• Section 2.12.2 describes basic concepts, include target detection; different space-time optimal filter formulations, including the maximum SINR filter, the minimum variance beamformer, and the generalized sidelobe canceler; the sample matrix inversion (SMI) approach to adaptive filtering; and, key performance metrics, such as SINR loss.

• Section 2.12.3 discusses clutter, RFI, target, and receiver space-time signal models.

• Section 2.12.4 discusses different adaptive filter topologies, including several reduced-rank methods, reduced-dimension STAP, and parametric adaptive matched filtering.

• Section 2.12.5 describes the utility of STAP to clutter suppression in airborne radar, as an example. This section also includes benchmark results for some of the algorithms discussed in Section 2.12.4.

• Much of current STAP research focuses on extending textbook discussion to real-world application. Section 2.12.6 describes various STAP challenges and research areas, including the impact of heterogeneous or nonstationary clutter on detection performance and mitigating techniques, application to airborne bistatic and conformal array radar, and knowledge-aided STAP. Each of these challenge areas predominantly focuses on issues surrounding effective estimation of the interference covariance matrix.

• A description of some additional, practical issues in STAP implementation is given in Section 2.12.7, including maximum likelihood angle estimation and adaptive matched filter normalization.

• In Section 2.12.8 we briefly discuss some multichannel radar data collection efforts.

• Summary comments are given in Section 2.12.9, along with notation and acronym lists in the appendices.

• STAP is really a catch-all phrase for a variety of weight calculation schemes and adaptive architectures.

• Radar terminology usually includes two time measurements: fast-time, corresponding to a sample rate consistent with the analog-to-digital converter clock speed and effectively measuring range; and, slow-time, where the pulse repetition interval (PRI) defines the sample rate and separating targets and clutter in Doppler frequency is the primary motivation. Space and slow-time are used to mitigate ground clutter returns in aerospace radar, whereas space and fast-time are used to cope with wideband RFI or multipath signals in aerospace or surface radar systems.

• Herein, we focus on space and slow-time adaptive processing for clutter suppression. The ideas discussed are easily extended to other radar degrees of freedom (DoFs), include fast-time, polarization, and multiple passes.

• Implementation of the optimal processor requires precise knowledge of the null-hypothesis covariance matrix and the target space-time steering vector. The adaptive filter is an approximation to the optimal processor, since neither covariance matrix nor steering vector are known in practice. There are two primary factors leading to differences between adaptive and optimal filter outputs: covariance matrix estimation error and unknown array manifold due to amplitude and phase error as a function of frequency. Adaptive SINR loss describes the impact of both of these errors on detection performance potential.

• Current STAP research focuses on coupling basic STAP theory, physics-based models, and the realities of the real-world operating environment to construct practical detection architectures.

• STAP is applied at the coherent processing interval (CPI) level, often typically prior to noncoherent—or post-detection—integration. Maximizing SINR is always effective at improving detection performance, since the separation between the null and alternative hypothesis output probability density functions increases.

• In many respects, STAP development is in its early stages, at least from an implementation perspective. It has been only recently that embedded computing power has improved to the point where more flexible, real-time STAP implementation is feasible.

2.12.2.2.1 Maximum SINR filter

A primary objective in STAP is to approximate as closely as possible the optimal weighting that maximizes SINR. By attempting to maximize output SINR, the STAP maximizes the probability of detection for a fixed false alarm rate, as discussed in the prior section. We now formulate the optimal weight vector.

The target snapshot is of the form, , where is a complex gain term proportional to the square root of the target cross section, is the space-time steering vector of dimension NM-by-one, is azimuth, is elevation, and is Doppler frequency. In the nonfluctuating target case, is a constant, whereas in the Swerling 1 case, , where is the target signal variance [1,7,8]. The space-time steering vector is the Kronecker product of the spatial steering vector, , and the temporal steering vector, . Generally, the spatial steering vector describes the phase variation among the spatial antenna channels (of common size and gain) to a signal with a direction of arrival of , whereas the temporal steering vector describes the phase variation over the slow-time aperture to a signal with a particular range-rate [9–11]. If the aperture in either space or time is uniformly sampled, the corresponding steering vector will be Vandermonde.

The output SINR of the FIR filter with weight vector, , is the ratio of target power, , to interference-plus-noise power, , given as

(12.4)

Using the definition for gives, , and the Schwarz Inequality lets us write (12.4) as

(12.5)

In this case, and . By inspection, we find that the right side of the inequality achieves the upper bound when , thereby providing the solution for the optimal weight vector. Through substitution, it follows that

(12.6)

for arbitrary scalar, , is the weighting that yields maximal output SINR. When the disturbance is uncorrelated noise, , where is the noise power; then, (12.6) becomes the space-time matched filter, viz. .

