Digital Signal Processing: Principles, Algorithms, and Applications, 5th edition

• John G. Proakis
• Dimitris G Manolakis

Overview

Digital Signal Processing presents the fundamental concepts and techniques of discrete-time signals, systems, and modern digital processing as well as related algorithms and applications for students in electrical engineering, computer engineering, and computer science departments.

Covering both time-domain and frequency-domain methods for the analysis of linear, discrete-time systems, the 5th Edition includes a new chapter on multirate digital filter banks and wavelets. Several new topics have been added to existing chapters, including the short-time Fourier Transform, the sparse FFT algorithm, ARMA model parameter estimation, and reverberation filters. Rigorous and challenging, it includes numerous examples and over 500 homework and computer problems that emphasize software implementation of digital signal processing algorithms.

1. Introduction

1.1 Signals, Systems, and Signal Processing
1.1.1 Basic Elements of a Digital Signal Processing System
1.1.2 Advantages of Digital over Analog Signal Processing
1.2 Classification of Signals
1.2.1 Multichannel and Multidimensional Signals
1.2.2 Continuous-Time Versus Discrete-Time Signals
1.2.3 Continuous-Valued Versus Discrete-Valued Signals
1.2.4 Deterministic Versus Random Signals
1.3 Summary
Problems

2. Discrete-Time Signals and Systems
2.1 Discrete-Time Signals
2.1.1 Some Elementary Discrete-Time Signals
2.1.2 Classification of Discrete-Time Signals
2.1.3 Simple Manipulations of Discrete-Time Signals
2.2 Discrete-Time Systems
2.2.1 Input-Output Description of Systems
2.2.2 Block Diagram Representation of Discrete-Time Systems
2.2.3 Classification of Discrete-Time Systems
2.2.4 Interconnection of Discrete-Time Systems
2.3 Analysis of Discrete-Time Linear Time-Invariant Systems
2.3.1 Techniques for the Analysis of Linear Systems
2.3.2 Resolution of a Discrete-Time Signal into Impulses
2.3.3 Response of LTI Systems to Arbitrary Inputs: The Convolution Sum
2.3.4 Properties of Convolution and the Interconnection of LTI Systems
2.3.5 Causal Linear Time-Invariant Systems
2.3.6 Stability of Linear Time-Invariant Systems
2.3.7 Systems with Finite-Duration and Infinite-Duration Impulse Response
2.4 Discrete-Time Systems Described by Difference Equations
2.4.1 Recursive and Nonrecursive Discrete-Time Systems
2.4.2 Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equations
2.4.3 Application of LTI Systems for Signal Smoothing
2.5 Implementation of Discrete-Time Systems
2.5.1 Structures for the Realization of Linear Time-Invariant Systems
2.5.2 Recursive and Nonrecursive Realizations of FIR Systems
2.6 Correlation of Discrete-Time Signals
2.6.1 Crosscorrelation and Autocorrelation Sequences
2.6.2 Properties of the Autocorrelation and Crosscorrelation Sequences
2.6.3 Correlation of Periodic Sequences
2.6.4 Input-Output Correlation Sequences
2.7 Summary
Problems
Computer Problems

3.The z-Transform and Its Application to the Analysis of LTI Systems
3.1 The z-Transform
3.1.1 The Direct z-Transform
3.1.2 The Inverse z-Transform
3.2 Properties of the z-Transform
3.3 Rational z-Transforms
3.3.1 Poles and Zeros
3.3.2 Pole Location and Time-Domain Behavior for Causal Signals
3.3.3 The System Function of a Linear Time-Invariant System
3.4 Inversion of the z-Transform
3.4.1 The Inverse z-Transform by Contour Integration
3.4.2 The Inverse z-Transform by Power Series Expansion
3.4.3 The Inverse z-Transform by Partial-Fraction Expansion
3.4.4 Decomposition of Rational z-Transforms
3.5 Analysis of Linear Time-Invariant Systems in the z-Domain
3.5.1 Response of Systems with Rational System Functions
3.5.3 Causality and Stability
3.5.4 Pole–Zero Cancellations
3.5.5 Multiple-Order Poles and Stability
3.5.6 Stability of Second-Order Systems
3.6 The One-sided z-Transform
3.6.1 Definition and Properties
3.6.2 Solution of Difference Equations
3.6.3 Response of Pole–Zero Systems with Nonzero Initial Conditions
3.7 Summary
Problems
Computer Problems

