Digital Signal Processing

Tarun Kumar Rawat is Assistant Professor of division of Electronics and Communication Engineering at Netaji Subhas Institute of Technology (NSIT), New Delhi. He has completed his M. Tech. and PhD from the University of Delhi in the area of Signal Processing.

• Dedication iii; Features of the Book iv; Preface vi; Brief Content s x; Discrete-Time Signals and Systems; 1.1 Introduction 1; 1.2 Signals, Systems, and Signal Processing 1.2.1 Basic Elements of a Digital Signal Processing Systems 1.2.2 Advantages of Digital Signal Processing (DSP) over Analog Signal Processing (ASP) 1.2.3 DSP Applications 1.3 Classification of Signals 1.3.1 Continuous-Time andDiscrete-Time Signals; 1.3.2 Continuous-Valued and Discrete-Valued Signals 1.3.3 Multichannel andMultidimensional Signals; 1.3.4 Deterministic and RandomSignals; 1.4 Discrete-Time Signal or Sequence; 1.4.1 Finite-Length and Infinite-Length Signal 8; 1.4.2Right-Sided, Causal, Left-Sided, and Anticausal Signals; 1.5 Basic Operations on Discrete-Time Signals; 1.5.1 Signal Addition Operation; 1.5.2 Scalar Addition Operation; 1.5.3 SignalMultiplication Operation; 1.5.4 ScalarMultiplication Operation; 1.6 Transformations of the Independent Variable (Time); 1.6.1 Time-Shifting; 1.6.2 Time-Scaling (Decimation and Interpolation; 1.6.3 Time-Reversal; 1.6.4 Combined Operations; 1.7 Some Basic Discrete-Time Signals; 1.7.1 Unit Step Signal; 1.7.2 UnitImpulse Signal (or Unit Sample Signal); 1.7.3 Unit Ramp Signal 24; 1.7.4 Discrete-Time Real Exponential Signal 25; 1.7.5 Discrete-Time Sinusoidal Signal 26; 1.7.6 Discrete-Time Complex ExponentialSignal 27; 1.8 Periodic and Aperiodic Signals 28; 1.8.1 Properties of Periodic Signals 28; 1.8.2 Periodicity of Discrete-Time Sinusoidal Signals 31; 1.9 Energy and Power Signals 41; 1.10 Even and Odd Signals 48; 1.10.1 Even and Odd Components of a Signal 49; 1.10.2 Properties of Discrete-Time Even and Odd Signals 49; 1.10.3 Conjugate-Symmetric and Conjugate-Antisymmetric Signals 54; 1.11 Bounded Signal, Absolutely Summable Signal, and Square-summable Signal 56; 1.12 Discrete-TimeSystems 56; 1.13 Basic SystemProperties 57; 1.13.1 Linear and Nonlinear Systems 57; 1.13.2 Time-Varying and Time-Invariant Systems (or Shift-Invariant Systems) 63; 1.13.3 Causal Systems 67; 1.13.4Stable Systems 68; 1.13.5 Systems with andWithoutMemory 70; 1.13.6 Invertibility and Inverse Systems 71; 1.13.7 Passive and Lossless Systems 72; 1.14 Examples 72; 1.15 MATLAB Programs 96; 1.16 Summary 102; 1.17 Multiple Choice Questions 103; 1.18 Problems 105; 1.19 Answers toMultiple Choice Questions 107; 2 Sampling and Quantization; 2.1 Introduction 2; 2.2 Sampling 2; 2.3 Sampling Theorem for Low-Pass Signals 3; 2.3.1 Aliasing or SpectrumFolding 6; 2.4Sampling Techniques 14; 2.5 Impulse Sampling or Ideal Sampling or Instantaneous Sampling 15; 2.6 Natural Sampling or Chopper Sampling 16; 2.7 Flat-Top Sampling 18; 2.7.1 Aperture Effect 21; 2.8 Reconstructionof a signal fromits Samples using Interpolation 22; 2.8.1 Zero-Order-Hold Interpolation 24; 2.8.2 First-Order-Hold Interpolation (or Linear Interpolation) 26; 2.9 Sampling of Sinusoidal Signals 27; 2.10 Sampling Theorem for Band-Pass Signals 32; 2.10.1 Reconstruction of Bandpass Signal 35; 2.11 Quantization 39; 2.11.1 UniformQuantizers 39; 2.11.2 Quantization Error and Quantization Noise 40; 2.11.3 Signal-to-Quantization Noise Ratio (SQNR) 42; 2.11.4 SQNR for Sinusoidal Signals 42; 2.11.5 NonuniformQuantizer (Lloyd-Max Quantizers) 43; 2.12 Sampling of Discrete-Time Signals 46; 2.12.1 Decimation or Down-Sampling 49; 2.12.2 Interpolation or Up-Sampling 51; 2.12.3Fractional Delays 54; 2.13 Relationship Between DTFT and CTFT 57; 2.14 Examples 59; 