analysis of the expected error performance of cooperative...
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Analysis of the Expected Error Performance ofCooperative Wireless Networks Employing
Distributed Space-Time Codes
Jan Mietzner1, Ragnar Thobaben2, and Peter A. Hoeher1
University of Kiel, Germany
1Information and Coding Theory Lab (ICT)2Institute for Circuits and System Theory (LNS)
{jm,rat,ph}@tf.uni-kiel.dehttp://www-ict.tf.uni-kiel.de
Globecom 2004, Dallas, Texas, USA
November 30, 2004
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1
From Co-located to Distributed Transmitters
Txn
Tx1
Tx2
Rx
Tx1 Txn
Rx
Tx2
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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2
Motivation for Distributed Space-Time Codes
I Benefits of multiple antennas for wireless communication systems:
– Performance of wireless systems often limited by fading due to multipath signal propagation
– System performance significantly improved by exploiting diversity
=⇒ Employ Space-time codes (STCs) to exploit spatial diversity
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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2
Motivation for Distributed Space-Time Codes
I Benefits of multiple antennas for wireless communication systems:
– Performance of wireless systems often limited by fading due to multipath signal propagation
– System performance significantly improved by exploiting diversity
=⇒ Employ Space-time codes (STCs) to exploit spatial diversity
I Concept of multiple antennas can be transferred to cooperative wireless networks:
– Multiple (single-antenna) nodes cooperate and perform a joint transmission strategy
=⇒ Nodes share their antennas using a distributed space-time code
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 5: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/5.jpg)
3
Cooperative Wireless Networks – Examples
I Simulcast networks for broadcasting or paging applications:
Conventionally, all nodes simultaneously transmit the same signal using the
same carrier frequency =⇒ Reduced probability of shadowing
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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3
Cooperative Wireless Networks – Examples
I Simulcast networks for broadcasting or paging applications:
Conventionally, all nodes simultaneously transmit the same signal using the
same carrier frequency =⇒ Reduced probability of shadowing
I Relay-assisted communication, e.g., in cellular systems, ad-hoc networks, sensor networks:
– Relay nodes receive signal from a source node and forwarded it to a destination node
– Fixed stations or other mobile stations (‘user cooperation diversity’)
=⇒ Distributed STCs are suitable for both types of networks
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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4
Cooperative Wireless Networks – General Setting
I n transmitting nodes (Tx1,...,Txn), one receiving node (Rx); single-antenna nodes
Txn
Tx1
Tx2
s1 (t)
Rx
a1
an
a 2
s 2(t
)
sn(t)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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5
Differences between Co-located and Distributed Transmitters
I Distributed STCs:
– No shadowing: Diversity degree n X
– Additionally: Diversity degree (n−ν) if any subset of ν Tx nodes obstructed (X)
I Higher probability of line-of-sight (LOS) component
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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5
Differences between Co-located and Distributed Transmitters
I Distributed STCs:
– No shadowing: Diversity degree n X
– Additionally: Diversity degree (n−ν) if any subset of ν Tx nodes obstructed (X)
I Higher probability of line-of-sight (LOS) component
I Transmitted signals si(t) subject to different average link gains ai, due to
different distances or shadowing =⇒ Reduced degree of diversity
Here: Focus on average link gains ai and associated diversity loss
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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6
Outline
I Error Performance of Distributed STCs
– Basic Assumptions
– Analytical Results
I Average Error Performance in a General Uplink Scenario
I Average Error Performance in an Uplink Scenario with Additional Constraint
I Conclusions
