ACRONYMS

An Approach to Knowledge-Aided Covariance Estimation WILLIAM L. MELVIN, Senior Member, IEEE GREGORY A. SHOWMAN, Member, IEEE Georgia Tech Research Institute

This paper introduces a parametric covariance estimation scheme for use with space-time adaptive processing (STAP) methods operating in heterogeneous clutter environments. The

AMF CFAR CMT CNR CPI CUT DoFs DTED EFA GMTI IID INU KA KAPE KASSPER LSE MCARM

approach blends both a priori knowledge and data observations within a parameterized model to capture instantaneous

MVDR

characteristics of the cell under test (CUT) and reduce covariance errors leading to detection performance loss. We justify this method using both measured and synthetic data. Performance potential for the specific operating conditions examined herein include 1) averaged behavior within roughly 2 dB of the optimal filter, 2) 1 dB improvement in exceedance characteristic relative to the optimal filter, highlighting improved instantaneous capability,

OS-CFAR PRF PRI SINR STAP TSD

Adaptive matched filter Constant false alarm rate Covariance matrix taper Clutter-to-noise ratio Coherent processing interval Cell under test Degrees of freedom Digital terrain elevation data Extended factored algorithm Ground moving target indication Independent and identically distributed Inertial navigation unit Knowledge-aided Knowledge-aided parametric covariance estimation Knowledge-aided sensor signal processing and expert reasoning Least squared error Multi-channel airborne radar measurements Minimum variance distortionless response Ordered-statistic constant false alarm rate Pulse repetition frequency Pulse repetition interval Signal-to-interference-plus-noise ratio Space-time adaptive processing Targets in the secondary data.

and 3) imperviousness to corruptive target-like signals in the secondary data (no additional signal-to-interference-plus-noise

I. INTRODUCTION

ratio (SINR) loss, compared with 10 dB or greater loss for the standard STAP implementation), with corresponding detections comparable to the optimal filter case.

Manuscript received March 24, 2005; revised November 16, 2005; released for publication March 7, 2006. IEEE Log No. T-AES/42/3/884471. Refereeing of this contribution was handled by W. D. Blair. This work sponsored by the Defense Advanced Research Projects Agency (DARPA) under U.S. Air Force Contract F30602-02-C-0011. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the U.S. Government. Authors’ address: Georgia Tech Research Institute, Sensors and Electromagnetic Applications Laboratory, 7220 Richardson Rd., Smyrna, GA, 30080, E-mail: ([email protected]).

c 2006 IEEE 0018-9251/06/$17.00 °

Space-time adaptive processing (STAP) is an important technique for improving detection performance of aerospace radar systems operating in strong clutter and interference environments [1—5]. The utility of STAP derives from its ability to tailor a multi-dimensional filter response to sensed characteristics of the disturbance environment, ideally maximizing output signal-to-interference plus noise ratio (SINR). It is well known that maximizing output SINR is tantamount to maximizing the probability of detection for a fixed probability of false alarm when the disturbance is multivariate Gaussian [1]. The common STAP implementation employs secondary (training) data to estimate the requisite null-hypothesis covariance matrix of the cell under test (CUT), thereby generating an approximation to the optimal (known covariance) filter response [6]. Typical STAP training schemes presume the availability of independent and identically distributed (IID) training data. Under such circumstances, convergence between adaptive and optimal filter responses is strictly a function of the quantity of training data [6]. This remarkable result further suggests convergence between estimated and actual

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covariance matrices. However, realistic clutter environments appear heterogeneous, or non-IID (independent and identically distributed) thereby violating intrinsic STAP assumptions [7—9]. For example, clutter reflectivity often varies spatially due to changing cultural characteristics, target-like signals frequently reside within the training data set, and clutter discretes (returns from stationary, manmade objects) are ever present. Such range-varying effects lead to covariance matrix estimation errors, adaptive filter mismatch, and consequently, degradation in output SINR or an increase in false alarm rate. Traditional STAP methods make little use of a priori knowledge. The processor employs the operating center frequency fc , the pulse repetition interval (PRI) T, and the location of each phase center da=p , to calculate the ideal space-time steering vector for hypothesized target Doppler frequency fd , and spatial frequency fs (or azimuth Áa and elevation µa for designs other than the uniform linear array). However, STAP generally does not exploit the anticipated structure of surface clutter returns, nor cultural information available from land-use databases and digital terrain elevation data (DTED) models. It is possible to leverage this unused information to enhance STAP performance in heterogeneous clutter environments. Indeed, the development of KA (KA) STAP is an important objective of DARPA’s Knowledge-Aided Sensor Signal Processing and Expert Reasoning (KASSPER) Program [10]. This paper introduces KA parametric covariance estimation (KAPE) to improve performance in heterogeneous clutter environments. The KAPE approach, which we now briefly overview, merges a priori information and measured observations to mitigate covariance matrix estimation errors. In-depth discussion is given in Section IV. Consider an M-channel array collecting N pulses over L range bins. The space-time data vector for range bin k is xk 2 C MNx1 . The two detection hypotheses corresponding to target absence (H0 ) or target presence (H1 ) are H0 : xk = ck + nk H1 : xk = ®T sT + xk=H0

(1)

where ck is the clutter space-time snapshot, nk is the vector of uncorrelated noise voltage contributions, and ®T sT is the target response. ®T is a complex voltage of unknown amplitude and uniformly distributed phase. Though not described, we presume the cancellation of jamming and interference via front-end signal processing, thereby allowing us to focus on mitigating the surface clutter component ck . As clutter and noise components are independent, the exact, 1022

null-hypothesis covariance matrix is given by H H Rk = E[xk=H0 xH k=H0 ] = E[ck ck ] + E[nk nk ]

= Mk + ¾n2 IMN :

(2)

Mk is the clutter covariance matrix and ¾n2 is the single-channel, single-pulse noise variance. As shown in [1], the optimal filter response, in the maximum SINR sense, follows as yopt=k = wH opt=k xk , ¡1 where wopt=k = ¯Rk ss-t (fs , fd ) is the optimal weight vector, ¯ is a scalar and ss-t (fs , fd ) is the space-time steering vector pointing to target spatial and Doppler frequencies fs and fd . Since Rk is unknown in practice, the processor employs secondary data, fxm gK m=1 , to calculate the covariance estimate as K X ˆ = 1 R xm xH k m: K

(3)

m=1

Equation (3) is the maximum likelihood solution if the vectors comprising the secondary data set are IID circular Gaussian with respect to the null-hypothesis condition of the kth range cell [6]. The adaptive ˆ ¡1 v , ˆ k = ¯ˆ R weight vector follows from (3) as w s-t k ˆ where ¯ is a constant and vs-t is a surrogate ss-t . (A surrogate steering vector is necessary since the array manifold, target angle, and target Doppler are not precisely known.) It is shown in [6] that the ratio of adaptive output SINR to optimal output SINR under the IID and matched steering vector case is beta distributed with expected value E[SINRjwˆ k =SINRjwk=opt ] = (K + 2 ¡ MN)=(K + 1): (4) Thus, for example, to achieve an average of 3 dB loss between adaptive and optimal implementations, K should be approximately 2NM. Given typical values for N and M, STAP training requirements are large relative to the normal window size of a constant false alarm rate (CFAR) algorithm, thus implying an increased likelihood of deviation from the IID condition due to changing clutter features over range. Reduced-rank and reduced-dimension STAP approaches reduce training requirements from 2NM to roughly twice the rank or twice the adaptive degrees of freedom (DoFs) [2—5]. Differences between vs-t and ss-t result from imperfect knowledge of target Doppler and direction of arrival, errors in the array manifold, and possibly small temporal errors, with corresponding losses generally small. However, covariance estimation errors can lead to substantial performance loss, ˆ ] deviates from R as a result of especially when E[R k k clutter heterogeneity. We discuss the impact of clutter heterogeneity on STAP in further detail in Section III.

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Mitigating the impact of covariance estimation errors is the primary focus of this paper. Next, suppose certain characteristics of the radar system and aerospace platform are adequately known, such as platform heading, speed and altitude, array normal direction, and, antenna phase steering. It is then plausible to construct a model for Rk in lieu of the calculation of (3) relying on K potentially heterogeneous space-time data vectors. Ideally this model captures localized features of the clutter environment, such as clutter amplitude and spectral characteristics within a narrow range window, while constraining the modeled clutter subspace to closely correspond with the received data. Through a straightforward linear transformation, the processor then uses the corresponding covariance model 2 RKAPE=k = MKAPE=k + ¾DL IMN to whiten dominant aspects of the clutter component. MKAPE=k is the modeled clutter covariance for the kth range cell and 2 ¾DL is the diagonal loading level. The prewhitened data vector and space-time snapshot are ¡1=2 x˜ k = RKAPE=k xk

(5)

¡1=2 v˜ s-t = RKAPE=k vs-t :

The whitening operation suppresses those components within xk matching the response embodied by MKAPE=k . It is then possible to apply a partially adaptive filter to the whitened component, yielding ¡1

¡1

ˆ˜ ¡1=2 ˆ˜ H x˜ = ¯˜ v˜ H R ˜ H ¡1=2 ˆ˜ ˜ yk = w k k s-t k xk = ¯vs-t RKAPE=k Rk RKAPE=k xk : (6) The covariance estimate for the whitened data is ˜

˜

K K ˆ˜ 1X 1 X ¡1=2 ¡1=2 H ˜ ˜ R x x = = RKAPE=m xm xH m RKAPE=m : k m m ˜ ˜ K m=1 K m=1

(7) The adaptive step provides the capability to remove any model mismatch, enables KA processing only in regions deemed troublesome (potentially reducing computational complexities corresponding to the parametric model generation), and provides flexibility in merging KA and traditional adaptive ¡1=2 signal processing. Also, since RKAPE=k is generally a rank reducing transformation achieved via suppression ˜ ¿ K; of the dominant, interference subspace, K thus, the adaptive processor can select appropriate training data in proximity to the CUT and more capably screen target-like signals and identify clutter discretes resulting from the improved contrast relative to the ambient surface clutter. If partial ˆ˜ adaptivity is undesirable, (6) applies with R set to k

the identity, IMN . This latter option is of most interest here.

