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Preface | p. xvii |
Introduction | p. 1 |
Signals, Systems, and Signal Processing | p. 2 |
Basic Elements of a Digital Signal Processing System | p. 4 |
Advantages of Digital over Analog Signal Processing | p. 5 |
Classification of Signals | p. 6 |
Multichannel and Multidimensional Signals | p. 6 |
Continuous-Time Versus Discrete-Time Signals | p. 9 |
Continuous-Valued Versus Discrete-Valued Signals | p. 10 |
Deterministic Versus Random Signals | p. 11 |
The Concept of Frequency in Continuous-Time and Discrete-Time Signals | p. 12 |
Continuous-Time Sinusoidal Signals | p. 12 |
Discrete-Time Sinusoidal Signals | p. 14 |
Harmonically Related Complex Exponentials | p. 17 |
Analog-to-Digital and Digital-to-Analog Conversion | p. 19 |
Sampling of Analog Signals | p. 21 |
The Sampling Theorem | p. 26 |
Quantization of Continuous-Amplitude Signals | p. 31 |
Quantization of Sinusoidal Signals | p. 34 |
Coding of Quantized Samples | p. 35 |
Digital-to-Analog Conversion | p. 36 |
Analysis of Digital Signals and Systems Versus Discrete-Time Signals and Systems | p. 36 |
Summary and References | p. 37 |
Problems | p. 37 |
Discrete-Time Signals and Systems | p. 41 |
Discrete-Time Signals | p. 42 |
Some Elementary Discrete-Time Signals | p. 43 |
Classification of Discrete-Time Signals | p. 45 |
Simple Manipulations of Discrete-Time Signals | p. 50 |
Discrete-Time Systems | p. 53 |
Input-Output Description of Systems | p. 54 |
Block Diagram Representation of Discrete-Time Systems | p. 57 |
Classification of Discrete-Time Systems | p. 59 |
Interconnection of Discrete-Time Systems | p. 67 |
Analysis of Discrete-Time Linear Time-Invariant Systems | p. 69 |
Techniques for the Analysis of Linear Systems | p. 69 |
Resolution of a Discrete-Time Signal into Impulses | p. 71 |
Response of LTI Systems to Arbitrary Inputs: The Convolution Sum | p. 73 |
Properties of Convolution and the Interconnection of LTI Systems | p. 80 |
Causal Linear Time-Invariant Systems | p. 83 |
Stability of Linear Time-Invariant Systems | p. 85 |
Systems with Finite-Duration and Infinite-Duration Impulse Response | p. 88 |
Discrete-Time Systems Described by Difference Equations | p. 89 |
Recursive and Nonrecursive Discrete-Time Systems | p. 90 |
Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equations | p. 93 |
Solution of Linear Constant-Coefficient Difference Equations | p. 98 |
The Impulse Response of a Linear Time-Invariant Recursive System | p. 106 |
Implementation of Discrete-Time Systems | p. 109 |
Structures for the Realization of Linear Time-Invariant Systems | p. 109 |
Recursive and Nonrecursive Realizations of FIR Systems | p. 113 |
Correlation of Discrete-Time Signals | p. 116 |
Crosscorrelation and Autocorrelation Sequences | p. 118 |
Properties of the Autocorrelation and Crosscorrelation Sequences | p. 120 |
Correlation of Periodic Sequences | p. 123 |
Input-Output Correlation Sequences | p. 125 |
Summary and References | p. 128 |
Problems | p. 129 |
The z-Transform and Its Application to the Analysis of LTI Systems | p. 147 |
The z-Transform | p. 147 |
The Direct z-Transform | p. 147 |
The Inverse z-Transform | p. 156 |
Properties of the z-Transform | p. 157 |
Rational z-Transforms | p. 170 |
Poles and Zeros | p. 170 |
Pole Location and Time-Domain Behavior for Causal Signals | p. 174 |
The System Function of a Linear Time-Invariant System | p. 177 |
Inversion of the z-Transform | p. 180 |
The Inverse z-Transform by Contour Integration | p. 180 |
The Inverse z-Transform by Power Series Expansion | p. 182 |
The Inverse z-Transform by Partial-Fraction Expansion | p. 184 |
Decomposition of Rational z-Transforms | p. 192 |
Analysis of Linear Time-Invariant Systems in the z-Domain | p. 193 |
Response of Systems with Rational System Functions | p. 194 |
