Fourier Transforms

An Introduction for Engineers

Robert M. Gray, Joseph W. Goodman (Autoren)

Buch | Hardcover

361 Seiten

1995
Springer (Verlag)
978-0-7923-9585-0 (ISBN)

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The Fourier transform is one of the most important mathematical tools in a wide variety of fields in science and engineering. In the abstract it can be viewed as the transformation of a signal in one domain (typically time or space) into another domain, the frequency domain. Applications of Fourier transforms, often called Fourier analysis or harmonic analysis, provide useful decompositions of signals into fundamental or "primitive" components, provide shortcuts to the computation of complicated sums and integrals, and often reveal hidden structure in data. Fourier analysis lies at the base of many theories of science and plays a fundamental role in practical engineering design. The origins of Fourier analysis in science can be found in Ptolemy's decomposing celestial orbits into cycles and epicycles and Pythagorus' de composing music into consonances. Its modern history began with the eighteenth century work of Bernoulli, Euler, and Gauss on what later came to be known as Fourier series. J. Fourier in his 1822 Theorie analytique de la Chaleur [16] (still available as a Dover reprint) was the first to claim that arbitrary periodic functions could be expanded in a trigonometric (later called a Fourier) series, a claim that was eventually shown to be incorrect, although not too far from the truth. It is an amusing historical sidelight that this work won a prize from the French Academy, in spite of serious concerns expressed by the judges (Laplace, Lagrange, and Legendre) re garding Fourier's lack of rigor.

1 Signals and Systems.- 1.1 Waveforms and Sequences.- 1.2 Basic Signal Examples.- 1.3 Random Signals.- 1.4 Systems.- 1.5 Linear Combinations.- 1.6 Shifts.- 1.7 Two-Dimensional Signals.- 1.8 Sampling, Windowing, and Extending.- 1.9 Probability Functions.- 1.10 Problems.- 2 The Fourier Transform.- 2.1 Basic Definitions.- 2.2 Simple Examples.- 2.3 Cousins of the Fourier Transform.- 2.4 Multidimensional Transforms.- 2.5 * The DFT Approximation to the CTFT.- 2.6 The Fast Fourier Transform.- 2.7 * Existence Conditions.- 2.8 Problems.- 3 Fourier Inversion.- 3.1 Inverting the DFT.- 3.2 Discrete Time Fourier Series.- 3.3 Inverting the Infinite Duration DTFT.- 3.4 Inverting the CTFT.- 3.5 Continuous Time Fourier Series.- 3.6 Duality.- 3.7 Summary.- 3.8 * Orthonormal Bases.- 3.9 * Discrete Time Wavelet Transforms.- 3.10 * Two-Dimensional Inversion.- 3.11 Problems.- 4 Basic Properties.- 4.1 Linearity.- 4.2 Shifts.- 4.3 Modulation.- 4.4 Parseval’s Theorem.- 4.5 The Sampling Theorem.- 4.6 The DTFT of a Sampled Signal.- 4.7 * Pulse Amplitude Modulation (PAM).- 4.8 The Stretch Theorem.- 4.9 * Downsampling.- 4.10 * Upsampling.- 4.11 The Derivative and Difference Theorems.- 4.12 Moment Generating.- 4.13 Bandwidth and Pulse Width.- 4.14 Symmetry Properties.- 4.15 Problems.- 5 Generalized Transforms and Functions.- 5.1 Limiting Transforms.- 5.2 Periodic Signals and Fourier Series.- 5.3 Generalized Functions.- 5.4 Fourier Transforms of Generalized Functions.- 5.5 * Derivatives of Delta Functions.- 5.6 * The Generalized Function ?(g(t)).- 5.7 Impulse Trains.- 5.8 Problems.- 6 Convolution and Correlation.- 6.1 Linear Systems and Convolution.- 6.2 Convolution.- 6.3 Examples of Convolution.- 6.4 The Convolution Theorem.- 6.5 Fourier Analysis of Linear Systems.- 6.6 The Integral Theorem.- 6.7Sampling Revisited.- 6.8 Correlation.- 6.9 Parseval’s Theorem Revisited.- 6.10 * Bandwidth and Pulsewidth Revisited.- 6.11 * The Central Limit Theorem.- 6.12 Problems.- 7 Two Dimensional Fourier Analysis.- 7.1 Properties of 2-D Fourier Transforms.- 7.2 Two Dimensional Linear Systems.- 7.3 Reconstruction from Projections.- 7.4 The Inversion Problem.- 7.5 Examples of the Projection-Slice Theorem.- 7.6 Reconstruction.- 7.7 * Two-Dimensional Sampling Theory.- 7.8 Problems.- 8 Memoryless Nonlinearities.- 8.1 Memoryless Nonlinearities.- 8.2 Sinusoidal Inputs.- 8.3 Phase Modulation.- 8.4 Uniform Quantization.- 8.5 Problems.- A Fourier Transform Tables.

Erscheint lt. Verlag	30.6.1995
Reihe/Serie	The Springer International Series in Engineering and Computer Science ; 322
Zusatzinfo	XX, 361 p.
Verlagsort	Dordrecht
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Informatik
	Naturwissenschaften ► Physik / Astronomie ► Mechanik
	Technik ► Elektrotechnik / Energietechnik
ISBN-10	0-7923-9585-9 / 0792395859
ISBN-13	978-0-7923-9585-0 / 9780792395850
Zustand	Neuware