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Signals and Systems,Collection Editor,Richard Baraniuk. Thanos Antoulas Don Johnson,Richard Baraniuk Ricardo Radaelli Sanchez. Steven Cox Justin Romberg,Benjamin Fite Phil Schniter. Roy Ha Melissa Selik,Michael Haag JP Slavinsky,Matthew Hutchinson. http cnx org content col10064 1 11,CONNEXIONS,Rice University Houston Texas.

2008 Richard Baraniuk, This selection and arrangement of content is licensed under the Creative Commons Attribution License. http creativecommons org licenses by 1 0,Table of Contents. 1 1 Signal Classi cations and Properties 1,1 2 Size of A Signal Norms 9. 1 3 Signal Operations 19,1 4 Useful Signals 22,1 5 The Impulse Function 25. 1 6 The Complex Exponential 28,1 7 Discrete Time Signals 31.

Solutions 35,2 1 System Classi cations and Properties 37. 2 2 Properties of Systems 41,3 Time Domain Analysis of Continuous Time Systems. 3 1 CT Linear Systems and Di erential Equations 47. 3 2 Continuous Time Convolution 53,3 3 Properties of Convolution 59. 3 4 BIBO Stability 65,4 Time Domain Analysis of Discrete Time Systems. 4 1 Discrete Time Systems in the Time Domain 69,4 2 Discrete Time Convolution 73.

4 3 Circular Convolution and the DFT 158, 4 4 Linear Constant Coe cient Di erence Equations 82. 4 5 Solving Linear Constant Coe cient Di erence Equations 83. Solutions 89,5 Linear Algebra Overview,5 1 Linear Algebra The Basics 91. 5 2 Eigenvectors and Eigenvalues 96,5 3 Matrix Diagonalization 101. 5 4 Eigen stu in a Nutshell 104,5 5 Eigenfunctions of LTI Systems 105. 5 6 Fourier Transform Properties 108,Solutions 109.

6 Continuous Time Fourier Series,6 1 Periodic Signals 111. 6 2 Fourier Series Eigenfunction Approach 112,6 3 Derivation of Fourier Coe cients Equation 115. 6 4 Fourier Series in a Nutshell 116,6 5 Fourier Series Properties 119. 6 6 Symmetry Properties of the Fourier Series 122, 6 7 Circular Convolution Property of Fourier Series 126. 6 8 Fourier Series and LTI Systems 127,6 9 Convergence of Fourier Series 130.

6 10 Dirichlet Conditions 132,6 11 Gibbs s Phenomena 134. 6 12 Fourier Series Wrap Up 137,Solutions 139,7 Discrete Fourier Transform. 7 1 Fourier Analysis 141,7 2 Fourier Analysis in Complex Spaces 142. 7 3 Matrix Equation for the DTFS 149,7 4 Periodic Extension to DTFS 150. 7 5 Circular Shifts 154,7 6 Circular Convolution and the DFT 158.

Solutions 163,8 Fast Fourier Transform FFT,8 1 DFT Fast Fourier Transform 165. 8 2 The Fast Fourier Transform FFT 166,8 3 Deriving the Fast Fourier Transform 167. Solutions 170,9 Convergence,9 1 Convergence of Sequences 171. 9 2 Convergence of Vectors 173,9 3 Uniform Convergence of Function Sequences 176. 10 Discrete Time Fourier Transform DTFT,10 1 Discrete Fourier Transformation 179.

10 2 Discrete Fourier Transform DFT 181,10 3 Table of Common Fourier Transforms 183. 10 4 Discrete Time Fourier Transform DTFT 184, 10 5 Discrete Time Fourier Transform Properties 185. 10 6 Discrete Time Fourier Transform Pair 185,10 7 DTFT Examples 186. Solutions 190,11 Continuous Time Fourier Transform CTFT. 11 1 Continuous Time Fourier Transform CTFT 191, 11 2 Properties of the Continuous Time Fourier Transform 192.

Solutions 196,12 Sampling Theorem,12 1 Sampling 197. 12 2 Reconstruction 201,12 3 More on Reconstruction 205. 12 4 Nyquist Theorem 207,12 5 Aliasing 209,12 6 Anti Aliasing Filters 212. 12 7 Discrete Time Processing of Continuous Time Signals 214. Solutions 217,13 Laplace Transform and System Design. 13 1 The Laplace Transforms 219,13 2 Properties of the Laplace Transform 222.

13 3 Table of Common Laplace Transforms 223, 13 4 Region of Convergence for the Laplace Transform 223. 13 5 The Inverse Laplace Transform 225,13 6 Poles and Zeros 227. 14 Z Transform and Digital Filtering,14 1 The Z Transform De nition 231. 14 2 Table of Common z Transforms 236, 14 3 Region of Convergence for the Z transform 237. 14 4 Inverse Z Transform 246,14 5 Rational Functions 249.

14 6 Di erence Equation 251, 14 7 Understanding Pole Zero Plots on the Z Plane 254. 14 8 Filter Design using the Pole Zero Plot of a Z Transform 258. 15 Appendix Hilbert Spaces and Orthogonal Expansions. 15 1 Vector Spaces 263,15 2 Norms 265,15 3 Inner Products 268. 15 4 Hilbert Spaces 270,15 5 Cauchy Schwarz Inequality 270. 15 6 Common Hilbert Spaces 277,15 7 Types of Basis 280. 15 8 Orthonormal Basis Expansions 283,15 9 Function Space 287.

