is assumed, however, for this development that the target bistatic cross-section is essentially constant in the directions of the multiple receivers and does not result in decorrelation of the target return signals in the various receivers. This condition may hold approximately if the angular separation of the various receivers is small. It is noted here that the noncoherent multistatic GLRT case has already been developed and evaluated earlier [1],[2] and the work presented here is also described in more detail in our recent report[3].

II. COHERENT MULTISTATIC GLRT TEST

The coherent generalized likelihood ratio test (GLRT), based on modeling the target return signal phase and magnitude in the receivers in accordance with the preceding discussion, is presented here. Arbitrarily choosing one of the receivers as the first (reference) receiver, the radar return signal complex amplitude for the mth receiver first pulse relative to the first receiver can be represented by a complex gain value

λπ /

)()()( (x)rj

mm

mexrxrxg 121 −=

where )( xrm is the distance of the mth receiver from the target whose position vector is denoted by x.

978-1-4244-2871-7/09/$25.00 ©2009 IEEE

)()()( xrxrxr mm 11 −= is the path length difference between the target and the first and the m’th receivers and 11 =g

The first pulse complex signal amplitude receive at the mth receiver is then smag where sa is the complex signal amplitude at the first receiver. Note that x)(gm is a function of the unknown target position vector x. The coherent GLRT statistic can now be derived in the usual manner by defining a modified concatenated spatial-temporal steering vector for the M receivers using the complex gains x)(gm as follows:

where the ),( xxd sm , m=1,2,……,M are the standard spatial-temporal steering vectors.

Under the assumption that the clutter observed in the M receivers are uncorrelated, the coherent GLRT statistic for detecting the target return can then be shown to be the following by using the above modified spatial-temporal steering vector in the well-known form for the adaptive matched filter (AMF) test statistic [4]:

Compute

and maximize with respect to xx , . mR is the interference plus noise covariance matrix of the m’th receiver. A detection is declared if this maximum exceeds a pre-set threshold and the resulting xx , are the maximum-likelihood estimators of the target position and velocity vectors.

III. COHERENT MULTISTATIC MINIMUM

VARIANCE DISTORTIONLESS RESPONSE (MVDR) METHOD

The MVDR method is well known for the single

receiver case [5]. For our multistatic model with unknown parameters, by using the modified form of the steering vector given above, the multistatic coherent MVDR test statistic can be shown to be the following:

Note that the covariance matrix mR also includes the signal rank 1 matrix if the search position vector x maps to the target bistatic range cell for the m’th receiver.

IV. COMPUTER SIMULATION RESULTS

a. Coherent GLRT Simulation Results

Figure 1 shows the overall radar scenario and the iso-range ellipses, with three receivers spanning a broad range of bi static angles. Figure 2 shows the corresponding coherent generalized likelihood ratio GLR function as a function of the unknown target position. The peak occurs in the vicinity of the true target location. Computing the GLR using a finer resolution in the x,y coordinates reveals ambiguous peaks occurring in a periodic manner which is due to the estimation error of the range differences to the receivers from the target being an integer multiple of the wavelength.

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Figure 1. Multistatic Geometry for Coherent GLRT

Identify applicable sponsor/s here. (sponsors)

[ ]TTsMM

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xxJ

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),(),()(),(

Figure 2: Combined Coherent GLR vs. (X,Y)

Location All Receivers

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oherentGLRT 315 Basic --Log Likelihood Image -true tgt pos 50.0 0.0 km vel -6.4 7.7 mps

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On the smaller scale and finer resolution, the combined coherent GLRT reveals the presence of additional peaks in

the vicinity of the true target location in (x,y). This is due to the fact that variation in x can result in the phase term in )x(gm changing by multiples of π2 for all m thereby rendering the GLR function constant. Zooming in on the target location (by plotting the combined GLR at finer sample resolution over a smaller area) reveals even more complex behavior (not shown here but detailed in our report[3] which show a succession of smaller areas around the target location. As the level of detail shown in these figures increases, more and more peaks become visible, and in the smallest, one meter square area a regular array of ambiguous peaks appears. Note that these ambiguous peaks tend to diminish in number and become attenuated if the number of receivers and the spatial diversity increases.

