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Orthogonal transforms for digital signal processing
Book Reviews ORTHOGONAL TRANSFORMS FOR DIGITAL SIGNAL PROCESSING,
by N. Ahmed and K. R. Rao. 275 pages, diagrams, 6x9 in. New York, Springer-Verlag Inc., 1975. Price $24.10.
In the present era the trend is towards more digital signal processing. Filtering and analysis with fast transforms is a primary area of digital processor application. Many engineers and scientists are familiar with Fourier decomposition of signals using the fast Fourier transform (FFT). There are many other orthogonal fast transforms besides the FIT. Orthogonal Transforms
for Digital Signal Processing
covers theory and application not only of the FFT, but also of many other fast transforms, including Walsh, Haar, slant and a class of generalized transforms. The book builds on a background common to most upper division students. Topics are covered in a clear, logical, and satisfactorily complete development. An understanding of basic matrix operations is assumed in the first part, and of elementary probability theory in later chapters. Assumptions of this understanding makes the book a concise and valuable tool for those practicing engineers and students with the proper background. The book is excellent for an advanced undergraduate or basic graduate course on orthogonal transforms. It is also suitable to supplement courses on discrete signal processing, filtering, data compaction, pattern recognition and image processing. Numerous problems illustrate, amplify, or prove statements in the text. The book contains ten chapters, the first seven of which are devoted to background, motivation, and development of orthogonal transforms. The last three chapters are more specialized in that they are directed towards applications of orthogonal transforms for digital signal processing. Use of the transforms is demonstrated with simple examples, as well as some interesting applications including processing elec-
222
trocardiogram data and data compaction. The first three chapters cover Fourier series, the Fourier transform, various theorems useful in signal processing, and the discrete Fourier transform. Chapter 4 develops the FFT using the approach of summing over each bit in the time (frequency) tags of the data. Other approaches, including matrix factorization and decimation in time or frequency, which may appeal more to some readers, are available in references stated in the book. The authors have chosen to have their development progress rapidly rather than go into numerous FFI algorithms. The development chosen for the FFT leads directly into Chap. 5 which treats both the class of orthogonal transforms on which Ahmed, Rao and co-workers have published and the Rademacher and Haar functions. Chapter 6 is on the WalshHadamard transforms. These chapters are useful and informative, although the significance of generalized transform phasespectra defined in analogy to FIT spectra might be questioned. The Walsh transform has attracted considerable interest in recent years. Orderings of this transform (Walsh, Paley, Hadamard), which can be confusing to the novice, are detailed in Chap. 5, and further developed in Chap. 6. Chapter 7 covers other important transforms, including the discrete cosine transform (DCT) which has recently been shown to be asymptotically equivalent to the Karhunen-Loev transform (KLT) if the data covariance matrix is a Toeplitz matrix (1). The Haar and slant transforms are also included. The generalized Wiener filtering is discussed next, including a demonstration that for the optimum discrete time filter the minimum mean-square-error is independent of the transform employed. The KLT and suboptimal diagonal filters are developed. Chapter 9 discusses data compaction and presents several practical examples. As an introduction to the subject of pattern recognition readers may like the intuitive physical reasoning which the
Journal ofTheFranklin Institute
Book
authors use to rapidly develop the concepts of training, classification, and reduction of dimensionality in feature space. Digital computer processing technology is developing so rapidly that technical literautre in the journals is considerably ahead of texts in some specific areas. One area which has needed an introductory text is that of orthogonal transforms. Furthermore, there has been a need for a text which shows how to apply the many transforms which have appeared in recent years. Orthogonal Transforms for Digital Signal Processing meets these needs. DOUGLAS F. ELLIOTT Autonetics Division Rockwell International Anaheim, California Reference
(1)K. Sam Shanmugan,
“Comments on Discrete Cosine Transform”, IEEE Trans. Comput., Vol. C-24, p. 759, July 1975.
