# Coherent Communications in Self-Organizing Networks with Distributed
Beamforming

Yi Shi and Yalin E. Sagduyu Yi Shi is with Virginia Tech, Blacksburg, VA
24061, USA, and Intelligent Automation, Inc., Rockville, MD 20855, USA
(e-mail: yshi@vt.edu). Yalin Sagduyu is with Intelligent Automation, Inc.,
Rockville, MD 20855, USA (e-mail: ysagduyu@i-a-i.com).This research was
supported under contract D17PC00022, from Defense Advanced Research Projects
Agency (DARPA).Distribution Statement A (Approved for Public Release,
Distribution Unlimited). The views, opinions and/or findings expressed are
those of the author and should not be interpreted as representing the official
views or policies of the Department of Defense or the U.S. Government.©2019
IEEE. Personal use of this material is permitted. Permission from IEEE must be
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###### Abstract

Coherent communications aim to support higher data rates and extended
connectivity at lower power consumption compared with traditional point-to-
point transmissions. The typical setting of coherent communication schemes is
based on a single data stream with multiple transmitters and a single
receiver. This paper studies the case of multiple concurrent data streams,
each with multiple transmitters and multiple receivers, in self-organizing
wireless networks. A distributed optimization solution based on joint network
formation and beamforming is developed for coherent group communications in a
network of nodes that need to communicate over long distances. This solution
significantly improves the power gain for a single data stream and the signal-
to-interference-ratio for multiple data streams compared to point-to-point
communications. These gains are in turn translated to improvements in
communication range, power efficiency, reliability, and throughput. In this
multi-layer network optimization solution, nodes coordinate with each other in
a distributed manner to form transmitter and receiver groups, and communicate
with each other coherently over long distances. The coherent beamforming
protocol is optimized for given transmitter and receiver groups, whereas the
network formation protocol is optimized to determine these groups. By using
single-antenna communication nodes, the proposed optimization solution
provides a major gain in network communications and outperforms other
benchmark combinations of beamforming and network formation protocols.

###### Index Terms:

Coherent communications, distributed protocols, beamforming, network
formation, power gain.

##  I Introduction

Wireless communications serve various mobile applications such as multimedia,
remote sensing, and inter-vehicle communications that demand high data rates
over long distances but they often need to run on battery-operated devices
with limited transmit powers. Many advanced techniques such as directional and
multiple-input-multiple-output (MIMO) communications have been developed to
support high-rate communications with low power consumption over long
distances, which cannot be met by traditional point-to-point transmissions
(from one transmitter to one receiver). Among these techniques, _coherent
communications_ use only a single omnidirectional antenna per node and has
emerged as a viable solution to improve the _power gain_ between distant
groups of nodes, which consequently leads to the better signal quality at
receivers [2, 3, 4, 5]. This power gain is up to $N^{2}M$ for $N$ transmitters
communicating with $M$ receivers when the number of data streams $K$ is $1$.
Examples for coherent communications include inter-vehicle communications,
air-to-ground and ground-to-air communications (such as involving UAVs). One
particular application that benefits from maximizing the received power is
communications under jamming where high SINR needs to be maintained to balance
destructive jamming effects.

A typical assumption on coherent communications is that there is one data
stream and there is one receiver for this data stream, i.e., $K=1$ and $M=1$.
In this paper, we consider the extension to multiple data streams ($K>1$)
operating simultaneously and potentially interfering with each other. There
are multiple transmitters ($N>1$) and multiple receivers ($M>1$). For each
data stream, a source node selects some of these transmitters to communicate
to some of the receivers selected by the corresponding destination. This
extension for self-organizing wireless networks requires joint optimization of
beamforming and network formation in a distributed manner.

In a wireless network where nodes are not physically connected, it is
necessary to coordinate nodes with each other to form transmitter and receiver
groups for _distributed coherent communications_. Without using any
centralized controller, coherent communication is supported with three
protocol steps. First, nodes self-organize themselves into transmitter and
receiver groups in an ad hoc network setting without a centralized controller,
where the source nodes is in the transmitter group and the destination node is
in the receiver group. Second, nodes exchange data and channel condition in
the transmitter group and transmit with coherent beamforming to the receiver
group. Third, nodes receive and exchange signals in the receiver group to
decode data at the destination. We define a time frame for coherent
beamforming with the above steps, which includes the data exchange in the
transmitter group, inter-group communications, and the data exchange in the
receiver group.

