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Introduction_1 - System and Signals Input Signal x(t System...

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10/20/10 1 System and Signals System h(t) Input Signal x(t) Response y(t) [ ] ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 jw Y t y jw H jw X jw Y OR t h t x t Y - = = = Signals Deterministic Random [o/p of radio receiver when there e.g. is no broadcasting, the o/p of speaker when no body is talking, etc] t A t x π 50 sin ) ( = Now, if the input is random How can we measure the output? We can use, for example, Convolution
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10/20/10 2 Energy and Power Signals: Let be the instantaneous voltage and current respectively, then the energy is : ) ( ) ( t i and t v Classification of is a voltage across The energy dissipated by the signal during the interval is given by: ) 1 ( ) ( resistor for t v ) ( t v 1 - 2 , 2 T T - = 2 2 2 ) ( T T T v dt t v E dt t v E resistor for dt t v R dt t i t v E v v - - - = = = 2 2 ) ( 1 ) ( 1 ) ( ) (
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10/20/10 3 And the average power is: - = 2 2 2 ) ( 1 T T T v dt t v T P Now is an energy signal if : is finite, i.e: , where: ) ( t v v E v E 0 2 2 2 2 ) ( lim V dt t v E T T T v - = = While is a power signal if: where, ) ( t v v P 0 - = 2 2 2 ) ( 1 lim T T T v dt t v T E ( Limited can be dropped )
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10/20/10 4 ( 29 signal power is t v finite A T T A dt t T A dt t A T P t A t v Ex T T T T T T v ) ( 2 2 2 cos 1 2 lim cos 1 lim cos ) ( : 2 2 2 2 2 2 2 2 2 = = + = = = - - In general, periodic signal is a power signal Parseval’s Theorem: - - -∞ = = = = = energy dw t v dt t v E signal power C dt t v T P v T n n 2 2 2 2 ) ( 2 1 ) ( ) ( 1 π
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10/20/10 5 energy per unit frequency ( jouls/Hz ). It is called energy spectral density ( or energy density ). Thus, to find the energy, we have to integrate over the frequency band of the signal. Notice, the energy depends on the magnitude squared of the signal ( independent of the phase of the signal ).
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