Optimal Relay Selection and Beamforming in MIMO Cognitive Multi-Relay Networks

Problem Statement System Model Signal Model Optimal Relay Selection and Beamforming Simulation Results

Optimal Relay Selection and Beamformingin MIMO Cognitive Multi-Relay Networks

Reference: Li, Quanzhong, et al. "Optimal relay selection and beamforming in MIMO cognitivemulti-relay networks." Communications Letters, IEEE 17.6 (2013): 1188-1191.

Mohamed Seif1

1Wireless Intelligent Networks Center (WINC), Nile University, Egypt

May 20, 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 workproposes an optimal relay selection and beamformingscheme subject to transmit power constraints at the relaysand the interference power constraints at the primaryusers.




System Model

Pair of a SU are transmitting in thepresence of 2 PUs, each equippedwith one antenna

K cognitive relays, each relay isequipped 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 thesystem

During the first time slot, the SU-TXtransmits signals to the relays

At the second time slot, the k th

selected relay, multiplies thereceived signal by a linearbeamforming matrix and forwards itthe SU-RX

TX RX

UE UE


K


1


Figure: CRN model




Signal Model

The received signal at the SU-RX isexpressed as:

y = h†2k Fk(h1k x + nr) + z

where,

E[∣x ∣2] = Psnr ∼ CN(0, σ2

r I)z ∼ CN(0, σ2

d )

Fk ∈ CN×N

h1k ∈ CN×1

h2k ∈ C1×N

TX RX

UE UE


K


1


Figure: CRN model




Signal Model

The SNR at the SU-RX is expressedas:

SNR =Ps ∣h†

2k Fh1k ∣2

σ2r ∥h

†2k F∥+σ2

d

TX RX

UE UE


K


1


Figure: CRN model




Signal Model

The transmit power of the SU-TXsatisfies that:Ps ∣g1m∣

2≤ Im, m ∈ {1,2}

then,

Ps = min(Ps,minm

Im∣g1m ∣

2 )

where,g1m ∈ C1×1

TX RX

UE UE


K


1


Figure: CRN model




Signal Model

The transmit power of the k th relayis: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


K


1


Figure: CRN model




Optimal Relay Selection and Beamforming

The optimization problem of relay selection and beamformingfor a MIMO congnitive multi-relay network is formulated as:

Problem Formulation

arg maxk

maxF

12 log2(1 +

Ps ∣h†2k Fh1k ∣

2

σ2r ∥h

†2k F∥

2+σ2

d

)

