J roberts 21807 if we convolve g n x n znxn gn y

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Unformatted text preview: yn k= hn xn D.2.3 hn Distributivity Property If we convolve x n with the sum of y n and z n we get xn (y n +z n )= xk k= (y n k +z n k ) or xn (y n +z n )= xkyn k+ k= =x n Therefore xn (y n +z n xkzn k . k= )= x =x n yn n D-5 y n +x n zn zn. M. J. Roberts - 2/18/07 D.2.4 Differencing Property Let y n = x n h n . Using the time-shifting property y n n0 = x n h n n0 = x n n0 hn the first backward difference of their convolution sum is y n 1 =x n yn hn xn hn 1 or yn yn 1= xmhn m m= xmhn m 1. m= Combining summations, yn 1= yn ( xm hn m m= or yn D.2.5 (h y n 1 =x n n hn m 1 hn 1 ) ) Sum Property Let y n = x n h n and let the sum of all the impulses in the functions y, x, and h be S y , S x and S h , respectively. Then yn= xmhn m m= and Sy = yn= n= xmhn m. n= m= Interchanging the order of summation, Sy = xm m= n= h n m = Sx Sh . = Sx = Sh D-6...
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