Phillip A. Regalia

The digital all-pass filter: a versatile signal processing building block

The properties of digital all-pass filters are reviewed and a broad overview of the diversity of applications in digital filtering is provided. Starting with the definition and basic properties of a scalar all-pass function, a variety of structures satisfying the all-pass property are assembled, with emphasis placed on the concept of structural losslessness. Applications are then outlined in notch filtering, complementary filtering and filter banks, multirate filtering, spectrum and group-delay equalization, and Hilbert transformations. In all cases, the structural losslessness property induces very robust performance in the face of multiplier coefficient quantization. Finally, the state-space manifestations of the all-pass property are explored, and it is shown that many all-pass filter structures are devoid of limit cycle behavior and feature very low roundoff noise gain. >

On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors

Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the well-known power method for matrices was proposed for its solution. This higher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, as its convergence is not guaranteed. The aim of this paper is to show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higher-order power method. Furthermore, a novel method for initializing the iterative process is developed which has been observed to yield an estimate that lies closer to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quantifiable. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied.

An improved lattice-based adaptive IIR notch filter

A novel lattice-based adaptive infinite impulse response (IIR) notch filter is developed which features independent tuning of the notch frequency and attenuation bandwidth. The internal structure is based on planar rotators, ensuring reliable numerical behaviour and high processing rates in CORDIC environments. A simple update law allows a simpler implementation than previously proposed designs. Rather than minimizing an output error cost function, the algorithm is designed to achieve a stable associated differential equation, resulting in a globally convergent unbiased frequency estimator in the single sinusoid case, independent of the notch filter bandwidth. Using a second-order structure in the multiple sinusoid case, unbiased estimation of one of the input frequencies is achieved by thinning the notch bandwidth. The tracking behavior is superior to conventional output error designs, and the estimation of extremal frequencies is less prone to overflow instability than previously reported structures. >

On the behavior of information theoretic criteria for model order selection

The Akaike (1974) information criterion (AIC) and the minimum description length (MDL) are two well-known criteria for model order selection in the additive white noise case. Our aim is to study the influence on their behavior of a large gap between the signal and the noise eigenvalues and of the noise eigenvalue dispersion. Our results are mostly qualitative and serve to explain the behavior of the AIC and the MDL in some cases of great practical importance. We show that when the noise eigenvalues are not clustered sufficiently closely, then the AIC and the MDL may lead to overmodeling by ignoring an arbitrarily large gap between the signal and the noise eigenvalues. For fixed number of data samples, overmodeling becomes more likely for increasing the dispersion of the noise eigenvalues. For fixed dispersion, overmodeling becomes more likely for increasing the number of data samples. Undermodeling may happen in the cases where the signal and the noise eigenvalues are not well separated and the noise eigenvalues are clustered sufficiently closely. We illustrate our results by using simulations from the effective channel order determination area.

Kronecker products, unitary matrices, and signal processing applications

Discrete unitary transforms are extensively used in many signal processing applications, and in the development of fast algorithms Kronecker products have proved quite useful. In this semitutorial paper, we briefly review properties of Kronecker products and direct sums of matrices, which provide a compact notation in treating patterned matrices. A generalized matrix product, which inherits some useful algebraic properties from the standard Kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula, is then introduced. The notation is intimately related to sparse matrix factorizations, and examples are included illustrating the utility of the new notation in signal processing applications. Finally, some novel characteristics of Hadamard transforms and polyadic permutations are derived in the framework of Kronecker products.

On the duality between fast QR methods and lattice methods in least squares adaptive filtering

The authors show that fast QR methods and lattice methods in least squares adaptive filtering are duals and follow from identical geometric principles. Whereas the lattice methods compute the residuals of a projection operation via the forward and backward prediction errors, the QR methods compute instead the weights used in the projections. Within this framework, the parameter identification problem is solved using fast QR methods by showing that the reflection coefficients and tap parameters of a least squares lattice filter operating in the joint process mode are immediately available as internal variables in the fast QR algorithms. This parameter set can be readily exploited in system identification, signal analysis, and linear predictive coding, for example. The relations derived also lead to a fast least squares algorithm of minimal complexity that is a hybrid between a QR and a lattice algorithm. The algorithm combines the order recursive properties of the lattice approach with the robust numerical behavior of the QR approach. >

Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification

Szegos (1984) theorem states that the asymptotic behavior of the eigenvalues of a Hermitian Toeplitz matrix is linked to the Fourier transform of its entries. This result was later extended to block Toeplitz matrices, i.e., covariance matrices of multivariate stationary processes. The present work gives a new proof of Szegos theorem applied to block Toeplitz matrices. We focus on a particular class of Toeplitz matrices, those corresponding to covariance matrices of single-input multiple-output (SIMO) channels. They satisfy some factorization properties that lead to a simpler form of Szegos theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses. This bound is interpreted in the context of blind channel identification using second-order algorithms, and more particularly in the case of band-limited channels.

Stable and efficient lattice algorithms for adaptive IIR filtering

Previous attempts at applying lattice structures to adaptive infinite-impulse-response (IIR) filtering have met with gradient computations of O(N/sup 2/) complexity. To overcome this computational burden, two new lattice-based algorithms are proposed for adaptive IIR filtering and system identification, with both algorithms of O(N) complexity. The first algorithm is a reinterpretation of the Steiglitz-McBride method (1965), while the second is a variation on the output error method. State space models are employed to make the derivations transparent, and the methods can be extended to other parameterizations if desired. The set of possible stationary points of the algorithms is shown to be consistent with the convergent points obtained from the direct-form versions of the Steiglitz-McBride and output error methods, whose properties are well studied. The derived algorithms are as computationally efficient as existing direct-form based algorithms, while overcoming the stability problems associated with time-varying direct-form filters. >

A blind multichannel identification algorithm robust to order overestimation

Active research in blind single input multiple output (SIMO) channel identification has led to a variety of second-order statistics-based algorithms, particularly the subspace (SS) and the linear prediction (LP) approaches. The SS algorithm shows good performance when the channel output is corrupted by noise and available for a finite time duration. However, its performance is subject to exact knowledge of the channel order, which is not guaranteed by current order detection techniques. On the other hand, the linear prediction algorithm is sensitive to observation noise, whereas its robustness to channel order overestimation is not always verified when the channel statistics are estimated. We propose a new second-order statistics-based blind channel identification algorithm that is truly robust to channel order overestimation, i.e., it is able to accurately estimate the channel impulse response from a finite number of noisy channel measurements when the assumed order is arbitrarily greater than the exact channel order. Another interesting feature is that the identification performance can be enhanced by increasing a certain smoothing factor. Moreover, the proposed algorithm proves to clearly outperform the LP algorithm. These facts are justified theoretically and verified through simulations.

Tunable digital frequency response equalization filters

Tunable digital frequency response equalization filters, which feature adjustable gain at specified frequencies while leaving the remainder of the spectrum unaffected, are advanced. The filter structure is such that the frequency response parameters are independently related to the multiplier coefficients, which permits simple frequency response adjustment by varying the coefficient values. The resulting structure exhibits low coefficient sensitivity characteristics.