Two Classes of Cosine-Modulated IIR/IIR and IIR/FIR NPR Filter Banks

This paper introduces two classes of cosine-modulated causal and stable filter banks (FBs) with near perfect reconstruction (NPR) and low implementation complexity. Both classes have the same infinite-length impulse response (IIR) analysis FB but different synthesis FBs utilizing IIR and finite-length impulse response (FIR) filters, respectively. The two classes are preferable for different types of specifications. The IIR/FIR FBs are preferred if small phase errors relative to the magnitude error are desired, and vice versa. The paper provides systematic design procedures so that PR can be approximated as closely as desired. It is demonstrated through several examples that the proposed FB classes, depending on the specification, can have a lower implementation complexity compared to existing FIR and IIR cosine-modulated FBs (CMFBs). The price to pay for the reduced complexity is generally an increased delay. Furthermore, two additional attractive features of the proposed FBs are that they are asymmetric in the sense that one of the analysis and synthesis banks has a lower computational complexity compared to the other, which can be beneficial in some applications, and that the number of distinct coefficients is small, which facilitates the design of FBs with large numbers of channels.     

Two-Channel FIR Filter Banks Utilizing the FRM Approach

The frequency-response masking (FRM) approach has been introduced as a means of generating narrow transition band linear-phase finite impulse response (FIR) filters with a low arithmetic complexity. This paper proposes an approach for synthesizing two-channel maximally decimated FIR filter banks utilizing the FRM technique. For this purpose, a new class of FRM filters is introduced. Filters belonging to this class are used for synthesizing nonlinear-phase analysis and synthesis filters for two types of two-channel filter banks. For the first type, there exist no phase distortion and aliasing errors, but this type suffers from a small amplitude distortion as for the well-known quadrature mirror filter (QMF) banks. Compared to conventional QMF filter banks, the proposed banks lower significantly the overall arithmetic complexity at the expense of a somewhat increased overall filter bank delay in applications demanding narrow transition bands. For the second type, there are also small aliasing errors, allowing one to reduce the arithmetic complexity even further. Efficient structures are introduced for implementing the proposed filter banks, and algorithms are described for maximizing the stopband attenuations of the analysis and synthesis filters in the minimax sense subject to the given allowable amplitude and/or aliasing errors. Examples are included illustrating the benefits provided by the proposed filter banks.     

Design of M‐Channel IIR Uniform DFT Filter Banks Using Recursive Digital Filters

In this paper, we propose a method for designing a class of M‐channel, causal, stable, perfect reconstruction, infinite impulse response (IIR), and parallel uniform discrete Fourier transform (DFT) filter banks. It is based on a previously proposed structure by Martinez et al. [1] for IIR digital filter design for sampling rate reduction. The proposed filter bank has a modular structure and is therefore very well suited for VLSI implementation. Moreover, the current structure is more efficient in terms of computational complexity than the most general IIR DFT filter bank, and this results in a reduced computational complexity by more than 50% in both the critically sampled and oversampled cases. In the polyphase oversampled DFT filter bank case, we get flexible stop‐band attenuation, which is also taken care of in the proposed algorithm.     

Design and Factorization of Two-Channel Perfect Reconstruction Filter Banks with Causal-Stable IIR Filters

In this paper, new design and factorization methods of two-channel perfect reconstruction (PR) filter banks (FBs) with casual-stable IIR filters are introduced. The polyphase components of the analysis filters are assumed to have an identical denominator in order to simplify the PR condition. A modified model reduction is employed to derive a nearly PR causal-stable IIR FB as the initial guess to obtain a PR IIR FB from a PR FIR FB. To obtain high quality PR FIR FBs for carrying out model reduction, cosine-rolloff FIR filters are used as the initial guess to a nonlinear optimization software for solving to the PR solution. A factorization based on the lifting scheme is proposed to convert the IIR FB so obtained to a structurally PR system. The arithmetic complexity of this FB, after factorization, can be reduced asymptotically by a factor of two. Multiplier-less IIR FB can be obtained by replacing the lifting coefficients with the canonical signal digitals (CSD) or sum of powers of two (SOPOT) coefficients.     