Transient and Steady-State Responses | p. 195 |
Causality and Stability | p. 196 |
Pole-Zero Cancellations | p. 198 |
Multiple-Order Poles and Stability | p. 200 |
Stability of Second-Order Systems | p. 201 |
The One-sided z-Transform | p. 205 |
Definition and Properties | p. 206 |
Solution of Difference Equations | p. 210 |
Response of Pole-Zero Systems with Nonzero Initial Conditions | p. 211 |
Summary and References | p. 214 |
Problems | p. 214 |
Frequency Analysis of Signals | p. 224 |
Frequency Analysis of Continuous-Time Signals | p. 225 |
The Fourier Series for Continuous-Time Periodic Signals | p. 226 |
Power Density Spectrum of Periodic Signals | p. 230 |
The Fourier Transform for Continuous-Time Aperiodic Signals | p. 234 |
Energy Density Spectrum of Aperiodic Signals | p. 238 |
Frequency Analysis of Discrete-Time Signals | p. 241 |
The Fourier Series for Discrete-Time Periodic Signals | p. 241 |
Power Density Spectrum of Periodic Signals | p. 245 |
The Fourier Transform of Discrete-Time Aperiodic Signals | p. 248 |
Convergence of the Fourier Transform | p. 251 |
Energy Density Spectrum of Aperiodic Signals | p. 254 |
Relationship of the Fourier Transform to the z-Transform | p. 259 |
The Cepstrum | p. 261 |
The Fourier Transform of Signals with Poles on the Unit Circle | p. 262 |
Frequency-Domain Classification of Signals: The Concept of Bandwidth | p. 265 |
The Frequency Ranges of Some Natural Signals | p. 267 |
Frequency-Domain and Time-Domain Signal Properties | p. 268 |
Properties of the Fourier Transform for Discrete-Time Signals | p. 271 |
Symmetry Properties of the Fourier Transform | p. 272 |
Fourier Transform Theorems and Properties | p. 279 |
Summary and References | p. 291 |
Problems | p. 292 |
Frequency-Domain Analysis of LTI Systems | p. 300 |
Frequency-Domain Characteristics of Linear Time-Invariant Systems | p. 300 |
Response to Complex Exponential and Sinusoidal Signals: The Frequency Response Function | p. 301 |
Steady-State and Transient Response to Sinusoidal Input Signals | p. 310 |
Steady-State Response to Periodic Input Signals | p. 311 |
Response to Aperiodic Input Signals | p. 312 |
Frequency Response of LTI Systems | p. 314 |
Frequency Response of a System with a Rational System Function | p. 314 |
Computation of the Frequency Response Function | p. 317 |
Correlation Functions and Spectra at the Output of LTI Systems | p. 321 |
Input-Output Correlation Functions and Spectra | p. 322 |
Correlation Functions and Power Spectra for Random Input Signals | p. 323 |
Linear Time-Invariant Systems as Frequency-Selective Filters | p. 326 |
Ideal Filter Characteristics | p. 327 |
Lowpass, Highpass, and Bandpass Filters | p. 329 |
Digital Resonators | p. 335 |
Notch Filters | p. 339 |
Comb Filters | p. 341 |
All-Pass Filters | p. 345 |
Digital Sinusoidal Oscillators | p. 347 |
Inverse Systems and Deconvolution | p. 349 |
Invertibility of Linear Time-Invariant Systems | p. 350 |
Minimum-Phase, Maximum-Phase, and Mixed-Phase Systems | p. 354 |
System Identification and Deconvolution | p. 358 |
Homomorphic Deconvolution | p. 360 |
Summary and References | p. 362 |
Problems | p. 363 |
Sampling and Reconstruction of Signals | p. 384 |
Ideal Sampling and Reconstruction of Continuous-Time Signals | p. 384 |
Discrete-Time Processing of Continuous-Time Signals | p. 395 |
Analog-to-Digital and Digital-to-Analog Converters | p. 401 |
Analog-to-Digital Converters | p. 401 |
Quantization and Coding | p. 403 |
Analysis of Quantization Errors | p. 406 |
Digital-to-Analog Converters | p. 408 |
Sampling and Reconstruction of Continuous-Time Bandpass Signals | p. 410 |
Uniform or First-Order Sampling | p. 411 |
Interleaved or Nonuniform Second-Order Sampling | p. 416 |
Bandpass Signal Representations | p. 422 |
