The basic concepts and building blocks in multirate digital signal processing (DSP), including the digital polyphase representation, are reviewed. Recent progress, as reported by several authors in this area, is discussed. Several applications are described, including subband coding of waveforms, voice privacy systems, integral and fractional sampling rate conversion (such as in digital audio), digital crossover networks, and multirate coding of narrowband filter coefficients. The M-band quadrature mirror filter (QMF) bank is discussed in considerable detail, including an analysis of various errors and imperfections. Recent techniques for perfect signal reconstruction in such systems are reviewed. The connection between QMF banks and other related topics, such as block digital filtering and periodically time-varying systems, is examined in a pseudo-circulant-matrix framework. Unconventional applications of the polyphase concept are discussed. >

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Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial

Perfect reconstruction finite impulse-response (FIR) filter banks are analyzed both in the z-transform and time domains, showing the alternatives between designs in the two domains. Various classes of perfect reconstruction schemes are indicated, and relations between previously known systems are given. Windowed modulated filter banks with low computational complexity and perfect reconstruction are shown. New factorizations of polyphase filter matrices leading in particular to linear-phase filters, are given. The computational complexity and the architecture of the new structures are indicated. >

/pdf/perfect-reconstruction-fir-filter-banks-some-properties-and-1318075m4q.pdf

Perfect reconstruction FIR filter banks: some properties and factorizations

A general analysis of multidimensional multirate filter banks is presented. The approach is applicable to discrete signal spaces of any dimension, to multirate systems based on arbitrary downsampling and upsampling lattices, and for filter banks with any number of channels. A new numerical design procedure is also presented for multidimensional multirate perfect reconstruction filter banks, which is based on methods of nonlinearly constrained numerical optimization. An error function that depends only on the analysis filter impulse response coefficients is minimized, subject to a set of quadratic equality constraints that involve both the analysis and synthesis filter coefficients. With this design framework, it is possible to design a wide variety of filter banks that have a number of desirable properties. The analysis and synthesis filters that result are finite impulse response (FIR) and of equal size. In addition, both paraunitary and nonparaunitary filter banks can be designed with this method. Unlike paraunitary filter banks, nonparaunitary filter banks are capable of performing analysis bank functions more general than band-splitting with flat passband filters. >

The analysis and design of multidimensional FIR perfect reconstruction filter banks for arbitrary sampling lattices

The theory and design of multidimensional perfect reconstruction filter banks are discussed. A necessary and sufficient condition for perfect reconstruction in multidimensional filter banks is derived, and an approach to filter-bank design using cascaded polyphase matrices and a statistical filter design method is presented. The approach is applicable to arbitrary (not necessarily rectangular) decimation and interpolation schemes and any number of channels. The results of a preliminary numerical design example are presented. >

https://ieeexplore.ieee.org/iel2/706/573/00015053.pdf

Design of perfect reconstruction multi-dimensional filter banks using cascaded Smith form matrices

This chapter develops the theory of multirate signal processing as used in subband coding systems. Multirate operations are reviewed, multirate filter banks are analyzed and structures and design methods for perfect reconstruction filter banks are indicated. Special attention is given to the multidimensional case. We do not consider how to actually code the subband signals, and it will be assumed that the subband signals are perfectly transmitted.

Multirate Filter Banks for Subband Coding

The concept of perfect reconstruction in filter banks is examined using the Smith form of the polyphase matrix. A class of M-channel FIR (finite-impulse response) perfect-reconstruction filter banks is introduced that includes the recently developed lossless filter banks. With the proposed filter banks, the synthesis filters are of the same complexity as the analysis filters. In addition, the synthesis filter bank is easily obtained from the analysis filter bank by inverting a set of M*M constant coefficient matrices. A statistical method for the design of such filter banks is presented. These can be used as building blocks in large tree-structured filter banks where the overall number of channels is any composite integer. An analysis of the computational and storage complexity of such tree structures is given. The results of the analysis are simple formulas for required storage and computation, which helps in the selection of efficient tree structures. >

The design of tree-structured M-channel filter banks using perfect reconstruction filter blocks

The design of multirate bandpass filters requires a search for the optimum multistage structure. The size of the search space grows exponentially as the filter bandwidth decreases, and excessive computer time is required to search it exhaustively. A fast branch-and-bound algorithm for searching this space is presented. In our tests, this algorithm decreases search time by an average factor which increases from 7.7 to 23.0 as the filter passband width decreases from 8 × 10-4to 3.2 × 10-6. It is also shown that when coefficient storage is the cost criterion, the actual filter structure resembles an unrealizable optimum filter structure for small bandwidth filters.

E. Viscito

Papers

The design of tree-structured M-channel filter banks using perfect reconstruction filter blocks

A fast algorithm for the design of narrow-band multirate digital filters