The success of the aforementioned KAPE approach predicates upon the accuracy of the parametrically modeled clutter covariance matrix MKAPE=k . KAPE employs data observations to estimate clutter amplitude and spectral spread and determine a best fit for the Doppler offset corresponding to errors in knowledge of the array normal. Inertial navigation unit (INU) and Global Positioning Satellite (GPS) data are used to generate the model response for clutter angle-Doppler properties. Determining the array manifold (the precise spatial steering vectors over angle given hardware imperfections) is a critical step significantly affecting whitening performance of the parametric clutter covariance model. For this reason, calibration-on-clutter techniques are proposed herein to estimate the unknown array manifold. We subsequently describe KAPE and its performance potential in further detail in later sections of the paper. This paper substantially builds upon our previous discussion in [11]—[13]. Related work is discussed in [14]—[21]. In [14] and [15], the authors describe a colored loading approach involving the addition of a scaled, simulated clutter covariance matrix to the covariance estimate of (3) and a diagonal loading term. Unity scaling is initially applied to each clutter patch comprising the simulated covariance matrix, which is thereafter scaled by an estimate of the clutter-to-noise ratio (CNR). The simulation directly employs a priori knowledge to generate a clutter covariance matrix. The stated objective of the colored loading method is faster convergence, thereby enabling localized selection of training data for covariance estimation of a subsequent adaptive processing stage. That localized training improves performance in heterogeneous clutter environments is an underlying presumption of the colored loading method. Reference [16] extends the results in [14] and [15] by employing multiple coherent processing intervals (CPIs) to generate improved estimates of the clutter reflectivity over range and angle prior to generating the simulated covariance matrix. A colored diagonal loading method is proposed in [17] to incorporate quiescent behavior into the minimum variance distortionless response (MVDR) beamformer formulation. Constraining the coloration to reside on the matrix diagonal enables certain responses but precludes the aforementioned prewhitening strategy necessary for surface clutter mitigation. A method described in [18] and [19] is termed nonlinear nonadaptive space-time processing and involves taking the minimum of the output of a bank of linear filters with preselected weights and variable spectral spread. The preselected weights define a space-time clutter filter and are generated based on knowledge of the pulse repetition frequency (PRF), platform velocity, and channel spacing. Thus, the approach presumes highly accurate knowledge of the space-time null

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location. The filter yielding minimal output is chosen as the best match to the spectral characteristics of the actual data. This approach was successfully applied to measured data. Stated objectives of nonlinear nonadaptive space-time processing include the potential for reduced computational burden over certain STAP implementations and robustness in the presence of heterogeneous clutter. In [20], the authors describe a structured covariance estimation approach leading to faster STAP convergence. The structured component corresponds to a ground clutter covariance matrix model generated through the use of prior knowledge. Similarly, [21] describes an approach merging synthetic aperture radar data used to estimate clutter backscatter characteristics and the aforementioned ground clutter model as a means of generating a clutter covariance matrix. In contrast to the colored loading approaches and nonlinear nonadaptive space-time processing method, KAPE uniquely blends both measurements and a priori knowledge to generate a model of the clutter covariance for each individual range bin of interest. The primary objective of KAPE is to reduce errors between the actual, yet unknown, covariance matrix and a model used to prewhiten the instantaneous clutter response. While radar application serves as the focus of this paper, the KAPE method may benefit other types of sensors operating in the presence of heterogeneous clutter. The remainder of the paper is organized as follows. We provide space-time signal models used to form the core of the clutter covariance model in Section II; measured data analysis proves the veracity of these models. Section III then motivates the use of KAPE by discussing the impact of heterogeneous clutter on STAP performance. Section IV provides details of the KAPE implementation. We conclude the paper with performance assessment using synthetic data examples in Section V. II. SPACE-TIME SIGNAL MODELS This section develops the space-time signal models characterizing the surface clutter signal component. We validate this model using measured data taken from the Multi-Channel Airborne Radar Measurements (MCARM) Program [22]. Two coordinate systems are necessary: a radar coordinate system and an array coordinate system. We use the radar coordinate system to calculate the Doppler response of a specific stationary point, whereas the array coordinate system is integral to the calculation of the spatial response. In the radar coordinate system, the x-axis aligns with true North, the y-axis points due West, and the z-axis is perpendicular and away from the Earth’s 1024

surface. The true North reference is convenient when employing navigation data and information from cultural databases. Define Á as azimuth measured clockwise from North (opposite convention) and µ as the elevation angle measured from nadir. A unit vector pointing to an arbitrary point p at (Áp , µp ) is kˆ r (Áp , µp ) = sin(µp ) cos(Áp )xˆ r ¡ sin(µp ) sin(Áp )yˆ r ¡ cos(µp )zˆ r

(8) where fxˆ r , yˆ r , zˆ r g are unit vectors aligned with the radar coordinate system axes. Presuming the point p is a stationary scatterer, the corresponding Doppler frequency is 2 fd = kˆ r (Áp , µp ) ¢ vp : (9) ¸ vp = vx xˆ r + vy yˆ r + vz zˆ r is the platform velocity in the radar coordinate system. Each of the components is available from the platform INU and/or GPS unit (it is most common to couple both navigation sources). The temporal steering vector describes the phase progression from pulse to pulse and is st (fd ) = [1 exp(j2¼fd T) exp(j2¼fd 2T) ¢ ¢ ¢ exp(j2¼fd (N ¡ 1)T)]T :

(10)

T is the PRI. The array coordinate system is also Cartesian and follows the right-hand rule: the z-axis points normal to the array face, the x-axis aligns in the horizontal direction, and the y-axis is directed upwards. In this coordinate system, elevation angle µa is measured from horizontal–in the y-z plane–and is positive in the upward direction; Áa is azimuth angle measured in the x-z plane and is positive in the counterclockwise direction from the z-axis. The wavenumber vector pointing normal to a propagating electromagnetic wave arriving from (Áa , µa ) is given by ka (Áa , µa ) =

2¼ (cos(µa ) sin(Áa )xˆ a + sin(µa )yˆ a ¸ + cos(µa ) cos(Áa )zˆ a ) (11)

where ¸ is wavelength. The spatial steering vector describes the array response to a point source with a specified direction of arrival and takes the form ss (Áa , µa ) = [exp(jka (Áa , µa ) ¢ da=1 ) exp(jka (Áa , µa ) ¢ da=2 ) ¢ ¢ ¢ exp(jka (Áa , µa ) ¢ da=M )]T

(12)

where da=p = dx=p xˆ a + dy=p yˆ a + dz=p zˆ a is the position vector of the pth phase center in the array coordinate system with Cartesian coordinates fdx=p , dy=p , dz=p g. The next step is to align the array in the radar coordinate system. We assign the array starting position aligned with the true North direction, where zˆ a aligns with xˆ r and xˆ a aligns with yˆ r . We then yaw, pitch, and roll the array in accord with a specified

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antenna deployment. The unit vector kˆ r corresponding to array coordinate system angles (Áa , µa ) is then ¸ ¡1 M M M k (Á , µ ): kˆ r = T 2¼ rcs-acs yaw pitch roll a a a

(13)

Trcs-acs = [0 1 0; 0 0 1; 1 0 0] is the transformation from radar coordinates to nonrotated array coordinates. Myaw , Mpitch , and Mroll are rotation matrices used to yaw, pitch, and roll the array from true North. These rotation matrices are 3 2 cos(´y ) 0 sin(´y ) 7 6 Myaw = 4 0 1 0 5 2

¡ sin(´y ) 0 cos(´y ) 1

0

6 Mpitch = 4 0

0

3

7 sin(´p ) 5

cos(´p )

(14)

0 ¡ sin(´p ) cos(´p ) 3 cos(´r ) ¡ sin(´r ) 0 7 6 Mroll = 4 sin(´r ) cos(´r ) 0 5 : 2

0

0

1

´y is the yaw angle measured counterclockwise from true North; ´p is the array pitch, where positive values tilt the array up; and, ´r is the roll angle, with positive values rotating the array clockwise (e.g., right wing down for aircraft flying due North). Equation (13) and (9) define the ground clutter angle-Doppler coupling. A simple, effective model for the clutter space-time snapshot involves summing the individual responses of Nc clutter patches over the iso-range of interest, including the Na range ambiguities. We assume each clutter patch is statistically independent, which is reasonable since the voltage of any given patch is generally uncorrelated with other patches. Thus, the surface clutter space-time snapshot takes the form ck =

Nc Na X X

m=0 n=1

®s-t (m, n; k) ¯ ss-t (Áa=m,n , µa=m,n , fd=m,n ; k) (15)

where (Áa=m,n , µa=m,n ) is the azimuth and elevation to the mnth patch; fd=m,n is the corresponding Doppler frequency; ®s-t (m, n; k) 2 C NMx1 is the vector containing the space-time voltages for each channel-pulse-range sample, with each element proportional to the square-root of the patch CNR; ss-t (Áa=m,n , µa=m,n , fd=m,n ; k) is the space-time snapshot; and, ¯ is the Hadamard product operation. The space-time steering vector is given as the Kronecker product of the temporal and spatial steering vectors, ss-t (Áa , µa , fd ) = st (fd ) − ss (Áa , µa ). The voltage vector can be written as ®s-t (m, n; k) = vk=m,n (tt (Áa=m,n , µa=m,n ; k) − ts (Áa=m,n , µa=m,n ; k)):

(16)

vk=m,n is the complex voltage of the mnth patch for the kth range, tt (Áa=m,n , µa=m,n ; k) is a random taper vector characterizing the voltage fluctuation over the temporal aperture, and ts (Áa=m,n , µa=m,n ; k) is a random spatial taper describing the voltage decorrelation over the spatial aperture. We recognize both tt and ts as covariance matrix taper (CMT) components [23]. Let 2 vk=m,n » CN(0, ¾k=m,n ). The covariance matrix of (16) then follows as H Dk (m, n) = E[®s-t (m, n; k)®s-t (m, n; k)] 2 Tt (Áa=m,n , µa=m,n ; k) = ¾k=m,n

− Ts (Áa=m,n , µa=m,n ; k) Tt (Áa=m,n , µa=m,n ; k) =

(17)

E[tt (Áa=m,n , µa=m,n ; k)tH t (Áa=m,n , µa=m,n ; k)]

Ts (Áa=m,n , µa=m,n ; k) = E[ts (Áa=m,n , µa=m,n ; k)tH s (Áa=m,n , µa=m,n ; k)]:

The expected value of the two outer products correspond to temporal and spatial CMTs. Plausible functions characterizing the elements of Tt , employed to model intrinsic clutter motion, include the Gaussian autocorrelation [28] and Billingsley model involving an exponential autocorrelation [24]. The model choice typically depends on the clutter environment: Gaussian fits best for regions with sea or freshwater, as it fully decorrelates, whereas exponential is more appropriate for wooded regions or fields. A sampled sinc or “angle dither” is appropriate for the elements of Ts and used to model wave-front dispersion [23]. Using (15) and (17), the clutter covariance matrix, Mk = E[ck cH k ], is given by Mk =