Sampling Using Bandpass Signal Representations | p. 426 |
Sampling of Discrete-Time Signals | p. 427 |
Sampling and Interpolation of Discrete-Time Signals | p. 427 |
Representation and Sampling of Bandpass Discrete-Time Signals | p. 430 |
Oversampling A/D and D/A Converters | p. 433 |
Oversampling A/D Converters | p. 433 |
Oversampling D/A Converters | p. 439 |
Summary and References | p. 440 |
Problems | p. 440 |
The Discrete Fourier Transform: Its Properties and Applications | p. 449 |
Frequency-Domain Sampling: The Discrete Fourier Transform | p. 449 |
Frequency-Domain Sampling and Reconstruction of Discrete-Time Signals | p. 449 |
The Discrete Fourier Transform (DFT) | p. 454 |
The DFT as a Linear Transformation | p. 459 |
Relationship of the DFT to Other Transforms | p. 461 |
Properties of the DFT | p. 464 |
Periodicity, Linearity, and Symmetry Properties | p. 465 |
Multiplication of Two DFTs and Circular Convolution | p. 471 |
Additional DFT Properties | p. 476 |
Linear Filtering Methods Based on the DFT | p. 480 |
Use of the DFT in Linear Filtering | p. 481 |
Filtering of Long Data Sequences | p. 485 |
Frequency Analysis of Signals Using the DFT | p. 488 |
The Discrete Cosine Transform | p. 495 |
Forward DCT | p. 495 |
Inverse DCT | p. 497 |
DCT as an Orthogonal Transform | p. 498 |
Summary and References | p. 501 |
Problems | p. 502 |
Efficient Computation of the DFT: Fast Fourier Transform Algorithms | p. 511 |
Efficient Computation of the DFT: FFT Algorithms | p. 511 |
Direct Computation of the DFT | p. 512 |
Divide-and-Conquer Approach to Computation of the DFT | p. 513 |
Radix-2 FFT Algorithms | p. 519 |
Radix-4 FFT Algorithms | p. 527 |
Split-Radix FFT Algorithms | p. 532 |
Implementation of FFT Algorithms | p. 536 |
Applications of FFT Algorithms | p. 538 |
Efficient Computation of the DFT of Two Real Sequences | p. 538 |
Efficient Computation of the DFT of a 2N-Point Real Sequence | p. 539 |
Use of the FFT Algorithm in Linear Filtering and Correlation | p. 540 |
A Linear Filtering Approach to Computation of the DFT | p. 542 |
The Goertzel Algorithm | p. 542 |
The Chirp-z Transform Algorithm | p. 544 |
Quantization Effects in the Computation of the DFT | p. 549 |
Quantization Errors in the Direct Computation of the DFT | p. 549 |
Quantization Errors in FFT Algorithms | p. 552 |
Summary and References | p. 555 |
Problems | p. 556 |
Implementation of Discrete-Time Systems | p. 563 |
Structures for the Realization of Discrete-Time Systems | p. 563 |
Structures for FIR Systems | p. 565 |
Direct-Form Structure | p. 566 |
Cascade-Form Structures | p. 567 |
Frequency-Sampling Structures | p. 569 |
Lattice Structure | p. 574 |
Structures for IIR Systems | p. 582 |
Direct-Form Structures | p. 582 |
Signal Flow Graphs and Transposed Structures | p. 585 |
Cascade-Form Structures | p. 589 |
Parallel-Form Structures | p. 591 |
Lattice and Lattice-Ladder Structures for IIR Systems | p. 594 |
Representation of Numbers | p. 601 |
Fixed-Point Representation of Numbers | p. 601 |
Binary Floating-Point Representation of Numbers | p. 605 |
Errors Resulting from Rounding and Truncation | p. 608 |
Quantization of Filter Coefficients | p. 613 |
Analysis of Sensitivity to Quantization of Filter Coefficients | p. 613 |
Quantization of Coefficients in FIR Filters | p. 620 |
Round-Off Effects in Digital Filters | p. 624 |
Limit-Cycle Oscillations in Recursive Systems | p. 624 |
Scaling to Prevent Overflow | p. 629 |
Statistical Characterization of Quantization Effects in Fixed-Point Realizations of Digital Filters | p. 631 |
Summary and References | p. 640 |
Problems | p. 641 |
Design of Digital Filters | p. 654 |
General Considerations | p. 654 |
Causality and Its Implications | p. 655 |
Characteristics of Practical Frequency-Selective Filters | p. 659 |
