Principles of Digital Signal Processing

Preface to the Second Edition
Preface to the First Edition
Contents
About the Author
1 Representation of Discrete Signals and Systems
	1.1 Introduction
	1.2 Terminologies Related to Signals and Systems
		1.2.1 Signal
		1.2.2 System
	1.3 Continuous and Discrete Time Signals
	1.4 Basic Discrete Time Signals
		1.4.1 The Unit Impulse Sequence
		1.4.2 The Basic Unit Step Sequence
		1.4.3 The Basic Unit Ramp Sequence
		1.4.4 Unit Rectangular Sequence
		1.4.5 Sinusoidal Sequence
		1.4.6 Discrete Time Real Exponential Sequence
	1.5 Basic Operations on Discrete Time Signals
		1.5.1 Addition of Discrete Time Sequence
		1.5.2 Multiplication of DT Signals
		1.5.3 Amplitude Scaling of DT Signal
		1.5.4 Time Scaling of DT Signal
		1.5.5 Time Shifting of DT Signal
		1.5.6 Multiple Transformation
	1.6 Classification of Discrete Time Signals
		1.6.1 Periodic and Non-periodic DT Signals
		1.6.2 Odd and Even DT Signals
		1.6.3 Energy and Power of DT Signals
	1.7 Discrete Time System
	1.8 Properties of Discrete Time System
		1.8.1 Linear and Nonlinear Systems
		1.8.2 Time Invariant and Time Varying DT Systems
		1.8.3 Causal and Non-causal DT Systems
		1.8.4 Stable and Unstable Systems
		1.8.5 Static and Dynamic Systems
		1.8.6 Invertible and Inverse Discrete Time Systems
2 Discrete and Fast Fourier Transforms (DFT and FFT)
	2.1 Introduction
	2.2 Discrete Fourier Transform (DFT)
		2.2.1 The Discrete Fourier Transform Pairs
		2.2.2 Four-Point, Six-Point and Eight-Point Twiddle Factors
		2.2.3 Zero Padding
	2.3 Relationship of the DFT to Other Transforms
		2.3.1 Relationship to the Fourier Series Coefficients of a Periodic Sequence
		2.3.2 Relationship to the Fourier Transform of an Aperiodic Sequence
		2.3.3 Relationship to the z-Transform
	2.4 Properties of DFT
		2.4.1 Periodicity
		2.4.2 Linearity
		2.4.3 Circular Shift and Circular Symmetric of a Sequence
		2.4.4 Symmetry Properties of the DFT
		2.4.5 Multiplication of Two DFTs and Circular Convolution
		2.4.6 Time Reversal of a Sequence
		2.4.7 Circular Time Shift of a Sequence
		2.4.8 Circular Frequency Shift
		2.4.9 Complex–Conjugate Properties
		2.4.10 Circular Correlation
		2.4.11 Multiplication of Two Sequences
		2.4.12 Parseval's Theorem
	2.5 Circular Convolution
		2.5.1 Method of Performing Circular Convolution
		2.5.2 Performing Linear Convolution Using DFT
	2.6 Fast Fourier Transform (FFT)
		2.6.1 Radix-2 FFT Algorithm
		2.6.2 Radix-4 FFT Algorithms
		2.6.3 Computation of IDFT through FFT
		2.6.4 Use of the FFT Algorithm in Linear Filtering and Correlation
	2.7 In-Plane Computation
3 Design of IIR Digital Filters
	3.1 Introduction
		3.1.1 Advantages
		3.1.2 Disadvantages
	3.2 IIR and FIR Filters
	3.3 Basic Features of IIR Filters
	3.4 Performance Specifications
	3.5 Impulse Invariance Transform Method
		3.5.1 Relation Between Analog and Digital Filter Poles
		3.5.2 Relation Between Analog and Digital Frequency
	3.6 Bilinear Transformation Method
		3.6.1 Relation Between Analog and Digital Filter Poles
		3.6.2 Relation Between Analog and Digital Frequency
		3.6.3 Effect of Warping on the Magnitude Response
		3.6.4 Effect of Warping on the Phase Response
	3.7 Specifications of the Lowpass Filter
	3.8 Design of Lowpass Digital Butterworth Filter
		3.8.1 Analog Butterworth Filter
		3.8.2 Frequency Response of Butterworth Filter
		3.8.3 Properties of Butterworth Filters
		3.8.4 Design Procedure for Lowpass Digital Butterworth Filters
	3.9 Design of Lowpass Digital Chebyshev Filter
		3.9.1 Analog Chebyshev Filter