Inserting the optimal weight vector, , into (12.4) yields the maximum output SINR,

(12.7)

For simplicity, and without loss in meaning, in subsequent sections we use .

The maximum SINR weight vector has the following interpretation. If , then is the whitened data vector, where and is the NM-by-NM identity matrix. This results because

(12.8)

Then, the filter employing the maximum SINR weight vector is seen as a cascade of whitening and warped matched filtering operations,

(12.9)

where is the warped matched filter accounting for the linear transformation of the target signal through the whitening process and is the original matched filter weight vector.

2.12.2.2.2 Minimum variance beamformer

The minimum variance (MV) space-time beam former is another common formulation leading to the adaptive processor [12,13]. The MV beamformer employs a weight vector yielding minimal output power subject to a linear constraint on the desired target spatial and temporal response:

(12.10)

where g is a complex scalar. We can determine the weight vector satisfying the problem statement in (12.10) using the method of Lagrange. Thus, define the following cost function

(12.11)

where is a complex Lagrangian multiplier and . Taking the gradient of (12.11) with respect to the conjugate of the weights and setting to zero gives

(12.12)

Solving (12.12) for the weight vector yields

(12.13)

Next, we find by applying the constraint in (12.10) that

(12.14)

Solving (12.14) for then gives

(12.15)

The minimum variance weight vector follows from (12.13) and (12.15) as

(12.16)

Equation (12.16) abides by the form , where

(12.17)

Hence, (12.16) likewise yields maximal SINR.

Setting is known as the distortionless response, and the corresponding weight vector yields the minimum variance distortionless response (MVDR) beamformer. The output power of the MVDR beamformer is given as

(12.18)

The MVDR spectrum, as given in Figure 12.1, follows from (12.18) by scanning the space-time steering vector in the denominator over the angle and Doppler locations of interest.

2.12.2.2.3 Generalized sidelobe canceler

The GSLC is a formulation that conveniently converts the minimum variance constrained beamformer described in the preceding section into an unconstrained form [12,14]. Many prefer the GSLC interpretation of STAP over the whitening filter-warped matched filter interpretation of Section 2.12.2.2.1; since the GSLC implements the MV beamformer for the single linear constraint [14], this structure likewise can be shown to maximize output SINR.

Figure 12.5 provides a block diagram of the GSLC. The top leg of the GSLC generates a quiescent response by forming a beam at the angle and Doppler of interest. A blocking matrix, , prevents the target signal from entering the lower leg; essentially, the blocking matrix forms a notch filter tuned to . With the desired signal absent from the lower leg, the processor tunes the weight vector to provide a minimal mean square error (MMSE) estimate of the interference in the top leg. In the final step, the processor differences the desired signal, , with the estimate from the lower leg, , to form the filter output, . Ideally, any large residual at the output corresponds to an actual target.

The desired signal is given as

(12.19)

The signal vector in the lower leg is

(12.20)

Forming the quiescent response of (12.19) uses a single degree of freedom (DoF), resulting in the reduced dimensionality of . By weighting the signal vector in the lower leg, the GSLC forms a MMSE estimate of as

(12.21)

where the MMSE weight vector follows from the well-known Wiener-Hopf equation [12,13],

(12.22)

The lower leg covariance matrix is

(12.23)

whilst the cross-correlation between lower and upper legs is

(12.24)

The GSLC filter output is then

(12.25)

Comparing (12.25) and (12.3), we identify

(12.26)

We compute the output SINR as the ratio of output signal power to interference-plus-noise power, as in (12.4). The signal-only output power of the GSLC is

(12.27)

and the output interference-plus-noise power is

(12.28)

The ratio of (12.27) and (12.28) is

(12.29)

where . We arrive at the denominator in (12.29) by using (12.22)–(12.24),

(12.30)

2.12.2.2.4 Rayleigh quotient

The optimal weight vector in Section 2.12.2.2.1 maximizes SINR. As seen from (12.4), for weight vector , SINR is given as the ratio of quadratics, . Thus, our problem is to find the weight vector that maximizes this function. In accord with [15], and similar to our discussion in Section 2.12.2.2.1, SINR is written as

(12.31)

where . The weight vector, , that maximizes this Rayleigh quotient form is known to be proportional to the eigenvector associated with the largest eigenvalue of the matrix in the numerator [13],

(12.32)

and so

(12.33)

Take the case where , as previously introduced in Section 2.12.2.2.1. The maximum eigenvector in (12.32) must abide by

(12.34)

The solution for in (12.34) is thus seen to be

(12.35)

and so, as expected,

(12.36)

(Note: inserting (12.35) in (12.34) gives times a scalar on each side of the equation, thereby satisfying the eigen-relation.)

