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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 ≥ . Ztransform: 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 . Realα 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 . Ztransform, y ( z ) = z z e aT . The splane pole is at s 1 = a , and the corresponding zplane pole is at z 1 = e aT . Roy Smith: ECE 147b 3 : 1 Sampling Sampled pole locations: (in detail)1.00.5 0 0.5 1.0 zplane 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
 RODWELL

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