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.     

Design of stable, causal, perfect reconstruction, IIR uniform DFTfilter banks

Design procedures for stable, causal and perfect reconstruction IIR parallel uniform DFT filter banks (DFT FBs) are presented. In particular a family of IIR prototype filters is a good candidate for DFT FB, where a tradeoff between frequency selectivity and numerical properties (as measured by the Weyl-Heisenberg frames theory) could be made. Some realizations exhibiting a simple and a massively parallel and modular processing structure making a VLSI implementation very suitable are shown. In addition, some multipliers in the filters (both the analysis and synthesis) could be made; powers or sum of powers of 2, in particular for feedback loops, resulting in a good sensitivity behavior. For these reasons as well as for the use of low order IIR filters (as compared with conventional FIR filters), the overall digital filter bank structure is efficient for high data rate applications. Some design examples are provided     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

Frame-theoretic analysis of oversampled filter banks

We provide a frame-theoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (FIR) uniform filter banks (FBs). Our analysis is based on a new relationship between the FBs polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FBs providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frame-theoretic procedure for the design of paraunitary FBs from given nonparaunitary FBs is formulated. We show that the frame bounds of an FB can be obtained by an eigen-analysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented     

Stable, causal, perfect reconstruction, IIR uniform DFT filterbanks

A simple design procedure for stable, causal and perfect reconstruction infinite impulse response parallel uniform discrete Fourier transform filter banks (DFT FBs) based on a new polyphase decomposition, the `polyphase-oversampled' FB, is presented. The proposed design results in causal and stable analysis and synthesis filters that are all derived from a single prototype filter, resulting in efficient realisations. A discussion of the FB numerical properties and some design examples are provided     

Frame-Theory-Based Analysis and Design of Oversampled Filter Banks: Direct Computational Method

This paper studies the frames corresponding to oversampled filter banks (FBs). For this class of FB frames, we present a state-space parameterization of perfect reconstruction FB frames and explicit and numerically efficient formulas to compute the tightest frame bounds, to obtain the dual FB frame, and to construct a tight (paraunitary) FB frame from a given untight (nonparaunitary) FB frame. The derivation uses well-developed techniques from modern control theory, which results in the unified formulas for generic infinite-impulse-response (IIR) and finite-impulse-response (FIR) FBs. These formulas involve only algebraic manipulations of real matrices and can be computed efficiently, reliably, and exactly without the approximation required in the existing methods for generic FBs. The results provide a unified framework for frame-theory-based analysis and systematic design of oversampled filter banks     

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     

A Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks

We address a problem to find optimal synthesis filters of oversampled uniform finite-impulse-response (FIR) filter banks (FBs) yielding perfect reconstruction (PR), when we are given an analysis FB, in the case where all the filters have the same length that is twice a factor of downsampling. We show that in this class of FBs, a synthesis FB that achieves PR can be found in closed form with elementary matrix operations, unlike conventional design methods with numerical optimization. This framework allows filter coefficients to be complex as well as real. Due to the extra degrees of freedom in a synthesis FB provided by oversampling, we can determine optimal coefficients of synthesis filters that meet certain criteria. We introduce in this paper two criteria: variance of additive noise and stopband attenuation. We show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear-phase FIR FBs and DFT-modulated FBs. Moreover, we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete Fourier transform with 50% overlapping.     

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     

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     

Low-Order IIR Filter Bank Design

The advantage of infinite-impulse response (IIR) filters over finite-impulse response (FIR) ones is that the former require a much lower order (much fewer multipliers and adders) to obtain the desired response specifications. However, in contrast with well-developed FIR filter bank design theory, there is no satisfactory methodology for IIR filter bank design. The well-known IIR filters are mostly derived by rather heuristic techniques, which work in only narrow design classes. The existing deterministic techniques usually lead to too high order IIR filters and thus cannot be practically used. In this paper, we propose a new method to solve the low-order IIR filter bank design, which is based on tractable linear-matrix inequality (LMI) optimization. Our focus is the quadrature mirror filter bank design, although other IIR filter related problems can be treated and solved in a similar way. The viability of our theoretical development is confirmed by extensive simulation.     