By plugging found in (12.33) into (12.4), it is straightforward to show that the maximum SINR is equal to . In this vein, using (12.35) in (12.34) gives

(12.37)

which is the desired result and matches the expression given in (12.7).

2.12.2.2.5 Generalized eigen-analysis

The second approach, described in [13], begins by noting that the solution to maximizing SINR is equivalent to finding the maximum eigenvalue and associated eigenvector for the generalized Eigen-problem

(12.38)

This can be solved as an ordinary eigen-equation by writing it as

(12.39)

Then, the weight vector that maximizes (12.4) is proportional to the eigenvector associated with the largest eigenvalue of ,

(12.40)

The maximum eigenvalue of is the maximum achievable SINR in (12.4).

If we again consider the case where , the generalized eigen-analysis equation of (12.39) becomes

(12.41)

which, we find, is satisfied when . Substituting this eigenvector, , into (12.41) gives,

(12.42)

from which it is seen the maximum eigenvalue, , is identical to the result in (12.37) and also the same as the maximum SINR expression found in Section 2.12.2.2.1.

2.12.2.2.6 Summary

Table 12.2 summarizes the weight vector formulation given in the prior sections.

2.12.2.4.1 Clairvoyant SINR loss

Clairvoyant SINR loss is the ratio of output SINR for a filter implementation, , where all parameters are known precisely, to the maximum SNR. Clairvoyant SINR is given as

(12.51)

A few cases are worth considering:

Case 1—Noise-Limited Condition: In this case, , where is the noise power. Then, the maximum SINR weight vector,

(12.52)

and so

(12.53)

As expected, the optimal weighting defaults to the matched filter and there is no loss relative to the bound defined by the uncorrelated noise.

Case 2—Color-Limited Condition, Matched Filter Weights: Clairvoyant loss characterizes the impact of interference on detection performance. Consider the case where the weight vector is set to the space-time matched filter, . The clairvoyant SINR loss follows from (12.51) as

(12.54)

where is the power spectral density (PSD), equal to the two-dimensional Fourier transform of the space-time covariance matrix [12]; Figure 12.1 provides example clutter-plus-noise PSD plots. Equation (12.54) shows the impact of interference on performance relative to the noise-limited case. Since the PSD is diffraction-limited (the mainlobe is determined by the size of the space-time aperture), (12.54) characterizes the performance impacts of using the deterministic matched filter. At those angles and Doppler frequencies away from the clutter angle-Doppler region of support, the clairvoyant SINR loss approaches the noise-limited case.

Case 3—Color-Limited Condition, Optimal Weights: Using the maximum SINR weight vector, , in (12.51), gives

(12.55)

where is a sample of the MVDR spectrum given in (12.18). Considering the MVDR plots in Figure 12.1, (12.55) suggests regions of loss confined to the sharp, super-resolution contours of the clutter MVDR response (note the inverse in (12.55)). For this reason, STAP is able to detect targets in close proximity to the center of the clutter angle-Doppler region of support. In contrast, (12.54) suggests that SINR loss extends to the full width of the diffraction-limited spectrum when using nonadaptive weights. Figure 12.25, given in Section 2.12.5.3, compares SINR loss for adaptive and nonadaptive solutions, confirming these observations.

2.12.2.4.2 Adaptive SINR loss

Adaptive SINR loss is the ratio of the output SINR for the filter using estimated quantities, , to the optimal filter output, , where all parameters are clairvoyantly known, viz.

(12.56)

Observe that when . Moreover, , since yields the maximum output SINR and so the denominator in (12.56) will always be greater than or equal to the numerator.

Substituting (12.43) and (12.44) into (12.56) yields

(12.57)

where is given by (12.7). Assuming uses (12.49) in its calculation, as (12.57) suggests, then determining is critical. This important matter was addressed in [17], where it is shown that (12.57) is Beta distributed, with mean value

(12.58)

Setting (12.58) equal to (or, 3 dB of loss) and solving for yields

(12.59)

It is popular to refer to the result in (12.59), where setting roughly equal to twice the processor’s DoFs yields 3 dB of loss, as the Reed-Mallett-Brennan (RMB) rule after its originators. In practice, 3 dB of loss is substantial, and so choosing to be at least five times the processor’s DoF is desirable.

