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CONTENTS

1 Introduction
2 Discrete-Time Signals and Systems
2.0 Introduction
2.1 Discrete-Time Signals
2.2 Discrete-Time Systems
2.2.1 Memoryless Systems
2.2.2 Linear Systems
2.2.3 Time-Invariant Systems
2.2.4 Causality
2.2.5 Stability
2.3 LTI Systems
2.4 Properties of Linear Time-Invariant Systems
2.5 Linear Constant-Coefficient Difference Equations
2.6 Frequency-Domain Representation of Discrete-Time Signals and Systems
2.6.1 Eigenfunctions for Linear Time-Invariant Systems
2.6.2 Suddenly Applied Complex Exponential Inputs
2.7 Representation of Sequences by Fourier Transforms
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems
2.9.1 Linearity of the Fourier Transform
2.9.2 Time Shifting and Frequency Shifting Theorem
2.9.3 Time Reversal Theorem
2.9.4 Differentiation in Frequency Theorem
2.9.5 Parseval��s Theorem
2.9.6 The Convolution Theorem
2.9.7 The Modulation or Windowing Theorem
2.10 Discrete-Time Random Signals
2.11 Summary
Problems
3 The z-Transform
3.0 Introduction
3.1 z-Transform
3.2 Properties of the ROC for the z-Transform
3.3 The Inverse z-Transform
3.3.1 Inspection Method
3.3.2 Partial Fraction Expansion
3.3.3 Power Series Expansion
3.4 z-Transform Properties
3.4.1 Linearity
3.4.2 Time Shifting
3.4.3 Multiplication by an Exponential Sequence
3.4.4 Differentiation of X(z)
3.4.5 Conjugation of a Complex Sequence
3.4.6 Time Reversal
3.4.7 Convolution of Sequences
3.4.8 Summary of Some z-Transform Properties
3.5 z-Transforms and LTI Systems
3.6 The Unilateral z-Transform
3.7 Summary
Problems
4 Sampling of Continuous-Time Signals
4.0 Introduction
4.1 Periodic Sampling
4.2 Frequency-Domain Representation of Sampling
4.3 Reconstruction of a Bandlimited Signal from Its Samples
4.4 Discrete-Time Processing of Continuous-Time Signals
4.4.1 Discrete-Time LTI Processing of Continuous-Time Signals
4.4.2 Impulse Invariance
4.5 Continuous-Time Processing of Discrete-Time Signals
4.6 Changing the Sampling Rate Using Discrete-Time Processing
4.6.1 Sampling Rate Reduction by an Integer Factor
4.6.2 Increasing the Sampling Rate by an Integer Factor
4.6.3 Simple and Practical Interpolation Filters
4.6.4 Changing the Sampling Rate by a Noninteger Factor
4.7 Multirate Signal Processing
4.7.1 Interchange of Filtering with Compressor/Expander
4.7.2 Multistage Decimation and Interpolation
4.7.3 Polyphase Decompositions
4.7.4 Polyphase Implementation of Decimation Filters
4.7.5 Polyphase Implementation of Interpolation Filters
4.7.6 Multirate Filter Banks
4.8 Digital Processing of Analog Signals
4.8.1 Prefiltering to Avoid Aliasing
4.8.2 A/D Conversion
4.8.3 Analysis of Quantization Errors
4.8.4 D/A Conversion
4.9 Oversampling and Noise Shaping in A/D and D/A Conversion
4.9.1 Oversampled A/D Conversion with Direct Quantization
4.9.2 Oversampled A/D Conversion with Noise Shaping
4.9.3 Oversampling and Noise Shaping in D/A Conversion
4.10 Summary
Problems
5 Transform Analysis of Linear Time-Invariant Systems
5.0 Introduction
5.1 The Frequency Response of LTI Systems
5.1.1 Frequency Response Phase and Group Delay
5.1.2 Illustration of Effects of Group Delay and Attenuation
5.2 System Functions��Linear Constant-Coefficient Difference Equations
5.2.1 Stability and Causality
5.2.2 Inverse Systems
5.2.3 Impulse Response for Rational System Functions
5.3 Frequency Response for Rational System Functions
5.3.1 Frequency Response of 1st-Order Systems
5.3.2 Examples with Multiple Poles and Zeros
5.4 Relationship between Magnitude and Phase
5.5 All-Pass Systems
5.6 Minimum-Phase Systems
