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Unformatted text preview: The Convolution Operation Convolution is a very natural mathematical operation which oc- curs in both discrete and continuous modes of various kinds. We often encounter it in the course of doing other operations without recognizing it. Suppose, for example, we have two polynomials, one of degree n , the other of degree m : p ( x ) = a x n + a 1 x n- 1 + + a n- 1 x + a n ; q ( x ) = b x m + b 1 x m- 1 + + b m- 1 x + b m . The ordinary product of these two polynomials is a polynomial of degree n + m , i.e., p ( x ) q ( x ) = a b x m + n +( a b 1 + a 1 b ) x m + n- 1 + + ( a b j + a 1 b j- 1 + + a j- 1 b 1 + a j b ) x m + n- j + + ( a n- 1 b m + a n b m- 1 ) x + a n b m . The coefficient of x m + n- j is a b j + a 1 b j- 1 + + a j- 1 b 1 + a j b = j X k =0 a k b j- k , the sum of products a k b for which k + = j . It is instructive to arrange the coefficients of p ( x ) and q ( x ) in the following manner: a a 1 a j- 1 a j a j +1 b j +1 b j b j- 1 b 1 b from which we see that the sum j k =0 a k b j- k is the sum of prod- ucts of overlapping a k , b in this diagram. The Matlab R opera- tion conv ( A, B ), where A and B are vectors of dimension n and m , works in essentially this way. 1 Convolution of functions is defined in a very similar way. Let us suppose f ( x ) and g ( x ) are Laplace transformable , piece- wise continuous functions defined on [0...
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