The mathematical basis for element and fourier beamspace MUSIC and root-MUSIC algorithms

Signal subspace spectral estimation algorithms of the MUSIC type [l-3] are particularly effective for estimating the direction of arrival (DOA) of incoming plane wave signals arising from point emission sources in the far field of spatial phased arrays. The subspace algorithms require eigenanalysis of an estimated form of the covariance matrix. A linear system of N distinct sensors with complex signal outputs can be mathematically formulated in terms of linear operators in an N-dimensional complex Hilbert space, e (iv]. The covariance matrices used in the analyses of the system response are N-dimensional Hermitian matrices. Eigenanalysis of the covariance matrix is required to separate the signal subspace from the noise subspace. The source DOAs are estimated from the array manifold vectors that exhibit minimum projections on the noise subspace. Eigenanalysis of N-dimensional matrices, which cannot take advantage of simplifying structures, typically requires 0 ( N3) operations. This computational burden often precludes the conventional application of signal subspace algorithms to large-order array systems. The computational complexity of signal subspace methods in the case of a large-order array system may be substantially reduced by operating in the beamspace domain [ 4-101. Here the spatial spectrum (angular region of interest) is divided into subbands (angular sectors). A particular spatial frequency subband is established by forming a number of overlapping beams to encompass the angular sector. In the simplest case, the beam outputs are computed as successive values of an N-point discrete Fourier transform (DFT) of the snapshot vector composed of the N sensor outputs at a particular sampling instant. Q successive N-point DFT beams subtend an angular fraction Q/N of the total spatial spectrum. Different subbands are selected by selecting different sets of Q successive N-point DFT beams; subbands may be processed simultaneously in parallel. Processing within each subband requires eigenanalysis of a Q-dimensional beamspace covariance matrix to determine the respective signal subspace and noise subspace associated with the subband. The DOAs of sources within the subband are determined as those for which the norm of the projection of the corresponding beamspace manifold on the noise subspace is minimum. In this way, a large problem of order N is reduced to a number of smaller problems of order Q. For a uniformly spaced linear array ( ULA) , the element space array manifold vectors possess a simple Vundernonde structure. The beamspace transformation causes an important change as the transformed array manifold vectors no longer exhibit a Vandermonde structure. This beamspace property was originally demonstrated in [lo] and is discussed in detail in Section IV of the present work. As a consequence of this change in structure, the class of algorithms that employ forward and backward spatial averaging is not applicable in beamspace. This renders the beamspace approach inoperative in the case of a single snapshot (single realization of a time series). In general, beamspace signal subspace methods have developed from a largely heuristic point of view. An alternative means of reducing computational complexity is motivated by the subbanding and multirate processing techniques employed in a number of application areas including speech processing, digital communications, etc. In this mode of operation, one ultimately works in a pseudo-element space as opposed to beamspace. This