5.6.1 Minimum-Phase and All-Pass Decomposition
5.6.2 Frequency-Response Compensation of Non-Minimum-Phase Systems
5.6.3 Properties of Minimum-Phase Systems
5.7 Linear Systems with Generalized Linear Phase
5.7.1 Systems with Linear Phase
5.7.2 Generalized Linear Phase
5.7.3 Causal Generalized Linear-Phase Systems
5.7.4 Relation of FIR Linear-Phase Systems to Minimum-Phase Systems
5.8 Summary
Problems
6 Structures for Discrete-Time Systems
6.0 Introduction
6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations
6.2 Signal Flow Graph Representation
6.3 Basic Structures for IIR Systems
6.3.1 Direct Forms
6.3.2 Cascade Form
6.3.3 Parallel Form
6.3.4 Feedback in IIR Systems
6.4 Transposed Forms
6.5 Basic Network Structures for FIR Systems
6.5.1 Direct Form
6.5.2 Cascade Form
6.5.3 Structures for Linear-Phase FIR Systems
6.6 Lattice Filters
6.6.1 FIR Lattice Filters
6.6.2 All-Pole Lattice Structure
6.6.3 Generalization of Lattice Systems
6.7 Overview of Finite-Precision Numerical Effects
6.7.1 Number Representations
6.7.2 Quantization in Implementing Systems
6.8 The Effects of Coefficient Quantization
6.8.1 Effects of Coefficient Quantization in IIR Systems
6.8.2 Example of Coefficient Quantization in an Elliptic Filter
6.8.3 Poles of Quantized 2nd-Order Sections
6.8.4 Effects of Coefficient Quantization in FIR Systems
6.8.5 Example of Quantization of an Optimum FIR Filter
6.8.6 Maintaining Linear Phase
6.9 Effects of Round-off Noise in Digital Filters
6.9.1 Analysis of the Direct Form IIR Structures
6.9.2 Scaling in Fixed-Point Implementations of IIR Systems
6.9.3 Example of Analysis of a Cascade IIR Structure
6.9.4 Analysis of Direct-Form FIR Systems
6.9.5 Floating-Point Realizations of Discrete-Time Systems
6.10 Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters
6.10.1 Limit Cycles Owing to Round-off and Truncation
6.10.2 Limit Cycles Owing to Overflow
6.10.3 Avoiding Limit Cycles
6.11 Summary
Problems
7 Filter Design Techniques
7.0 Introduction
7.1 Filter Specifications
7.2 Design of Discrete-Time IIR Filters from Continuous-Time Filters
7.2.1 Filter Design by Impulse Invariance
7.2.2 Bilinear Transformation
7.3 Discrete-Time Butterworth, Chebyshev and Elliptic Filters
7.3.1 Examples of IIR Filter Design
7.4 Frequency Transformations of Lowpass IIR Filters
7.5 Design of FIR Filters by Windowing
7.5.1 Properties of Commonly Used Windows
7.5.2 Incorporation of Generalized Linear Phase
7.5.3 The KaiserWindow Filter Design Method
7.6 Examples of FIR Filter Design by the KaiserWindow Method
7.6.1 Lowpass Filter
7.6.2 Highpass Filter
7.6.3 Discrete-Time Differentiators
7.7 Optimum Approximations of FIR Filters
7.7.1 Optimal Type I Lowpass Filters
7.7.2 Optimal Type II Lowpass Filters
7.7.3 The Parks?CMcClellan Algorithm
7.7.4 Characteristics of Optimum FIR Filters
7.8 Examples of FIR Equiripple Approximation
7.8.1 Lowpass Filter
7.8.2 Compensation for Zero-Order Hold
7.8.3 Bandpass Filter
7.9 Comments on IIR and FIR Discrete-Time Filters
7.10 Design of an Upsampling Filter
7.11 Summary
Problems
8 The Discrete Fourier Transform
8.0 Introduction
8.1 Representation of Periodic Sequences: The Discrete Fourier Series
8.2 Properties of the DFS
8.2.1 Linearity
8.2.2 Shift of a Sequence
8.2.3 Duality
8.2.4 Symmetry Properties
8.2.5 Periodic Convolution
8.2.6 Summary of Properties of the DFS Representation of Periodic Sequences
8.3 The Fourier Transform of Periodic Signals
8.4 Sampling the Fourier Transform
8.5 Fourier Representation of Finite-Duration Sequences
8.6 Properties of the DFT
8.6.1 Linearity
8.6.2 Circular Shift of a Sequence
8.6