Discrete-Time Signal Processing / Edition 3

by Alan Oppenheim

Overview

Discrete-time signal processing continues to be a dynamic and rapidly growing field with a wide range of applications including speech and data communication, acoustics, radar, sonar, seismology, remote sensing, instrumentation, consumer electronics, and many others. Discrete-time Signal Processing, by the authors of the classic text Digital Signal Processing (Prentice Hall, 1975), is a completely up-to-date, thorough, and coherent treatment of the fundamentals of this field. Considerable emphasis is placed on illustrative examples and intuitive interpretation. The authors include more than 400 carefully prepared problems to help the reader develop a thorough foundation in the use of this material.

Product Details

ISBN-13:	2900131988421
Publication date:	08/18/2009
Pages:	1144
Product dimensions:	6.50(w) x 1.50(h) x 9.50(d)

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Chapter 2: Discrete-Time Signals and Systems

HISTORICAL PERSPECTIVE

Discrete-time signal processing has advanced in uneven steps over a long period of time. Looking back at the development of the field provides a valuable perspective on fundamentals that will remain central to the field long into the future. Since the invention of calculus in the 17th century, scientists and engineers have developed models to represent physical phenomena in terms of functions of continuous variables and differential equations. Numerical techniques have been used to solve these equations when analytical solutions are not possible. Indeed, Newton used finite-difference methods that are special cases of some of the discrete-time systems that we present in this text. Mathematicians of the 18th century, such as Euler, Bernoulli, and Lagrange, developed methods for numerical integration and interpolation of functions of a continuous variable. Interesting historical research by Heideman, Johnson, and Burrus (1984) showed that Gauss discovered the fundamental principle of the fast Fourier transform (discussed in Chapter 9) as early as 1805 - even before the publication of Fourier's treatise on harmonic series representation of functions.

typically done with analog systems that were implemented with electronic circuits or even with mechanical devices. Even though digital computers were becoming available in business environments and in scientific laboratories, they were expensive and had relatively limited capabilities. About that time, the need for more sophisticated signal processing in some application areas created considerable interest in discrete-time signal processing. One of the first uses of digital computers in digital signal processing was in oil prospecting, where seismic data could be recorded on magnetic tape for later processing. This type of signal processing could not generally be done in real time; minutes or even hours of computer time were often required to process only seconds of data. Even so, the flexibility of the digital computer and the potential payoffs made this alternative extremely inviting.

in a different way. Because of the flexibility of digital computers, it was often useful to simulate a signal processing system on a digital computer before implementing it in analog hardware. In this way, a new signal processing algorithm or system could be studied in a flexible experimental environment before committing economic and engineering resources to constructing it. Typical examples of such simulations were the vocoder simulations carried out at Lincoln Laboratory and Bell Laboratories. In the implementation of an analog channel vocoder, for example, the filter characteristics affected the perceived quality of the coded speech signal in ways that were difficult to quantify objectively. Through computer simulations, these filter characteristics could be adjusted and the perceived quality of a speech coding system evaluated prior to construction of the analog equipment.

computer offered tremendous advantages in flexibility. However, the processing could not be done in real time. Consequently, a prevalent attitude was that the digital computer was being used to approximate, or simulate, an analog signal processing system. In keeping with that style, early work on digital filtering was very much concerned with ways in which a filter could be programmed on a digital computer so that with analog-to-digital conversion of the signal, followed by digital filtering, followed by digital-to-analog conversion, the overall system approximated a good analog filter. The notion that digital systems might, in fact, be practical for the actual real-time implementation of signal processing in speech communication, radar processing, or any of a variety of other applications seemed at the most optimistic times to be highly speculative. Speed, cost, and size were, of course, three of the important factors in favor of the use of analog components.

