Block adaptive filtering algorithm based on the preconditioned conjugate gradient method
Introduction
Recently, the conjugate gradient algorithms have been proposed for adaptive filtering applications 2, 3. In particular, the conjugate gradient algorithm was employed in nonlinear adaptive filtering problems [3]. The algorithms operate on the sample-by-sample basis and use a gradient average window to generate some direction vectors conjugate to the initial direction vector. In the papers, it was also shown that the conjugate gradient algorithms can outperform the LMS and RLS algorithms in terms of speed of convergence.
The conjugate gradient algorithms were implemented for the block adaptive filter by block processing in 6, 7. The algorithms were derived using the least-squares solution so that an estimate of the block mean-square error (BMSE) is minimized along the conjugate direction vectors that are generated to be mutually conjugate. The block conjugate gradient algorithms are based on the nested iteration technique [9] that is realized by performing filter tap updates repeatedly for the same block of data until a terminating condition is satisfied. However, the convergence rate of the block conjugate gradient algorithms basically depends on the eigenvalue spread of the input autocorrelation matrix.
In this paper, using the preconditioning and the nestetd iteration technique, we propose a block adaptive filtering algorithm whose rate of convergence is independent of the input signal conditioning. The Toeplitz-preconditioned BCG (TBCG) algorithm is formulated by combining the BCG algorithm with a Toeplitz preconditioner that is assumed to be a symmetric Toeplitz matrix. The Toeplitz preconditioner is directly obtained from the data, and the deconvolution is solved by the Levinson-type algorithms 1, 4. The convergence rate of the proposed algorithm is very fast and comparable to the RLS algorithm.
In Section 2, the TBCG algorithm is formulated based on the nested iteration technique as a least-squares approach. Computational complexities and simulation results are presented in 3 Computational complexities, 4 Computer simulations, respectively. Finally, conclusions are drawn in Section 5.
Section snippets
Formulation of TBCG algorithm
Hereafter, boldfaced upper- and lower-case symbols denote matrices and vectors, respectively. Superscript T corresponds to the transpose of a vector or a matrix and the expectation operation, respectively. We use k for the iteration index and j for the block index. We assume that all inputs are stationary and persistently exciting. It is also assumed that the data block length L is greater than or equal to the filter order N.
In order to formulate the block adaptive algorithm based on the nested
Computational complexities
Table 1 shows computational complexities of the various algorithms. We can see that the TBCG algorithm yields a moderate increase in multiplications as compared with the NOBA and BCG algorithms. In the table the fast transversal filter (FTF) is very efficient as a fast RLS algorithm, but it has a tendency to be numerically unstable. It is worthwhile to note that significant computational saving in the TBCG algorithm can be achieved if the algorithm is implemented by using the FFT. All the
Computer simulations
In order to study the convergence behavior of the proposed algorithm, computer simulations have been done in the system identification used in [7]. As additive noise, Gaussian zero-mean white noise with the variance of 10−5 was added to the desired signals. The input signal was generated using the Gaussian zero-mean white noise with unit variance so that its eigenvalue spread may be about 682, or 2300. The ensemble averaging was also carried out over 50 independent runs.
Fig. 1 shows convergence
Conclusions
In this paper, we have proposed a new block adaptive filtering algorithm based on the preconditioned conjugate gradient method. Through computer simulations, it has been shown that the proposed algorithm is very fast in convergence speed comparable to the RLS algorithm. Unlike the NLMS and BCG algorithms, the proposed algorithm is robust to the change of eigenvalue spread.
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