14 Digital Signal Processing 14.1 Fourier transforms Introduction The Classical Fourier Trans Fourier Series Representation of CT Period dic signals generalized Complex Fourier Transform. DT Fourier Transform Relationship between the CT and DT Spectra. Discrete Fourier Transform 14.2 Fourier Transforms and the Fast Fourier transform The Discrete Time Fourier Transform(DTFr).Relationship to the Z-Transform. Properties. Fourier Transforms of Finite Time W. Kenneth Jenkins Sequences. Frequency Response of LTI Discrete Systems. Th University of Illinois Discrete Fourier Transform. Properties of the DFT. Relation etween DFT and Fourier Transform. Power, Amplitude, and Phase Alexander D, Poularikas Spectra.Observations. Data Windowing. Fast Fourier Transform. Computation of the Inverse DFT Bruce w. bomar 14.3 Design and Implementation of Digital Filters Finite Impulse Response Filter Design. Infinite Impulse Response University of Tennessee Space Filter Design. Finite Impulse Response Filter Implementation Infinite Impulse Response Filter Implementation L. Montgomery Smith 14.4 Signal Restoration University of Tennessee Space ntroduction Attribute Sets: Closed Attribute Sets. Closed Convex Sets. Closed Project tors.Algebraic Properties of Matrices. Structural Pr James A Cadzow Nonnegative Sequence Approximation.Exponential Signals and vanderbilt University the Data Matrix. Recursive Modeling of Data 14.1 Fourier Transforms W. Kenneth Jenkins Introduction The Fourier transform is a mathematical tool that is used to expand signals into a spectrum of sinusoidal components to facilitate signal analysis and system performance. In certain applications the Fourier transform is used for spectral analysis, or for spectrum shaping that adjusts the relative contributions of different frequency components in the filtered result. In other applications the Fourier transform is important for its ability to decompose the input signal into uncorrelated components, so that signal processing can be more effectively implemented on the individual spectral components. Decorrelating properties of the Fourier transform are important in frequency domain adaptive filtering, subband coding, image compression, and transform coding thods such as the Fourier series and the Fourier integral are used for continuous-time (CT) signals and systems, i.e., systems in which the signals are defined at all values of t on the continuum <t<oo. A more recently developed set of discrete Fourier methods, including the discrete-time(DT) Fourier transform and the discrete Fourier transform(DFT), are extensions of basic Fourier concepts for DT signals and systems. A DT signal is defined only for integer values of n in the range -oo< n oo. The class of D c 2000 by CRC Press LLC© 2000 by CRC Press LLC 14 Digital Signal Processing 14.1 Fourier Transforms Introduction • The Classical Fourier Transform for CT Signals • Fourier Series Representation of CT Periodic Signals • Generalized Complex Fourier Transform • DT Fourier Transform • Relationship between the CT and DT Spectra • Discrete Fourier Transform 14.2 Fourier Transforms and the Fast Fourier Transform The Discrete Time Fourier Transform (DTFT) • Relationship to the Z-Transform • Properties • Fourier Transforms of Finite Time Sequences • Frequency Response of LTI Discrete Systems • The Discrete Fourier Transform • Properties of the DFT • Relation between DFT and Fourier Transform • Power, Amplitude, and Phase Spectra • Observations • Data Windowing • Fast Fourier Transform • Computation of the Inverse DFT 14.3 Design and Implementation of Digital Filters Finite Impulse Response Filter Design • Infinite Impulse Response Filter Design • Finite Impulse Response Filter Implementation • Infinite Impulse Response Filter Implementation 14.4 Signal Restoration Introduction • Attribute Sets: Closed Subspaces • Attribute Sets: Closed Convex Sets • Closed Projection Operators • Algebraic Properties of Matrices • Structural Properties of Matrices • Nonnegative Sequence Approximation • Exponential Signals and the Data Matrix • Recursive Modeling of Data 14.1 Fourier Transforms W. Kenneth Jenkins Introduction The Fourier transform is a mathematical tool that is used to expand signals into a spectrum of sinusoidal components to facilitate signal analysis and system performance. In certain applications the Fourier transform is used for spectral analysis, or for spectrum shaping that adjusts the relative contributions of different frequency components in the filtered result. In other applications the Fourier transform is important for its ability to decompose the input signal into uncorrelated components, so that signal processing can be more effectively implemented on the individual spectral components. Decorrelating properties of the Fourier transform are important in frequency domain adaptive filtering, subband coding, image compression, and transform coding. 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 the signals are defined at all values of t on the continuum –• < t < •. A more recently developed set of discrete Fourier methods, including the discrete-time (DT) Fourier transform and the discrete Fourier transform (DFT), are extensions of basic Fourier concepts for DT signals and systems. A DT signal is defined only for integer values of n in the range –• < n < •. The class of DT W. Kenneth Jenkins University of Illinois Alexander D. Poularikas University of Alabama in Huntsville Bruce W. Bomar University of Tennessee Space Institute L. Montgomery Smith University of Tennessee Space Institute James A. Cadzow Vanderbilt University