Optimal Power Allocation for Relay Assisted Cognitive ... - IEEE Xplore

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Telecommunications Field of Study, School of Engineering and Technology,. Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand.
Optimal Power Allocation for Relay Assisted Cognitive Radio Networks L.K. Saliya Jayasinghe∗ , Nandana Rajatheva† Telecommunications Field of Study, School of Engineering and Technology, Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand [email protected]∗ , [email protected]

Abstract—In this paper, we study the optimal power allocation of wireless relay nodes which are used in the secondary user (SU) communication of a cognitive radio (CR) network. We consider the behavior of transmitting powers of SUs where those powers are limited to the tolerable interference as seen by the primary user (PU) communication. To improve the performance of the secondary communication based on minimizing the outage probability, we re-formulate the power allocation problem with a new set of constraints. These are obtained by considering the co-channel interference generated by the SU communication to the PU communication. The power allocation problem is solved for both regenerative and non regenerative relay models under Rayleigh fading conditions. SU communication with N number of relays is discussed and compared. The outage probability of SU communication is limited by the interference power threshold (IPT) constraints of PUs and is affected significantly by the IPT levels of PUs.

I. I NTRODUCTION Cognitive radio (CR) is a promising technology that enables solutions for complicated problems in wireless communication which are believed to be infeasible in the past. Demand for data rates gradually increase due to the rapid development of services needed which makes the efficient use of available spectrum, a sine qua non, in wireless system implementations. Recent surveys from the Federal Communications Commission (FCC) indicate that in about 90 percent of the time, many licensed frequency bands remain idle [1]. The concept of CR introduced as a method to improve the spectrum utilization by allowing a secondary user (SU) to utilize a licensed band when the primary user (PU) is absent. SU transmission is possible with the same spectrum via different types of strategies. A common approach is to dynamically sense a frequency hole vacated by a PU and use that frequency slot for communication [2]. Additionally, interesting techniques are introduced to use the same frequency band by controlling the power levels of the SU transmissions to limit the interferences to PU communication [3]. In some cases PUs utilize different licensed spectrum bands in multiple geographic areas and hence the SU communication between those geographical areas [3] is made possible by maintaining a certain level of interference to the PUs. However, the required SU transmitting power can be much higher in direct communication and thus causing severe interference to the PUs. The circumstances where the SUs limit their power and use the same spectrum as the PUs simultaneously are discussed in [3] where the use of a cooperative relay scheme

between the SUs to reduce SU transmitting power is proven to be an effective way of ensuring lower interference to the PUs [4]. The relay based transmission is introduced to the secondary communication in [4] and the positive improvements of that method also are discussed in their research work. It is found that cooperative relay communication enhances the system performance, reduces unwanted interference and saves power at the source [5]. The performance of cooperative relay networks is discussed with different parameters and channel environments in [6, 7] where improvements in the overall transmission rate and the diversity are obtained. The power allocation schemes for relays to achieve better performance are discussed for the regenerative, non regenerative, etc. cases in [8–10]. In cognitive radio with the relay based secondary communication, it is essential to improve the performance of secondary system subject to pre-determined interference levels to PUs.We should consider optimum power allocation to achieve this objective. To the best of authors knowledge the power allocations problem where a relay is used in the secondary transmission is not previously considered in the open literature. Therefore, we study this issue in Rayleigh fading channels. To improve the secondary communication, the power has to be distributed in an optimal way among the source and relay nodes to minimize the outage probability while maintaining a reliable primary link. We formulate an optimization problem for power allocation in the relays which are used in the SU communication where we also introduce additional IPT constraints for the SU which guarantee predetermined interference levels at the PUs. Both regenerative and non regenerative relay models are considered. The rest of the paper is organized as follows. In section II, we describe the system and the channel model. In addition to that, the basic formulation of optimization problem is given there. In section III, we study the case of a single relay node for both regenerative and non regenerative cases. N relay nodes scheme for the secondary communication is discussed in section IV and the Numerical results are presented in section V. Finally, conclusions are given in section VI.

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II. SYSTEM MODEL AND PROBLEM FORMULATION In the problem formulation, we consider a communication system with N relay nodes and it is shown in Fig.1. rN

m

..

d

..

I a-b Ua

I b-a

..