4Frequency Analysis of Signals
4.1 The Concept of Frequency in Continuous-Time and Discrete-Time Signals
4.1.1 Continuous-Time Sinusoidal Signals
4.1.2 Discrete-Time Sinusoidal Signals
4.1.3 Harmonically Related Complex Exponentials
4.1.4 Sampling of Analog Signals
4.1.5 The Sampling Theorem
4.2 Frequency Analysis of Continuous-Time Signals
4.2.1 The Fourier Series for Continuous-Time Periodic Signals
4.2.2 Power Density Spectrum of Periodic Signals
4.2.3 The Fourier Transform for Continuous-Time Aperiodic Signals
4.2.4 Energy Density Spectrum of Aperiodic Signals
4.3 Frequency Analysis of Discrete-Time Signals
4.3.1 The Fourier Series for Discrete-Time Periodic Signals
4.3.2 Power Density Spectrum of Periodic Signals
4.3.3 The Fourier Transform of Discrete-Time Aperiodic Signals
4.3.4 Convergence of the Fourier Transform
4.3.5 Energy Density Spectrum of Aperiodic Signals
4.3.6 Relationship of the Fourier Transform to the z-Transform
4.3.7 The Cepstrum
4.3.8 The Fourier Transform of Signals with Poles on the Unit Circle
4.3.9 Frequency-Domain Classification of Signals: The Concept of Bandwidth
4.3.10 The Frequency Ranges of Some Natural Signals
4.4 Frequency-Domain and Time-Domain Signal Properties
4.5 Properties of the Fourier Transform for Discrete-Time Signals
4.5.1 Symmetry Properties of the Fourier Transform
4.5.2 Fourier Transform Theorems and Properties
4.6 Summary
Problems
Computer Problems

5Frequency-Domain Analysis of LTI Systems
5.1 Frequency-Domain Characteristics of Linear Time-Invariant Systems
5.1.1 Response to Complex Exponential and Sinusoidal Signals: The Frequency Response Function
5.1.2 Steady-State and Transient Response to Sinusoidal Input Signals
5.1.3 Steady-State Response to Periodic Input Signals
5.1.4 Steady-State Response to Aperiodic Input Signals
5.2 Frequency Response of LTI Systems
5.2.1 Frequency Response of a System with a Rational System Function
5.2.2 Computation of the Frequency Response Function
5.3 Correlation Functions and Spectra at the Output of LTI Systems
5.4 Linear Time-Invariant Systems as Frequency-Selective Filters
5.4.1 Ideal Filter Characteristics
5.4.2 Lowpass, Highpass, and Bandpass Filters
5.4.3 Digital Resonators
5.4.4 Notch Filters
5.4.5 Comb Filters
5.4.6 Reverberation Filters
5.4.7 All-Pass Filters
5.4.8 Digital Sinusoidal Oscillators
5.5 Inverse Systems and Deconvolution
5.5.1 Invertibility of Linear Time-Invariant Systems
5.5.2 Minimum-Phase, Maximum-Phase, and Mixed-Phase Systems
5.5.3 System Identification and Deconvolution
5.5.4 Homomorphic Deconvolution
5.6 Summary
Problems
Computer Problems

6Sampling and Reconstruction of Signals
6.1 Ideal Sampling and Reconstruction of Continuous-Time Signals
6.2 Discrete-Time Processing of Continuous-Time Signals
6.3 Sampling and Reconstruction of Continuous-Time Bandpass Signals
6.3.1 Uniform or First-Order Sampling
6.3.2 Interleaved or Nonuniform Second-Order Sampling
6.3.3 Bandpass Signal Representations
6.3.4 Sampling Using Bandpass Signal Representations
6.4 Sampling of Discrete-Time Signals
6.4.1 Sampling and Interpolation of Discrete-Time Signals
6.4.2 Representation and Sampling of Bandpass Discrete-Time Signals
6.5 Analog-to-Digital and Digital-to-Analog Converters
6.5.1 Analog-to-Digital Converters
6.5.2 Quantization and Coding
6.5.3 Analysis of Quantization Errors
6.5.4 Digital-to-Analog Converters
6.6 Oversampling A/D and D/A Converters
6.6.1 Oversampling A/D Converters
6.6.2 Oversampling D/A Converters
6.7 Summary
Problems
Computer Problems