Na X Nc X

2 ¾k=m,n Ts-t (Áa=m,n , µa=m,n ; k)

m=0 n=1

¯ ss-t (Áa=m,n , µa=m,n ; k)sH s-t (Áa=m,n , µa=m,n ; k)

(18)

Ts-t (Áa=m,n , µa=m,n ; k) = Tt (Áa=m,n , µa=m,n ; k) − Ts (Áa=m,n , µa=m,n ; k): As we discuss in subsequent sections, the corresponding models of (15) and (18) provide a basis for the KAPE technique. We remove fd=m,n from the arguments in (18) since the angles Áa=m,n and µa=m,n to the stationary scatterer uniquely describe the corresponding Doppler frequency. A. Clutter Model Validation using Measured Data The MCARM program provides multi-channel airborne radar data to prove the performance potential of STAP. Details of MCARM are given in [22]. The data were collected at L-band using a twenty-four channel, port-mounted antenna; twenty-two channels in an eleven-over-eleven configuration comprised the main array, with sum and difference channels also available. The system employed modest waveform

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Fig. 3. MVDR spectrum for MCARM simulation. Fig. 1. Receiver test file eigenspectrum used for noise floor calculation.

bandwidth of 800 kHz, leading to fairly coarse range cells. The peak transmit power of 15 kw led to fairly strong clutter returns for the relatively short collection ranges. Along with a collection of the space-time data vectors over L range bins, fxm gLm=1 , each CPI also included an extensive set of header data, including INU/GPS data on platform velocity and attitude and array normal direction. In most cases, L is typically on the order of 500 useable samples. We consider data collected over the Delaware-Maryland-Virginia (DelMarVa) Peninsula region in the United States. Using the signal models of this section and auxiliary information collected with the MCARM radar data, we compare features of the synthetic data against actual measurements. We find, except for anomalies attributed to target-like signals, very good match between measurements and the aforementioned models. Fig. 1 shows the eigenspectrum for a receiver test file. In this case, transmit energy is diverted through the receive chain for calibration purposes; time delay between transmit pulses provides long

records of sampled receiver noise in each channel. We thus form a sample covariance matrix from the noise samples and estimate the noise floor as the minimum eigenvalue. The single channel, single pulse noise variance is found to be approximately ¡84 dB. Fig. 2(a) and Fig. 2(b) show MVDR-like spectra, ¡1 ˆ ˆ ¡1 calculated as (vH s-t R vs-t ) , where R is the sample covariance matrix calculated with training data taken over different range bin intervals corresponding to nearer and mid-ranges. The clutter ridge is evident as the “s-shaped” curve. Clutter power off of the ridge is due to near-field scattering, a likely consequence of the port-mounted antenna fore of the aircraft wing [22]. The near-field scattering only has a mild impact on detection performance in this case, as the transmit direction is boresight (zero degrees). Fig. 3 shows the MVDR spectrum resulting from the synthetic covariance matrix of (18) plus a scaled identity matrix representing the uncorrelated noise component. Observe the extremely good match between the simulated and measured data clutter ridges. The adaptive filter must null along the ridge evident in Fig. 2 to effectively mitigate the impact of clutter on target detection performance. The

Fig. 2. Measured MCARM MVDR spectra with training data selected over (a) bins 200 to 300 and (b) bins 350 to 450. 1026

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Fig. 4. Estimated SINR loss for boresight direction, nearer range, compared with simulated response.

close resemblance between the simulated response of Fig. 3 and the measured data characteristics in Fig. 2 suggests the predictability of the actual clutter response given a priori knowledge of platform attitude and clutter reflectivity and intrinsic clutter motion (ICM). It is precisely this idea that gives rise to the proposed KAPE method: take advantage of a priori knowledge allowing for relatively accurate characterization of clutter angle-Doppler response while locally estimating key parameters (e.g., array manifold, amplitude, and spectral spread) from the measurements. Fig. 4 shows the estimated SINR loss when training over two different intervals: range bins 200 to 300 comprise the first interval, while bins 300 to 400 comprise the second. We also superimpose the simulated SINR loss curve. Observe the accurate null placement and depth, but mismatch in spectral width. The lobing structure evident in the measured data does not conform to anticipated clutter characteristics. Rather, this lobing is a result of target-like signals corrupting the covariance estimate employed in the SINR loss calculation; this point was confirmed via identification of roadways at the subject ranges through the use of mapping data [12]. The lobing was found to disappear after excising those range bins presumed to overlay the road network [12]. Fig. 5 shows estimated SINR loss at further ranges, including an interval relatively devoid of roadways. In this case the match between estimated SINR loss and simulation is excellent. Some spurious nulls are still evident, such as the one just below 100 Hz when training over the furthest range interval. In the next section we investigate the impact of heterogeneous signal environments on STAP performance. Our analysis of target-like signals in the secondary data leads to loss characteristics very similar to the lobing and spurious nulls in Fig. 4 and Fig. 5. Performance loss is a result of target nulling. This anomalous behavior further motivates the use

Fig. 5. Estimated SINR loss for boresight direction, farther range, compared with simulated response.

of a priori knowledge as a means of constraining the adaptive filter response to null only in those locations where ground clutter returns presumably exist. III. STAP IN HETEROGENEOUS CLUTTER ENVIRONMENTS As described in Section I, the ability to accurately estimate the null-hypothesis covariance matrix is critical to adaptive algorithm success. Statistically homogeneous training data facilitates effective STAP implementation. However, real-world clutter is heterogeneous: clutter properties change over angle and range. A statistically heterogeneous training data set results in covariance matrix estimation error, which consequently increases mismatch between adaptive and optimal filter responses. The impact of heterogeneous training data on STAP performance is discussed in detail in [7]—[9] and some of the references therein. This section of the paper adds some new results describing the impact of heterogeneous clutter on STAP performance. We employ numerical simulation of a notional, short aperture, X-band radar to facilitate this analysis; these simulation parameters are then subsequently used to evaluate the performance of the proposed KAPE method in Section V. As described in [7]—[9], a variety of heterogeneous effects lead to degraded STAP performance. The optimal filter response incorporates precise statistical knowledge to tailor the filter null depth, width, and location. STAP, the practical implementation of the optimal filter, discerns clutter characteristics through a training step. Mismatch between the true, but unknown, test cell covariance matrix and the estimated value then leads to inappropriate filter amplitude and spectral response. Table I classifies sources of signal heterogeneity and qualitatively describes the impact on STAP performance.

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TABLE I Heterogeneous Signal Effects Heterogeneity Type Distributed Amplitude Variation

Causes

Impact

Shadowing and obscuration, range-angle dependent change in clutter reflectivity, urban centers, land-sea interfaces, etc.

Overnulling, leading to signal cancellation; undernulling, leading to increased clutter residue.

Clutter Discretes Stationary, manmade objects; sea spikes.

Distributed Spectral Variation

Variable intrinsic clutter motion due to soft scatterers (trees, windblown fields, etc.), ocean waves, weather effects.

Increased false alarm rate when in cell under test; some slight SINR loss when in training data. Inappropriate null width leads to either signal cancellation, when too wide, or increased clutter residue when too narrow.

Dense surface vehicle Target-Like environments. Signals in the Secondary Data (TSD)

Whitens desired target response, thereby substantially degrading detection probability.

Angle-Doppler Clutter Loci Variation

Inappropriate null location, depth and width leads to increased clutter residue and potential for signal cancellation.

Sensor collection geometry.

TABLE II X-band Multi-Channel Radar Simulation Parameters Parameter

Parameter

Center frequency: 10 GHz

Array configuration: sidelooking

PRF: 1 kHz

Clutter: constant gamma model, ¡12 dB reflectivity, 0.25 m2 /s2 spectral spread

Pulses, N : 32

Nominal swath center: 100 km

Array size: 2.1 m by 0.44 m

Platform velocity: 200 m/s

Spatial channels, M : 6 (non-overlapping)

Platform height: 6000 m

It is further desirable to quantify the performance loss due to heterogeneity. Such analysis is a necessary component in the design of ameliorating techniques and assists in prioritizing the challenges at hand. For this reason, we simulate an X-band radar operating under certain heterogeneous clutter conditions. The corresponding results significantly augment past discussion on this topic. Salient parameters for the X-band multi-channel radar simulation are given in Table II. Fig. 6 shows the asymptotic SINR loss for different degrees of amplitude mismatch–defined in 1028

Fig. 6. SINR loss resulting from amplitude mismatch.

this case as a power difference–between the CUT and training data. The asymptotic SINR loss isolates the performance degradation due to heterogeneity and is the ratio of the output SINR for the filter employing a known, but mismatched, covariance matrix, to the output SINR for the optimal filter; finite sample effects are conveniently excluded. The case of 0 dB clutter power difference corresponds to the matched condition (i.e., the true and asymptotic covariance matrices are equal), whereas negative or positive power differences define regions of either undernulling or overnulling. The nominal CNR at the specified range of 100 km and beam center is approximately 25 dB. Losses are noticeable after the mismatch exceeds §5 dB, starting at approximately 1—2 dB and approaching 7 dB of loss in excess of the unavoidable finite sample support loss [6]. The range of clutter power difference shown in the figure is not unlike characteristics of measured data. Finally, the value of spectral spread influences the loss calculation; the 0:25 m2 =s2 spectral spread parameter used in this instance is modest, and results using the Billingsley model are similar in nature. Fig. 7 captures the impact of spectral mismatch. We define spectral mismatch as the difference between the parameters describing spectral characteristics of the CUT and training data regions. In this case, a Gaussian autocorrelation spectral spread parameter [28] of 1 m2 =s2 characterizes the CUT. Negative values of spectral spread difference correspond to the scenario where the spectral width of the data comprising the training data, given by the spectral spread parameter, is less than the CUT, whereas the converse holds for positive values. In the negative difference region the corresponding filter null width applied to the CUT is inadequate, with the ensuing SINR loss a result of clutter residue. Signal cancellation is the primary culprit leading to losses in the positive difference region, with some slight undernulling (due to a decreased concentration of clutter power at a specified Doppler) also contributing to degraded performance. Losses approach 2—3 dB

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Fig. 7. SINR loss resulting from spectral mismatch.

in the more significantly mismatched regions, with the exception of the increased loss in the vicinity of mainbeam clutter in the negative difference region. The penalty for setting the null slightly too wide is modest. An increase in false alarm rate is the predominant effect of clutter discretes. The filter null depth inadequately suppresses the clutter discrete signal, leading to significant clutter residue crossing the detection threshold. To assess the impact of clutter discretes on performance, we employ a Poisson distribution to seed discretes into a data cube consisting of 200 range bins, where the distributed clutter component appears homogeneous. The radar cross section (RCS) and density of the clutter discretes is: 20 dBsm with density of 10 discretes/km2 ; 30 dBsm with density of 1 discrete/km2 ; 40 dBsm with density of 0.1 discretes/km2 ; 50 dBsm with density of 0.01 discretes/km2 ; and, 60 dBsm with density of 0.001 discretes/km2 . Next, we apply the post-Doppler extended factored algorithm (EFA) [29] (with three temporal DoFs and a Hanning Doppler weighting) to the distributed clutter-plus-noise data cube for cases with and without clutter discretes present. Training data are chosen using a sliding window about the CUT, with two guard cells and a total of 72 data vectors (four times the space-time DoFs, three temporal and six spatial) comprising the calculation of the covariance matrix estimate (for near and far ranges, data are chosen asymmetrically to ensure adequate training data). Fig. 8 shows the EFA output with adaptive matched filter (AMF) normalization [25]. Observe the shape of the exceedance characteristic for the case with no clutter discretes, indicating the Gaussian nature of the voltages at the filter output. In contrast, when clutter discretes are present, a heavy tailed distribution is evident at the filter output. This heavy tail leads to increased false alarm rate for threshold settings using the Gaussian assumption and can also result in lost probability of detection due to an upward bias of the CFAR detection threshold. As mentioned above, the heavy tail results from undernulling. In this example

Fig. 8. Exceedance plot comparing post-Doppler STAP output with and without clutter discretes.

