D 3 d m j roberts 21807 d16 scaling property

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Unformatted text preview: ( ) h (t ) d ) ( ) and (1 / a ) y ( at ) = x ( at ) h ( at ) . If we do a similar get (1 / a ) y ( at ) = x ( at ) h ( at ) . Therefore, in general, if it follows that z t = 1 / a y at proof for a < 0 w e ( ) ( ) h (t ) then y t =x t () ( ) h ( at ) . y at = a x at D.2 Discrete-Time Convolution Properties D.2.1 Commutativity Property The commutativity of DT convolution can be proven by starting with the definition of convolution xn hn= and letting q = n k . Then we have xn xn qhq= hn= q= D.2.2 xkhn k k= h q x n q =h n q= Associativity Property D-4 xn M. J. Roberts - 2/18/07 If we convolve g n = x n ( zn=xn gn y n with z n we get yn ) zn= xkyn k zn k= gn or zn= gn xkyq q= k zn q k= gq Exchanging the order of summation, (x yn n ) zn= Let n q = m and let h n = y n (x ) yn n yq xk k= kzn q q= z n . Then zn= ( zmy n k xk k= m= zn y n k =y n k ) m z n =h n k or (x n yn ) zn= x k h n k =x n zn...
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