lecture3_small

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Sampling Sampling: period = T y(k) d

@

y(t)

T

Example (single pole signal) −at e , t≼0 Consider, y(t) = 0 t<0 Laplace transform: Sampled signal:

with a > 0.

1 . s+a

k

y(k) = y(t)

= e−akT = e−aT .

y(s) =

t=kT

Z-transform,

y(z) =

z . z − e−aT

The s-plane pole is at s1 = −a, and the corresponding z-plane pole is at z1 = e−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) = eâˆ’ÎąkT sin(βkT ),

y(t) = eâˆ’Îąt sin(βt),

t ≼ 0, with ι > 0.

Poles: s1,2 = âˆ’Îą Âą jβ. k ≼ 0.

z −1 eâˆ’ÎąT sin(βT ) y(z) = . −1 1 − z 2eâˆ’ÎąT cos(βT ) + z −2 e−2ÎąT

Z domain poles given by: z 2 − 2eâˆ’ÎąT cos(βT )z + e−2ÎąT = 0. q z1,2 = eâˆ’ÎąT cos(βT ) Âą e2ÎąT cos2 (βT ) − e−2ÎąT p = eâˆ’ÎąT cos(βT ) Âą j 1 − cos2 (βT )

Imaginary z-plane

eâˆ’ÎąT

βT Real

= eâˆ’ÎąT (cos(βT ) Âą j sin(βT )) = eâˆ’ÎąT eÂąjβT = e(−ι¹jβ)T .

Roy Smith: ECE 147b 3: 2


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lecture3_small by Steve Rogers - Issuu