Optimal Relay Selection and Beamforming in MIMO Cognitive Networks

Problem Statement System Model Signal Model Optimal Relay Selection and Beamforming Simulation Results
Optimal Relay Selection and Beamforming
in MIMO Cognitive Multi-Relay Networks
Mohamed Seif1
1Wireless Intelligent Networks Center (WINC), Nile University, Egypt
May 19, 2015
Mohamed Seif Nile University
Optimal Relay Selection and Beamforming in MIMO Cognitive Multi-Relay Networks 1

Outline
1 Problem Statement
2 System Model
3 Signal Model
4 Optimal Relay Selection and Beamforming
5 Simulation Results

Problem Statement
For a MIMO cognitive multi-relay network, this work
proposes an optimal relay selection and beamforming
scheme subject to transmit power constraints at the relays
and the interference power constraints at the primary
users.

System Model
Pair of a SU are transmitting in the
presence of 2 PUs, each equipped
with one antenna
K cognitive relays, each relay is
equipped with N antennas
No direct link between the SU nodes
Intference from the PUs is neglected
TX RX
UE UE
Primary User
Network
K
Secondary User Network
1
Desired Link Interference Link
Figure: CRN model

System Model
TDD mode is considered for the
system
During the ﬁrst time slot, the SU-TX
transmits signals to the relays
At the second time slot, the kth
selected relay, multiplies the
received signals and forwards it the
SU-RX
TX RX
UE UE
Primary User
Network
K
1
Figure: CRN model

Signal Model
The received signal at the SU-RX is
expressed as:
y = h†
2k Fk (h1k x + nr ) + z
where,
E[ x 2
] = Ps
nr ∼ CN(0,σ2
r I)
z ∼ CN(0,σ2
d )
Fk ∈ CN×N
TX RX
UE UE
Primary User
Network
K
1
Figure: CRN model

Signal Model
The SNR at the SU-RX is expressed
as:
SNR =
Ps h†
2k
Fh1k
2
σ2
r h†
2k
F +σ2
d
TX RX
UE UE
Primary User
Network
K
1
Figure: CRN model

Signal Model
The transmit power of the SU-TX
satisﬁes that:
Ps g1m
2
≤ Im, m ∈ {1,2}
then,
Ps = min(Ps,min
m
Im
g1m
2 )
where,
g1m ∈ C1×1
TX RX
UE UE
Primary User
Network
K
1
Figure: CRN model

Signal Model
The transmit power of the kth
relay
is:
PRk
= Ps Fh1k
2
+ σ2
r F
2
and satisﬁes that,
Ps g†
2mFh1k
2
+ σ2
r g†
2m
2
≤ Im,
m ∈ {1,2}
where,
g2m ∈ CN×1
TX RX
UE UE
Primary User
Network
K
1
Figure: CRN model

The optimization problem of relay selection and beamforming
for a MIMO congnitive multi-relay network is formulated as:
Problem Formulation
arg max
k
max
F
1
2 log2(1 +
Ps h†
2k
Fh1k
2
σ2
r h†
2k
F
2
+σ2
d
)
s.t. Ps Fh1k
2
+ σ2
r F 2
≤ PR
Ps g†
2mFh1k
2
+ σ2
r g†
2mF
2
≤ Im
k ∈ {1,2,...,K}, m ∈ {1,2,...,M}

Problem Formulation
arg max
k
max
F
1
2log2(1 +
Ps h†
2k
Fh1k
2
σ2
r h†
2k
F
2
+σ2
d
)
PrecoderDesign
s.t. Ps Fh1k
2
+ σ2
r F 2
≤ PR
Ps g†
2mFh1k
2
+ σ2
r g†
2mF
2
≤ Im
k ∈ {1,2,...,K}, m ∈ {1,2,...,M}

Problem Formulation
arg max
k
RelaySelection
max
F
1
2log2(1 +
Ps h†
2k
Fh1k
2
σ2
r h†
2k
F
2
+σ2
d
)
PrecoderDesign
s.t. Ps Fh1k
2
+ σ2
r F 2
≤ PR
Ps g†
2mFh1k
2
+ σ2
r g†
2mF
2
≤ Im
k ∈ {1,2,...,K}, m ∈ {1,2,...,M}

Beamforming Optimization Problem
P1: max
f
f†(Pshh†)f
f†(σ2
r H2H†
2
)f+σ2
d
s.t. f†
(PsH1H†
1 + σ2
r I)f ≤ PR
f†
(PsG1mG†
1m + σ2
r G2mG†
2m)f ≤ Im
m ∈ {1,2,...,M}
where,
f=vec(F)
h = h∗
1k ⊗ h1k
H1 = h∗
1k ⊗ I, H2 = I ⊗ h2k
G1m = h∗
1k ⊗ g2m, G2m = I ⊗ g2m

