FFT-based RLS in signal processing
ICASSP'93 Proceedings of the 1993 IEEE international conference on Acoustics, speech, and signal processing: digital speech processing - Volume III
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The authors consider the solution of least squares problems $\min \|b - Tx\|_2$ by the preconditioned conjugate gradient method, for $m$-by-$n$ complex Toeplitz matrices $T$ of rank $n$. A circulant preconditioner $C$ is derived using the $T$. Chan optimal preconditioner on $n$-by-$n$ Toeplitz row blocks of $T$. For Toeplitz $T$ that are generated by $2\pi$-periodic continuous complex-valued functions without any zeros, the authors prove that the singular values of the preconditioned matrix $TC^{-1}$ are clustered around 1, for sufficiently large $n$. The paper shows that if the condition number of $T$ is of $O(n^{\alpha}), \alpha 0$, then the least squares conjugate gradient method converges in at most $O(cd\alpha \log n + 1)$ steps. Since each iteration requires only $O(m \log n)$ operations using the Fast Fourier Transform, it follows that the total complexity of the algorithm is then only $O(\alpha m \log^{2} n + m \log n)$. Conditions for superlinear convergence are given and regularization techniques leading to superlinear convergence for least squares computations with ill-conditioned Toeplitz matrices arising from inverse problems are derived. Numerical examples are provided illustrating the effectiveness of the authors' methods.