Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Discrete-Time Signal Processing
(A Review)
–1–
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Sampling
xc(t)
xc(t)
C-D
CT
x(n)
0
T 2T 3T 4T
fclk
x  n  = x c  t = nT 
T
t
xc(nT)
x(n)
FT
x c  t  ⇔ X c  jΩ 
DT
0
X z = e
jω
∞
 =  x n  z
n=-∞
–2–
-n
1
?
2
3
4
↔ X c  jΩ 
n
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Zero-Order Hold (ZOH)
xc(nT)
xc(t)
0
0
t
T 2T 3T 4T
xc(nT)
T
xSH(t)
T
t
u(t) - u(t-T)
T
1/T
w
0
T 2T 3T 4T
T
0
t
T
1 ∞
x SH  t  =  x c  nT  u  t - nT  - u  t - nT - T  
T n=-∞
–3–
t
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Sampling


FT lim x SH  t  = FT  x s  t  
Area fixed
xSH(t)
w→0

1 ∞

= FT  lim   x c  nT  u  t - nT  - u  t - nT - w    

 w→0  w n=-∞
0
1 ∞
1 -s nT+w   
1
= lim   x c  nT   e -snT - e 
 
w→0
s
s
 w n=-∞

t
w→0
 1- e -sw  ∞
= lim 
x c  nT  e -snT


w→0
 sw  n=-∞
Impulse train
xs(t)
∞
xc(t)·δ(t-nT)
=  x c  nT  e
n=-∞
-snT
∞
=  x  n  z -n
n=-∞
CT → Ω, DT → ω,
0
T 2T 3T 4T
T
t
X z = e
s = jΩ, z = e jω , ω = ΩT.
jω
∞
 =  x n  z
n=-∞
–4–
-n
↔ X s  jΩ 
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Sampling
xc(t)
xs(t)
0
t
δ(t-nT)
s(t)
0
xc(t)·δ(t-nT)
0
X s  jΩ  =
∞
X  e jω  =  x  n  z -n
T
t
1
X c  jΩ   S  jΩ 
2π
2π
2π
δ
Ω
kΩ
,
Ω
=
 
s
s
T k
T
1
X s  jΩ  =  X c  Ω - kΩs 
T k
FT
n=-∞
= X s  jΩ  Ω= ω =
T
x s  t  = x c  t  s  t  = x c  t  δ  t - nT 
t
T
T 2T 3T 4T
st⇔
1
X c  Ω - kΩs 

ω
T k
Ω=
T
–5–
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Spectrum of Sampled Signal (Ωs>2ΩN)
Xc(jΩ)
-ΩN ΩN
Ω
S(jΩ)
-3Ωs
-2Ωs
-Ωs
Ωs
0
2Ωs
3Ωs
Ω
2Ωs
3Ωs
Ω
Xs(jΩ)
-3Ωs
-2Ωs
-Ωs
0
Ωs
The spectrum of the sampled signal is periodic in Ωs=2π/T.
–6–
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Spectrum of Sampled Signal (Ωs<2ΩN)
Xc(jΩ)
ΩN
-ΩN
Ω
S(jΩ)
-3Ωs
-2Ωs
-Ωs
Ωs
0
2Ωs
3Ωs
Ω
2Ωs
3Ωs
Ω
Xs(jΩ)
-3Ωs
•
•
-2Ωs
-Ωs
0
Ωs
Aliasing (folding) results in irreversible signal distortion.
Can only be avoided by using sufficiently high sample rate, or bandlimit the input signal with a coarse, continuous-time filter – AAF.
–7–
Data Converters
EECT 7327
Discrete-Time Signal Processing
Professor Y. Chiu
Fall 2014
Reconstruction Filter (Nyquist)
Xs(jΩ)
-3Ωs
-2Ωs
-Ωs
Ωs
0
2Ωs
3Ωs
Ω
hr  t  =
Hr(jΩ)
sin  πt / T 
πt / T
∞
-Ωs/2 Ωs/2
Ω
x r  t  =  x  n  hr  t - nT 
n=-∞
∞
Xr(jΩ)
=  x n
n=-∞
-ΩN ΩN
sin  π  t - nT  / T 
π  t - nT  / T
Ω
Reconstruction filter = “smoothing” filter = “interpolation” filter
–8–