Space-time adaptive processing

Doppler-Bearing response of a 2-dimensional beam-former

Space-time adaptive processing (STAP) is a signal processing technique most commonly used in radar systems. It involves adaptive array processing algorithms to aid in target detection. Radar signal processing benefits from STAP in areas where interference is a problem (i.e. ground clutter, jamming, etc.). Through careful application of STAP, it is possible to achieve order-of-magnitude sensitivity improvements in target detection.

STAP involves a two-dimensional filtering technique using a phased-array antenna with multiple spatial channels. Coupling multiple spatial channels with pulse-Doppler waveforms lends to the name "space-time." Applying the statistics of the interference environment, an adaptive STAP weight vector is formed. This weight vector is applied to the coherent samples received by the radar.

History

The theory of STAP was first published by Lawrence E. Brennan and Irving S. Reed in the early 1970s. At the time of publication, both Brennan and Reed were at Technology Service Corporation (TSC). While it was formally introduced in 1973 ^[1], it has theoretical roots dating back to 1959 ^[2].

Motivation

For ground-based radar, cluttered returns tend to be at DC, making them easily discriminated by MTI^[3]. Thus, a notch filter at the zero-Doppler bin can be used ^[2]. Airborne platforms with ownship motion experience relative ground clutter motion dependent on the angle, resulting in angle-Doppler coupling at the input^[2]. In this case, 1D filtering is not sufficient, since clutter can overlap the desired target's Doppler from multiple directions^[2].

Basic theory

STAP is essentially filtering in the space-time domain^[2]. This means that we are filtering over multiple dimensions, and multi-dimensional signal processing techniques must be employed ^[4]. The goal is to find the optimal space-time weights in $NM$ -dimensional space, where $N$ is the number of antenna elements (our spatial degrees of freedom) and $M$ is the number of pulse-repetition interval (PRI) taps (our time degrees of freedom), to maximize the signal-to-interference and noise ratio (SINR)^[2]. Thus, the goal is to suppress noise, clutter, jammers, etc, while keeping the desired radar return. It can be thought of as a 2-D finite-impulse response (FIR) filter, with a standard 1-D FIR filter for each channel (steered spatial channels from an electronically steered array), and the taps of these 1-D FIR filters corresponding to multiple returns (spaced at PRI time)^[1]. Having degrees of freedom in both the spatial domain and time domain is crucial, as clutter can be correlated in time and space, while jammers tend to be correlated spatially (along a specific bearing)^[1].

A simple example of STAP is shown in the first figure, for $N=M=10$ . This is an idealized example of a steering pattern, where the response of the array has been steered to the ideal target response, $s$ ^[2]. Unfortunately, in practice, this is oversimplified, as the interference to be overcome by steering the nulls shown is not deterministic, but statistical in nature^[2]. This is what requires STAP to be an adaptive technique.

The basic functional diagram is shown to the right. For each antenna, a down conversion and analog-to-digital conversion step is typically completed. Then, a 1-D FIR filter with PRI length delay elements is used for each steered antenna channel. The lexicographically ordered weights $W_{1}$ to $W_{NM}$ are the degrees of freedom to be solved in the STAP problem. That is, STAP aims to find the optimal weights for the antenna array. It can be shown, that for a given $MNxMN$ interference covariance matrix, $\mathbf {R}$ , the optimal weights maximizing the SINR is

$\mathbf {W} =\kappa \mathbf {R} ^{-1}s$

where $\kappa$ is a scalar that does not affect the SINR^[2]. The main difficulty of STAP is solving for the typically unknown interference covariance matrix, $\mathbf {R}$ ^[1]. Other difficulties arise when the interference covariance matrix is ill-conditioned, making the inversion numerically unstable^[5].

Sample matrix inversion is a common STAP algorithm, which uses the estimated (sample) interference covariance matrix in place of the actual interference covariance matrix ^[6]. This is because the actual interference covariance matrix is not known in practice^[1].

Modern applications

Despite nearly 40 years of existence, STAP has modern applications.

MIMO Communications

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 $\mathbf {\hat {S}}$ at a MIMO receiver, we can linearly weight our space-time input $\mathbf {\widetilde {Z}}$ with weighting matrix $\mathbf {\widetilde {W}}$ as follows

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

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

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

MIMO Radar

STAP has been extended for MIMO radar to improve spatial resolution for clutter, using modified SIMO radar STAP techniques^[7]. 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^[7], which typically involving exploiting the block diagonal structure of the MIMO interference covariance matrix to break the large matrix inversion problem into smallers ones. In comparison with SIMO radar systems, which will have $M$ transmit degrees of freedom, and $NL$ receive degrees of freedom, for a total of $M+NL$ , MIMO radar systems have $MNL$ degrees of freedom, allowing for much greater adaptive spatial resolution for clutter mitigation^[7].

References

^ ^a ^b ^c ^d ^e Melvin, W.L., A STAP Overview, IEEE AES Systems Magazine – Special Tutorials Issue, Vol. 19, No. 1, January 2004, pp. 19–35.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Guerci, J.R., Space-Time Adaptive Processing for Radar, Artech House Publishers, 2003. ISBN 1-58053-377-9.
^ Richards, M.A., Scheer, J.A., and Holm, W.A., Principles of Modern Radar, SciTech Publishing, 2010. ISBN 1-89112-152-9.
^ Dudgeon, D.E. and Mersereau, R.M., Multidimensional Digital Signal Processing, Prentice-Hall Signal Processing Series, 1984. ISBN 0-13604-959-1.
^ ^a ^b ^c ^d Bliss, D.W. and Govindasamy, S., Adaptive Wireless Communications: MIMO Channels and Networks, Cambridge University Press, 2013. ISBN 1-10703-320-9.
^ Van Trees, H. L., Optimum Array Processing, Wiley, NY, 2002.
^ ^a ^b ^c Li, J. and Stoica, P., MIMO Radar Signal Processing, John Wiley & Sons, 2009. ISBN 0-47017-898-1.

[Melvin_STAP-1] Melvin, W.L., A STAP Overview, IEEE AES Systems Magazine – Special Tutorials Issue, Vol. 19, No. 1, January 2004, pp. 19–35.

[Guerci_STAP-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Guerci, J.R., Space-Time Adaptive Processing for Radar, Artech House Publishers, 2003. ISBN 1-58053-377-9.

[Richards_Principles-3] Richards, M.A., Scheer, J.A., and Holm, W.A., Principles of Modern Radar, SciTech Publishing, 2010. ISBN 1-89112-152-9.

[Dudgeon_Multi-4] Dudgeon, D.E. and Mersereau, R.M., Multidimensional Digital Signal Processing, Prentice-Hall Signal Processing Series, 1984. ISBN 0-13604-959-1.

[Bliss_Adaptive-5] Bliss, D.W. and Govindasamy, S., Adaptive Wireless Communications: MIMO Channels and Networks, Cambridge University Press, 2013. ISBN 1-10703-320-9.

[VanTrees_Array-6] Van Trees, H. L., Optimum Array Processing, Wiley, NY, 2002.

[Li_MIMO-7] Li, J. and Stoica, P., MIMO Radar Signal Processing, John Wiley & Sons, 2009. ISBN 0-47017-898-1.

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