Filter Bank Fusion Frames

In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific finite impulse response filters whose frequency responses are well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas

Oversampled Filter Banks

Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given. Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIR and give a parameterization of a class of filter banks satisfying this property. Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuous-time transforms and allow for very flexible design. We investigate relations of these filter banks to continuous-time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two-channel filter banks that yield highly regular wavelets with a given number of vanishing moments

Oversampled Filter Banks

Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in t(2)(Z), These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a. necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given, Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIE and give a parameterization of a class of filter banks satisfying this property, Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuous-time transforms and allow for very flexible design. We investigate relations of these filter banks to continuous-time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two-channel filter banks that yield highly regular wavelets with a given number of vanishing moments

Paraunitary oversampled filter bank design for channel coding

Oversampled filter banks (OSFBs) have been considered for channel coding, since their redundancy can be utilised to permit the detection and correction of channel errors. In this paper, we propose an OSFB-based channel coder for a correlated additive Gaussian noise channel, of which the noise covariance matrix is assumed to be known. Based on a suitable factorisation of this matrix, we develop a design for the decoder's synthesis filter bank in order to minimise the noise power in the decoded signal, subject to admitting perfect reconstruction through paraunitarity of the filter bank. We demonstrate that this approach can lead to a significant reduction of the noise interference by exploiting both the correlation of the channel and the redundancy of the filter banks. Simulation results providing some insight into these mechanisms are provided

Computation of the para-pseudoinverse for oversampled filter banks: Forward and backward Greville formulas

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2008 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience. In frame expansions, the least square signal reconstruction operator is called the dual frame, which can be obtained by choosing the synthesis filter bank as the para-pseudoinverse of the analysis bank. In this paper, we study the computation of the dual frame by exploiting the Greville formula, which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended. Here, we first develop the backward Greville formula to handle the case of row deletion. Based on the forward Greville formula, we then study the computation of para-pseudoinverse for extended filter banks and Laplacian pyramids. Through the backward Greville formula, we investigate the frame-based error resilient transmission over erasure channels. The necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived. A postfiltering structure is also presented to implement the para-pseudoinverse when the transform coefficients in one subband are completely lost

Generic Feasibility of Perfect Reconstruction with Short FIR Filters in Multi-channel Systems

We study the feasibility of short finite impulse response (FIR) synthesis for perfect reconstruction (PR) in generic FIR filter banks. Among all PR synthesis banks, we focus on the one with the minimum filter length. For filter banks with oversampling factors of at least two, we provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR almost surely. The prescribed length is as short or shorter than the analysis filters and has an approximate inverse relationship with the oversampling factor. Our results are in form of necessary and sufficient statements that hold generically, hence only fail for elaborately-designed nongeneric examples. We provide extensive numerical verification of the theoretical results and demonstrate that the gap between the derived filter length prescriptions and the true minimum is small. The results have potential applications in synthesis FB design problems, where the analysis bank is given, and for analysis of fundamental limitations in blind signals reconstruction from data collected by unknown subsampled multi-channel systems.Comment: Manuscript submitted to IEEE Transactions on Signal Processin

Design of Near Perfect Reconstruction Oversampled Filter Banks for Subband Adaptive Filters

In this brief, a design algorithm for real-valued and complex-valued oversampled filter banks which yield a low level of inband alias and enable simple subband adaptiv structures is presented. The filter banks are either based on complex modulation of a real-valued low-pass prototype or on the direct or modulated setups of real-valued filter banks. If real-valued filter banks are required, then the different channels will have different subsampling ratios so that the bandpass sampling theorem is not violated. This brief also presents design examples of real-valued and complex-valued filter banks