Source Localization: Subspace Methods
The location parameters are estimated directly without having to search for peaks as in the frequency-wavenumber spectrum (the approach described in Chapter 4). In open space, the direction of arrival (DOA), that is, the azimuth or elevation or both, is estimated using the subspace properties of the spatial covariance matrix or spectral matrix. Multiple signal classifi cation (MUSIC) is a well-known algorithm where we defi ne a positive quantity that becomes infi nity whenever the assumed parameter(s) is equal to the true parameter. We shall call this quantity a spectrum even though it does not possess the units of power as in the true spectrum. The MUSIC algorithm in its original form does involve scanning and searching, often very fi ne scanning lest we may miss the peak. Later extensions of the MUSIC, like root MUSIC, ESPRIT, etc. have overcome this limitation of the original MUSIC algorithm. When a source is located in a bounded space,such as a duct, the wavefront reaching an array of sensors is necessarily nonplanar due to multipath propagation in the bounded space. In this case, all three position parameters can be measured by means of an array of sensors. But the complexity of the problem of localization is such that a good prior knowledge of the channel becomes mandatory for successful localization. In active systems, since one has control over the source, it is possible to design waveforms that possess the property that is best suited for localization; for example, a binary phase shift key (BPSK) signal with its narrow autocorrelation function is best suited for time delay estimation. Source tracking of a moving source is another important extension of the source localization problem. With a linear array, one can only estimate the azimuth but not the elevation, which is required for localization in 3D space. In the last section, we take the problem of elevation estimation using a circular array and an L-shaped array.