Weighted least-squares approximation of FIR by IIR digital filters

This paper presents a method for the weighted least-squares approximation of finite impulse response (FIR) filters by infinite impulse response (IIR) filters. It is shown, how a solution to this approximation problem can be obtained by solving a related pure least-squares approximation problem. For the latter, we utilize a generalized version of a previously published technique with low computational complexity and guaranteed stability of the IIR filters. Unlike the well-established model-reduction approaches that are carried out in the state space, our method works directly with the numerator and denominator coefficients of the transfer functions. Thus, the influence of finite-precision arithmetic on the results is small. This makes our approach applicable for the approximation of large-order FIR filters and allows the usage of arbitrarily shaped weighting functions. It is shown that our method can successfully be employed to achieve a uniform approximation     

Extending the concept of IIR filtering to array processing using approximate spatial IIR structure

Various array processing techniques applied to uniform linear arrays are involuntarily realized using structures that are analogous to finite impulse response filters. This observation leads to the following question: “can we extend infinite impulse response (IIR) filtering to array processing?”. In this paper, we introduce the concept of IIR array in spatial domain. Note that IIR array here does not mean time-domain IIR filtering for array beamforming which is commonly understood. This paper is dedicated to the study of an alternate approach for array signal processing which defines IIR structure in spatial domain. To illustrate the applicability of the concept of IIR array, we propose a new direction-of-arrival estimation technique as well as a beamformer with the spatial domain IIR array implementation. The performance of the proposed methods are comparable to the existing techniques. These illustrations are intended to introduce a new approach which potentially can offer more degrees of freedom to control the performance of the array and reduce the complexity of the system for a desired performance.     

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.     

Design of perfect-reconstruction nonuniform recombination filter banks with flexible rational sampling factors

This paper studies the design of a class of perfect-reconstruction (PR) nonuniform filter banks (FBs) called recombination nonuniform FBs (RNFBs). They are constructed by merging subbands in a uniform FB with sets of transmultiplexers (TMUXs). It generalizes the RNFBs previously proposed by the authors to allow more general choice of the sampling factors. The spectral inversion and spurious response suppression problems of these new RNFBs using cosine modulation are analyzed, and a simple design method based on a matching condition is proposed. It is also found that the FB and the TMUX in the recombination structure can be designed separately to satisfy the matching condition. In addition, real-time adaptive merging of the channels to provide dynamic nonuniform frequency partitioning is feasible. Another advantage of the RNFBs is that the recombination and processing of the subband signal can be done at the decimated domain of the uniform FB, which greatly reduces its implementation complexity. Design examples show that high quality nonuniform PR FBs with low implementation complexity and variable time-frequency resolution can be obtained by the proposed method.     

Design of low-order linear-phase IIR filters via orthogonalprojection

This paper presents an indirect linear-phase IIR filter design technique based on a reduction of linear-phase FIR filters. The desired filter is obtained by minimizing the L₂ norm of the difference between the original FIR filter and the lower order IIR filter. We first establish a relationship between the Hankel singular values of the discarded part of the FIR filter and the L₂ norm of the corresponding filter approximation error based on model truncation. This result motivates us to propose a simple finite search method that will achieve better approximation results than commonly used truncation methods such as the balanced truncation (BT) and the impulse response gramian (IRG) methods. We then develop an iterative algorithm for finding an optimal IIR filter based on a matrix projection of the original FIR filter. The convergence of the proposed algorithm is established. Filters designed using the proposed algorithm are compared with those obtained by other techniques with respect to the amplitude response and group delay characteristics in the passband. Numerical examples show that the proposed algorithm offers the best performance     

Frequency-Response Masking-Based Design of Nearly Perfect-Reconstruction Two-Channel FIR Filterbanks With Rational Sampling Factors

In order to ensure a good filterbank (FB) performance in cases where there are significant changes in the subband signals, the filters in such FBs must have very narrow transition bandwidths. When using conventional finite-impulse response (FIR) filters as building blocks for generating these FBs, this implies that their orders become very high, thereby resulting in a high overall arithmetic complexity. For considerably reducing the overall complexity, this contribution exploits the frequency-response masking (FRM) technique for synthesizing FIR filters for the above-mentioned FBs, where rational sampling factors are used. Comparisons between various optional methods of utilizing the FRM technique for designing FBs under consideration shows that the most efficient one, from both the design and the implementation viewpoints, are FBs that are constructed such that the bandedge-shaping or periodic filters are evaluated at the input sampling rate and the masking filters at the output sampling rate. This is shown by means of illustrative examples.      

Chromatic Dispersion Compensation Using Digital IIR Filtering With Coherent Detection

Digital infinite impulse response (IIR) filtering is proposed as a means for compensating chromatic dispersion in homodyne-detected optical transmission systems with subsequent digital signal processing. Compared to finite impulse response (FIR) filtering, IIR filtering achieves dispersion compensation (DC) using a significantly smaller number of taps. DC of 80 and 160 km in a 10-Gb/s binary phase-shift-keying is experimentally compared for the two filtering schemes. IIR filtering can achieve performance similar to the FIR filtering scheme.     

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison