We consider the relay network shown in Figure 1 consisting of three relays assisting a source *S* transmitting towards a destination *D*. All the nodes are equipped with a single antenna, and we indicate with *h*_{
i
}∈ ℂ, *g*_{
i
}∈ ℂ, *i* = 1, 2, 3, the flat-fading channel between *S* and relay *i* and between relay *i* and *D*, respectively. *S* transmits a signal *x*, and the signal received by relay *i* can be written as

{y}_{i}={h}_{i}x+{w}_{i},

(1)

where *w*_{
i
}is the additive white complex Gaussian noise with zero mean. We indicate the signal transmitted by relay *i* with *z*_{
i
}and consider a unitary power constraint for each node in the network, i.e., E[|*x*|^{2}] ≤ 1, E[|*z*_{
i
}|^{2}] ≤ 1, *i* = 1, 2, 3. By assuming channel state information of the relay-destination link at the relay, the transmission performed by relays can be done coherently adjusting the phases of *z*_{
i
}, *i* = 1, 2, 3. For this reason and without loss of generality, in the following we only consider the channel gains

{H}_{i}={\left|{h}_{i}\right|}^{2},

(2a)

{G}_{i}={\left|{g}_{i}\right|}^{2}.

(2b)

Without loss of generality, we assume that all noises have unitary variance, while the effective signal to noise ratio (SNR) at the receiver is obtained by a proper scaling of the channel gains, see (19). We assume that a *central controller* knows all channels, correspondingly allocates resources to nodes, including power, and constellation sizes. Hence, the determined network spectral efficiency can be assumed as a bound for cases where only a partial channel state information is available.

Nodes operate in half-duplex mode, i.e., they cannot transmit and receive simultaneously. Each relay alternates a phase in which it receives data from the source and a phase when it transmits to the destination. No communication among relays is allowed. Moreover, we assume no direct transmission from *S* to *D* because of shadowing or the long distance between *S* and *D*. Let the time used for two consecutive phases be unitary. The odd phases are assigned a time *λ*, and the even phases are assigned a time 1 - *λ*, where the parameter *λ* will be optimized. The scheduling of transmission is then fully characterized by the variable

{\delta}_{i}=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill \text{relay}\phantom{\rule{2.77695pt}{0ex}}i\phantom{\rule{2.77695pt}{0ex}}\text{transmits}\phantom{\rule{2.77695pt}{0ex}}\text{during}\phantom{\rule{2.77695pt}{0ex}}\text{even}\phantom{\rule{2.77695pt}{0ex}}\text{phases},\hfill \\ \hfill 1,\hfill & \hfill \text{relay}\phantom{\rule{2.77695pt}{0ex}}i\phantom{\rule{2.77695pt}{0ex}}\text{transmits}\phantom{\rule{2.77695pt}{0ex}}\text{during}\phantom{\rule{2.77695pt}{0ex}}\text{odd}\phantom{\rule{2.77695pt}{0ex}}\text{phases},\hfill \end{array}\right.

(3)

for *i* = 1, 2, 3. We will see that the scheduler must know the channel gains for all relays and all phases in order to compute *λ* and assign *δ*_{
i
}.

We first review transmission schemes proposed in the literature where all relays receive in odd phases and transmit in even phases, i.e., *δ*_{
i
}= 0, *i* = 1, 2, 3. In detail, we consider the amplify-and-forward (AF), the decode-and-forward (DF), and the broadcast-multiaccess (BM) techniques. Let us denote the link spectral efficiency for a given SNR *η* as

C\left(\eta \right)={\text{log}}_{2}\left(1+\eta \right).

