Power Allocation Based on SEP Minimization in Two-Hop Decode-and-Forward Relay Networks

# Power Allocation Based on SEP Minimization in Two-Hop Decode-and-Forward Relay Networks

Arash Khabbazibasmenj,   and  Sergiy A. Vorobyov, This work is supported in parts by the Natural Science and Engineering Research Council (NSERC) of Canada and the Alberta Ingenuity Foundation, Alberta, Canada. The authors are with the ECE Dept., University of Alberta, 9107-116 St., Edmonton, Alberta, T6G 2V4 Canada. Emails: {khabbazi, vorobyov}@ece.ualberta.ca.Corresponding author: Sergiy A. Vorobyov, Dept. Elect. and Comp. Eng., University of Alberta, 9107-116 St., Edmonton, Alberta, T6G 2V4, Canada; Phone: +1 780 492 9702, Fax: +1 780 492 1811. Email: vorobyov@ece.ualberta.ca.
###### Abstract

The problem of optimal power allocation among the relays in a two-hop decode-and-forward cooperative relay network with independent Rayleigh fading channels is considered. It is assumed that only the relays that decode the source message correctly contribute in data transmission. Moreover, only the knowledge of statistical channel state information is available. A new simple closed-form expression for the average symbol error probability is derived. Based on this expression, a new power allocation method that minimizes the average symbol error probability and takes into account the constraints on the total average power of all the relay nodes and maximum instant power of each relay node is developed. The corresponding optimization problem is shown to be a convex problem that can be solved using interior point methods. However, an approximate closed-form solution is obtained and shown to be practically more appealing due to significant complexity reduction. The accuracy of the approximation is discussed. Moreover, the so obtained closed-form solution gives additional insights into the optimal power allocation problem. Simulation results confirm the improved performance of the proposed power allocation scheme as compared to other schemes.

Cooperative systems, convex optimization, decode-and-forward relay networks, power allocation.

## I Introduction

Cooperative relay networks enjoy the advantages of the multiple-input multiple-output (MIMO) systems such as, for example, high data rate and low probability of outage by exploiting the inherent spatial diversity without applying multiple antennas at the nodes. In cooperative relay networks, after receiving the source message, relay nodes process and then retransmit it to the destination. Different cooperation protocols such as decode-and-forward (DF), amplify-and-forward (AF), coded cooperation, and compress-and-forward can be used for processing the message at the relay nodes [1], [2]. The benefits of cooperative relay networks can be further exploited by optimal power allocation among the source and relay nodes. Specifically, based on the knowledge of the channel state information (CSI) at the relays and/or destination, the system performance can be improved by optimally allocating the available power resources among the relays [1][3].

Different power allocation schemes have been proposed in the literature [6][15]. These schemes differ from each other due to the different considerations on the network topology, assumptions on the available CSI, use of different cooperation protocols for relay nodes, and use of different performance criteria [1]. Most of the existing power allocation methods require the knowledge of instantaneous CSI to enable optimal power distribution [1], [6][9]. The application of such methods is practically limited due to the significant amount of feedback needed for transmitting the estimated channel coefficients and/or the power levels of different nodes. This overhead problem becomes even more severe when the rate of change of the channel fading coefficients is fast.

In this paper, we aim at avoiding the overhead problem by considering only statistical CSI which is easy to obtain. Recently, some power allocation methods based on statistical CSI have been proposed[10][14]. The optimal power allocation problem among multiple AF relay nodes that minimizes the total power given a required symbol error probability (SEP) at the destination is studied in [11]. The problem of optimal power distribution in a three node DF relay network which aims at minimizing the average SEP is studied in [12]. The authors of [13] study the power allocation problem in a multi-relay DF cooperative network in which the relay nodes cooperate and each relay coherently combines the signals received from previous relays in addition to the signal received from the source to minimize the average SEP. The power allocation problem aiming at minimizing the average SEP in a cooperative network consisting of two DF relay nodes in Nakamgi- fading channel has been studied in [14]. All of the aforementioned power allocation methods are based on minimizing or bounding the asymptotic approximate average SEP which is valid at high signal-to-noise ratios (SNRs) and is not applicable at low and moderate SNRs.