2.15 MATLAB Programs 63; 2.16 Summary 67; 2.17 Multiple Choice Questions 69; 2.18 Problems 70; 2.19 Answers to Multiple Choice Questions 71; 3 Convolution and Correlation; 3.1 Introduction 3; 3.2 The Discrete-Time LTI systems: The Convolution Sum 4; 3.2.1 The Impulse Response (or Unit Sample Response) 4; 3.2.2 The Convolution Sum 4; 3.2.3 GraphicalMethod for the ConvolutionSum 8; 3.2.4 Analytical Method (using convolution sumexpression) 12; 3.3 Properties of Convolution Sum 16; 3.3.1 Commutative Property 16; 3.3.2 Associative Property 17; 3.3.3 Distributive Property 18; 3.3.4Shift Property 19; 3.3.5 Convolution with an Impulse 20; 3.3.6 Width Property 20; 3.3.7 SumProperty 22; 3.4 Convolution of Finite-Length Signals 24; 3.4.1 TabulationMethod 25; 3.4.2 Multiplication Method 27; 3.5 System Response to Periodic Inputs 28; 3.6 Relations between LTI system Properties and the Impulse Response 31; 3.6.1 LTI Systems with and withoutMemory 31; 3.6.2 Causality for LTI Systems 32; 3.6.3 Stability for LTI Systems 33; 3.6.4 Invertibility for LTI Systems 34; 3.6.5 The Unit Step Response of an LTI Systems 35; 3.7 Correlation of Signals 40; 3.7.1 Crosscorrelation Sequence of Discrete-Time Energy Signals 40; 3.7.2 Crosscorrelation Sequence of PowerSignals 41; 3.7.3 Autocorrelation Sequence of Discrete-time Signals 42; 3.7.4 Properties of Crosscorrelation and Autocorrelation Sequences 44; 3.8 Examples 46; 3.9 MATLAB Programs 56; 3.10 Summary 63; 3.11 Multiple Choice Questions 64; 3.12 Problems 65; 3.13 Answers toMultiple Choice Questions 67; 4 Discrete-Time Fourier Series; 4.1 Introduction 4; 4.2 Discrete-Time Fourier Series (DTFS) 5; 4.2.1 Evaluation of DTFS Coefficient 6; 4.2.2 Magnitude and PhaseSpectrum of Discrete-Time Periodic Signals (Fourier; Spectra); 4.3 Properties of DTFS 14; 4.3.1 Linearity 15; 4.3.2 Time Shifting 15; 4.3.3 Frequency Shifting 16; 4.3.4 Time Reversal 16; 4.3.5 Time Scaling orTime Expansion 17; 4.3.6 Periodic Convolution 18; 4.3.7 Multiplication 19; 4.3.8 First Difference 20; 4.3.9 Running Sum or Accumulation 20; 4.3.10 Conjugation and Conjugate Symmetry 21; 4.3.11 Parseval's Relation 24; 4.4 Systems with Periodic Inputs; 4.5 Examples 26; 4.6 MATLAB Programs 37; 4.7 Summary 41; 4.8 Multiple Choice Questions 42; 4.9 Problems 43; 4.10 Answers toMultiple Choice Questions 44; 5 Discrete-Time Fourier Transform; 5.1 Introduction 5;5.2 Fourier Transform Representation of Aperiodic Discrete-Time Signals 6; 5.3 Periodicity of the DTFT 9; 5.4 Convergence of DTFT 9; 5.4.1 Gibbs phenomenon 11; 5.5 Properties of Discrete-Time Fourier Transform 24;5.5.1 Linearity 25; 5.5.2 Time Shifting 25; 5.5.3 Frequency Shifting 27; 5.5.4 Time Reversal 28; 5.5.5 Time Expansion 29; 5.5.6 Differencing in Time Domain 29; 5.5.7 Differentiation in Frequency Domain 30; 5.5.8 Convolution Property 32; 5.5.9 Accumulation Property 34; 5.5.10 Multiplication (orModulation orWindowing) Property 35; 5.5.11 Conjugation and Conjugate Symmetry 36; 5.5.12 Parseval's Relation 41; 5.6 Some Important Results 42; 5.7 Fourier Transformof Periodic Signals 47; 5.8 SignalTransmission Through LTI Systems 50; 5.8.1 Response to Complex Exponentials 51; 5.8.2 Response to Sinusoidal Signals 53; 5.8.3 Response to a Causal Exponential Sequence 55; 5.8.4 Linear and NonlinearPhase 62; 5.8.5 Phase Delay and Group Delay 63; 5.9 Ideal and Practical Filters 70; 5.9.1 Paley-Wiener Criterion 73; 5.10 Energy Spectral Density (ESD) 78; 5.10.1 Relationship Between Input and Output Energy Spectral Densities of an; LTI System 79; 5.10.2 Relation of ESD to Autocorrelation 79; 5.11 Power Spectral Density (PSD) 79; 5.11.1 Relationship Between Input and Output Power Spectral Densities of an; LTI System 80; 5.11.2 Relation of