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Unformatted text preview: Lecture 7: CT Convolution Tie Liu September 19, 2011 1 CT Convolution DT LTI systems: x [ n ] = summationdisplay k = x [ k ] [ n k ] = y [ n ] = summationdisplay k = x [ k ] h [ n k ] CT LTI systems: Need to express x ( t ) as a linear combination of a basic signal and its time shifts. Can we use ( t ) = braceleftbigg 1 , t = 0 , t negationslash = 0 No, that would require uncountably infinite many time shifts of ( t ). And we do not know how to add uncountably infinite many numbers in general. Solution: Approximate! Let x ( t ) = x ( k ) when t (( k 1 / 2) , ( k + 1 / 2)]. When 0, we have x ( t ) x ( t ) for all t R . t x ( t ) x ( t ) - 2 - 2 - 3 3 1 Can we express x ( t ) as a linear combination of a basic signal and its time shifts? Yes, consider ( t ) = braceleftbigg 1 , t ( 2 , 2 bracketrightbig , otherwise Note that ( t k ) are non-overlapping for different k and together they fill up the entire time axis. We thus have x ( t ) = summationdisplay k = x ( k ) ( t k ) where here is used to compensate 1 / in the definition of ( t ). t ( t ) - 2 - 2 - 3 3 ( t- ) ( t- 2 ) ( t- 3 ) ( t + ) ( t + 2) ( t + 3 ) 1 Let h ( t ) be the response to ( t ). By the LTI properties, the output signal)....
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- Fall '09