I a-b

s

r1

r3

I b-a

shadowing, noise power, and similar parameters [11], and pn is the transmitted power of the nth relay. we select the outage probability of secondary communication as our objective function and formulate an optimization problem by considering total power constraint and IPT constraints.

n

subject to

Ub

r2

min Pout N ⎧ p s + ⎪ n=1 pn = PT ⎪ ⎪ ⎨ pn ≤ Pmax : n = 1, 2, ..., N ps ≤ Pmax ⎪ ⎪ s hsn ≤ Tb ⎪ p ⎩  N n=1 pn hrn m ≤ Ta

b

a Secondary transmission using Cooperative scheme Intereferences for PUs

Fig. 1. N relay nodes used in secondary communication. Source node and relay nodes are transmit signal with different frequency sets.

Ua and Ub are PUs and their coverage areas are shown as a and b. s and d are two SUs which use relays r1 to rN for their communication. Frequency sets are selected as shown in Fig.1 to satisfy the condition that s or d does not interfere with their nearest PU. : Available spectrum for s Ia : Available spectrum for d Ib Ia−b = Ia - Ib : Idle channel set in a, but busy in b s to r1 transmission uses Ia−b frequency set and r1 to r2 ,.., rN −1 to rN , rN to d transmissions use Ib−a frequency set. In addition to this, we assume that the SUs can sense the spectrum and distribute their channel state information (CSI) to each other. SU source and destination nodes have the capability to sense and locate the most affected PUs in their PU coverage areas. m and n are primary users of Ua and Ub . n is the most affected user at Ub due to the transmission of source s and m be the most affected user in Ua . IPT level is the highest that can be tolerated by a PU. The transmit power of source s is denoted as ps and that of relay node rn by pn . The link gain between node i and node j is hij . The IPT levels on m and n are Ta and Tb . Total power allocated for source and relays is denoted as PT . Interferences from nodes which are used in the secondary communication should fulfill the following requirements to have stable PU communication. ps hsn ≤ Tb N 

pn h r n m ≤ T a

(1) (2)

n=1

The signal envelopes of the links are assumed to have Rayleigh distributions. Therefore, the Signal-to-noise ratio (SNR) of the channel is exponentially distributed. In general, the average SNR of the links γ n can be written as γ n = Gn pn . Gn contains parameters such as the antenna gains, path loss,

Outage probabilities that we obtain for this are convex functions and our constraints are linear. The convexity of the feasibility set also satisfied due to linearity. Therefore, we can solve this as a convex optimization problem and obtain a unique solution. III. SINGLE RELAY TRANSMISSION Here, we consider the power allocation of the source and the relay node for both regenerative and non regenerative relay systems. A. Regenerative Relay scheme Outage probability is given by the following equation [12], [13], 1 1 Pout = 1 − e−γth ( G1 p1 + G2 p2 ) (3) where p1 and p2 are source and relay node transmit powers. Minimizing Pout in this case is equivalent to minimizing γth ( G11p1 + G21p2 ) term. Therefore the problem reduces to, 1 G1 p1 ⎧ ⎪ ⎨ subject to ⎪ ⎩

min

γth (

+

1 ) G2 p2

2

n=1 pn = PT pn ≤ Pmax : n = 1, 2 p1 hsn ≤ Tb p2 hrm ≤ Ta

Since this is a convex problem, Lagrangian function given as 1 1 + ) + μ(p1 + p2 − PT ) + λ1 (p1 − G1 p1 G2 p2 Ta Tb ) + λ2 (p2 − ) + λ3 (p1 − pmax ) + λ4 (p2 − pmax ). hsn hrm (4)

Λ(p1 , p2 ) = γth (

By using Karush-Kuhn-Tucker (KKT) conditions, we obtain the solutions as follows. Neglecting all inequality constraints,we have the following optimal solution.  p∗1

= PT [1 +

G1 −1 ] G2

 and

p∗2

= PT [1 +

G2 −1 ] G1

(5)