P1: max
f
f†(Pshh†)f
f†(σ2
r H2H†
2
)f+σ2
d
s.t. f†
(PsH1H†
1 + σ2
r I)f ≤ PR
f†
(PsG1mG†
1m + σ2
r G2mG†
2m)f ≤ Im
m ∈ {1,2,...,M}
Difﬁcult to Solve!
where,
f=vec(F)
h = h∗
1k ⊗ h1k
H1 = h∗
1k ⊗ I, H2 = I ⊗ h2k
G1m = h∗
1k ⊗ g2m, G2m = I ⊗ g2m

P2: max
W≽0
tr(A1W)
tr(A2W)+σ2
d
s.t. tr(A3W) ≤ PR
tr(BmW) ≤ Im
m ∈ {1,2,...,M}
where,
A1 = Pshh†
A2 = σ2
r G2mG†
2m
A3 = PsH1H†
1
+ σ2
r I
W = ff†

P2: max
W≽0
tr(A1W)
tr(A2W)+σ2
d
s.t. tr(A3W) ≤ PR
tr(BmW) ≤ Im
m ∈ {1,2,...,M}
rank (W)=1,

P2: max
W≽0
tr(A1W)
tr(A2W)+σ2
d
s.t. tr(A3W) ≤ PR
tr(BmW) ≤ Im
m ∈ {1,2,...,M}
rank (W)=1, then rank of W has been relaxed
where,
A1 = Pshh†
A2 = σ2
r G2mG†
2m
A3 = PsH1H†
1
+ σ2
r I
W = ff†

(Charnes-Cooper transformation)
P3: max
S≽0,ν≥0
tr(A1S)
s.t. tr(A1S) + σ2
d ν = 1
tr(A3S) ≤ νPR
tr(BmS) ≤ νIm, m ∈ {1,2,...,M}
where,
W = S
ν
tr(A2W) + σ2
d = 1

P4: max
W
tr(A3W)
s.t. max
W≽0
tr(A1W)
tr(A2W)+σ2
d
≥ γ − ∆γ
tr(BmW) ≤ Im, m ∈ {1,2,...,M}
where,
γ = max
S≽0,ν≥0
tr(A1S) (P3)
0 ≤ ∆γ < γ

Solution of P4 is tight to P2 by (1 − ∆γ
γ ) (Proof Hint)
Solution of P4 has rank one (Proof Hint)
then,
W =
√
λiff†
, λi ≠ 0

Solution of P4 is tight to P2 by (1 − ∆γ
γ ) (Proof Hint)
Solution of P4 has rank one (Proof Hint)
then,
W =
√
λiff†
, λi ≠ 0
Relay Selection
arg max
k
RelaySelection
max
F
1
2log2(1 +
Ps h†
2k
Fh1k
2
σ2
r h†
2k
F
2
+σ2
d
)
PrecoderDesign

Simulation Setup
Symbol Description Realization
M Number of PUs 2
K Number of relays ∼
N Number of antennas per relay ∼
PR Transmitted power at the relay ∼
σ2
r Noise power at the relay per anetenna unit power
P Number of iterations 50
σ2
d Noise power at SU-RX unit power
Table: Parameters of Simulation

Simulation Results
0 5 10 15 20
0.5
1
1.5
2
2.5
3
P
R
(dB)
R
ave
(bps/Hz)
ORSB, N=3, K=3
Figure: Average capacity versus the maximum allowable transmit
power of the relay

Simulation Results
0 5 10 15 20
0.5
1
1.5
2
2.5
3
P
R
(dB)
R
ave
(bps/Hz)
ORSB, N=3, K=3
ORSB, N=3, K=2
ORSB, N=3, K=1
k=1,2,3
Figure: Effect of number of relays

Simulation Results
0 5 10 15 20
0.5
1
1.5
2
2.5
3
P
R
(dB)
R
ave
(bps/Hz)
ORSB, N=4, K=3
ORSB, N=5, K=3
ORSB, N=6, K=3
ORSB, N=3, K=3
ORSB, N=2, k=3
N=2,3,4,5,6
Figure: Effect of number of antennas

Simulation Results
0 5 10 15 20
0.5
1
1.5
2
2.5
3
P
R
(dB)
R
ave
(bps/Hz)
ORSB, N=3, K=3, I1=I2=20dB
ORSB, N=3, K=3, I1=I2=10dB
ORSB, N=3, K=3, I1=I2=5dB
ORSB, N=3, K=3, I1=I2=0dB
I=0,5,10,20 (dB)
Figure: Effect of Interference Thresholds

Thank You!

Optimal Relay Selection and Beamforming in MIMO Cognitive Networks

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