PSD to Autocorrelation 81; 5.12 Examples 81; 5.13MATLAB Programs 98; 5.14 Summary 106; 5.15 Multiple Choice Questions 107; 5.16 Problems 108; 5.17 Answers to Multiple Choice Questions 110; 6 The z-Transform; 6.1 Introduction 6; 6.2 Bilateral(Two-sided) z-Transform 7; 6.2.1 Inverse z-Transform 7; 6.3 Relationship Between z-Transform and Discrete-Time Fourier Transform 8; 6.4 z-plane 9; 6.4.1 Poles and Zeros 10; 6.5 Region-of-Convergence for z-Transforms 11; 6.6 Properties of ROC 15; 6.7 Relationship Between Laplace Transform and z-transform (s to z-plane Mapping) 26; 6.8 Relationship Between z-transformand DTFS 29; 6.9 Properties of the z-Transform 29; 6.9.1 Linearity 29; 6.9.2 Time Shifting 31; 6.9.3 Scaling in the z-Domain 32;6.9.4 Time Reversal 34; 6.9.5 Differentiation in the z-Domain 35; 6.9.6 Time Expansion 40; 6.9.7 Convolution Property 43; 6.9.8 Correlation Property 45; 6.9.9 Accumulation Property 47; 6.9.10 FirstDifference 49; 6.9.11 Conjugation and Conjugate Symmetry 50; 6.10 z-Transformof Causal Periodic Signals 51; 6.11 Inversion of the z-Transform53; 6.11.1 Contour Integration Method (or ResidueMethod) 53; 6.11.2 Power Series Expansion Method (or Long Division Method) 57; 6.11.3 Partial Fraction ExpansionMethod62; 6.12 Analysis and Characterization of LTI Systems using the z-Transform 71; 6.12.1 The Transfer Function and Difference-Equation System Description 72; 6.12.2 Impulse Response and Stepresponse 72; 6.12.3 Causality 77; 6.12.4 Stability 79; 6.12.5 Stability of a Causal LTI System 80; 6.13 The Unilateral (One-Sided) z-Transform 85; 6.14 Properties of unilateral Z-Transform 89; 6.14.1Linearity 89; 6.14.2 Scaling in the z-Domain 89; 6.14.3 Differentiation in the z-Domain 90; 6.14.4 Time Expansion 90; 6.14.5 Conjugation Property 90; 6.14.6 Convolution Property 90; 6.14.7 Accumulation Property 91; 6.14.8 Time-Delay (Right-Shift) Property 92; 6.14.9 Time-Advance (Left-Shift) Property 95; 6.14.10 First Difference 97; 6.14.11 Initial-Value Theorem 97; 6.14.12 Final-Value Theorem 99; 6.14.13 Solving Difference Equations using the Unilateral z-Transform 102; 6.14.14 Zero-InputResponse and Zero-State Response 106; 6.14.15Natural Response and Forced Response 107; 6.14.16Transient Response and Steady-State Response 108; 6.15 Block Diagrams Representation 114; 6.15.1 CascadeInterconnection 114; 6.15.2 Parallel Interconnection 115; 6.15.3 Feedback Interconnection 116; 6.16 Some Application of z-Transformin Signal Processing 116; 6.16.1 Pole-Zero Description of Discrete-Time Systems 116; 6.16.2 Frequency Response Estimation 117; 6.16.3 Frequency Used in Discrete-Time Systems 117; 6.16.4 Causality and Stability Considerations 119; 6.16.5 Difference Equations 119; 6.16.6 Applications in Digital Filter Design 120; 6.16.7 Realization Structures for Digital Filters120; 6.17 Examples 120; 6.18 MATLAB Programs 137; 6.19 Summary 140; 6.20 Multiple Choice Questions 140; 6.21 Problems 142; 6.22 Answers toMultiple Choice Questions 144; 7 Filter Concepts; 7.1Introduction 7; 7.2 Frequency Response and Filter Characteristics 8; 7.2.1 Phase Delay and Group delay 9; 7.2.2 Geometric Evaluation of Frequency Response 9; 7.3 Zero-Phase Filter 12; 7.3.1 Zero Locations of Zero-Phase FIR Transfer Functions 13; 7.4 Linear-Phase Filter 14; 7.5 Simple FIR Digital Filters 14; 7.5.1 Lowpass FIR Digital Filter 14; 7.5.2 Highpass FIR Digital Filter 16; 7.5.3 Bandpass FIR Digital Filter 18; 7.5.4 Bandstop (Notch) FIR Digital Filter 20; 7.6 Simple IIR DigitalFilters 22; 7.6.1 Lowpass IIR Digital Filter 23; 7.6.2 Highpass IIR Digital Filter 24; 7.6.3 Bandpass IIR Digital Filter 26; 7.6.4 Bandstop IIR Digital Filter 27; 7.7 Allpass Filters 28; 7.7.1 Properties ofan allpass filter 30; 7.8 