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Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction

Collaborative Relay Beamforming for Secrecy

Channel Model Main Results

Channel Model Beamforming under Total Power Constraint

Junwei Zhang and Mustafa Cenk Gursoy

High-SNR Regime Low-SNR Regime

Beamforming under Individual Power Constraint Semidefinite Relaxation (SDR) Approach Second-order Cone Program (SOCP) Approach Suboptimal Design

Numerical Results Conclusion 1 / 21

University of Nebraska-Lincoln

May 2010

Outline Collaborative Relay Beamforming for Secrecy

1

Channel Model

Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results

Channel Model

Main Results 2

Channel Model

3

Beamforming under Total Power Constraint High-SNR Regime

Beamforming under Total Power Constraint High-SNR Regime Low-SNR Regime

Low-SNR Regime 4



5

Numerical Results

6

Conclusion


Introduction

Channel Model Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results

Channel Model Beamforming under Total Power Constraint High-SNR Regime Low-SNR Regime



We consider a secrecy communication channel with a source S, a destination D, an eavesdropper E, and M relays {Rm }M m=1 Related Studies Physical layer security has been studied in Gaussian channel, fading channel, Multi-anterna and Multi-user channel Collaborative-relay beamforming in perfect CSI and imperfect CSI. Secrecy relay beamfoming under total relay power constraint.

Main Results Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




DF Relaying under Totoal Power Constraint We obtain a closed-form solution for the achievable secrecy rate. We investigate the beamforming structure in the high- and low-SNR regimes.

DF Relaying under individual Power Constraint There is no analytical solutions available. We use the semidefinite relaxation (SDR) approach to approximate the problem as a convex semidefinite programming (SDP) problem. We also provide an alternative method by formatting the original optimization problem as a convex second-order cone programming (SOCP) problem.

Channel Model Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results

Channel Model Beamforming under Total Power Constraint

The received signal at Rm yr,m = gm xs + ηm Relay Rm first decodes the message xs and normalizes it as p xs0 = xs / Ps The Relay output xr = wm xs0 ,

High-SNR Regime Low-SNR Regime



The received signals at the destination D and eavesdropper E are yd =

M X

hm wm xs0 + n0 = h† wxs0 + n0 ,

m=1

ye =

M X m=1

zm wm xs0 + n1 = z† wxs0 + n1

and

Secrecy Rate Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy

The secrecy rate is now given by

Introduction Channel Model Main Results




Rs = I (xs ; yd ) − I (xs ; ye ) = log(1 + Γd ) − log(1 + Γe ) ! PM N0 + | m=1 hm wm |2 = log PM N0 + | m=1 zm wm |2 Our Goal is the joint optimization of {wm } with the aid perfect CSI and hence identify the optimum collaborative relay beamforming (CRB) direction that maximizes the secrecy rate .

Analytical Solution Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




We consider a total relay power constraint in the following form: ||w||2 = w† w ≤ PT . The optimization problem can now be formulated as follows: ! PM N0 + | m=1 hm wm |2 Rs (h, z, PT ) = max log PM w† w≤PT N0 + | m=1 zm wm |2 PM N0 + | m=1 hm wm |2 = log max PM 2 w† w≤PT N0 + | m=1 zm wm | = log max

w† w≤PT

= log max

w† w≤PT

N0 w† ( P I + hh† )w T N0 w† ( P I + zz† )w T

w† (N0 I + PT hh† )w w† (N0 I + PT zz† )w

= log λmax (N0 I + PT hh† , N0 I + PT zz† ) wopt = ςu where u is the eigenvector that corresponds to λmax (N0 I + PT hh† , N0 I + PT zz† ) .

