Linear Systems and Signals

Preface

B. Background B.1 Complex Numbers B.1.1 A Historical Note B.1.2 Algebra of Complex Numbers B.2 Sinusoids B.2.1 Addition of Sinusoids B.2.2 Sinusoids in Terms of Exponentials B.3 Sketching Signals B.3.1 Monotonic Exponentials B.3.2 The Exponentially Varying Sinusoid B.4 Cramer's Rule B.5 Partial Fraction Expansion B.5.1 Method of Clearing Fractions B.5.2 The Heaviside "Cover-Up" Method B.5.3 Repeated Factors of Q(x) B.5.4 A Combination of Heaviside "Cover-Up" and Clearing Fractions B.5.5 Improper F(x) with m = n B.5.6 Modified Partial Fractions B.6 Vectors and Matrices B.6.1 Some Definitions and Properties B.6.2 Matrix Algebra B.7 MATLAB: Elementary Operations B.7.1 MATLAB Overview B.7.2 Calculator Operations B.7.3 Vector Operations B.7.4 Simple Plotting B.7.5 Element-by-Element Operations B.7.6 Matrix Operations B.7.7 Partial Fraction Expansions B.8 Appendix: Useful Mathematical Formulas B.8.1 Some Useful Constants B.8.2 Complex Numbers B.8.3 Sums B.8.4 Taylor and Maclaurin Series B.8.5 Power Series B.8.6 Trigonometric Identities B.8.7 Common Derivative Formulas B.8.8 Indefinite Integrals B.8.9 L'Hopital's Rule B.8.10 Solution of Quadratic and Cubic Equations 1. Signals and Systems 1.1 Size of a Signal 1.1.1 Signal Energy 1.1.2 Signal Power 1.2 Some Useful Signal Operations 1.2.1 Time Shifting 1.2.2 Time Scaling 1.2.3 Time Reversal 1.2.4 Combined Operations 1.3 Classification of Signals 1.3.1 Continuous-Time and Discrete-Time Signals 1.3.2 Analog and Digital Signals 1.3.3 Periodic and Aperiodic Signals 1.3.4 Energy and Power Signals 1.3.5 Deterministic and Random Signals 1.4 Some Useful Signal Models 1.4.1 The Unit Step Function 1.4.2 The Unit Impulse Function 1.4.3 The Exponential Function 1.5 Even and Odd Functions 1.5.1 Some Properties of Even and Odd Functions 1.5.2 Even and Odd Components of a Signal 1.6 Systems 1.7 Classification of Systems 1.7.1 Linear and Nonlinear Systems 1.7.2 Time-Invariant and Time-Varying Systems 1.7.3 Instantaneous and Dynamic Systems 1.7.4 Causal and Noncausal Systems 1.7.5 Continuous-Time and Discrete-Time Systems 1.7.6 Analog and Digital Systems 1.7.7 Invertible and Noninvertible Systems 1.7.8 Stable and Unstable Systems 1.8 System Model: Input-Output Description 1.8.1 Electrical Systems 1.8.2 Mechanical Systems 1.8.3 Electromechanical Systems 1.9 Internal and External Descriptions of a System 1.10 Internal Description: The State-Space Description 1.11 MATLAB: Working with Functions 1.11.1 Anonymous Functions 1.11.2 Relational Operators and the Unit Step Function 1.11.3 Visualizing Operations on the Independent Variable 1.11.4 Numerical Integration and Estimating Signal Energy 1.12 Summary