Fig. 9. Impact of TSD on STAP performance.

we observe an 8.5 dB loss in threshold sensitivity for a false alarm rate of 10¡4 . Target-like signals in the secondary data (TSD) lead to significant STAP performance degradation as a result of signal cancellation [8]. Dense target environments often characterize the ground moving target indication (GMTI) radar operating environment. As an example of TSD effects, Fig. 9 shows SINR loss when applying EFA, in the same configuration as just described, to a data cube comprised of homogeneous distributed clutter and Poisson-seeded TSD. Vehicles with 10 dBsm RCS are uniformly seeded throughout the mainbeam and near sidelobe regions, over a range interval in the vicinity of 100 km, with the following radial velocities and densities: 16 km/hr and density of 1 vehicle per km2 ; ¡16 km/hr and density of 3 vehicles per km2 ; 19 km/hr and density of 0.8 vehicles per km2 ; and, 13 km/hr and density of 2 vehicles per km2 . Also, vehicles with 22 dBsm RCS, for example, representative of tractor trailer trucks, are also seeded within the same range-angle sectors and

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with same radial velocities, but reduced Poisson densities of 0.4 trucks per km2 , 0.09 trucks per km2 , 0.1 trucks per km2 and 0.1 trucks per km2 . All vehicles exhibit radial ground accelerations of 0.1 m/s2 . The legend of Fig. 9 indicates the training region for two cases: training data taken over range bins 1 to 200, or over bins 1 to 36 (twice the processor’s DoF). These two cases approximate either global or local training. We also show EFA performance when employing the known (“clairvoyant”) clutter-plus-noise only covariance matrix, and when training over 200 homogeneous clutter-only data vectors (“no TSD”). As seen from this example, losses due to TSD are significant and neither training strategy provides adequate compensation. Targets moving slower than the settings in this example seriously degrade MDV, as seen in the measured MCARM example. Among the preceding heterogeneous clutter effects, TSD is most serious. Losses routinely approach ten decibels or more. The mechanism for failure in this case is signal whitening, with adaptive filter null placement in locations inconsistent with anticipated clutter behavior. Clutter discretes are of second most consequence; Fig. 8 shows the dramatic effect of clutter discretes on output exceedance characteristics. Inadequate null depth is the culprit leading to the deleterious effect of clutter discretes. Finally, distributed clutter variability can lead to losses of 2 dB to 7 dB for amplitude mismatch, and isolated losses approaching 12 dB for spectral mismatch. SINR loss due to amplitude mismatch is a result of either undernulling (when training on weak clutter) or overnulling (training data incorporates clutter power greater than the CUT). When the spectral spread of the training set is less than the spectral spread characterizing the CUT, SINR loss is most severe; this finding suggests the value of purposely setting the notch width slightly larger than necessary to impart robustness, a finding consistent with [23]. Given the aforementioned losses, it is desirable to consider solutions other than the traditional sample matrix inversion approach based on the estimate of (3), which is prone to error in heterogeneous clutter environments. In the next section we introduce the KAPE approach, blending both measurements and a priori knowledge to construct a range-varying covariance estimate whose characteristics are consistent with anticipated clutter angle-Doppler properties and null depth and width based on highly localized parameter estimates. IV. KNOWLEDGE-AIDED PARAMETRIC COVARIANCE ESTIMATION Fig. 10 shows a flow diagram identifying the various KAPE steps. The first step involves estimating 1030

Fig. 10. KAPE flow diagram.

the array manifold using calibration-on-clutter techniques suitable for short dwells. Next, KAPE employs the set of calibrated steering vectors to generate an estimate of clutter signal power over a specified range-angle sector. In the third and fourth steps, KAPE applies a bank of whitening filters to the space-time data and selects the best match. The fifth step (shown in two parts in Fig. 10) centers on estimating the clutter spectral spread and then applying the corresponding CMT as a means of tailoring the filter width to the measured clutter response. The final step involves prewhitening the data on a range cell basis. By characterizing clutter properties over a very narrow range interval, KAPE actually can achieve better instantaneous performance than the optimal, maximum SINR filter. We subsequently describe each of the steps comprising the KAPE method. When the clutter patch voltage is perfectly correlated over the space-time aperture, tt = ts = 1, thereby allowing us to simplify (15) as ck =

Na X Nc X

®k=m,n ss-t (Áa=m,n , µa=m,n , fd=m,n ; k) (19)

m=0 n=1

where ®k=m,n is a Gaussian-distributed scalar with scaling proportional to the square root of the clutter RCS. An approximation to (19) is ck ¼

Nc X

®k=n vs-t (Áa=n , µa=n ; k) = Vk ak

n=1

c ak = (®k=n )N n=1 ,

Vk = [vs-t (1) vs-t (2) ¢ ¢ ¢ vs-t (Nc )]:

(20) c fvs-t (Áa=n , µa=n ; k)gN are the collection of n=1 hypothesized (estimated) space-time steering vectors to each ground clutter patch, where we again note the angle to the stationary scatterer determines the Doppler frequency, hence leading to removal of fd=m,n from the argument. Equation (20) provides the basis for the first several steps of the KAPE method. Ignoring range ambiguities in this approximation is justified as follows: 1) range ambiguous returns exhibit nearly identical angle-Doppler responses

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in the near-sidelooking case; 2) the angle-Doppler response for the forward-looking configuration is very similar when the slant range divided by the platform altitude is greater than five [3]; and, the combination of antenna elevation pattern and judicious selection of PRI suppress range ambiguous returns when variation does exist. Arrays errors are a practical occurrence. The true spatial steering vector is ³ss (Áa , µa ) = "(Áa , µa ) ¯ ss (Áa , µa ):

TABLE III Summary of Calibration-on-Clutter Methods Method

Premise

Maximum Eigenvector, Doppler Centroid (EV-centroid). Form spatial covariance using Doppler filter encompassing main beam clutter peak.

Boresight (or steered boresight) clutter signal maximally projects onto maximum eigenvector.

(21)

"(Áa , µa ) is the angle dependent array error vector and ss (Áa , µa ) is the ideal spatial steering vector defined previously. We further presume "(Áa , µa ) = "1 (Áa , µa ) ¯ "o , where "1 (Áa , µa ) is the angle-dependent component and "o is angle independent. This model applies to the narrowband condition, requiring modification as frequency variation among receiver transfer functions becomes significant. Angle-dependent errors result from factors such as erroneous element placement. The angle-independent error characterizes the bulk of the mismatch between the ideal and actual steering vector as measured at the output of the analog-to-digital converters, at least for the angle of interest. Our objective centers on estimating "o ; this is an essential aspect of KAPE, since we must ensure the modeled clutter covariance matrix accurately matches the actual measurements. We can exploit the clutter signal itself to estimate "o . This approach is sometimes referred to as “cal-on-clutter.” Table III summarizes the cal-on-clutter methods considered in this paper. Further detail on each of the approaches is described in the Appendix. Accurate knowledge of the array manifold enables KAPE to effectively whiten the distributed clutter component. In traditional STAP applications, the unknown array manifold degrades the matched filtering operation, but the covariance estimate contains the necessary whitening information to suppress clutter (in the absence of complications resulting from heterogeneous clutter, as described previously in Section III). The estimate "ˆ o is used to construct a set of spatial steering vectors, Vk , critical to several KAPE steps, including estimates of clutter power and formation of candidate whitening filters. Each column of Vk is given by vs-t (Áa=n , µa=n ; k) = vt (Áa=n , µa=n ; k) − ("ˆ o ¯ vs (Áa=n , µa=n ; k))

(22) where vt (Áa=n , µa=n ; k) and vs (Áa=n , µa=n ; k) are hypothesized, though otherwise ideal, temporal and spatial steering vectors to the ground clutter patch at the specified angle. Given the collection of calibrated space-time steering vectors, we then wish to estimate the voltage vector, ak . A least squares estimate (LSE) of ak

Comments Each eigenvector spans multiple space-time signals, thus leading to performance degradation. Requires adequate Doppler centroid estimate.

Maximum Eigenvector, Same premise as above. All Data (EV-all). Form spatial covariance matrix by averaging over slow-time and fast-time.

Each eigenvector spans multiple space-time signals, thus leading to performance degradation.

Cross-Correlate Adjacent Channel Pairs (Ch-pr).

The phase history between adjacent channels is very similar, with the complex channel gain error accounting for most of the difference.

Performance improves as dwell increases, since mismatch exists at beginning and end of phase history collection.

Channel Pair Detrend (Ch-pr-detrend).

The spatial phase variation between adjacent channels as a function of Doppler appears linear across the main beam, with a y-intercept approximately equal to the complex channel gain error.