(4)

### 2.1. Amplify-and-forward

In AF each relay simply retransmits a scaled version of the received signal, i.e., *z*_{
i
}= *γ*_{
i
}*y*_{
i
}, observing the unitary power constraint {\gamma}_{i}^{2}\left({H}_{i}+1\right)\le 1. The signal received at node *D* can be written as

r=\sum _{i=1}^{3}\sqrt{{G}_{i}}{z}_{i}+{w}_{D}=x\sum _{i=1}^{3}{\gamma}_{i}\sqrt{{G}_{i}{H}_{i}}+{w}_{D}+\sum _{i=1}^{3}{w}_{i}{\gamma}_{i}\sqrt{{G}_{i}},

(5)

where *w*_{
D
}is complex Gaussian with zero mean and unitary variance. Note that with AF no optimization of the time-allocation *λ* is performed. Indeed, each relay retransmits the whole received signal from *S* towards *D*, therefore equal-time has to be assigned to both phases, i.e., *λ* = 1/2. The spectral efficiency is obtained by (4), where from (5) the signal power is {\left|{\sum}_{i=1}^{3}\sqrt{{G}_{i}{H}_{i}}{\gamma}_{i}\right|}^{2} and the noise power is 1+{\sum}_{i=1}^{3}{G}_{i}{\gamma}_{i}^{2}. Scaling factors *γ*_{
i
}are selected in order to maximize the network spectral efficiency, i.e.,

{R}^{\left(\text{AF}\right)}=\underset{{\gamma}_{i}\ge 0}{\text{max}}\frac{1}{2}C\left(\frac{{\left|\sum _{i=1}^{3}\sqrt{{G}_{i}{H}_{i}}{\gamma}_{i}\right|}^{2}}{1+\sum _{i=1}^{3}{G}_{i}{\gamma}_{i}^{2}}\right)

(6a)

s.t.

{\gamma}_{i}^{2}\left({H}_{i}+1\right)\le 1,\phantom{\rule{1em}{0ex}}i=1,2,3.

(6b)

Note that the factor 1/2 in (6a) is due to the equal duration of phases.

Moreover, we observe that (6) is a non-linear non-convex optimization problem in the three variables {*γ*_{
i
}}.

### 2.2. Decode-and-forward

With DF a relay decodes the information received from *S*, which is transmitted coherently with the other relays towards *D*[4]. A bottleneck of this scheme is the channel between *S* and each relay, as the spectral efficiency is strongly limited by the worst channel min_{
i
}{*H*_{
i
}}. For this reason, we also consider a selection of the relays involved in the operations. For each subset \mathcal{S}\subseteq \left\{1,2,3\right\}, in order to maximize the information rate from *S* to *D*, we impose the equality of the spectral efficiency in both phases, i.e.,

{R}^{\left(\text{DF}\right)}\left(\mathcal{S}\right)=\lambda C\left(\underset{i\in \mathcal{S}}{\text{min}}\left\{{H}_{i}\right\}\right)=\left(1-\lambda \right)C\left({\left(\sum _{i\in \mathcal{S}}\sqrt{{G}_{i}}\right)}^{2}\right).

(7)

The optimal value of *λ* is obtained by solving (7) for each subset \mathcal{S}, i.e.,

\lambda =\frac{C\left({\left(\sum _{i\in \mathcal{S}}\sqrt{{G}_{i}}\right)}^{2}\right)}{C\left(\underset{i\in \mathcal{S}}{\text{min}}\left\{{H}_{i}\right\}\right)+C\left({\left(\sum _{i\in \mathcal{S}}\sqrt{{G}_{i}}\right)}^{2}\right)}.

(8)

The network spectral efficiency is then computed optimizing the choice of \mathcal{S}, yielding

{R}^{\left(\text{DF}\right)}=\underset{\mathcal{S}\subseteq \left\{1,2,3\right\}}{\text{max}}{R}^{\left(\text{DF}\right)}\left(\mathcal{S}\right)=\underset{\mathcal{S}\subseteq \left\{1,2,3\right\}}{\text{max}}\frac{C\left(\underset{i\in \mathcal{S}}{\text{min}}\left\{{H}_{i}\right\}\right)C\left({\left(\sum _{i\in \mathcal{S}}\sqrt{{G}_{i}}\right)}^{2}\right)}{C\left(\underset{i\in \mathcal{S}}{\text{min}}\left\{{H}_{i}\right\}\right)+C\left({\left(\sum _{i\in \mathcal{S}}\sqrt{{G}_{i}}\right)}^{2}\right)}.