b. Coherent MVDR Simulation Results

Figure 4 shows the MVDR statistic as a function of hypothetical x-y target location, with the hypothetical z location component and target velocity fixed at the true values. In figure 5, the data is plotted as a three dimensional surface. These results are for the three receiver scenario shown in Figure 1. These results appear generally quite similar to the coherent GLRT results. The coarse 75 m resolution plots show three ridges (corresponding to constant bi-static range loci with respect to each of the receivers) with a strong peak at the common intersection point of these loci. Zooming in on this peak at progressively finer resolutions reveals a complex structure of multiple ambiguous sub-peaks, just as occurred with the coherent GLRT.

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MVDR Statistic (dB) vs Hypothetical Target X, Y (Tgt Spd = 20.0 m/s)Max Value = 31.8676 at x,y = (50000,0) meters

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Figure 4. Combined Coherent MVDR

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MVDR Statistic (dB) vs Hypothetical Target X, Y (Tgt Spd = 20.0 m/s)Max Value = 31.8676 at x,y = (50000,0) meters

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Figure 5. Combined Coherent MVDR – Finer Resolution

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MVDR Statistic (dB) vs Hypothetical Target X, Y (Tgt Spd = 20.0 m/s)Max Value = 31.8676 at x,y = (50000,0) meters

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Figure 6. Combined Coherent MVDR- Very Fine

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MVDR Statistic (dB) vs Hypothetical Target XDot, YDot (Tgt Spd = 20.0 m/s)Max Value = 21.5875 at vx,vy = (-27.8558,27.8209) meters/sec

runMVDR08 2LLRWGS2N RCVR-1-2-3 hi-res BasicScenario-clutter-med speed-velocity search

Y Velocity (meters/sec)

Figure 7. Coherent MVDR as a function of x-velocity and y-velocity for fixed target x,y

The combined MVDR response as a function of postulated target velocity is shown in figure 7 (with the location held fixed at the true target location) for the same

scenario as the previous MVDR versus location results. In contrast to the GLRT, there are two ridges (corresponding to the target and to clutter) instead of the single target ridge that occurs in the GLRT. This is because the MVDR responds to both target and clutter, whereas the GLRT attempts to null the interfering clutter. In the combined multi-receiver response, this results in the appearance of two peaks: one corresponding to clutter, and one to the target.

V . ANALYTICAL PERFORMANCE EVALUATION OF THE COHERENT GLRT STATISTIC

Under the condition of perfect estimation of the underlying parameters in the coherent GLRT statistic, the probability density function (pdf) of the GLRT under the signal plus noise hypothesis is non-central chi-squared with 2 real degrees of freedom and noncentrality parameter (integrated SNR)

∑=

−=M

msmm

Hsmm dRdga

1

1221λ

where 1a is the complex signal amplitude at the first receiver. Under the noise only hypotheses, the pdf is central chi-squared with 2 real degrees of freedom. This characterizes completely the probability density functions of the statistic under the two hypotheses and enables one to determine the probability of detection and probability of false alarms as a function of integrated signal-to-noise ratio given by λ by using well-known methods given in [6]. It is expected that numerical results will be presented in the near future.

REFERENCES

1. A. G. Jaffer, B.W. Evans and P.A. Zulch, “Generalized Likelihood Ratio Methods for Pre-Detection Fusion in Multistatic Radar Systems”, Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, 6-7 June, 2006

2. A. Jaffer, B. Evans, P. Zulch and M. Rangaswamy, “Combined GLRT Processing Methods for Multistatic Radar Systems”, Proceedings Tri-Service Radar Symposium, May 2007

3. A. Jaffer and B. Evans, “Generalized likelihood Ratio Processing Methods for Multistatic Radar Systems, Raytheon Co. Final Technical Report for AFRL Contract No. FA8750-05-C-0231, May 14, 2008

4. J.R. Guerci, Space-Time Adaptive Processing for Radar, ARTECH House, 2003.

5. S. Haykin, Adaptive Filter Theory, 3rd Edition, Prentice-Hall, 1996.

6. A.D. Whalen, Detection of Signals in Noise, Academic Press, 1971.