3015_Discussion_4_Solutions

3015_Discussion_4_Solutions - University of Minnesota Dept....

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University of Minnesota Dept. of Electrical and Computer Engineering EE3015 – Signals and Systems Discussion Session #4: DT convolution 1. Calculate y [ n ] = x [ n ] * h [ n ] where x [ n ] = u [ n -1] h [ n ] = u [ n +2] – u [ n -3] Signals x [ n ] and h [ n ] are We first graph x [ k ] To obtain h [ n - k ] first we graph h [ k ], then shift it by n to obtain h [ n + k ], and reflect it (with respect to k ) to obtain h [ n-k ] k - n -2 - n - n +2 1 h [ n+k ] k -4 -3 -2 -1 0 1 2 3 4 5 1 h [ k ] k -3 -2 -1 0 1 2 3 4 5 6 7 1 x [ k ] n -4 -3 -2 -1 0 1 2 3 4 5 1 h [ n ] n -3 -2 -1 0 1 2 3 4 5 6 7 1 x [ n ]
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Next we multiply both x [ k ] and h [ n - k ] and evaluate the sum for different values of n For n +2<1 n < -1 −∞ = = = k n y 0 0 ] [ For n +2 1, n -2<1 -1 n < 3 2 1 1 2 1 1 ] [ 2 1 + = + + = = + = n n n y n k For n -2 1 n 3 5 1 ) 2 ( 2 1 1 ] [ 2 2 = + + = = + = n n n y n n k < + < = 3 , 5 3 1 2 1 0 ] [ n n n n n y Notice that y[-2]=-2+2=0 and y[3]=3+2=0, so the result is consistent. k n -2 n n +2 h [ n-k ] 1 k -3 -2 -1 0 1 2 3 4 5 6 7 1 x [ k ] k n -2 n n +2 h [ n-k ] k n -2 n n +2 h [ n-k ] k n -2 n n +2 h [ n-k ] k n -2 n n +2 h [ n-k ] k n -2 n n +2 h [ n-k ] n y [ n ] -3 -2 -1 0 1 2 3 4 5 6 7 1 5 3 4 2
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Calculate y [ n ] = x [ n ] * h [ n ] where now x [ n ] = u [ n+2 ]- u [ n -6] h [ n ] = n 2 1 ( u [ n +1] – u [ n -4]) Signals x [ n ] and h [ n ] are We first graph x [ k ] To obtain h [ n - k ] first we graph h [ k ], then shift it by n to obtain h [ n + k ], and reflect it (with respect to k ) to obtain h [ n-k ] Next we multiply both x [ k ] and h [ n - k ] and evaluate the sum for different values of n k -4 -3 -2 -1 0 1 2 3 4 5 1 h [ k ] 1/2 1/4 1/8 2 k -3 -2 -1 0 1 2 3 4 5 6 7 1 x [ k ] k n-3 n n+1 h [ n-k ] 1 1/2 1/4 1/8 2 k - n -1 - n - n +3 h [ n+k ] 1 1/2 1/4 1/8 2 n -4 -3 -2 -1 0 1 2
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This note was uploaded on 12/20/2009 for the course EE 3015 taught by Professor Staff during the Fall '08 term at Minnesota.

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3015_Discussion_4_Solutions - University of Minnesota Dept....

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