The IID assumption inherent in the calculation of (12.57) sets the bound on performance for the adaptive processor given homogeneous training samples. We address the impact of non-IID clutter conditions in further detail in Section 2.12.5.

2.12.2.4.3 Improvement factor

Improvement factor (IF) is given as the ratio of the output SINR to the input (element-level) SINR [18,19]. IF is given as

(12.60)

where is the input (element-level) interference power.

Case 1—Noise-Limited Condition: In the noise-limited case, and , and considering the matched filter, , then

(12.61)

As expected, the improvement factor equals the space-time integration gain in this instance.

Case 2—Clutter-Limited Condition, Matched Filter Weights: In the clutter-limited case, with , the improvement factor is

(12.62)

The presence of interference in the PSD degrades the improvement factor. If we let and in (12.62), we arrive at (12.61).

Case 3—Clutter-Limited Condition, Optimal Weights: Using the maximum SINR weight vector, , in (12.60), gives

(12.63)

This expression indicates reduced improvement for those angle-Doppler locations aligning with the clutter, and otherwise good performance in proximity to the clutter response owing to the super-resolution characteristic of (see Figure 12.1).

2.12.2.4.4 Optimal and adaptive filter patterns

The filter gain pattern follows directly by evaluating the filter response over the angles (or spatial frequencies) and Doppler frequencies of interest. Define the space-time steering matrix,

(12.64)

for all , and . The optimal gain pattern is

(12.65)

where is the optimal weighting given by (12.43). The adaptive gain pattern follows similarly as

(12.66)

with (12.44) yielding .

Example gain patterns corresponding to the clutter environments shown in Figure 12.1 are given in Figure 12.2. This gain pattern corresponds to (12.65).

2.12.3.1.1 Distributed clutter

Distributed clutter returns result from the integrated response of all scatterers within the range resolution cell. A model for distributed clutter involves discretizing the range resolution cell into fine angle bins called clutter patches. The location of the angle bin relative to the platform velocity vector determines the clutter patch Doppler frequency. The clutter snapshot then follows as the sum of the returns from each of the individual patches.

Figure 12.6 depicts the discretized clutter patch model for distributed clutter. The antenna gain varies around the range resolution annulus on the Earth’s surface in accordance with the array normal and steering direction. It is common to model the clutter complex voltage as a Gaussian random variable due to the constructive and destructive sum of the returns from the sub-scatterers comprising each resolution cell. The patch clutter-to-noise ratio (CNR) at the channel-level (assuming matched channels) is

(12.67)

where is the transmit power, is the transmit gain in the pth direction of interest, r is the slant range, is the pth clutter patch radar cross section, is the receive channel gain, is the center wavelength, is Boltzman’s constant, is standard operating temperature, is the receiver bandwidth, is noise figure, is RF system loss, and is signal processing gain (the pulse compression ratio, in this case) [1,7,8]. The clutter RCS is a function of the patch reflectivity, , and area, , where the area is determined to be a fraction of the resolution cell,

(12.68)

The constant gamma model is a common choice used to model reflectivity and is given as , where is a normalized reflectivity term dependent on the terrain type and is the grazing angle to the clutter patch [20]. The clutter patch power then follows by multiplying (12.67) by the receiver noise, . The clutter voltage for the kth range bin and pth patch voltage is then

(12.69)

where [9–11,18,19]. (If the response is non-fluctuating, as in the case of a clutter discrete subsequently discussed, then is a complex scalar with unity magnitude and uniformly distributed phase.)

The patch voltage in (12.69) varies over the space-time aperture: there is a phase change from channel-to-channel due to direction of arrival, described by the spatial steering vector; and, a phase change from pulse-to-pulse due to the change in range-rate between the clutter patch and the radar platform, described by the temporal steering vector. The clutter space-time snapshot then follows by summing the voltages over the P patches for each of Q range ambiguities, where is the first (unambiguous) range return

(12.70)

It is common to assume each patch is statistically independent, in which case the clutter covariance matrix corresponding to (12.70) is

(12.71)

where is the clutter patch power and follows from (12.69). Equations (12.70) and (12.71) form the basic ground clutter models.

The clutter voltage decorrelates, mainly in time due to intrinsic clutter motion (ICM). Windswept vegetation and moving water are two cases where ICM results. The two basic models describing clutter temporal decorrelation include the Billingsley model [21] and the Gaussian model [7,10]. The Billingsley model is most common for overland surveillance; it allows a certain amount of the clutter power to decorrelate, say due to leaves fluctuating in the breeze, whilst some of the clutter power is persistent, thus modeling the tree trunk, for example. The Billingsley model leads to an exponential autocorrelation function. In contrast, the Gaussian power spectrum of [7] leads to a Gaussian autocorrelation; the Gaussian model leads to complete decorrelation over a specified time interval and is most appropriate in marine or riverine environments. Dispersion (group delay) across the array is the common source of spatial decorrelation; a sinc autocorrelation model is commonly chosen to model this effect, since it is the inverse transform of a rectangular function in the frequency domain [22,23].