2. Time-Domain Analysis of Continuous-Time Systems 2.1 Introduction 2.2 System Response to Internal Conditions: The Zero-Input Response 2.2.1 Some Insights into the Zero-Input Behavior of a System 2.3 The Unit Impulse Response 2.4 System Response to External Input: Zero-State Response 2.4.1 The Convolution Integral 2.4.2 Graphical Understanding of Convolution Operation 2.4.3 Interconnected Systems 2.4.4 A Very Special Function for LTIC Systems: The Everlasting Exponential 2.4.5 Total Response 2.5 System Stability 2.5.1 External (BIBO) Stability 2.5.2 Internal (Asymptotic) Stability 2.5.3 Relationship Between BIBO and Asymptotic Stability 2.6 Intuitive Insights into System Behavior 2.6.1 Dependence of System Behavior on Characteristic Modes 2.6.2 Response Time of a System: The System Time Constant 2.6.3 Time Constant and Rise Time of a System 2.6.4 Time Constant and Filtering 2.6.5 Time Constant and Pulse Dispersion (Spreading) 2.6.6 Time Constant and Rate of Information Transmission 2.6.7 The Resonance Phenomenon 2.7 MATLAB: M-Files 2.7.1 Script M-Files 2.7.2 Function M-Files 2.7.3 For-Loops 2.7.4 Graphical Understanding of Convolution 2.8 Appendix: Determining the Impulse Response 2.9 Summary

3. Time-Domain Analysis of Discrete-Time Systems 3.1 Introduction 3.1.1 Size of a Discrete-Time Signal 3.2 Useful Signal Operations 3.3 Some Useful Discrete-Time Signal Models 3.3.1 Discrete-Time Impulse Function 3.3.2 Discrete-Time Unit Step Function 3.3.3 Discrete-Time Exponential 3.3.4 Discrete-Time Sinusoid cos 3.3.5 Discrete-Time Complex Exponential 3.4 Examples of Discrete-Time Systems 3.4.1 Classification of Discrete-Time Systems 3.5 Discrete-Time System Equations 3.5.1 Recursive (Iterative) Solution of Difference Equation 3.6 System Response to Internal Conditions: The Zero-Input Response 3.7 The Unit Impulse Response h[n] 3.7.1 The Closed-Form Solution of h[n] 3.8 System Response to External Input: The Zero-State Response 3.8.1 Graphical Procedure for the Convolution Sum 3.8.2 Interconnected Systems 3.8.3 Total Response 3.9 System Stability 3.9.1 External (BIBO) Stability 3.9.2 Internal (Asymptotic) Stability 3.9.3 Relationship Between BIBO and Asymptotic Stability 3.10 Intuitive Insights into System Behavior 3.11 MATLAB: Discrete-Time Signals and Systems 3.11.1 Discrete-Time Functions and Stem Plots 3.11.2 System Responses Through Filtering 3.11.3 A Custom Filter Function 3.11.4 Discrete-Time Convolution 3.12 Appendix: Impulse Response for a Special Case 3.13 Summary

4. Continuous-Time System Analysis Using the Laplace Transform 4.1 The Laplace Transform 4.1.1 Finding the Inverse Transform 4.2 Some Properties of the Laplace Transform 4.2.1 Time Shifting 4.2.2 Frequency Shifting 4.2.3 The Time-Differentiation Property 4.2.4 The Time-Integration Property 4.2.5 The Scaling Property 4.2.6 Time Convolution and Frequency Convolution 4.3 Solution of Differential and Integro-Differential Equations 4.3.1 Comments on Initial Conditions at 0- and at 0+ 4.3.2 Zero-State Response 4.3.3 Stability 4.3.4 Inverse Systems 4.4 Analysis of Electrical Networks: The Transformed Network 4.4.1 Analysis of Active Circuits 4.5 Block Diagrams 4.6 System Realization 4.6.1 Direct Form I Realization 4.6.2 Direct Form II Realization 4.6.3 Cascade and Parallel Realizations 4.6.4 Transposed Realization 4.6.5 Using Operational Amplifiers for System Realization 4.7 Application to Feedback and Controls 4.7.1 Analysis of a Simple Control System 4.8 Frequency Response of an LTIC System 4.8.1 Steady-State Response to Causal Sinusoidal Inputs 4.9 Bode Plots 4.9.1 Constant Ka1a2/b1b3 4.9.2 Pole (or Zero) at the Origin 4.9.3 First-Order Pole (or Zero) 4.9.4 Second-Order Pole (or Zero) 4.9.5 The Transfer Function from the Frequency Response 4.10 Filter Design by Placement of Poles and Zeros of H(s) 4.10.1 Dependence of Frequency Response on Poles and Zeros of H(s) 4.10.2 Lowpass Filters 4.10.3 Bandpass Filters 4.10.4 Notch (Bandstop) Filters 4.10.5 Practical Filters and Their Specifications 4.11 The Bilateral Laplace Transform 4.11.1 Properties of Bilateral Laplace Transform 4.11.2 Using the Bilateral Transform for Linear System Analysis 4.12 MATLAB: Continuous-Time Filters 4.12.1 Frequency Response and Polynomial Evaluation 4.12.2 Butterworth Filters and the Find Command 4.12.3 Using Cascaded Second-Order Sections for Butterworth Filter Realization 4.12.4 Chebyshev Filters 4.13 Summary