LORD KELVIN AND THE AGE OF THE EARTH, by Joe D. Burchfield. 26O+xii pages, bibliography, index, 6x 9 in., New York, Science &story Publications, 1975. Price, $15.00. Sir William Thomson, Baron Kelvin, was one of the most prolific and versatile nineteenth century scientists, making fundamental contributions to the fields of mathematics, thermodynamics, electrodynamics, engineering, and the theory of matter. He also had a lifelong amateur interest in geology. The great impact of Kelvin’s physical approach to geochronology on British geology and biology is charted in Burchfield’s fascinating book under review. It was the application of the first two laws of thermodynamics to the solar system which convinced Lord Kelvin that the
Vol. 303, No. 2, February 1977
Reviews
popular uniformitarian school of geology could not be right regarding the age of the earth. Arguments from the probable age of the sun, the cooling of the earth, and the slowing of the earth’s rate of rotation from tidal friction seemed to indicate an age on the order of a hundred million years, much younger than that assumed by Lye11 and his followers. Worse, Kelvin repeatedly revised his estimates downward during a forty-year period, until by the turn of the century he was speaking of only some twenty million years. Although pockets of opposition existed throughout this time, most geologists modified their own estimates (derived from purely geological data, such as rates of deposition of strata) to reach at least approximate agreement with the physicists’ figures. The discovery of the nature of radioactive decay at the beginning of the twentieth century transformed this situation at a stroke, since it revealed an unexpected source of heat energy which threw all thermodynamic arguments awry. Burchfield also treats in some detail the influence of Kelvin’s work on the development and reception of Charles Darwin’s evolutionary theory, an important topic which has hitherto been nearly completely neglected by historians of biology. Darwin’s mechanism of natural selection required huge tracts of time in which to operate, and as a consequence Kelvin’s evidence for a foreshortened history of the earth constituted for many years one of Darwin’s “sorest troubles”, as he wrote in a letter to A. R. Wallace. To the end, of his life, Kelvin remained an implacable foe of Darwinist biology, apparently because natural selection seemed to exclude evidence of divine design in nature. Burchfield has placed two important limitations on his work: he does not discuss Continental opinion on these controversies, nor has he considered the influence of religious thought as deserving of any more than the most cursory mention. The author defends his deletion of these topics by denying their importance.
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by N. Ahmed and K. R. Rao. 275 pages, diagrams, 6x9 in. New York, Springer-Verlag Inc., 1975. Price $24.10.
In the present era the trend is towards more digital signal processing. Filtering and analysis with fast transforms is a primary area of digital processor application. Many engineers and scientists are familiar with Fourier decomposition of signals using the fast Fourier transform (FFT). There are many other orthogonal fast transforms besides the FIT. Orthogonal Transforms
for Digital Signal Processing
covers theory and application not only of the FFT, but also of many other fast transforms, including Walsh, Haar, slant and a class of generalized transforms. The book builds on a background common to most upper division students. Topics are covered in a clear, logical, and satisfactorily complete development. An understanding of basic matrix operations is assumed in the first part, and of elementary probability theory in later chapters. Assumptions of this understanding makes the book a concise and valuable tool for those practicing engineers and students with the proper background. The book is excellent for an advanced undergraduate or basic graduate course on orthogonal transforms. It is also suitable to supplement courses on discrete signal processing, filtering, data compaction, pattern recognition and image processing. Numerous problems illustrate, amplify, or prove statements in the text. The book contains ten chapters, the first seven of which are devoted to background, motivation, and development of orthogonal transforms. The last three chapters are more specialized in that they are directed towards applications of orthogonal transforms for digital signal processing. Use of the transforms is demonstrated with simple examples, as well as some interesting applications including processing elec-
222
