Sampling Sampling: period = T y(k) d
@
y(t)
T
Example (single pole signal) â&#x2C6;&#x2019;at e , tâ&#x2030;Ľ0 Consider, y(t) = 0 t<0 Laplace transform: Sampled signal:
with a > 0.
1 . s+a
k
y(k) = y(t)
= eâ&#x2C6;&#x2019;akT = eâ&#x2C6;&#x2019;aT .
y(s) =
t=kT
Z-transform,
y(z) =
z . z â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;aT
The s-plane pole is at s1 = â&#x2C6;&#x2019;a, and the corresponding z-plane pole is at z1 = eâ&#x2C6;&#x2019;aT . Roy Smith: ECE 147b 3: 1
Sampling Example: (second order) Now consider a damped sinusoidal signal, Laplace transform: Sampled signal:
Z-transform:
y(s) =
β , (s + ι)2 + β 2
y(k) = eâ&#x2C6;&#x2019;ÎąkT sin(βkT ),
y(t) = eâ&#x2C6;&#x2019;Îąt sin(βt),
t â&#x2030;Ľ 0, with Îą > 0.
Poles: s1,2 = â&#x2C6;&#x2019;Îą Âą jβ. k â&#x2030;Ľ 0.
z â&#x2C6;&#x2019;1 eâ&#x2C6;&#x2019;ÎąT sin(βT ) y(z) = . â&#x2C6;&#x2019;1 1 â&#x2C6;&#x2019; z 2eâ&#x2C6;&#x2019;ÎąT cos(βT ) + z â&#x2C6;&#x2019;2 eâ&#x2C6;&#x2019;2ÎąT
Z domain poles given by: z 2 â&#x2C6;&#x2019; 2eâ&#x2C6;&#x2019;ÎąT cos(βT )z + eâ&#x2C6;&#x2019;2ÎąT = 0. q z1,2 = eâ&#x2C6;&#x2019;ÎąT cos(βT ) Âą e2ÎąT cos2 (βT ) â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;2ÎąT p = eâ&#x2C6;&#x2019;ÎąT cos(βT ) Âą j 1 â&#x2C6;&#x2019; cos2 (βT )
Imaginary z-plane
eâ&#x2C6;&#x2019;ÎąT
βT Real
= eâ&#x2C6;&#x2019;ÎąT (cos(βT ) Âą j sin(βT )) = eâ&#x2C6;&#x2019;ÎąT eÂąjβT = e(â&#x2C6;&#x2019;ι¹jβ)T .
Roy Smith: ECE 147b 3: 2