Direktna sinteza digitalnih filtara za podopsežno kodovanje

In this PhD thesis two-channel QMF IIR filter banks have been discussed. As a part of this research, direct synthesis of a new IIR filter class, as a lowpass protoype function which satisfies condition for complementary decomposition, has been proposed. Since attenuation characteristic of the new filter class lies in the area between the Butterworth’s and the Elliptic’s attenuation characteristics of the same order, new class has been referred to as Transitional Butter-Elliptic filters. Transfer function of the the lowpass prototype function was separated into two allpass filters by complementary decomposition. Then, lowpass and highpass filters, as power symmetric filter pair, are obtained using those allpass filters. The main advantage of complementary IIR filters based on parallel connection of two allpass filters, is its smaller sensitivity to digital word length change compared to standard filters and low-cost implementation since the same hardware is used for the lowpass and the highpass transfer function. Within the analysis of the new filter class, transfer functions of the odd order with one, two, three and four extremal values of attenuation characteristics in the passband and in the stopband, has been discussed. All types of Transitional filters have been compared with the Butterworth’s and the Elliptic’s filters of the same order. It is shown that in comparison with Butterworth filter, the Transitional filter has better selectivity but lower selectivity than Elliptic filter. However, the proposed new filter class has pure imaginary poles and accuracy in complementary decomposition is at very high level, which the Elliptic filter cannot achieve. Also, the practical realization of the new filter class is easier than for the Elliptic filter since the new class has conjugate-complex zeros on the unit circle and one multiple zero at z = −1. Further, poles and zeros of the Elliptic filter are accumulated near the passband edge and the distance between them decreases as the filter order increases. This requires a very long digital word length for high-order filter realization. The new class filter has approximately equidistant arrangement of poles so it can be realized with shorter digital word length than it is required for the Elliptic filter realization. For Transitional filters class with one extremum of multiplicity L in the passband and in the stopband, closed-form expression for the transfer function coefficients has been obtained. Application of these kind of filters is significant since for sample rate conversion the lowestcomplexity filter is required. With appropriate selection of filters in analysis and synthesis section, the conditions for ii removing the aliasing effects are fulfilled. Magnitude distortions in IIR filter bank can also be completely eliminated and therefore output signal of the QMF bank has only phase distortion. In order to obtain nearly perfect reconstruction of the output signal, the allpass equalizer has been proposed for group delay compensation. Also, effect of the equalizer order on the accuracy of signal reconstruction has been analyzed

Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z)(for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined

A new approach to the realization of low-sensitivity IIR digital filters

A new implementation of an IIR digital filter transfer function is presented that is structurally passive and, hence, has extremely low pass-band sensitivity. The structure is based on a simple parallel interconnection of two all-pass sections, with each section implemented in a structurally lossless manner. The structure shares a number of properties in common with wave lattice digital filters. Computer simulation results verifying the low-sensitivity feature are included, along with results on roundoff noise/dynamic range interaction. A large number of alternatives is available for the implementation of the all-pass sections, giving rise to the well-known wave lattice digital filters as a specific instance of the implementation

Tree-structured complementary filter banks using all-pass sections

Tree-structured complementary filter banks are developed with transfer functions that are simultaneously all-pass complementary and power complementary. Using a formulation based on unitary transforms and all-pass functions, we obtain analysis and synthesis filter banks which are related through a transposition operation, such that the cascade of analysis and synthesis filter banks achieves an all-pass function. The simplest structure is obtained using a Hadamard transform, which is shown to correspond to a binary tree structure. Tree structures can be generated for a variety of other unitary transforms as well. In addition, given a tree-structured filter bank where the number of bands is a power of two, simple methods are developed to generate complementary filter banks with an arbitrary number of channels, which retain the transpose relationship between analysis and synthesis banks, and allow for any combination of bandwidths. The structural properties of the filter banks are illustrated with design examples, and multirate applications are outlined

Design of nonuniform near allpass complementary FIR filters via a semi-infinite programming technique

In this paper, we consider the problem of designing a set of nonuniform near allpass complementary FIR filters. This problem can be formulated as a quadratic semi-infinite programming problem, where the objective is to minimize the sum of the ripple energy for the individual filters, subject to the passband and stopband specifications as well as to the allpass complementary specification. The dual parameterization method is used for solving the linear quadratic semi-infinite programming problem

Theory and design of uniform DFT, parallel, quadrature mirror filter banks

In this paper, the theory of uniform DFT, parallel, quadrature mirror filter (QMF) banks is developed. The QMF equations, i.e., equations that need to be satisfied for exact reconstruction of the input signal, are derived. The concept of decimated filters is introduced, and structures for both analysis and synthesis banks are derived using this concept. The QMF equations, as well as closed-form expressions for the synthesis filters needed for exact reconstruction of the input signalx(n), are also derived using this concept. In general, the reconstructed. signalhat{x}(n)suffers from three errors: aliasing, amplitude distortion, and phase distortion. Conditions for exact reconstruction (i.e., all three distortions are zero, andhat{x}(n)is equal to a delayed version ofx(n))of the input signal are derived in terms of the decimated filters. Aliasing distortion can always be completely canceled. Once aliasing is canceled, it is possible to completely eliminate amplitude distortion (if suitable IIR filters are employed) and completely eliminate phase distortion (if suitable FIR filters are employed). However, complete elimination of all three errors is possible only with some simple, pathalogical stable filter transfer functions. In general, once aliasing is canceled, the other distortions can be minimized rather than completely eliminated. Algorithms for this are presented. The properties of FIR filter banks are then investigated. Several aspects of IIR filter banks are also studied using the same framework

The role of lossless systems in modern digital signal processing: a tutorial

A self-contained discussion of discrete-time lossless systems and their properties and relevance in digital signal processing is presented. The basic concept of losslessness is introduced, and several algebraic properties of lossless systems are studied. An understanding of these properties is crucial in order to exploit the rich usefulness of lossless systems in digital signal processing. Since lossless systems typically have many input and output terminals, a brief review of multiinput multioutput systems is included. The most general form of a rational lossless transfer matrix is presented along with synthesis procedures for the FIR (finite impulse response) case. Some applications of lossless systems in signal processing are presented

On power-complementary FIR filters

Conditions are derived, under which two linear-phase FIR filter transfer functions H(z)and G(z) have the power-complementary property, i.e., |H(e^{j omega})|^{2} + |G(e^{jomega})|^{2} = 1. It is shown that, the constraint of linear phase on the transfer functions strongly restricts the class of frequency responses that can be realized by a power-complementary pair

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

Digital filter design using root moments for sum-of-all-pass structures from complete and partial specifications

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