2010 5th International Symposium on Telecommunications (IST'2010)
Power Minimization in Uni-directional Relay Networks with Cognitive Radio Capabilities Ardalan Alizadeh and Seyed Mohammad-Sajad Sadough Cognitive Telecommunication Research Group Department of Electrical and Computer Engineering Shahid Beheshti University G.C. Evin 1983963113, Tehran, Iran Email:
[email protected],
[email protected]
Abstract—In this work, we consider a uni-directional (one way) relay network with a pair of primary transmitter and receiver which are relay-assisted by multiple cognitive radio (CR) terminals. A CR base station is deployed to sense the absence/presence of the primary user based on the received signal-to-noise ratio (SNR) at the fusion center. The primary network utilizes a two-step one-way amplify-and-forward (AF) relaying scheme and uses the CRs as its relay nodes. In the first step of AF relaying, the primary transmitter sends its signal to the CRs/relays. A distributed beamforming is then performed in the second step of relaying. In this phase, the CR base station sets the beamforming weights of CR terminals to satisfy the SNR requirements at the primary transmitter and at the CR base station. We aim at minimizing the power dissipated in the cognitive (relay) nodes together with ensuring an accurate spectrum sensing process. This optimization problem is efficiently solved by using semidefinite relaxation (SDR) techniques. Simulation results are provided to compare the relay transmit power in our investigated network with that in a conventional AF relay network.
receiver that form the primary network. In the first step, the transmitter sends its signal towards the relays. Each relay processes its received signal and multiplies it by a derived beamforming weight in the second step. Recently, distributed beamforming has been proposed for relaying schemes to satisfy some system requirements for the network. For instance, different optimal beamforming problems have been presented in [6] to minimize the transmit power and to maximize the received signal-to-noise ratio (SNR). In this paper, we complete our previous work presented in [7] for uni-directional primary relay networks where the main idea is to enable a CR capability for a conventional relay network. In this model, the CR base station uses the combination of signals from CR/relays (during the second step) for its sensing purpose. We derive the optimal beamforming weights so as to minimize the required power at the relay network subject to SNR constraints on the primary receiver and on the cognitive base station. Our proposed scheme provides two desired features in each CR network: i) the SNR constraint at the CR network ensures an accurate spectrum sensing, and ii) the power dissipated at the CR network is minimized. We show that the investigated beamforming optimization problem can be formulated as a non-convex quadratically constrained quadratic program (QCQP) [8]. Then, the semidefinite relaxation (SDR) is deployed to change the problem into a convex form and obtain an efficient solution for the so-obtained convex problem. Then, the problem is solved by semidefinite programming (SDP).
Cognitive radio, one-way relaying, convex optimization, distributed beamforming, relay networks, semidefinite relaxation.
I.
INTRODUCTION
Cognitive radio (CR) [1] is a new design paradigm to eliminate the problem of scarce spectrum resources in wireless communications. CR network allows a secondary (unlicensed) user to utilize a frequency band already allocated to some primary users (PU). To prevent harmful interference at the primary receiver, the CR network senses the licensed spectrum constantly to detect the absence/presence of the primary transmitter. Recently, the hidden terminal problem has become a great challenge for implementing CRs. This happens when the CR node is shadowed or is in severe multipath fading [2]. Cooperation among different cognitive users is an efficient solution to overcome the hidden terminal limitation. The basic idea behind cooperative communications is presented in many references [3]-[5]. In these networks, each terminal works as a relay and collaborates with other users to increase the sensing/detection reliability. In a two-step one-way amplifyand-forward (AF) scheme, there is a pair of transmitter and
II.
A. Cognitive Relay Network As shown in Fig. 1, our network is composed of a primary transmitter and receiver, nr decentralized CR terminals (which are also used as relay nodes, similar to [7]). We assume that there is no direct link between the primary transmitter and receiver. In this scenario, the CR network uses the licensed spectrum opportunistically when the primary network is off. We consider a two-step AF relaying scheme for the primary
This work is supported by the Iranian Education & Research Institute for ICT (ERICT) under the grant number 8974/500.