To enable coherent group communications, we develop a distributed optimization
solution that combines a _distributed transmit beamforming protocol_ for given
transmitters and receivers at the PHY layer (that sets the initial phase value
at each transmitter to achieve phase coherence at receivers) and a
_distributed network formation protocol_ at the network layer (that selects
and maintains a transmitter group for each source and a receiver group for
each destination). Our approach is first building a basis in theoretical
system model analysis on power gain, then designing a distributed coherent
beamforming protocol (given a transmitter group and a receiver group), and
finally developing joint beamforming and network formation protocols through a
_multi-layer network optimization_ solution across PHY and network layers.

  * •

_Distributed coherent beamforming protocol._ We develop a distributed coherent
beamforming protocol for given a group of $N$ transmitters and a group of $M$
receivers. The coherent communication gain is defined as the ratio of the
total received energy by coherent communications and the received energy by
point-to-point transmissions. Existing coherent beamforming protocols are
mainly designed for $M=1$ (single receiver) and the optimal solution with the
maximum gain can be achieved if phase coherence is achieved at the receiver.
For the general case of $M>1$, it is not clear which phase coherent solution
can achieve the maximum coherent communication gain. To solve this problem, we
first derive a closed-form expression of the coherent communication gain for
any solution and then analyze its upper bound and assess the conditions to
achieve this upper bound. We construct an optimization problem to maximize the
coherent communication gain. We then present several coherent beamforming
protocols and compare their performance in simulation. We show that _iterative
optimization_ protocol achieves the best gain that is very close to the
theoretical upper bound.

  * •

_Analysis and simulation on coherent beamforming._ We analyze the
synchronization requirement for coherent beamforming and show that only
transmitters should be synchronized. We analyze the impact of Doppler spread
and show the tolerable Doppler spread as a function of distance between
transmitter and receiver groups. Then we analyze the effects of group size,
inter-group distance, and channel model for the developed coherent beamforming
protocols. We show that the iterative optimization protocol achieves the best
performance among the developed protocols and achieves near-optimal
performance by comparing with the upper bound. Then we study the performance
over distance and find the advantage of coherent beamforming as the inter-
group distance increases.

  * •

_Distributed network formation protocol._ We develop a protocol at the network
layer that selects transmitters and receivers for each source-destination pair
to improve the coherent communication gain. Existing designs for coherent
communications mainly focus on the case of a single source-destination pair
($K>1$). Since there is only one source and one destination, all nodes close
to the source operate as transmitters and all nodes close to the destination
operate as receivers. With multiple source-destination pairs, nodes need to be
assigned to sources/destinations. Moreover, there is interference from data
streams for other sources. We use the signal-to-interference-ratio (SIR) as
the performance measure and define an objective function based on multiple
SIRs for all destinations. While the exhaustive search on network formation
can find the optimal solution that maximizes this gain, this solution can only
be used as a benchmark due to its high complexity. Therefore, we design and
compare low-complexity and distributed protocols for network formation and
beamforming. We show that the best solution is achieved when each transmitter
is assigned to the closest source, each receiver is assigned to the closest
destination, and coherent beamforming is performed towards the best selection
of receiver.

Both analytical and simulation results show significant power gains of a
coherent system than a non-coherent system and demonstrate the feasibility of
the proposed multi-layer network optimization solution using single-antenna
communications.

The rest of the paper is organized as follows. Section II discusses the
related work. Section III describes the system architecture. Section IV
analyzes the requirements for coherent communications. Section V designs the
protocol for coherent beamforming from a set of transmitters to a set of
receivers. Section VI designs the network formation protocol that determines a
set of transmitters and a set of receivers for each coherent communication
group and the beamforming protocol for each group. Section VII concludes the
paper. Appendix provides proofs of Theorems 1–3 and Corollary 1.