Given the aforementioned discussion, define the length-NM space-time correlation taper as , with correlation matrix . is the temporal correlation matrix and is the spatial correlation matrix. The resulting clutter covariance matrix is

(12.72)

Generating space-time clutter snapshots exhibiting the correlation described by typically involves shaping white noise by multiplying the matrix square root of and by random vectors of length N or M, where the elements of each vector are uncorrelated, zero mean, unity variance Gaussian variates.

2.12.3.1.2 Discrete clutter

Clutter discretes are strong returns that occur relatively infrequently and correspond to stationary, point-like objects, such as parked vehicles, utility poles, water towers, etc. Generally, the RCS of the clutter discrete is large relative to the RCS of a typical, resolved clutter patch. In some cases, the discrete clutter can appear extended, such as in the case of a fence line or train track.

Figure 12.7 shows an example of discrete-like returns in a spotlight synthetic aperture radar (SAR) image of the Mojave Desert Airport. The hangar edges and fence lines, as well as some aircraft on the tarmac, are evident in the figure. Figure 12.8 shows an exceedance plot characterizing the typical impact of clutter discretes on detection performance. (EFA is a post-Doppler STAP method described in Section 2.12.4 and AMF refers to a STAP normalization discussed in Section 2.12.7.) In a homogeneous environment, the output of the STAP is well-behaved and the cumulative distribution follows the expected exponentially-shaped exceedance; in contrast, with discretes present, the tails of the exceedance plot are extended, indicating a requirement to raise the detection threshold to maintain a constant false alarm rate.

A plausible discrete model involves laying out the spatial locations of discrete clutter of varying densities. One way to do this is to specify the density per km2 for a particular discrete RCS and employ a Poisson distribution to identify the random clutter discrete locations. An example of a range-angle seeding of discrete clutter of varying RCS from 20 dBsm to 60 dBsm is show in Figure 12.9.

Since discrete clutter is stationary, the angular position uniquely specifies Doppler frequency. In other words, clutter discretes reside along the angle-Doppler region of support corresponding to stationary objects. As in the case of distributed clutter, the discrete-to-noise ratio (DNR) follows in a form similar to (12.67) for the lth discrete at range, , and angle , as

(12.73)

As the discrete is considered point-like, the RCS is taken as nonfluctuating, whereas the phase is uniformly distributed. The corresponding discrete space-time snapshot is

(12.74)

where u is a uniformly-distributed random variable between and is the range realization for the lth clutter discrete. The discrete clutter is additive. The corresponding covariance matrix for each discrete is

(12.75)

• Clutter and jamming tend to be of low numerical rank and the processor explicitly removes only those signal subspaces corresponding to interference.

• The eigendecomposition maximally compresses the interference into the fewest basis vectors [12].

• The RMB rule applies to the RR-STAP case, with rank replacing DoFs in the formulation, viz. training over twice the rank yields roughly 3 dB loss on average [27].

2.12.4.1.1 Adaptive beamformer pattern synthesis

It is shown in [28] that (12.43) can be written

(12.94)

where is the noise-floor eigenvalue level and is the projection of the mth eigenvector onto the quiescent pattern. As seen from (12.94), the STAP response appears as a notching of the space-time beampattern given by by the weighted, interference eigenvectors. When , no subtraction occurs, since the corresponding eigenvector lies in the noise subspace.

Naturally, (12.94) applies to the SMI formulation of (12.44), and can provide robustness in the presence of training sample support limitations if the interference rank is known (which is usually only practical for strong interferers). Specifically, running the sum in (12.94) only over subspaces leads to adaptive pattern robustness, since low sample support tends to predominantly perturb the noise subspace; the perturbed noise subspace estimate leads to poor adaptive sidelobe performance when the sum in (12.94) is run over all values of the index, m.

2.12.4.1.2 Principle components inverse

The idea behind principle components inverse (PCI) is to apply an orthogonal projection to , and then apply a matched filter to the remaining transformed data [29]. The PCI formulation starts by writing (12.94) applied to the data as

(12.95)

Then, for the eigenvalues where ,

(12.96)

We see that

(12.97)

where, for the case of as an example, (12.97) shows the coherent removal of and . The term in brackets in (12.97) is called an orthogonal projection for this reason.