Requires a good Doppler centroid to avoid bias in the y-intercept estimate. Has performed well with measured data (results not shown in this paper).

min kxk ¡ Vk ak k22 :

(23)

satisfies ak

The pseudo-inverse provides the solution to (23), taking the form ¡1 H ak = (VH k Vk ) Vk xk :

(24)

The pseudoinverse calculation generally involves the computationally costly singular value decomposition. Alternative, suboptimal approximations, in order of decreasing computational burden and reduced estimation performance, include the weighted space-time spectrum and the weighted spatial spectrum averaged over the temporal aperture. The former approximation takes the form, ak = (Wst ¯ Vk )H xk ,

Wst = (bt − bs )1TNc (25)

where bt 2 R Nx1 and bs 2 R Mx1 are temporal and spatial weightings and Wst is the space-time weighting

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matrix. The latter approach appears as v u N u1 X diag(Bn B¤n ) ak = t N n=1

Bn = diag((Ws ¯ Vs )H xs=k (n))

(26)

Ws = bs 1TNc with xs=k (n) 2 C Mx1 representing the spatial snapshot for the nth pulse and the collection of spatial steering vectors of interest comprising the columns of Vs . The performance of (25) and (26) relative to the LSE of (24) depends on radar system parameters, especially array characteristics; equation (26) performs especially poorly when spatial ambiguity is present. An important issue when implementing (24)—(26) concerns the angular spacing between steering vector locations. Oversampling is not useful since the aforementioned estimation approaches for ak are nonparametric, and consequently diffraction limited. The diffraction-limited clutter patch resolution is limited to the lesser of the Doppler angular beamwidth or array spatial beamwidth, thereby providing guidance on steering vector separation. The third step of the KAPE approach involves forming candidate whitening filters. A whitening filter is given by the inverse square root of the modeled clutter-plus-noise covariance matrix. Based on the discussion in the preceding section, the parametric clutter covariance model corresponding to (20) follows as M0KAPE=k = Vk Pk VH k,

c Pk = diag(j[ak ]n j2 )N n=1

(27) where Pk is a matrix of instantaneous power estimates at angles corresponding to the pointing directions of the steering vectors comprising the columns of Vk . Key factors influencing performance at this stage include accuracy of array normal and platform velocity vectors and knowledge of terrain height. A separate, coupled INU/GPS measurement unit attached to the antenna array provides an estimate of the boresight direction. The ownship navigation unit yields an estimate of the platform velocity vector. DTED provides local terrain height characteristics; information on the height of discrete features is not included and accuracy at discontinuous surfaces is diminished. Through slight variation of parameters comprising the model of (27), we form a set of candidate filters to compensate mild imperfections in prior knowledge or errors in the model. This approach is consistent with the overall KAPE objective of blending both prior knowledge and measurements via the parametric model. Again, the angular spacing between steering vectors is critical to effective implementation. Since (27) is a parametric model, thereby possessing 1032

superresolution potential, it is important to oversample relative to the diffraction-limited case. A good rule-of-thumb, based on numerical simulation, is to set the angular spacing of the steering vectors in (27) to about twenty to thirty percent of the diffraction-limited resolution. As a result of oversampling, it is then necessary to equally distribute the estimated, diffraction-limited patch power–embodied along the diagonal of Pk –among the clutter subpatches. Errors in the estimated array normal vector lead to differences between anticipated and actual mainbeam clutter Doppler centroids and the general shape of the clutter ridge. The platform velocity vector is generally known with sufficient accuracy using modern navigation equipment; small errors between measured and actual velocity vectors generally lead to imperceptible differences between the true and modeled clutter ridge locations. KAPE can incorporate knowledge of both radar platform and terrain height into the parametric covariance model. In the case of a perfectly sidelooking array, clutter returns fall along the same clutter ridge over all range, regardless of height variation (i.e., neglecting DTED information has no impact on performance). When aircraft yaw is present, unaccounted terrain height variation will lead to small errors between estimated and actual clutter ridge locations, mainly at near slant range. The extreme case of a forward-looking array exhibits sensitivity to errors in knowledge of terrain height; the corresponding error exhibits range dependence, especially for those regions where the slant range divided by the platform altitude is less than five [3]. By dithering the filtering notch in the Doppler domain, KAPE mitigates the aforementioned error sources. The nth Doppler dither vector is given by hn = s¢=n − 1M , where s¢=n = [1 exp(j2¼¢f=n T) exp(j2¼¢f=n 2T) ¢ ¢ ¢ exp(j2¼¢f=n (N ¡ 1)T)]T

(28)

and ¢f=n is the difference in Doppler frequency between the dithered and modeled values (e.g., if the modeled mainbeam clutter is at 5 Hz and we wish to move the null to 15 Hz due to slight error in array normal, ¢f=n = 10 Hz). The Doppler dither is applied as H M00KAPE=k,n = hn hH n ¯ Vk Pk Vk

(29)

thereby effectively shifting the clutter ridge up or down depending on the choice of ¢f=n . Ideally, ¢f=n is specified as the Doppler shift resulting from a fractional beamwidth error in array normal pointing direction. Since effective mainbeam clutter cancellation is the most critical consideration for GMTI radar, use of a Doppler dither vector is sensible.

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The basis for the KAPE approach is the availability of fairly accurate prior knowledge describing the anticipated stationary clutter response; the MCARM analysis in Section II justifies this presumption. In the first three steps, KAPE then estimates the array manifold error vector, clutter amplitude response for each iso-range and specified angular sector, and provides a mechanism for parametrically adjusting the clutter notch null position. The modeled covariance at this stage of the processing is given as R0KAPE=k,n

=

M00KAPE=k,n

2 + ¾DL IMN

(30)

2 is the previously defined diagonal where ¾DL loading level. The fourth step requires selecting the whitening filter best matched to clutter properties, effectively determining the M00KAPE=k,n best matching the mainbeam clutter response. A plausible test of whitening filter goodness is given by the AMF output [25] averaged over a localized region defined by L0 range cells,

´n =

L0 X

00 2 ¡1 2 jvH 1 s-t (MKAPE=k,n + ¾DL INM ) xj j : 00 2 ¡1 L0 vH s-t (MKAPE=k,n + ¾DL INM ) vs-t j=1

Tt = toeplitz(1 exp(¡T2 ½) exp(¡(2T)2 ½) ¢ ¢ ¢ exp(¡((N ¡ 1)T)2 ½)

(33)

where ½ = 8¼2 ¾v2 =¸2 and ¾v2 is the variance of the clutter spectral spread in m2 /s2 [28]. The Gaussian autocorrelation function corresponds to a Gaussian power spectrum; it completely decorrelates as the temporal aperture increases. The temporal CMT for the Billingsley model [24] is µ Tt = toeplitz

·

r 1 1T + 1 1 + r1 N 1 + r1

1 1 1 1 £ 1 ¢¢¢ 1 + ¯m 1 + ¯m 2 2 1 + ¯m 3 2 1 + ¯m (N ¡ 1)2

(31)

¸¶

(34)

The AMF test is sensitive to the steered angle-Doppler location given by the choice of vs-t . In the present case, steering to the center of mainbeam clutter is an appropriate choice. Moreover, the denominator term normalizes the output to a value of unity for those signal components matching the covariance model. We recognize the term in parentheses in (31) as R0KAPE=k,n for a particular choice of the Doppler dither vector, hn . Choosing an hn yielding minimal AMF output is the objective; we denote the corresponding clutter ¯ 00 covariance model as M KAPE=k and the associated, best ¯ selection of Doppler dither vector as h. An alternate measure of whitening filter efficacy is given by the quadratic form, L0

1 X H 00 2 °n = 0 xj (MKAPE=k,n + ¾DL INM )¡1 xj : L

CMT. Spatial and temporal CMTs are discussed in Section II, with the space-time CMT defined in (18). At issue here is the appropriate temporal autocorrelation model for the particular land type and dwell length: the Gaussian model applies over water, whereas the Billingsley model is suitable over land. Cultural databases can assist in selecting the best model. In the Gaussian case,

(32)

where r1 = (1=489:8)!11:55 Äfc1:21 , !1 is windspeed in miles per hour, and Äfc = fc =1E9 is the radar center frequency in gigahertz. Also, ¯m = (4¼T¯1 =¸)2 with ¯1 = 0:1048(log10 (!1 ) + 0:4147) if !1 > 0:4 and ¯1 = 0 otherwise. As evident from the form of the equation, the Billingsely model is based on extensive measured data analysis. This model includes both ac and dc terms; the leaves of soft scatterers blowing in the wind decorrelate exponentially, while the trunk remains highly correlated. An approach for setting Ts based on angle dithering is given in [23]. We then accomplish step 5 and step 5a of Fig. 10 in a manner similar to the selection procedure for the best whitening filter. Specifically, we seek the smallest temporal and spatial spread parameters minimizing the AMF test statistic,

j=1

Equation (32) is a test of covariance structure: when the covariance structure embodied in xj matches the covariance model, the quadratic form takes on its minimal value. Relative to the AMF test, (32) is independent of a particular steered direction–thus providing a global assessment of covariance match, with reduced sensitivity to any particular direction–and is simpler to implement. A version of (32) has been used to screen STAP training data [26—27]. After selecting the best M00KAPE=k,n –which accounts for null placement and depth–the KAPE then must estimate the clutter spectral spread and subsequently tailor the filter notch width accordingly via the selection of an appropriate space-time

0

´˜ n,p =

L 1 X L0 j=1

£

2 2 ¡1 2 ¯ 00 jvH s-t (Ts-t=n,p (¾v=n , !1=n ; ¢Á=p ) ¯ MKAPE=k + ¾DL INM ) xj j : ¯ 00 vH (T (¾2 , ! ; ¢ ) ¯ M + ¾2 I )¡1 v s-t

s-t=n,p

v=n

1=n

Á=p

KAPE=k

DL NM

s-t

(35) 2 Ts-t=n,p (¾v=n , !1=n ; ¢Á=p )

is the space-time CMT and 2 or !1=n , depending on the is a function of either ¾v=n chosen temporal autocorrelation function, and the spatial dither parameter ¢Á=p [23]. The notch width should adequately capture the clutter spectral extent, but should not be overly wide as to suppress slow moving targets; it is for this reason that we require the smallest parameter leading to the minimum AMF

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output. Moreover, by averaging over a limited range extent, the demarcation should generally be clear: at a certain filter notch width, clutter is appropriately mitigated, after which further increases in spectral spread parameters prove detrimental. A quadratic form similar to (32) can also be used. However, we prefer the AMF test of (35) due to its angle dependence and the realization that setting the filter width in the look direction is most critical. We denote the best selection of the spectral spread parameters, dependent on the ¯ . autocorrelation model, as ¾¯ v2 , !¯ 1 and ¢ Á Accomplishing the prior steps yields the RKAPE=k needed to whiten the ground clutter signal in the kth range bin. As defined previously, RKAPE=k = 2 IMN , where MKAPE=k + ¾DL ¯ )¯M ¯ 00 MKAPE=k = Ts-t (¾¯ v2 , !¯ 1 ; ¢ Á KAPE=k

