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Prentice Hall, 1995. — 406 p.One of the by-products of the computer revolution has been the emergence of completely new fields of study. Each year, as integrated circuits have become faster, cheaper, and more compact, it has become possible to find feasible solutions to problems of ever-increasing complexity. Because it demands massive amounts of digital storage and comparable quantities of numerical computation, multidimensional digital signal processing is a problem area which has only recently begun to emerge. Despite this fact, it has already provided the solutions to important problems ranging from computer-aided tomography (CAT), a technique for combining x-ray projections from different orientations to create a three-dimensional recon- reconstruction of a portion of the human body, to the design of passive sonar arrays and the monitoring of the earth's resources by satellite. In addition to its many glamorous and humble applications, however, multidimensional digital signal processing also possesses a firm mathematical foundation, which allows us not only to understand what has already been accomplished, but also to explore rationally new problem areas and solution methods as they arise.

Simply stated, a signal is any medium for conveying information, and signal processing is concerned with the extraction of that information. Thus ensembles of time-varying voltages, the density of silver grains on a photographic emulsion, or lists of numbers in the memory of a computer all represent examples of signals. A typical signal processing task involves the transfer of information from one signal to another. A photograph, for example, might be scanned, sampled, and stored in the memory of a computer. In this case, the information is transferred from a variable silver density, to a beam of visible light, to an electrical waveform, and finally to a sequence of numbers, which, in turn, are represented by an arrangement of magnetic domains on a computer disk. The CAT scanner is a more complex example; information about the structure of an unknown object is first transferred to a series of electromagnetic waves, which are then sampled to produce an array of numbers, which, in turn, are processed by a computational algorithm and finally displayed on the phosphor of a cathode ray tube (CRT) screen or on photographic film. The digital processing which is done cannot add to the information, but it can rearrange it so that a human observer can more readily interpret it; instead of looking at multiple shadows the observer is able to look at a cross-sectional view.

Whatever their form, signals are of interest only because of the information they contain. At the risk of overgeneralizing we might say that signal processing is concerned with two basic tasks —information rearrangement and information reduction. We have already seen two examples of information rearrangement — computer-aided tomography and image scanning. To those we could easily add other examples: image enhancement, image deblurring, spectral analysis, and so on. Information reduction is concerned with the removal of extraneous information. Someone observing radar returns is generally interested in only a few bits of information, specifically, the answer to such questions as: Is anything there? If so, what? Friend or foe? How fast is it going, and where is it headed? However, the receiver is also giving the observer information about the weather, chaff, birds, nearby build- buildings, noise in the receiver, and so on. The observer must separate the relevant from the irrelevant, and signal processing can help. Other examples of information-lossy signal processing operations include noise removal, parameter estimation, and feature extraction.