In our initial conference contribution [15], a power allocation method for multi-relay DF cooperative network with Rayleigh fading channels that minimizes the exact average SEP has been proposed. However, the assumption of correct decoding in relay nodes used in [15] limits its practical applicability. For obtaining a more practically suitable power allocation method, we consider in this paper the case when relay nodes may not be able to decode the source signal correctly. We derive the optimal power allocation in a multi node DF relay network with Rayliegh fading channels in which only relays which have decoded the source message correctly contribute in the data relaying. More specifically, after receiving the source message, only the relays which decode the message correctly retransmit it to the destination. A new exact and simple closed-form expression for average SEP is derived. Then a new power allocation strategy is developed by minimizing the exact average SEP rather than its high SNR approximation under the constraints on the total average power of all relays and maximum powers of individual relays. Only the knowledge of the average channel gains, i.e., the knowledge of the variances of the channel coefficients, is assumed to be available. We show that the corresponding optimization problem is convex and, thus, can be solved using the well established interior point methods. In order to find better insights into the power allocation problem, we derive an approximate closed-form solution to the problem and discuss the accuracy of the approximation used. We also show by simulations that the exact numerical and approximate solutions provide close average SEP performance and that the proposed power allocation scheme outperforms other schemes.

The paper is organized as follows. System model is introduced in Section II. A simple closed-form expression for the average SEP is derived in Section III, while power allocation (exact and approximate) strategies based on SEP minimization are derived in Section IV. Simulation results are given in Section V followed by conclusion. All technical proofs are given in Appendix. This paper is reproducible research [16] and the software needed to generate the simulation results will be made available together with the paper. It can be also requested by the reviewers if needed.

## Ii System Model

Consider a wireless relay network with a single source communicating with a single destination through relay nodes as it is shown in Fig. 1. The relays are assumed to be half-duplex, that is, the relays either transmit or receive the signal at the same frequency at any given time instant. Therefore, every data transmission from the source to the destination occurs in two phases. In the first phase, the source node transmits its message to the destination and the relay nodes, while in the second phase, relay nodes retransmit the source message to the destination. The channels between the source and the relay nodes, between the relay nodes and the destination, and the direct path are assumed to be flat Rayligh fading and independent from each other. The source and relay nodes use the -phase shift keying (M-PSK) modulation111Note that other types of modulation can be straightforwardly adopted and PSK modulation is considered only because of space limitation. for data transmission where is the size of the constellation. Relay nodes use DF cooperation protocol for processing the received signal from the source node. The received signal at the destination and the th relay node in the first phase can be expressed, respectively, as

 ys,d = √p0hs,dx+ns,d (1) ys,i = √p0hs,ix+ns,i,     i=1,…,N (2)

where is the source message of unit power, is the transmit power of the source node, and denote the channel coefficients between the source and the destination and between the source and the th relay node, respectively, and and are the complex additive white Gaussian noises (AWGNs) in the destination and in the th relay, respectively. Since the channel fading is Rayleigh distributed, the channel coefficients are modeled as independent complex Gaussian random variables with zero mean and variances and , respectively, for the channels between the source and the destination and between the source and the th relay node. The additive noises are zero mean and have variance .

After decoding the received signal from the source (2), only the relay nodes which have decoded the source message correctly retransmit it to the destination through orthogonal channels using time division multiple access (TDMA) or frequency division multiple access (FDMA). By means of an ideal cyclic redundancy check (CRC) code applied to the transmitted information from the source, relays can determine whether they have decoded the received signal correctly or not [12]. Then the probability that th relay node can decode the received signal correctly conditioned on the instantaneous CSI can be expressed as [17]

 αi(hs,i)=1−1π∫M−1Mπ0e−gPSKsin2(θ)N0|hs,i|2p0dθ (3)

where .

Let denote a vector that indicates whether each relay has decoded the source message correctly or not. Specifically, , if th relay node decodes the source message correctly, and , otherwise. Here stands for the transpose. In the rest of the paper, we refer to the vector as the vector of decoding state at the relay nodes. Since consists of only binary values, there are in total different combinations that the vector can take. Moreover, there is a one-to-one correspondence between the binary representation of decimal numbers and different values that the vector can take. For example, in a cooperative network with two relays, if the relay enumerated as first decodes the source message correctly and the relay enumerated as second decodes it incorrectly, the corresponding decoding state vector is and the corresponding representation in decimal is . For simplicity, we represent hereafter each possible combination of vector by its corresponding decimal number and denote this combination as . For example, corresponds to the situation when all relay nodes decode the source message incorrectly.

