Role of anticausal inverses in multirate filter-banks. II. The FIR case, factorizations, and biorthogonal lapped transforms

For pt. I see ibid., vol.43, no.5, p.1090, 1990. In part I we studied the system-theoretic properties of discrete time transfer matrices in the context of inversion, and classified them according to the types of inverses they had. In particular, we outlined the role of causal FIR matrices with anticausal FIR inverses (abbreviated cafacafi) in the characterization of FIR perfect reconstruction (PR) filter banks. Essentially all FIR PR filter banks can be characterized by causal FIR polyphase matrices having anticausal FIR inverses. In this paper, we introduce the most general degree-one cafacafi building block, and consider the problem of factorizing cafacafi systems into these building blocks. Factorizability conditions are developed. A special class of cafacafi systems called the biorthogonal lapped transform (BOLT) is developed, and shown to be factorizable. This is a generalization of the well-known lapped orthogonal transform (LOT). Examples of unfactorizable cafacafi systems are also demonstrated. Finally it is shown that any causal FIR matrix with FIR inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix

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

Vector space framework for unification of one- and multidimensional filter bank theory

A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem

The role of integer matrices in multidimensional multirate systems

The basic building blocks in a multidimensional (MD) multirate system are the decimation matrix M and the expansion matrix L. For the D-dimensional case these are D×D nonsingular integer matrices. When these matrices are diagonal, most of the one-dimensional (ID) results can be extended automatically. However, for the nondiagonal case, these extensions are nontrivial. Some of these extensions, e.g., polyphase decomposition and maximally decimated perfect reconstruction systems, have already been successfully made by some authors. However, there exist several ID results in multirate processing, for which the multidimensional extensions are even more difficult. An example is the development of polyphase representation for rational (rather than integer) sampling rate alterations. In the ID case, this development relies on the commutativity of decimators and expanders, which is possible whenever M and L are relatively prime (coprime). The conditions for commutativity in the two-dimensional (2D) case have recently been developed successfully in [1]. In the MD case, the results are more involved. In this paper we formulate and solve a number of problems of this nature. Our discussions are based on several key properties of integer matrices, including greatest common divisors and least common multiples, which we first review. These properties are analogous to those of polynomial matrices, some of which have been used in system theoretic work (e.g., matrix fraction descriptions, coprime matrices, Smith form, and so on)

Properties of the MIMO radar ambiguity function

MIMO (multiple-input multiple-output) radar is an emerging technology which has drawn considerable attention. Unlike the traditional SIMO (single-input multiple-output) radar, which transmits scaled versions of a single waveform in the antenna elements, the MIMO radar transmits independent waveforms in each of the antenna elements. It has been shown that MIMO radar systems have many advantages such as high spatial resolution, improved parameter identifiability, and enhanced flexibility for transmit beampattern design. In the traditional SIMO radar, the range and Doppler resolutions can be characterized by the radar ambiguity function. It is a major tool for studying and analyzing radar signals. Recently, the ambiguity function has been extended to the MIMO radar case. In this paper, some mathematical properties of the MIMO radar ambiguity function are derived. These properties provide insights into the MIMO radar waveform design

Precoded FIR and Redundant V-BLAST Systems for Frequency-Selective MIMO Channels

The vertical Bell labs layered space-time (V-BLAST) system is a multi-input multioutput (MIMO) system designed to achieve good multiplexing gain. In recent literature, a precoder, which exploits channel information, has been added in the V-BLAST transmitter. This precoder forces each symbol stream to have an identical mean square error (MSE). It can be viewed as an alternative to the bit-loading method. In this paper, this precoded V-BLAST system is extended to the case of frequency-selective MIMO channels. Both the FIR and redundant types of transceivers, which use cyclic-prefixing and zero-padding, are considered. A fast algorithm for computing a cyclic-prefixing-based precoded V-BLAST transceiver is developed. Experiments show that the proposed methods with redundancy have better performance than the SVD-based system with optimal powerloading and bit loading for frequency-selective MIMO channels. The gain comes from the fact that the MSE-equalizing precoder has better bit-error rate performance than the optimal bitloading method

MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms

The concept of multiple-input multiple-output (MIMO) radars has drawn considerable attention recently. Unlike the traditional single-input multiple-output (SIMO) radar which emits coherent waveforms to form a focused beam, the MIMO radar can transmit orthogonal (or incoherent) waveforms. These waveforms can be used to increase the system spatial resolution. The waveforms also affect the range and Doppler resolution. In traditional (SIMO) radars, the ambiguity function of the transmitted pulse characterizes the compromise between range and Doppler resolutions. It is a major tool for studying and analyzing radar signals. Recently, the idea of ambiguity function has been extended to the case of MIMO radar. In this paper, some mathematical properties of the MIMO radar ambiguity function are first derived. These properties provide some insights into the MIMO radar waveform design. Then a new algorithm for designing the orthogonal frequency-hopping waveforms is proposed. This algorithm reduces the sidelobes in the corresponding MIMO radar ambiguity function and makes the energy of the ambiguity function spread evenly in the range and angular dimensions

Corrosion inhibitors for water-base slurry in multiblade sawing

The use of a water-base slurry instead of the standard PC oil vehicle was proposed for multiblade sawing (MBS) silicon wafering technology. Potential cost savings were considerable; however, significant failures of high-carbon steel blades were observed in limited tests using a water-based slurry during silicon wafering. Failures were attributed to stress corrosion. A specially designed fatigue test of 1095 steel blades in distilled water with various corrosion inhibitor solutions was used to determine the feasibility of using corrosion inhibitors in water-base MBS wafering. Fatigue tests indicate that several corrosion inhibitors have significant potential for use in a water-base MBS operation. Blade samples tested in these specific corrosion-inhibitor solutions exhibited considerably greater lifetime than those blades tested in PC oil