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basicsig_06

# basicsig_06 - Basic Properties Of Signals Types Of Signals...

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Basic Properties Of Signals Types Of Signals There are 3 basic types of signals. They are * Continuous-Time * Discrete-Time * Digital A description of each follows. Continuous-Time Signal : A continuous-time signal is defined for all values of time over a specified time interval between t 1 and t 2 . Example : The signal ( ) sin(2 ) x t t t π = Defined over the interval . A plot of this signal is shown below. 0 to 1 t = Discrete-Time Signal : A discrete-time signal is defined only for integer values, n , over some specified interval from integer value n 1 to integer value n 2 . Example: The signal ( ) ( ) .9 n x n = Defined for integer values for . A plot of this signal is shown below. 0 to 15 n = Digital Signal : A digital signal is the same as a discrete-time signal, except its amplitude values are restricted to a finite set of values L . In other words a digital signal is both discrete in time and amplitude. When L =2, the digital signal is called a binary signal.

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2 Classification Of Signals Based On Their Properties 1. Power Vs Energy — Continuous Signals v ( t ) induces current i ( t ) i (t) causes resistor to dissipate energy in the form of heat Energy Dissipated over time interval t 1 to t 2 = t 1 + T ( ) 2 1 2 Energy Dissipated t T t v t E dt R = = Average Power over time interval t 1 to t 2 = t 1 + T Usual Notation: ( ) 2 1 2 1 t T T t v t E P dt T T R = = Normalize By Setting R=1 Then ( ) ( ) 1 1 1 1 2 2 t T t T T t t E v t dt i t dt + + = = ( ) ( ) 1 1 1 1 2 2 1 1 t T t T T t t P v t dt i t dt T T + + = = Generalizations 1. Allow signal v ( t ) to have complex values ( ) Example : = j t v t Ae ω When the signal is complex the above expressions must be modified
3 ( ) ( ) 1 1 1 1 2 2 t T t T T t t E v t dt i t dt + + = = ( ) ( ) 1 1 1 1 2 2 1 1 t T t T T t t P v t dt i t dt T T + + = = 2. If the signal x ( t ) is neither a voltage nor current the above expressions are still called the energy or power of the signal Limiting Values — Important Quantities ( ) T →∞ lim lim T T T T E E P P →∞ →∞ = = Here, is the total energy dissipated and is the average power. E P Classification of Signals Is Based On Limiting Values Finite Energy Signal E <∞ Ex: ( ) for 0 0 for 0 t x t e t t = = < Let t 1 = - T /2, then ( ) 2 2 2 2 2 2 0 0 0 1 1 1 2 2 T T T t t t T T E e dt e dt e e = = =− =−

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4 1 lim 2 T T E E →∞ = = < ∞ And 1 lim lim 1 0 T T T T E P e T T →∞ →∞ = = = Finite Power Signal 0 P < <∞ Ex: ( ) ( ) 0 10cos for all x t t t ω = Again let t 1 = - T /2, then ( ) ( ) 2 2 2 2 0 2 2 100cos T T T T T E x t dt t dt ω = = Note: Some useful identities ( ) ( ) 2 2 1 1 cos cos 2 2 2 1 1 sin cos 2 2 2 x x x x = + = Using the first identity
5 ( ) ( ) ( ) ( ) ( ) [ ] ( ) 2 0 2 2 2 0 2 2 2 2 0 0 0 0 0 2 2 0 0 1 1 100 cos 2 2 2 50 50 cos 2 25 25 50 sin 2 50 sin sin 50 50 sin T T T T T T T T T T T E t dt dt t dt t t T T T T T ω ω ω ω ω ω ω ω ω = + = + = + = + + = + As T approaches , the second term oscillates between values 0 0 50 50 and ω ω These values are finite. However the first term grows without bound. Thus E =∞ But ( ) 0 0 50 50 sin T T E P T T T ω ω

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basicsig_06 - Basic Properties Of Signals Types Of Signals...

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