The received signal from the th relay node which is able to decode the source message correctly, that is, , at the destination can be modeled as

 yi,d=√pihi,dx+ni,d (4)

where is the transmitted power of the th relay node, is the channel coefficient between the th relay node and the destination, and is the AWGN with zero mean and variance . The channel coefficient is modeled as an independent complex Gaussian random variable with zero mean and variance due to the Rayleigh fading assumption. It is worth stressing that the assumption that the channel coefficients are independent from each other is applicable for relay networks because the distances between different relay nodes are typically large enough. It is assumed that the destination knows perfectly the instantaneous CSI from the relays to the destination and the instantaneous CSI of the direct link. The knowledge of instantaneous CSI for the links between the source and the relay nodes is not needed. Then the maximal ratio combining (MRC) principle can be used at the destination to combine received signals form the source and relay nodes. As a result of MRC, the received SNR at the destination conditioned on the decoding state at the relay nodes, i.e., , can be expressed as

 γD(ϕk)=γs+∑{i|ϕk(i)=1}γi (5)

where and are the received SNRs from the source and the th relay node at the destination, respectively. Here and , are exponential random variables with means and , respectively. Moreover, and , are all statically independent.

## Iii Average SEP

For the considered case when the data transmission is performed using the M-PSK modulation, the SEP of the signal at the destination conditioned on the channel states and the decoding state at the relay nodes can be written as [17]

 Pe{CS,ϕk} = 1π∫M−1Mπ0e−gPSKsin2(θ)γD(ϕk)dθ. (6)

Using the total probability rule, the SEP conditioned on the channel states can be expressed as

 Pe{CS}=2N−1∑k=0Pr{ϕk}Pe{CS,ϕk} (7)

where is the probability of the decoding state that can be calculated as

 Pr{ϕk}=∏{i|ϕk(i)=1}αi⋅∏{i|ϕk(i)=0}(1−αi) (8)

where is the probability of correct decoding in th relay node (3). Note that for obtaining (8), the independency of the AWGNs at the relay nodes has been exploited.

The average SEP can be obtained by averaging (7) over , , , and , and using the fact that is statistically independent from . The latter follows from the statistical independence between the channel coefficients and the fact that depends only on and depends only on and . Then the average SEP can be expressed as

 Pe=1π2N−1∑k=0E{Pr{ϕk}}E⎧⎨⎩∫M−1Mπ0e−gPSKsin2(θ)γD(ϕk)dθ⎫⎬⎭ (9)

where denotes the expectation operation.

The second expectation in (9) can be computed as

 E⎧⎨⎩∫M−1Mπ0e−gPSKsin2(θ)γD(ϕk)dθ⎫⎬⎭ (10) = ∫M−1Mπ0E⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩e−gPSKsin2(θ)N0⎛⎝∑{i|ϕk(i)=1}pi|hi,d|2+p0|hs,d|2⎞⎠⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭dθ = ∫M−1Mπ0∏{i|ϕk(i)=1}∪{i=0}sin2(θ)sin2(θ)+bipidθ

where , and . Similarly, the first expectation in (9) can be computed as

 E{Pr{ϕk}} = E⎧⎨⎩∏{i|ϕk(i)=1}αi⋅∏{i|Φk(i)=0}(1−αi)⎫⎬⎭ (11) = ∏{i|Φk(i)=1}βi⋅∏{i|Φk(i)=0}(1−βi)

where

 βi = E⎧⎨⎩1−1π∫M−1Mπ0e−gPSKsin2(θ)N0|hs,i|2p0dθ⎫⎬⎭ (12) = 1−1π∫M−1Mπ0sin2(θ)sin2(θ)+cip0dθ

and , .

Substituting (10) and (11) in (9), the average SEP can be equivalently expressed as

 Pe = 1π2N−1∑k=0⎛⎝∏{i|ϕk(i)=1}βi⋅∏{i|ϕk(i)=0}(1−βi) (13) × ∫M−1Mπ0∏{i|ϕk(i)=1}∪{i=0}sin2(θ)sin2(θ)+bipidθ⎞⎠.