5. Discrete-Time System Analysis Using the z-Transfor 5.1 The z-Transform 5.1.1 Inverse Transform by Partial Fraction Expansion and Tables 5.1.2 Inverse z-Transform by Power Series Expansion 5.2 Some Properties of the z-Transform 5.2.1 Time-Shifting Properties 5.2.2 z-Domain Scaling Property (Multiplication by yn) 5.2.3 z-Domain Differentiation Property (Multiplication by n) 5.2.4 Time-Reversal Property 5.2.5 Convolution Property 5.3 z-Transform Solution of Linear Difference Equations 5.3.1 Zero-State Response of LTID Systems: The Transfer Function 5.3.2 Stability 5.3.3 Inverse Systems 5.4 System Realization 5.5 Frequency Response of Discrete-Time Systems 5.5.1 The Periodic Nature of Frequency Response 5.5.2 Aliasing and Sampling Rate 5.6 Frequency Response from Pole-Zero Locations 5.7 Digital Processing of Analog Signals 5.8 The Bilateral z-Transform 5.8.1 Properties of the Bilateral z-Transform 5.8.2 Using the Bilateral z-Transform for Analysis of LTID Systems 5.9 Connecting the Laplace and z-Transforms 5.10 MATLAB: Discrete-Time IIR Filters 5.10.1 Frequency Response and Pole-Zero Plots 5.10.2 Transformation Basics 5.10.3 Transformation by First-Order Backward Difference 5.10.4 Bilinear Transformation 5.10.5 Bilinear Transformation with Prewarping 5.10.6 Example: Butterworth Filter Transformation 5.10.7 Problems Finding Polynomial Roots 5.10.8 Using Cascaded Second-Order Sections to Improve Design 5.11 Summary

6. Continuous-Time Signal Analysis: The Fourier Series 6.1 Periodic Signal Representation by Trigonometric Fourier Series 6.1.1 The Fourier Spectrum 6.1.2 The Effect of Symmetry 6.1.3 Determining the Fundamental Frequency and Period 6.2 Existence and Convergence of the Fourier Series 6.2.1 Convergence of a Series 6.2.2 The Role of Amplitude and Phase Spectra in Waveshaping 6.3 Exponential Fourier Series 6.3.1 Exponential Fourier Spectra 6.3.2 Parseval's Theorem 6.3.3 Properties of the Fourier Series 6.4 LTIC System Response to Periodic Inputs 6.5 Generalized Fourier Series: Signals as Vectors 6.5.1 Component of a Vector 6.5.2 Signal Comparison and Component of a Signal 6.5.3 Extension to Complex Signals 6.5.4 Signal Representation by an Orthogonal Signal Set 6.6 Numerical Computation of Dn 6.7 MATLAB: Fourier Series Applications 6.7.1 Periodic Functions and the Gibbs Phenomenon 6.7.2 Optimization and Phase Spectra 6.8 Summary