High-SNR Analysis Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




we can see that ˜ 2) lim (Rs − log(PT )) = log(max |h† ψ|

PT →∞

˜ ψ

where ψ˜ is a unit vector on the null space of z† . Based on this observation, we can choose at high SNRs the beamforming vectors to lie in the null spaces of the eavesdropper’s channel vector, that is PM | m=1 zm wm |2 = z† w = 0. In this case, the eavesdropper can not receive any data from the relays, and secrecy is automatically guarantied. No secrecy coding is needed at the relays. Now, for large PT , the optimization problem becomes  2  M X hm wm  s.t z† w = 0 max log  w† w≤PT m=1

˜ 2 ). = log(PT ) + log(max |h† ψ| ˜ ψ

(1)

Hence, null space beamforming is optimal at high SNR. Furthermore, the optimal null space beamforming vector can be obtained explicitly.

Optimal Null Space Beamforming Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




⊥ we can write w = H⊥ z v, where Hz denotes the projection matrix † onto the null space of z . The power constraint † ⊥ † w † w = v † H⊥ z Hz v = v v ≤ PT . Then, the optimization problem can be recast as  2  M X max log 1 + hm wm  w† w≤PT m=1 † † = log 1 + max (w hh w) w† w≤PT † ⊥† † ⊥ = log 1 + max (v Hz hh Hz v) v† v≤PT † † ⊥ hh H ) = log 1 + PT λmax (H⊥ z z † ⊥ ⊥† = log 1 + PT h Hz Hz h .

Therefore, the optimum null space beamforming weights vector w is †

wopt,n = H⊥ v = ς1 H⊥ H⊥ h

Low-SNR Analysis Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




In the low SNR regime, in which both Ps , PT → 0, we can see that lim

Ps →0

Rs = λmax (hh† − zz† ) PT

Thus, in the low SNR regime, the direction of the optimal beamforming vector approaches that of the eigenvector that corresponds to the largest eigenvalue of hh† − zz† .

Problem Statement Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




We now impose |wm |2 ≤ pm ∀m or equivalently |w|2 ≤ p where | · |2 denotes the element-wise norm-square operation and p is a column vector that contains the components {pm }. In what follows, the problem of interest will be the maximization of the term inside log function of Rs under individual power constraints: max 2

|w| ≤p

N0 + |

PM

N0 + |

PM

= max 2

|w| ≤p

m=1

hm wm |2

2 m=1 zm wm | † †

N0 + w hh w N0 + w† zz† w

SDR Approach Collaborative Relay Beamforming for Secrecy

Using the definition X , ww† , we can rewrite the optimization problem in as


N0 + tr(hh† X) X N0 + tr(zz† X) s.t diag(X) ≤ p rank X = 1, and

max





X0

or equivalently as max X,t

t

s.t tr(X(hh† − tzz† )) ≥ N0 (t − 1), diag(X) ≤ p, rank X = 1, and X 0 where tr(·) represent the trace of a matrix and X 0 means that X is a symmetric positive semi-definite matrix. The optimization problem in is not convex and may not be easily solved.

SDR Approach Collaborative Relay Beamforming for Secrecy

Let us now ignore the rank constraint in That is, using a semidefinite relaxation (SDR), we aim to solve the following optimization problem:


max X,t

s.t tr(X(hh† − tzz† )) ≥ N0 (t − 1), and diag(X) ≤ p, and X 0.



Beamforming under Individual Power Constraint

Note that the optimization problem is quasiconvex. In fact, for any value of t, the feasible set in is convex. Let tmax be the maximum value of t obtained by solving the optimization problem . We can use a simple bisection algorithm to solve the quasiconvex optimization problem by solving a convex feasibility problem find

Semidefinite Relaxation (SDR) Approach Second-order Cone Program (SOCP) Approach Suboptimal Design


t

X

such that tr(X(hh† − tzz† )) ≥ N0 (t − 1), and diag(X) ≤ p, and X 0 at each step.

SDR Approach Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




Once the maximum feasible value for tmax is obtained, one can solve min X

tr(X)

s.t tr(X(hh† − tmax zz† )) ≥ N0 (tmax − 1), and diag(X) ≤ p, and X 0 to get the solution Xopt . To solve the convex feasibility problem, one can use the well-studied interior-point based methods as well. We use the well-developed interior point method based package SeDuMi , which produces a feasibility certificate if the problem is feasible, and its popular interface Yalmip .