Minimum-Phase, Maximum-Phase and Non-minimum ( or Mixed) Phase Systems 32; 7.8.1 Invertibility and Inverse System 36; 7.8.2 Minimum-phase and Allpass Decomposition 37; 7.9 System Identification and Deconvolution 38; 7.9.1 The Cepstrumand Homomorphic Deconvolution 39; 7.10 Averaging Filters 41; 7.10.1 Zeros of Averaging Filters 42; 7.11 Comb Filters 42; 7.12 Digital Resonators 45; 7.13 Notch Filters 47; 7.14 Digital Sinusoidal Oscillators (Sinusoid Generators) 50;7.14.1 Digital Sine-Cosine Generators 52; 7.15 Digital Differentiator 54; 7.16 Digital Hilbert Transformer 57; 7.17 Examples 59; 7.18 MATLAB Programs 72; 7.19 Summary 82; 7.20 Multiple Choice Questions83; 7.21 Problems 85; 7.22 Answers toMultiple Choice Questions 86; 8 Discrete Fourier Transform (DFT); 8.1 Introduction 8; 8.2 Frequency Domain Sampling (Sampling of DTFT) 9; 8.3 The Discrete Fourier Transform(DFT) and its Inverse 13; 8.3.1 Derivation of Inverse DFT (IDFT) 16; 8.3.2 Magnitude and Phase of DFT 17; 8.3.3 Zero Padding 24; 8.4 DFT as a Linear Transformation (Matrix Formulation) 28; 8.4.1 Twiddle factor (WN) and its Properties 30; 8.5 Properties of the DFT 33;8.5.1 Periodicity 34; 8.5.2 Linearity 34; 8.5.3 Circular Time Shifting 35; 8.5.4 Circular Frequency Shifting 40; 8.5.5 Circular Time Reversal (Circular Folding or Circular Flipping) 43; 8.5.6 Conjugation and ConjugateSymmetry (Symmetry Properties) 47; 8.5.7 Duality 51; 8.5.8 Circular Convolution (Multiplication of Two DFTs) 54; 8.5.9 Circular Correlation 62; 8.5.10 Multiplication (orModulation) Property 63; 8.5.11 Parseval's Relation 63; 8.6 Some Important Results 64; 8.7 Linear Convolution Using the DFT (Linear Convolution Using Circular Convolution) 68; 8.7.1 Circular Convolution as Linear Convolution with Aliasing 71; 8.8 Filtering of Long Data Sequences Using DFT (Linear Convolution of a Finite LengthSequence with an Infinite length Sequence) (or Fast Convolution) (or Block Convolution) 72; 8.8.1 Overlap-Save Method 73; 8.8.2 Overlap-AddMethod 75; 8.9 The Discrete Cosine Transform(DCT) 79; 8.9.1Relationship between the DFT and DCT 81; 8.10 DiscreteWalsh Transform(DWT) 83; 8.10.1 Discrete Hadamard Transform(DHT) 84; 8.11 Relationship of the DFT to Other Transforms 85; 8.11.1 Relationship to Discrete-Time Fourier Series (DTFS) 85; 8.11.2 Relationship to Discrete-Time Fourier Transform(DTFT) 86; 8.11.3 Relationship to z-Transform86; 8.12 SpectrumAnalysis Using DFT 87; 8.12.1 Relationship Between DFT and Continuous-Time Fourier Transform (CTFT) 87; 8.12.2 Relationship Between theFrequency Bin k and its Associated Analog Frequency O or f 88; 8.12.3 Selection of Parameters for Signal Processing with the DFT 90; 8.12.4 High Density Spectrum(Zero-Padding) 91; 8.12.5 Spectral Leakage 92;8.12.6 Spectral Estimation UsingWindow Functions 93; 8.13 Spectral Analysis of Nonstationary Signals 95; 8.13.1 Short-Time Fourier Transform(STFT) 97; 8.14 Examples 97; 8.15 MATLAB Programs 109; 8.16 Summary 119; 8.17 Multiple Choice Questions 120; 8.18 Problems 121; 8.19 Answers toMultiple Choice Questions 122; 9 Fast Fourier Transform (FFT); 9.1 Introduction 9; 9.2 Computational Complexity of the Direct Computation of the DFT 9; 9.2.1 Symmetry and Periodicity Properties ofthe Twiddle Factor (WN) 10; 9.2.2 Radix-2 FFT Algorithms 11; 9.3 Decimation-In-Time (DIT) FFT Algorithm 11; 9.3.1 Computational Advantage of the DIT-FFT 16; 9.3.2 In-Place Computation 17; 9.3.3 Bit-Reversal 17; 9.4Decimation-In-Frequency (DIF) FFT Algorithm 21; 9.4.1 Computational Cost 26; 9.5 Comparison Between DIT and DIF Algorithms 27; 9.6 Inverse DFT Using FFT Algorithms 34; 9.7 A Linear Filtering Approach to Computation of the DFT 37; 9.7.1 The Goertzel Algorithm 37; 9.7.2 The Chirp-z