Overview

The analysis of signals and systems using transform methods is an important aspect of the examination of processes and problems in a wide range of applications. Whereas the initial impetus in the development of methods appropriate for handling discrete sets of data occurred mainly in an electrical engineering context, the same techniques are now in use in such disciplines as cardiology, speech analysis, optics and other branches of science and engineering. This introduction to discrete transform techniques is aimed at engineering and science students who have completed the first year of their degree courses. The textbook assumes a familiarity with Fourier series, although a review of the basic theory is provided to assist readers. Worked examples are provided throughout the text to enable the students to relate techniques to practical problems.

Full Product Details

Author:   J.M. Firth
Publisher:   Chapman and Hall
Imprint:   Chapman and Hall
Edition:   Softcover reprint of the original 1st ed. 1992
Dimensions:   Width: 14.00cm , Height: 1.00cm , Length: 21.60cm
Weight:   0.261kg
ISBN:  

9780412429903

ISBN 10:   041242990
Pages:   187
Publication Date:   30 April 1992
Audience:   College/higher education ,  Undergraduate
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Fourier series, integral theorem, and transforms: a review.- 1.1 Fourier series.- 1.2 Fourier exponential series.- 1.3 The Fourier integral theorem.- 1.4 Odd and even functions.- 1.5 The Fourier transform.- 1.6 The Fourier sine and cosine transforms.- 1.7 The Laplace transform.- 1.8 Laplace transform properties and pairs.- 1.9 Transfer functions and convolution.- Summary.- Problems.- 2 The Fourier transform. Convolution of analogue signals.- 2.1 Duality.- 2.2 Further properties of the Fourier transform.- 2.3 Comparison with the Laplace transform, and the existence of the Fourier transform.- 2.4 Transforming using a limit process.- 2.5 Transformation and inversion using duality and other properties.- 2.6 Some frequently occurring functions and their transforms.- 2.7 Further Fourier transform pairs.- 2.8 Graphical aspects of convolution.- Summary.- Problems.- 3 Discrete signals and transforms. The Z-transform and discrete convolution.- 3.1 Sampling, quantization and encoding.- 3.2 Sampling and ‘ideal’ sampling models.- 3.3 The Fourier transform of a sampled function.- 3.4 The spectrum of an ‘ideally sampled’ function.- 3.5 Aliasing.- 3.6 Transform and inversion sums; truncation.- 3.7 Windowing: band-limited signals and signal energy.- 3.8 The Laplace transform of a sampled signal.- 3.9 The Z-transform.- 3.10 Input-output systems and transfer functions.- 3.11 Properties of the Z-transform.- 3.12 Z-transform pairs.- 3.13 Inversion.- 3.14 Discrete convolution.- Summary.- Problems.- 4 Difference equations and the Z-transforms.- 4.1 Forward and backward difference operators.- 4.2 The approximation of a differential equation.- 4.3 Ladder networks.- 4.4 Bending in beams: trial methods of solution.- 4.5 Transforming a second-order forward difference equation.- 4.6 Thecharacteristic polynomial and the terms to be inverted.- 4.7 The case when the characteristic polynomial has real roots.- 4.8 The case when the characteristic polynomial has complex roots.- 4.9 The case when the characteristic equation has repeated roots.- 4.10 Difference equations of order N > 2.- 4.11 A backward difference equation.- 4.12 A second-order equation: comparison of the two methods.- Summary.- Problems.- 5 The discrete Fourier transform.- 5.1 Approximating the exponential Fourier series.- 5.2 Definition of the discrete Fourier transform.- 5.3 Establishing the inverse.- 5.4 Inversion by conjugation.- 5.5 Properties of the discrete Fourier transform.- 5.6 Discrete correlation.- 5.7 Parseval’s theorem.- 5.8 A note on sampling in the frequency domain, and a further comment on window functions.- 5.9 Computational effort and the discrete Fourier transform.- Summary.- Problems.- 6 Simplification and factorization of the discrete Fourier transform matrix.- 6.1 The coefficient matrix for an eight-point discrete Fourier transform.- 6.2 The permutation matrix and bit-reversal.- 6.3 The output from four two-point discrete Fourier transforms.- 6.4 The output from two four-point discrete Fourier transforms.- 6.5 The output from an eight-point discrete Fourier transform.- 6.6 ‘Butterfly’ calculations.- 6.7 ‘Twiddle’ factors.- 6.8 Economies.- Summary.- Problems.- 7 Fast Fourier transforms.- 7.1 Fast Fourier transform algorithms.- 7.2 Decimation in time for an eight-point discrete Fourier transform: first stage.- 7.3 The second stage: further periodic aspects.- 7.4 The third stage.- 7.5 Construction of a flow graph.- 7.6 Inversion using the same decimation-in-time signal flow graph.- 7.7 Decimation in frequency for an eight-point discrete Fourier transform.-Summary.- Problems.- Appendix A: The Fourier integral theorem.- Appendix B: The Hartley transform.- Appendix C: Further reading.

Reviews

A useful book for anyone dealing with the transform mentioned above [Fourier transform] or with digital signal processing. Useful worked examples are given throughout the text.' International Journal of Electrical Engineering Education

`A useful book for anyone dealing with the transform mentioned above [Fourier transform] or with digital signal processing. Useful worked examples are given throughout the text.' International Journal of Electrical Engineering Education

`A useful book for anyone dealing with the transform mentioned above [Fourier transform] or with digital signal processing. Useful worked examples are given throughout the text.' International Journal of Electrical Engineering Education

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