Time-varying Filter Banks Research Articles

Modelling of time-varying discrete-time systems

A theory to analyse linear time-varying (LTV) discrete-time systems from the point of digital signal processing of view is established here. The LTV discrete-time system is changing with time slowly in comparison with the sampling frequency of the input signal is supposed. Three models were set up to simulate and analyse the LTV system. Models 1 and 2 can be used to design and analyse the LTV system, while the filter bank representation relates the LTV system with the time-varying filter bank. Based on the established models, the input–output relationship of a time-varying system in the frequency-domain was analysed in detail.

Analysis and Properties of Time-Varying Modified DFT Filter Banks

We provide a comprehensive analysis of the time-varying modified DFT (MDFT) filter bank based on the general time-varying filter bank theories (Wang, 2005, 2006, 2008, 2009) in both the time domain and frequency domain. We give firstly the description of the time-invariant MDFT filter bank including its perfect reconstruction (PR) condition, its mechanism of aliasing error cancellation and the relationship with the cosine-modulated filter bank in detail. Then, the time-varying MDFT filter bank is analyzed according to the time-domain description. Finally, the window switching method is introduced to design the prototype filter in the time-varying MDFT filter bank with examples. The error analysis shows that the introduced design approach is useful in practice.

A Two-Channel Overlapped Block Transform for Image Compression

In this correspondence, a new two-channel overlapped block transform is introduced by prefiltering the Haar transform. The prefiltering employs plane rotations with one single rotation angle to achieve high computational efficiency. The resulting transform is equivalent to a periodically time-varying (PTV) filter bank. The proposed transform or PTV filter bank has several advantages for image coding. First, symmetric extension is very easily performed at the boundaries in spite of the fact that the filters are asymmetric. Second, arithmetic complexity can be reduced to simple shifts and adds. Finally, the performance for image compression is shown to be high, both in terms of visual quality and PSNR. As a result, the proposed transform is a competitive alternative to the 5/3 and the 9/7 filters, which are currently popular for subband/wavelet image coders.

Time-Varying FIR Transmultiplexers With Minimum Redundancy

In this paper, we derive conditions for existence of zero-forcing equalizers for FIR transmultiplexers. We extend theoretical results from the literature for zero-forcing equalization in transmultiplexer systems, and derive new conditions for a more general configuration that includes filter-bank based systems with long responses and time-varying filter banks. The time-varying filter banks can be used to model code division multiple access systems with long codes, as well as other practical problems. The results can be applied to both downlink and uplink scenarios, due to the general framework considered for the analysis. The derived relations allow the use of relatively simple equalizers, lead to transmitters using small amount of redundancy, and also allow the transmission through channels with long impulse responses. Experimental results obtained via computer simulations validate the derived expressions. The simulations were carried out for zero-forcing design and also for least-squares and minimum mean squared error designs that use the relations derived for existence of zero-forcing equalizers. The results show that the performance of all equalizers improve if the zero-forcing conditions derived in this paper are followed.

Time-varying discrete-time signal expansions as time-varying filter banks

The time-varying discrete-time signal expansion was analysed based on the theory of time-varying filter banks in detail. A general definition of time-varying discrete-time wavelet transforms is provided. Usually, a time-varying discrete-time signal expansion can be implemented using a time-varying filter bank. Using the time-varying filter bank theory, the authors developed a useful algorithm to calculate the dual basis function in a biorthogonal time-varying discrete-time signal expansion. Example is given to show the usage of the algorithm. In the last part, the authors provide a detailed analysis of the general time-varying discrete-time wavelet transform. Some useful properties of the time-varying discrete-time wavelet transform including their proofs are given. The relationship between the tree-structured implementation and the non-uniform filter bank implementation is discussed.

