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SP_IV_P03_4p - Introduction to Signal Processing 1...

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Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha D. Signal Transmission Through Linear Time-Invariant Systems If ( ) x t and ( ) y t are the input and output of an LTIC system with impulse response ( ) h t then recall that the output of the system ( zero-state response ) can be evaluated as the convolution of the input ( ) x t and the impulse response ( ) h t : Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha ( ) ( ) ( ) * y t x t h t = . where ( ) ( ) ( ) y t x h t d τ τ τ −∞ = ( ) ( ) ( ) * y t x t h t = ( ) h t ( ) x t Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Let ( ) x t , ( ) y t , and ( ) h t have corresponding Fourier transform functions ( ) X ω , ( ) Y ω , and ( ) H ω respectively: ( ) ( ) x t X ω ( ) ( ) y t Y ω ( ) ( ) h t H ω Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha Now, by using the time-convolution property, we have the following: ( ) ( ) ( ) Y H X ω ω ω = . ( ) ( ) ( ) Y X H ω ω ω = ( ) H ω ( ) X ω
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