978-1-4244-8185-9/10/$26.00 ©2010 IEEE
CONSIDERED SCENARIO AND SYSTEM MODEL
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transmission. During the first step, the primary transmitter transmits its signal to the CR/relays. In the second step, each CR multiplies its received signal with a complex weighting vector and broadcasts it to the receiver. Obviously, this signal is also received at CR base station. During the first step, the received signal vector x of size nr × 1 at the CR terminals can be written as:
x = P1 f1s + n1
(1) where P1 is the transmitted power of primary transmitter, s is the information symbols transmitted by primary transmitter, n1 is the nr × 1 complex noise vector at the CR users, and T
f1 = ⎡⎣f 11 f 21 ... f n r 1 ⎤⎦
(2) Figure 1. Architecture of the considered cognitive one-way relay network.
is the vector of the complex channel coefficients between the transmitter and the CR terminals, and (.)T stands for the transpose operator. During the second step, the i -th CR terminal ( i = 1, 2,..., nr ) multiplies its received signal by a complex weight wi* and broadcasts it in the network. The nr × 1 complex vector t denotes the transmitted signal as:
t = Wx
(3)
W = diag { ⎡⎢ w1* w2* ... wn*r ⎤⎥ } , ⎣ ⎦
(4)
is the vector of complex channel coefficients between the CR users and the CR base station, and n3 is the received noise at the CR base station. Similar to (6), we can rewrite z as: z =
and where diag{a} is a diagonal matrix with diagonal elements equal to vector a . B. Received Signals at the Primary Network The received complex signal, denoted by y , can be written as:
(
)
P1 f1s + n1 + n2
⎧⎪ SNR BS ≥ γ2 , ⎪⎨ ⎪⎪ SNR BS < γ2 , ⎩
(5)
where n2 is the additive noise corresponding to the link between the relay and the primary receiver. Note that since aT diag ( b ) = bT diag ( a ) , we can rewrite (5) as: y =
(9)
where F3 = diag (f3 ) . We assume that the noise process is zeromean and spatially white with variance σ 2 which means that E [n1n1H ] = σ 2 I and E [| n 2 |2 ] = E [| n3 |2 ] = σ 2 . Since P1 is defined as the transmit power of primary transmitter, this implies that E [| s |2 ] = 1 . As shown in Fig. 2, the energy summation of the received signal z during the sensing period is used as a metric for deciding about the presence/absence of the primary signal [9]. It can be shown that the probability of detection increases with the increase of the SNR at the CR base station. Therefore, we can express the sensing hypothesis as:
where
y = f2T Wx + n2 = f2T W
P1 wH F3 f1s + wH F3 n1 + n 3 ,
H1 H0
(10)
where SNR BS is the received SNR at the CR base station and γ 2 is a predefined SNR threshold ensuring a target detection probability.
P1 wH F2 f1s + wH F2 n1 + n2
(6) where Fk = diag ( fk ) for k = 1, 2 , w = diag{W } , and (.)H denotes Hermitian transpose. Here, diag{A} is a vector formed by the diagonal elements of the square matrix A . The weight vector w is set after solving our optimization problem.
III.
H
OPTIMAL DISTRIBUTED BEAMFORMING FOR RELAY POWER MINIMIZATION
The optimization problem that minimizes the relaying power under the constraints that the received SNR at the primary receiver and at the CR base station are kept above a certain threshold is stated as follows:
C. Spectrum Sensing Based on Received SNR The signal z received at the CR base station can be written
minPr subject to: SNR PR ≥ γ1 , SNR BS ≥ γ2 (11)
as:
w
z =
f3T Wx
+ n3 =
f3T W
(
)
P1 f1s + n1 + n3
(7)
where Pr is the transmitted power from the CR/relay terminals. By assuming that the information symbol s and the relay noise n1 are independent variables/vectors and complex channel coefficients are constant during the relaying scheme, the total transmitted power of the relay terminals can be expressed as:
where T f3 = ⎡⎢ f13 f23 ... fn 3 ⎤⎥ r ⎣ ⎦
(8)
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||2 || w
min w
subject to: −1/2H
c1 :
H D P1 w 2
2
Figure 2. Energy detection metric evaluation by data fusion at the CR base station.
