ex6 - CS221 Exercise Set #6 1 CS 221, Autumn 2007 Exercise...

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CS221 Exercise Set #6 1 CS 221, Autumn 2007 Exercise Set #6 1. Convolution (a) Suppose f, g are two functions with bounded support; i.e., there exist constants l f , u f , l g , u g such that f [ i ] = 0 unless i [ l f , u f ] and g [ i ] = 0 unless i [ l g , u g ]. Give bounds on the support of h = f * g , that is, numbers l h , u h such that h [ n ] = 0 unless n [ l h , u h ]. (b) Let f [ i ] = 2 i for i [0 , 4]; f [ i ] = 0, otherwise. Let g [ i ] = | i | for i [ - 2 , 2]; g [ i ] = 0, otherwise. What is the discrete convolution h = f * g ? (c) Let k [ i ] be the kernel [0 . 5 , 0 . 5]. Let f [ i ] = 2 i . What is the discrete convolution k * f ? Show associativity of convolution explicitly in the case k * ( k * f ) = ( k * k ) * f . 2. Properties of the convolution operator In this problem, we will investigate some properties of the convolution operator. For simplicitly, we consider just the 1-D convolution, de±ned for input image X and kernel K as: Y [ n ] = ( X * K
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ex6 - CS221 Exercise Set #6 1 CS 221, Autumn 2007 Exercise...

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