EFFICIENT STRUCTURES FOR QUADRATIC TIME-FREQUENCY AND TIME-SCALE ARRAY PROCESSORS Anil M. Rao and Douglas L. Jones Coordinated Science Laboratory University of Illinois 1308 W. Main Street Urbana, IL 61801 e-mail: anilrao@dsp.csl.uiuc.edu, d-jones@csl.uiuc.edu phone: (217) 244-6823 ABSTRACT Sensor arrays are able to enhance desired signal reception while si- multaneously suppressing undesired components through the use of directionality. In many important applications, the return signal is best modeled as being nonstationary and may lose coherence between sensors, severely limiting the performance of traditional array processors based on matched-field beamforming. Quadratic array processing is optimal for many stochastic signals of interest, but direct implementation poses a significant computational bur- den making it impractical in many situations. Recently it has been shown that quadratic time-frequency representations and time-scale representations (TFRs and TSRs) provide a structured detection framework for detecting certain nonstationary signals in the pres- ence of nonstationary noise using a partially coherent sensor array, making quadratic array processing a viable alternative to subop- timal matched-filter techniques. In this paper we discuss restric- tions on the decorrelation between sensors which lead to efficient TFR/TSR-based processors known as banded and subarray beam- formers. 1. INTRODUCTION The detection of signals in noise is a classical hypothesis testing problem. The use of a sensor array can considerably enhance sig- nal detection by providing a large gain in the SNR and, more im- portantly, sensor arrays allow for target or signal source localiza- tion. Due to the need for fast processing of target data, array pro- cessing structures in most radar/sonar applications have been lim- ited to simple beamforming followed by matched filtering which, from a detection theoretic point of view, is the optimal structure only for deterministic returns or narrowband, stationary, perfectly coherent random returns [2]. Unfortunately, such simple models for the return signal are inadequate in many important situations. In active sensing situations such as radar and sonar, a known wave- form is generated which in turn propagates through a medium and is reflected by some target back to the array. That transmitted sig- nal undergoes a delay and frequency shift in the narrowband case and a delay and scale offset in the wideband case. In addition, the angle at which the return signal arrives may be unknown. More- over, changing target and environmental characteristics, combined This work was supported by the Office of Naval Research, contract no. N00014-95-1-0674 and N00014-95-1-0907 with other types of disturbances, cause the signals that arrive at the array to be regarded as random, and at times the physical phenom- ena responsible for the randomness in the signal make it plausible to assume that the signals are Gaussian (perhaps nonstationary) random processes [3]. In addition, inhomogeneities in the medium or movement of the sensors may cause the signal to lose coherence as it propagates between sensors. Though the signal correlation structure is known, it may still contain unknown parameters such as a delay, frequency shift, scale offset, or angle of arrival. These uncertainties, combined with the nonstationary nature of the sig- nal and noise processes, make TFRs and TSRs powerful tools in designing the optimal detector in the array environment. Optimal detection for Gaussian signals involves a detection structure that is quadratic in the observations [4], and matrix-based processors are necessary to deal with loss of signal coherence. Un- fortunately, quadratic and matrix-based processing pose a consid- erable computational burden. In many modern systems prompt ac- quisition and processing of target data is imperative, making gen- eral quadratic array processors impractical. In the past, the only alternative has been to resort to suboptimal structures involving beamforming followed by matched filtering and its simple time- frequency generalizations based on the ambiguity function when the signal structure is known but contains an unknown time delay or frequency shift. However, research efforts in the use of time- frequency and time-scale representations for efficient implementa- tion of certain quadratic detectors make practical implementation of quadratic array processors look promising. Recently it has been shown that quadratic TFRs and TSRs provide a natural, structured implementation of the optimal quadratic array processor even in the case of partial signal coherence [1]. This identification of struc- ture allows for efficient implementation, making the quadratic ar- ray processor viable in practice. In this paper, we explicitly de- scribe efficient implementation structures for the TFR/TSR-based optimal array processor developed in [1] by exploiting the time- frequency and time-scale structure. This paper is organized as follows: Section 2 sets up the ar- ray configuration and the signal detection problem while Section 3 reviews the time-frequency/time-scale implementation of the opti- mal array processor. In Section 4 we describe restrictions on the loss of signal coherence that allows for a sliding auto-TFR/TSR detector structure known as banded beamforming. In Section 5 we discuss the subarray structure for the TFR/TSR-based processor and in Section 6 we conclude the paper.