(9)

Note that (9) is an integer optimization problem, which can be easily solved by an exhaustive search among all the subsets of relay nodes.

### 2.3. Broadcast-multiaccess

Only for this section, we assume for simplicity that *H*_{1} ≥ *H*_{2} ≥ *H*_{3}. In the BM scheme the first phase is a Gaussian broadcast channel [11], where *S* transmits three messages *M*_{1}, *M*_{2}, and *M*_{3} at rates *R*_{1}, *R*_{2}, and *R*_{3}, respectively, where *M*_{1} is decoded only by relay 1, *M*_{2} is decoded by both relay 1 and relay 2, and *M*_{3} is decoded by all three relays. The second phase is a Gaussian multiple-access channel with correlated information [9], where relays send different but not independent information. We indicate with *α*_{1}, *α*_{2}, and *α*_{3} the powers used by *S* to transmit the messages *M*_{1}, *M*_{2}, and *M*_{3}, respectively, *γ*_{11}, *γ*_{12}, and *γ*_{1} the powers used by relay 1 to transmit *M*_{1}, *M*_{2}, and *M*_{3}, respectively, *γ*_{21} and *γ*_{2} the powers used by relay 2 to transmit *M*_{2} and *M*_{3}, respectively, and *γ*_{3} the power used by relay 3 to transmit *M*_{3}. Note that here we are extending the BM scheme with two relays of [8] to the case of three relays. Therefore, the network spectral efficiency is the solution of the following optimization problem:

{R}^{\left(\text{BM}\right)}=\underset{{\alpha}_{i},{\gamma}_{u},\lambda \ge 0}{\text{max}}{R}_{1}+{R}_{2}+{R}_{3}

(10a)

s.t.

{R}_{1}\le \lambda C\left({\alpha}_{1}{H}_{1}\right),

(10b)

{R}_{2}\le \lambda C\left(\frac{{\alpha}_{2}{H}_{2}}{1+{\alpha}_{1}{H}_{2}}\right),

(10c)

{R}_{3}\le \lambda C\left(\frac{{\alpha}_{3}{H}_{3}}{1+{\alpha}_{1}{H}_{3}+{\alpha}_{2}{H}_{3}}\right),

(10d)

{R}_{1}\le \left(1-\lambda \right)C\left({\gamma}_{11}{G}_{1}\right),

(10e)

{R}_{1}+{R}_{2}\le \left(1-\lambda \right)C\left({\gamma}_{11}{G}_{1}+{\left(\sqrt{{\gamma}_{12}{G}_{1}}+\sqrt{{\gamma}_{21}{G}_{2}}\right)}^{2}\right),

(10f)

\begin{array}{c}{R}_{1}+{R}_{2}+{R}_{3}\le \left(1-\lambda \right)C\left({\gamma}_{11}{G}_{1}+{\left(\sqrt{{\gamma}_{12}{G}_{1}}+\sqrt{{\gamma}_{21}{G}_{2}}\right)}^{2}+\right.\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\right)close=")">{\left(\sqrt{{\gamma}_{1}{G}_{1}}+\sqrt{{\gamma}_{2}{G}_{2}}+\sqrt{{\gamma}_{3}{G}_{3}}\right)}^{2}& ,\end{array}\n

(10g)

{\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3}\le 1,

(10h)

{\gamma}_{11}+{\gamma}_{12}+{\gamma}_{1}\le 1,

(10i)

{\gamma}_{21}+{\gamma}_{2}\le 1,

(10j)

Note that (i) constraints (10b)-(10d) represent the rate-region of the Gaussian broadcast channel between *S* and the relays, (ii) constraints (10e)-(10g) represent the rate-region of the Gaussian multiple-access channel with correlated information between the relays and *D*, and (iii) constraints (10h)-(10k) represent the power constraints at node *S* and at the relays, respectively. Similarly to (6), (10) is a non-linear non-convex optimization problem.