15 10 Haar Wavelet Basis 288, 15 11 Orthonormal Bases in Real and Complex Spaces 295. 15 12 Plancharel and Parseval s Theorems 297, 15 13 Approximation and Projections in Hilbert Space 298. Solutions 301,16 Homework Sets,16 1 Homework 1 303. 16 2 Homework 1 Solutions 307,17 Viewing Embedded LabVIEW Content 319. Glossary 320,Attributions 330,1 1 Signal Classi cations and Properties 1.

1 1 1 Introduction, This module will lay out some of the fundamentals of signal classi cation This is basically a list of de nitions. and properties that are fundamental to the discussion of signals and systems It should be noted that some. discussions like energy signals vs power signals have been designated their own module for a more complete. discussion and will not be included here,1 1 2 Classi cations of Signals. Along with the classi cation of signals below it is also important to understand the Classi cation of Systems. Section 2 1,1 1 2 1 Continuous Time vs Discrete Time. As the names suggest this classi cation is determined by whether or not the time axis x axis is discrete. countable or continuous Figure 1 1 A continuous time signal will contain a value for all real numbers. along the time axis In contrast to this a discrete time signal Section 1 7 is often created by using the. sampling theorem to sample a continuous signal so it will only have values at equally spaced intervals along. the time axis, 1 This content is available online at http cnx org content m10057 2 17. 2 Signal Energy vs Signal Power http cnx org content m10055 latest. 3 The Sampling Theorem http cnx org content m0050 latest. 2 CHAPTER 1 SIGNALS,Figure 1 1,1 1 2 2 Analog vs Digital.

The di erence between analog and digital is similar to the di erence between continuous time and discrete. time In this case however the di erence is with respect to the value of the function y axis Figure 1 2. Analog corresponds to a continuous y axis while digital corresponds to a discrete y axis An easy example. of a digital signal is a binary sequence where the values of the function can only be one or zero. Figure 1 2,1 1 2 3 Periodic vs Aperiodic, Periodic signals Section 6 1 repeat with some period T while aperiodic or nonperiodic signals do not. Figure 1 3 We can de ne a periodic function through the following mathematical expression where t can. be any number and T is a positive constant,f t f T t 1 1. The fundamental period of our function f t is the smallest value of T that the still allows 1 1 to be. Figure 1 3 a A periodic signal with period T0 b An aperiodic signal. 1 1 2 4 Causal vs Anticausal vs Noncausal, Causal signals are signals that are zero for all negative time while anticausal are signals that are zero for. all positive time Noncausal signals are signals that have nonzero values in both positive and negative time. Figure 1 4,4 CHAPTER 1 SIGNALS, Figure 1 4 a A causal signal b An anticausal signal c A noncausal signal. 1 1 2 5 Even vs Odd, An even signal is any signal f such that f t f t Even signals can be easily spotted as they are.

symmetric around the vertical axis An odd signal on the other hand is a signal f such that f t. f t Figure 1 5,Figure 1 5 a An even signal b An odd signal. Using the de nitions of even and odd signals we can show that any signal can be written as a combination. of an even and odd signal That is every signal has an odd even decomposition To demonstrate this we. have to look no further than a single equation,f t f t f t f t f t 1 2. By multiplying and adding this expression out it can be shown to be true Also it can be shown that. f t f t ful lls the requirement of an even function while f t f t ful lls the requirement of an. odd function Figure 1 6,Example 1 1,6 CHAPTER 1 SIGNALS. Figure 1 6 a The signal we will decompose using odd even decomposition b Even part e t. f t f t c Odd part o t 21 f t f t d Check e t o t f t. 1 1 2 6 Deterministic vs Random, A deterministic signal is a signal in which each value of the signal is xed and can be determined by a. mathematical expression rule or table Because of this the future values of the signal can be calculated. from past values with complete con dence On the other hand a random signal 4. has a lot of uncertainty, about its behavior The future values of a random signal cannot be accurately predicted and can usually.

only be guessed based on the averages of sets of signals Figure 1 7. Figure 1 7 a Deterministic Signal b Random Signal,1 1 2 7 Right Handed vs Left Handed. A right handed signal and left handed signal are those signals whose value is zero between a given variable. and positive or negative in nity Mathematically speaking a right handed signal is de ned as any signal. where f t 0 for t t1 and a left handed signal is de ned as any signal where f t 0 for. t t1 See Figure 1 8 for an example Both gures begin at t1 and then extends to positive or. negative in nity with mainly nonzero values, 4 Introduction to Random Signals and Processes http cnx org content m10649 latest. 5 Random Processes Mean and Variance http cnx org content m10656 latest. and properties that are fundamental to the discussion of signals and systems It should be noted that some discussions like energy signals vs power signals 2 have been designated their own module for a more complete discussion and will not be included here 1 1 2 Classi cations of Signals Along with the classi cation of signals below it is also important to understand the Classi cation of

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