7The Discrete Fourier Transform: Its Propertiesand Applications
7.1 Frequency-Domain Sampling: The Discrete Fourier Transform
7.1.1 Frequency-Domain Sampling and Reconstruction of Discrete-Time Signals
7.1.2 The Discrete Fourier Transform (DFT)
7.1.3 The DFT as a Linear Transformation
7.1.4 Relationship of the DFT to Other Transforms
7.2 Properties of the DFT
7.2.1 Periodicity, Linearity, and Symmetry Properties
7.2.2 Multiplication of Two DFTs and Circular Convolution
7.3 Linear Filtering Methods Based on the DFT
7.3.1 Use of the DFT in Linear Filtering
7.3.2 Filtering of Long Data Sequences
7.4 Frequency Analysis of Signals Using the DFT
7.5 The Short-Time Fourier Transform
7.6 The Discrete Cosine Transform
7.6.1 Forward DCT
7.6.2 Inverse DCT
7.6.3 DCT as an Orthogonal Transform
7.7 Summary
Problems
Computer Problems

8Efficient Computation of the DFT: Fast Fourier Transform Algorithms
8.1 Efficient Computation of the DFT: FFT Algorithms
8.1.1 Direct Computation of the DFT
8.1.2 Divide-and-Conquer Approach to Computation of the DFT
8.1.6 Implementation of FFT Algorithms
8.1.7 Sparse FFT Algorithm
8.2 Applications of FFT Algorithms
8.2.1 Efficient Computation of the DFT of Two Real Sequences
8.2.2 Efficient Computation of the DFT of a 2N-Point Real Sequence
8.2.3 Use of the FFT Algorithm in Linear Filtering and Correlation
8.3 A Linear Filtering Approach to Computation of the DFT
8.3.1 The Goertzel Algorithm
8.3.2 The Chirp-z Transform Algorithm
8.4 Quantization Effects in the Computation of the DFT
8.4.1 Quantization Errors in the Direct Computation of the DFT
8.4.2 Quantization Errors in FFT Algorithms
8.5 Summary
Problems
Computer Problems

9Implementation of Discrete-Time Systems
9.1 Structures for the Realization of Discrete-Time Systems
9.2 Structures for FIR Systems
9.2.1 Direct-Form Structure
9.2.3 Frequency-Sampling Structures
9.2.4 Lattice Structure
9.3 Structures for IIR Systems
9.3.1 Direct-Form Structures
9.3.2 Signal Flow Graphs and Transposed Structures
9.3.4 Parallel-Form Structures
9.3.5 Lattice and Lattice-Ladder Structures for IIR Systems
9.4 Representation of Numbers
9.4.1 Fixed-Point Representation of Numbers
9.4.2 Binary Floating-Point Representation of Numbers
9.4.3 Errors Resulting from Rounding and Truncation
9.5 Quantization of Filter Coefficients
9.5.1 Analysis of Sensitivity to Quantization of Filter Coefficients
9.5.2 Quantization of Coefficients in FIR Filters
9.6 Round-Off Effects in Digital Filters
9.6.1 Limit-Cycle Oscillations in Recursive Systems
9.6.2 Scaling to Prevent Overflow
9.6.3 Statistical Characterization of Quantization Effects in Fixed-Point Realizations of Digital Filters
9.7 Summary
Problems
Computer Problems

