A fractional delay filter is a device for bandlimited interpolation between samples. It finds applications in numerous fields of signal processing, including communications, array processing, speech processing, and music technology. We present a comprehensive review of FIR and allpass filter design techniques for bandlimited approximation of a fractional digital delay. Emphasis is on simple and efficient methods that are well suited for fast coefficient update or continuous control of the delay value. Various new approaches are proposed and several examples are provided to illustrate the performance of the methods. We also discuss the implementation complexity of the algorithms. We focus on four applications where fractional delay filters are needed: synchronization of digital modems, incommensurate sampling rate conversion, high-resolution pitch prediction, and sound synthesis of musical instruments.

Splitting the unit delay [FIR/all pass filters design]

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.

https://www.pnas.org/content/pnas/102/40/14150.full.pdf

Nonlinear system theory: Another look at dependence

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described. >

Adaptive polynomial filters

As a robust nonlinear similarity measure in kernel space, correntropy has received increasing attention in domains of machine learning and signal processing. In particular, the maximum correntropy criterion (MCC) has recently been successfully applied in robust regression and filtering. The default kernel function in correntropy is the Gaussian kernel, which is, of course, not always the best choice. In this paper, we propose a generalized correntropy that adopts the generalized Gaussian density (GGD) function as the kernel, and present some important properties. We further propose the generalized maximum correntropy criterion (GMCC) and apply it to adaptive filtering. An adaptive algorithm, called the GMCC algorithm, is derived, and the stability problem and steady-state performance are studied. We show that the proposed algorithm is very stable and can achieve zero probability of divergence (POD). Simulation results confirm the theoretical expectations and demonstrate the desirable performance of the new algorithm.

Generalized Correntropy for Robust Adaptive Filtering

A family of nonlinear filters based on order statistics is presented. A mathematical tool derived through robust estimation theory, order statistics has allowed engineers to develop nonlinear filters with excellent robustness properties. These filters are well suited to digital image processing because they preserve the edges and the fine details of an image much better than conventional linear filters. The probabilistic and deterministic properties of the best known and most widely used filter in this family, the median filter, are discussed. In addition, the authors consider filters that, while not based on order statistics, are related to them through robust estimation theory. A table that ranks nonlinear filters under a variety of performance criteria is included. Most of the topics treated are very active research areas, and the applications are varied, including HDTV, multichannel signal processing of geophysical and ECG/EEG data, and a variety of telecommunications applications. >

/pdf/order-statistics-in-digital-image-processing-1rhdihn5hi.pdf

Order statistics in digital image processing

Volterra Series Expansions. Realization of Truncated Volterra Filters. Multidimensional Volterra Filters. Parameter Estimation. Frequency-Domain Methods for Volterra System Identification. Adaptive Truncated Volterra Filters. Recursive Polynomial Systems. Inversion and Time Series Analysis. Applications of Polynomial Filters. Some Related Topics and Recent Developments. Appendices. References. Index.

Polynomial Signal Processing

1. "Analysis and Optimization of Weighted Order Statistics and Stack Filters" S. Siren, P. Kuosmainen, K. Egiazarian, M. Gabbouj and J. Astola 2. "Image Enhancement and Analysis with Weighted Medians" G. Arce and J. L. Paredes 3. "Spatial-Rank Order Selection Filters" R. Hardie and K. Barner 4. "Signal-Dependent Rank-Ordered Mean (SD-ROM) Filter" E. Abreu 5. "Nonlinear Mean Filters and Their Applications in Image Filtering and Edge Detection" M. Pappas and I. Pitas 6. "Two-Dimensional Teager Filters" S. Thurnhofer 7. "Polynomial and Rational Operators for Image Processing and Analysis" G. Ramponi 8. "Nonlinear Partial Differential Equations in Image Processing" G. Sapiro 9. "Region-Based Filtering of Images and Video Sequences: a Morphological Viewpoint" P. Salembier 10. "Differential Morphology" P. Maragos 11. "Coordinate Logic Filters" B.G. Mertzios and K. Tsirikolias 12. "Nonlinear Filters Based on Fuzzy Models" F. Russo 13. "Digital Halftoning" G. Arce and D. Lau 14. "Intrinsic Dimensionality: Nonlinear Image Operators and Higher Order Statistics" C. Zetzsche and G. Krieger

Nonlinear image processing

In this paper, we propose an extension of the well-known FLANN filter using trigonometric expansions to include suitable cross-terms, i.e., products of input samples with different time shifts. It is shown that in some applications of nonlinear active noise control, the generalized FLANN filter can offer performance as good as that of high-order Volterra filters with a reduced complexity.

A Generalized FLANN Filter for Nonlinear Active Noise Control

Fourier nonlinear filters

In this paper, a bounded-input bounded-output (BIBO) stability condition for the recursive functional link artificial neural network (FLANN) filter, based on trigonometric expansions, is derived. This filter is considered as a member of the class of causal shift-invariant recursive nonlinear filters whose output depends linearly on the filter coefficients. As for all recursive filters, its stability should be granted or, at least, tested. The relevant conclusion we derive from the stability condition is that the recursive FLANN filter is not affected by instabilities whenever the recursive linear part of the filter is stable. This fact is in contrast with the case of recursive polynomial filters where, in general, specific limitations on the input range are required. The recursive FLANN filter is then studied in the framework of a feedforward scheme for nonlinear active noise control. The novelty of our study is due to the simultaneous consideration of a nonlinear secondary path and an acoustical feedback between the loudspeaker and the reference microphone. An output error nonlinearly Filtered-U normalized LMS adaptation algorithm, derived for the elements of the above-mentioned class of nonlinear filters, is then applied to the recursive FLANN filter. Computer simulations show that the recursive FLANN filter, in contrast to other filters, is able to simultaneously deal with the acoustical feedback and the nonlinearity in the secondary path.

Giovanni L. Sicuranza

Papers

Polynomial Signal Processing

Nonlinear image processing

A Generalized FLANN Filter for Nonlinear Active Noise Control

Fourier nonlinear filters

On the BIBO Stability Condition of Adaptive Recursive FLANN Filters With Application to Nonlinear Active Noise Control