Dirk T. M. Slock

Numerically stable fast transversal filters for recursive least squares adaptive filtering

A solution is proposed to the long-standing problem of the numerical instability of fast recursive least squares transversal filter (FTF) algorithms with exponential weighting, an important class of algorithms for adaptive filtering. A framework for the analysis of the error propagation in FTF algorithms is first developed; within this framework, it is shown that the computationally most efficient 7N form is exponentially unstable. However, by introducing redundancy into this algorithm, feedback of numerical errors becomes possible; a judicious choice of the feedback gains then leads to a numerically stable FTF algorithm with a complexity of 8N multiplications and additions per time recursion. The results are presented for the complex multichannel joint-process filtering problem. >

Fractionally spaced equalization of linear polyphase channels and related blind techniques based on multichannel linear prediction

We consider the problem of linear equalization of polyphase channels and its blind implementation. These channels may result from oversampling the single output of a transmission channel or/and by receiving multiple outputs of an antenna array. A number of previous contributions in the field of blind channel identification have shown that polyphase channels can be blindly identified using only second-order statistics (SOS) of the output. In this work, we are mostly interested in the blind linear equalization of these channels. After some elaboration on the specifics of the equalization problem for polyphase channels, we show how optimal settings of various well-known types of linear equalization structures can be obtained blindly using only the outputs SOS by using multichannel linear prediction or related techniques.

Normalized sliding window constant modulus and decision-directed algorithms: a link between blind equalization and classical adaptive filtering

By minimizing a deterministic criterion of the constant modulus (CM) type or of the decision-directed (DD) type, we derive normalized stochastic gradient algorithms for blind linear equalization (BE) of QAM systems. These algorithms allow us to formulate CM and DD separation principles, which help obtain a whole family of CM or DD BE algorithms from classical adaptive filtering algorithms. We focus on the algorithms obtained by using the affine projection adaptive filtering algorithm (APA). Their increased convergence speed and ability to escape from local minima of their cost function make these algorithms very promising for BE applications.

Numerically stable fast recursive least-squares transversal filters

The problem of numerical stability of fast recursive least-squares transversal filter (FTF) algorithms is addressed. The prewindowing case with exponential weighting is considered. A framework for the analysis of the error propagation in these algorithms is developed. Within this framework, it is shown that the computationally most efficient 7N form (dealt with by G. Carayanmis et al. (1983) and by J.M. Cioffi (1984)) is exponentially unstable. By introducing redundancy in this algorithm, feedback of numerical errors becomes possible. This leads to a numerically stable FTF algorithm with complexity 9N. The results are presented for the complex multichannel joint-process filtering problem.<<ETX>>

A Toeplitz displacement method for blind multipath estimation for long code DS/CDMA signals

The problem of blind channel identification for direct-sequence/code-division multiple-access (DS/CDMA) multiuser systems is explored. For wideband DS/CDMA signals, multipath distortion is well modeled by a finite-impulse response filter. In this work, a blind channel identification technique based on second-order statistics is investigated. The method exploits knowledge of the spreading code of the user of interest via matched filtering, as well as properties of spreading codes. The current scheme focuses on a method appropriate for randomized long sequence DS/CDMA. This access scheme poses special challenges as the spreading codes are time varying. An analytical approximation of the mean-squared error is derived using perturbation techniques. The performance of the algorithm is studied via simulation and through the mean-squared error approximation, which is observed to be tight.

Modular and numerically stable fast transversal filters for multichannel and multiexperiment RLS

The authors present scalar implementations of multichannel and multiexperiment fast recursive least squares algorithms in transversal filter form, known as fast transversal filter (FTF) algorithms. By processing the different channels and/or experiments one at a time, the multichannel and/or multiexperiment algorithm decomposes into a set of intertwined single-channel single-experiment algorithms. For multichannel algorithms, the general case of possibly different filter orders in different channels is handled. Geometrically, this modular decomposition approach corresponds to a Gram-Schmidt orthogonalization of multiple error vectors. Algebraically, this technique corresponds to matrix triangularization of error covariance matrices and converts matrix operations into a regular set of scalar operations. Modular algorithm structures that are amenable to VLSI implementation on arrays of parallel processors naturally follow from the present approach. Numerically, the resulting algorithm benefits from the advantages of triangularization techniques in block processing. >