The inequality constraints are relevant at higher PT values, for lower values there is no use of those to obtain the solution. b a and hTrm are always less In general we can assume both hTsn

than pmax . Therefore we neglect pmax and by considering the IPT constraints, we get



p1 =

⎧ Tb G1 G2 Ta G1 −1 ⎪ PT [1 + : PT < min[ hsn (1 + ⎪ G2 ] G2 ), hrm (1 + G1 )]) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Tb Tb Tb G1 G1 ⎪ ⎪ : if hsn (1 + ⎪ hsn G2 ) = min[ hsn (1 + G2 ), ⎪ ⎪ ⎪ T G2 G1 Ta ⎪ b (1 + ⎪ (1 + )] and ) < PT < ⎪ hrm G1 hsn G2 ⎪ ⎪ ⎪ Tb Ta ⎪ ⎨ + hrm hsn ⎪ ⎪ ⎪ ⎪ PT − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Tb

Ta hrm

: if Ta hrm

Ta hrm

1

Ta hrm

+



Tb G1 ) = min[ hsn (1 + G2 ), G2 Ta and hrm (1 + G1 ) < P T < Tb Ta hrm + hsn

G2 G1

(1 + G2 (1 + G )]

: PT ≥

hsn



Tb hsn

We can clearly see that with different PT values, the optimal power allocation has different solutions. Outage probability variation with PT is considered in the numerical results section. B. Non regenerative Relay scheme Non regenerative relay receives the signal from source s and amplifies and forwards it to the destination d. Outage probability for a dual hop systems is given by [13], Pout = 1 − √

2C 2C −γ ( 1 + 1 ) K1 ( √ )e th G1 p1 G2 p2 G1 p1 G2 p2 G1 p1 G2 p2 (6)

K1 is

first order modified Bessel function and C = γth or C = γth 2 + γth . Therefore the problem formulation for non-regenerative relay is given by, min

1− √

2C 2C −γ ( 1 + 1 ) K1 ( √ )e th G1 p1 G2 p2 G1 p1 G2 p2 G1 p1 G2 p2 ⎧ 2 ⎪ n=1 pn = PT ⎨ pn ≤ Pmax : n = 1, 2 subject to ⎪ ⎩ p1 hsn ≤ Tb p2 hrm ≤ Ta

Tb hsn

=

N N   γth γth + + μ(ps + pn − PT ) + λs Gn pn Gs ps n=1 n=1

(ps −

N  Tb ) + λ0 ( pn hrn m − Ta ) hsn n=1 (8)

Problem is solved using KKT conditions and the optimal solutions have different expressions depending on PT and gain parameters. 1) Case where PT has a lower value, both interference inequalities do not have a contribution to the optimal solution. Optimal solution given as(λs ∗ = 0 and λ0 ∗ = 0), 

N 



1 √ + Gk 



ps = pn

Ta hrm

Tb hsn Ta hrm

K0 (.) is the zeroth-order modified Bessel function of the second kind and the first expression for p1 ∗ is obtained when all the constraints are relaxed. The first solution is a transcendental function of p1 ∗ and therefore numerical techniques have



Gn −1 ] : n = 1, 2, ..., N Gs (9)

Gn Gs

(10)

The following two inequalities are needed to be satisfied by other parameters to obtain above optimal solution. PT [1 +



Gn

N  k=1,k=n

(7)

T

PT −

Λ(ps , pn ) =

k=1,k=n

⎧ K0 ( √ 2C ) ⎪ G1 p1 ∗ G2 p2 ∗ 1 ⎪ [ G1 + γCth G1 × ⎪ 2C G2 (PT −p1 ∗ )2 G1 K1 ( √ ⎪ ) ⎪ G1 p1 ∗ G2 p2 ∗ ⎪ ⎪ ⎪ ⎪ (√ ∗ 1 − √ ∗3 1 )]−1/2 ⎨ p1 (PT −p1 ∗ ) p1 (P −p1 ∗ )3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Outage probability for regenerative relay scheme = in Rayleigh faded case is given as Pout −γth −γth N G p G p s s n n 1−e . n=1 e Minimizing the outage probability is similar to maxi−γth −γth N mizing e Gs ps n=1 e Gn pn and it reduces to minimizing N γth γth n=1 Gn pn + Gs ps . Pmax constraint can be neglected without causing much damage to original problem. This is possible, since node powers are limited by the IPT values. Lagrangian function is given as,

pn = PT [1+ Gn

We assume both and are always less than pmax . Therefore, we neglect the effect of pmax constraints and the optimal solution is given by,

p∗1

IV. N R ELAY T RANSMISSION



We have use the following expression [14] of Bessel functions to solve this using KKT conditions. d z Kv (z) + vKv (z) = −zKv−1 (z) dz

to be used to solve it. Techniques such as Newton algorithm can be considered. Additionally, the closed form solution that we have for the regenerative model can be approximated to the non regenerative case. These details are discussed in the numerical results section.

