Note for Signals And Systems - SS By JNTU Heroes

Topics ( 7 )

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Contents 1 Fundamentals of Signals 1.1 What is a Signal? . . . . . . . . . . . . . . . . . . 1.2 Review on Complex Numbers . . . . . . . . . . . 1.3 Basic Operations of Signals . . . . . . . . . . . . 1.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . 1.5 Even and Odd Signals . . . . . . . . . . . . . . . 1.6 Impulse and Step Functions . . . . . . . . . . . . 1.7 Continuous-time Complex Exponential Functions 1.8 Discrete-time Complex Exponentials . . . . . . . . . . . . . . . 5 5 6 7 11 15 17 22 24 . . . . 27 28 33 37 41 3 Fourier Series 3.1 Eigenfunctions of an LTI System . . . . . . . . . . . . . . . . . . . . 3.2 Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . 3.3 Properties of Fourier Series Coefficients . . . . . . . . . . . . . . . . . 43 43 47 54 4 Continuous-time Fourier Transform 4.1 Insight from Fourier Series . . . . . . . 4.2 Fourier Transform . . . . . . . . . . . . . 4.3 Relation to Fourier Series . . . . . . . . 4.4 Examples . . . . . . . . . . . . . . . . . 4.5 Properties of Fourier Transform . . . . . 4.6 System Analysis using Fourier Transform 57 57 59 61 64 66 69 2 Fundamentals of Systems 2.1 System Properties . . . . . . . . . . . . . 2.2 Convolution . . . . . . . . . . . . . . . . 2.3 System Properties and Impulse Response 2.4 Continuous-time Convolution . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 CONTENTS 5 Discrete-time Fourier Transform 5.1 Review on Continuous-time Fourier Transform 5.2 Deriving Discrete-time Fourier Transform . . . 5.3 Why is X(ejω ) periodic ? . . . . . . . . . . . . 5.4 Properties of Discrete-time Fourier Transform 5.5 Examples . . . . . . . . . . . . . . . . . . . . 5.6 Discrete-time Filters . . . . . . . . . . . . . . 5.7 Appendix . . . . . . . . . . . . . . . . . . . . 6 Sampling Theorem 6.1 Analog to Digital Conversion . . . . . 6.2 Frequency Analysis of A/D Conversion 6.3 Sampling Theorem . . . . . . . . . . . 6.4 Digital to Analog Conversion . . . . . 7 The 7.1 7.2 7.3 7.4 z-Transform The z-Transform . . . . . . . . . . . z-transform Pairs . . . . . . . . . . . Properties of ROC . . . . . . . . . . System Properties using z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 73 74 76 77 78 80 81 . . . . 83 83 84 89 90 . . . . 95 95 100 102 104

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Chapter 1 Fundamentals of Signals 1.1 What is a Signal? A signal is a quantitative description of a physical phenomenon, event or process. Some common examples include: 1. Electrical current or voltage in a circuit. 2. Daily closing value of a share of stock last week. 3. Audio signal: continuous-time in its original form, or discrete-time when stored on a CD. More precisely, a signal is a function, usually of one variable in time. However, in general, signals can be functions of more than one variable, e.g., image signals. In this class we are interested in two types of signals: 1. Continuous-time signal x(t), where t is a real-valued variable denoting time, i.e., t ∈ R. We use parenthesis (·) to denote a continuous-time signal. 2. Discrete-time signal x[n], where n is an integer-valued variable denoting the discrete samples of time, i.e., n ∈ Z. We use square brackets [·] to denote a discrete-time signal. Under the definition of a discrete-time signal, x[1.5] is not defined, for example. 5

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6 1.2 CHAPTER 1. FUNDAMENTALS OF SIGNALS Review on Complex Numbers We are interested in the general complex signals: x(t) ∈ C and x[n] ∈ C, where the set of complex numbers is defined as C = {z | z = x + jy, x, y ∈ R, j = √ −1.} A complex number z can be represented in Cartesian form as z = x + jy, or in polar form as z = rejθ . Theorem 1. Euler’s Formula ejθ = cos θ + j sin θ. (1.1) Using Euler’s formula, the relation between x, y, r, and θ is given by ( ( p x = r cos θ r = x2 + y 2 , and θ = tan−1 xy . y = r sin θ Figure 1.1: A complex number z can be expressed in its Cartesian form z = x + jy, or in its polar form z = rejθ .