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# Note for Signals and Stochastic Process - SSP By CHEVELLA ANILKUMAR

• Signals and Stochastic Process - SSP
• Note
• Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP
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Chapter 1 Signals 1.1 Introduction In typical applications of science and engineering, we have to process signals, using systems. While the applications can be varied large com-munication systems to control systems but the basic analysis and design tools are the same. In a signals and systems course, we study these tools: convolution, Fourier analysis, z-transform, and Laplace transform. The use of these tools in the analysis of linear time-invariant (LTI) systems with determin-istic signals. For most practical systems, input and output signals are continuous and these signals can be processed using continuous systems. However, due to advances in digital systems technology and numerical algorithms, it is advantageous to process continuous signals using digital systems by converting the input signal into a digital signal. Therefore, the study of both continuous and digital systems is required. As most practical systems are digital and the concepts are relatively easier to understand, we describe discrete signals and systems rst, im-mediately followed by the corresponding description of continuous signals and systems. 3

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4 Dept. of ECE, Vardhaman College of Engineering 1.2 Classi cation of the Signals A Signals can be classi ed into several categories depending upon the criteria and for its classi cation. Broadly the signals are classi ed into the following categories 1. Continuous, Discrete and Digital Signals 2. Periodic and Aperiodic Signals 3. Even and Odd Signals 4. Complex Symmetry and Complex asymmetry Signals 5. Power and Energy Signals 1.2.1 Continuous-time and Discrete-time Signals Continuous-Time (CT) Signals: They may be de ned as continuous in time and continuous in amplitude as shown in Figure 1.1. Ex: Speech, audio signals etc.. Discrete Time (DT) Signals: Discretized in time and Continuous in amplitude. They may also be de ned as sampled version of continuous time signals. Ex: Rail tra c signals. Digital Signals: Discretized in time and quantized in amplitude. They may also be de ned as quantized version of discrete signals. Figure 1.1: Description of Continuous, Discrete and Digital Signals Dr.J.V.R.Ravindra

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Dept. of ECE, Vardhaman College of Engineering 1.2.2 5 Periodic Signals A CT signal x(t) is said to be periodic if it satis es the following condition x (t) = x (t + T0) (1.1) The smallest positive value of T0 that satis es the periodicity condition Eq.(1.1), is referred as the fundamental period of x(t). The reciprocal of fundamental period of a signal is fundamental frequency f0. Likewise, a DT signal x[n] is said to be periodic if it satis es x [n] = x [n + N0] (1.2) at all time k and for some positive constant N0. The smallest positive value of N0 that satis es the periodicity condition Eq.(1.2) is referred to as the fundamental period of x [n]. Note: All periodic signals are ever lasting signals i.e. they start at 1 and end at +1 as shown in Figure 1.2. Figure 1.2: A typical periodic signal Ex.1.1 Consider a periodic signal is a sinusoidal function represented as x (t) = A sin (!0t + ) The time period of the signal T0 is 2 =!0. Ex.1.2 CT tangent wave: x (t) = tan (10t) is a periodic signal with period T = =10 Note: Amplitude and phase di erence will not e ect the time period. i.e. 2 sin (3t), 4 sin (3t), Dr.J.V.R.Ravindra

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6 Dept. of ECE, Vardhaman College of Engineering 4 sin (3t + 32 ) will have the same time period. Ex.1.3 CT complex exponential: x (t) = e with period T = Ex.1.4 CT sine wave of limited duration: ( x (t) = j(2t+7) sin 4 t 2 0 is a periodic signal t 2 otherwise is a aperiodic signal. Ex.1.5 CT linear relationship x (t) = 2t + 5 is an aperiodic signal. Note: An arbitrary DT sinusoidal sequence x [n] = A sin ( 0n + ) is periodic i 0=2 is a rational number. Ex.1.6 x [n] = cos (4 n) is a periodic signal whereas x [n] = cos (4n) is not a periodic signal. Linear Combination of `n' signals: A signal g(t) a periodic sig-nal with time period T and is a linear combination of `n' distinct signals x1 (t) ; x2 (t) ; x3 (t) xn (t) whose time periods are T1; T2; T3 Tn respectively. Then T = LCM (T1; T2; T3; Tn) or T1 T1 T1 T2 ; T3 Tn must be rational Ex.1.7 Consider the following signal g (t) = 2 cos (4 t) 4 sin (5 t) Calculating the ratio of the two fundamental periods yields T 1 T 1/ = 2 2 2 /5 which is a rational number. Hence, the linear combination g(t) is periodic and its period is T = 1=2. Ex.1.8 Consider the following signal g T(t) = 2 cos (4 t) 4 cos (3t) is aperiodic signal, because the ratio 1 is irrational. T2 Dr.J.V.R.Ravindra