s.t. Ps ∥Fh1k∥2+ σ2

r ∥F∥2≤ PR


2+ σ2

r ∥g†2mF∥

2≤ Im

k ∈ {1,2, . . . ,K}, m ∈ {1,2, . . . ,M}






Problem Formulation

arg maxk

maxF

12 log2(1 +


2

σ2r ∥h

†2k F∥

2+σ2

d

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶PrecoderDesign


r ∥F∥2≤ PR


2+ σ2

r ∥g†2mF∥

2≤ Im

k ∈ {1,2, . . . ,K}, m ∈ {1,2, . . . ,M}






Problem Formulation

arg maxk

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶RelaySelection

maxF

12 log2(1 +


2

σ2r ∥h

†2k F∥

2+σ2

d

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶PrecoderDesign


r ∥F∥2≤ PR


2+ σ2

r ∥g†2mF∥

2≤ Im

k ∈ {1,2, . . . ,K}, m ∈ {1,2, . . . ,M}





Beamforming Optimization Problem

P1: maxf

f †(Pshh†

)ff †(σ2

r H2H†2)f+σ

2d

s.t. f †(PsH1H†1 + σ

2r I)f ≤ PR

f †(PsG1mG†1m + σ

2r G2mG†

2m)f ≤ Imm ∈ {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: maxf

f †(Pshh†

)ff †(σ2

r H2H†2)f+σ

2d

s.t. f †(PsH1H†1 + σ

2r I)f ≤ PR

f †(PsG1mG†1m + σ

2r G2mG†

2m)f ≤ Imm ∈ {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: maxW≽0

tr(A1W)tr(A2W)+σ2

d

s.t. tr(A3W ) ≤ PRtr(BmW ) ≤ Im

m ∈ {1,2, . . . ,M}

where,A1 = Pshh†

A2 = σ2r G2mG†

2m

A3 = PsH1H†1 + σ

2r I

W = ff †






P2: maxW≽0

tr(A1W)tr(A2W)+σ2

d


m ∈ {1,2, . . . ,M}

rank (W )=1, then rank of W has been relaxed

where,A1 = Pshh†

A2 = σ2r G2mG†

2m

A3 = PsH1H†1 + σ

2r I

W = ff †






P2: maxW≽0

tr(A1W)tr(A2W)+σ2

d


m ∈ {1,2, . . . ,M}rank (W )=1, then rank of W has been relaxed

where,A1 = Pshh†

A2 = σ2r G2mG†

2m

A3 = PsH1H†1 + σ

2r I

W = ff †





Beamforming Optimization Problem(Charnes-Cooper transformation)

P3: maxS≽0,ν≥0

tr(A1S)

s.t. tr(A1S) + σ2dν = 1

tr(A3S) ≤ νPRtr(BmS) ≤ νIm, m ∈ {1,2, . . . ,M}

where,W =

Sν

tr(A2W) + σ2d =

1ν






P4: maxW≽0

tr(A3W )

s.t. tr(A1W)tr(A2W)+σ2

d≥ γ −∆γ

tr(BmW ) ≤ Im, m ∈ {1,2, . . . ,M}

where,γ = max

S≽0,ν≥0tr(A1S) (P3)

0 ≤∆γ < γ






Solution of P4 is tight to P2 by (1 − ∆γγ ) (Proof Hint)

Solution of P4 has rank one (Proof Later ,)

then,f =√λiUiU

†i , λi ≠ 0

where, Ui ⇐ W = U∆U†

Relay Selection

arg maxk

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶RelaySelection

maxF

12 log2(1 +


2

σ2r ∥h

†2k F∥

2+σ2

d

)






Solution of P4 is tight to P2 by (1 − ∆γγ ) (Proof Hint)

Solution of P4 has rank one (Proof Later ,)

then,f =√λiUiU

†i , λi ≠ 0

where, Ui ⇐ W = U∆U†

Relay Selection

arg maxk

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶RelaySelection

maxF

12 log2(1 +


2

σ2r ∥h

†2k F∥

2+σ2

d

)




Simulation Setup

Symbol Description RealizationM Number of PUs 2K Number of relays ∼

N Number of antennas per relay ∼

PR Transmitted power at the relay ∼

σ2r Noise power at the relay per anetenna Normalized

P Number of iterations 50σ2

d Noise power at SU-RX Normalized

Table: Parameters of Simulation




Simulation Results

0 5 10 15 200.5

1

1.5

2

2.5

3

PR

(dB)

Rav

e(bps

/Hz)

ORSB, N=3, K=3

Figure: Average capacity versus the maximum allowable transmitpower of the relay




Simulation Results

0 5 10 15 200.5

1

1.5

2

2.5

3

PR

(dB)

Rav

e(bps

/Hz)

ORSB, N=3, K=3ORSB, N=3, K=2ORSB, N=3, K=1

k=1,2,3

Figure: Effect of number of relays




Simulation Results

0 5 10 15 200.5

1

1.5

2

2.5

3

PR

(dB)

Rav

e(bps

/Hz)

ORSB, N=4, K=3ORSB, N=5, K=3ORSB, N=6, K=3ORSB, N=3, K=3ORSB, N=2, k=3

N=2,3,4,5,6

Figure: Effect of number of antennas




Simulation Results

0 5 10 15 200.5

1

1.5

2

2.5

3

PR

(dB)

Rav

e(bps

/Hz)

ORSB, N=3, K=3, I1=I2=20dBORSB, N=3, K=3, I1=I2=10dBORSB, N=3, K=3, I1=I2=5dBORSB, N=3, K=3, I1=I2=0dB

I=0,5,10,20 (dB)

Figure: Effect of Interference Thresholds




Thank You!



Engineering

Optimal Relay Selection and Beamforming in MIMO Cognitive Multi-Relay Networks