Space-time adaptive processing

The optimum solution is using all degrees of freedom by processing the adaptive filter on the antenna elements. For adaptive direct methods, Sample Matrix Inversion (SMI) uses the estimated (sample) interference covariance matrix in place of the actual interference covariance matrix.[8] This is because the actual interference covariance matrix is not known in practice.[1] If it is known by some means, then it need not be estimated, and the optimal weights are fixed. This is sometimes called the data-independent variation. The data-dependent variation estimates the interference covariance matrix from the data. In MIMO communications systems, this can be done via a training sequence.[5] The clairvoyant detector is given when the covariance matrix is known perfectly and defined as:

${\displaystyle \mathbf {R_{k}} ={\mbox{E}}\left[x_{k}x_{k}^{H}\right]{\Bigr |}_{H_{0}}}$

where ${\displaystyle x_{k}}$ is the space-time snapshot statistic for the ${\displaystyle k^{th}}$ range cell under the interference only hypothesis, ${\displaystyle H_{0}}$.[1] For SMI, the interference covariance matrix for the ${\displaystyle k^{th}}$ range cell consisting of the statistics from interfering noise, clutter, and jammers is estimated as follows:[4]

${\displaystyle \mathbf {\hat {R_{k}}} ={\frac {1}{P}}\sum _{m=0}^{P-1}x_{m}x_{m}^{H}}$

where ${\displaystyle x_{m}}$ is the training data obtained from the input processor for the ${\displaystyle m^{th}}$ range cell. Therefore, space-time snapshots surrounding the desired range cell are averaged. Note that the desired range cell space-time snapshot is typically excluded (as well as a number of additional cells, or "guard cells") to prevent whitening of the statistics.[1]

The main problem with direct methods is the great computational complexity associated with the estimation and inversion of matrices formed from many degrees of freedom (large number of elements and or pulses).[1] In addition, for methods where ${\displaystyle \mathbf {R} }$ must be estimated using data samples, the number of samples required to achieve a particular error is heavily dependent on the dimensionality of the interference covariance matrix.[4] As a result, for high dimensional systems, this may require an unachievable number of unambiguous range cells.[1] Further, these adjacent data cells must contain stationary statistics as a function of range which is rarely a good assumption for the large number of cells required (${\displaystyle 2NM}$ for 3 dB SINR degradation from optimal, clairvoyant STAP).[2][1]

Reduced rank methods aim to overcome the computational burdens of the direct method by reducing the dimensionality of the data or the rank of the interference covariance matrix.[2] This can be accomplished by forming beams and performing STAP on the beamspace.[7] Both pre and post Doppler methods can be used in the beamspace. Post Doppler methods may also be used on the full antenna element input as well to reduce the data in this dimension only. A popular example is displaced phase center antenna (DPCA), which is a form of data-independent STAP in the beamspace, pre-Doppler.[7] The goal is to perform beamforming such that the beam appears stationary as the airborne radar is in motion over discrete time periods so the clutter appears without Doppler.[2] However, phase errors can cause significant degradation since the algorithm is not adaptive to the returned data.[2] Many other methods may be used to reduce the rank of the interference covariance matrix, and so all methods in the reduced rank category can be thought of as simplifying the covariance matrix to be inverted:

${\displaystyle \mathbf {R} \Rightarrow \mathbf {\widetilde {R}} }$

Post-Doppler methods decompose the STAP problem from an ${\displaystyle MN\times MN}$ adaptive filtering problem to ${\displaystyle M}$ individual adaptive filters of length ${\displaystyle N}$ (an ${\displaystyle N\times N}$ adaptive filter problem).[2] By performing fixed Doppler processing, the adaptive filters become spatial only.[2] Since the target response is already steered to a specified angle-Doppler location, the dimensionality can be reduced by pre-processing multiple Doppler bins and angles surrounding this point.[4] In addition to reducing the dimensionality of the adaptive processor, this in turn reduces the number of required training data frames when estimating the interference covariance matrix since this quantity is dimension dependent.[2]

Since these methods reduce the data dimensionality, they are inherently sub-optimal.[1] There are a number of techniques to compare the performance of reduced-rank methods and estimated direct methods to clairvoyant STAP (direct with perfect knowledge of interference covariance matrix and target steering vector), mostly based around SINR loss.[1] One such example is

${\displaystyle L_{s,2}={\frac {{\mbox{SINR}}{\Bigr |}_{W={\hat {W}}}}{{\mbox{SINR}}{\Bigr |}_{W=W_{opt}}}}}$

where we've taken the ratio of the SINR evaluated with the sub-optimal weights ${\displaystyle {\hat {W}}}$ and the SINR evaluated with the optimal weights ${\displaystyle W_{opt}}$.[1] Note in general this quantity is statistical and the expectation must be taken to find the average SINR loss. The clairvoyant SINR loss may also be calculated by taking the ratio of the optimal SINR to the system SNR, indicating the loss due to interference.[1]

