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ece431notesForLec_2007_09_24

ece431notesForLec_2007_09_24 - LinearConvolution The...

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Unformatted text preview: LinearConvolution The linearconvolution oftwosequences x n and y n isdefinedby ( x * y ) n : = ∞ ∑ m =- ∞ x m y n- m . Ifoneofthesequences,say x m ,hasfiniteduration,say x m = 0for m < M 1 and m > M 2 , then ( x * y ) n : = M 2 ∑ m = M 1 x m y n- m . Letusfurthersupposethatweinterestedintheconvolutiononlyfor N 1 ≤ n ≤ N 2 ,sayfor plottingpurposes.Thenincomputingtheconvolution,thevaluesofthesubscriptof y ,that is, n- m ,rangefrom N 1- M 2 to N 2- M 1 .Thepointhereisthattocompute ( x * y ) n for n = N 1 ,..., N 2 requiresonlythevalues x M 1 ,..., x M 2 and y N 1- M 2 ,..., y N 2- M 1 .Forexample, if N 1 = M 1 = 0,thenweneed y- M 2 ,..., y N 2 . [Drawpictures] Letusnowdefineˆ y n : = y n for n = N 1- M 2 ,..., N 2- M 1 andˆ y n : = 0otherwise.A graphicalargumentwith y n- m ,ˆ y n- m ,and x m showsthat ( x * y ) n and ( x * ˆ y ) n areequalfor n = N 1 ,..., N 2 .Fortunately,theconvolutionoftwofinite-lengthsequencesiseasilydone withtheM ATLAB command z=conv(x,yhat) .However,sinceM....
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