Before starting our problem formulation, we explain the system model used throughout this work to build our optimization problem and coalition formation algorithm.

### System model

Our approach can be viewed as an adaptive transmit diversity scheme. The transmission protocol starts, as in standard Glossy, with one node initiating the flood. This initial phase is a simple multicast transmission where one node transmits a message, and the nodes within a reception range receive it. After a successful reception of the message by, let us say, *K* nodes, these nodes become transmitters in the next round. In standard Glossy, all *K* nodes proceed to transmit the same information simultaneously in a non-cooperative way. Here, we propose to define *K*_{1}⊆*K* of cooperating nodes which acquire CSI from the receiver through limited feedback, in order to perform multicast beamforming. The remaining *K*_{2}=*K*−*K*_{1} nodes do not cooperate and operate in standard Glossy. Please note that *K*_{1}=*∅* is standard Glossy. An example of the system model is shown in Fig. 2, where *K*=5 nodes, which have successfully received and decoded the message from the last transmission, send the same data *x* to *L*=3 receiving nodes. First, CSI is fed back from the receivers to the transmitters through a limited feedback channel as shown in Fig. 2 using random vector quantization as thoroughly explained in Section 2.2. Then, each transmitting node constructs an estimated channel matrix *H* using the acquired feedback and proceeds to calculate an energy-efficient coalition to which all or subset of the *K* nodes (*K*_{1} nodes) can participate. The formation of this coalition is based on a greedy approach that can be executed at each transmitting node independently. This is explained in more details in Section 5. Afterwards, the information is divided into *K*=5 symbols where the first three symbols [*x*_{1},*x*_{2},*x*_{3}] are cooperatively transmitted by *K*_{1} nodes one to three, while symbols *x*_{4} and *x*_{5} are transmitted by *K*_{2} nodes four and five, respectively. The coalition of the three nodes solves an energy efficiency optimization problem that will be formulated and discussed in Section 3 and cooperatively transmits the information forming a virtual antenna array that uses energy-efficient robust beamforming by applying the precoding matrix *W*, while the remaining two nodes operate in standard Glossy mode and transmit using maximum power *p*. The received signal at any receiver *l* can be characterized as follows:

$$ y_{l} =\boldsymbol{x}^{T}\boldsymbol{U} \boldsymbol{h}_{l} + z_{l} $$

(1)

where cooperation is controlled by the *K* ×*K* matrix \( \boldsymbol {U}= \text {blockdiag }(\boldsymbol {W}, \sqrt {p}\boldsymbol {I})\), *x*^{T}=[*x*_{1}...*x*_{5}] are the data symbols, *W* is the *K*_{1}*xK*_{1} beamforming matrix, \(\boldsymbol {h}_{l}^{T} = [h_{1l},... h_{5l}]\) is the channel vector between receiver *l* and the transmitting nodes, and *z*_{l} is the receiver noise. The [ ]^{T} describes the transpose operation. After the signal reception, interference cancelation methods (i.e., minimum mean square error (MMSE) and successive interference cancelation (SIC)) are applied at each receiving node before the original message *x* can be retrieved. The maximum rate achieved by applying such processing at any receiver node *l* to the received signals can then be calculated as follows:

$$ {} R_{l}^{*} \,=\, \log \left(1\,+\,\frac{\left|\boldsymbol{x}^{T}\boldsymbol{U}\boldsymbol{h}_{l}\right|^{2}}{{\sigma_{l}^{2}}}\right) \,=\, \log \left(1\,+\, \frac{\boldsymbol{h}_{l}^{H} \boldsymbol{U}^{H} E\left[\boldsymbol{x}^{*} \boldsymbol{x}^{T}\right] \boldsymbol{U} \boldsymbol{h}_{l}}{{\sigma_{l}^{2}}}\right) $$