10Design of Digital Filters
10.1 General Considerations
10.1.1 Causality and Its Implications
10.1.2 Characteristics of Practical Frequency-Selective Filters
10.2 Design of FIR Filters
10.2.1 Symmetric and Antisymmetric FIR Filters
10.2.2 Design of Linear-Phase FIR Filters Using Windows
10.2.3 Design of Linear-Phase FIR Filters by the Frequency-Sampling Method
10.2.4 Design of Optimum Equiripple Linear-Phase FIR Filters
10.2.5 Design of FIR Differentiators
10.2.6 Design of Hilbert Transformers
10.2.7 Comparison of Design Methods for Linear-Phase FIR Filters
10.3 Design of IIR Filters From Analog Filters
10.3.1 IIR Filter Design by Approximation of Derivatives
10.3.2 IIR Filter Design by Impulse Invariance
10.3.3 IIR Filter Design by the Bilinear Transformation
10.3.4 Characteristics of Commonly Used Analog Filters
10.3.5 Some Examples of Digital Filter Designs Based on the Bilinear Transformation
10.4 Frequency Transformations
10.4.1 Frequency Transformations in the Analog Domain
10.4.2 Frequency Transformations in the Digital Domain
10.5 Summary
Problems
Computer Problems

11Multirate Digital Signal Processing
11.1 Introduction
11.2 Decimation by a Factor D
11.3 Interpolation by a Factor I
11.4 Sampling Rate Conversion by a Rational Factor I /D
11.5 Implementation of Sampling Rate Conversion
11.5.1 Polyphase Filter Structures
11.5.2 Interchange of Filters and Downsamplers/Upsamplers
11.5.3 Sampling Rate Conversion with Cascaded Integrator Comb Filters
11.5.4 Polyphase Structures for Decimation and Interpolation Filters
11.5.5 Structures for Rational Sampling Rate Conversion
11.6 Multistage Implementation of Sampling Rate Conversion
11.7 Sampling Rate Conversion of Bandpass Signals
11.8 Sampling Rate Conversion by an Arbitrary Factor
11.8.1 Arbitrary Resampling with Polyphase Interpolators
11.8.2 Arbitrary Resampling with Farrow Filter Structures
11.9 Applications of Multirate Signal Processing
11.9.1 Design of Phase Shifters
11.9.2 Interfacing of Digital Systems with Different Sampling Rates
11.9.3 Implementation of Narrowband Lowpass Filters
11.9.4 Subband Coding of Speech Signals
11.10 Summary
Problems
Computer Problems

12MultirateDigital Filter Banks and Wavelets
12.1 Multirate Digital Filter Banks
12.1.1 DFT Filter Banks
12.1.2 Polyphase Structure of the Uniform DFT Filter Bank
12.1.3 An Alternative Structure of the Uniform DFT Filter Bank
12.2 Two-Channel Quadrature Mirror Filter Bank
12.2.1 Elimination of Aliasing
12.2.2 Polyphase Structure of the QMF Bank
12.2.3 Condition for Perfect Reconstruction
12.2.4 Linear Phase FIR QMF Bank
12.2.5 IIR QMF Bank
12.2.6 Perfect Reconstruction in Two-Channel FIR QMF Bank
12.2.7 Two-Channel Paraunitary QMF Bank
12.2.8 Orthogonal and Biorthogonal Two-channel FIR Filter Banks
12.2.9 Two-Channel QMF Banks in Subband Coding
12.3 M-Channel Filter Banks
12.3.1 Polyphase Structure for the M-Channel Filter Bank
12.3.2 M-Channel Paraunitary Filter Banks
12.4 Wavelets and Wavelet Transforms
12.4.1 Ideal Bandpass Wavelet Decomposition
12.4.2 Signal Spaces and Wavelets
12.4.3 Multiresolution Analysis and Wavelets
12.4.4 The Discrete Wavelet Transform
12.5 From Wavelets to Filter Banks
12.5.1 Dilation Equations
12.5.2 Orthogonality Conditions
12.5.3 Implications of Orthogonality and Dilation Equations
12.6 From Filter Banks to Wavelets
12.7 Regular Filters and Wavelets
12.8 Summary
Problems
Computer Problems