TransformAlgorithm 43; 9.7.3 Dual-Tome Multi-Frequency (DTMF) Tone Detection Using the Goertzel Algorithm 46; 9.8 Examples 48; 9.9 MATLAB Programs 55; 9.10 Summary 58; 9.11 Multiple Choice Questions 59; 9.12 Problems60; 9.13 Answers toMultiple Choice Questions 61; 10 Realization of Digital Filters; 10.1 Introduction 10; 10.1.1 FIR Filter or All Zero (AZ) or Moving Average (MA) system: 11; 10.1.2 IIR Filter or AllPole (AP) or Autoregressive (AR) system 11; 10.1.3 IIR Filter or Pole-Zero (PZ) or Autoregressive, Moving average (ARMA) system 12; 10.2 Nonrecursive and Recursive Structures 13; 10.3 Factors that Influence the Choice of Structure 13; 10.4 Block Diagram Representation and Signal Flow Graph 14; 10.4.1 Basic Building Blocks 14; 10.4.2 Advantages in Representing the Digital Filter in Block Diagram Form 14; 10.4.3 Canonic and Noncanonic Structures 15; 10.4.4 Equivalent Structures (TransposedStructure) 15; 10.5 FIR Filter Structures 15; 10.5.1 Direct Form (Transversal or Tapped-Delay Line) Structure 16; 10.5.2 Cascade-Form Structure 17; 10.5.3 Linear-Phase Structure 18; 10.5.4 PolyphaseStructure 20; 10.5.5 Conversion of Nonrecursive Structure into Recursive Structure 24; 10.5.6 Frequency Sampling Structure 26; 10.6 Basic Structures for IIR Systems 31; 10.6.1 Direct Form I Structure 32; 10.6.2 Direct-Form II Structure 34; 10.6.3 Cascade Form Structure 40; 10.6.4 Parallel From Structure 42; 10.6.5 Polyphase Structure 47; 10.7 Lattice Structures 51; 10.7.1 Advantages of Lattice Structures 53; 10.8 Lattice Structures for FIR Systems (All-Zero Systems) 53; 10.8.1 Conversion ofLattice Coefficients to Direct-Form Filter Coefficients 57; 10.8.2 Conversion of Direct-Form Filter Coefficients to Lattice Coefficients 59; 10.9 Lattice Structures for All-Pole (AP) IIR Systems 65;10.9.1 Stability of an All-Pole System (Schur-Cohn Stability Test) 68; 10.10Lattice Structures for Pole-Zero (PZ) IIR Systems (or Lattice-Ladder Structure) 70; 10.11 Examples 75; 10.12 MATLAB Programs 83; 10.13Summary 86; 10.14 Multiple Choice Questions 86; 11 Finite Impulse Response (FIR) Digital Filter; 11.1 Introduction to Digital Filters 11; 11.1.1 Advantages and Disadvantages of Digital Filters 11; 11.1.2 Types of Digital Filters: FIR and IIR Filters 12; 11.1.3Difference Between FIR and IIR Filters 13; 11.2 Desirability of Linear-Phase Filters 14; 11.2.1 Effect of Phase Distortion 15; 11.2.2 Condition for a Filter to have a Linear-Phase Response 16; 11.3 Frequency Response ofLinear-Phase FIR Filters 21; 11.3.1 Type 1: Symmetric Impulse Response with Odd Length (M odd) 24; 11.3.2 Type 2: Symmetric Impulse Response with Even Length (M Even) 26; 11.3.3 Type 3: Antisymmetric Impulse Response with Odd Length (M Odd) 28; 11.3.4 Type 4: Antisymmetric Impulse Response with Even Length (M Even) 31; 11.3.5 Location of Zeros of Linear Phase FIR Transfer Functions 34; 11.4 Filter Specifications 38; 11.4.1 Absolute Specifications 39; 11.4.2 Relative Specifications 40; 11.4.3Continuous-time (Analog) Filter Specifications 42; 11.4.4 Estimation of FIR Filter Order 43; 11.5 Impulse Responses of Ideal Filters 45; 11.5.1 Impulse Response of an Ideal Lowpass Filter 45; 11.5.2Impulse Response of an Ideal Highpass Filter 46; 11.5.3 Impulse Response of an Ideal Bandpass Filter 48; 11.5.4 Impulse Response of an Ideal Bandstop Filter 48; 11.6 Design Techniques for Linear-Phase FIR Filters 49; 11.7 Fourier Series Method 50; 11.7.1 Gibbs Phenomenon 52; 11.8 Windowing Method 62; 11.8.1 Rectangular Window 63; 11.8.2 Triangular (or Bartlett) Window 76; 11.8.3 Hann (or Hanning) Window 81; 11.8.4 Hamming Window 86; 11.8.5 Blackman Window 97; 11.8.6 Kaiser Window 118;11.8.7 Advantages and Disadvantages of the Window Method 123; 