##  II Related Work

Coherent communications were studied mainly for a _single data stream_ ($K=1$)
with _multiple transmitters_ ($N>1$) and a _single receiver_ ($M=1$). Most of
these works considered applications in sensor networks [6]. In [7], a coherent
beamforming scheme was designed for multiple-input-single-output (MISO)
communications with finite-rate feedback to minimize the transmit power. In
[8], an open-loop coherent beamforming scheme was designed for MISO
communications. The impact of phase errors on distributed coherent beamforming
was analyzed in [9]. For wireless sensor networks, a distributed beamforming
scheme was designed in [10] to improve network connectivity and energy
efficiency. In [11], coherent beamforming was applied for MISO to minimize
transmit power, thereby showing the benefits of coherent communications.
Distributed transmit beamforming was designed and implemented with software-
defined radio (USRP) [12, 13]. In [14], a distributed transmit beamforming
scheme was developed for wideband communications using channel reciprocity and
relative calibration. A distributed beamforming scheme was designed in [15]
for noise-resilient wideband communications, and the aggregate feedback was
considered in [16]. Coherent beamforming was also studied for radar systems
[17]. We consider a MIMO scheme in this paper. It can be regarded as a
summation of multiple MISO schemes if we do not consider the time to schedule
multiple MISO transmissions. MIMO scheme can transmit data to $M$ receivers in
one time slot while the corresponding MISO scheme needs $M$ time slots to
transmit data to all $M$ receivers. As a consequence, the scheme of multiple
MISO transmissions would significantly reduce the throughput that can be
achieved by MIMO transmissions.

There are many studies on self-organizing networks and they are mostly focused
on clustering in wireless networks, e.g., [18, 19, 20, 21]. An adaptive fuzzy
clustering protocol was designed in [18] to prolong the sensor network
lifetime. In [19], a self-organizing and adaptive clustering solution was
developed to improve the sensor network performance such a delivery ratio,
delay, and lifetime. Both a centralized and a distributed clustering algorithm
were designed in [20] for sensor networks. Unequal clustering (considering
smaller size clusters near to the sink and larger size clusters relatively far
away from the sink) was designed in [21] along with a routing algorithm. This
unequal clustering achieves better traffic load balance and thus improves
performance such as lifetime, delivery ratio, and convergence rate. Some
related works considered noncoherent beamforming and clustering, e.g., [22,
23, 24]. [22] and [23] aimed to maximize the total data rate via beamforming
and clustering for download traffic. In [24], an energy efficient solution was
designed to meet Quality of Service (QoS) requirements of users. These
clustering solutions are not designed to optimize the performance of coherent
communications.

There are only a few works on joint coherent beamforming and network
formation. [25] aimed to minimize the transmission power of nodes for sensor
networks under the finite-rate feedback by considering only the MISO case,
i.e., a single receiver ($M=1$). In [1], we considered the preliminary study
of coherent beamforming and network formation for the case of multiple
transmitters and multiple receivers, which includes MISO as a special case. In
this paper, we consider additional settings, including the impact of Doppler
spread, additional channel model (the two-ray model), new beamforming
protocols, and extension of joint beamforming and network formation protocols
to better characterize the gains of distributed coherent group communications.

##  III System Model

We consider a coherent communications scenario, where $N$ transmitters
transmit data coherently to $M$ receivers. The distance between transmitter
$i$ and receiver $j$ is $d_{ij}$. In Section V, we consider a single data
steam ($K=1$). In Section VI, we extend the setting to simultaneous operation
of $K>1$ data streams such that $N$ transmitters transmit data coherently to
$M$ receivers for all data streams. Among $M$ transmitters, there is one
source node that has data to transmit to one destination node, which is one of
$M$ receivers. Nodes perform distributed protocols in three main steps as
shown in Fig. 1.

  1. 1.

_Clustering_ : Nodes cluster each other into (disjoint) transmitter and
receiver groups.

  2. 2.

_Transmit protocols_ :

    1. (a)

_Data exchange in the transmitter group_ : The source node broadcasts data in
the transmitter group.

    2. (b)

_Inter-group communications_ : The transmitter group sends data to the
receiver group in three steps:

      1. i.

_channel estimation_ : Nodes in the transmitter group send pilot signals and
all channels are estimated at the receivers.

      2. ii.

_feedback_ : The channel state information is sent back to nodes in the
transmitter group.

      3. iii.

_data transmission/beamforming_ : All nodes in the transmitter group transmit
with appropriate amplitudes and phases to the receiver group.

  3. 3.

_Data exchange in the receiver group_ : The destination node collects the
received samples from other nodes in the receiver group.

Figure 1: Steps of distributed coherent group communications.