2.12.4.1.3 Eigencanceler

Haimovich [30] gives two different developments; one of these essentially leads to (12.96) via an alternate route.

The minimum power eigencanceler (MPE) weighting results from

(12.98)

where is a constraint matrix and is a desired response vector. The solution to (12.98) is shown in [30] to be

(12.99)

For the case where and , (12.99) takes the form

(12.100)

with a scalar. The weight vector in (12.100) lies entirely in the noise subspace—as required by (12.98)—and will, by virtue of the orthonormal property of the set of eigenvectors, annihilate the clutter subspace.

The minimum norm eigencanceler (MNE) is formulated as

(12.101)

The MNE weight vector is then shown in [30] to be

(12.102)

Again, for the case where and , and using (12.87), we see (12.102) can be written in the same form as PCI,

(12.103)

with a scaling that replaces in (12.96). Using (12.87), the expression in (12.103) can be written

(12.104)

which closely relates to the MPE solution in (12.100) with the inverse eigenvalue weighting given by absent. It is known that limited training sample support leads to perturbed estimates of the noise eigenvalues appearing along the diagonal of . For this reason, the MNE provides robustness relative to MPE when training data samples are lacking.

2.12.4.1.4 Hung turner projection

The Hung Turner Projection (HTP) is applicable to RFI suppression [31]. While clutter mitigation is paramount to our discussion, it is worth taking a moment to describe the HTP as a general tool in adaptive filter implementation. Reducing computational burden associated with the eigendecomposition leading to the eigenvalues and eigenvectors of the null-hypothesis covariance matrix is a primary goal.

The basic idea behind the HTP is to use the Gram-Schmidt method to characterize the interference subspace. Specifically, suppose we identify J snapshots where J RFI sources are present,

(12.105)

Next, the modified Gram-Schmidt technique is applied to (12.105) to generate the orthonormal basis, , where . Then, in accord with [31], the HTP weight vector is chosen as , with

(12.106)

and where is the quiescent weight vector.

2.12.4.1.5 Diagonal loading

Diagonal loading involves adding a scaled diagonal matrix to the covariance matrix estimate of (12.49) [32],

(12.107)

where is the diagonal loading level and and follow from the decomposition of the covariance matrix estimate. It is common to set , with most typical. Primary benefits of diagonal loading include improved conditioning of the covariance matrix estimate (which improves the numerical stability of inversion routines) and compression of the range corresponding to the smallest eigenvalues. This latter benefit leads to a reduction in spurious sidelobes of the adapted pattern at the expense of null depth. Effectively, diagonal loading removes some of the adaptivity of the system to better condition the filter’s sidelobe response.

While often considered ad hoc, it turns out diagonal loading is a component of the optimal solution to the constrained optimization,

(12.108)

The Lagrangian for (12.108) is readily found to be

(12.109)

where and are Lagrangian multipliers. Taking the gradient of (12.109) with respect to the conjugate of the weight vector and setting the result to zero yields [13]

(12.110)

Solving for the weight vector gives

(12.111)

Using the linear constraint, it is seen that

(12.112)

The resulting weight vector is

(12.113)

From (12.113) we see that diagonal loading plays a key role in the solution to the constrained optimization. In this case, is the diagonal loading level, , mentioned previously. If the loading level is set to zero, (12.113) defaults to the distortionless MV beamformer solution.

2.12.4.1.6 Cross spectral metric

Consider the GSLC structure of Figure 12.5. A principal components approximation to is

(12.114)

with, ideally, , and where and represent the mth eigenvalue and eigenvector of the GSLC lower leg covariance matrix. Then,

(12.115)

The computational burden of the GSLC does not generally justify its use in practice. Improved statistical convergence is a potential advantage of the principal components decomposition of the GSLC (PC-GSLC).

The cross spectral metric operates in a fashion similar to the PC-GSLC method, except now the selection of the basis defining the rank-reducing transformation is target signal-dependent [33]. The error signal at the output of the GSLC follows as the difference between (12.19) and (12.21), viz.

(12.116)

The corresponding MMSE is given as the expected value of the magnitude squared of (12.116),

(12.117)

In this case, , where no target signal energy is present in the upper leg (an assumption that carries through in the calculation of the cross correlation vector, ). When choosing only P terms in the summation of (12.117), where , the processor attains the lowest possible mean square error (MSE) by choosing subspaces with the largest cross-spectral metric terms, given as

(12.118)

The target signal influences the selection via . Thus, the idea behind the CSM approach is to select only those subspaces from that most greatly enhance performance.