Analysis of M-channel time-varying filter banks

In this paper we present a new theory to analyze M-channel time-varying filter banks in the transform-domain. Through studying the behavior of time-varying building blocks at a fixed time point we have established two models to analyze the time-varying building block. Model 1 gives us the possibility to analyze the spectra changing in the time-varying building blocks. Model 2 shows that the time-varying building block at a fixed time point can be realized by a time-invariant filter bank. Such observation gives us chance to analyze the time-varying filter bank using the theory for time-invariant filter banks. Setting the time-varying building blocks into M-channel time-varying filter banks we find that an M-channel time-varying filter bank at a fixed time point can be implemented with an N-channel time-invariant vector filter bank or a L-channel single filter bank if the filter length is equal to L=NM. Based on the established models, in this paper we analyze the M-channel time-varying filter bank using the classical z-transform in detail. The perfect reconstruction conditions for an M-channel time-varying filter bank are given based on the polyphase-domain analysis. In the modulation-domain analysis we provide the condition for a distortion-free M-channel time-varying filter bank. The relationship between the time- and transform-domain analysis is presented. Finally, a design example is given to show how to use the established theory by designing time-varying filter banks.

The Most General Time-Varying Filter Bank and Time-Varying Lapped Transforms

In this paper, a model-based method is used to analyze the most general time-varying filter bank, where all parameters in the filter bank change with time, including filter length, decimator factor, and the number of channel. Through modeling the time-varying filter bank at a fixed time point as a time-invariant filter bank, we change the study of the most general time-varying filter bank to study of the established model. Based on the model, the most general time-varying filter bank is analyzed in both polyphase domain and modulation domain. The general perfect reconstruction (PR) conditions of the most general time-varying filter bank are given using time-varying polyphase matrices in the polyphase domain. In the modulation domain, the time-varying distortion-matrix is presented, and the distortion-free condition of the most general time-varying filter bank is given. Based on the established theory for the most general time-varying filter bank, in the second part of this paper, the time-varying cosine-modulated lapped transform (TV-MLT) as an example is analyzed in detail. An interesting design approach called the window-switching technique, used to design the time-varying prototype windows for a perfectly reconstructed TV-MLT, is introduced. The window-switching technique provides us with the possibility to design a perfectly reconstructed TV-MLT without complex mathematical optimization computation, and simply by graphically building the prototype windows in the transition period based on the given diagrams of the prototype windows before and after the transform changing.

Time-varying cosine-modulated filter banks

In applications of the linear time-invariant (LTI) cosine-modulated filter banks, the idea to adapt analysis filter banks in order to match the signal statistical properties has existed for a long time. A typical example is found in audio coding, where audio signals are segmented as different block sizes using different analysis filter banks with the aim to eliminate the pre-echo phenomenon. Such a technique usually called window-switching. However, there was not any reasonable theoretical support for such techniques. In this work, we have generally studied time-varying cosine-modulated filter banks. The general PR conditions for an LTV cosine-modulated filter bank are given. Applying the theory in analyzing a special time-varying cosine-modulated filter bank, namely time-varying modulated lapped transforms (LTV MLTs), we have provided a solid theoretical support for the well-known window-switching techniques.

Wavelet packet-based subband adaptive equalization

This paper proposes adaptive algorithms for channel equalization in the wavelet packet transform domain. Adaptation to highly correlated time-varying channels is achieved by the use of the orthogonal tree-structured time-varying filter bank associated to this transform. The filter bank structure, that is, the type of decomposition, is obtained, for a fixed number of bands, as the one that leads to a quasi-optimum convergence rate of the adaptive filters in the different subbands. Then, we propose an NLMS-type adaptive strategy for two possible implementations, a block algorithm, BWPKNLMS, and a non-block approach, WPKNLMS. A theoretical analysis of both schemes is also provided: we obtain the mean squared error after the adaptive algorithm has converged and we find the expression for the optimal solution, to finally show that faster convergence is achieved. Experimental results for the learning curves show the efficiency of the proposed schemes and illustrate the previously presented theoretical results.

An adaptive tiling of the time–frequency plane with application to multiresolution-based perceptive audio coding

The research efforts recently devoted to the theoretical treatment of time-varying filter bank structures have created an adequate environment for the development of adaptive analysis tools, useful in many signal processing problems. In this paper we present a complete adaptive analysis system suitable for the compression of audio signals; we would like to remark that our goal is not to develop a complete audio coder, just the appropriate analysis/synthesis scheme. For basis selection, we propose a search algorithm and a new cost function based on perceptual considerations. In addition, a procedure that determines the optimum length of the analysis segment is described. Examples are given of the performance of these algorithms on some representative audio signals. When the search method is compared with previous basis selection algorithms, we find and show that, for the application of audio compression, the perceptually driven basis selection algorithm leads to a much higher compression efficiency and better quality of the reconstructed signal.