We require the inter-group channels (namely, channels among nodes in the same
group) to remain static during the coherent transmission duration. The nodes
in the transmitter group exchange the data prior to the coherent transmission
duration. The nodes in the receiver group exchange their signals for coherent
combining after the coherent transmission from the transmitter group.
Therefore, the intra-group channels do not need to be static during the intra-
group data exchange period. To perform coherent communications, it is
necessary to collect channel information, e.g., channel gain and phase shift
[26, 27, 28], between each pair of transmitter and receiver in channel
sensing. A simple channel sensing scheme is time multiplexed training, where
transmitters transmit the training signal with length $T_{t}$ sequentially.
Since each transmitter-receiver pair has a different propagation delay
$t_{ij}$, the previous transmission may interfere with the current one.
Therefore, guard period $T_{g}$ between transmissions is used to avoid this
type of interference. The total training time $N(T_{t}+T_{g})$ increases
linearly with the number of transmitters, $N$. This overhead reduces the
average data rate achieved by coherent beamforming over all three steps.

##  IV Coherent Communications Requirement

###  IV-A Synchronization Requirement

For coherent communications, it would be the best if we can synchronize all
transmitters and receivers. However, this may be a challenging task,
especially when transmitters and receivers are far away from each other. It
turns out that such a strict requirement is not needed. Instead, the only
requirement to support coherent communications is to synchronize transmitters,
as stated in Theorem 1. The proof of Theorem 1 is given in Appendix A.

######  Theorem 1.

To achieve coherent beamforming at a receiver, the only requirement for
synchronization is that transmitters are synchronized.

Theorem 1 shows that there is no need to synchronize transmitters and
receivers together for coherent communications. Further, it is easy to see
that phase coherence at a receiver can be achieved without the consideration
of other receivers. Thus, synchronization among receivers is also not needed.
The problem of synchronization between transmitters has been extensively
studied, see e.g., [29, 30], and thus in this paper we assume that such
synchronization is available.

Figure 2: Doppler spread vs. range.

###  IV-B Doppler Spread Tolerance

We characterize the _Doppler spread_ that can be tolerated to transmit
coherently, accounting for channel estimation, quantization, and feedback.
Doppler spread and coherence time are inversely proportional. Doppler spread
measures the spectral broadening due to the rate of channel changes.

We use the following parameters to assess the tolerable Doppler spread.
Transmitter power is $P_{t}=10$ dBm, receiver noise figure is $N_{r}=3$ dBm,
noise floor is $N_{0}=-174$ dBm, signaling bandwidth is $W=1$ MHz, distance
$D$ between groups varies from $1$km to $10$km, number of nodes within a group
is $N=10$, center frequency is $f_{c}=2.4$GHz, and speed of light is $c=3\cdot
10^{8}$m/s. Free-space propagation is used to compute the path loss. The SNR
required for accurate channel estimation is $\gamma_{t}=20$ dBm and the SNR
required for channel state feedback to be received successfully at the
transmitter group is $\gamma_{f}=10$ dBm. Number of bits for each channel
estimate is $B_{c}=16$, number of bits for general header is $B_{h}=10$, and
overhead percentage is selected as either $F_{o}=10\%$ or $50\%$ (remaining
time is dedicated to data transmission using beamforming).

The calculation of tolerable Doppler spread is as follows. For wavelength
$\lambda=\frac{c}{f_{c}}$, received power in dBm is
$P_{r}=P_{t}+20\log\frac{\lambda}{4\pi D}$ and received SNR in dBm is
$\gamma=P_{r}-10\log(W)-N_{r}-N_{0}$. Then the processing gain needed for
training in dBm is $g_{t}=\gamma_{t}-\gamma$ and the processing gain needed
for feedback in dBm is $g_{f}=\gamma_{f}-\gamma$. Number of training bits is
$B_{t}=\max\\{10^{g_{t}/10},N\\}$ and the number of total feedback bits is
$B_{f}=\max\\{10^{g_{f}/10},1\\}NB_{c}+B_{h}N$. Then total overhead (in bits)
is given by $B_{o}=B_{t}+B_{f}$. This overhead is translated to time duration
$T_{o}=\frac{B_{o}}{W}+\frac{3D}{c}$ (by accounting for the three-step
transmission of pilot signal, feedback, and data). Then the minimum coherence
time is given by $T_{c}=\frac{T_{o}}{F_{o}}$ when overhead percentage is
$F_{o}$. From $T_{c}$, tolerable Doppler spread is computed as
$S=\frac{1}{T_{c}}$. Figure 2 shows that the tolerable Doppler spread
decreases as the distance between transmitter and receiver groups increases.

##  V Coherent Beamforming

After channel gain and phase shift are obtained between any transmitter and
receiver, and synchronization is achieved among transmitters, we now present
the coherent beamforming protocol to maximize the power gain at receivers.