A direct-form implementation of the CSM method is given in [34]. In this case, the objective involves maximizing SINR by choosing the appropriate, signal-dependent, reduced-rank subspace (hence, the analogous metric is called the SINR metric in [34]). Thus, decomposing output SINR in (12.7) using the inverse of the eigendecomposition in (12.86) gives

(12.119)

Maximizing SINR requires choosing the largest terms in (12.119) of the form,

(12.120)

where DFP signifies “direct form processor” (i.e., the approach using a weight vector of the form given in (12.43) or (12.44)). Thus, with limited sample support and uncertainty over the interference rank, selecting the largest CSM terms of (12.120) leads to maximal output SINR. Commonly, then, the CSM-DFP chooses the components in the noise subspace, but this is an artifact of the development and not useful: terms with the smallest eigenvalues are generally chosen, with any signal-dependent correlation meant to aid in selecting the most significant terms lost in the formulation. To avoid this scenario, [34] suggests choosing those terms maximally impacting the weight vector. In other words, considering (12.94), selecting the P largest terms of

(12.121)

most greatly impacts the adaptive process. Equation (12.121) is signal-dependent and forces the selection to lie in the dominant subspace.

2.12.4.1.7 Multi-Stage Wiener Filter

The Multi-Stage Wiener Filter (MWF) is a truncated decomposition of the GSLC of Figure 12.5 [35]. Its operation is best understood from the diagram in Figure 12.11, which shows a two-stage decomposition. From this figure we see that the estimation stage is broken into a series of smaller problems, where the weight vector is calculated as a scalar using the Wiener-Hopf equation [12,13] in each stage. The filter, , is a normalized cross-correlation vector between its input, , and the output of the filter in the preceding leg, ; the intent of choosing in this manner is to maximize the correlation between the interference signal in each leg. Specifically,

(12.122)

where . The blocking matrix, , lies in the null space of (expect for , where . The weights, , are scalars equal to the Wiener weight minimizing the mean square error between and . In this figure, for the MWF output to equal the GSLC output, the processor must calculate all quantities, including the vector weight, . The MWF truncates the number of stages by simply dropping the lower leg of the last stage, setting the vector weight to all zeros (i.e., ) so that , where is the number of MWF stages. So, for .

In the adaptive version of the MWF, the processor replaces the known covariance matrices and steering vectors with estimates. The quantities, , and, naturally, , are all expressible as linear transformations applied to and ; in the adaptive filter, and . The implementation requires first calculating and applying ; next, the processor implements the first stage, calculating , and ; with these first stage quantities in hand, the processor calculates the second stage quantities, , and , then third stage quantities, and so forth, working out to the last stage; then, as noted above, the error signal in the last stage is set to , where is the last stage; and, finally, the scalar weights, , are solved from the outer (last) stage into the first stage.

While the computational loading of the MWF is generally high, the target signal-dependent nature of the stage decomposition identifies the interference subspace most greatly impacting the MMSE through the cross-correlation process, estimates it, and subtracts it out. This allows the MWF to converge towards the optimal solution with minimal use of available training data.

• The clutter exhibits a highly structured angle-Doppler region of support, as shown in Figure 12.13. In this figure, T1 and T2 are target locations and C1–C4 are clutter regions. Target T1 predominantly competes with clutter located at C3, and a spatial null within the corresponding Doppler filter suffices to mitigate clutter. Clutter at other locations—such as at C1, C2, and C4—has no significant bearing on target detection. Alternately, target T2 is impacted by clutter in the region of C1 and C4; a two-dimensional null, positioned in angle and Doppler, is needed to suppress mainlobe and near-in sidelobe clutter.

• The RMB rule of Section 2.12.2.4.2 applies to RD-STAP, with . It is not uncommon to have , whereas is reasonable. Thus, limited and potentially heterogeneous or nonstationary training sample support is reduced by a factor of 10–100 or more.

• The implicit inverse of the sample covariance matrix is , being reduced to in the RD-STAP case. For each halving of the processor’s DoFs, computational burden decreases by a factor of eight. Real-time operation requires RD-STAP methods.