trocardiogram data and data compaction. The first three chapters cover Fourier series, the Fourier transform, various theorems useful in signal processing, and the discrete Fourier transform. Chapter 4 develops the FFT using the approach of summing over each bit in the time (frequency) tags of the data. Other approaches, including matrix factorization and decimation in time or frequency, which may appeal more to some readers, are available in references stated in the book. The authors have chosen to have their development progress rapidly rather than go into numerous FFI algorithms. The development chosen for the FFT leads directly into Chap. 5 which treats both the class of orthogonal transforms on which Ahmed, Rao and co-workers have published and the Rademacher and Haar functions. Chapter 6 is on the WalshHadamard transforms. These chapters are useful and informative, although the significance of generalized transform phasespectra defined in analogy to FIT spectra might be questioned. The Walsh transform has attracted considerable interest in recent years. Orderings of this transform (Walsh, Paley, Hadamard), which can be confusing to the novice, are detailed in Chap. 5, and further developed in Chap. 6. Chapter 7 covers other important transforms, including the discrete cosine transform (DCT) which has recently been shown to be asymptotically equivalent to the Karhunen-Loev transform (KLT) if the data covariance matrix is a Toeplitz matrix (1). The Haar and slant transforms are also included. The generalized Wiener filtering is discussed next, including a demonstration that for the optimum discrete time filter the minimum mean-square-error is independent of the transform employed. The KLT and suboptimal diagonal filters are developed. Chapter 9 discusses data compaction and presents several practical examples. As an introduction to the subject of pattern recognition readers may like the intuitive physical reasoning which the
Journal ofTheFranklin Institute
Book
authors use to rapidly develop the concepts of training, classification, and reduction of dimensionality in feature space. Digital computer processing technology is developing so rapidly that technical literautre in the journals is considerably ahead of texts in some specific areas. One area which has needed an introductory text is that of orthogonal transforms. Furthermore, there has been a need for a text which shows how to apply the many transforms which have appeared in recent years. Orthogonal Transforms for Digital Signal Processing meets these needs. DOUGLAS F. ELLIOTT Autonetics Division Rockwell International Anaheim, California Reference
(1)K. Sam Shanmugan,
“Comments on Discrete Cosine Transform”, IEEE Trans. Comput., Vol. C-24, p. 759, July 1975.
LORD KELVIN AND THE AGE OF THE EARTH, by Joe D. Burchfield. 26O+xii pages, bibliography, index, 6x 9 in., New York, Science &story Publications, 1975. Price, $15.00. Sir William Thomson, Baron Kelvin, was one of the most prolific and versatile nineteenth century scientists, making fundamental contributions to the fields of mathematics, thermodynamics, electrodynamics, engineering, and the theory of matter. He also had a lifelong amateur interest in geology. The great impact of Kelvin’s physical approach to geochronology on British geology and biology is charted in Burchfield’s fascinating book under review. It was the application of the first two laws of thermodynamics to the solar system which convinced Lord Kelvin that the
Vol. 303, No. 2, February 1977
Reviews
popular uniformitarian school of geology could not be right regarding the age of the earth. Arguments from the probable age of the sun, the cooling of the earth, and the slowing of the earth’s rate of rotation from tidal friction seemed to indicate an age on the order of a hundred million years, much younger than that assumed by Lye11 and his followers. Worse, Kelvin repeatedly revised his estimates downward during a forty-year period, until by the turn of the century he was speaking of only some twenty million years. Although pockets of opposition existed throughout this time, most geologists modified their own estimates (derived from purely geological data, such as rates of deposition of strata) to reach at least approximate agreement with the physicists’ figures. The discovery of the nature of radioactive decay at the beginning of the twentieth century transformed this situation at a stroke, since it revealed an unexpected source of heat energy which threw all thermodynamic arguments awry. Burchfield also treats in some detail the influence of Kelvin’s work on the development and reception of Charles Darwin’s evolutionary theory, an important topic which has hitherto been nearly completely neglected by historians of biology. Darwin’s mechanism of natural selection required huge tracts of time in which to operate, and as a consequence Kelvin’s evidence for a foreshortened history of the earth constituted for many years one of Darwin’s “sorest troubles”, as he wrote in a letter to A. R. Wallace. To the end, of his life, Kelvin remained an implacable foe of Darwinist biology, apparently because natural selection seemed to exclude evidence of divine design in nature. Burchfield has placed two important limitations on his work: he does not discuss Continental opinion on these controversies, nor has he considered the influence of religious thought as deserving of any more than the most cursory mention. The author defends his deletion of these topics by denying their importance.
223