N 

√ hrn m PT [1+ Gn

n=1

1 √ + Gk N  k=1,k=n



Gn −1 ] Gs





1 √ + Gk

Gn Tb < (11) Gs hsn

Gn −1 ] < Ta (12) Gs

2) Case where the ps is limited by the IPT threshold. Then b and the source power reaches its maximum power hTsn ∗ ∗ node powers are given by (λs = 0 and λ0 = 0), pn ∗ = (PT −

√ Tb )[1 + Gn hsn ps ∗ =

N  k=1,k=n

Tb hsn

1 √ ]−1 Gk

(13)

(14)

3) Both m and n users sense maximum interference from the source and relay nodes, when both IPT constraints are satisfied with equality (λs ∗ = 0 and λ0 ∗ = 0). PT has no effect in the optimal solution. ps ∗ =

Tb hsn

(15)



N 

Gn



k=1,k=n

1 hrn m hrk m √ ]−1 Gk (16)

4) Most probable case in a real environment is where all relay nodes get power levels to interfere primary user m. The ps does not reach its maximum. In other words, this simply means λs ∗ = 0 and λ0 ∗ = 0. The closed form expressions are not available for this and the following equations are obtained by simplifying KKT conditions. pn ∗ = p1 ∗ [

Gn hrn m Gn p21 hr m + (1 − n )]−1/2 G1 hr1 m Gs p2s hr1 m

0

10

Non Regenerative,no IPTs Regenerative,no IPTs Non Regenerative,IPTs Regenerative,IPTs

Outage Probability

pn ∗ = Ta [hrn m +

(17)

The above can be simplified into pn ∗ = Kn p1 ∗ and Kn is a function of p1 ∗ and ps ∗ . Then the following two equations are dependent only on p1 ∗ and ps ∗ . Therefore the optimal solutions can be obtained by carrying out a numerical technique. ∗

p1 [hr1 m +

N 

hri m Ki ] = Ta

−1

10

(18)

−2

10

0

5

10 15 Total power,[dB]

20

25

Fig. 3. Outage probability comparison between non regenerative and regenerative relay cases

i=2

p1 ∗ [1 +

N 

Ki ] + P s ∗ = P T

(19)

With IPT constraints Without IPT constraints

i=2

V. NUMERICAL RESULTS

−1

Numerical results are obtained for three different instances. Power allocation with and without IPT constraints and uniform allocation are considered. Optimal outage probabilities for these cases are plotted with total power PT . In addition to that, numerical results are obtained for both non regenerative and regenerative cases by assuming certain parameters. We take γth = 3, G1 = 1, G2 = 10, Tb = 10, hsn = 0.2 , Ta = 20 and hrm = 0.2. Also in the equal power allocation case, powers are equally divided between the source and the relay node. Fig.2. shows the outage variation with the total power for regenerative relay based secondary communication. The case without IPT constraints is also plotted in the same graph. Fig.3 describes the outage variation of regenerative and non regenerative models. 0

10

Outage Probability

Optimal − With IPT Optimal − Without IPT Equal power − Without IPT

−1

10

−2

10

0

5

10 15 Total power,[dB]

20

25

Fig. 2. Single regenerative relay based secondary communication outage probability behavior

Bit Error Rate

10

−2

10

−3

10

0

5

10

15 20 Ptotal dB

25

30

35

Fig. 4. Average BER of a regenerative relay system with BPSK modulation. With and without IPT constraints cases are considered.

