There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Digital processing of speech signals

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]

Wave digital filters (WDFs) are modeled after classical filters, preferably in lattice or ladder configurations or generalizations thereof. They have very good properties concerning coefficient accuracy requirements, dynamic range, and especially all aspects of stability under finite-arithmetic conditions. A detailed review of WDF theory is given. For this several goals are set: to offer an introduction for those not familiar with the subject, to stress practical aspects in order to serve as a guide for those wanting to design or apply WDFs, and to give insight into the broad range of aspects of WDF theory and its many relationships with other areas, especially in the signal-processing field. Correspondingly, mathematical analyses are included only if necessary for gaining essential insight, while for all details of more special nature reference is made to existing literature.

Wave digital filters: Theory and practice

The design of FIR digital filters with a complex-valued desired frequency response using the Chebyshev error is investigated. The complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem. A standard linear programming algorithm for the Chebyshev solution of overdetermined equations is used to solve the real approximation problem. Additional constraints are introduced which allow weighting of the phase and/or group delay of the approximation. Digital filters are designed which have nearly constant group delay in the passbands. The desired constant group delay which gives the minimum Chebyshev error is found to be smaller than that of a linear phase filter of the same length. These filters, in addition to having a smaller, approximately constant group delay, have better magnitude characteristics than exactly linear phase filters with the same length. The filters have nearly equiripple magnitude and group delay.

Design of FIR filters in the complex domain

This simplified introduction to wavelets starts with a short historical background. The continuous wavelet transform is presented. The multiresolution concept is concisely described. It is followed by the filter banks method for wavelet analysis and reconstruction. Wavelet packets are briefly presented. Some typical applications are mentioned.

A short introduction to wavelets and their applications

This paper proposes a method of obtaining a two-dimensional wave digital filter of the recursive type from a doubly terminated LC-ladder network in two variables by replacing each series or shunt arm element of the ladder by its equivalent digital two-port. A number of realizations of the wave digital two-ports, which are canonic in multipliers, have been obtained. An example of a circularly symmetric low-pass two-dimensional digital filter is considered using these realizations. The sensitivity of this filter with respect to the multiplier coefficient changes due to finite word length is compared with that of the direct realization. It is found that the wave digital filter appears to be a more desirable form of implementation than the conventional cascade form.

Two-dimensional wave digital filters using doubly terminated two-variable LC-ladder configurations

Based on Schussler's theorem, some new properties of polynomials containing zeros inside the unit circle are obtained. These properties give rise to (i) a new stability test of 1-D discrete systems, and (ii) some necessary coefficient conditions that have to be satisfied by the denominator polynomial of a stable 1-D discrete system.

Implementation of a stability test of 1-D discrete system based on Schussler's theorem and some consequent coefficient conditions

In this paper a closed-form relationship between the original coefficients of a 2-D analog transfer function and the coefficients of the transformed function after application of the double bilinear transformation in an alternative manner is given The extension to the n -dimensional case, as well as the 1 -dimensional case are presented Examples are given to illustrate the usefulness of the derived relationship

On the algebra of multiple bilinear transformations

In this paper, based on a new stability test, two 1-D polynomials in z 1 and z 2 having all their zeros inside the unit circles in z 1 and z 2 planes are generated and assigned to the denominator of a separable denominator 2-D transfer function while the numerator is left to a 2-D nonseparable polynomial in z 1 and z 2 . Now any suitable nonlinear optimization technique can be used to calculate the parameters of the 2-D filter in such a way so as to minimize the general mean square error between the ideal and the designed magnitude and group delay responses of the 2-D filter subject to the coefficient constraints given by the new stability test. To illustrate the method, an example is given.

Design of 2-D recursive digital filters with constant group delay characteristics using separable denominator transfer function and a new stability test

V. Ramachandran

Papers

A short introduction to wavelets and their applications

Two-dimensional wave digital filters using doubly terminated two-variable LC-ladder configurations

Implementation of a stability test of 1-D discrete system based on Schussler's theorem and some consequent coefficient conditions

On the algebra of multiple bilinear transformations

Design of 2-D recursive digital filters with constant group delay characteristics using separable denominator transfer function and a new stability test