11.9 Half-Band FIR Filters 124; 11.10 Design of FIR Digital Differentiators 127; 11.11 Design of FIR Hilbert Transformers 135; 11.12Frequency Sampling Method 142; 11.12.1 Type-I Design 143; 11.12.2 Type-II Design 151; 11.12.3 Transition-band Optimization 155; 11.13 The Optimal Method 159; 11.13.1 Minimax Criterion 160; 11.13.2 Alternation Theorem 166; 11.13.3 Parks-McClellan Algorithm 168; 11.13.4 Disadvantages of Optimal Method 172; 11.14 Comparison of Design Methods for Linear-Phase FIR Filters 172; 11.15 Examples 172; 11.16 MATLAB Programs 175; 11.17 Summary 193; 11.18 Multiple Choice Questions 195; 11.19 Problems197; 11.20Answers to Multiple Choice Questions 198; 12 Infinite Impulse Response (IIR) Digital Filter; 12.1 Introduction 12; 12.2 Design of IIR Filters from Analog Filters 13; 12.2.1 Filter DesignSteps 13; 12.3 IIR Filter Design by Approximation of Derivatives 14; 12.3.1 Backward Difference Algorithm 14; 12.4 Impulse-Invariant Method 16; 12.4.1 Relationship Between Analog and Digital Filter Poles 18; 12.4.2 Relationship Between Analog and Digital Frequency 22; 12.4.3 Advantages and Disadvantages of Impulse-Invariant Method 22; 12.5 The Matched z-Transformation 33; 12.6 Step -Invariant Method 35; 12.7 Bilinear Transformation Method 37; 12.7.1 Relationship Between Analog and Digitalfrequency 39; 12.7.2 Effect of Warping on the Magnitude and Phase Response 40; 12.7.3 Prewarping 40; 12.7.4 Advantages and Disadvantages of Bilinear Transformation Method 41; 12.7.5 Comparison ofImpulse-Invariant and Bilinear Transformations 41; 12.7.6 Relationship Between Impulse-Invariant and Bilinear Transformations 41; 12.8 IIR Filter Specifications 50; 12.9 Specifications of the Lowpass Filter 51; 12.9.1 Properties of Magnitude-squared Response jHa(j-)j2 51; 12.10Analog Butterworth Filter 52; 12.10.1 Determination of Filter Parameters (Order N and Cutoff Frequency -c) of; Butterworth Filter 60; 12.10.2 Design Procedure for Lowpass Digital Butterworth Filters 63; 12.11AnalogChebyshev Filter 77; 12.11.1Type I Chebyshev Filter 77; 12.11.2 Determination of Filter Parameters (Order N and Cut-off Frequency -c) of Chebyshev Filter 81; 12.11.3 Design Procedure for Lowpass DigitalChebyshev Filters 85; 12.11.4Type II Chebyshev Filter (Inverse Chebyshev Filter) 92; 12.11.5 Difference Between Butterworth and Chebyshev Filter 93; 12.12Frequency (or Spectral) Transformation in the Analog Domain 94; 12.12.1 Lowpass to Lowpass Transformation 94; 12.12.2 Lowpass to Highpass Transformation 95; 12.12.3 Lowpass to Bandpass Transformation 96; 12.12.4 Lowpass to Bandstop Transformation 97; 12.13Frequency (or Spectral) Transformation in the Digital Domain 106; 12.13.1 Lowpass toLowpass Transformation 107; 12.13.2 Lowpass to Highpass Transformation 108; 12.13.3 Lowpass to Bandpass Transformation 108; 12.13.4 Lowpass to Bandstop Transformation 109; 12.14Applications 113; 12.14.1Digital Audio Equalizer 113; 12.14.2 Generation and Detection of Dual-Tone Multifrequency Tones Using Goertzel Algorithm 114; 12.14.3 60-Hz Hum Eliminator and Heart Rate Detection Using Electrocardiography (ECG) 117; 12.14.4 Timing (or Clock) Recovery for Synchronous Digital Communication System121; 12.15 Examples 122; 12.16 MATLAB Programs 126; 12.17Summary 133; 12.18 Multiple Choice Questions 134; 12.19 Problems 136; 12.20Answers to Multiple Choice Questions 138; 13 Analysis ofFinite Word Length Effects; 13.1 Introduction 13; 13.2 Representation of Numbers 14; 13.2.1 Fixed-Point Representation of Numbers 15; 13.2.2 Binary Floating-Point Representation of Numbers 18; 13.3 Quantization21; 13.3.1 Truncation 22; 13.3.2 Rounding 22; 13.4 Quantization of Fixed-Point Numbers 26; 13.4.1 Effect of Truncation 26; 13.4.2 Effect