Outage probability reduces with the total power PT in each case that we consider here. Interestingly, in the cognitive case, the outage probability does not reduce below a certain level with PT . This happens due to the IPT constraints that we introduce into the optimization problem. When the secondary communication reaches its minimum outage probability, PUs experience the highest interference from the secondary transmission. Therefore, there is a limit in the outage probability that the secondary transmission can achieve. Also by referring to Fig.3, we can see that the regenerative model can be approximated to the non regenerative case. Plots obtained for both are very much related. Fig. 4 compares the average bit-error rate (BER) for the regenerative relay with the binary phase shift keying (BPSK) modulation. Here we have consider the optimal power allocation cases with and without IPT constraints. It is clear that have BER also behaves similar to the outage probability as seen in Fig.4. Fig. 5 shows the multi-hop transmission comparison up

0

10

Outage Probability

Four relays Three relays Two relays Single relay

−1

10

−2

10

0

5

10

15

20

25

Total power,[dB]

Fig. 5. Outage probability comparison between N realy model with 1,2,3 and 4 relays 0

10

Outage Probability

Optimal power allocation Equal power allocation Optimal power allocation − no IPTs

0

5

10

15

20

25

Total power,[dB]

Fig. 6. Outage probability of comparison with 3 series relays. Optimal power allocation with IPT constraints, equal power allocation and optimal power allocation without considering IPT constraints compared

to four relays. Relays are in series and the regenerative relay model with Rayleigh fading channel environment is selected for the comparison. When the number of relays are increased, outage probability becomes higher. Fig. 6 compares the outage behavior in the three relay nodes model. Optimal power allocation shows much better outage performance when compared to equal power allocation and the IPT constraints effect is visible as in the previous case. VI. CONCLUSIONS In this paper, we consider the optimum power allocation for relay based secondary transmission schemes. Cooperative relay network cases are reformulated by introducing additional interference power threshold constraints. These are convex optimization problems and we consider the outage probability of the secondary communication as the objective function. Then the problem is solved for the single relay model. Both

regenerative and non regenerative cases are considered where they behave in a similar manner even though the objective function has different expressions. It is observed from the numerical results that the outage probability cannot be reduced after a certain level, even though there is sufficient total power. Optimal solutions for the general case with N relay nodes are obtained. They have different expressions depending on the other parameters like channels gains and total power of relay nodes. Performance of the secondary user communication improves with the stable primary user transmission at the same time. Finally, by considering results that we have, the optimal power allocation guarantees that the primary user will not be affected due to the secondary transmission while the secondary user achieves its maximum possible performance under some limitations. R EFERENCES [1] FCC, “Spectrum Policy Task Force,” ET Docket 02-135, Nov. 2002. [2] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Select. Areas Commun., vol. 23, no. 2, pp. 201-220, Feb. 2005. [3] J. Jia and Q. Zhang, “A Non-Cooperative Power Control Game for Secondary Spectrum Sharing,” in Proc. of IEEE International Conference on Communications (ICC) 2007, Jun. 2007. [4] X. Gong, W. Yuan, W. Liu, W. Cheng, and S. Wang, “A Cooperative Relay Scheme for Secondary Communication in Cognative Radio Networks,” IEEE Global Telecommunications Conference, Dec 2008, Dec. 2008. [5] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10,pp.7480, Oct. 2006. [6] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080. Dec. 2004. [7] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [8] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-and-forward relay networks: optimal power allocation versus selection,” Wireless Communications, IEEE Transactions on August 2007, Aug. 2007. [9] Z. Qi, Z. Jingmei, S. Chunju, W. Ying, Z. Ping, and H. Rong, “Power allocation for regenerative relay channel with Rayleigh fading,” Vehicular Technology Conference, 2004. VTC 2004-Spring. 2004 IEEE 59th May 2004, May 2004. [10] X. Deng and A. M. Haimovich, “Power allocation for cooperative relaying in wireless networks,” Communications Letters, IEEE , vol.9, no.11, pp. 994-996, Nov. 2005. [11] T. S. Rappaport, Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1996. [12] M. O. Hasna and M. S. Alouini, “Optimal power allocation for relayed transmissions over Rayleigh-fading channels,” Wireless Communications, IEEE Transactions on Nov. 2004, vol.3, no.6, pp. 1999-2004, Nov. 2004. [13] M. O. Hasna and M. S. Alouini, “Performance analysis of two-hop relayed transmissions over Rayleigh fading channels,” Vehicular Technology Conference, 2002. Proceedings. VTC 2002-Fall. 2002 IEEE 56th , vol.4, no., pp. 1992-1996, 2002. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 1994.