The perfect-reconstruction QMF bank: New architectures, solutions, and optimization strategies

n this paper, a scheme for perfect reconstruction in M channel, maximally decimated QMF banks is first presented, for arbitrary M. The solutions are such that the analysis and synthesis filters are FIR and of the same length. Based on the theory, lattice structures for the two-channel case are derived, which offer an efficient design as well as implementation procedure for two-channel perfect reconstruction systems. Such lattice implementations are robust in the sense that the perfect-reconstruction property is preserved in spite of coefficient quantization

A Spectral Factorization Approach to Pseudo-QMF Design

A new approach to the design of M-channel pseudoquadrature-mirror-filter (QMF) banks is presented. In this approach, the prototype filter is obtained as a spectral factor of a 2Mth band filter. This completely eliminates the need for optimization whereas in conventional pseudo-QMF designs, the main computational effort is in optimization of the prototype. As in the conventional approach, the aliasing cancellation (AC) constraint ensures that all the significant aliasing terms are canceled. The overall transfer function T(z) of the analysis/synthesis system has a linear phase and an approximately “flat” magnitude response in the frequency region ε ≤ ω ≤ (π - ε), where ε depends on the transition bandwidth of the prototype and 0 < ε < (π/2M). Three design examples are included

Polyphase networks, block digital filtering, LPTV systems, and alias-free QMF banks: a unified approach based on pseudocirculants

The relationship between block digital filtering and quadrature mirror filter (QMF) banks is explored. Necessary and sufficient conditions for alias cancellation in QMF banks are expressed in terms of an associated matrix, derived from the polyphase components of the analysis and synthesis filters. These conditions, called the pseudocirculant conditions, make it possible to unite QMF banks with the framework of block digital filtering directly. Absence of amplitude distortion in an alias-free QMF bank translates into the 'losslessness' property of the pseudocirculant matrix involved

Alias-free, real coefficient m-band QMF banks for arbitrary m

Based on a generalized framework for alias free QMF banks, a theory is developed for the design of uniform QMF banks with real-coefficient analysis filters, such that aliasing can be completely canceled by appropriate choice of real-coefficient synthesis filters. These results are then applied for the derivation of closed-form expressions for the synthesis filters (both FIR and IIR), that ensure cancelation of aliasing for a given set of analysis filters. The results do not involve the inversion of the alias-component (AC) matrix

Linear phase cosine modulated maximally decimated filter banks with perfect reconstruction

We propose a novel way to design maximally decimated FIR cosine modulated filter banks, in which each analysis and synthesis filter has a linear phase. The system can be designed to have either the approximate reconstruction property (pseudo-QMF system) or perfect reconstruction property (PR system). In the PR case, the system is a paraunitary filter bank. As in earlier work on cosine modulated systems, all the analysis filters come from an FIR prototype filter. However, unlike in any of the previous designs, all but two of the analysis filters have a total bandwidth of 2π/M rather than π/M (where 2M is the number of channels in our notation). A simple interpretation is possible in terms of the complex (hypothetical) analytic signal corresponding to each bandpass subband. The coding gain of the new system is comparable with that of a traditional M-channel system (rather than a 2M-channel system). This is primarily because there are typically two bandpass filters with the same passband support. Correspondingly, the cost of the system (in terms of complexity of implementation) is also comparable with that of an M-channel system. We also demonstrate that very good attenuation characteristics can be obtained with the new system

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

Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks

A lattice structure and an algorithm are presented for the design of two-channel QMF (quadrature mirror filter) banks, satisfying a sufficient condition for perfect reconstruction. The structure inherently has the perfect-reconstruction property, while the algorithm ensures a good stopband attenuation for each of the analysis filters. Implementations of such lattice structures are robust in the sense that the perfect-reconstruction property is preserved in spite of coefficient quantization. The lattice structure has the hierarchical property that a higher order perfect-reconstruction QMF bank can be obtained from a lower order perfect-reconstruction QMF bank, simply by adding more lattice sections. Several numerical examples are provided in the form of design tables

Classical sampling theorems in the context of multirate and polyphase digital filter bank structures

The recovery of a signal from so-called generalized samples is a problem of designing appropriate linear filters called reconstruction (or synthesis) filters. This relationship is reviewed and explored. Novel theorems for the subsampling of sequences are derived by direct use of the digital-filter-bank framework. These results are related to the theory of perfect reconstruction in maximally decimated digital-filter-bank systems. One of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error. The reconstruction filters turn out to be multiplierless and of the FIR (finite impulse response) type. These ideas are extended to the case of two-dimensional signals by use of a Kronecker formalism. The subsampling of bandlimited sequences is also considered. A sequence x(n ) with a Fourier transform vanishes for |ω|&ges;Lπ/M, where L and M are integers with L<M, can in principle be represented by reducing the data rate by the amount M/L. The digital polyphase framework is used as a convenient tool for the derivation as well as mechanization of the sampling theorem

Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property

Based on the concept of losslessness in digital filter structures, this paper derives a general class of maximally decimated M-channel quadrature mirror filter banks that lead to perfect reconstruction. The perfect-reconstruction property guarantees that the reconstructed signalhat{x} (n)is a delayed version of the input signal x (n), i.e.,hat{x} (n) = x (n - n_{0}). It is shown that such a property can be satisfied if the alias component matrix (AC matrix for short) is unitary on the unit circle of the z plane. The number of channels M is arbitrary, and when M is two, the results reduce to certain recently reported 2-channel perfect-reconstruction QMF structures. A procedure, based on recently reported FIR cascaded-lattice structures, is presented for optimal design of such FIR M-channel filter banks. Design examples are included