Tele4653 l8
8
TELE4653 Digital Modulation & Coding Detection Theory Wei Zhang [email protected] School of Electrical Engineering and Telecommunications The University of New South Wales
Transcript of Tele4653 l8
TELE4653 Digital Modulation &Coding
Detection Theory
Wei Zhang
School of Electrical Engineering and Telecommunications
The University of New South Wales
Q Function
Denote N (µ, σ2) the PDF of Gaussian RV with mean µ and
variance σ2. The Q function is defined as
Q(x) = Pr[N (0, 1) > th] =1√2π
∫
th∞e−
t2
2 dt. (1)
If X ∼ N (µ, σ2), then
Pr[X > th] = Q
(
th− µ
σ
)
(2)
Pr[X < th] = Q
(
µ− th
σ
)
(3)
Useful bound: Q(x) ≤ 1
2e−x2/2.
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.1/6
Union Bound of Pe
When sm is sent, an error occurs when the received r is not in
Dm. Assume that the messages are equiprobable.
Pe =1
M
M∑
m=1
Pe|m (4)
where Pe|m =
∫
Dcm
p(r|sm)dr (5)
=M∑
1≤m′≤M,m
′6=m
∫
Dm
′
p(r|sm)dr (6)
Note that the decision region Dm′ under ML is
Dm′ ={
r ∈ RN : P [r|sm′ ] > P [r|sk],∀1 ≤ k ≤ M and k 6= m
′
}
(7)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.2/6
Union Bound of Pe
Define Dmm′ as
Dmm′ ={
r ∈ RN : P [r|sm′ ] > P [r|sm]
}
(8)
Note that Dmm′ is the decision region for m′ in a binary
equiprobable system with signals sm and sm′ . Obviously,
Dm′ ⊆ Dmm′ . Therefore,
Pe|m ≤M∑
1≤m′≤M,m
′6=m
∫
Dmm
′
p(r|sm)dr
=
M∑
1≤m′≤M,m
′6=m
Q
√
d2mm′
2N0
(9)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.3/6
Union Bound of Pe
Finally,
Pe ≤ 1
M
M∑
m=1
M∑
1≤m′≤M,m
′6=m
Q
√
d2mm′
2N0
(10)
≤ 1
2M
M∑
m=1
M∑
1≤m′≤M,m
′6=m
exp
(
−d2mm′
4N0
)
(11)
where we used Q(x) ≤ 1
2e−x2/2.
Define dmin the minimum distance of a constellation, then
Pe ≤ (M − 1)Q
√
d2min
2N0
≤ M − 1
2exp
(
−d2min
4N0
)
(12)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.4/6
A Lower Bound of Pe
Note that Dcmm′ ⊆ Dc
m. Hence,
Pe =1
M
M∑
m=1
∫
Dcm
p(r|sm)dr
≥ 1
M
M∑
m=1
∫
Dc
mm′
p(r|sm)dr (13)
=1
M
M∑
m=1
∫
Dmm′
p(r|sm)dr (14)
=1
M
M∑
m=1
Q
√
d2mm′
2N0
(15)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.5/6
A Lower Bound of Pe
Denoting by Nmin the number of the points in the constellation
that are at the distance dmin from at least one point in the
constellation, we obtain
Pe ≥ 1
M
M∑
m=1
maxm′ 6=m
Q
(
dmm′√2N0
)
(16)
≥ Nmin
MQ
(
dmin√2N0
)
. (17)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.6/6
RepeatersTo boost the signal strength which may be degraded by the
channel attenuation, repeaters are used.
Regenerative repeaters: detect and regenerate a noise-free
signal. For PAM with K repeaters,
Pe ≈ KQ
(
√
2EbN0
)
(18)
Analogue repeaters: amplifies the signal, and also boosts
the noise. For PAM with K repeaters,
Pe ≈ Q
(
√
2EbKN0
)
(19)
TELE4653 - Digital Modulation & Coding - Lecture 8. May 3, 2010. – p.7/6
