Digital Signal Processing

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1 Digital Signal Processing System Analysis and Design Paulo S. R. Diniz Eduardo A. B. da Silva and Sergio L. Netto Federal University of Rio de Janeiro CAMBRIDGE UNIVERSITY PRESS

2 Preface page xv Introduction 1 Discrete-time systems Introduction Discrete-time signals Discrete-time systems Linearity Time invariance Causality Impulse response and convolution sums Stability Difference equations and time-domain response Sampling of continuous-time signals Basic principles Sampling theorem Discrete-time signals and systems with MATLAB Summary Exercises The z and Fourier transforms Introduction Definition of the z transform Inverse z transform Computation based on residue theorem Computation based on partial-fraction expansions Computation based on polynomial division Computation based on series expansion 52 vii

3 viii 2.4 Properties of the z transform Linearity Time-reversal Time-shift theorem Multiplication by an exponential Complex differentiation Complex conjugation Real and imaginary sequences Initial value theorem Convolution theorem Product of two sequences Parseval's theorem Table of basic z transforms Transfer functions Stability in the z domain Frequency response Fourier transform Properties of the Fourier transform Linearity Time-reversal Time-shift theorem Multiplication by an exponential Complex differentiation Complex conjugation Real and imaginary sequences Symmetric and antisymmetric sequences Convolution theorem Product of two sequences Parseval's theorem Transfer functions with MATLAB Summary Exercises 81 Discrete transforms Introduction Discrete Fourier transform Properties of the DFT Linearity Time-reversal 93

4 ix Time-shift theorem Circular frequency-shift theorem (modulation theorem) Circular convolution in time Correlation Real and imaginary sequences Symmetric and antisymmetric sequences Parseval's theorem Relationship between the DFT and the z transform Digital filtering using the DFT Linear and circular convolutions Overlap-and-add method Overlap-and-save method Fast Fourier transform Radix-2 algorithm with decimation in time Decimation in frequency Radix-4 algorithm Algorithms for arbitrary values of N Alternative techniques for determining the DFT Other discrete transforms Discrete cosine transform A family of sine and cosine transforms Discrete Hartley transform Hadamard transform Other important transforms Signal representations Discrete transforms with MATLAB Summary Exercises 144 Digital filters us 4.1 Introduction Basic structures of nonrecursive digital filters Direct form Cascade form Linear-phase forms Basic structures of recursive digital filters Direct forms Cascade form 162

5 4.3.3 Parallel form Digital network analysis State-space description Basic properties of digital networks Tellegen's theorem Reciprocity Interreciprocity Transposition Sensitivity Digital filter forms with MATLAB Summary Exercises 184 FIR filter approximations i sx 5.1 Introduction Ideal characteristics of standard filters Lowpass, highpass, bandpass, and bandstop filters Differentiators Hilbert transformers Summary FIR filter approximation by frequency sampling FIR filter approximation with window functions Rectangular window Triangular windows Hamming and Hanning windows Blackman window Kaiser window Dolph-Chebyshev window Maximally flat FIR filter approximation FIR filter approximation by optimization Weighted-least-squares method Chebyshev method WLS-Chebyshev method FIR filter approximation with MATLAB Summary Exercises 248

6 xi IIR filter approximations Introduction Analog filter approximations Analog filter specification Butterworth approximation Chebyshev approximation Elliptic approximation Frequency transformations Continuous-time to discrete-time transformations Impulse-invariance method Bilinear transformation method Frequency transformation in the discrete-time domain Lowpass to lowpass transformation Lowpass to highpass transformation Lowpass to bandpass transformation Lowpass to bandstop transformation Variable cutoff filter design Magnitude and phase approximation Basic principles Multi-variable function minimization method Alternative methods Time-domain approximation IIR filter approximation with MATLAB Summary Exercises 307 Finite-precision effects Introduction Binary number representation Fixed-point representations Floating-point representation Product quantization Signal scaling Coefficient quantization Deterministic sensitivity criterion Statistical forecast of the wordlength

7 xii 7.6 Limit cycles Granular limit cycles Overflow limit cycles Elimination of zero-input limit cycles Elimination of constant-input limit cycles Forced-response stability of digital filters with nonlinearities due to overflow 7.7 Summary 7.8 Exercises 8 Multirate systems Introduction Basic principles Decimation Interpolation Examples of interpolators Rational sampling-rate changes Inverse operations Decimation and interpolation for efficient filter implementation Narrowband FIR filters Wideband FIR filters with narrow transition bands Multirate systems with MATLAB Summary Exercises Filter banks and wavelets 9.1 Introduction 9.2 Filter banks Decimation of a bandpass signal Inverse decimation of a bandpass signal Critically decimated M-band filter banks 9.3 Perfect reconstruction Noble identities Polyphase decompositions Commutator models M-band filter banks in terms of polyphase components Perfect reconstruction M-band filter banks Transmultiplexers

8 xiii 9.4 General 2-band perfect reconstruction filter banks QMF filter banks CQF filter banks Block transforms Cosine-modulated filter banks Lapped transforms Fast algorithms and biorthogonal LOT Generalized LOT Wavelet transforms Hierarchical filter banks Wavelets Scaling functions Relation between x(t) and x(n) Relation between the wavelets and the filter coefficients Regularity Examples Filter banks and wavelets with MATLAB Summary Exercises Efficient FIR structures Introduction Lattice form Filter banks using the lattice form Polyphase form Frequency-domain form Recursive running sum form Realizations with reduced number of arithmetic operations Prefilter approach Interpolation approach Frequency response masking approach Quadrature approach Efficient FIR structures with MATLAB Summary Exercises Efficient IIR structures Introduction 491

9 xiv 11.2 IIR parallel and cascade filters Parallel form Cascade form Error spectrum shaping Closed-form scaling State-space sections Optimal state-space sections State-space sections without limit cycles Lattice filters Wave filters Motivation Wave elements Lattice wave digital filters Efficient IIR structures with MATLAB Summary Exercises Implementation of DSP systems Introduction Basic elements Properties of the two's-complement representation Serial adder Serial multiplier Parallel adder Parallel multiplier Distributed arithmetic implementation Programmable logic devices ASIC implementation Digital signal processors Analog Devices DSPs Motorola DSPs Texas Instruments DSPs Summary Exercises 592 References 594 Index 604

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