Pr = E ⎡⎢ tH t ⎤⎥ ⎣ ⎦ P1 f1H s + n1H WH ⎤⎥ = E ⎡⎢ W P1 f1s + n1 ⎣ ⎦ ⎡ 2⎤ H H H ⎤ H ⎡ = P1Wf1f1 W E ⎢ s ⎥ + WE ⎢ n1n1 ⎥ W ⎣ ⎦ ⎣ ⎦ = wH ( P1F1F1H + σ 2 I ) w.
(
)(
)
SNR PR =
σ 2 + σ 2 wH D2 w
(12)
D2 D w H −1/2 D P1w h2 h2 D w −1/2 2 2 H −1/2H D σ +σ w D3 D w
and
P1 wH h 2 h2H w
σ 2 + σ 2 wH D3 w
F2 F2H
(13)
.
(14)
F3 F3H
w
IV.
c2 :
−1/2H
D2 D−1/2
−1/2H
−1/2 h 2 hH − γ2 D 2 D
)
D3 D−1/2
)
CONVEX RELAXATION OF THE OPTIMIZATION PROBLEM
W
subject to:
P1wH h1h1H w ≥ γ1, and σ 2 + σ 2 wH D2 w P1 w h2 h H 2 w 2 σ + σ 2 wH D3 w
(
−1/2H
h1h1H D−1/2 − γ1D
In this section, the optimal solution of non-convex relay power minimization problem is obtained by a direct convex relaxation of (17) using semidefinite programming [8]. By using xH Mx = Tr(M(xxH )) , the optimization problem (17) becomes: ) min Tr(W
F2 f1H
subject to:
H
≥ γ2 .
The optimization problem is a quadratically constrained quadratic program (QCQP). Since the constraints may be nonconvex, the QCQP is NP-hard. To overcome this problem, we propose to use a relaxation technique to efficiently solve this optimization problem by convex programming.
wH (P1D1 + σ 2 I)w
c1 :
−1/2H
are positive semidefinite. Then the constraints are generally non-convex.
, D3 = , h1 = and where D2 = h2 = F3 f1H . Then, the minimization of the relay transmitted power can be rewritten as: min
−1/2H
B = P1D
and SNR BS =
(
A = P1D
.
≥ γ1, and (17)
−1/2
where D = P1D1 + σ 2 I. The objective function is a convex function in quadratic form but in general the constraints may be non-convex. Given the nature of the wireless channels, and depending on the SNR thresholds and transmit power P1 , it is not guaranteed that the matrices
In the optimization problem (11), SNR PR , SNRBS , γ1 and γ2 are the SNR at the primary receiver and CR base station, and minimum required SNR of primary receiver and CR base station, respectively. Similar to (12), the received SNR in the second transmission step at primary receiver and CR base station can be expressed as: P1 wH h1h1H w
−1/2H
D σ +σ w H
c2 :
H
h1h1H D−1/2 w
) ≥ γ1, Tr((P1H1′ − γ1D2′ )W
(15)
) ≥ γ2 , Tr((P1H2′ − γ2 D3′ )W = ww ; 0 and Rank(W ) = 1, H , W W
≥ γ2 ,
−1/2H
where D1 = F1F1H . To solve the problem (15), let us write as: w = D1/2 w w
(18)
where H1′ = D −1/2H
D2′ = D
(16)
−1/2H
h1h1H D−1/2 , H2′ = D −1/2H
D2 D−1/2 and D3′ = D
−1/2 h2 hH , 2 D
D3 D−1/2 . P1 is the
relay transmit power sets such that the optimization problem is feasible. The problem is relaxed into a convex SDP by relaxing
where wH w = 1 . Then the optimization problem can be written as:
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Figure 3. Average minimum relay transmit power Pr versus σ 2f3 for different values of σ 2f1 = σ 2f2 for the relay transmit power minimization method with an initial transmit power equal to 12 dBW for γ1 = γ2 = 10dB .