of Rounding 29; 13.5 Quantization of Floating-Point Numbers 30; 13.5.1 Effect of Truncation 31; 13.5.2 Effect of Rounding 33; 13.6 Coefficient Quantization Error 34; 13.6.1 Coefficient Sensitivity Analysis of a Second-order Direct Form II Structure 37; 13.6.2 Analysis of Coefficient Quantization Effects in FIR Filters 43; 13.7 Quantization in SamplingAnalog Signals (A/D Conversion Noise Analysis) 44; 13.7.1 Quantization NoiseModel 46; 13.7.2 Signal-to-Quantization Noise Ratio 47; 13.7.3 Effect of Input Scaling on SNR 48; 13.7.4 Propagation of InputQuantization Noise to Digital Filter Output 48; 13.8 Product Quantization Error 57; 13.8.1 Direct Form I Structure 58; 13.8.2 Direct Form II Structure 61; 13.8.3 Product Quantization using Double-Length Accumulator 78; 13.9 Limit Cycles in IIR Digital Filters 82; 13.9.1 Zero-Input Limit Cycle 83; 13.9.2 Dead Band 84; 13.9.3 Overflow Limit Cycle 88; 13.9.4 Saturation to Avoid Overflow 89; 13.9.5 Scaling to Prevent Overflow 90; 13.10Quantization Effects in Floating-Point Realizations of IIRDigital Filters 101; 13.10.1First Order IIR Digital Filter 102; 13.11Quantization Effects in Realizations of FIR Digital Filters 103; 13.11.1 Quantization Effects in Fixed-Point Realizations of FIR DigitalFilters 104; 13.11.2 Quantization Effects in Floating-Point Realizations of FIR Digital Filters 105; 13.12Quantization Effects in Discrete Fourier Transform (DFT) Computations 107; 13.12.1 Quantization Effects in Direct Computation of the DFT 107; 13.12.2 Quantization Effects in Fixed-Point FFT Algorithms 110; 13.13 Examples 112; 13.14 MATLAB Programs 116; 13.15Summary 117; 13.16 Multiple Choice Questions 119; 13.17 Problems 121; 13.18Answers toMultiple Choice Questions 123; 14Multirate Digital Signal Processing; 14.1 Introduction 14; 14.1.1 Advantages ofMultirate DSP 14; 14.2 Decimation 15; 14.2.1 Time-Domain Characterization 15; 14.2.2 Frequency-Domain Characterization 16; 14.2.3Aliasing Effect 19; 14.2.4 Anti-aliasing Filter Specifications 20; 14.3 Interpolation 22; 14.3.1 Time-Domain Characterization 22; 14.3.2 Frequency-Domain Characterization 23; 14.3.3 Anti-imaging Filter Specifications 25; 14.4 Sampling Rate Conversion by a Rational factor LM 28; 14.5 Identities (Cascaded Equivalences) 33; 14.5.1 Identities for the Downsampling 33; 14.5.2 Identities for the Upsampling 35; 14.5.3 Upsampler and Downsampler Cascade 37; 14.6 Computational Requirements 41; 14.6.1Efficient Direct Form Structure of a Decimator 42; 14.6.2 Efficient Direct Form Structure of an Interpolator 43; 14.7 The Polyphase Decomposition 44; 14.7.1 FIR Filter Structures Based on PolyphaseDecomposition 44; 14.7.2 Decimation with Polyphase Filters 46; 14.7.3 Interpolation with Polyphase Filters 49; 14.7.4 Efficient Rational Sampling Rate Converter with Polyphase Filters 50; 14.7.5 The Polyphase Identity 53; 14.8 Application ofMultirate DSP: Design of Narrowband Filters 54; 14.8.1 Interpolated FIR (IFIR) Filters 55; 14.9 Multistage Implementation of Sampling Rate Conversion 59; 14.9.1 Multistage Implementation of Decimator 60; 14.9.2 Multistage Implementation of Interpolator 67;14.9.3 Multistage Implementation of Rational Sampling Rate Converter 67; 14.9.4 The IFIR Approach forMultistage Implementation 68; 14.10Nyquist Filters (or Lth-Band Filters) 74; 14.10.1Half-Band Filters 74; 14.10.2Lth-Band Filters 76; 14.11Digital Filter Banks 79; 14.11.1UniformDFT Filter Bank 80; 14.11.2Polyphase Implementations of UniformDFT Filter Bank 81; 14.11.3Application of Multirate DSP: Subband Coding of Speech and Audio Signals 83; 14.12Two-Channel Quadrature-Mirror Filter (QMF) Bank 85; 14.12.1Analysis of Two-Channel QMF Bank 86; 14.13Application ofMultirate DSP: Transmultiplexer 