Fast transversal filters with data sequence weighting

A fast O(N) algorithm is presented for the adaptive recursive least-squares design of transversal filters. Data sequence weighting is introduced to allow for arbitrarily time-varying weighting strategies, facilitating the tracking of arbitrary nonstationary phenomena. A novel weighting adaptation mechanism is presented that has various desirable features and optimal performance under certain conditions. The algorithm is derived for the prewindowed given data case, and the underlying principles of the derivation are clearly exposed. Exact and soft-constraint initialization are discussed, and an improved restart procedure is proposed. >

Fast subsampled-updating stabilized fast transversal filter (FSU SFTF) RLS algorithm for adaptive filtering

We present a new, doubly fast algorithm for recursive least-squares (RLS) adaptive filtering that uses displacement structure and subsampled-updating. The fast subsampled-updating stabilized fast transversal filter (FSU SFTF) algorithm is mathematically equivalent to the classical fast transversal filter (FTF) algorithm. The FTF algorithm exploits the shift invariance that is present in the RLS adaptation of an FIR filter. The FTF algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the Levinson (1947) algorithm. In the subsampled-updating approach, we accumulate the rotation matrices over some time interval before applying them to the filters. It turns out that the successive rotation matrices themselves can be obtained from a Schur-type algorithm that, once properly initialized, does not require inner products. The various convolutions that appear In the algorithm are done using the fast Fourier transform (FFT). The resulting algorithm is doubly fast since it exploits FTF and FFTs. The roundoff error propagation in the FSU SFTF algorithm is identical to that in the SFTF algorithm: a numerically stabilized version of the classical FTF algorithm. The roundoff error generation, on the other hand, seems somewhat smaller. For relatively long filters, the computational complexity of the new algorithm is smaller than that of the well-known LMS algorithm, rendering it especially suitable for applications such as acoustic echo cancellation.

Modular and numerically stable multichannel FTF algorithms

The authors present scalar implementations of multichannel fast recursive least squares algorithms in transversal filter form (so-called FTF). By processing the different channels sequentially, i.e one at a time, the processing of any channel reduces to that of the single-channel algorithm. This sequential processing decomposes the multichannel algorithm into a set of intertwined single-channel algorithms. Geometrically, this corresponds to a modified Gram-Schmidt orthogonalization of multichannel error vectors. Algebraically, this technique corresponds to matrix triangularization of multichannel error covariance matrices and converts matrix operations into a regular set of scalar operations. Algorithm structures that are amenable to VLSI implementation on arrays of parallel processors follow naturally from this approach. Numerically, the resulting algorithm benefits from the advantages of triangularization techniques in block processing. Stabilization techniques for control of numerical error propagation in the update recursions are incorporated.<<ETX>>

The fast subsampled-updating fast Newton transversal filter (FSU FNTF) algorithm for adaptive filtering

The fast Newton transversal filter (FNTF) algorithm starts from the recursive least-squares algorithm for adapting a finite impulse response filter. The FNTF algorithm approximates the Kalman gain by replacing the sample covariance matrix inverse by a banded matrix of total bandwidth 2M+1 (AR(M) assumption for the input signal). In this algorithm, the approximate Kalman gain can still be computed using an exact recursion that involves the prediction parts of two fast transversal filter (FTF) algorithms of order M. We introduce the subsampled updating (SU) approach in which the FNTF filter weights and the Kalman gain are provided at a subsampled rate, say every L samples. Because of its low computational complexity, the prediction part of the FNTF algorithm is kept as such. A Schur type procedure is used to compute various filter outputs at the intermediate time instants, while some products of vectors with Toeplitz matrices are carried out with the FFT. This leads to the fast subsampled-updating FNTF (FSU FNTF) algorithm, an algorithm that is mathematically equivalent to the FNTF algorithm in the sense that exactly the same filter output is produced. However it shows a significantly smaller computational complexity for large filter lengths at the expense of some relatively small delay. The FSU FNTF algorithm (like the FNTF algorithm) has good convergence and tracking properties. This renders the FSU FNTF algorithm very interesting for applications such as acoustic echo cancellation.