7. Continuous-Time Signal Analysis: The Fourier Transform 7.1 Aperiodic Signal Representation by the Fourier Integral 7.1.1 Physical Appreciation of the Fourier Transform 7.2 Transforms of Some Useful Functions 7.2.1 Connection Between the Fourier and Laplace Transforms 7.3 Some Properties of the Fourier Transform 7.4 Signal Transmission Through LTIC Systems 7.4.1 Signal Distortion During Transmission 7.4.2 Bandpass Systems and Group Delay 7.5 Ideal and Practical Filters 7.6 Signal Energy 7.7 Application to Communications: Amplitude Modulation 7.7.1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation 7.7.2 Amplitude Modulation (AM) 7.7.3 Single-Sideband Modulation (SSB) 7.7.4 Frequency-Division Multiplexing 7.8 Data Truncation: Window Functions 7.8.1 Using Windows in Filter Design 7.9 MATLAB: Fourier Transform Topics 7.9.1 The Sinc Function and the Scaling Property 7.9.2 Parseval's Theorem and Essential Bandwidth 7.9.3 Spectral Sampling 7.9.4 Kaiser Window Functions 7.10 Summary

8. Sampling: The Bridge from Continuous to Discrete 8.1 The Sampling Theorem 8.1.1 Practical Sampling 8.2 Signal Reconstruction 8.2.1 Practical Difficulties in Signal Reconstruction 8.2.2 Some Applications of the Sampling Theorem 8.3 Analog-to-Digital (A/D) Conversion 8.4 Dual of Time Sampling: Spectral Sampling 8.5 Numerical Computation of the Fourier Transform: The Discrete Fourier Transform 8.5.1 Some Properties of the DFT 8.5.2 Some Applications of the DFT 8.6 The Fast Fourier Transform (FFT) 8.7 MATLAB: The Discrete Fourier Transform 8.7.1 Computing the Discrete Fourier Transform 8.7.2 Improving the Picture with Zero Padding 8.7.3 Quantization 8.8 Summary

9. Fourier Analysis of Discrete-Time Signals 9.1 Discrete-Time Fourier Series (DTFS) 9.1.1 Periodic Signal Representation by Discrete-Time Fourier Series 9.1.2 Fourier Spectra of a Periodic Signal x[n] 9.2 Aperiodic Signal Representation by Fourier Integral 9.2.1 Nature of Fourier Spectra 9.2.2 Connection Between the DTFT and the z-Transform 9.3 Properties of the DTFT 9.4 LTI Discrete-Time System Analysis by DTFT 9.4.1 Distortionless Transmission 9.4.2 Ideal and Practical Filters 9.5 DTFT Connection with the CTFT 9.5.1 Use of DFT and FFT for Numerical Computation of DTFT 9.6 Generalization of the DTFT to the z-transform 9.7 MATLAB: Working with the DTFS and the DTFT 9.7.1 Computing the Discrete-Time Fourier Series 9.7.2 Measuring Code Performance 9.7.3 FIR Filter Design by Frequency Sampling 9.8 Summary

10. State-Space Analysis 10.1 Mathematical Preliminaries 10.1.1 Derivatives and Integrals of a Matrix 10.1.2 The Characteristic Equation of a Matrix: The Cayley-Hamilton Theorem 10.1.3 Computation of an Exponential and a Power of a Matrix 10.2 Introduction to State Space 10.3 A Systematic Procedure to Determine State Equations 10.3.1 Electrical Circuits 10.3.2 State Equations from a Transfer Function 10.4 Solution of State Equations 10.4.1 Laplace Transform Solution of State Equations 10.4.2 Time-Domain Solution of State Equations 10.5 Linear Transformation of State Vector 10.5.1 Diagonalization of Matrix A 10.6 Controllability and Observability 10.6.1 Inadequacy of the Transfer Function Description of a System 10.7 State-Space Analysis of Discrete-Time Systems 10.7.1 Solution in State-Space 10.7.2 The z-Transform Solution 10.8 MATLAB: Toolboxes and State-Space Analysis 10.8.1 z-Transform Solutions to Discrete-Time State-Space Systems 10.8.2 Transfer Functions from State-Space Representations 10.8.3 Controllability and Observability of Discrete-Time Systems 10.8.4 Matrix Exponentiation and the Matrix Exponential 10.9 Summary