lecture7 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Discrete-Time Convolution

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7 - 2 Discrete-time Convolution Output y [ n ] for input x [ n ] Any signal can be decomposed into sum of discrete impulses Apply linearity properties of homogeneity then additivity Apply shift-invariance Apply change of variables [ ] [ ] { } n x T n y = [ ] [ ] [ ] - = -∞ = m m n m x T n y δ [ ] [ ] [ ] { } m n T m x n y m - = -∞ = δ [ ] [ ] [ ] m n h m x n y m - = -∞ = [ ] [ ] [ ] m n x m h n y m - = -∞ =
7 - 3 Discrete-time Convolution Filtering viewpoint Hold impulse response h [ n ] in place and change variables Flip and slide input signal x [ n ] about impulse response Example of finite impulse response (FIR) filter Impulse response has finite extent (non-zero duration) x [ n ] y [ n ] [ ] [ ] [ ] m n x m h n y m - = -∞ = y [ n ] = h [0] x [ n ] + h [1] x [ n -1] = ( x [ n ] + x [ n -1] ) / 2 n h [ n ] 2 1 Averaging filter impulse response 0 1 2 3

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7 - 4 Convolution in Both Domains Continuous-time convolution of x ( t ) and h ( t ) For each value of t , we compute a different (possibly) infinite integral. Discrete-time definition is the continuous-time definition with integral replaced by summation Linear time-invariant (LTI) system Output signal in time domain is convolution of impulse response and input signal
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