Multidimensional Signals and Systems

Two-Dimensional Discrete signals

Multidimensional systems

Frequency-Domain Characterization of signals and systems

Sampling Continuous 2-D signals

Processing Continuous signals with Discrete Systems

Discrete Fourier Analysis of Multidimensional Signals

Discrete Fourier Series Representation of Rectangularly Periodic Sequences

Multidimensional Discrete Fourier Transform

Calculation of the Discrete Fourier Transform

Discrete Fourier Transforms for General Periodically Sampled Signals

Interrelationship between M-dimensional and One-Dimensional DFTs

Design and Implementation of Two-dimensional FIR Filters

FIR Filters

Implementation of FIR Filters

Optimal FIR Filter Design

Design of FIR Filters for Special Implementations

FIR Filters for Hexagpnally Sampled Signals

Multidimensional Recursive Systems

Finite-Order Difference Equations

Multidimensional z-Transforms

Stability of Recursive Systems

Two-Dimensional Complex Cepstrum

Design and Implementation of Two-dimensional IIR Filters

Classical 2-D IIR Filter Implementations

Iterative Implementations for 2-D IIR Filters

Signal Flowgraphs and State-Variable Realizations

Space-Domain Design Techniques

Frequency-Domain Design Techniques

Design Techniques for Specialized Structures

Stabilization Techniques

Processing Signals Carried by Propagating Waves

Beamforming

Discrete-Time Beamforming

Further Considerations for Array Processing Applications

Multidimensional Spectral Estimation

Inverse Problems

Constrained Iterative Signal Restoration

Seismic Wave Migration

Reconstruction of Signals from Their Projections

Projection of Discrete Signals

Simply stated, a signal is any medium for conveying information, and signal processing is concerned with the extraction of that information. Thus ensembles of time-varying voltages, the density of silver grains on a photographic emulsion, or lists of numbers in the memory of a computer all represent examples of signals. A typical signal processing task involves the transfer of information from one signal to another. A photograph, for example, might be scanned, sampled, and stored in the memory of a computer. In this case, the information is transferred from a variable silver density, to a beam of visible light, to an electrical waveform, and finally to a sequence of numbers, which, in turn, are represented by an arrangement of magnetic domains on a computer disk. The CAT scanner is a more complex example; information about the structure of an unknown object is first transferred to a series of electromagnetic waves, which are then sampled to produce an array of numbers, which, in turn, are processed by a computational algorithm and finally displayed on the phosphor of a cathode ray tube (CRT) screen or on photographic film. The digital processing which is done cannot add to the information, but it can rearrange it so that a human observer can more readily interpret it; instead of looking at multiple shadows the observer is able to look at a cross-sectional view.

Whatever their form, signals are of interest only because of the information they contain. At the risk of overgeneralizing we might say that signal processing is concerned with two basic tasks —information rearrangement and information reduction. We have already seen two examples of information rearrangement — computer-aided tomography and image scanning. To those we could easily add other examples: image enhancement, image deblurring, spectral analysis, and so on. Information reduction is concerned with the removal of extraneous information. Someone observing radar returns is generally interested in only a few bits of information, specifically, the answer to such questions as: Is anything there? If so, what? Friend or foe? How fast is it going, and where is it headed? However, the receiver is also giving the observer information about the weather, chaff, birds, nearby build- buildings, noise in the receiver, and so on. The observer must separate the relevant from the irrelevant, and signal processing can help. Other examples of information-lossy signal processing operations include noise removal, parameter estimation, and feature extraction.

Multidimensional Signals and Systems

Two-Dimensional Discrete signals

Multidimensional systems

Frequency-Domain Characterization of signals and systems

Sampling Continuous 2-D signals

Processing Continuous signals with Discrete Systems

Discrete Fourier Analysis of Multidimensional Signals

Discrete Fourier Series Representation of Rectangularly Periodic Sequences

Multidimensional Discrete Fourier Transform

Calculation of the Discrete Fourier Transform

Discrete Fourier Transforms for General Periodically Sampled Signals

Interrelationship between M-dimensional and One-Dimensional DFTs

Design and Implementation of Two-dimensional FIR Filters

FIR Filters

Implementation of FIR Filters

Optimal FIR Filter Design

Design of FIR Filters for Special Implementations

FIR Filters for Hexagpnally Sampled Signals

Multidimensional Recursive Systems

Finite-Order Difference Equations

Multidimensional z-Transforms

Stability of Recursive Systems

Two-Dimensional Complex Cepstrum

Design and Implementation of Two-dimensional IIR Filters

Classical 2-D IIR Filter Implementations

Iterative Implementations for 2-D IIR Filters

Signal Flowgraphs and State-Variable Realizations

Space-Domain Design Techniques

Frequency-Domain Design Techniques

Design Techniques for Specialized Structures

Stabilization Techniques

Processing Signals Carried by Propagating Waves

Beamforming

Discrete-Time Beamforming

Further Considerations for Array Processing Applications

Multidimensional Spectral Estimation

Inverse Problems

Constrained Iterative Signal Restoration

Seismic Wave Migration

Reconstruction of Signals from Their Projections

Projection of Discrete Signals

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