		3.9.2 Determination of the Order of the Chebyshev Filter
		3.9.3 Unnormalized Chebyshev Lowpass Filter Transfer Function
		3.9.4 Frequency Response of Chebyshev Filter
		3.9.5 Properties of Chebyshev Filter (Type I)
		3.9.6 Design Procedures for Lowpass Digital Chebyshev IIR Filter
	3.10 Frequency Transformation
		3.10.1 Analog Frequency Transformation
		3.10.2 Digital Frequency Transformation
	3.11 IIR Filter Design by Approximation of Derivatives
	3.12 Frequency Response from Transfer Function H(z)
	3.13 Structure Realization of IIR System
		3.13.1 Direct Form-I Structure
		3.13.2 Direct Form-II Structure
		3.13.3 Cascade Form Realization
		3.13.4 Parallel Form Realization
		3.13.5 Transposed Direct Form Realization
		3.13.6 Transposition Theorem and Transposed Structure
		3.13.7 Lattice Structure of IIR System
		3.13.8 Conversion from Direct Form to Lattice Structure
		3.13.9 Lattice–Ladder Structure
4 Finite Impulse Response (FIR) Filter Design
	4.1 Introduction
		4.1.1 LTI System as Frequency Selective Filters
	4.2 Characteristic of Practical Frequency Selective Filters
	4.3 Structures for Realization of the FIR Filter
		4.3.1 Direct Form Realization
		4.3.2 Cascade Form Realization
		4.3.3 Linear Phase Realization
		4.3.4 Lattice Structure of an FIR Filter
	4.4 FIR Filters
		4.4.1 Characteristics of FIR Filters with Linear Phase
		4.4.2 Frequency Response of Linear Phase FIR Filter
	4.5 Design Techniques for Linear Phase FIR Filters
		4.5.1 Fourier Series Method of FIR Filter Design
		4.5.2 Window Method
		4.5.3 Frequency Sampling Method
5 Finite Word Length Effects
	5.1 Introduction
	5.2 Representation of Numbers in Digital System
		5.2.1 Fixed Point Representation
		5.2.2 Floating Point Representation
	5.3 Methods of Quantization
		5.3.1 Truncation
		5.3.2 Rounding
	5.4 Quantization of Input Data by Analog to Digital Converter
		5.4.1 Output Noise Power Due to the Quantization Error Signal
	5.5 Quantization of Filter Coefficients
	5.6 Product Quantization Error
	5.7 Limit Cycles in Recursive System
		5.7.1 Zero-Input Limit Cycles
		5.7.2 Overflow Limit Cycle Oscillation
	5.8 Scaling to Prevent Overflow
6 Multi-rate Digital Signal Processing
	6.1 Introduction
	6.2 Advantages and Applications of Multi-rate Signal Processing
	6.3 Downsampling (Decimator)
	6.4 Upsampling (Interpolator)
	6.5 Sampling Rate Conversion by Non-integer Factors Represented by Rational Number
	6.6 Characteristics of Filter and Downsampler
	6.7 Linearity and Time Invariancy of Decimator and Interpolator
		6.7.1 Linearity of Decimator
		6.7.2 Linearity of an Interpolator
		6.7.3 Time Invariancy of a Decimator
		6.7.4 Time Invariancy of an Interpolator
	6.8 Spectrum of Downsampled Signal
	6.9 Effect of Aliasing in Downsampling
	6.10 Spectrum of Upsampling Signal
		6.10.1 Anti-imaging Filter
	6.11 Efficient Transversal Structure for Decimator
	6.12 Efficient Transversal Structure for Interpolator
	6.13 Identities
	6.14 Polyphase Filter Structure of a Decimator
		6.14.1 The Polyphase Decomposition
		6.14.2 Polyphase Structure of a Decimator Using z-Transform
		6.14.3 Polyphase Structure of an Interpolator
		6.14.4 Polyphase Structure of an Interpolator Using z-Transform
	6.15 Polyphase Decomposition of IIR Transfer Function
	6.16 Cascading of Upsampler and Downsampler
	6.17 Multi-stage Rating of Sampling Rate Conversion
	6.18 Implementation of Narrow Band Lowpass Filter
	6.19 Adaptive Filters
		6.19.1 Concepts of Adaptive Filtering
		6.19.2 Adaptive Noise Canceller
		6.19.3 Main Components of the Adaptive Filter
		6.19.4 Adaptive Algorithms
Index