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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7
Basic Assumptions
I Transmitting nodes Tx1,...,Txn perfectly synchronized in time and frequency
I All nodes employ a single antenna
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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7
Basic Assumptions
I Transmitting nodes Tx1,...,Txn perfectly synchronized in time and frequency
I All nodes employ a single antenna
I Frequency-flat block-fading channel model (Rayleigh):
Channel coefficients hi ∼ CN (0, ai), i = 1, ..., n Normalization:P
i ai := n
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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7
Basic Assumptions
I Transmitting nodes Tx1,...,Txn perfectly synchronized in time and frequency
I All nodes employ a single antenna
I Frequency-flat block-fading channel model (Rayleigh):
Channel coefficients hi ∼ CN (0, ai), i = 1, ..., n Normalization:P
i ai := n
I Same average transmitter power P/n for all transmitting nodes Txi; no shadowing
I Congenerous antennas at Tx nodes, omnidirectional antenna at Rx node
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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7
Basic Assumptions
I Transmitting nodes Tx1,...,Txn perfectly synchronized in time and frequency
I All nodes employ a single antenna
I Frequency-flat block-fading channel model (Rayleigh):
Channel coefficients hi ∼ CN (0, ai), i = 1, ..., n Normalization:P
i ai := n
I Same average transmitter power P/n for all transmitting nodes Txi; no shadowing
I Congenerous antennas at Tx nodes, omnidirectional antenna at Rx node
=⇒aj
ai=
„di
dj
«ρ(according to Friis formula)
di: Length of transmission link i, ρ: Path-loss exponent (2 ≤ρ≤ 4)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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8
Analytical Results for the Error Performance
I Average signal-to-noise ratio (SNR) for transmission link i: aiEs/nN0
(Es/N0: Overall received SNR)
I In the sequel, Alamouti’s Tx diversity scheme (n=2) and binary transmission
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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8
Analytical Results for the Error Performance
I Average signal-to-noise ratio (SNR) for transmission link i: aiEs/nN0
(Es/N0: Overall received SNR)
I In the sequel, Alamouti’s Tx diversity scheme (n=2) and binary transmission
I Using Proakis’ theoretical results for diversity reception, one obtains the bit error rate (BER):
Pb(a1) =1
2
»a1 (1− µ(a1))
a1 − a2
+a2 (1− µ(a2))
a2 − a1
–,
where a1 ∈ [0, 2], a2 = 2− a1 and µ(ai) =1q
1 +2N0aiEs
(i = 1, 2)
Specifically, Pb(a1)=Pb(2−a1) holds for all a1
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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9
Analytical Results for the Error Performance
Pb(a1) vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1.8, a
2=0.2
Distr. Alamouti, a1=1.6, a
2=0.4
Distr. Alamouti, a1=1.4, a
2=0.6
Distr. Alamouti, a1=1, a
2=1
Multiple-antenna system (colocated antennas)
Single transmitting nodeSNR Es/N0
SNR Es/2N0 + Es/2N0 = Es/N0
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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9
Analytical Results for the Error Performance
Pb(a1) vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1.8, a
2=0.2
Distr. Alamouti, a1=1.6, a
2=0.4
Distr. Alamouti, a1=1.4, a
2=0.6
Distr. Alamouti, a1=1, a
2=1
Multiple-antenna system (colocated antennas)
Single transmitting nodeSNR Es/N0
SNR Es/2N0 + Es/2N0 = Es/N0
I Best performance for a1 = a2 = 1
(diversity degree of two)
I Worst performance for a1=2 and
a2=0 (diversity degree of one)
I Even for large a1, significant gains
w.r.t. single transmission node
(diversity degree still close to two)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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9
Analytical Results for the Error Performance
Pb(a1) vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1.8, a
2=0.2
Distr. Alamouti, a1=1.6, a
2=0.4
Distr. Alamouti, a1=1.4, a
2=0.6
Distr. Alamouti, a1=1, a
2=1
Multiple-antenna system (colocated antennas)
Single transmitting nodeSNR Es/N0
SNR Es/2N0 + Es/2N0 = Es/N0
I Best performance for a1 = a2 = 1
(diversity degree of two)
I Worst performance for a1=2 and
a2=0 (diversity degree of one)
