Signal processing and linear systems

TLDR: This chapter discusses the Fourier Transform and its applications to Discrete-Time Signal Systems, as well as some of the techniques used to design and implement these systems in the real-time world.

Abstract: BACKGROUND B1 Complex Numbers B2 Sinusoids B3 Sketching Signals B4 Cramer's Rule B5 Partial Fraction Expansion B6 Vectors and Matrices B7 Miscellaneous CHAPTER 1 INTRODUCTION TO SIGNALS AND SYSTEMS 11 Size of a Signal 12 Classification of Signals 13 Some Useful Signal Operations 14 Some Useful Signal Models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model: Input-Output Description CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS 21 Introduction 22 System Response to Internal Conditions: Zero-Input Response 23 The Unit Impulse Response h(t) 24 System Response to External Input: Zero-State Response 25 Classical Solution of Differential Equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21: Determining the Impulse Response CHAPTER 3 SIGNAL REPRESENTATION BY FOURIER SERIES 31 Signals and Vectors 32 Signal Comparison: Correlation 33 Signal Representation by Orthogonal Signal Set 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D[n 37 LTIC System response to Periodic Inputs 38 Appendix CHAPTER 4 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM 41 Aperiodic Signal Representation by Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Application to Communications: Amplitude Modulation 48 Angle Modulation 49 Data Truncation: Window Functions CHAPTER 5 SAMPLING 51 The Sampling Theorem 52 Numerical Computation of Fourier Transform: The Discrete Fourier Transform (DFT) 53 The Fast Fourier Transform (FFT) 54 Appendix 51 CHAPTER 6 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE LAPLACE TRANSFORM 61 The Laplace Transform 62 Some Properties of the Laplace Transform 63 Solution of Differential and Integro-Differential Equations 64 Analysis of Electrical Networks: The Transformed Network 65 Block Diagrams 66 System Realization 67 Application to Feedback and Controls 68 The Bilateral Laplace Transform 69 Appendix 61: Second Canonical Realization CHAPTER 7 FREQUENCY RESPONSE AND ANALOG FILTERS 71 Frequency Response of an LTIC System 72 Bode Plots 73 Control System Design Using Frequency Response 74 Filter Design by Placement of Poles and Zeros of H(s) 75 Butterworth Filters 76 Chebyshev Filters 77 Frequency Transformations 78 Filters to Satisfy Distortionless Transmission Conditions CHAPTER 8 DISCRETE-TIME SIGNALS AND SYSTEMS 81 Introduction 82 Some Useful Discrete-Time Signal Models 83 Sampling Continuous-Time Sinusoids and Aliasing 84 Useful Signal Operations 85 Examples of Discrete-Time Systems CHAPTER 9 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS 91 Discrete-Time System Equations 92 System Response to Internal Conditions: Zero-Input Response 93 Unit Impulse Response h[k] 94 System Response to External Input: Zero-State Response 95 Classical Solution of Linear Difference Equations 96 System Stability 97 Appendix 91: Determining Impulse Response CHAPTER 10 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS 101 Periodic Signal Representation by Discrete-Time Fourier Series 102 Aperiodic Signal Representation by Fourier Integral 103 Properties of DTFT 104 DTFT Connection with the Continuous-Time Fourier Transform 105 Discrete-Time Linear System Analysis by DTFT 106 Signal Processing Using DFT and FFT 107 Generalization of DTFT to the Z-Transform CHAPTER 11 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM 111 The Z-Transform 112 Some Properties of the Z-Transform 113 Z-Transform Solution of Linear Difference Equations 114 System Realization 115 Connection Between the Laplace and the Z-Transform 116 Sampled-Data (Hybrid) Systems 117 The Bilateral Z-Transform CHAPTER 12 FREQUENCY RESPONSE AND DIGITAL FILTERS 121 Frequency Response of Discrete-Time Systems 122 Frequency Response From Pole-Zero Location 123 Digital Filters 124 Filter Design Criteria 125 Recursive Filter Design: The Impulse Invariance Method 126 Recursive Filter Design: The Bilinear Transformation Method 127 Nonrecursive Filters 128 Nonrecursive Filter Design CHAPTER 13 STATE-SPACE ANALYSIS 131 Introduction 132 Systematic Procedure for Determining State Equations 133 Solution of State Equations 134 Linear Transformation of State Vector 135 Controllability and Observability 136 State-Space Analysis of Discrete-Time Systems ANSWERS TO SELECTED PROBLEMS SUPPLEMENTARY READING INDEX Each chapter ends with a Summary