¯ ) ¯ h¯ h¯ H ¯ M0 = Ts-t (¾¯ v2 , !¯ 1 ; ¢ KAPE=k Á

¯ ) ¯ h¯ h¯ H ¯ V P VH : (36) = Ts-t (¾¯ v2 , !¯ 1 ; ¢ Á k k k If a subsequent adaptive step is not applied, the test statistic calculations of (35) can be put to good use by directly applying a detection threshold. Otherwise, the processor passes both the prewhitened data vector, ¡1=2 x˜ k , and the whitening filter transformation, RKAPE=k , ¡1=2

to the adaptive stage (RKAPE=k is necessary in the subsequent space-time steering vector calculations). V. PERFORMANCE ASSESSMENT The purpose of this section is to evaluate the performance potential of the KAPE method and identify any specific sensitivity to either model mismatch or other practical effects. Numerical examples use the radar parameters given in Table II with a 15 mi/h Billingsley model replacing the Gaussian intrinsic clutter motion model. Array calibration is the first step in applying KAPE. Residual errors between the actual and estimated array manifold limit clutter suppression. Achieving 30 dB of clutter cancellation requires maintaining channel phase errors below roughly 2 deg and amplitude errors below 0.1 dB. It is important, then, to evaluate the performance of the array calibration-on-clutter methods proposed in the prior section. Fig. 11 shows the residual phase errors after applying the four calibration techniques described in Table III to simulated data with 20 deg initial rms phase error and 3 dB rms initial amplitude error. With the exception of the channel pair detrend method, all other approaches capably reduce the phase errors within the required §2 deg interval. The cause of degraded performance for the channel pair detrend method in this case is presumably error in the Doppler centroid; however, in other analyses with X-band 1034

Fig. 11. Residual phase error after calibration-on-clutter procedure.

measured data, we have found very good performance for this strategy. Fig. 12 shows the residual amplitude error after applying the calibration methods. The algorithms based on adjacent channel pair processing and maximum eigenvector method applied after Doppler centroiding lead to very good performance. Slight degradation is seen for the channel pair detrend method, while the maximum eigenvector method using all of the data fails to meet our required residual level below 0.1 dB. This latter strategy exhibits degraded amplitude estimation performance because signal components in addition to the mainbeam clutter response contribute to the elements of the maximum eigenvector (i.e., the maximum eigenvector in this case spans not only the spatial component corresponding to the center of mainbeam clutter, but also clutter signals with spatial steering vectors off of the array boresight). Interestingly, the phase estimation performance of EV-all is quite good; this occurs as a result of phase averaging of components to either side of the mainbeam clutter response. In calculating Fig. 11 and Fig. 12, we average over 100 trials and apply each method using an auxiliary data set whose size is equal to twice the available space-time DoFs. Next, KAPE is applied to 200 realizations of identically distributed, multi-channel, synthetic radar data whose general characteristics are shown in Table II. Initial array manifold errors are Gaussian distributed with 10 deg rms phase and 3 dB rms amplitude offset. Fig. 13 compares the SINR loss between KAPE and the optimal filter for the four calibration-on-clutter methods summarized in Table III, as well as the case with no calibration. This SINR loss is specifically given as the ratio of output SINR using the KAPE method to the output SINR for the optimal filter. The calculation takes account of the precise array manifold description and employs the known clutter-plus-noise covariance

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Fig. 12. Residual amplitude errors after calibration-on-clutter procedure.

Fig. 13. SINR loss between KAPE and optimal space-time processor using averaged KAPE clutter covariance matrix.

matrix. Additionally, the calculation uses the mean KAPE covariance matrix model (i.e., the average of the covariance model generated for each realization). From this figure we find very admirable results for the KAPE method for all four calibration methods. The losses, though generally small, are a result of calibration residue and an artifact of KAPE tracking the instantaneous variation of the Raleigh-distributed clutter signals. Observe the poor performance achieved without array calibration; this latter curve further validates the efficacy of the different calibration-on-clutter techniques and emphasizes their critical role in effectively implementing KAPE. Since KAPE tracks the variation in Rayleigh clutter, it potentially provides better performance at any given instant than the optimal filter (whose construction is based on expected values). If, for example, the clutter signal is in a fade, the KAPE adjusts the null depth accordingly to potentially improve sensitivity to nearby targets. Similarly, the clutter null depth increases when clutter

Fig. 14. Instantaneous loss between KAPE and optimal space-time processor.

Fig. 15. Output exceedance comparison between KAPE and optimal space-time processor.

coherently sums to a strong value. Fig. 14 shows the instantaneous SINR loss between the KAPE and the optimal filter over 200 trials. As suggested, these losses are misleading since KAPE is tracking the instantaneous variation of the clutter signal while the optimal filter response is based on expected values. An exceedance plot, shown in Fig. 15 for our present example, supports this conjecture: the exceedance characteristic for the KAPE method is approximately 1 dB better than for the optimal filter. The improved sensitivity in this case translates to the potential for effective target detection and false alarm rate management, as our next example shows. The impact of TSD on detection performance is a strong motivation for pursuing the KAPE method. The measured data result in Fig. 4 and the simulation analysis in Fig. 9 present compelling evidence of the seriousness of TSD. Using the same radar parameters, we seeded target-like signals into the data cube in a manner representative of a typical GMTI scenario; this seeding is a different realization of the same parameter set leading to the findings in

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Fig. 16. SINR loss for KAPE and EFA relative to the optimal space-time processor in TSD environment.

Fig. 17. KAPE detections using OS-CFAR.

Fig. 9. Fig. 16 shows the SINR loss for the KAPE method relative to the optimal space-time filter, as well as the SINR loss between adaptive and optimal implementations of the EFA method of [29]. We employ the channel pair (Ch-pr) calibration method to estimate the unknown array manifold. The KAPE and EFA approaches use the estimated error vector "ˆ o in their implementations. As anticipated, TSD leads to no ill effect in terms of the KAPE SINR loss curve taken over the 200 realizations. On the other hand, the two EFA implementations–with training over all 200 realizations, K = 1 : 200, and over bins forty to seventy-five, K = 40 : 75–show substantial loss consistent with calculations leading to Fig. 9. By design, the KAPE approach yields a structured covariance matrix model whose various components lie largely in the true clutter subspace. As no desired target component corrupts this model, the corresponding KAPE weight vector does not lie in the null space of the target steering vector. Thus, KAPE prevents signal whitening despite the presence of TSD. 1036

Fig. 18. Detections using space-time filter with known covariance matrix followed by OS-CFAR.

Further evidence of KAPE resilience to TSD and the benefits of improved instantaneous performance may be seen by comparing Fig. 17 to Fig. 18. Fig. 17 shows the ordered-statistic CFAR (OS-CFAR) detection map after applying KAPE. Target truth is also shown on this map; it is important to keep in mind that a detectable signal is not present at each truth location, since targets abide by a fluctuating, Swerling I model and are otherwise subject to constraints imposed by the radar range equation. The RCS for seeded cars was taken as 10 dBsm and 22 dBsm was used for trucks. No centroiding is applied to the OS-CFAR output, so Doppler sidelobes also appear as detections. Results in Fig. 17 are best understood by comparing with the OS-CFAR output applied after the space-time filter employing the known covariance and calibrated steering vector (also using the Ch-pr method), as shown in Fig. 18. The KAPE approach enables detection of one additional target not identified in the known covariance filter case, as shown by the arrow in Fig. 17. Moreover, an additional false alarm is apparent for the known covariance case, as identified by the arrow in Fig. 18. Though not shown here, we also generated the detection map for the optimal filter–constructed with the known covariance matrix and precise steering vector incorporating known array errors–and found the same performance as in Fig. 17 (same number of detections with no additional false alarms). It is thus remarkable that, while operating on a finite data set of fluctuating clutter signals corrupted with TSD and characterized by an erroneous array manifold, KAPE provides performance as good or better than the known covariance case and comparable to the optimal filter capability. Fig. 19 depicts KAPE SINR loss relative to the optimal filter for different implementations. As in the prior discussion, we use the Ch-pr calibration-on-clutter method, specifically considering

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concern of the radar community. It is widely held that judicious exploitation of a priori knowledge is central to robust STAP deployment under a diverse set of operating conditions [10]. In this vein, this paper introduced and analyzed a KA parametric covariance estimation method. Rather than estimating the whole covariance matrix, the KAPE approach involves estimating parameters of a covariance matrix model. Specific advantages of this method include:

Fig. 19. SINR loss relative to optimal filter for different KAPE implementations.

the following three cases formulated to include the AMF-normalized, beamformed output: 1) KAPE with known values of frequency offset for step 4 and employing step 5 with the known spectral spread parameters (a 15 mp/h Billingsley model in this case), 2) KAPE using only steps 1 through 4, where step 4 employs the known frequency offset, and 3) the fully automated KAPE implementation involving step 1—step 6. As seen from this figure, employing all six steps leads to performance comparable to the scenario where the precise frequency offset and spectral spread parameters are known, and exceeding the performance for the case where no spectral spread CMT is used (step 1 through step 4 only). We further point out the calibration-on-clutter approach by itself leads to an 8.87 dB improvement relative to the filter using the ideal, but erroneous, steering vector for this particular random draw of array errors. KAPE performance enhancement results from the capability to instantaneously track clutter variability, which is invaluable in complex, heterogeneous clutter environments. VI. SUMMARY STAP is an important method for improving the performance of aerospace radar operating in clutter-limited environments. Estimating the null-hypothesis covariance matrix for the CUT is a fundamental requirement of any practical STAP implementation. Seminal STAP results presume availability of finite, IID training data to bound performance potential [6]. It is known, however, that many realistic clutter environments are heterogeneous, thereby leading to covariance estimation errors and a consequently mismatched STAP response relative to optimal. Improving STAP performance in complex, heterogeneous clutter environments is a fundamental

1) conversion of a higher-dimensional matrix estimation problem to the estimation of several model parameters, thereby reducing the potential for error in complex signal environments; 2) improved instantaneous filter response through highly-localized training data selection; 3) avoidance of anomalous filter behavior–due, for example, to target-like signals in the training data–through constraints enforced by a physical model leading to nulls consistent with anticipated characteristics of surface clutter returns; 4) better conditioning of output clutter residue. Herein we developed a clutter covariance model, subsequently validated using measured, multi-channel airborne radar data. This model provides a foundation for the KAPE method. Unknown parameters of the model include the array manifold (precise spatial steering vector for a specified direction of arrival), as well as the clutter amplitude, exact clutter null placement, and spectral width for a given range-angle cell. Critical KAPE steps, summarized in Fig. 10 and described in detail in Section IV, thus focus on accurately estimating these unknowns. We introduced four calibration-on-clutter methods to estimate the array manifold, a least squares estimator to ascertain clutter amplitude, a filter bank strategy to test plausible space-time clutter whitening filters and select the best match, and a subsequent filter bank approach implementing a sequence of CMTs to estimate clutter spectral width. Using numerical simulation, we evaluated the performance of the KAPE method. Key results included validation of all four calibration-on-clutter methods. Using SINR loss we found the average KAPE performance closely approached the optimal situation, to within roughly 2 dB of optimal for the best performing calibration strategy. We further showed instantaneous SINR losses, which at first glance appear fairly substantial. However, these losses are a consequence of KAPE’s ability to accurately track the range variation of the clutter return voltage; the SINR loss calculation is relative to the optimal condition, which necessarily involves an expectation operation and thus fails to capture instantaneous behavior. For this reason, the instantaneous SINR loss calculation is misleading, and so instead we compared the exceedance characteristic of KAPE

