Estimation of Multiple Sinusoidal Frequencies Using Truncated Least-Squares Methods
Liu, K.J. Ray
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Tufts and Kumaresan (1982) first proposed using a SVD-based method to solve the forward-backward linear prediction (FBLP) least-squares problem for resolving closely spaced frequencies of multiple sinusoids from limited amount of data samples. By imposing an excessive order in the FBLP model and then truncating small singular values to zero, this truncated SVD (TSVD) method yields a low SNR threshold and greatly suppresses spurious frequencies. However, the massive computation required by SVD makes it unsuitable for real time super-resolution applications. We propose to use truncated QR methods which are amenable to VLSI implementations, such as systolic arrays, with slightly degraded performances as compared to the TSVD method. Three truncated QR methods for sinusoidal frequency estimation will be considered: (1) truncated QR without column pivoting (TQR); (2) truncated QR with re-ordered columns (TQRR); and (3) truncated QR with column pivoting (TQRP). It is demonstrated that the benefit of the TSVD method for high frequency resolution is achievable under the truncated QR methods with much lower computational cost. Other attractive features of the proposed methods include the ease of updating which is difficult for the SVD method, and numerical stability. Thus, the TQR methods offer efficient ways for identifying sinusoids closedly clustered in frequencies under stationary and nonstationary conditions. Some results based on the truncated normal equation approach as well as on sufficient conditions for perfect truncations based on truncated QR and SVD methods are considered. Based on the FBLP model, computer simulations and comparisons are provided for different truncation methods under various SNR's. Comparisons of asymptotic performance with large data samples are also given.