(2)

where *E*[ ] is the expectation operation, and *E*[*x*^{∗}*x*^{T}] is the transmit covariance matrix as shown in [28]. By applying MMSE and SIC techniques as demonstrated in ([29], ch. 10), it can be equivalently described as:

$$ {\begin{aligned} R_{l}^{*} \,=\, \log \left(1+ \frac{ \left[\begin{array}{l}h_{1l}\\h_{2l}\\h_{3l}\\ \end{array}\right]^{H} \boldsymbol{W}^{H} \boldsymbol{C_{3}} \boldsymbol{W} \left[\begin{array}{l}h_{1l}\\h_{2l}\\h_{3l}\\ \end{array}\right]}{{\sigma_{l}^{2}}} \,+\, \frac{|h_{l4}|^{2} E \left[|x_{4}|^{2}\right]}{{\sigma_{l}^{2}}} \,+\, \frac{|h_{l5}|^{2} E \left[|x_{5}|^{2} \right]}{{\sigma_{l}^{2}}}\right) \end{aligned}} $$

(3)

where *h*_{il} is the channel coefficient from transmitter *i* to receiver *l*, *C*_{3} is the covariance matrix of data in the cooperating transmitters and is equal to the identity matrix *I*, the [ ]^{∗} and [ ]^{H} describe the conjugate and conjugate transpose operations, respectively. Using the definition of Gram matrix ([30], p. 55), we can rewrite the achievable rate equation as follows:

$$ {} R_{l}^{*} = \log \left(1+ \frac{\left[\begin{array}{l}h_{1l}\\h_{2l}\\h_{3l}\\ \end{array}\right]^{H} \boldsymbol{G} \left[\begin{array}{l}h_{1l}\\h_{2l}\\h_{3l}\\ \end{array}\right]}{{\sigma_{l}^{2}}} + \frac{|h_{l4}|^{2} p}{{\sigma_{l}^{2}}} + \frac{|h_{l5}|^{2} p}{{\sigma_{l}^{2}}}\right) $$

(4)

where

$$ \boldsymbol{G}= \boldsymbol{W}^{H}\boldsymbol{W} $$

(5)

is a positive semi-definite matrix. This can be generalized to the case of *K*=*K*_{1}+*K*_{2} transmitters with *K*_{1} cooperating nodes and *K*_{2} non-cooperating nodes as follows:

$$ R_{l}^{*} = \log \left(1+ \frac{\boldsymbol{h}_{l} \boldsymbol{G} \boldsymbol{h}_{l}^{H} }{{\sigma_{l}^{2}}} + \sum_{i =K_{1}+1}^{K}\frac{|h_{{il}}|^{2} p}{{\sigma_{l}^{2}}}\right) $$

(6)

where \(\boldsymbol {h}_{l}^{T} = [h_{1},h_{2}... h_{K_{1}}]\), *G* is a *K*_{1}×*K*_{1} positive semi-definite matrix, and *p* is the maximum transmit power of the node. Moreover, for the case where all transmitting nodes are cooperating, the maximum achievable rate can be described as:

$$ R_{l}^{*} = \log \left(1+ \frac{\boldsymbol{h}_{l} \boldsymbol{G} \boldsymbol{h}_{l}^{H} }{{\sigma_{l}^{2}}}\right) $$

(7)

with *K* ×*K* positive semi-definite matrix *G*.

#### Channel model

In order to examine the effect of limited feedback on a complex communication scenario such as Glossy, we must specify our channel characteristics. We start with the signal received at receiver *l*, described in (1). Channel vectors of receivers 1 to *L* with \(\mathbf {h}_{l}^{T} = [h_{1l}, h_{2l},....h_{{Kl}}]\) and \(h_{{kl}} \in \mathbb {C}\) is the channel gain from transmitter *k* to receiver *l* modeled as quasi-static, independent, and experience flat-fading on the block length, which is a valid assumption for low-power wireless devices such as Zigbee ([31], E 5.3). These coefficients are calculated according to *h*_{kl}=*w*_{kl}*g*_{kl}, where small-scale fading *w*_{kl} follows a complex normal distribution \(\mathcal {CN}(0,1)\), and large-scale fading *g*_{kl} depends on the distance *d*_{kl} between transmitter *k* and receiver *l* ([31], E 5.3) as follows:

$$ g_{{kl}} = \left\{ \begin{array}{ll} 40.2+20 \log(d_{{kl}}) &, d_{{kl}} \leq 8~{\,\mathrm{m}}\\ 58.5+33 \log({d{kl}/8}) &,d_{{kl}} > 8~{\,\mathrm{m}} \end{array} \right. $$

