# Discrete Transforms [electronic resource] / by J.M. Firth

- Author:
- Firth, J.M.
- Published:
- Dordrecht : Springer Netherlands : Imprint: Springer, 1992.
- Edition:
- 1st ed. 1992.
- Physical Description:
- 187 pages : online resource
- Additional Creators:
- SpringerLink (Online service)

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- 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 The characteristic 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.
- Summary:
- The analysis of signals and systems using transform methods is a very important aspect of the examination of processes and problems in an increasingly 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 (for example in the design of digital filters), the same techniques are in use in such disciplines as cardiology, optics, speech analysis and management, as well as in other branches of science and engineering. This text is aimed at a readership whose mathematical background includes some acquaintance with complex numbers, linear differenÂ tial equations, matrix algebra, and series. Specifically, a familiarity with Fourier series (in trigonometric and exponential forms) is assumed, and an exposure to the concept of a continuous integral transform is desirable. Such a background can be expected, for example, on completion of the first year of a science or engineering degree course in which transform techniques will have a significant application. In other disciplines the readership will be past the second year undergraduate stage. In either case, the text is also intended for earlier graduates whose degree courses did not include this type of material and who now find themselves, in a professional capacity, requiring a knowledge of discrete transform methods.
- Subject(s):
- ISBN:
- 9789401123587
- Digital File Characteristics:
- PDF

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