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Unformatted text preview: ECE 181b Homework 6 May 26, 2006 In this homework you will explore some of the properties of convolution and we will look into some aspects of the principal components analysis. Henceforth we will adopt the following conventions (READ CAREFULLY): • Boldface letters indicate points or vectors. • For vectors/points in P 3 we use capital letters, Sans font (e.g. X ), for vectors/points in P 2 we use lower case letters, Sans font (e.g. x ). • For vectors/points in R 3 we use capital letters, normal font (e.g. X ), for vectors/points in R 2 we use lower case letters, normal font (e.g. x ). • All the quantities related to the second camera are identified using the superscript prime ( ). • Square brackets indicate that the signal argument (indicating either a time instant or a spatial location) is an integer. Caveat. To receive full credit your answers must be clearly justified. Make sure to include the relevant steps that were required to obtain the numerical answer. 1 Discrete Convolution Recall that the expression for the discrete convolution of two signals f and g in 2D is given by: ( f * g )[ l,m ] = ∞ X h =∞ ∞ X k =∞ f [ h,k ] g [ l h,m k ] = ∞ X h =∞ g [ h,k ] f [ l h,m k ] (1) Question 1 Consider the expression (1) and two finite support 2D signals such that f 6 = 0 only for M f ≤ h ≤ M f , N f ≤ k ≤ N f and g 6 = 0 only for M g ≤ h ≤ M g , N g ≤ k ≤ N g . Assume that: 1 • M g M f and N g N f (i.e. the support of g is much smaller than the support of f ) • the signals are zero padded near the borders • a multiplication by zero counts as a multiplication by any other number....
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This note was uploaded on 12/29/2011 for the course ECE 181b taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
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