Distributed Power Allocation Algorithm for Amplify-and-Forward Relaying Networks

INTRODUCTION

---

Recently, there has been tremendous research interest in relay-assisted cooperative communication. Conservation of transmit power can improve the communication reliability to some extent. Provided some statistical channel knowledge at the transmitter and relays(mean channel gain information). Annavajjala et al. (2007) derived the optimum transmit power allocation that minimizes the outage probability of the mutual information for wireless relaying protocols at high SNR for AF, DF and DSTC protocols operating over a Rayleigh fading channel. To optimally allocate the power, the algorithm Annavajjala et al. (2007) mentioned, however, requires centralized controller. In practical applications, due to the distributed characteristic of relays, it is more preferred to have a distributed alternative. Motivated by this, we transform the initial optimization problem into a dual one that consists of separable subproblems. The optimal power allocation then can be obtained by solving one subproblem at each relay at a globally iterative manner and information ows in a sequential manner from one relay to the adjacent relay except a single node broadcast by the source node. Annavajjala et al. (2007) considered three relaying protocols, while in this letter, for simplicity, we consider only the AF protocol. The distributed algorithm for other relaying protocols can be obtained similarly.

SYSTEM MODEL AND PROBLEM DESCRIPTION

Consider an AF relay network of Annavajjala et al. (2007), with a single source-destination pair and N relay nodes. The channels between all nodes are assumed to be quasi-static and at fading. For simplicity, we assume that there is no direct link between source and destination though the proposed algorithm can be easily adapted to the case with direct link. Half-duplex constraints are imposed on relay nodes as that of Annavajjala et al. (2007). The channel gain from the source to the relay j is with variance Ω^j₂ and the channel gain from the relay j to the destination is with variance Ω^j₃. With repetition-based AF protocol, the total bandwidth is divided into N+1 equi-width, disjoint channel, so that the bandwidth available for each relay in the second phase is W/(N+1). The single-sided power spectral density of additive Gaussian noise is assumed to be 1. Let Ps and P_r,j denote the transmit power of the source and the transmit power of relay j. The outage probability with information rate R is induced by Annavajjala et al. (2007) as:

Where:

is independent of power allocation and α_j = Ω_j²/Ω_j³. Annavajjala et al. (2007) proved that the power allocation optimization problem can be described as:

(1)

where, PT is the total transmit power. Compared with the results of Annavajjala et al. (2007), the product has been transformed into sum in Eq. 1. Because log (A) is concave, we multiply the cost function by -1 in (1) such that this transformation do not change the convex characteristics of the cost function. Thus, Eq. 1 is equivalent to results of Annavajjala et al. (2007).

While the centralized optimization algorithm is proposed by Annavajjala et al. (2007), we proposed a distributed alternative that does not require a central controller.

Distributed power allocation algorithm: The Lagrange cost function of Eq. 1 can be written as:

(2)

where, λ>0 is the Lagrange parameter.

The dual optimization problem is:

(3)

Where:

(4)

An observation is that if Ps is given with some value, the evaluation of Eq. 3 is amenable to parallel computation with a separate relay calculating each component g_i(λ) of g(λ) due to the separable structure of the problem. When given the initial value λ₀ and P_s(0), the proposed distributed power allocation algorithm can be described as:

(5)

where, λ_j^(t) denote a local estimate of λ, at relay k at time t, μ>0 is a step-size parameter, Ps is updated at source once in every iteration and the result is distributed to all relays by a single node broadcasting (Bertsekas and Tsitsiklis, 1989). P^(t)_{r, j} in Eq. 6 is computed as:

Fig. 1(a-b):	Data flow in the proposed distributed algorithm

(6)

Equation 6 and P_s^(t) in Eq. 5 are obtained by noting that L(P_s, {P_r,j}, λ) is convex [1] and ∂g_i(λ)/∂P_r,j and ∂g(λ)/∂P_s if the minimum in Eq. 3 is attained. In this proposed scheme, we define a cycle visiting each node over the network as that of (Lopes and Sayed (2007) such that relay j has access only to λ^(t)_j-1, which is an estimate of λ at its immediate neighbor in the defined cycle, except for P_s^(t) that is broadcasted by the source. The algorithm is further illustrated Fig. 1: at time t we start with initial condition (6) at relay 1, iterate cyclicly across the relays and get λ_t at relay N. Ps is then updated and distributed to all relays. The algorithm ends if the difference between λ_t-1 and t is smaller than one certain threshold.