92; 14.14 Examples 93; 14.15 MATLAB Programs 96; 14.16Summary 102; 14.17 Multiple Choice Questions102; 14.18 Problems 104; 14.19Answers toMultiple Choice Questions 105; 15 Optimum Linear Filters (Wiener Filters); 15.1 Introduction 15; 15.1.1 Mean-Square Error Criterion 15; 15.2 The FIRWiener Filter17; 15.2.1 Filtering 20; 15.2.2 Linear Prediction 27; 15.3 Noncausal IIRWiener Filter 32; 15.3.1 Filtering 35; 15.4 Innovations Representation 37; 15.4.1 Spectral Factorization 38; 15.5 Causal IIRWiener Filter 39; 15.6 Deconvolution (or Deblurring) 44; 15.7 Examples 47; 15.8 MATLAB Programs 49; 15.9 Summary 51; 15.10 Multiple Choice Questions 52; 15.11 Problems 52; 15.12Answers toMultiple Choice Questions 53; 16 Power Spectrum Estimation; 16.1 Introduction 16; 16.1.1Parseval's Theorem 17; 16.1.2 Performance of Estimators 19; 16.2 Nonparametric (or Classical) Methods 20; 16.2.1 The Periodogram 21; 16.2.2 TheModified Periodogram 24; 16.2.3 Bartlett's Method: Periodogram Averaging 25;16.2.4 Welch's Method: AveragingModified Periodograms 28; 16.2.5 Blackman-Tukey Approach: Periodogram Smoothing 30; 16.3 Parametric (or Nonclassical) Methods 34; 16.3.1 Autoregressive SpectrumEstimation 36; 16.3.2 Computation of Model Parameters Yule-Walker Equations 37; 16.3.3 Least-squares (LS)Method and Linear Prediction 39; 16.3.4 Moving Average SpectrumEstimation 41; 16.3.5 AutoregressiveMoving Average SpectrumEstimation 42; 16.4 Eigenvalues and Eigenvectors of the Autocorrelation Matrix42; 16.4.1 Properties of Eigenvalues and Eigenvectors 43; 16.5 Eigenanalysis Algorithms for SpectrumEstimation 47; 16.5.1 HarmonicModel 47; 16.5.2 Eigen-Decomposition of the Autocorrelation Matrix48; 16.5.3 Pisarenko Harmonic Decomposition Method 59; 16.5.4 MUSIC Algorithm 62; 16.6 Examples 64; 16.7 MATLAB Programs 66; 16.8 Summary 72; 16.9 Multiple Choice Questions 73; 16.10 Problems 74; 16.11Answers toMultiple Choice Questions 74; 17 Introduction to Digital Signal Processors (DSPs); 17.1 Introduction 17; 17.2 Evolution of Digital Signal Processors 18; 17.2.1 DSP Algorithms mold DSP Architectures 18; 17.2.2 FastMultipliers 18; 17.2.3 Multiple Execution Units 18;17.2.4 Efficient memory Accesses 19; 17.2.5 Data Format 20; 17.2.6 Zero-Overhead Looping 20; 17.2.7 Streamlined I/O 21; 17.2.8 Specialized Instruction Set 21; 17.3 Digital Signal Processor Architecture 21; 17.3.1 VonNeumann Architecture 22; 17.3.2 Harvard Architecture 23; 17.3.3 Super Harvard Architecture (SHARC) 24; 17.4 Digital Signal Processor Hardware Units 24; 17.4.1 Multiplier and Accumulator (MAC) Unit 24; 17.4.2 Shifters 24; 17.4.3 Address Generators 26; 17.5 Fixed-Point and Floating-Point Format 26; 17.6 Fixed-Point Digital Signal Processor 27; 17.7 Floating-Point Digital Signal Processor 27; 17.8 Pipelining 28; 17.9 Memory Access schemes in DSPs 29; 17.9.1 Multiple Access Memory 29; 17.9.2Multiport memory 29; 17.10 Very Long InstructionWord (VLIW) Architecture 29; 17.10.1Advantages 30; 17.10.2Disadvantages 31; 17.11AddressingModes 31; 17.11.1Implied Addressing 31; 17.11.2ImmediateAddressing 31; 17.11.3Memory-Direct Addressing 32; 17.11.4Register-Direct Addressing 32; 17.11.5Register-Indirect Addressing 32; 17.11.6Bit-Reversed Addressing 32; 17.11.7Circular Addressing 32; 17.12The TMS320 Family 33; 17.12.1 TMS320 C2X Generation 33; 17.12.2 TMS320 C3X Generation 34; 17.12.3 TMS320 C4X Generation 34; 17.12.4 TMS320 C5X Generation 35; 17.12.5Overview of TMS 320 C6713 DSP 35; 17.13 Interfacing; 17.13.1 External Memory Interfacing; 17.13.2 Serial-port Interfacing; 17.13.3Parallel-port Interfacing; 17.13.4 Host-port Interfacing 1063; Summary; Multiple Choice Questions; Questions 36; Answers to Multiple Choice Questions; Bibliography; Index