Figure 4. Average relay transmit power versus the minimum required SNR at the primary receiver in dB for the conventional and the cognitive relay networks and different QoS requirement at the CR base station, nr = 10 .
the rank-one constraint. It can be solved efficiently by convex optimization programs, such as CVX [10]. Note that the optimization problem in (17) cannot always guarantee that the has a optimal solution has a rank equal to one. However, if W rank greater than one, the rank reduction program in [11] can be used efficiently. V.
SIMULATION RESULTS
Here, we provide some numerical results to compare our considered mixed network with a conventional relay network, i.e., a network without any cognitive capability. Throughout the simulations, we consider three channel vectors f1 , f2 and f3 with elements distributed as complex zero-mean Gaussian random variables with variances σ f , σ f and σ f . We assume that the noise power of the three channels (denoted by σ 2 ) is equal to one. We have used a number of 100 random channel realizations for averaging the solution of our optimization problem. Figure 3 shows the average minimum relay transmit power Pr versus σ 2 for different values of σ 2f = σ f2 and for the f3 relay transmit power minimization with a transmitted power of 12 dBW for γ1 = γ2 = 10 dB . As observed, when the quality of the transmitter-CR, CR-receiver and CR-base station links are improved, the minimum relay transmit power is decreased. Figure 4 illustrates the effect of sensing constraint on power consumption of relays. Note that, when the required SNR threshold at the CR base station is not very high (5 dB for instance), the average CR transmit power is close to the average relay transmit power in a conventional network (i.e., without cognitive capabilities). Otherwise, we observe that the 1
2
3
1
Figure 5. Average relay transmit power versus an equal SNR threshold at the primary receiver and CR base station for different number of CR/Relays.
2
CR network requires more power in order to satisfy the desired constraint and to provide the target SNR. The average relay power consumption for different number of CR/relays is presented in Fig. 5. It is observed that when the number of relays increases, the total relaying power is reduced. For instance, the relay power consumption for 10 CR terminals is approximately equal to that required for 5 relays in a conventional relay network.
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VI.
[4]
——, “User cooperation diversity– part II: Implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, pp. 1939 – 1948, November 2003. [5] K. B. Letaief and W. Zhang, “Cooperative communications for cognitive radio networks,” Invited paper, Proceedings of IEEE, vol. 97, no. 5, May 2009. [6] V. Havary-Nassab, S. Shahbazpanahi, A. Grami, and Z.-Q. Luo, “Distributed beamforming for relay networks based on second-order statistics of the channel state information,” IEEE Transactions on Signal Processing, vol. 56, no. 9, pp. 4306 – 4316, September 2008. [7] A. Alizadeh, S. M. S. Sadough, and N. T. Khajavi, “Optimal Beamforming in cognitive two-way relay netwroks,” IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC, 2010, pp. 2329-2333. [8] S. Boyde and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [9] J. Ma, G. Zhao, and Y. Li, “Soft combination and detection for cooperative spectrum sensing in cognitive radio networks,” IEEE Transactions on Wireless Communications, vol. 7, no. 11, Nov 2008. [10] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version 1.21. http://cvxr.com/cvx, October 2010. [11] Y. Huang and S. Zhang, “New results on hermitian matrix rank-one decomposition,” http://www.se.cuhk.edu.hk/ywhuang/dcmp/paper, Aug 2009.
CONCLUSION
This paper formulated the problem of power minimization in a one-way relay network with CR capabilities. Given a set of minimum required SNR at the receivers (constraints), an optimal beamforming weight vector was obtained to minimize the transmitted power by CRs/relays. The constraints are chosen in order to ensure accurate spectrum sensing at the cognitive base station. The optimization problem was transformed to a non-convex QCQP which simply can be solved by SDP relaxation. Simulation results demonstrated that the minimum relay transmit power is decreased when the quality of the channels are improved. REFERENCES [1]
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S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Area in Communications, vol. 23, pp. 201–220, 2005. F. Digham, M. S. Alouini, and M. K. Simon, “on the energy detection of unknown signals over fading channels,” IEEE International Conference on Communications”, vol. 5, pp. 3575–3579 , 2003. A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity– part I: System description,” IEEE Transactions on Communications, vol. 51, pp. 1927 – 1938, November 2003.
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