Technique for design of two-channel approximately linear phase QMFfilter bank and its application to image compression

The two-channel QMF filter bank based on allpass sections is one of the best known circuits for building up a multi-channel filter bank for signal compression. An analysis-synthesis combination can satisfy two of the three perfect reconstruction (PR) conditions. The third, the phase condition, can be met to any desired accuracy. However, when applied to images, PR is possible because non-causality can be allowed in the synthesis bank. The development of explicit formulae for the coefficients of such filters is considered. The resulting QMF designs can be used in a wavelet structure for image compression problems using rectangular or diamond symmetry. Several design examples are given, including a comparison of performance with FIR filter banks for such problems     

Some results in the theory of crosstalk-free transmultiplexers

The crosstalk-free transmultiplexer (CF-TMUX) focuses on crosstalk cancellation (CC) rather than on suppressing it. The authors present an analysis of the CF-TMUX based on the polyphase component matrices of the filter banks used in TDM→FDM and FDM→TDM conversions, respectively. Thus a necessary and sufficient condition for complete CC is obtained. It is shown that the filters for a CF-TMUX are the same as the filters for a 1-skewed alias free QMF bank. In addition, if the QMF bank satisfies the perfect reconstruction (PR) property, then the TMUX also satisfies PR. The relation between CF-TMUX filters and alias-free QMF banks is used to obtain a direct design procedure for CF-TMUX filters (both FIR and IIR). It is also shown that approximately crosstalk-free TMUX filters can be obtained from any approximately alias-free QMC bank. Design examples and comparison tables are included     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

Minimax design of two-channel nonuniform-division filterbanks using IIR allpass filters

The design of two-channel linear-phase nonuniform-division filter (NDF) banks constructed by infinite impulse response (IIR) digital allpass filters (DAFs) in the sense of L/sub /spl infin// error criteria is considered. First, the theory of two-channel NDF bank structures using two IIR DAFs is developed. Then, the design problem is appropriately formulated to result in a simple optimization problem. Utilizing a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the IIR DAFs. The resulting two-channel NDF banks can possess approximately linear-phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations. Several simulation examples are presented for illustration and comparison.     

Rational sampling filter banks based on IIR filters

The problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response (IIR) filter banks is addressed. Near perfect reconstruction (NPR) is considered. The method uses the modulation of different IIR prototypes. The cancellation of the main aliasing components constrains the prototypes to be dependent on each other. By using this approach, linear-phase prototypes are needed, and noncausal filtering is required. Numerical examples of filter bank design are given, and the computational complexity is compared with the finite impulse response (FIR) case     

Designs and architectures of filter bank trees for spectrally efficient multi-user communications: review, modifications and extensions of wavelet packet filter bank trees

A review is presented first of the evolution of transmultiplexers since about 1966, in the context of a long progression of theoretical advances and developments leading to recent proposals to fundamentally improve OFDM type systems using principles of perfect reconstruction filter (PRF) banks. The equivalence of transmultiplexers to OFDM type multi-user systems is discussed. The desirable goals for performance and implementation of transmultiplexers or multiband, multiuser communication systems that are addressed and met in this paper using filter bank trees are set down. Then modifications and extensions are presented of the designs and architectures of wavelet packet based synthesis and analysis pairs of filter bank trees (Sablatash and Lodge in Digital Signal Process 13: 58–92, 2003) that can be used as transmultiplexers. These exhibit a number of advantages over the previous designs and address three shortcomings of the designs used to illustrate basic principles in Sablatash and Lodge (Digital Signal Process 13:58–92, 2003). The first of these is the asymmetry of the magnitude frequency responses of the multiplexer channels, which is addressed using a symmetric design for a lowpass and highpass quadrature mirror filter (QMF) pair described herein. The second is the problem of minimizing the total delay of the signal in passing through the analysis and synthesis filter banks. This is addressed using an architecture involving DFT polyphase synthesis filter banks to replace the wideband VSB filters at the roots of the two identical synthesis filter bank trees, but results in the multiplexer having fewer levels. In this way a tradeoff is effected of lower delay and complexity with fewer levels of bandwidth on demand. At the receiver matching DFT polyphase analysis filters and the other matching analysis filters are implemented. The third shortcoming is the difficulty in designing a synchronization scheme if the filters in the synthesis and analysis filter banks have non-linear phase. This is addressed by designing linear phase filters that do not affect the ISI to any significant degree for communication purposes, although exact perfect reconstruction is lost, but greatly ease and improve the design of the synchronization scheme. Relationships of this paper and its advantages over recent research studies and IEEE 802.22 standards proposals using PR filter banks for multi-user systems to greatly improve on OFDM systems are discussed. Financial support under Industry Canada’s Spectrum Research Funding is gratefully acknowledged.     