(8)

which corresponds to a path loss exponent of 2 for the first 8 m and a path loss exponent of 3.3 for distances larger than 8 m. To ensure *g*_{kl}>0 according to (8), we consider only the topologies where the distance between any pair of nodes exceeds 0.1 m, which is reasonable in real deployments [32]. We denote the normalized channel vector \(\tilde {\mathbf {h}_{l}} = \frac {\mathbf {h}_{l}}{||\mathbf {h}_{l}||}\), where \(\tilde {\mathbf {h}_{l}}\) is a unit vector that describes only the channel direction. Moreover, in our model, transmitters are assumed to know the channel quality information (CQI), represented by ||**h**_{l}||, perfectly, while each receiver *l* is assumed to know local CSI **h**_{l} through pilot training [33, 34] at the beginning of transmission which is not the focus of this work. CSI is then quantized and shared with transmitting nodes through a limited feedback channel as described in Section 2.2. Finally, *z*_{l} is the additive white Gaussian noise with a complex normal distribution \(z_{l} \sim \mathcal {CN}(0,{\sigma _{l}^{2}})\).

### CSI feedback

Quantization is done using randomly generated vector quantization codebooks independently at each receiver. A codebook \(\mathcal {C}\) contains 2^{B}*K*-dimensional unit norm vectors randomly drawn from the isotropic distribution on the *K*-dimensional unit sphere where \(\mathcal {C} \overset {\Delta }{=} \{\mathbf {c}_{1},\mathbf {c}_{2},....\mathbf {c}_{2^{B}} \} \). In addition to being mathematically tractable, random vector quantization (RVQ) performs close to optimal quantization as feedback bits *B*→*∞* [35]. This can be independently applied at each receiver *l* where the closest vector **c**_{i} to the normalized real channel \(\tilde {\mathbf {h}_{l}}\) is chosen. The index *i* is then broadcasted to active transmitters using *B* feedback bits. The RVQ process is performed such that the closest vector **c**_{i} to the normalized real channel \(\tilde {\mathbf {h}_{l}}\) has to be chosen. We consider the Euclidean distance, defined as the norm of the difference between **c**_{i} and \(\tilde {\mathbf {h}_{l}}\), as a measure of closeness. Each receiver must choose the closest vector **c**_{i} to the normalized channel \(\tilde {\mathbf {h}_{l}}\) independently. This is done by solving the following problem:

$$ \begin{aligned} & i_{l} = \text{arg} & & \min\limits_{1 \leq i \leq 2^{B}} ||\tilde{\mathbf{h}_{l}}- \mathbf{c}_{i}||^{2}\\ \end{aligned} $$

(9)

Since the norm of the error vector is:

$$ \begin{aligned} & ||\delta_{l}||^{2} & & = ||\tilde{\mathbf{h}_{l}}-\hat{\mathbf{h}_{l}} ||^{2}\\ \end{aligned} $$

(10)

where the \(\hat {\mathbf {h}_{l}}\) is the quantized channel vector based on the Euclidean distance, this manner of choosing the quantized channel vector results in the minimum error norm possible. Unfortunately, the statistical behavior of this error is not available in closed form.