natural tendency to experiment with increasingly sophisticated signal processing algorithms. Some of these algorithms grew out of the flexibility of the digital computer and had no apparent practical implementation in analog equipment. Thus, many of these algorithms were treated as interesting, but somewhat impractical, ideas. The development of such signal processing algorithms made the notion of all-digital implementation of signal processing systems even more tempting. Active work began on the investigation of digital vocoders, digital spectrum analyzers, and other all-digital systems, with the hope that eventually such systems would become practical.

was further accelerated by the disclosure by Cooley and Tukey (1965) of an efficient algorithm for computation of Fourier transforms. This class of algorithms has come to be known as the fast Fourier transform, or FFT. The FFT was significant for several reasons. Many signal processing algorithms that had been developed on digital computers required processing times several orders of magnitude greater than real time. Often this was because spectrum analysis was an important component of the signal processing and no efficient means were available for implementing it. The fast Fourier transform algorithm reduced the computation time of the Fourier transform by orders of magnitude, permitting the implementation of increasingly sophisticated signal processing algorithms with processing times that allowed interactive experimentation with the system. Furthermore, with the realization that the fast Fourier transform algorithms might, in fact, be implementable in special-purpose digital hardware, many signal processing algorithms that previously had appeared to be impractical began to appear to have practical implementations.

that it was an inherently discrete-time. concept. It was directed toward the computation of the Fourier transform of a discrete-time signal or sequence and involved a set of properties and mathematics that was exact in the discrete-time domain-it was not simply an approximation to a continuous-time Fourier transform. This had the effect of stimulating a reformulation of many signal processing concepts and algorithms in terms of discrete-time mathematics, and these techniques then formed an exact set of relationships in the discrete-time domain. Following this shift away from the notion that signal processing on a digital computer was merely an approximation to analog signal processing techniques, there emerged a strong interest in discrete-time signal processing as an important field of investigation in its own right.

occurred in the field of microelectronics. The invention and subsequent proliferation of the microprocessor paved the way for low-cost implementations of discrete-time signal processing systems. Although the first microprocessors were too slow to implement most discrete-time systems in real time, by the mid-1980s integrated Circuit technology had advanced to a level that permitted the implementation of very fast fixed-point and floating- point microcomputers with architectures specially designed for implementing discrete-time signal processing algorithms. With this technology came, for the first time, the possibility of widespread application of discrete-time signal processing techniques.

FUTURE PROMISE

Microelectronics engineers continue to strive for increased circuit densities and production yields, and as a result, the complexity and sophistication of microelectronic systems are continually increasing. Indeed, complexity and capability of DSP chips have grown exponentially since the early 1980s and show no sip of slowing down. As wafer-scale integration techniques become highly developed, very complex discrete-time signal processing systems will be implemented with low cost, miniature size, and low power 'consumption. Consequently, the importance of discrete-time signal processing will almost certainly continue to increase and the future development of the field is likely to be even more dramatic than the course of development that we have just described. Discrete-time signal processing techniques are already promoting revolutionary advances in some fields of application. A notable example is in the area of telecommunications, where discrete-time signal processing techniques, microelectronic technology, and fiber optic transmission combine to change the nature of communication systems in truly revolutionary ways. A similar impact can be expected in many other areas of technology.

its fundamentals are well formulated. Our goal in this book is to provide a coherent treatment of the theory of discrete-time linear systems, filtering, sampling, and discrete-time Fourier analysis. The topics presented should provide the reader with the knowledge necessary for an appreciation of the wide scope of applications for discrete-time signal processing and a foundation for contributing to future developments in this exciting field of technology....

Table of Contents

1. Introduction.
2. Discrete-Time Signals and Systems.

Introduction. Discrete-time Signals: Sequences. Discrete-time Systems. Linear Time-Invariant Systems. Properties of Linear Time-Invariant Systems. Linear Constant-Coefficient Difference Equations. Frequency-Domain Representation of Discrete-Time Signals and Systems. Representation of Sequence by Fourier Transforms. Symmetry Properties of the Fourier Transform. Fourier Transform Theorems. Discrete-Time Random Signals. Summary.

