R23-1 - ENEE 241 02 READING ASSIGNMENT 23 Thu 05/07 Lecture...

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* READING ASSIGNMENT 23 Thu 05/07 Lecture Topics: linear convolution of sequences and vectors; linear convolution as circular convolution; block convolution Textbook References: sections 4.6, 4.7.1, 4.7.2 and 4.7.4 Key Points: The input x = x [ · ] and output y = y [ · ] of a linear time-invariant system are related by the convolution sum y [ n ] = X k = -∞ h [ k ] x [ n - k ] def = ( h * x )[ n ] The sequence h = h [ · ] is the system’s response to a unit impulse δ = δ [ · ]. If h and x are ﬁnite-duration sequences with activity intervals 0 : K - 1 and 0 : L - 1 respectively, then y = h * x is also a ﬁnite-duration sequence with activity interval 0 : K + L - 2. This serves as an implicit deﬁnition for the linear convolution of two vectors: y [0 : K + L - 2] = h [0 : K - 1] * x [0 : L - 1] If b = b [0 : K - 1] and s = s [0 : L - 1], then b * s = [ b ; 0 L - 1 ] ~ [ s ; 0 K - 1 ] Thus linear convolution of vectors can be implemented by means of circular convolution—and thus optionally, DFT’s. If s (1) and s (2) have lengths L 1 and L 2 , respectively, then b * [ s (1) ; s (2) ] = [ b * s (1) ; 0 L 2 ] + [ 0 L 1 ; b * s (2) ] Theory and Examples: 1 . In the preceding lectures, we examined in detail the response of an FIR ﬁlter to two types of inputs: inﬁnite-duration (two-sided) exponentials and periodic sequences. We will now broaden our scope to arbitrary input sequences x [ · ], and discuss the implications of the input-output relationship y [ n ] = M X k =0 b k x [ n - k ] , n Z We note that the same value for y [ n ] is obtained if we pad b with inﬁnitely many zero coeﬃcients on both sides, i.e., using the sequence h [ · ] deﬁned h [ n ] = b n , 0 n M ; 0 , otherwise, instead of

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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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R23-1 - ENEE 241 02 READING ASSIGNMENT 23 Thu 05/07 Lecture...

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