#### Feedback overhead

Feedback is the operation of sharing quantized CSI between receivers and transmitters. Each receiver node *l* must quantize and share its CSI using the feedback channel. Thanks to the broadcast nature of the wireless medium and assuming error-free feedback channels, it is safe to assume that any wireless signal received successfully by the furthest node, can be received by all other nodes, and the power needed for reliable transmission is at least:

$$ P_{l} = (2^{B}-1) \max\limits_{1 \leq j \leq K, j \neq l} |d_{{lj}}|^{2} $$

(11)

where *d*_{lj} is the distance from receiver *l* to transmitter *j*, operator |.| denotes the Euclidean norm, and *P*_{l} is defined as the transmission power needed to distribute CSI from receiver *l* to all transmitters. The total feedback energy overhead is computed as follows:

$$ P_{\text{coop}} = (2^{B}-1) \sum\limits_{l=1}^{L} \max\limits_{1 \leq j \leq K, j \neq l} |d_{{lj}}|^{2} $$

(12)

where we can see the exponential dependency of *P*_{coop} on the number of feedback bits *B* and linear dependency on the distances between receivers and transmitters. This means that increasing the number of feedback bits *B* will result in a higher cooperation over head in terms of *P*_{coop} but will also result in a more accurate estimation of the channels, leading to less transmit power. Therefore, there must be an optimum number of feedback bits *B* that minimizes the total energy consumption and maximizes energy efficiency. This brings us to our problem statement, which we present in Section 2.3.

### Problem statement

In order to efficiently manage the network resources and achieve high energy efficiency in the state-of-the-art Glossy [36, 37], we seek to analyze the feasibility of using power control and transmission cooperation schemes such as coherent multicast beamforming. To achieve our goal, we first look at the problem of limited feedback, where receiving nodes try to share their CSI with transmitting nodes using *B* feedback bits. Increasing the number of feedback bits *B* used in CSI sharing results in a better representation of the channels and lower quantization error. However, the consumed energy overhead by reliably transmitting these feedback bits increases as well. This results in a clear trade-off between the cooperation gain achieved by coherent multicast beamforming among the *K* transmitting nodes and the consumed energy overhead due to CSI feedback. Therefore, there must exist an optimal *B*, dependent on network parameters such as topology, number of transmitters *K*, and density of the network, that maximizes energy efficiency. This allows us to solve a programming problem, that depends on the CSI uncertainty parameter *ε*_{l}, and in turn optimal *B*, that maximizes energy efficiency under quality of service constraints (QoS) and guarantees successful reception as follows:

$$ {{} \begin{aligned} &\min\limits_{\boldsymbol{G}} & &{tr}\,\left({\boldsymbol{G}}\right) \end{aligned}} $$

(13)

$$ {{} \begin{aligned} & \text{subject to} & &\!\!\!\!\!\!\!\left(\hat{\mathbf{h}_{l}} \,+\, \mathbf{\delta}\right) \boldsymbol{G}\left(\hat{\mathbf{h}_{l}} \,+\, \mathbf{\delta}\right)^{H} \geq \frac{\left(2^{R}-1\right){{\sigma_{l}^{2}}}}{||\mathbf{h}_{l}||^{2}} ~;~\forall l \in \mathcal{L}~, \end{aligned}} $$

(14)

$$ {\begin{aligned} &&&\forall \mathbf{\delta} : ||\mathbf{\delta}||\leq \epsilon,\\ \end{aligned}} $$

(15)

$$ {\begin{aligned} &&&\boldsymbol{G}\succeq 0. \end{aligned}} $$

(16)

where all transmitters are assumed to be cooperating and minimizing the beamforming transmit powers *tr* (*G*) and the feedback power under three inequality constraints. Inequality (14) ensures the achievable rate robustness against CSI uncertainty, inequality (15) limits the maximum CSI uncertainty error, and inequality (16) insures that cooperation multicast beamforming covariance matrix *G* is positive semi-definite. This programming problem results in maximizing the energy efficiency and is solved and discussed in more details in Sections 3 and 4. Lastly, we look at the problem of choosing cooperating nodes. We investigate how the choice of cooperating transmitters *K*_{1} affects energy efficiency, whether there exists an optimum coalition and how to form it.