3. The z-Transform.

Introduction. The z-Transform. Properties of the Region of Convergence for the z-Transform. The Inverse z-Transform. z-Transform Properties. Summary.

4. Sampling of Continuous-Time Signals.

Introduction. Periodic Sampling. Frequency-Domain Representation of Sampling. Reconstruction of a Bandlimited Signal from its Samples. Discrete-Time Processing of Continuous-Time Signals. Continuous-Time Processing of Discrete-Time Signals. Changing the Sampling Rate Using Discrete-Time Processing. Practical Considerations. Oversampling and Noise Shaping. Summary.

5. Transform Analysis of Linear Time-Invariant Systems.

Introduction. The Frequency Response of LTI Systems. System Functions for Systems Characterized by Linea. Frequency Response for Rational System Functions. Relationship Between Magnitude and Phase. All-Pass Systems. Minimum-Phase Systems. Linear Systems with Generalized Linear Phase. Summary.

6. Structures for Discrete-Time Systems.

Introduction. Block Diagram Representation of Linear Constant-Coefficient Difference Equations. Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations. Basic Structures for IIR Systems. Transposed Forms. Basic Network Structures for FIR Systems. Overview of Finite-Precision Numerical Effects. The Effects of Coefficient Quantization. Effects of Roundoff Noise in Digital Filters. Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters. Summary.

7. Filter Design Techniques.

Introduction. Design of Discrete-Time IIR Filters from Continuous-Time Filters. Design of FIR Filters by Windowing. Examples of FIR Filter Design by the Kaiser Window Method. Optimum Approximations of FIR Filters. Examples of FIR Equiripple Approximation. Comments on IIR and FIR Digital Filters. Summary.

8. The Discrete Fourier Transform.

Introduction. Representation of Periodic Sequences: the Discrete Fourier Series. Summary of Properties of the DFS Representation of Periodic Sequences. The Fourier Transform of Periodic Signals. Sampling the Fourier Transform. Fourier Representation of Finite-Duration Sequences: The Discrete-Fourier Transform. Properties of the Discrete Fourier Transform. Summary of Properties of the Discrete Fourier Transform. Linear Convolution Using the Discrete Fourier Transform. The Discrete Cosine Transform (DCT). Summary.

9. Computation of the Discrete Fourier Transform.

Introduction. Efficient Computation of the Discrete Fourier Transform. The Goertzel Algorithm Decimation-in-Time FFT Algorithms. Decimation-in-Frequency FFT Algorithms. Practical Considerations Implementation of the DFT Using Convolution. Summary.

10. Fourier Analysis of Signals Using the Discrete Fourier Transform.

Introduction. Fourier Analysis of Signals Using the DFT. DFT Analysis of Sinusoidal Signals. The Time-Dependent Fourier Transform. Block Convolution Using the Time-Dependent Fourier Transform. Fourier Analysis of Nonstationary Signals. Fourier Analysis of Stationary Random Signals: the Periodogram. Spectrum Analysis of Random Signals Using Estimates of the Autocorrelation Sequence. Summary.

11. Discrete Hilbert Transforms.

Introduction. Real and Imaginary Part Sufficiency of the Fourier Transform for Causal Sequences. Sufficiency Theorems for Finite-Length Sequences. Relationships Between Magnitude and Phase. Hilbert Transform Relations for Complex Sequences. Summary.

Appendix A: Random Signals.

Discrete-Time Random Process. Averages. Properties of Correlation and Covariance Sequences. Transform Representation of Random Signals.

Appendix B: Continuous-Time Filters.

Butterworth Lowpass Filters. Chebyshev Filters. Elliptic Filters.

Editorial Reviews

Emphasizes the fundamentals of processing signals using digital techniques and their application to practical problems. Topics include: the latest methods and applications for sampling of continuous-time signals; transform analysis of LTI systems, and digital filter design. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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