DESIGN OF POLYPHASE WAVE DIGITAL FILTERS WITH APPROXIMATELY LINEAR PHASE

It is well known that IIR digital filters require quite fewer computations,comparedwith FIR filters,in order to meet stringent magnitude specifications when the phase distortioncan be tolerated.An approximately linear phase,however,can be also obtained with the IIRfilter by making use of a technique without increasing the complexity.Based on a certain numberof attenuation zeros in the pass band,a new approach is developed for the design of polyphasewave digital filters with exact magnitude responses and Chebyshev approximation of the desiredphase responses.The minimum number of attenuation zeros is estimated,and some examples areincluded.     

Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks     

Analysis and design of periodically time-varying IIR filters, withapplications to transmultiplexing

Fundamental constraints on the form of infinite impulse response (IIR) periodically time-varying (PTV) filters are identified, and a design technique with well-defined error and stability characteristics based on those constraints is presented. The design technique is based on the selection of poles and zeros within the time-invariant filter banks of equivalent PTV filter analysis structures. A simple example is presented to illustrate the design method, which implements the IIR PTV as a time-invariant all-feedback IIR filter of the form 1/D(z^P) cascaded with an finite impulse response (FIR) PTV filter. An application of IIR PTV filters to telecommunications transmultiplexing is presented to illustrate the design method and for comparison to an existing PTV design method. The computational complexity of the resulting system compares favorably with that of existing transmultiplexers     

Lagrange multiplier approaches to the design of two-channelperfect-reconstruction linear-phase FIR filter banks

The authors present two approaches to the design of two-channel perfect-reconstruction linear-phase finite-impulse-response (FIR) filter banks. Both approaches analyze and design the impulse responses of the analysis filter bank directly. The synthesis filter bank is then obtained by simply changing the signs of odd-order coefficients in the analysis filter bank. The approach deals with unequal-length filter banks. By designing the lower length filters first, one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect-reconstruction constraint equations. The second approach generalizes the first, and covers the design for all parts of linear phase perfect reconstruction constraint equations     

Polyphase Conditions and Structures for 2-D Quincunx FIR Filter Banks Having Quadrantal or Diagonal Symmetries

In this brief, we derive conditions on the polyphase matrix of 2-D finite-impulse response (FIR) quincunx filter banks, for the filters in the filter bank to have quadrantal or diagonal symmetry. These conditions provide a framework for synthesizing polyphase structures which structurally enforce the symmetry. This is demonstrated by constructing examples of small parameterized matrix structures which satisfy the above conditions, thus giving perfect reconstruction FIR quincunx filter banks with quadrantal or diagonally symmetric short-kernel (i.e., short-support) filters. It is also shown that cascades of the above constructed small structures can be used to construct filters of higher order.     

Interpolated LIR Mth-band filter design with allpass subfilters

We propose a new allpass-based structure for the IIR Mth-and 2Mth-band filters. These filters consist of M allpass filters and an interpolation filter (sum of two allpasses). Consequently, the proposed structure is very efficient in implementation. By choosing the allpass phase appropriately, the resulting phase response of the IIR Mth-band filter becomes approximately linear. An example is designed and compared with FIR Mth-band filters     

Using fractional delay to control the magnitudes and phases of integrators and differentiators

The use of fractional delay to control the magnitudes and phases of integrators and differentiators has been addressed. Integrators and differentiators are the basic building blocks of many systems. Often applications in controls, wave-shaping, oscillators and communications require a constant 90deg phase for differentiators and -90deg phase for integrators. When the design neglects the phase, a phase equaliser is often needed to compensate for the phase error or a phase lock loop should be added. Applications to the first-order, Al-Alaoui integrator and differentiator are presented. A fractional delay is added to the integrator leading to an almost constant phase response of -90deg. Doubling the sampling rate improves the magnitude response. Combining the two actions improves both the magnitude and phase responses. The same approach is applied to the differentiator, with a fractional sample advance leading to an almost constant phase response of 90deg. The advance is, in fact, realised as the ratio of two delays. Filters approximating the fractional delay, the finite impulse response (FIR) Lagrange interpolator filters and the Thiran allpass infinite impulse response (IIR) filters are employed. Additionally, a new hybrid filter, a combination of the FIR Lagrange interpolator filter and the Thiran allpass IIR filter, is proposed. Methods to reduce the approximation error are discussed.     

General analysis of two-band QMF banks

The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions