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Unformatted text preview: Sampling Example: (second order) Now consider a damped sinusoidal signal, y ( t ) = e- αt sin( βt ) , t ≥ , with α > 0. Laplace transform: y ( s ) = β ( s + α ) 2 + β 2 , Poles: s 1 , 2 =- α ± jβ . Sampled signal: y ( k ) = e- αkT sin( βkT ) , k ≥ . Z-transform: y ( z ) = z- 1 e- αT sin( βT ) 1- z- 1 2e- αT cos( βT ) + z- 2 e- 2 αT . Z domain poles given by: z 2- 2e- αT cos( βT ) z + e- 2 αT = 0. z 1 , 2 = e- αT cos( βT ) ± radicalBig e 2 αT cos 2 ( βT )- e- 2 αT = e- αT parenleftBig cos( βT ) ± j radicalbig 1- cos 2 ( βT ) parenrightBig = e- αT (cos( βT ) ± j sin( βT )) = e- αT e ± jβT = e (- α ± jβ ) T . Real-α T e Imaginary z- plane β T Roy Smith: ECE 147b 3 : 2 Sampling Sampling: period = T a27 a64 a100 y ( t ) y ( k ) T Example (single pole signal) Consider, y ( t ) = braceleftbigg e- at , t ≥ t < with a > . Laplace transform: y ( s ) = 1 s + a . Sampled signal: y ( k ) = y ( t ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle t = kT = e- akT = ( e- aT ) k . Z-transform, y ( z ) = z z- e- aT . The s-plane pole is at s 1 =- a , and the corresponding z-plane pole is at z 1 = e- aT . Roy Smith: ECE 147b 3 : 1 Sampling Sampled pole locations: (in detail)-1.0-0.5 0 0.5 1.0 z-plane N = 2 N = 4 N = 5 N = 8 N = 20 ω = 0.9π/ T n ω = 0.8π/ T n ω...
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This note was uploaded on 04/06/2010 for the course ECE 145 taught by Professor Rodwell during the Spring '07 term at UCSB.
- Spring '07