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Coprime and nested arrays are sparse sensor arrays that provide O(MN ) degrees of freedom using only O(M + N ) sensors. The signals sampled by these sparse arrays contain the aliasing artifact; the min processor is used to disambiguate this aliasing artifact. The min processor beampattern has the same main lobe width and resolution as the beampattern of a full uniform linear array (ULA) with many more sensors. However, the peak side lobe (PSL) height of the min processor beampattern is higher than the PSL height of a full ULA with the same taper and resolution if the sparse arrays are not extended. Extending the sparse arrays, while keeping the intersensor spacing fixed, can reduce the PSL height to the level of the full ULA. Until now, the analytical expressions of the extension factor required to match the PSL height have not been found for the min processor. Also, the analytical expressions of the probability density function (PDF), mean, and variance of the min processor output for non-uniform tapers have not been derived. In this paper, we derive both of these analytical expressions. We consider the uniform, Hamming, Hann, Blackman-Harris, and Dolph-Chebyshev tapers for both coprime and nested arrays. We use the derived PDF in evaluating detection performance metrics such as receiver operation characteristic, Kullback Leibler divergence, and symmetric Kullback Leibler divergence. Our results show that min processing is superior to conventional beamforming and product processing in Gaussian signal detection for a given extended coprime or nested array.INDEX TERMS Blackman-Harris taper, coprime arrays, Dolph-Chebyshev taper, Hamming taper, Hann taper, min processing, nested arrays, sparse arrays, uniform taper.

Linear sparse arrays (coprime and nested arrays) have been studied extensively as a means of performing direction of arrival (DoA) estimation while bypassing Nyquist sampling theorem. However, rectangular sparse arrays have few studies, most of which are based in lattice theory. Although the multiple signal classification (MUSIC) alogrithm can be applied to lattice theory-based sparse arrays, fewer DoAs can be estimated for these arrays than for a full array with the same aperture. One contribution of this paper is the formulation of symmetry-imposed rectangular coprime and nested array designs that have wider contiguous lags than lattice-based arrays. Also, existing algorithms for rectangular arrays employ the direct sample covariance matrix estimate, which has low accuracy. Another contribution of this paper is an alternate method for estimating the covariance matrix that ensures the matrix is block-Toeplitz. Using the proposed covariance matrix estimate, we integrate a MUSIC-based DoA estimation method that applies to both full arrays and sparse arrays. The results show that the covariance estimates produced by the proposed method have higher accuracy than the traditional sample covariance estimates. The results also demonstrate that symmetry-imposed sparse arrays have higher resolution than full rectangular arrays with the same number of sensors.

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