Setting for notation simplicity, the average SEP expression in (13) can be simplified as

 Pe = 1π2N−1∑k=0⎛⎝∏{i|ϕk(i)=0}(1−βi)∏{i|ϕk(i)=1}∪{i=0}βi × ∫M−1Mπ0∏{i|Φk(i)=1}∪{i=0}sin2(θ)sin2(θ)+bipidθ⎞⎠ = 1π∫M−1Mπ0⎛⎝2N−1∑k=0∏{i|ϕk(i)=0}(1−βi) × ∏{i|ϕk(i)=1}∪{i=0}βi⋅sin2(θ)sin2(θ)+bipidθ⎞⎠. (15)

Finally, the expression (15) can be further simplified as

 Pe=1π∫M−1Mπ0N∏i=0((1−βi)+βisin2(θ)sin2(θ)+bipi)dθ. (16)

To verify the latter result, note that (15) can be obtained by simply expanding (16). Moreover, after finding the integral in (12), , can be expressed as

 βi = 1−√cip0cip0+1⎛⎜ ⎜⎝tan−1(cot(M−1Mπ)√cip0cip0+1)π−12⎞⎟ ⎟⎠ (17) − ⎛⎜ ⎜⎝12−tan−1(cot(M−1Mπ))π⎞⎟ ⎟⎠,i=1,…,N.

Here has a meaning of statistical average of the correct decoding probability in the th relay node (3) with respect to the channel coefficients. It is worth noting that the average SEP expression (16) is used later for finding the optimal power distribution among the relay nodes.

Using partial fraction decomposition, it is possible to further simplify (16) and find a closed-form expression for the average SEP. It is worth noting that for simplicity and because of space limitations, we hereafter assume that , , . However, if the condition , , does not hold, the following average SEP derivation approach remains unchanged, while the only change is the need of using another form of the partial fraction decomposition. Thus, it is straightforward to derive closed-form expression for the average SEP in the case when , , does not hold by using the same steps as we show next.

Toward this end, we first rewrite the integral inside (III), denoted hereafter as , by changing the variable as

 I = ∫M−1Mπ0∏{i|ϕk(i)=1}∪{i=0}sin2(θ)sin2(θ)+bipidθ (18) = ∫∞cot(M−1Mπ)11+x2∏{i|ϕk(i)=1}∪{i=0}1(1+bipi)+bipix2dx.

Then applying the partial fraction technique to the right-hand side of (18) and using the fact that , , , we obtain

 I = ∫∞cot(M−1Mπ)11+x2 (19) − ∑{i|ϕk(i)=1}∪{i=0}∏({j|ϕk(j)=1}∪{j=0})−{j≠i}11−bjpjbipi(1+1bipi)+x2dx.

The integral in (19) is the summation of the terms that all have the form of . By using the fact that , (19) can be calculated as

 I = ∑{i|ϕk(i)=1}∪{i=0}((tan−1(cot(M−1Mπ)√bipibipi+1)−π2) (20) × √bipibipi+1∏({j|ϕk(j)=1}∪{j=0})−{j≠i}11−bjpjbipi) + (π2−tan−1(cot(M−1Mπ)))

Substituting the so obtained expression (20) in (III) and also expanding the resulted expression, the average SEP can be equivalently expressed as

 Pe = 2N−1∑k=0∏{i|ϕk(i)=1}∪{i=0}βi⋅N∏{i|ϕk(i)=0}(1−βi) (21) × ∑{i|ϕk(i)=1}∪{i=0}((tan−1(cot(M−1Mπ)√bipibipi+1)π−12) × √bipibipi+1∏({j|ϕk(j)=1}∪{j=0})−{j≠i}11−bjpjbipi) + 2N−1∑k=0∏{i|ϕk(i)=1}∪{i=0}βi×N∏{i|ϕk(i)=0}(1−βi) × ⎛⎜ ⎜⎝12−tan−1(cot(M−1Mπ))π⎞⎟ ⎟⎠.

