Linear Systems and Signals

Preface Each chapter ends with a Summary and References. Each MATLAB section ends with Problems. B. Background B.1. Complex Numbers B.2. Sinusoids B.3. Sketching Signals B.4. Cramer's Rule B.5. Partial Fraction Expansion B.6. Vectors and Matrices B.7. Miscellaneous MATLAB Session B: Elementary Operations B.M.1. MATLAB Overview B.M.2. Calculator Operations B.M.3. Vector Operations B.M.4. Simple Plotting B.M.5. Element-by-Element Operations B.M.6. Matrix Operations B.M.7. Partial Fraction Expansions 1. Signals and Systems 1.1. Size of a Signal 1.2. Some Useful Signal Operations 1.3. Classification of Signals 1.4. Some Useful Signal Models 1.5. Even and Odd Functions 1.6. Systems 1.7. Classification of Systems 1.8. System Model: Input-Output Description 1.9. Internal and External Descriptions of a System 1.10. Internal Description: The State-Space Description MATLAB Session 1: Working with Functions 1.M.1. Inline Functions 1.M.2. Relational Operators and the Unit Step Function 1.M.3. Visualizing Operations on the Independent Variable 1.M.4. Numerical Integration and Estimating Signal Energy 2. Time-Domain Analysis of Continuous-Time Systems 2.1. Introduction 2.2. System Response to Internal Conditions: The Zero-Input Response 2.3. The Unit Impulse Response h(t) 2.4. System Response to External Input: Zero-State Response 2.5. Classical Solution of Differential Equations 2.6. System Stability 2.7. Intuitive Insights into System Behavior 2.8. Appendix 2.1: Determining the Impulse Response MATLAB Session 2: M-Files 2.M.1. Script M-Files 2.M.2. Function M-Files 2.M.3. For Loops 2.M.4. Graphical Understanding of Convolution 3. Time-Domain Analysis of Discrete-Time Systems 3.1. Introduction 3.2. Useful Signal Operations 3.3. Some Useful Discrete-Time Signal Models 3.4. Examples of Discrete-Time Systems 3.5. Discrete-Time System Equations 3.6. System Response to Internal Conditions: The Zero-Input Response 3.7. The Unit Impulse Response h[n] 3.8. System Response to External Input: The Zero-State Response 3.9. Classical Solution of Linear Difference Equations 3.10. System Stability: The External (BIBO) Stability Criterion 3.11. Intuitive Insights into System Behavior 3.12. Appendix 3.1: Impulse Response for a Special Case When aN = 0 MATLAB Session 3: Discrete-Time Signals and Systems 3.M.1. Discrete-Time Functions and Stem Plots 3.M.2. System Responses Through Filtering 3.M.3. A Custom Filter Function 3.M.4. Discrete-Time Convolution 4. Continuous-Time System Analysis Using the Laplace Transform 4.1. The Laplace Transform 4.2. Some Properties of the Laplace Transform 4.3. Solution of Differential and Integro-Differential Equations 4.4. Analysis of Electrical Networks: The Transformed Network 4.5. Block Diagrams 4.6. System Realization 4.7. Application to Feedback and Controls 4.8. Frequency-Response of an LTIC System 4.9. Bode Plots 4.10. Filter Design by Placement of Poles and Zeros of H(s) 4.11. The Bilateral Laplace Transform MATLAB Session 4: Continuous-Time Filters 4.M.1. Frequency Response and Polynomial Evaluation 4.M.2. Design and Evaluation of a Simple RC Filter 4.M.3. A Cascaded RC Filter and Polynomial Expansion 4.M.4. Butterworth Filters and the FIND Command 4.M.5. Butterworth Filter Realization Using Cascaded Second.Order Sections 4.M.6. Chebyshev Filters 5. Discrete-Time System Analysis Using the z -Transform 5.1. The z -Transform 5.2. Some Properties of the z -Transform 5.3. z -Transform Solution of Linear Difference equations 5.4. System Realization 5.5. Frequency Response of Discrete-Time Systems 5.6. Frequency Response from Pole-Zero Location 5.7. Digital Processing of Analog Signals 5.8. Connection Between the Laplace and the z -Transform 5.9. The Bilateral z -Transform MATLAB Session 5: Discrete-Time IIR Filters 5.M.1. Frequency Response and Pole-Zero Plots 5.M.2. Transformation Basics 5.M.3. Transformation by First-Order Backward Difference 5.M.4. Bilinear Transformation 5.M.5. Bilinear Transformation with Prewarping 5.M.6. Example: Butterworth Filter Transformation 5.M.7. Problems Finding Polynomial Roots 5.M.8. Improved Design Using Cascaded Second-Order Sections 6. Continuous-Time Signal Analysis: The Fourier Series 6.1. Periodic Signal Representation by Trigonometric Fourier Series 6.2. Existence and Convergence of the Fourier Series 6.3. Exponential Fourier Series 6.4. LTIC System Response to Periodic Inputs 6.5. Generalized Fourier Series: Signals as Vectors 6.6. Numerical Computation of Dn MATLAB Session 6: Fourier Series Applications 6.M.1. Periodic Functions and the Gibbs Phenomenon 6.M.2. Optimization and Phase Spectra 7. Continuous-Time Signal Analysis: The Fourier Transform 7.1. Aperiodic Signal Representation by Fourier Integral 7.2. Transforms of Some Useful Functions 7.3. Some Properties of the Fourier Transform 7.4. Signal Transmission Through LTIC Systems 7.5. Ideal and Practical Filters 7.6. Signal Energy 7.7. Application to Communications: Amplitude Modulation 7.8. Data Truncation: Window Functions MATLAB Session 7: Fourier Transform Topics 7.M.1. The Sinc Function and the Scaling Property 7.M.2. Parseval's Theorem and Essential Bandwidth 7.M.3. Spectral Sampling 7.M.4. Kaiser Window Functions 8. Sampling: The Bridge from Continuous to Discrete 8.1. The Sampling Theorem 8.2. Signal Reconstruction 8.3. Analog-to-Digital (A/D) Conversion 8.4. Dual of Time-Sampling: The Spectral Sampling 8.5. Numerical Computation of the Fourier Transform: The Discrete Fourier Transform (DFT) 8.6. The Fast Fourier Transform (FFT) MATLAB Session 8: The Discrete Fourier Transform 8.M.1. Computing the Discrete Fourier Transform 8.M.2. Improving the Picture with Zero-Padding 8.M.3. Quantization 9. Fourier Analysis of Discrete-Time Signals 9.1. Discrete-Time Fourier Series (DTFS) 9.2. Aperiodic Signal Representation by Fourier Integral 9.3. Properties of DTFT 9.4. LTI Discrete-Time System Analysis by DTFT 9.5. DTFT Connection with the CTFT 9.6. Generalization of the DTFT and the z -Transform MATLAB Session 9: Working with the DTFS and the DTFT 9.M.1. Computing the Discrete-Time Fourier Series 9.M.2. Measuring Code Performance 9.M.3. FIR Filter Design by Frequency Sampling 10. State-Space Analysis 10.1. Introduction 10.2. A Systematic Procedure for Determining State Equations 10.3. Solution of State Equations 10.4. Linear Transformation of State Vectors 10.5. Controllability and Observability 10.6. State-Space Analysis of Discrete-Time Systems MATLAB Session 10: Toolboxes and State-Space Analysis 10.M.1. z -Transform Solutions to Discrete-Time State-Space Systems 10.M.2. Transfer Functions from State-Space Representations 10.M.3. Controllability and Observability of Discrete-Time Systems 10.M.4. Matrix Exponentiation and the Matrix Exponential Index