Bài giảng Signals and Systems - Chapter: Sampling - Đặng Quang Hiếu

ET2060 - Signals and Systems
Sampling theorem
Dr. Quang Hieu Dang
Hanoi University of Science and Technology
School of Electronics and Telecommunications
Autumn 2012
Sampling theorem
x(t)
sampling
−−−−−−→
Ts
x(nTs)
normalization
−−−−−−−−−−→ x [n]
t
x(t)
b
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nTs
x(nTs )
“If a function x(t) contains no frequencies higher than B hertz, it
is completely determined by giving its ordinates at a series of
points spaced 1/(2B) seconds apart.” – Claude Shannon.
Proof of the sampling theorem (1)
Ω
X (jΩ)
2piB−2piB
Let X (jΩ) be the spectrum of x(t). We have:
x(t) =
1
2pi
∫
∞
−∞
X (jΩ)e jΩtdΩ =
1
2pi
∫ 2piB
−2piB
X (jΩ)e jΩtdΩ
Substitute t = n2B where n ∈ Z into the equation:
x(n/2B) =
1
2pi
∫ 2piB
−2piB
X (jΩ)e jΩ
n
2B dΩ
Proof of the sampling theorem (2)
Ω
X˜ (jΩ)
2piB−2piB 6piB−6piB
X˜ (jΩ) =
∞∑
n=−∞
cne
j 2pi
4piB
nΩ =
∞∑
n=−∞
cne
jΩ n
2B
cn =
1
4piB
∫ 2piB
−2piB
X˜ (jΩ)e−j
2pi
4piB
nΩdΩ =
1
4piB
∫ 2piB
−2piB
X (jΩ)e−jΩ
n
2B dΩ
x(n/2B)→ cn =
1
2B
x(−n/2B)→ X˜ (jΩ)→ X (jΩ)→ x(t) QED!!!
Another approach
Sampling operation can be considered as multiplying signal x(t)
with a periodic unit impulse function (with period Ts).
xs(t) = x(t)p(t)
t
x(t)
u u u u u u u u u u u u u u u u u u u u u
t
p(t)
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t
xs(t)
Spectrum of the sampled signal
Xs(jΩ) =
1
2pi
[X (jΩ)∗P(jΩ)], where P(jΩ) =
2pi
Ts
∞∑
k=−∞
δ(Ω−k
2pi
Ts
)
=⇒ Xs(jΩ) =
1
Ts
∞∑
k=−∞
X (j(Ω − kΩs)),Ωs =
2pi
Ts
Ω
X (jΩ)
1
u u u u u
Ω
P(jΩ)
2pi
Ts
Ωs−Ωs
Ω
Xs(jΩ)
1
Ts
Ωs−Ωs 2Ωs−2Ωs
Signal reconstruction
Let signal xs(t) pass through an ideal low pass filter with cutoff
frequency Ωc = Ωs/2 > B
H(jΩ) =
{
Ts , |Ω| ≤ Ωc
0, |Ω| > Ωc
h(t) =
Ts sin(Ωct)
pit
Then the original signal x(t) can be reconstructed as
x(t) = xs(t) ∗ h(t) =
∞∑
n=−∞
x(nTs)h(t − nTs)
=
∞∑
n=−∞
x(nTs)
ΩcTs
pi
sin(Ωc(t − nTs))
Ωc(t − nTs)