EEL5173 Fall 2009 Lecture _5

# EEL5173 Fall 2009 Lecture _5 - (2) Real Translation (a)...

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(b) Advance: y ( k ) = x ( k + N ) with N >0, not causal . In particular, { } ) 0 ( ) ( ) 1 ( zx z zX k x Z = + {} { } ∑∑ = = = = = = + = = = = = + = + = = 1 0 0 1 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N i i N N i N i i i N N i i N N i N i N k i k k z i x z z X z z i x z i x z z i x z z i x z N k x N k x Z k y Z z Y (2) Real Translation (a) Delay:

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(3) The Initial-value Theorem Given X (z), can we determine x (0) without finding the sequence first ? {} + + + + + = = = = z as x z k x z x z x x z k x k x Z z X k k k ), 0 ( ) ( ) 2 ( ) 1 ( ) 0 ( ) ( ) ( ) ( 2 1 0 L L ) ( lim ) 0 ( z X x z = 1 ) cos( 2 ) sin( ) ( 2 + = θ z z z z X 0 1 ) cos( 2 ) sin( lim ) ( lim ) 0 ( 2 = + = = z z z z X x z z Ex.
(4) The Final-value Theorem Given X (z), can we determine x ( ) without finding the sequence first ? ) ( ) 1 ( lim ) ( 1 z X z x z = Theorem. If x ( ) exists, Theorem. The final value x ( ) exists if and only if the region of convergence of ( z -1) X ( z ) contains the point z = 1, i.e., if all poles of ( z X ( z ) are inside the unit circle centered at the origin.

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{} ) )( 1 ( ) 2 ( ) ( ) ( 3 1 + + = = z z z z k x Z z X ) ( ) 2
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## This note was uploaded on 07/15/2011 for the course EEL 5173 taught by Professor Tung during the Fall '09 term at FSU.

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EEL5173 Fall 2009 Lecture _5 - (2) Real Translation (a)...

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