2025-L15su09

# 2025-L15su09 - ECE2025 Summer 2009 Lecture 15...

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1 6/26/2009 1 ECE2025 Summer 2009 Lecture 15 Continuous-Time Signals and Systems 29 Jun 09 6/26/2009 3 ANNOUNCEMENTS ± June 29-30 ± Turn in Homework 6 ± July 1-2: ± Turn in Lab #6 ± Do Lab #7 ± 6/26/2009 4 READING ASSIGNMENTS ± This Lecture: ± Chapter 9, Sects 9-1 to 9-5 ± Other Reading: ± Recitation: Ch. 9, all ± Next Lecture: Chapter 9, Sects 9-6 to 9-8 6/26/2009 5 Recap: z-Transform with three input types ± Given: ± Find the output, y[n] ± When H ( z ) = 5 1 + 0.8 z 1 x [ n ] = cos(0.2 π n ) x [ n ] = u [ n ] x [ n ] = cos(0.2 n ) u [ n ]

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2 6/26/2009 6 π 089 . 0 2 . 0 2 . 0 919 . 2 8 . 0 1 5 ) ( j j j e e e H = + = SINUSOID ANSWER ± Given: ± The input: ± Then y[n] x [ n ] = cos(0.2 n ) y [ n ] = M cos(0.2 n + ψ ) H ( z ) = 5 1 + 0.8 z 1 6/26/2009 7 Step Response Y ( z ) = H ( z ) X ( z ) = 5 1 + .8 z 1 1 1 z 1 Y ( z ) = A 1 + .8 z 1 + B 1 z 1 = ( A + B ) + (.8 B A ) z 1 1 + .8 z 1 () 1 z 1 ( A + B ) = 5a n d (.8 B A ) = 0 Y ( z ) = A 1 + .8 z 1 + B 1 z 1 Partial Fraction Expansion 6/26/2009 8 Step Response Y ( z ) = 20 9 1 + .8 z 1 + 25 9 1 z 1 y [ n ] = 20 9 ( .8) n u [ n ] + 25 9 u [ n ] y [ n ] 25 9 as n →∞ 6/26/2009 9 SINUSOID Starting at n=0 ± Given: ± The input: ± Then y[n] H ( z ) = 5 1 + 0.8 z 1 ] [ ) 2 . 0 cos( ] [ n u n n x = ]} [ ] [ ) 8 . 0 ( 5 { ]} [ ] [ { ] [ 2 . 0 n u e n u n x n h n y n j n = =
3 6/26/2009 10 CALCULATE the RESPONSE H ( z ) Use the Z-Transform Method And PARTIAL FRACTIONS 6/26/2009 11 GENERAL INVERSE Z (pole) n 6/26/2009 12 SPLIT Y(z) to INVERT ± Need SUM of Terms: 6/26/2009 13 INVERT Y(z) to y[n] ± Use the Z-Transform Table

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4 6/26/2009 14 SINUSOID Starting at n=0 ]} [ ] [ ) 8 . 0 ( 5 { ] [ 2 . 0 n u e n u n y n j n π = 1 2 . 0 1 1 1 8 . 0 1 5 ) ( ) ( ) ( + = = z e z z X z H z Y j 1 2 . 0 1 1 2 . 0 8 . 0 1 5 1 25 . 1 1 5 1 8 . 0 81 . 2 8 . 0 1 8 . 0 19 . 2 1 8 . 0 1 ) ( 2 . 0 2 . 0 + + + + + = + + = z e j z j z e z z Y j j e e j j ) ( 2 . 0 j e H 6/26/2009 15 SINUSOID Starting at n=0 1 2 . 0 1 1 2 . 0
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2025-L15su09 - ECE2025 Summer 2009 Lecture 15...

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