Design of FIR Filters | p. 660 |
Symmetric and Antisymmetric FIR Filters | p. 660 |
Design of Linear-Phase FIR Filters Using Windows | p. 664 |
Design of Linear-Phase FIR Filters by the Frequency-Sampling Method | p. 671 |
Design of Optimum Equiripple Linear-Phase FIR Filters | p. 678 |
Design of FIR Differentiators | p. 691 |
Design of Hilbert Transformers | p. 693 |
Comparison of Design Methods for Linear-Phase FIR Filters | p. 700 |
Design of IIR Filters From Analog Filters | p. 701 |
IIR Filter Design by Approximation of Derivatives | p. 703 |
IIR Filter Design by Impulse Invariance | p. 707 |
IIR Filter Design by the Bilinear Transformation | p. 712 |
Characteristics of Commonly Used Analog Filters | p. 717 |
Some Examples of Digital Filter Designs Based on the Bilinear Transformation | p. 727 |
Frequency Transformations | p. 730 |
Frequency Transformations in the Analog Domain | p. 730 |
Frequency Transformations in the Digital Domain | p. 732 |
Summary and References | p. 734 |
Problems | p. 735 |
Multirate Digital Signal Processing | p. 750 |
Introduction | p. 751 |
Decimation by a Factor D | p. 755 |
Interpolation by a Factor I | p. 760 |
Sampling Rate Conversion by a Rational Factor I/D | p. 762 |
Implementation of Sampling Rate Conversion | p. 766 |
Polyphase Filter Structures | p. 766 |
Interchange of Filters and Downsamplers/Upsamplers | p. 767 |
Sampling Rate Conversion with Cascaded Integrator Comb Filters | p. 769 |
Polyphase Structures for Decimation and Interpolation Filters | p. 771 |
Structures for Rational Sampling Rate Conversion | p. 774 |
Multistage Implementation of Sampling Rate Conversion | p. 775 |
Sampling Rate Conversion of Bandpass Signals | p. 779 |
Sampling Rate Conversion by an Arbitrary Factor | p. 781 |
Arbitrary Resampling with Polyphase Interpolators | p. 782 |
Arbitrary Resampling with Farrow Filter Structures | p. 782 |
Applications of Multirate Signal Processing | p. 784 |
Design of Phase Shifters | p. 784 |
Interfacing of Digital Systems with Different Sampling Rates | p. 785 |
Implementation of Narrowband Lowpass Filters | p. 786 |
Subband Coding of Speech Signals | p. 787 |
Digital Filter Banks | p. 790 |
Polyphase Structures of Uniform Filter Banks | p. 794 |
Transmultiplexers | p. 796 |
Two-Channel Quadrature Mirror Filter Bank | p. 798 |
Elimination of Aliasing | p. 799 |
Condition for Perfect Reconstruction | p. 801 |
Polyphase Form of the QMF Bank | p. 801 |
Linear Phase FIR QMF Bank | p. 802 |
IIR QMF Bank | p. 803 |
Perfect Reconstruction Two-Channel FIR QMF Bank | p. 803 |
Two-Channel QMF Banks in Subband Coding | p. 806 |
M-Channel QMF Bank | p. 807 |
Alias-Free and Perfect Reconstruction Condition | p. 808 |
Polyphase Form of the M-Channel QMF Bank | p. 808 |
Summary and References | p. 813 |
Problems | p. 813 |
Linear Prediction and Optimum Linear Filters | p. 823 |
Random Signals, Correlation Functions, and Power Spectra | p. 823 |
Random Processes | p. 824 |
Stationary Random Processes | p. 825 |
Statistical (Ensemble) Averages | p. 825 |
Statistical Averages for Joint Random Processes | p. 826 |
Power Density Spectrum | p. 828 |
Discrete-Time Random Signals | p. 829 |
Time Averages for a Discrete-Time Random Process | p. 830 |
Mean-Ergodic Process | p. 831 |
Correlation-Ergodic Processes | p. 832 |
Innovations Representation of a Stationary Random Process | p. 834 |
Rational Power Spectra | p. 836 |
Relationships Between the Filter Parameters and the Autocorrelation Sequence | p. 837 |
Forward and Backward Linear Prediction | p. 838 |
Forward Linear Prediction | p. 839 |
Backward Linear Prediction | p. 841 |
The Optimum Reflection Coefficients for the Lattice Forward and Backward Predictors | p. 845 |
Relationship of an AR Process to Linear Prediction | p. 846 |
Solution of the Normal Equations | p. 846 |