I Even for large a1, significant gains
w.r.t. single transmission node
(diversity degree still close to two)
I Results hold approximately also, e.g.,
for TR-STBCs and delay diversity
I Generalizations are possible:
– n>2 Tx nodes (e.g., OSTBCs)
– Rice fading, shadowing
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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10
Outline
I Error Performance of Distributed STCs
I Average Error Performance in a General Uplink Scenario
– General Uplink Scenario
– Derivation of the Mean Bit Error Rate
I Average Error Performance in an Uplink Scenario with Additional Constraint
I Conclusions
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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11
General Uplink Scenario
I Assumptions:
– n = 2 Tx nodes (MS1, MS2), one Rx node (BS), distributed Alamouti scheme
(MS1 and MS2 may be mobile relays)
– Coverage area A of BS is a disk of radius r
A
MS1
MS2
d2
d1 =c r
rBS (Rx)
ϕ1
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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11
General Uplink Scenario
I Assumptions:
– n = 2 Tx nodes (MS1, MS2), one Rx node (BS), distributed Alamouti scheme
(MS1 and MS2 may be mobile relays)
– Coverage area A of BS is a disk of radius r
A
MS1
MS2
d2
d1 =c r
rBS (Rx)
ϕ1
– For MS1 a fixed distance d1 to BS is assumed
where d1 := c r, c ≤ 1 (angle ϕ1 arbitrary)
– MS2 is located anywhere within A, according to a
uniform distribution
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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11
General Uplink Scenario
I Assumptions:
– n = 2 Tx nodes (MS1, MS2), one Rx node (BS), distributed Alamouti scheme
(MS1 and MS2 may be mobile relays)
– Coverage area A of BS is a disk of radius r
A
MS1
MS2
d2
d1 =c r
rBS (Rx)
ϕ1
– For MS1 a fixed distance d1 to BS is assumed
where d1 := c r, c ≤ 1 (angle ϕ1 arbitrary)
– MS2 is located anywhere within A, according to a
uniform distribution
I The mean BER can be calculated as
P̄b =
Z 2
0
pA1(a1)Pb(a1) da1
=⇒ pA1(a1) required
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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12
Derivation of the Mean Bit Error Rate
I Let q := d2/d1 (corresponding random variable Q)
I Since d1 is fixed, the pdf of Q is given by
pQ(q) = d1 · pD2(d1q) = c r ·
∂
∂d2
P (D2≤d2)˛̨̨d2=c r q
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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12
Derivation of the Mean Bit Error Rate
I Let q := d2/d1 (corresponding random variable Q)
I Since d1 is fixed, the pdf of Q is given by
pQ(q) = d1 · pD2(d1q) = c r ·
∂
∂d2
P (D2≤d2)˛̨̨d2=c r q
I With P (D2≤d2) = πd22/πr
2 one obtains pQ(q) = 2c2q
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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12
Derivation of the Mean Bit Error Rate
I Let q := d2/d1 (corresponding random variable Q)
I Since d1 is fixed, the pdf of Q is given by
pQ(q) = d1 · pD2(d1q) = c r ·
∂
∂d2
P (D2≤d2)˛̨̨d2=c r q
I With P (D2≤d2) = πd22/πr
2 one obtains pQ(q) = 2c2q
I Using a1/a2 = (d2/d1)ρ = qρ and a2 = 2− a1
=⇒ a1 is a function of q: a1 =2 qρ
1 + qρ
=⇒ The pdf pA1(a1) can be determined using pQ(q) = 2c2q
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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13
Derivation of the Mean Bit Error Rate
I One obtains
pA1(a1) =
(1 + ξ(a1))2
2ρ ξ(a1)(ρ−1)/ρ· pQ(ξ(a1)
1/ρ)
=
8><>:c2 (1 + ξ(a1))
2
ρ ξ(a1)(ρ−2)/ρ, for a1 ∈ [0, a1max]
0 else ,
where
ξ(a1) :=a1
2− a1
and a1max = a1max(c, ρ) :=2
(1+cρ)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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14
Average Error Performance
pA1(a1) vs. a1 (ρ = 2, 3, 4; c = 0.5)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a1
p A1(a
1)
ρ = 2ρ = 3ρ = 4
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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14
Average Error Performance
pA1(a1) vs. a1 (ρ = 2, 3, 4; c = 0.5)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a1
p A1(a
1)
ρ = 2ρ = 3ρ = 4
P̄b vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1, a
2=1
Average BER resulting for ρ=2Average BER resulting for ρ=3Average BER resulting for ρ=4
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14
Average Error Performance
