# lecture19n - Stanford University Winter 2008-2009 Signal...

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Stanford University Winter 2008-2009 Signal Processing and Linear Systems I Lecture 19: Overview and Conclusions March 10, 2010 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 1 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 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 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 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 3 Linearity and Time Invariance As y s t em 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 ) Asy s temi s 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 ( t τ ) . EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 4
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 . EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 5 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 ! ) EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 6

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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 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 7 y ( t ° −∞ h ( τ ) e j ω ( t τ ) d τ = e j ω t ° −∞ h ( τ ) e j ωτ d τ = H ( j ω ) e j ω t H ( j ω ) is the continuous time Fourier transform of the time function h ( t ) . Complex exponential is eigenfunction of LTI system with eigenvalue H ( j ω ) EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 8
If the input is a sum of complex exponentials, x ( t )= ° n = −∞ D n e jn ω 0 t , ! " n = ! D n e jn # 0 t y ( t ) h ( t ) y ( t ± −∞ h ( τ ) ² ° n = −∞ D n e ω 0 ( t τ ) ³ d τ = ° n = −∞ D n e ω 0 t ± −∞ h ( τ ) e ω 0 τ d τ = ° n = −∞ ( D n H ( ω 0 )) e ω 0 t Same frequencies as input, di f erent complex multipliers.

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lecture19n - Stanford University Winter 2008-2009 Signal...

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