Eigenvalues of Covariance Matrix for Two-Source Array Processing

Abstract

SummaryWe begin this report with a survey of direction-finding algorithms restricted to the narrowband case in element space. We then set out to search for a simple satisfactory stress measure to direction-finding algorithmns applied to scenarios with two sources. The word "stress" for a scenario is used to indicate the difficulty that an algorithm has in determining the direction of arrival of each of the two sources. A scenario is said to be "stressful" to an algorithm if it causes difficulty in resolving the sources or in determining their directions of arrival. The apparent lack of understanding of the effect of phase difference between two signals in direction-finding made such an ideal stress measure illusive. Regardless of all its deficiencies, eigenvalue ratio is still a useful measu~c in summarizing stressing factors in many aspects for two-arrival direcion-finding scenario.Eigenvalue weighting appears in some noise subspace methods, in parametric signal subspace fitting methods, and in nonparametric subspace adaptive nulliiig beamforming. For a two-source array-processing scenario, normalized largc and small eigenvalues A, and A 2 are reduced to forms depending only on a real triplet: phase-dependent variable , phase-independent variable 77, and power ratio 21. The pairs ( , /) are conined to an isosceles-like region. We characterize * this isoscelks-like region and the many-to-one mapping from the Caitesian product of the temporal and spatial correlation unit-disks onto this region, * the behavior of the eigenvalues and their ratios as functions of the real triplet both analytically and graphically with respect to direction-finding.The main contribution of this work is a manageable presentation of a compact map showing Al, A 2 , and A as functions of , r, and 11 over all possible scenarios. This enables one to see A 2Ir 2 the relative positions among different scenarios. We also present some easy-to-remember formulas that enable one to exercise "back-of-envelope" assessment of scenarios. The small eigenvalue is shown to diminish qualitatively and quantitatively for two-arrival scenarios increasingly stressed with high temporal and/or spatial correlations. The special case of equal-strength signal arrivals (a = 1), also important in low-angle radar tracking, shares many rich structures of general a. The equal-strength case also has several additional unique features for signal eigenvalue ratio '\, which is important in direction-finding: e an extra 6 dB over the vertical axis =0 as compared to the general E, case, r 2 * a 6-dB increase for equipower arrivals highly correlated both temporally and spatially from changing the angle difference between the two unit-disk vectors from 90°(orthogonal waveforms) to 0°(in-phase waveforms), * the lower left corner of the isosceles-like triangle is a point of discontinuity for the eigenvalue ratio -'The developed results are used to assess some scenarios used by Cadzow and Ottersten.