pA1(a1) vs. a1 (ρ = 2, 3, 4; c = 0.5)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a1
p A1(a
1)
ρ = 2ρ = 3ρ = 4
P̄b vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1, a
2=1
Average BER resulting for ρ=2Average BER resulting for ρ=3Average BER resulting for ρ=4
For large path-loss exponent ρ, probability that a1 ≈ 1 comparably small
=⇒ Significant average loss compared to co-located antennas (a1 = a2 = 1)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 31: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/31.jpg)
15
Outline
I Error Performance of Distributed STCs
I Average Error Performance in a General Uplink Scenario
I Average Error Performance in an Uplink Scenario with Additional Constraint
– Uplink Scenario with Additional Constraint
– Mean Bit Error Rate
I Conclusions
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 32: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/32.jpg)
16
Uplink Scenario With Additional Constraint
I Assumptions:
– Constraint for MS2: Distance d12 between MS2 and MS1 significantly smaller than d1
=⇒MS2 within disk A′ of radius r12�d1 around MS1, according to uniform distribution
=⇒ Constraint reasonable when MS1 and MS2 act as mutual relays:MS1 and MS2 only cooperate if d12≤r12, so as to avoid error propagation
– Distance d1 between MS1 and BS normalized to one
BS (Rx) MS1
r12
MS2
d12
d1 =1
A′
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
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16
Uplink Scenario With Additional Constraint
I Assumptions:
– Constraint for MS2: Distance d12 between MS2 and MS1 significantly smaller than d1
=⇒MS2 within disk A′ of radius r12�d1 around MS1, according to uniform distribution
=⇒ Constraint reasonable when MS1 and MS2 act as mutual relays:MS1 and MS2 only cooperate if d12≤r12, so as to avoid error propagation
– Distance d1 between MS1 and BS normalized to one
BS (Rx) MS1
r12
MS2
d12
d1 =1
A′I Derivation of pA1
(a1) and
P̄b as before, via the pdf pQ(q)
(However, deriving pQ(q) is
more involved)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 34: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/34.jpg)
17
Average Error Performance
pA1(a1) vs. a1 (ρ = 2, 3, 4)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
a1
p A1(a
1)
ρ = 2ρ = 3ρ = 4
Radius r12 = 0.3 (solid), r12 = 0.9 (dashed)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 35: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/35.jpg)
17
Average Error Performance
pA1(a1) vs. a1 (ρ = 2, 3, 4)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
a1
p A1(a
1)
ρ = 2ρ = 3ρ = 4
Radius r12 = 0.3 (solid), r12 = 0.9 (dashed)
P̄b vs. Es/N0 in dB
0 2 4 6 8 10 12 14 16 18 2010−4
10−3
10−2
10−1
100
Es/N
0 (dB)
BE
R
Distr. Alamouti, a1=2, a
2=0
Distr. Alamouti, a1=1, a
2=1
Average BER resulting for ρ=2Average BER resulting for ρ=3Average BER resulting for ρ=4
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 36: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/36.jpg)
18
Conclusions
I Wireless systems with distributed transmitters: Specific differences compared to systems with
co-located antennas
I Here: Focus on different average link gains =⇒ Reduced diversity degree
I Two typical uplink scenarios considered =⇒ Analytical derivation of the mean BER
=⇒ In most scenarios performance loss < 2 dB at a BER of 10−3
=⇒ Most significant performance loss for large path-loss exponents (e.g. ρ = 4)
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel
![Page 37: Analysis of the Expected Error Performance of Cooperative ...ece.ubc.ca/~janm/Presentations/globecom04.pdf · Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel](https://reader033.vdocuments.mx/reader033/viewer/2022042009/5e713bc7303e0d075006d08f/html5/thumbnails/37.jpg)
19
Appendix: Expressions for the Uplink Scenario with Additional Constraint
I Probability P (D2≤d2), where 1−r12 ≤ d2 ≤ 1+r12 :
P (D2≤d2) =1
π
d22
r212
„ϕB(d2)−
1
2sin(2ϕB(d2))
«+
1
π
„ϕM(d2)−
1
2sin(2ϕM(d2))
«,
where ϕB(d2) = arccos
1 + d2
2 − r212
2 d2| {z }=: ψ(d2)
!and ϕM(d2) = arccos
1− d2
2 + r212
2 r12| {z }=: ζ(d2)
!
I Pdf pQ(q), q=d2/d1=d2: (→ from pQ(q) one obtains pA1(a1))
pQ(q) =∂
∂d2
P (D2≤d2)˛̨̨d2=q
=1
π
2q
r212
„ϕB(q)−
1
2sin(2ϕB(q))
«+ ...
... +1
π
1− q2 − r212
2 r212
p1− ψ2(q)
„1− cos(2ϕB(q))
«+
1
π
q
r12
p1− ζ2(q)
„1− cos(2ϕM(q))
«
Jan Mietzner, Ragnar Thobaben, and Peter A. Hoeher, University of Kiel