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relative to the optimal filter, finding approximately a 1 dB enhancement, as a more accurate depiction of performance. Finally, we identified the most compelling advantage of KAPE: its robustness to target-like signals in the secondary data, or TSD. By design, the KAPE covariance model does not include any target component, thus avoiding any target signal whitening. This is an important benefit of the proposed approach, as TSD most significantly degrades STAP detection performance and is a very practical effect, as demonstrated herein with measured and simulated data examples (see Fig. 4, Fig. 5 and Fig. 9). Through numerical analysis, we used a Poisson distribution to seed TSD into a radar data cube for testing purposes. Analysis showed KAPE behavior is unaffected by TSD (i.e., no anomalous target nulling), whereas standard STAP methods suffer significant loss for both global and local training. We further confirmed this point by comparing the output detection maps for KAPE and the filter constructed using the known covariance matrix. KAPE detects an additional target not identified when applying the filter based on the known covariance matrix, whereas we find an additional false alarm in the latter case. Moreover, for this example KAPE performs as well as the optimal filter despite the presence of TSD. The potential for performance improvement in heterogeneous clutter environments follows directly as a result of KAPE’s ability to instantaneously track the fluctuating clutter voltage signal over range. Drawbacks of the KAPE method include its reliance on effective calibration strategies and increased computational burden. Clutter cancellation is constrained by knowledge of the array manifold, needed to implement the covariance matrix model; uncompensated array errors lead to increased clutter residue. Also, KAPE has unique computational demands, as it constructs the covariance matrix model and whitening filter on a range cell basis. Moreover, the range-varying whitening filter must be stored in memory to form the matched filter. For the implementation described in this paper, computational burden is on the order of the space-time optimal filter application with a range-varying covariance matrix. It is plausible to form covariance models over range sectors to alleviate some of this computational burden, while a post-Doppler KAPE implementation will greatly mitigate the required processing load, just as in the traditional STAP case. Also, KAPE should be highly parallelizable, since each range cell is effectively manipulated independently after the calibration step. Finally, ranged folded clutter with significantly different angle-Doppler support is a cause for concern. When noise jamming is present, a front-end adaptive processing step should precede KAPE. This is possible since the jammer and clutter 1038

signals are additive and statistically independent. The processor uses a clutter-free listening interval to train and set the adaptive jammer cancellation weights. Moreover, benefits of this strategy include mitigation of computational burden and training support requirements arising from the subsequently reduced adaptive spatial DoFs. An adaptive processing implementation following KAPE prewhitening requires specialized training to prevent TSD and other aspects of heterogeneous clutter residual over range from reintroducing some of the challenges resolved by KAPE [26—27]. KAPE enables a more effective implementation, however, as it provides improved contrast to better screen and remove potential target components from the adaptive training interval and facilitates localized training which may prove beneficial when tailoring the adaptive response to variations in the distributed clutter environment. Future work will include testing KAPE using measured data. Given the excellent match between measured data characteristics and the proposed clutter model shown in this paper, we anticipate effective KAPE performance. Additionally, the development of a post-Doppler KAPE implementation to alleviate computational burden is an important next step. APPENDIX–CALIBRATION-ON-CLUTTER METHODS Array calibration has two components: 1) determining complex channel error gain, and 2) estimating array pointing direction with respect to the platform velocity vector. We find that an integrated inertial navigation unit and GPS receiver effectively measure requisite parameters leading to high-fidelity prediction of clutter angle-Doppler loci, as seen from Fig. 2 and Fig. 3. Random array errors, on the other hand, limit the attainable level of clutter precancellation. For this reason, in situ calibration is a necessary component of the KAPE scheme. Table III summarizes four calibration methods considered in this paper. We provide further detail on each of the methods in this appendix. Each of the four methods was tested against synthetic data, with results summarized in Fig. 11 and Fig. 12. A. Eigenvector Techniques (EV-all and EV-centroid) The objective of the calibration-on-clutter methods summarized in Table III is to estimate the angle-independent error component "o as a means of forming the collection of steering vectors defined in (22). The central concept behind all eigen-based methods for array calibration lies in the following fact: qk=m 2 span(sj )Pm=1

(37)

where qk=m is the mth eigenvector of the P-dimensional, dominant subspace characterizing

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the clutter-plus-noise covariance matrix, and sj is the jth received signal steering vector of appropriate dimension. (The index k, dropped for notational convenience in the remainder of this appendix, identifies the range cell number.) Given either adequate resolution, sufficient magnitude differential or substantial physical separation between signal sources, (37) suggests the eigenvector corresponding to the largest eigenvalue takes a form similar to qmjm=1 =

P X p=1

am,p sj ¼ a1,1 s1

(38)

where am,p is the pth coefficient describing the linear combination of signal sources equating to the mth eigenvector (m = 1 in this case). Indeed, eigen-based spectral estimators take advantage of the form in (38). Assuming a uniform linear array in a perfectly sidelooking configuration,1 we then might anticipate an error-free, space-time eigenvector corresponding to mainbeam clutter of the form q1 ¼ a1,1 1

(39)

where 1 is a column vector with all entries set to unity p and a1,1 = 1= NM. The next step involves comparing (39) to the corresponding eigenvector of the sample covariance matrix: qˆ 1 ¼ aˆ 1,1 ("o ¯ 1):

(40)

"o is the vector of complex gains characterizing the error between the actual and ideal array response; arriving at "ˆ o involves simply comparing (40) to (39). A spatial covariance matrix estimate is required prior to the eigen-decomposition. A calculation similar to (3) is used in the EV-all method of Table III, where training data are taken over both slow-time and fast-time. Averaging over both temporal domains minimizes statistical error and avoids estimating a Doppler centroid. The spatial covariance matrix takes the form N

K

1 XX ˆ R xs=k (n)xH s=k = s=k (n) NK

(41)

n=1 k=1

with xs=k (n) the spatial snapshot for the nth pulse and kth range. An eigen-decomposition of (41) yields qˆ 1 as the eigenvector corresponding to the maximum eigenvalue. Since clutter returns result from the response of a distribution of scatterers spread over range and angle, the likelihood of the most dominant space-time eigenvector simply being proportional to a single signal vector is low. In other words, the approach to estimating the actual array manifold suggested by (39) and (40) doesn’t always provide acceptable 1 This

approach is valid for any known array configuration. The uniform linear array is a simple case to consider.

performance because (38) is not strictly valid. A better approach involves isolating the dominant clutter return through Doppler processing. Specifically, given the transformation matrix T leading to the post-Doppler, spatial snapshot ³k = TH xk x (42) the resulting covariance matrix estimate is K X Hˆ ³ˆ = 1 R TH xm xH k m T = T Rk T: K

(43)

m=1

ˆ is given by (3). T is a Doppler transformation R k converting the pulse information within each channel to the frequency domain and takes the form T = st (fˆd=centroid ) − IM . In this case, fˆd=centroid is an estimate of the Doppler centroid corresponding to mainbeam clutter and IM is the MxM identity matrix. Choosing T aligning with the center of the mainbeam clutter Doppler frequency results in a spatial response predominated by the (virtual) broadside direction (this is valid for a steered array as well). The next step ³ˆ to involves comparing the largest eigenvector of R k

the ideal broadside2 spatial response embodied by qˆ 1 , resulting from the eigen-decomposition of (43), to arrive at "ˆ o . This is the essence of the EV-centroid approach of Table III. Performance improves since the projection of the space-time data onto the space given by T isolates the dominant signal component. Nevertheless, drawbacks remain: the eigenvector still is not directly proportional to the true, but unknown, spatial response of mainbeam clutter; and, errors in calculating the Doppler centroid (leading to T) bias the result (i.e., the anticipated error-free spatial steering vector is incorrect). B. Adjacent Channel Pair Processing (Ch-pr and Ch-pr-detrend) Channel pair processing focuses on comparing the phase histories of adjacent channels and attributing any large deviation to the relative complex error. That the phase histories in adjacent channels should appear highly similar, with the exception of error attributable to differences in receive channel hardware, is the rationale behind the approach. Specifically, adjacent channel pair processing–identified as Ch-pr in Table III–involves cross correlating the slow-time data for nearby channels and then assigning the phase error as the mean of the phase of the corresponding result (perfectly matched channels naturally yield a phase error of 0 radians). The estimate for the amplitude error is the ratio of L2 -norms between channel data. This process yields the relative error. 2 When the array is steered off broadside, the ideal spatial response is still proportional to 1 assuming the presteer spatial weights remain on the receive data.

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The absolute error leading to an estimate for "o follows by comparing pairs of relative errors. For each channel m define the range-pulse matrix 2 (x (m : M : M(N ¡ 1) + m))T 3 1

6 (x (m : M : M(N ¡ 1) + m))T 7 7 6 2 7 Zm = 6 7 6 .. 5 4 .

