lecture3_small

# lecture3_small - Sampling Example: (second order) Now...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 04/06/2010 for the course ECE 145 taught by Professor Rodwell during the Spring '07 term at UCSB.

### Page1 / 6

lecture3_small - Sampling Example: (second order) Now...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online