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 ) xt 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 xn = 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 vt Ed t R == Average Power over time interval t 1 to t 2 = t 1 + T Usual Notation: 2 1 2 1 t T T t E Pd t TT R Normalize By Setting R=1 Then () () 11 22 tT T tt E v td t i td t ++ ∫∫ T Pv t d t i t d t Generalizations 1. Allow signal v ( t ) to have complex values ( ) Example : = jt vt Ae ω When the signal is complex the above expressions must be modified
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3 () 11 22 tT T tt E v td t i t ++ == ∫∫ T P v t i t TT 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 EEP 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 xt e t t =≥ =< Let t 1 = - T /2, then 2 2 00 0 1 T ttt T T Ee d t e d t e e −− = = =−
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4 1 lim 2 T T EE →∞ == < And 1 lim lim 1 0 T T TT E Pe →∞ →∞ = Finite Power Signal 0 P ⇒< < Ex: ( ) ( ) 0 10cos for all xt t t ω = Again let t 1 = - T /2, then () 22 0 100cos T Ex t d t t d t −− ∫∫ Note: Some useful identities 2 2 11 cos cos 2 sin cos 2 xx =+ =− Using the first identity
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5 () [] 2 0 2 22 0 00 0 0 0 11 100 cos 2 50 50 cos 2 25 25 50 sin 2 50 sin sin 50 50 sin T T T TT Et d t dt t dt tt T T T ω ωω −−  =+   = + + ∫∫ As T approaches , the second term oscillates between values 50 50 and These values are finite. However the first term grows without bound. Thus E =∞ But 0 0 50 50 sin T T E PT == + Therefore 50 P =< In this case the average power is finite and the total energy dissipated is .
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basicsig_06 - Basic Properties Of Signals Types Of Signals...

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