(44)

(xL (m : M : M(N ¡ 1) + m))T

where Zm 2 C LxN and L and N correspond to range and pulse dimensions, as previously defined. We next initialize the error vector estimate such that all estimates relate to the first entry: "ˆ o,rel = [1 0TM¡1 ]T . Further define the collection of phase history data in the mth and prior channel as the matrix, Ym 2 C 2xLN , given as (45) Ym = [Zm¡1 (:) Zm (:)]T : ˆ The entries of the 2 £ 2 covariance matrix, R Ch-pr=m = H Ym Ym , contain the information necessary to relate the error in the mth channel to the reference. The relative error is given by v uˆ u RCh-pr=m (2, 2) ˆ exp(j arg(R "ˆ o,rel (m) = t Ch-pr=m (2, 1))): ˆ (1, 1) R Ch-pr=m

(46) ˆ ˆ The absolute error follows as "o (m) = "o,rel (m) ¢ "ˆ o,rel (m ¡ 1). These calculations are applied for all m = 2 : M to construct the estimated error vector. Short dwells affect this approach, since mismatch at the ends of the phase histories bias the calculation of the mean phase. The approach, however, does not require a priori knowledge of the array configuration, and its implementation is low complexity (the calculation is tantamount to comparing two complex numbers), but it is strictly applicable only to the sidelooking deployment. The Ch-pr-detrend method employs multi-channel, range-Doppler processing to identify and exploit the approximately linear phase across the spatial beamwidth as a function of Doppler. Imagine comparing the phase between adjacent channels for different Doppler filters about the mainbeam clutter Doppler. A plot of relative (spatial) phase versus Doppler is approximately linear, with zero phase at the mainbeam clutter Doppler (assuming presteered data). Beating the estimated phase function against the ideal linear phase function yields a line in the Doppler-phase plane of approximately zero slope but appears offset from the abscissa when array errors are present. The offset corresponds to the complex error gain. (Averaging over Doppler enhances the estimate of this relative error.) Absolute error follows by comparing relative errors, as in the case of the Ch-pr method. 1040

We describe the essence of the Ch-pr-detrend method as follows. Starting with (44), we form a range-Doppler map in the mth channel by taking the fast Fourier transform along the columns of Zm . We denote the mth channel range-Doppler map Fm . The frequency domain cross-correlation between channels m and m ¡ 1 over range and Doppler is given as Cm = Fm ¯ F¤m¡1 , where the superscript * denotes conjugation. We further require an estimate of the Doppler centroid for mainbeam clutter, fˆd=centroid . For an ideal array, the cross-correlation matrix should yield zero value at the Doppler frequency corresponding to the center of mainbeam clutter. This point corresponds to the array (virtual) boresight. Any non-zero phase bias or amplitude offset is attributable to the complex gain error between the two channel pairs. Naturally, since noise biases the phase estimate, it is desirable to first estimate and then remove the linear phase variation over Doppler for the various ranges of interest, subsequently averaging the residual phase deviation from zero. This detrending of the linear phase variation in the frequency domain is tantamount to resampling the phase history data to align the channel responses; the array complex gain error is then the residual offset. Define ©m as the matrix of estimated linear phase variation between channel m and m ¡ 1 over Doppler for each range. The detrended cross-correlation matrix is then C0m = Cm ¯ ©¤m . The relative array error is then 0 0 11 ˜ L X N X kFm (:)k2 1 "ˆ o,rel (m) = exp @j arg @ C0m (k, n)AA kFm¡1 (:)k2

˜ LN

k=1 n=1

(47) ˜ is the number of Doppler filters. As in the where N Ch-pr case, the absolute error follows as "ˆ o (m) = "ˆ o,rel (m)"ˆ o,rel (m ¡ 1). Advantages of this latter approach include its simplicity and compatibility with typical GMTI processing. Additionally, fitting the data with an estimate of linear spatial phase over a restricted angular sector generally obviates the need for highly accurate array layout information. Calculating the Doppler centroid is the primary disadvantage, with error in this calculation biasing the error estimate. REFERENCES [1]

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Brennan, L. E., and Reed, I. S. Theory of adaptive radar. IEEE Transactions on Aerospace and Electronic Systems, 9, 2 (Mar. 1973), 237—252. Guerci, J. R. Space-Time Adaptive Processing for Radar. Norwood, MA: Artech House, 2003. Klemm, R. Space-Time Adaptive Processing: Principles and Applications, IEE Radar, Sonar, Navigation and Avionics 9, IEE Press, 1998.

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Ward, J. Space-time adaptive processing for airborne radar. Technical Report ESC-TR-94-109, Lincoln Laboratory, Dec. 1994. Melvin, W. L. A STAP overview. IEEE AES Systems Magazine–Special Tutorials Issue, 19, 1 (Jan. 2004), 19—35. Reed, I. S., Mallett, J. D., and Brennan, L. E. Rapid convergence rate in adaptive arrays. IEEE Transactions on Aerospace and Electronic Systems, 10, 6 (Nov. 1974), 853—863. Melvin, W. L. Space-time adaptive radar performance in heterogeneous clutter. IEEE Transactions on Aerospace and Electronic Systems, 36, 2 (Apr. 2000), 621—633. Melvin, W. L., and Guerci, J. R. Adaptive detection in dense target environments. In Proceedings 2001 IEEE Radar Conference, Atlanta, GA, May 1—3, 2001, 187—192. Melvin, W. L. STAP in heterogeneous clutter environments. In R. Klemm (Ed.), The Applications of Space-Time Processing, IEE Radar, Sonar, Navigation and Avionics 9, IEE Press, 2004. Guerci, J. R. Knowledge-aided sensor signal processing and expert reasoning. In Proceedings 2002 Knowledge-Aided Sensor Signal Processing and Expert Reasoning (KASSPER) Workshop, Washington, D.C., Apr. 3, 2002, CD ROM. Melvin, W. L., and Showman, G. A. A KA STAP architecture. In Proceedings 2003 DARPA/AFRL KASSPER Workshop, Las Vegas, NV, Apr. 14—16, 2003. Melvin, W. L., Showman, G. A., and Guerci, J. R. A KA GMTI detection architecture. In Proceedings 2004 IEEE Radar Conference, Philadelphia, PA, Apr. 26—29, 2004, ISBN 0-7803-8235-8. Melvin, W. L., Showman, G. A., and Guerci, J. R. Knowledge-aided clutter mitigation. In Record of the 50th Annual Tri-Service Radar Symposium, Sandia National Laboratories, Albuquerque, NM, June 2004. Bergin, J. S., Teixeria, C. M., Techau, P. M., and Guerci, J. R. Space-time beamforming with KA constraints. In Proceedings 2003 Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, Mar. 11—13, 2003. Bergin, J. S., Teixeira, C. M., Techau, P. M., and Guerci, J. R. STAP with KA data prewhitening. In Proceedings 2004 IEEE Radar Conference, Philadelphia, PA, Apr. 26—29, 2004, ISBN 0-7803-8235-8. Page, D., Scarborough, S., Owirka, G., and Crooks, S. Improving KA STAP performance using past CPI data. In Proceedings 2004 IEEE Radar Conference, Philadelphia, PA, Apr. 26—29, 2004, ISBN 0-7803-8235-8.

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Hiemstra, J. D. Colored diagonal loading. In Proceedings 2002 IEEE Radar Conference, Long Beach, CA, Apr. 22—25, 2002, ISBN 0-7803-7358-8. Farina, A., Lombardo, P., and Pirri, M. Nonlinear nonadaptive space-time processing for airborne early warning radar. IEE Proceedings–Radar, Sonar Navigation, 145, 1 (Feb. 1998), 9—18. Farina, A., Lombardo, P., and Pirri, M. Nonlinear STAP processing. IEE Electrical and Communcations Engineering Journal, Feb. 1999, 41—48. Gerlach, K., and Picciolo, M. L. Airborne/spacebased radar STAP using a structured covariance matrix. IEEE Transactions on Aerospace and Electronic Systems, 39, 1 (Jan. 2003), 269—281. Goodman, N. A., and Gurram, P. R. STAP training through KA predictive modeling. In Proceedings 2004 IEEE Radar Conference, Philadelphia, PA, Apr. 26—29, 2004, ISBN 0-7803-8235-8. Fenner, D. K., and Hoover, W. F. Test results of a space-time adaptive processing system for airborne early warning radar. In Proceedings 1996 IEEE National Radar Conference, Ann Arbor, MI, May 13—16, 1996, 88—93. Guerci, J. R. Theory and application of covariance matrix tapers for robust adaptive beamforming. IEEE Transactions on Signal Processing, 47, 4 (Apr. 1999), 977—985. Billingsley, J. B. Low-Angle Radar Land Clutter: Measurements and Empirical Models. Norwich, NY: William Andrew Publishing, 2002. Robey, F. C., Fuhrman, D. R., Kelly, E. J., and Nitzberg, R. A CFAR adaptive matched filter detector. IEEE Transactions on Aerospace and Electronic Systems, 28, 1 (Jan. 1992), 208—216. Chen, P., Melvin, W. L., and Wicks, M. C. Screening among multivariate normal data. Journal of Multivariate Analysis, 69 (Mar. 1999), 10—29. Melvin, W. L., Wicks, M. C., and Brown, R. D. Assessment of multichannel airborne radar measurements for analysis and design of space-time processing architectures and algorithms. In Proceedings 1996 IEEE National Radar Conference, Ann Arbor, MI, May 13—16, 1996, 130—135. Skolnik, M. I. Introduction to Radar Systems (2nd ed.). NY: McGraw Hill, 1980. DiPietro, R. C. Extended factored space-time processing for airborne radar. In Proceedings 26th Asilomar Conference, Pacific Grove, CA, Oct. 1992, 425—430.

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William L. Melvin (S’90–M’94–SM’99) received the Ph.D. in electrical engineering from Lehigh University, Bethlehem, PA, in 1994, as well as the M.S.E.E. and B.S.E.E. degrees from this same institution in 1992 and 1989, respectively. He is a principal research engineer at the Georgia Tech Research Institute (GTRI), Director of the Adaptive Sensor Technology Project Office within GTRI’s Sensors and Electromagnetic Applications Laboratory, and an adjunct professor in Georgia Tech’s Electrical and Computer Engineering Department. Dr. Melvin has published widely in his technical areas of interest, which include digital signal and array processing, modeling and simulation, and aerospace radar systems engineering. He served as a guest editor for two recent special editions appearing in the IEEE Transactions on Aerospace and Electronic Systems, one on space-time adaptive processing (2000), and a second on KA signal processing (2006). He also acted as the technical co-chair of the 2001 IEEE Radar Conference and 2004 IEEE Southeastern Symposium on System Theory, and frequently serves as a reviewer for IEEE publications and as a session chair at IEEE conferences. He received a “Best Paper” award at the 1997 IEEE Radar Conference and was recently selected as the 2006 IEEE AESS Fred Nathanson Memorial Radar Award Young Engineer of the Year. Additionally, in 2002 he represented the U.S. as a speaker on the NATO-sponsored “Military Applications of Space-Time Adaptive Processing” lecture series and has provided tutorials and invited talks at a number of IEEE conferences and local IEEE section meetings.

Gregory A. Showman (S’94–M’00) was born in Sanford, FL, on November 14, 1962. He received the B.S. degree in applied physics from the University of California at Davis in 1985, and the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1994 and 2000. From 1985 to 1992 he worked as a physicist and anti-warfare analyst at the Naval Weapons Center, China Lake. He began his employment at the Georgia Tech Research Institute (GTRI) as a graduate research assistant in 1992 and accepted a faculty position in 2000. He is currently a Senior Research Engineer and Associate Director of the Adaptive Sensor Technology Project Office within GTRI’s Sensors and Electromagnetic Applications Laboratory, where he develops and analyzes signal processing algorithms for airborne and space-based radar, with emphasis on multi-dimensional adaptive filtering, synthetic aperture imaging, and electronic protection applications. 1042

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