13Linear Prediction and Optimum Linear Filters
13.1 Random Signals, Correlation Functions, and Power Spectra
13.1.1 Random Processes
13.1.2 Stationary Random Processes
13.1.3 Statistical (Ensemble) Averages
13.1.4 Statistical Averages for Joint Random Processes
13.1.5 Power Density Spectrum
13.1.6 Discrete-Time Random Signals
13.1.7 Time Averages for a Discrete-Time Random Process
13.1.8 Mean-Ergodic Process
13.1.9 Correlation-Ergodic Processes
13.1.10 Correlation Functions and Power Spectra for Random Input Signals to LTI Systems
13.2 Innovations Representation of a Stationary Random Process
13.2.1 Rational Power Spectra
13.2.2 Relationships Between the Filter Parameters and the Autocorrelation Sequence
13.3 Forward and Backward Linear Prediction
13.3.1 Forward Linear Prediction
13.3.2 Backward Linear Prediction
13.3.3 The Optimum Reflection Coefficients for the Lattice Forward and Backward Predictors
13.3.4 Relationship of an AR Process to Linear Prediction
13.4 Solution of the Normal Equations
13.4.1 The Levinson–Durbin Algorithm
13.5 Properties of the Linear Prediction-Error Filters
13.6 AR Lattice and ARMA Lattice-Ladder Filters
13.6.1 AR Lattice Structure
13.6.2 ARMA Processes and Lattice-Ladder Filters
13.7 Wiener Filters for Filtering and Prediction
13.7.1 FIR Wiener Filter
13.7.2 Orthogonality Principle in Linear Mean-Square Estimation
13.7.3 IIR Wiener Filter
13.7.4 Noncausal Wiener Filter
13.8 Summary
Problems
Computer Problems

14.1.1 System Identification or System Modeling
14.1.3 Suppression of Narrowband Interference in a Wideband Signal
14.2 Adaptive Direct-Form FIR Filters - The LMS Algorithm
14.2.1 Minimum Mean-Square-Error Criterion
14.2.2 The LMS Algorithm
14.2.4 Properties of the LMS Algorithm
14.3 Adaptive Direct-Form Filters - RLS Algorithms
14.3.1 RLS Algorithm
14.3.2 The LDU Factorization and Square-Root Algorithms
14.3.3 Fast RLS Algorithms
14.3.4 Properties of the Direct-Form RLS Algorithms
14.4.2 Other Lattice Algorithms
14.5 Stability and Robustness of Adaptive Filter Algorithms
14.6 Summary
Problems
Computer Problems

15Power Spectrum Estimation
15.1 Estimation of Spectra from Finite-Duration Observations of Signals
15.1.1 Computation of the Energy Density Spectrum
15.1.2 Estimation of the Autocorrelation and Power Spectrum of Random Signals: The Periodogram
15.1.3 The Use of the DFT in Power Spectrum Estimation
15.2 Nonparametric Methods for Power Spectrum Estimation
15.2.1 The Bartlett Method: Averaging Periodograms
15.2.2 The Welch Method: Averaging Modified Periodograms
15.2.3 The Blackman and Tukey Method: Smoothing the Periodogram
15.2.4 Performance Characteristics of Nonparametric Power Spectrum Estimators
15.2.5 Computational Requirements of Nonparametric Power Spectrum Estimates
15.3 Parametric Methods for Power Spectrum Estimation
15.3.1 Relationships Between the Autocorrelation and the Model Parameters
15.3.2 The Yule–Walker Method for the AR Model Parameters
15.3.3 The Burg Method for the AR Model Parameters
15.3.4 Unconstrained Least-Squares Method for the AR Model Parameters
15.3.5 Sequential Estimation Methods for the AR Model Parameters
15.3.6 Selection of AR Model Order
15.3.7 MA Model for Power Spectrum Estimation
15.3.8 ARMA Model for Power Spectrum Estimation
15.3.9 Some Experimental Results
15.4 ARMA Model Parameter Estimation
15.5 Filter Bank Methods
15.5.1 Filter Bank Realization of the Periodogram
15.5.2 Minimum Variance Spectral Estimates
15.6 Eigenanalysis Algorithms for Spectrum Estimation
15.6.1 Pisarenko Harmonic Decomposition Method
15.6.2 Eigen-decomposition of the Autocorrelation Matrix for Sinusoids in White Noise
15.6.3 MUSIC Algorithm
15.6.4 ESPRIT Algorithm
15.6.5 Order Selection Criteria
15.6.6 Experimental Results
15.7 Summary
Problems
Computer Problems

A. Random Number Generators
B.Tables of Transition Coefficients for the Design of Linear-Phase FIR Filters

References and Bibliography