2.12.4.2.1 Post-Doppler STAP

Of the various RD-STAP methods, post-Doppler STAP techniques are most popular. Post-Doppler STAP is shown in the lower left and right of Figure 12.12, where the lower left box corresponds to the so-called “channel space,” post-Doppler methods that operate on space-Doppler data, and the lower right box characterizes the post-Doppler, (spatial) beamspace methods. The Extended Factored Algorithm (EFA) is an example of a “channel space,” post-Doppler method [36]. In EFA, each channel is Doppler filtered, and then the adaptive weighting of (12.128) is applied to the transformed data vector,

(12.129)

where is the spatial snapshot corresponding to the qth Doppler bin output (i.e., the qth Doppler bin output from channel 1 through M stacked in a vector). Equation (12.129) shows, as an example, the case of five adjacent Doppler bins. The EFA output for the kth range bin and nth Doppler bin is then given by , where corresponds to the nth Doppler bin output. Figure 12.14 depicts the EFA processing architecture, where is the EFA weight applied to the mth channel and qth Doppler bin. This figure shows the case of processing three adjacent Doppler bins; EFA typically employs three or five adjacent bins. The special case of one Doppler bin—where the processor only generates a spatial null—is known as Factored Time-Space (FTS) [10].

Figure 12.15 describes EFA operation in the angle-Doppler domain. Each horizontal, rectangular box corresponds to a Doppler filter extent. The circles correspond to null locations EFA is able to generate. EFA can generate up to nulls, where is the number of temporal DoFs (the number of Doppler bins used in the adaptive combiner). The target location, T1, is shown close to mainlobe clutter. Nulling along the section of the clutter ridge in proximity to T1 is required for target detection. EFA is able to effectively suppress the clutter local to the target position. Its performance benchmarks very close to that of the space-time optimal solution, as will be shown in Section 2.12.5. FTS can generate up to spatial nulls within the Doppler bin; as seen in this case, given the closeness of T1 to mainlobe clutter, only nulling sidelobe clutter is insufficient.

References [37,38] describe post-Doppler, (spatial) beamspace methods. The Joint Domain Localized (JDL) algorithm generalizes EFA by applying a beamspace transformation prior to Doppler filtering in Figure 12.14. JDL then combines spatial beams and Doppler bins to suppress clutter, in a manner similar to a two-dimensional sidelobe canceler. For example, a common configuration is and ; in this three beam configuration, the processor uses the eight angle-Doppler beams surrounding the center beam to estimate and coherently remove the clutter signal in this ninth angle-Doppler direction of interest, thereby exploiting the structure of the clutter local to the target position. JDL performance tends to benchmark well relative to the space-time optimal solution. A special case of JDL is given in [38], where the Doppler filtering is applied to sum and difference beams, and then “auxiliary” beams about the sum beam and Doppler bin of interest are adapted to coherently remove the clutter signal. This latter method has been called Sigma-Delta STAP. Reference [39] also includes germane discussion.

Additional post-Doppler STAP methods based on PRI designs are given in [10].

2.12.4.2.2 Pre-Doppler STAP

Pre-Doppler STAP—sometimes called adaptive displaced phase center antenna (ADPCA)—involves processing overlapped sub-CPIs to mitigate clutter, then applying Doppler processing to the aggregated output [10,40]. It has particular application in those situations where the clutter response decorrelates during the course of the CPI, such as when the antenna array rotates. Figure 12.16 shows the pre-Doppler STAP, or ADPCA, architecture for the three pulse case.

The corresponding ADPCA data snapshot for three pulses is simply

(12.130)

The ADPCA weight vector is

(12.131)

with the peak clutter Doppler times the PRI. The -by- ADPCA covariance matrix estimate, , is an approximation to

(12.132)

where is given in (12.130). The covariance inverse in the ADPCA weight vector provides a dynamic response to whiten ground clutter returns. The steering vector term in parentheses suppresses mainlobe clutter with the binominal weights, identified as , while forming a beam in a specified angular direction; the steering vector incorporates an additional linear phase variation over the temporal pulses to steer the Doppler null in cases where clutter is not centered at 0 Hz.

The performance of ADPCA is generally not as good as post-Doppler STAP techniques, thus limiting its use in practice. Performance benchmark results are given in Section 2.12.5.

• FTS—uses all eleven channels and a Hanning weighting on the Doppler filters. Eleven adaptive spatial DoFs.

• EFA—uses all eleven spatial channels and three adjacent, Hanning-weighted Doppler filters. Thirty-three adaptive space-Doppler DoFs.

• JDL—uses three adjacent, uniform weighted spatial beams and three adjacent, Hanning-weighted Doppler filters. Nine total adaptive angle-Doppler DoFs. The beam spacing is three degrees (a little less than one-third of the full aperture 3 dB beamwidth).

• ADPCA—uses three adjacent pulses and all spatial channels for cancellation, then Doppler filters the resulting outputs using a Hanning weight. Thirty-three adaptive space-time DoFs. The implementation steers the temporal gain response to center the temporal steering vector null at −100 Hz, the center of mainlobe clutter.

• PAMF—uses a fourth-order filter model. No weighting is used on the Doppler steering vector.