The performance and implementation of the inverse QR algorithm for recursive least squares filtering are examined. This algorithm is based upon an inverse QR decomposition and therefore avoids the highly serial backsubstitution step required in previously derived direct QR approaches. A parallel implementation of the method is discussed and computational wavefronts are displayed. A brief comparison is made with an alternative array structure.>

Analytical expressions for the theoretical probability density function (pdf) of the LMS (least mean square) adaptive filter weights are obtained for the steady state. The LMS update is formulated as a stochastic differential equation and the weight PDF is derived as the solution to the Fokker-Planck equation. The application of systems identification is examined, and closed-form solutions are obtained for the steady-state PDF for the LMS weights. Only the steady-state PDFs are found for a system identification problem, although the theory developed can also be applied to finding the time-varying PDFs during the adaptation phase.>

The performance of recursive-least-squares (RLS) algorithm based on an inverse QR decomposition is reported. Theoretical analysis provides performance measures in a finite precision environment. The performance measure is derived in terms of the biases that are present in steady-state along the diagonal entries of the matrix used in the approach. An analytical expression has been derived for this bias as a function of wordlength, forgetting factor, and signal statistics. This result is further used to show that the diagonal entries will not reduce to zero or become negative, thereby ensuring stability of the algorithm. All analytical results are verified by corresponding simulation results.>

An examination is made of the statistical behavior of RLS (recursive least squares) algorithms incorporating soft constrained initialization during the initialization period. In particular, it is shown that the mean and variance of the elements of the weight vector produced by this method are defined, and expressions are given for these quantities. Simulation results for various cases are presented to support the accuracy of these theoretical results.>

A method is presented for minimum mean square error (MMSE) pole-zero modeling which computes the parameter estimates at the absolute minimum of the MSE surface. Homotopy continuation methods are used to systematically find the set of all stationary points from which the global minimum is selected. The number of stationary points is expressed as a function of the number of data samples used in computing the estimate. These methods are applied to IIR (infinite impulse response) filter design and autoregressive moving-average (ARMA) estimation for first order models and are compared to previous approaches.>

The authors identify several specific sources of instability in the conventional recursive least squares (CRLS) algorithm. They explore why the algorithm is numerically unstable under fixed-point precision and examine how it can be stabilized. A form of the CRLS algorithm that maintains symmetry in the inverse autocorrelation matrix estimate is first presented, and divergence phenomena are discussed. Several sources of instability are identified as well as practical techniques for combating each instability source. Experimental results that demonstrate the usefulness of these techniques are presented. Initial simulation results indicate that by compensating for the source of numerical instability, it is possible to obtain numerical stability with negligible loss in performance.>

Sensitivity results are presented for the LS weight vector of an all-zero adaptive equalizer in an ill-conditioned signal environment. The ill-conditioned data results from severe amplitude distortion introduced by the data channel. Specifically, a theoretical upper bound for the relative change in the magnitude of the weight vector due to additive noise is derived as a function of the channel characteristics and equalizer order, and simulation results are presented which indicate the derived bound to be applicable over a wide range of equalizer orders.>

This paper investigates analytically and experimentally the rate (transmitted bandwidth) versus signal distortion introduced by transmitting and reconstructing the signal from the quantized LMS prediction residual. The contribution of this paper is the analytical determination of the received signal distortion as a function of quantizer wordlength (which determines the transmitted bit rate), LMS filter parameters and input signal correlation properties. This distortion is computable for both the transient adaptation stage and in the steady state converged stage.

Numerical properties of the hyperbolic rotation method for windowed RLS filtering are examined. This matrix-oriented approach is important from two standpoints: (1) it provides the LS predictor for a sliding-window block of data, and (2) it is amenable to parallel implementation. It is shown how a hyperbolic rotation matrix may be constructed to update the LS Cholesky factor as a function of the previous Cholesky factor and the data in the sliding window. Finally, it is shown that the hyperbolic rotation method is stable for observation matrices which are not rank-deficient.

New analytical results are derived for quantifying the performance degradation induced by finite precision arithmetic upon the autocorrelation method and Burg's algorithm for linear prediction. Analysis shows that reflection coefficients computed by the FP autocorrelation method have more severe degradation than those computed by the FP Burg algorithm. Finally, experimental results are presented which show very close agreement between the analytical derivations and experimental results.

A new architecture for a single instruction stream, multiple data stream (SIMD) implementation of the LMS adaptive algorithm is investigated. This is denoted as a ring architecture, due to its physical configuration, and it effectively solves the latency problem often associated with prediction error feedback in adaptive filters. The multiprocessor ring efficiently updates the filter input vector by operating as a pipeline structure, while behaving as a parallel structure in computing the filter output and applying the weight adaptation algorithm. Last, individual processor timing and capacity considerations are examined.

