Mithat C. Dogan

Applications of cumulants to array processing .I. Aperture extension and array calibration

An interpretation for the use of cumulants in nar- rowband array processing problems is proposed. It is shown how fourth-order cumulants of multichannel observations increase the directional information compared with second-order statistics. Based on the interpretation, it is shown how cumulants can be used to increase the effective aperture of an arbitrary antenna array. The amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far- field sources is also investigated. It is proven that the presence of a doublet and use of fourth-order cumulants is sdcient to accomplish this task. The proposed approach is computationally efficient and more general than covariance-based algorithms that have addressed the calibration problem under constraints. A class of beamforming techniques is proposed to recover the source waveforms. Proposed estimation procedures are based on cumulants, which bring insensitivity to the spatial correlation structure of additive Gaussian measurement noise. Simulations are provided to illustrate the use of the proposed algorithms.

Applications of cumulants to array processing. IV. Direction finding in coherent signals case

For pt.III see ibid., vol.45, no.9, p.2253-64, 1997. We present a subspace-based direction finding method for coherent signal environments using an antenna array. Our method, which uses fourth-order statistics, is capable of resolving more signals than a comparable second-order statistics-based subspace method and is applicable to a larger class of arrays. The maximum number of signals resolvable with our method may exceed the number of sensors in the array. Only a uniform linear subarray is needed; the rest of the array may have arbitrary and unknown response and does not require calibration. On the other hand, the comparable second-order statistics-based method is limited to uniform linear arrays only. No search procedure is needed in our method. Simulation experiments supporting our conclusions are provided.

Cumulant-based blind optimum beamforming

Sensor response, location uncertainty, and use of sample statistics can severely degrade the performance of optimum beamformers. We propose blind estimation of the source steering vector in the presence of multiple, directional, correlated or coherent Gaussian interferers via higher order statistics. In this way, we employ the statistical characteristics of the desired signal to make the necessary discrimination, without any a-priori knowledge of array manifold and direction-of-arrival (DOA) information about the desired signal. We then improve our method to utilize the data in a more efficient manner. In any application, only sample statistics are available, so we propose a robust beamforming approach that employs the steering vector estimate obtained by cumulant-based signal processing. We further propose a method that employs both covariance and cumulant information to combat finite sample effects. We analyze the effects of multipath propagation on the reception of the desired signal. We show that even in the presence of coherence, cumulant-based beamformer still behaves as the optimum beamformer that maximizes the signal-to-interference-plus-noise ratio (SINR). Finally, we propose an adaptive version of our algorithm simulations demonstrate the excellent performance of our approach in a wide variety of situations. >

Applications of cumulants to array processing. II. Non-Gaussian noise suppression

The main motivation of using higher order statistics in signal processing applications has been their insensitivity to additive colored Gaussian noise. The main objection to those methods is their possible vulnerability to non-Gaussian noise. The authors investigate the effects of non-Gaussian ambient noise on cumulant-based direction-finding systems using the interpretation for the information provided by cumulants for array processing applications described in Dogan and Mendek. they first demonstrate the suppression of uncorrelated non-Gaussian noise that has spatially varying statistics. Then, they indicate methods to suppress spatially colored non-Gaussian noise using cumulants and an additional sensor whose measurement noise component is independent of the noise components of the original array measurements. They also indicate the noise suppression properties of the virtual-ESPRIT algorithm proposed in Dogan and Mendel. In addition, they propose a method that combines second- and fourth-order statistics together in order to suppress spatially colored non-Gaussian noise. Finally, they also illustrate how to suppress spatially colored non-Gaussian noise when the additional sensor measurement is not available. Simulations are presented to verify the results. >

Applications of cumulants to array processing: direction-finding in coherent signal environment

Dogan and Mendel (see IEEE Trans. on Signal Processing, 1994) have developed the virtual-ESPRIT algorithm (VESPA) for direction-finding and recovery of independent sources. VESPA can calibrate an array of unknown configuration and arbitrary response by using just one additional pair of identical sensors (instead of a copy of the entire array or storage of the entire array response for every possible scenario, which is required by existing alternatives). We present an approach that generalizes VESPA to handle the case of highly correlated or coherent sources. Unlike existing methods, our method is not restricted to linear arrays, and no search procedure is needed. Just as in VESPA, it is still possible to detect more sources than sensors, and suppress both Gaussian as well as non-Gaussian noise. A simulation experiment supporting our conclusions is provided.<<ETX>>

Real-time robust pitch detector

The authors propose a cumulant-based method to perform voice-unvoiced decision and pitch period estimation. The approach is based on the nature of excitation for different states of speech. The authors accomplished this goal by analyzing cumulant-related time sequences obtained via nonlinear processing of the speech signal. Experimental results indicating the performance of the proposed method, especially in the pitch estimation problem in which there are high power harmonics are presented.<<ETX>>

Joint array calibration and direction-finding with virtual-ESPRIT algorithm

The amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far-field sources is investigated. The authors prove that the presence of a doublet and use of fourth-order cumulants is sufficient to accomplish this task. Their approach is based on the interpretation of cumulants for array processing. The cumulant-based algorithm is computationally efficient and more general than constrained covariance-based algorithms. Simulations indicate excellent results by the proposed algorithm.<<ETX>>

Antenna array noise reconditioning by cumulants

The main motivation of using higher order statistics in signal processing applications has been their insensitivity to additive colored Gaussian noise. The main objection to those methods is their vulnerability to nonGaussian noise. The authors first give an interpretation for the information provided by cumulants for array processing applications. Then, they investigate the possibility of suppressing the effects of nonGaussian noise. It is shown that existing arrays can be reconditioned by an additional sensor and by using cumulants. It is also shown that existing cumulant-based algorithms are less sensitive to colored nonGaussian noise than their covariance-based counterparts above a threshold SNR.<<ETX>>

Blind deconvolution (equalization): some new results

Abstract An interesting equation developed by Giannakis and Mendel (1989), and referred to by others (e.g., Friedlander and Porat (1989)) as the ‘GM equation’, links higher-order statistics to second-order statistics. In this paper, we show how this equation leads to a new universal relationship between a system and its inverse, and can be employed to obtain a new closed-form formula for a Shalvi-Weinstein-like super-exponential blind deconvolution algorithm.