102_1_Lecture19 - UCLA Spring 2009-2010 Systmes and Signals...

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UCLA Spring 2009-2010 Systmes and Signals Lecture 19: Overview and Conclusions June 03 2010 EE102: Systems and Signals; Spr 08-09 Lee 1
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Overview and Conclusions Several key ideas: Linearity and time invariance Convolution systems Complex exponentials, and transfer functions Representation of signals by linear combinations of complex exponentials Fourier series Fourier transform Convolution becomes multiplication (and the reverse) Communications and modulation EE102: Systems and Signals; Spr 08-09 Lee 2
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Sampling and reconstruction Even unstable signals can have transforms Laplace transform Solving for the evolution of dynamic systems Steady state frequency response Feedback and automatic control EE102: Systems and Signals; Spr 08-09 Lee 3
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Linearity and Time Invariance A system F is linear if for any two signals x 1 , x 2 and scalars a 1 , a 2 , F ( a 1 x 1 + a 2 x 2 ) = a 1 F ( x 1 ) + a 2 F ( x 2 ) . Alternatively, F is linear if these properties hold: 1. homogeneity: F ( ax ) = aF ( x ) 2. superposition: F ( x + ˜ x ) = F ( x ) + F x ) A system is time-invariant if a time shift in the input only produces the same time shift in the output. y ( t ) = Fx ( t ) implies that y ( t - τ ) = Fx ( t - τ ) . EE102: Systems and Signals; Spr 08-09 Lee 4
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LTI Systems If a system is linear and time-invariant, it is completely characterized by its impulse response H t 0 0 0 t 0 ! h ( t ) h ( t - ! ) ! ( t - " ) ! ( t ) t t For an input x ( t ) and impulse response h ( t ) , the output is given by y ( t ) = -∞ x ( τ ) h ( t - τ ) d τ . This is a convolution integral . EE102: Systems and Signals; Spr 08-09 Lee 5
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Graphically, this can be represented as: t 0 t h ( t ) 0 t 0 t 0 ! ( t ) t 0 t 0 t 0 t 0 x ( t ) y ( t ) ( x ( ! ) d ! ) " ( t - ! ) ( x ( ! ) d ! ) h ( t - ! ) ! ( t - " ) ! ! ! ! ! x ( t ) x ( t ) = Z ! - ! x ( " ) # ( t - " ) d " y ( t ) = Z ! - ! x ( " ) h ( t - " ) d " Input Output h ( t - ! ) EE102: Systems and Signals; Spr 08-09 Lee 6
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Graphical Interpretation of Convolution The convolution integral is y ( t ) = -∞ x ( τ ) h ( t - τ ) d τ where t t x ( t ) h ( t ) If we consider h ( t - τ ) to be a function of τ , then h ( t - τ ) is delayed to time t , and reversed . EE102: Systems and Signals; Spr 08-09 Lee 7
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τ t h ( t - τ ) τ t h ( t - τ ) This is multiplied point by point with the input, τ t h ( t - τ ) x ( τ ) τ t x ( τ ) h ( t - τ ) Then integrate over τ to find y ( t ) for this t . y ( t ) = -∞ x ( τ ) h ( t - τ ) d τ EE102: Systems and Signals; Spr 08-09 Lee 8
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“Flip and drag” h ( t ) . 3 -1 0 1 2 1 2 3 -1 0 1 2 1 2 h ( ! ) ! h ( t - ! ) h ( - ! ) 3 -1 0 1 2 1 2 x ( ! ) ! 3 -1 0 1 2 1 2 ! ! EE102: Systems and Signals; Spr 08-09 Lee 9
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3 -1 0 1 2 1 2 x ( ! ) ! h ( t - ! ) t < 0 3 -1 0 1 2 1 x ( ! ) ! h ( t - ! ) 0 < t < 1 3 -1 0 1 2 1 x ( ! ) h ( t - ! ) 1 < t < 2 3 -1 0 1 2 1 2 x ( ! ) ! h ( t - ! ) 2 < t < 3 3 -1 0 1 2 1 2 x ( ! ) ! h ( t - ! ) t > 3 3 -1 0 1 2 1 2 ! y ( t ) = ( x * h )( t ) EE102: Systems and Signals; Spr 08-09 Lee 10
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Complex Exponentials and Transfer Functions Complex exponentials have a special relationship with convolution systems. For an LTI system with impulse response h ( t ) , output is the convolution of input and impulse response. y ( t ) = -∞ h ( τ ) x ( t - τ ) d τ If the input is a complex exponential x ( t ) = e j ω t y ( t ) * h ( t ) e j ! t EE102: Systems and Signals; Spr 08-09 Lee 11
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y ( t ) = -∞ h ( τ ) e j ω ( t - τ ) d τ = e j ω t -∞ h ( τ ) e - j ωτ d τ Δ = H ( j ω ) e j ω t H ( j ω ) is the continuous time Fourier transform of the time function h ( t ) .
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