Modulated filter banks with arbitrary system delay: efficient implementations and the time-varying case

We present a new method for the design and implementation of modulated filter banks with perfect reconstruction. It is based on the decomposition of the analysis and synthesis polyphase matrices into a product of two different types of simple matrices, replacing the polyphase filtering part in a modulated filter bank. Special consideration is given to cosine-modulated as well as time-varying filter banks. The new structure provides several advantages. First of all, it allows an easy control of the input-output system delay, which can be chosen in single steps of input sampling rate, independent of the filter length. This property can be used in audio coding applications to reduce pre-echoes. Second, it results in a structure that is nearly twice as efficient as performing the polyphase filtering directly. Perfect reconstruction is a structurally inherent feature of the new formulation, even for nonlinear operations or time-varying coefficients. Hence, the structure is especially suited for the design of time-varying filter banks where both the number of bands as well as the prototype filters can be changed while maintaining perfect reconstruction and critical sampling. Further, a proof of effective completeness is given, and the design of equal magnitude-response analysis and synthesis filter banks is described. Filter design can be performed by nonconstrained optimization of the matrix coefficients according to a given cost function. Design and audio-coding application examples are given to show the performance of the new filter bank.

Speech analysis and synthesis using an AM–FM modulation model

In this paper, the AM–FM modulation model is applied to speech analysis, synthesis and coding. The AM–FM model represents the speech signal as the sum of formant resonance signals each of which contains amplitude and frequency modulation. Multiband filtering and demodulation using the energy separation algorithm are the basic tools used for speech analysis. First, multiband demodulation analysis (MDA) is applied to the problem of fundamental frequency estimation using the average instantaneous frequency as estimates of pitch harmonics. The MDA pitch tracking algorithm is shown to produce smooth and accurate fundamental frequency contours. Next, the AM–FM modulation vocoder is introduced, which represents speech as the sum of resonance signals. A time-varying filterbank is used to extract the formant bands and then the energy separation algorithm is used to demodulate the resonance signals into the amplitude envelope and instantaneous frequency signals. Efficient modeling and coding (at 4.8–9.6 kbits/sec) algorithms are proposed for the amplitude envelope and instantaneous frequency of speech resonances. Finally, the perceptual importance of modulations in speech resonances is investigated and it is shown that amplitude modulation patterns are both speaker and phone dependent.

Symmetric delay factorization: generalized framework for paraunitary filter banks

The symmetric delay factorization (SDF) was introduced to synthesize linear-phase paraunitary filter banks (LPPUFBs) with uniform order (i.e., filter length equal to NM for arbitrary N) and real-valued coefficients. The SDF presents the advantage of decomposing the polyphase transfer matrix (PTM) into only orthogonal matrices, even at the boundary of finite-duration signals, simplifying significantly the design of time-bounded filter banks (TBFBs) or of time-varying filter banks (TVFBs). However, the symmetric delay factorization applies only to LPPUFBs. On the other hand, lattice structures, as well as finite-size lattice structures, are proposed for classes of nonlinear-phase paraunitary filter banks, as the modulated lapped transform (MLT) and the extended tapped transform (ELT). This paper describes a new minimal and complete symmetric delay factorization valid for a larger class of paraunitary filter banks, encompassing paraunitary cosine modulated filter banks, with nonlinear phase basis functions, as well as for a set of LPPUFBs including the linear-phase lapped orthogonal transforms (LOTs) and the generalized tapped orthogonal transforms (GenLOTs). The derivations for filter banks with even and odd numbers of channels are formulated in a unified form. This approach opens new perspectives in the design of time-varying filter banks used for image and video compression, especially in the framework of region or object-based coding.