The Levinson-Durbin Algorithm | p. 847 |
The Schur Algorithm | p. 850 |
Properties of the Linear Prediction-Error Filters | p. 855 |
AR Lattice and ARMA Lattice-Ladder Filters | p. 858 |
AR Lattice Structure | p. 858 |
ARMA Processes and Lattice-Ladder Filters | p. 860 |
Wiener Filters for Filtering and Prediction | p. 863 |
FIR Wiener Filter | p. 864 |
Orthogonality Principle in Linear Mean-Square Estimation | p. 866 |
IIR Wiener Filter | p. 867 |
Noncausal Wiener Filter | p. 872 |
Summary and References | p. 873 |
Problems | p. 874 |
Adaptive Filters | p. 880 |
Applications of Adaptive Filters | p. 880 |
System Identification or System Modeling | p. 882 |
Adaptive Channel Equalization | p. 883 |
Echo Cancellation in Data Transmission over Telephone Channels | p. 887 |
Suppression of Narrowband Interference in a Wideband Signal | p. 891 |
Adaptive Line Enhancer | p. 895 |
Adaptive Noise Cancelling | p. 896 |
Linear Predictive Coding of Speech Signals | p. 897 |
Adaptive Arrays | p. 900 |
Adaptive Direct-Form FIR Filters-The LMS Algorithm | p. 902 |
Minimum Mean-Square-Error Criterion | p. 903 |
The LMS Algorithm | p. 905 |
Related Stochastic Gradient Algorithms | p. 907 |
Properties of the LMS Algorithm | p. 909 |
Adaptive Direct-Form Filters-RLS Algorithms | p. 916 |
RLS Algorithm | p. 916 |
The LDU Factorization and Square-Root Algorithms | p. 921 |
Fast RLS Algorithms | p. 923 |
Properties of the Direct-Form RLS Algorithms | p. 925 |
Adaptive Lattice-Ladder Filters | p. 927 |
Recursive Least-Squares Lattice-Ladder Algorithms | p. 928 |
Other Lattice Algorithms | p. 949 |
Properties of Lattice-Ladder Algorithms | p. 950 |
Summary and References | p. 954 |
Problems | p. 955 |
Power Spectrum Estimation | p. 960 |
Estimation of Spectra from Finite-Duration Observations of Signals | p. 961 |
Computation of the Energy Density Spectrum | p. 961 |
Estimation of the Autocorrelation and Power Spectrum of Random Signals: The Periodogram | p. 966 |
The Use of the DFT in Power Spectrum Estimation | p. 971 |
Nonparametric Methods for Power Spectrum Estimation | p. 974 |
The Bartlett Method: Averaging Periodograms | p. 974 |
The Welch Method: Averaging Modified Periodograms | p. 975 |
The Blackman and Tukey Method: Smoothing the Periodogram | p. 978 |
Performance Characteristics of Nonparametric Power Spectrum Estimators | p. 981 |
Computational Requirements of Nonparametric Power Spectrum Estimates | p. 984 |
Parametric Methods for Power Spectrum Estimation | p. 986 |
Relationships Between the Autocorrelation and the Model Parameters | p. 988 |
The Yule-Walker Method for the AR Model Parameters | p. 990 |
The Burg Method for the AR Model Parameters | p. 991 |
Unconstrained Least-Squares Method for the AR Model Parameters | p. 994 |
Sequential Estimation Methods for the AR Model Parameters | p. 995 |
Selection of AR Model Order | p. 996 |
MA Model for Power Spectrum Estimation | p. 997 |
ARMA Model for Power Spectrum Estimation | p. 999 |
Some Experimental Results | p. 1001 |
Filter Bank Methods | p. 1009 |
Filter Bank Realization of the Periodogram | p. 1010 |
Minimum Variance Spectral Estimates | p. 1012 |
Eigenanalysis Algorithms for Spectrum Estimation | p. 1015 |
Pisarenko Harmonic Decomposition Method | p. 1017 |
Eigen-decomposition of the Autocorrelation Matrix for Sinusoids in White Noise | p. 1019 |
MUSIC Algorithm | p. 1021 |
ESPRIT Algorithm | p. 1022 |
Order Selection Criteria | p. 1025 |
Experimental Results | p. 1026 |
Summary and References | p. 1029 |
Problems | p. 1030 |
Random Number Generators | p. 1041 |
Tables of Transition Coefficients for the Design of Linear-Phase FIR Filters | p. 1047 |
References and Bibliography | p. 1053 |
Answers to Selected Problems | p. 1067 |
Index | p. 1077 |
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