There are also model based methods that attempt to force or exploit the structure of the covariance interference matrix. The more generally applicable of these methods is the covariance taper matrix structure.[2] The goal is to compactly model the interference, at which point it can then be processed using principal component techniques or diagonal-loading SMI (where a small magnitude, random diagonal matrix is added to attempt to stabilize the matrix prior to inverting).[2] This modeling has an added benefit of decorrelating interference subspace leakage (ISL), and is resistant to internal clutter motion (ICM).[2] The principal component method firsts applies principal component analysis to estimate the dominant eigenvalues and eigenvectors, and then applies a covariance taper and adds an estimated noise floor:

${\displaystyle \mathbf {{\widetilde {R}}_{PC-CMT}} =\left(\sum _{m=0}^{P-1}\lambda _{m}v_{m}v_{m}^{H}\right)\circ T+\sigma _{n}^{2}}$

where ${\displaystyle \lambda _{m}}$ is the ${\displaystyle m^{th}}$ eigenvalue estimated using PCA, ${\displaystyle v_{m}}$ is the associated ${\displaystyle m^{th}}$ eigenvector estimated using PCA, ${\displaystyle A\circ B}$ implies element-by-element multiplication of matrices ${\displaystyle A}$ and ${\displaystyle B}$, ${\displaystyle T}$ is the estimated covariance matrix taper, and ${\displaystyle \sigma _{n}^{2}}$ is the estimated noise floor.[2] The estimation of the covariance taper ${\displaystyle T}$ can be complicated, depending on the complexity of the underlying model attempting to emulate the interference environment. The reader is encouraged to see [2] for more information on this particular subject. Once this taper is sufficiently modeled, it may also be applied to the more simple SMI adaptation of CMT as follows:

${\displaystyle \mathbf {{\widetilde {R}}_{SMI-CMT}} =\mathbf {{\widetilde {R}}_{SMI}} \circ T+\delta I}$

where ${\displaystyle \mathbf {{\widetilde {R}}_{SMI}} }$ is the typical SMI estimated matrix seen in the approximate direct method, ${\displaystyle \delta }$ is the diagonal loading factor, and ${\displaystyle I}$ is the identity matrix of the appropriate size. It should be seen that this is meant to improve the standard SMI method where SMI uses a smaller number of range bins in its average than the standard SMI technique. Since fewer samples are used in the training data, the matrix often requires stabilization in the form of diagonal loading.[2]

More restrictive examples involve modeling the interference to force Toeplitz structures, and can greatly reduce the computational complexity associated with the processing by exploiting this structure.[2] However, these methods can suffer due to model-mismatch, or the computational savings may be undone by the problem of model fitting (such as the nonlinear problem of fitting to a Toeplitz or block-Toeplitz matrix) and order estimation.[2]

For dispersive channels, multiple-input multiple-output communications can formulate STAP solutions. Frequency-selective channel compensation can be used to extend traditional equalization techniques for SISO systems using STAP.[5] To estimate the transmitted signal ${\displaystyle \mathbf {\hat {S}} }$ at a MIMO receiver, we can linearly weight our space-time input ${\displaystyle \mathbf {\widetilde {Z}} }$ with weighting matrix ${\displaystyle \mathbf {\widetilde {W}} }$ as follows

${\displaystyle \mathbf {\hat {S}} =\mathbf {\widetilde {W}} ^{\mathrm {T} }\mathbf {\widetilde {Z}} }$

to minimize the mean squared error (MSE).[5] Using STAP with a training sequence ${\displaystyle \mathbf {\widetilde {S}} }$, the estimated optimal weighting matrix (STAP coefficients) is given by:[5]

${\displaystyle \mathbf {\widetilde {W}} \approx \left(\mathbf {\widetilde {Z}} \mathbf {\widetilde {Z}} ^{\mathrm {T} }\right)\mathbf {\widetilde {Z}} \mathbf {\widetilde {S}} ^{\mathrm {T} }}$

STAP has been extended for MIMO radar to improve spatial resolution for clutter, using modified SIMO radar STAP techniques.[9] New algorithms and formulations are required that depart from the standard technique due to the large rank of the jammer-clutter subspace created by MIMO radar virtual arrays,[9] which typically involves exploiting the block diagonal structure of the MIMO interference covariance matrix to break the large matrix inversion problem into smaller ones. In comparison with SIMO radar systems, which will have ${\displaystyle M}$ transmit degrees of freedom, and ${\displaystyle NL}$ receive degrees of freedom, for a total of ${\displaystyle M+NL}$, MIMO radar systems have ${\displaystyle MNL}$ degrees of freedom, allowing for much greater adaptive spatial resolution for clutter mitigation.[9]