Moreover, we can modify the first term of (21) by taking the product into the summation part of that term and expanding the product such that is multiplied by the corresponding term in the summation and also modify the second term of (21) by using the following equality which is easy to prove

 2N−1∑k=0∏{i|ϕk(i)=1}∪{i=0}βi⋅N∏{i|ϕk(i)=0}(1−βi)=1. (22)

By doing so, the average SEP can be rewritten as

 Pe = 2N−1∑k=0∏{i|Φk(i)=0}(1−βi) (23) × ∑{i|ϕk(i)=1}∪{i=0}(βi(tan−1(cot(M−1Mπ)√bipibipi+1)π−12) × √bipibipi+1∏({j|Φk(j)=1}∪{j=0})−{j≠i}βj1−bjpjbipi) + ⎛⎜ ⎜⎝12−tan−1(cot(M−1Mπ))π⎞⎟ ⎟⎠.

Finally, after rearranging and factoring out the terms, we can obtain the following closed form expression for the average SEP

 Pe = N∑i=0(βi(tan−1(cot(M−1Mπ)√bipibipi+1)π−12)√bipibipi+1 (24) × ∏{j|j=0,…,N}−{j≠i}⎛⎜ ⎜⎝βj1−bjpjbipi+(1−βj)⎞⎟ ⎟⎠) + ⎛⎜ ⎜⎝12−tan−1(cot(M−1Mπ))π⎞⎟ ⎟⎠.

Note that (23) can be obtained by simply expanding (24). It is interesting to mention that if we consider similar conditions as in [15], in which all relays are capable of decoding the source message correctly, i.e., , and also there is no direct link between source and destination, i.e., , the closed-form average SEP (24) simplifies to the one that was obtained in [15]. The closed-form expression (24) is simple and does not include any other functions rather than basic analytic functions.

## Iv Power allocation

In this section, we address in details the problem of optimal power allocation among the relay nodes such that the average SEP is minimized. Only statistical information on the channel states is used. Moreover, we assume that the power of the source node is fixed and, in turn, the average probabilities of the correct decoding of the relays, i.e., , , are also fixed. The relevant figure of merit for the performance of relay network is then the average SEP that is derived above in closed form. It enables us to apply power allocation also in the case when the rate of change of channel fading is high. Since only statistical CSI is available and relay nodes retransmit only if they decode the source message correctly, the averaged power of the relay nodes over CSI and probability of correct decoding need to be considered. It is easy to see that the average power used by the th relay node equals . Indeed, since is the statistical average of the probability of correct decoding in the th relay node, the transmitted power of the th relay node weighted by gives average power used by the th relay node during a single transmission.

In the following, we develop a power allocation strategy by minimizing the average SEP (16), while satisfying the constraint on the total average power used in relay nodes and the constraints on maximum instantaneous powers per every relay , . With the knowledge of the average channel gains , , , and specifications on , , , and , the power allocation problem can be formulated as

 minp Pe(p) (25) s.t.N∑i=1βipi=pR (26) 0≤pi≤pmaxi,i=1,…,N. (27)

where is the average SEP (16) written as a function of powers at the relay nodes . For notation simplicity, we use hereafter the following equation for the average SEP in (16)

 Pe(p)=1π∫M−1Mπ0g(θ,p)dθ (28)

where

 g(θ,p)≜N∏i=0((1−βi)+βisin2(θ)sin2(θ)+bipi). (29)

Note that the optimization problem (25)–(27) is infeasible if and it has the trivial solution, that is, , if . Thus, it is assumed that . The following theorem about convexity of the optimization problem (25)–(27) is in order.

Theorem 1: The optimization problem (25)–(27) is convex.

See the proof in Appendix A.

Although this problem does not have a simple closed-form solution, an accurate approximate closed-form solution can be found as it is shown in the rest of the paper. A numerical algorithm for finding the exact solution of the optimization problem (25)–(27) based on the interior-point methods is summarized in Appendix B. Despite the higher complexity of the numerical method as compared to the approximate closed-form solution, numerical method can provide an exact solution, which can be used as a benchmark to evaluate the accuracy of the approximate solution.

### Iv-a Approximate Closed-Form Solution

The optimization problem (25)–(27) is strictly feasible because as it has been assumed earlier. Thus, the Slater’s condition holds and since the problem is convex, the Karush-Kuhn-Tucker (KKT) conditions are the necessary and sufficient optimality conditions [18]. Indeed, since , then is a strictly feasible point for the optimization problem (25)–(27).

