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01 Fourier Series, Fourier Transforms, and the DFT

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Fourier methods are commonly used for signal analysis and system design in modern telecommunications, radar, and image processing systems. | Jenkins W.K. Fourier Series Fourier Transforms and the DFT Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton CRC Press LLC 1999 1999 by CRC Press LLC 1 Fourier Series Fourier Transforms and the DFT W. Kenneth Jenkins University of Illinois Urbana-Champaign 1.1 Introduction 1.2 Fourier Series Representation of Continuous Time Periodic Signals Exponential Fourier Series The Trigonometric Fourier Series Convergence of the Fourier Series 1.3 The Classical Fourier Transform for Continuous Time Signals Properties of the Continuous Time Fourier Transform Fourier Spectrum of the Continuous Time Sampling Model Fourier Transform of Periodic Continuous Time Signals The Generalized Complex Fourier Transform 1.4 The Discrete Time Fourier Transform Properties of the Discrete Time Fourier Transform Relationship between the Continuous and Discrete Time Spectra 1.5 The Discrete Fourier Transform Properties of the Discrete Fourier Series Fourier Block Processing in Real-Time Filtering Applications Fast Fourier Transform Algorithms 1.6 Family Tree of Fourier Transforms 1.7 Selected Applications of Fourier Methods Fast Fourier Transform in Spectral Analysis Finite Impulse Response Digital Filter Design Fourier Analysis of Ideal and Practical Digital-to-Analog Conversion 1.8 Summary References 1.1 Introduction Fourier methods are commonly used for signal analysis and system design in modern telecommunications radar and image processing systems. Classical Fourier methods such as the Fourier series and the Fourier integral are used for continuous time CT signals and systems i.e. systems in which a characteristic signal s t is defined at all values of t on the continuum 1 t 1 .A more recently developed set of Fourier methods including the discrete time Fourier transform DTFT and the discrete Fourier transform DFT are extensions of basic Fourier concepts that apply to discrete time DT signals. A characteristic DT signal s n is defined only for .