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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 16 Thu 04/09 Lecture Topics: multiplication and circular convolution; periodic extension of a vector Textbook References: sections 3.6, 3.7 Key Points: The circular convolution of two N-length vectors a and b is the N-length vector a ~ b whose n th entry is given by a T P n Rb . If a b denotes the element-by-element product of a and b , then x y 1 N X ~ Y x ~ y X Y If x is formed by concatenating M copies of the L-point vector s , then the DFT X is obtained by inserting M- 1 zeros between consecutive entries of S , and scaling the result by M . Theory and Examples: 1 . The element-by-element product x y of two N-point vectors x and y is defined by ( x y )[ n ] = x [ n ] y [ n ] , n = 0 : N- 1 Its DFT can be obtained directly from X and Y using a procedure known as circular convo- lution. 2 . If s = x y , then for every frequency index k , S [ k ] = N- 1 X n =0 x [ n ] y [ n ] v- kn The product y [ n ] v- kn is the n th entry of the vector y v ( N- k ) = F- k y We can therefore write S [ k ] = x T F- k y Using the synthesis equation; known facts about V ; and the analysis equation, we obtain S [ k ] = 1 N ( VX ) T F- k y = 1 N X T VF- k y = 1 N X T P k Vy = 1 N X T P k RWy = 1 N X T P k RY 1 3 . The computation of the dot product a T P n Rb for every index n = 0 : N- 1 is known as circular convolution. The result is the vector a ~ b defined by...
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