In this paper, the adaptive Least Mean Squares (LMS) algorithm is used to separate speaker-produced "information" from interferer-produced "noise" on the basis of the difference in power levels associated with the two phenomena. This method exploits the property of LMS that it rapidly adapts for the dominant excitation modes while simultaneously adapting very slowly for the weaker modes of excitation. This selective convergence property of LMS is next analyzed using an eigenvalue-eigenvector approach which easily displays the signal separation property. Lastly, computer simulations are presented which verify the analysis above for representative synthetic speech waveforms.

A new architecture for a Single-Instruction Multiple Data (SIMD) implementation of the LMS adaptive algorithm is investigated. This is denoted as a ring architecture, due to its physical configuration, and it effectively solves the latency problem often associated with prediction error feedback in adaptive filters. The multiprocessor ring efficiently updates the filter input vector by operating as a pipeline structure, while behaving as a parallel structure in computing the filter output and applying the weight adaptation algorithm. Lastly, individual processor timing and capacity considerations are examined.

The Least Mean Squares (LMS) Algorithm may be used to adapt the coefficients of an adaptive prediction filter for image source encoding. It is shown that if one allows a small amount of distortion in the received image, then the LMS algorithm allows a dramatic reduction in the required channel transmission bandwidth. Results are presented which show LMS provides up to a 1-2 bits per symbol reduction in transmitted bit rate (compared to DPCM) when distortion levels are approximately the same in both cases. Further, LMS can equivalently be used in fixed channel bandwidth environments to decrease the image distortion. When compared to results for fixed coefficient DPCM, one is able to reduce reconstructed image distortion by as much as 7-10 dB using LMS.

One of the sources of degradation in LPC-synthesized speech is the mechanical quality due to a single impulse excitation per pitch period. This paper presents a simple method for extracting the amplitudes and locations of a multiple impulse excitation model. These multipulse parameters are obtained very easily from the autoregressive (LPC) residual. Additionally, a method is developed which allows a more accurate recomputation of the autoregressive coefficients based upon incorporating the multipulse excitation.

The Least Mean Squares (LMS) adaptive algorithm is applied to the problem of reducing the mean square power of the information sequence required for transmission of video or teleconferencing image sequences. The results of processing actual teleconferencing sequences are presented and compared with the results of using a fixed coefficient predictive algorithm. Operating on actual teleconferencing sequences averaging 2% - 12% moving area, the LMS adaptive filter is shown to provide approximately 1-4 dB reduction in transmitted sequence power when compared to a fixed coefficient filter. Further, the performance of the LMS adaptive system is shown to be quite insensitive to the value of the feedback gain parameter, α, for a wide range of α. This allows the user to achieve a substantial performance gain (compared to the fixed coefficient system) without encountering problems of adaptive filter stability.

In many applications of Wiener filtering to noise cancellation an external reference noise input is required. However, if an external reference is not available, it is still possible to suppress additive noise using a Wiener linear prediction filter (LPF) provided the signal bandwidth is significantly less than the noise bandwidth. The purpose of this paper is to provide an analytical basis for bounding the performance of a digital LPF when applied to the problem of cancelling broadband additive noise from narrow-band signals.

In this paper, a study of adaptive lattice algorithms as applied to channel equalization is presented. The orthogonalization properties of the lattice algorithms make them appear promising for equalizing channels which exhibit heavy amplitude distortion. Furthermore, unlike the majority of other orthogonalization algorithms, the number of operations per update for the adaptive lattice equalizers is linear with respect to the number of equalizer taps.

The optimal minimum-mean-square-error discrete filters are obtained for finite bandwidth signals in noise represented by a rational pole-zero model in the spectral domain. An analytical technique called the method of undetermined coefficients is used to solve the Wiener equation for the optimal prediction coefficients. These coefficients are then used to examine (a) the frequency response of the Wiener prediction filter, and (b) the maximum entropy spectral estimator. Special cases considered include the limiting case of narrow bandwidth signals and large prediction filter lengths.

A high resolution technique for estimating the frequencies, powers, and the number of sinusoids in noise is examined. Unlike the Pisarenko decomposition method which may only be applied to sinusoids in uncorrelated noise, the technique examined in this paper may be applied even when the noise is correlated. This technique consists of applying Prony's algorithm to a segment of the autocorrelation function, r(k), for which|k|is greater than Γcorr, where Γcorris the decorrelation time of the noise.