Equirotational stack parameterization in subspace estimation and tracking

We study the following “equirotational” stack (ES) parameterization of subspaces: E_=[E/ES/ES²/./././ES ^n-1] where E is the N×r, N>r core basis matrix, and S is the r×r subrotor. The fact that successive submatrices in the basis stack E_ are just identically rotated versions of each other is usually a direct consequence of uniform sampling. Uniformly sampled complex exponential sequences can always be represented perfectly in subspaces of this kind. Early notions of ES subspace parameterization appear in array processing, particularly in direction finding using multiple invariance ESPRIT and regular array geometries (uniform spatial sampling). Another potential application area is spatiotemporal array data analysis. Even an application of ES subspace parameterization in time series analysis and adaptive filtering is not unreasonable. We present a class of fast algorithms for total least squares (TLS) estimation and tracking of the parameters E and S. Using these new algorithms, signal subspaces can be estimated with a much higher accuracy, provided only that the subspaces of the given signals are ES parameterizable. This is always the case for uniformly sampled narrowband signals. The achievable gain in estimated subspace SNR is then 10 log₁₀(4N/r) dB over conventional (unparameterized) subspace tracking, where the potential ES structure of the underlying data cannot be exploited. Consequently, we make the point that our algorithms offer a significant performance gain in all major application areas with uniformly sampled narrowband signals in noise over the previously used conventional (unparameterized) subspace estimators and trackers