Discrete-Time Signal Processing

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009









Discrete-Time

Signal Processing







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Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Sampling

xc(t)





xc(t) C-D x(n) CT

0 T 2T 3T 4T t

T

fclk





xn   xc t  nT  xc(nT)

x(n)



xc t   X c  j

FT



DT

0 1 2 3 4 n





    xnz



X c  j 

?

j n

X ze 

n  







–2–

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Zero-Order Hold (ZOH)

xc(t) xc(nT)









0 t 0 T 2T 3T 4T t

T

xc(nT)

T u(t) - u(t-T)

xSH(t)

T

1/T

w

0 T 2T 3T 4T t 0 T t

T







1 

xSH t    xc nT   u t  nT   u t  nT  T 

T n  



–3–

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Sampling

xSH(t)

Area fixed

 

FT lim xSH t   FT  xs t 

w 0

 1  

 FT  lim   xc nT   u t  nT   u t  nT  w 

 w 0 w 

  n   

1  1 

 lim   xc nT    e  snT  e  s nT  w   

0 t

1

w→0

 n  

w 0 w

s s 

 1  e  sw  

 lim     xc nT  e  snT

Impulse train w 0  

 sw  n  

 

xs(t) xc(t)·δ(t-nT)   x nT e

n  

c

 snT

  xn  z

n  

n









CT  , DT   ,

0 T 2T 3T 4T

T t s  j, z  e j ,   T .



    xnz



X ze j n

 X s  j 

n  





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Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Sampling

xc(t)







xs(t) xc(t)·δ(t-nT)

0 t



δ(t-nT)

s(t)

0 T 2T 3T 4T t

T







0

T t

xs t   xc t   s t   xc t    t  nT 



X s  j   X c  j   S  j 

1

   xnz



X e j  n

2

n   2 2

s t       k ,

FT

s s 

 X s  j       X c   k s  

1 T k T

T k

X s  j    X c   k s 

T  1

T

T k



–5–

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Spectrum of Sampled Signal (Ωs>2ΩN)

Xc(jΩ)







-ΩN ΩN Ω

S(jΩ)







-3Ωs -2Ωs -Ωs 0 Ωs 2Ωs 3Ωs Ω

Xs(jΩ)







-3Ωs -2Ωs -Ωs 0 Ωs 2Ωs 3Ωs Ω









The spectrum of the sampled signal is periodic in Ωs=2π/T.



–6–

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Spectrum of Sampled Signal (Ωs<2ΩN)

Xc(jΩ)







-ΩN ΩN Ω

S(jΩ)







-3Ωs -2Ωs -Ωs 0 Ωs 2Ωs 3Ωs Ω

Xs(jΩ)







-3Ωs -2Ωs -Ωs 0 Ωs 2Ωs 3Ωs Ω





• Aliasing (folding) results in irreversible signal distortion.

• Can only be avoided by using sufficiently high sample rate, or band-

limit the input signal with a coarse, continuous-time filter – AAF.

–7–

Advanced Analog IC Design Discrete-Time Signal Processing Professor Y. Chiu

ECE 581 Fall 2009







Reconstruction Filter (Nyquist)

Xs(jΩ)





sin t / T 

-3Ωs -2Ωs -Ωs 0 Ωs 2Ωs 3Ωs Ω hr t  

t / T

Hr(jΩ)





xr t    xnh t  nT 

-Ωs/2 Ωs/2 Ω r

n  

Xr(jΩ) 

sin t  nT  / T 

  xn 

n    t  nT  / T

-ΩN ΩN Ω







Reconstruction filter = “smoothing” filter = “interpolation” filter



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