Let us introduce the Lagrangian

 L(p,λ,γ,ν) = 1π∫M−1Mπ0g(θ,p)dθ−N∑i=1λipi + N∑i=1γi(pi−pmaxi)+ν(N∑i=1βipi−pR) (30)

where and are vectors of non-negative Lagrange multipliers associated with the inequality constraints , and , , respectively, and is the Lagrange multiplier associated with the equality constraint . Then the KKT conditions can be obtained as

 λi≥0,     γi≥0,    i=1,⋯,N (31) λipi=0,γi(pi−pmaxi)=0,i=1,⋯,N (32) 1π∫M−1Mπ0−βibisin2(θ)g(θ,p)sin4(θ)+(2−βi)bipisin2(θ)+(1−βi)b2ip2idθ +νβi−λi+γi=0,i=1,⋯,N (33) 0≤pi≤pmaxi,     i=1,⋯,N (34) N∑i=1βipi=pR. (35)

Although the exact optimal solution of the problem (25)–(27) can be found through solving the system (31)–(35) numerically or as it is summarized in Appendix B by solving the original problem directly, a near optimum closed-form solution for the system (31)–(35), and thus, optimization problem (25)–(27) can be found by approximating the gradient of the Lagrangian, that is, the left hand side of (33). Specifically, it can be verified that for fixed , the function is strictly increasing/decreasing with respect to in the intervals and , correspondingly. Under the condition that the number of relays is large enough, the slope of the increment and decrement in the aforementioned intervals is high and the Chebyshev-type approximation on the conditions (33) is highly accurate. This approximation of the conditions (33) is of the form

 −βibiπ(1+(2−βi)bipi+(1−βi)b2ip2i) ×∫M−1Mπ0g(θ,p)dθ+βiν−λi+γi=0, (36) i=1,⋯,N

where the fact that is used in the ratio

 −βibisin2(θ)sin4(θ)+(2−βi)bipisin2(θ)+(1−βi)b2ip2i (37)

By rearranging the denominator of (37), substituting the approximation (36) in (33), dividing the equations (31)–(33) by the positive quantity , and also multiplying these equations by , we obtain

 λ′i≥0,     γ′i≥0,    i=1,⋯,N (38) λ′ipi=0,γ′i(pi−pmaxi)=0,i=1,⋯,N (39) −βibiβi(1+bipi)+(1−βi)(1+bipi)2+βiν′−λ′i+γi′=0 i=1,⋯,N (40) 0≤pi≤pmaxi,     i=1,⋯,N (41) N∑i=1βipi=pR. (42)

where , , and   , . It is possible to eliminate from the set of equations (38)–(42) in order to find a simpler set of smaller number of equations. By doing so, the approximate KKT conditions can be equivalently rewritten as

 γi′≥0,i=1,⋯,N (43) γi′(pi−pimax)=0,i=1,⋯,N (44) pi(βiν′+γi′−βibiβi(1+bipi)+(1−βi)(1+bipi)2)=0, i=1,⋯,N (45) βibiβi(1+bipi)+(1−βi)(1+bipi)2≤βiν′+γi′, i=1,⋯,N (46) 0≤ pi ≤pimax,i=1,⋯,N (47) N∑i=1βipi=pR (48)

Note that (40) is rewritten as (IV-A) since or, equivalently, is positive. Moreover, (IV-A) is obtained by solving (40) with respect to and then substituting the result in . The following result gives a closed-form solution for the system (43)–(48), and thus, it also gives an approximate solution for the power allocation optimization problem (25)–(27).

Theorem 2: For a set of DF relays, the approximate power allocation , i.e., the approximate solution of the optimization problem (25)-(27), is

 pi=⎛⎜ ⎜⎝−βi+√β2i+4(1−βi)bi/ν′2bi(1−βi)−1bi⎞⎟ ⎟⎠pmaxi0 (49)

where is determined so that .

See the proof in Appendix C.

It is interesting to mention that the power allocation scheme of [15] is a special case of the power allocation method given by (49) when . Indeed, it can be checked that for , we have

 pi = ⎛⎜ ⎜⎝limβi→1−βi+√β2i+4(1−βi)bi/ν′2bi(1−βi)−1bi⎞⎟ ⎟⎠pmaxi0 (50) = (1ν′