Coding gain in paraunitary analysis/synthesis systems

A formal proof that bit allocation results hold for the entire class of paraunitary subband coders is presented. The problem of finding an optimal paraunitary subband coder, so as to maximize the coding gain of the system, is discussed. The bit allocation problem is analyzed for the case of the paraunitary tree-structured filter banks, such as those used for generating orthonormal wavelets. The even more general case of nonuniform filter banks is also considered. In all cases it is shown that under optimal bit allocation, the variances of the errors introduced by each of the quantizers have to be equal. Expressions for coding gains for these systems are derived

Linear phase paraunitary filter banks: theory, factorizations and designs

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. In this paper, we study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. We begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, we develop a minimal factorizdion for a large class of such systems. This factorization will be proved to be complete for even M. Further, we structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. We then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory

Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain

Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself

A new algorithm for time-frequency spread coders using multirate filters

Filter banks have been shown to be efficient in several emerging signal communication applications. A new class of time-frequency spread coders for transmultiplexer systems using multirate filter banks is presented. As compared with conventional filter banks designed with stopband attenuation and passband flatness criteria, the user coders with the new algorithms are designed with time and frequency spread criteria. In new the algorithms, the filters are achieved through a cascade of lattice structures and delay chains. Along with the time-frequency property and the reconstruction property, the intercode/intracode correlation property are included in the design algorithms to be optimized. The designed coders are used in the application of digital watermarking of images and demonstrate good performance in JPEG encoding at different qualities.published_or_final_versio

Role of anticausal inverses in multirate filter-banks. I. System-theoretic fundamentals

In a maximally decimated filter bank with identical decimation ratios for all channels, the perfect reconstructibility property and the nature of reconstruction filters (causality, stability, FIR property, and so on) depend on the properties of the polyphase matrix. Various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse. In this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties (i.e., properties of state-space descriptions, McMillan degree, degree of determinant, and so forth). We find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory. We study their properties both for the FIR and IIR cases. Techniques for implementing anticausal IIR inverses based on state space descriptions are outlined. It is found that causal FIR matrices with anticausal FIR inverses (cafacafi) have a key role in the characterization of FIR filter banks. In a companion paper, these results are applied for the factorization of biorthogonal FIR filter banks, and a generalization of the lapped orthogonal transform called the biorthogonal lapped transform (BOLT) developed

Orthogonal Wavelets via Filter Banks: Theory and Applications

Wavelets are used in many applications, including image processing, signal analysis and seismology. The critical problem is the representation of a signal using a small number of computable functions, such that it is represented in a concise and computationally efficient form. It is shown that wavelets are closely related to filter banks (sub band filtering) and that there is a direct analogy between multiresolution analysis in continuous time and a filter bank in discrete time. This provides a clear physical interpretation of the approximation and detail spaces of multiresolution analysis in terms of the frequency bands of a signal. Only orthogonal wavelets, which are derived from orthogonal filter banks, are discussed. Several examples and applications are considered

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can only capture well edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [8] and applied to image denoising in [13]. Despite several desirable properties, the directional tensor product complex tight framelets constructed in [8,13] are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall satisfactorily answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented