– 1 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

Discrete-Time Signal Processing(A Review)

Sampling

– 2 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

CT

fclk

xc(t) x(n)C-D

xc(t)

t0

x(n)

n0 1 2 3 4

xc(nT)

T 2T 3T 4T T

cx n =x t=nT

∞ ?jω -n

cn=-∞

X z =e = x n z ↔ X jΩ

DT FT

c cx t ⇔ X jΩ

Zero-Order Hold (ZOH)

– 3 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

xc(t)

t0

xSH(t)

t0 T 2T 3T 4T T

xc(nT)T

xc(nT)

t0 T 2T 3T 4T

t0

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

T

1/T

T

w

∞

SH cn=-∞

1x t = x nT u t-nT -u t-nT -TT

Sampling

– 4 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

SH sw→0∞

cw→0 n=-∞∞

-s nT+w-snTcw→0 n=-∞

-sw ∞-snT

cw→0 n=-∞∞ ∞

-snT -nc

n=-∞ n=-∞

FT limx t =FT x t

1=FT lim x nT u t-nT -u t-nT - ww1 1 1=lim x nT e - ew s s1-e=lim x nT esw

= x nT e = x n z

xSH(t)

t0 w→0

Area fixed

Impulse train

∞

jω -ns

n=-∞X z =e = x n z ↔ X jΩ

xs(t)

t0 T 2T 3T 4T T

xc(t)·δ(t-nT)

jωCT →Ω, DT →ω,

s =jΩ, z =e , ω=ΩT.

Sampling

– 5 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

s c c

s c

FTs s

k

s c sk

x t =x t s t =x t δ t-nT1X jΩ = X jΩ S jΩ2π

2π 2πs t ⇔ δ Ω-kΩ , Ω =T T1X jΩ = X Ω-kΩT

∞jω -n

n=-∞

ωs c sΩ= ωT k Ω=T

X e = x n z

1=X jΩ = X Ω-kΩT

xc(t)

t0

s(t)

t0 T

δ(t-nT)

xs(t)

t0 T 2T 3T 4T T

xc(t)·δ(t-nT)

Spectrum of Sampled Signal (Ωs>2ΩN)

– 6 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

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

Xc(jΩ)

Xs(jΩ)

Ω

Ω

Ω

ΩN-ΩN

Ωs-Ωs-2Ωs-3Ωs 2Ωs 3Ωs0

Ωs-Ωs-2Ωs-3Ωs 2Ωs 3Ωs0

S(jΩ)

Spectrum of Sampled Signal (Ωs<2ΩN)

– 7 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

• 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.

Xc(jΩ)

Ω

Ω

ΩN-ΩN

Ωs-Ωs-2Ωs-3Ωs 2Ωs 3Ωs0

Xs(jΩ)

ΩΩs-Ωs-2Ωs-3Ωs 2Ωs 3Ωs0

S(jΩ)

Reconstruction Filter (Nyquist)

– 8 –

Data Converters Discrete-Time Signal ProcessingProfessor Y. Chiu

EECT 7327Fall 2014

∞

r rn=-∞

∞

n=-∞

x t = x n h t-nT

sin π t-nT /T= x n π t-nT /T

r

sin πt/Th t = πt/T

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

Xr(jΩ)

Xs(jΩ)

Ω

Ω

ΩN-ΩN

Ωs-Ωs-2Ωs-3Ωs 2Ωs 3Ωs0

Hr(jΩ)

ΩΩs/2-Ωs/2