SOCP Approach Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




We can reformulate the problem as a second order cone problem. This provides us with another way of solving the optimization. The optimization problem is equivalent to max w,t

s.t

t

N0 + |h† w|2 ≥t N0 + |z† w|2

and

|w|2 ≤ p.

Observe that an arbitrary phase rotation can be added to the beamforming vector without affecting the constraint. Thus, h† w can be chosen to be real without loss of generality.

! 2 †

1 † 2

q z w

|h w| ≥

. 1 1 − N

t 0 t

SOCP Approach Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




Since h† w can be assumed to be real, we may take the square root of the above equation. The constraint becomes a second-order cone constraint, which is convex. The optimization problem now becomes max t w,t

r

1 †

h w≥ s.t

t

† q z w 1 − 1t N0

!

and

|w|2 ≤ p

The optimal solution of can be obtained by repeatedly checking the feasibility and using a bisection search over t with the aid of interior point methods for second order cone program. Once the maximum feasible value tmax is obtained, we can then solve the following second order cone problem (SOCP) to obtain optimum beamforming vector: min w

r s.t

||w||2

 

z† w

r †

  h w≥ 1

1 − N tmax 0

tmax 1

and

|w|2 ≤ p

Suboptimal Approach Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




As shown above, the design of the beamformer under individual relay power constraints requires an iterative procedure in which, at each step, a convex feasibility problem is solved. We now propose a suboptimal beamforming vector that can be obtained without significant computational complexity. We choose aP simplified beamformer as wsim = θwopt , ||wopt ||2 = PT = pi , and we choose θ=

1 √ |wopt,k |/ pk

where wopt,k and pk are the kth entries of wopt and p respectively, and we choose k as k = arg max

1≤i≤M

|wopt,i |2 pi

Eavesdropper Has a Weaker Channel Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy

12 11

Introduction




10 secrecy rate (bits/symbol)


9 8 7 6 Total Power constraint individual power constraint, SDR approach individual power constraint, SOCP approch individual power constraint, suboptimal

5 4

0

20

40

60

80

100

P

T

Figure: Second-hop secrecy rate vs. the total relay transmit power PT for different cases. Eavesdropper has a weaker channel.

Eavesdropper Has a Stronger Channel Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy

10 9

Introduction




8 secrecy rate (bits/symbol)


7 6 5 4

Total Power constraint individual power constraint, SDR approach individual power constraint, SOCP approch individual power constraint, suboptimal

3 2

0

20

40

60

80

100

P

T

Figure: Second-hop secrecy rate vs. the total relay transmit power PT for different cases. Eavesdropper has a stronger channel.

NO. of Relay Changes Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy

12.5

12 Channel Model Main Results




secrecy rate (bits/symbol)

Introduction 11.5

11

10.5

10 Total Power constraint individual power constraint, SDR approach individual power constraint, SOCP approch individual power constraint, suboptimal

9.5

9

2

4

6

8

10 12 NO. of Relays

14

16

18

20

Figure: Second-hop secrecy rate vs. number of relays for different cases.

Conclusion Collaborative Relay Beamforming for Secrecy Junwei Zhang and Mustafa Cenk Gursoy Introduction Channel Model Main Results




In DF, under total power constraints, we analytically determine the beamforming structure in the high- and low-SNR regimes. In DF, under individual power constraints, not having analytical solutions available, we provide an optimization framework to obtain the optimal beamforming that maximizes the secrecy rate. We use the semidefinite relaxation (SDR) approach to approximate the problem as a convex semidefinite programming (SDP) problem which can be solved efficiently. We also provide an alternative method by formatting the original optimization problem as a convex second-order cone programming (SOCP) problem that can be efficiently solved by interior point methods. Also, we describe a simplified suboptimal beamformer design under individual power constraints