Time-invariant and time-varying multirate filter banks : application to image coding

This paper reviews various concepts and solutions of time-invariant and time-varying multirate filter banks. It discusses their performance for image and video coding at low bit rates, and their applicability in the mpeg-4 framework. Time-invariant multirate filter banks, and methods of design with different criteria appropriate for signal compression are first presented. Several procedures of quantization, namely scalar and lattice vector quantization, with bit allocation optimized in the rate-distortion sense, are used for the encoding of the subband signals. A technique of rate-constrained lattice vector quantization (rc-lvq), combined with a three components entropy coding, allow, together with distortion psychovi-sual weighting mechanisms to obtain significant visual improvements versus scalar quantization or the zerotree technique. However, time-invariant multirate filter banks, although efficient in terms of compression, are not well suited for content-based functionalities. Content-based features may require the ability to manipulate and thus encode a given region in the scene independently of the neighbouring regions, hence the use of transformations that can be adapted to arbitrary size bounded supports. Also, to increase the compression efficiency, one may want to adapt the transformation to the region characteristics, and thus use transform switching mechanisms, with soft or hard transitions. Three main classes of transformations can address these problems: shape-adaptive block transforms, transforms relying on signal extensions and transforms relying on time-varying multirate filter banks. These various solutions, with their methods of design, are reviewed. Emphasis is put on an extension of the SDF (symmetric delay factorization) technique which opens new perspectives in the design of time-bounded and time-varying filter banks. A region-adapted rate-distortion quantization algorithm has been used in the evaluation of the transformations compression efficiency. The coding results illustrate the interest of these techniques for compression but also for features such as quality scalability applied to selected regions of the image.

Balanced multiwavelets theory and design

This article deals with multiwavelets, which are a generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we are interested. First, we review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we have a close look at the problems encountered when using multiwavelets in applications, and we propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets.

Time-varying biorthogonal filter banks: a state-space approach

Using state-space representations of biorthogonal time-varying filter banks, it is possible to come up with a theory for the transitional behavior between two stationary filter banks. The transition interval depends on the size of the common subspace spanned by the controllability operators of the initial and final filters on the decomposition sides, and the common subspace spanned by the observability operators of the filters on the reconstruction sides. When the respective operators span the same spaces, we can derive conditions under which the transition between the filter banks can be so controlled that both the decomposition and the reconstruction functions gently embrace. For such filters, the transition interval can be made arbitrarily short. If it is zero, then the special case of instantaneous transition is reached.

Cosine-modulated two-dimensional time-varying filter banks

Two-dimensional (2D) filter banks are widely used for analysis and synthesis systems applied to subband coding of images. In recent years, time-varying filter banks that vary their structure appropriately for a local property of signals have been studied for application to the compression of image data. In time-varying filter banks, each different part of the signals is processed with different filter coefficients or different numbers of channels (a different sampling matrix). Those signals must be perfectly reconstructed even across transitions of the filter banks. In this paper, we consider time-varying systems of cosine-modulated 2D filter banks for arbitrary sampling lattices. We show perfect reconstruction (PR) conditions for time-varying filter banks that vary filter coefficients or a sampling matrix. Some applications are presented to show the effectiveness of this method. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(11): 46–54, 1997

Cosine‐modulated two‐dimensional time‐varying filter banks

Two-dimensional (2D) filter banks are widely used for analysis and synthesis systems applied to subband coding of images. In recent years, time-varying filter banks that vary their structure appropriately for a local property of signals have been studied for application to the compression of image data. In time-varying filter banks, each different part of the signals is processed with different filter coefficients or different numbers of channels (a different sampling matrix). Those signals must be perfectly reconstructed even across transitions of the filter banks. In this paper, we consider time-varying systems of cosine-modulated 2D filter banks for arbitrary sampling lattices. We show perfect reconstruction (PR) conditions for time-varying filter banks that vary filter coefficients or a sampling matrix. Some applications are presented to show the effectiveness of this method. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(11): 46–54, 1997

Adaptive lapped transform-based image coding

A structure is presented for implementing perfectly reconstructing time-varying linear-phase paraunitary filterbanks (LPPUFB) using a novel adaptive lapped transform (ALT). The ALT is based on the generalized lapped orthogonal transform (GenLOT). A time-varying filterbank is constructed through the factorization of the GenLOT into cascaded matrix stages. Variable-length lapped transforms are subsequently generated by cascading a number of these matrix stages to build filters of specific length. An embedded ALT image codec is presented.

The modulated lapped transform, its time-varying forms, and its applications to audio coding standards

The modulated lapped transform (MLT) is used in both audio and video data compression schemes. This paper describes its properties and how it can be used to generate a time-varying filterbank. Examples of its implementation in two audio coding standards are presented.