Essentials of Signals and Systems

2.12.9 Product in the Time Domain 97 Worked Exercise: The Fourier Transform of a Physical Sinusoidal Wave 98 2.12.10 The Energy of a Signal and Parseval's Theorem 101 2.13 The Fourier Series 102 Worked Exercise: The Fourier Series of a Square Wave 108 2.14 Exercises 109 3 Representation of Systems 113 3.1 Introduction and Properties 113 3.1.1 Linearity 114 3.1.2 Time Invariance 114 Worked Exercise: Example of a Time Invariant System 116 Worked Exercise: An Example of a Time Variant System 117 3.1.3 Causality 117 3.2 Operators Representing Linear and Time Invariant Systems 118 3.3 Linear Systems as Matrices 119 3.4 Operators in Dirac Notation 121 3.5 Statement of the Problem 123 3.6 Eigenvectors and Eigenvalues of LTI Operators 123 3.7 General Method of Solution 124 3.7.1 Step 1: Defining the Problem 124 3.7.2 Step 2: Finding the Eigenvalues 125 3.7.3 Step 3: The Representation in the Basis of Eigenvectors 126 3.7.4 Step 4: Solving the Equation and Returning to the Original Basis 129 Worked Exercise: Input is an Eigenvector 130 Worked Exercise: Input is an Explicit Linear Combination of Eigenvectors 131 Worked Exercise: An Arbitrary Input 132 3.8 The Physical Meaning of Eigenvalues: The Impulse and Frequency Responses 133 Worked Exercise: Impulse and Frequency Responses of a Harmonic Oscillator 136 Worked Exercise: How can the Frequency Response be Measured? 139 Worked Exercise: The Transient of a Harmonic Oscillator 142 Worked Exercise: Charge and Discharge in an RC Circuit 145 3.9 Frequency Conservation in LTI Systems 147 3.10 Frequency Conservation in Other Fields 148 3.10.1 Snell's Law 149 3.10.2 Wavefunctions and Heisenberg's Uncertainty Principle 150 3.11 Exercises 152 4 Electric Circuits as LTI Systems 157 4.1 Electric Circuits as LTI Systems 157 4.2 Phasors, Impedances, and the Frequency Response 158 Worked Exercise: An RLC Circuit as a Harmonic Oscillator 163 4.3 Exercises 164 5 Filters 165 5.1 Ideal Filters 165 5.2 Example of a Low-pass Filter 167 5.3 Example of a High-pass Filter 170 5.4 Example of a Band-pass Filter 171 5.5 Exercises 172 6 Introduction to the Laplace Transform 175 6.1 Motivation: Stability of LTI Systems 175 6.2 The Laplace Transform as a Generalization of the Fourier Transform 179 6.3 Properties of Laplace Transforms 181 6.4 Region of Convergence 182 6.5 Inverse Laplace Transform by Inspection 185 Worked Exercise: Example of Inverse Laplace Transform by Inspection 185 Worked Exercise: Impulse Response of a Harmonic Oscillator 187 6.6 Zeros and Poles 188 Worked Exercise: Finding the Zeros and Poles 189 Worked Exercise: Poles of a Harmonic Oscillator 190 6.7 The Unilateral Laplace Transform 191 6.7.1 The Differentiation Property of the Unilateral Fourier Transform 193 Worked Exercise: Differentiation Property of the Unilateral Fourier Transform Involving Higher Order Derivatives 195 Worked Exercise: Example of Differentiation Using the Unilateral Fourier Transform 196 Worked Exercise: Discharge of an RC Circuit 197 6.7.2 Generalization to the Unilateral Laplace Transform 198 6.8 Exercises 199 Interlude: Discrete Signals and Systems: Why do we Need Them? 203 7 The Sampling Theorem and the Discrete Time Fourier Transform (DTFT) 205 7.1 Discrete Signals 205 7.2 Fourier Transforms of Discrete Signals and the Sampling Theorem 207 7.3 The Discrete Time Fourier Transform (DTFT) 216 Worked Exercise: Example of a Matlab Routine to Calculate the Dtft 218 Worked Exercise: Fourier Transform from the DTFT 221 7.4 The Inverse DTFT 223 7.5 Properties of the DTFT 224 7.5.1 'Time' shifting 225 7.5.2 Difference 226 7.5.3 Sum 228 7.5.4 Convolution in the 'Time' Domain 229 7.5.5 Product in the Time Domain 230 7.5.6 The Theorem that Should not be: Energy of Discrete Signals 231 7.6 Concluding Remarks 235 7.7 Exercises 235 8 The Discrete Fourier Transform (DFT) 239 8.1 Discretizing the Frequency Domain 239 8.2 The DFT and the Fast Fourier Transform (fft) 246 Worked Exercise: Getting the Centralized DFT Using the Command fft 250 Worked Exercise: Getting the Fourier Transform with the fft 254 Worked Exercise: Obtaining the Inverse Fourier Transform Using the ifft 256 8.3 The Circular Time Shift 258 8.4 The Circular Convolution 259 8.5 Relationship Between Circular and Linear Convolutions 264 8.6 Parseval's Theorem for the DFT 269 8.7 Exercises 270 9 Discrete Systems 275 9.1 Introduction and Properties 275 9.1.1 Linearity 276 9.1.2 'Time' invariance 276 9.1.3 Causality 276 9.1.4 Stability 276 9.2 Linear and Time Invariant Discrete Systems 277 Worked Exercise: Further Advantages of Frequency Domain 279 9.3 Digital Filters 283 9.4 Exercises 285 10 Introduction to the z-transform 287 10.1 Motivation: Stability of LTI Systems 287 10.2 The z-transform as a Generalization of the DTFT 289 Worked Exercise: Example of z-transform 290 10.3 Relationship Between the z-transform and the Laplace Transform 292 10.4 Properties of the z-transform 293 10.4.1 'Time' shifting 294 10.4.2 Difference 294 10.4.3 Sum 294 10.4.4 Convolution in the Time Domain 294 10.5 The Transfer Function of Discrete LTI Systems 295 10.6 The Unilateral z-transform 295 10.7 Exercises 297 References with Comments 299 Appendix A: Laplace Transform Property of Product in the Time Domain 301 Appendix B: List of Properties of Laplace Transforms 303 Index 305