###  V-A Coherent Communication and Benchmark Scenarios

We consider a coherent communication scenario for one data stream from
transmitter $1$ to receiver $1$. There are $N-1$ additional transmitters and
$M-1$ additional receivers that also participate in coherent communications.
Suppose transmitters use a sine wave for the signal with the same signal
amplitude $A$ and signal period $T$, i.e., signal from transmitter $n$ is
$A\sin(\frac{2\pi}{T}t+\theta_{n})$, where $\theta_{n}$ is the initial phase
value that will be determined by a coherent beamforming protocol. Denote
channel gain and phase shift from transmitter $n$ to receiver $m$ as
$h_{nm}^{2}$ and $\theta_{nm}$, respectively, which are obtained in channel
estimation. Thus, receiver $m$ receives the signal
$Ah_{nm}\sin(\frac{2\pi}{T}t+\theta_{n}+\theta_{nm})$ from transmitter $n$.
The total received energy during a period at all receivers is

$\displaystyle
E_{NM}=\sum_{m=1}^{M}\int_{t=0}^{T}\left(\sum_{n=1}^{N}Ah_{nm}\sin\left(\frac{2\pi}{T}t+\theta_{n}+\theta_{nm}\right)\right)^{2}dt.$

We define the benchmark scenario as the case of $N=1$ and $M=1$ (i.e., point-
to-point), where coherent communication is not applied. The received energy
during a period is

$\displaystyle
E_{11}=\int_{t=0}^{T}\left(Ah_{11}\sin\left(\frac{2\pi}{T}t+\theta_{1}\right)\right)^{2}dt=\frac{A^{2}h_{11}^{2}T}{2}\;.$

###  V-B Coherent Communication Gain

We define the ratio of the total received power under the coherent
communication scenario and the received power $\frac{A^{2}h_{11}^{2}T}{2}$
under the benchmark scenario as the coherent communication gain
$G(\bm{\theta})$ that is a function of beamforming parameters $\bm{\theta}$
(where the $i$th entry of vector $\bm{\theta}$ is the phase angle $\theta_{i}$
of node $i$). Theorem 2 presents the coherent communication gain. The proof of
Theorem 2 is given in Appendix B.

######  Theorem 2.

The power gain of coherent communications for one data stream with $N$
transmitters and $M$ receivers is

$\displaystyle
G(\bm{\theta})=\frac{1}{h_{11}^{2}}\sum_{m=1}^{M}\beta_{m}(\bm{\theta})\;,$
(1)

where $\beta_{m}(\bm{\theta})$ is given by

$\displaystyle\beta_{m}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\left(\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})\right)^{2}+$
$\displaystyle\left(\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})\right)^{2}\;.$

The upper bound on the coherent communication gain is stated in Corollary 1.
Its proof is given in Appendix C.

######  Corollary 1.

$G(\bm{\theta})$ has an upper bound $G_{\text{UB}}=N^{2}M$ if
$h_{nm}^{2}\approx h_{11}^{2}$ for $m=1,\cdots,M$ and $n=1,\cdots,N$.

Note that we assume the same transmit power at each transmitter and thus we
define $A$ as the signal amplitude of a transmitter. For the general case of
different transmit powers, we can define $A_{n}$ as the signal amplitude of
transmitter $n$ and replace each $h_{nm}$ by $A_{n}h_{nm}$. Then the result in
Theorem 2 is changed to

$\displaystyle G(\bm{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{A_{1}^{2}h_{11}^{2}}\sum_{m=1}^{M}\beta_{m}(\bm{\theta})\;,$
$\displaystyle\beta_{m}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\left(\sum_{n=1}^{N}A_{n}h_{nm}\cos(\theta_{n}+\theta_{nm})\right)^{2}+$
$\displaystyle\left(\sum_{n=1}^{N}A_{n}h_{nm}\sin(\theta_{n}+\theta_{nm})\right)^{2}\;.$

The condition in Corollary 1 is changed to $A_{n}^{2}h_{nm}^{2}\approx
A_{1}^{2}h_{11}^{2}$ for $m=1,\cdots,M$ and $n=1,\cdots,N$.

Note that this formulation can be extended to include interference effects and
provide the capability of distributed interference cancellation (such as
studied in [31]). We compare the improvement in data rate relative to a long-
haul point-to-point link without coherent communications (such as studied in
[32, 33, 34]). We set the coherence time to be $100$ milliseconds. We assume
the optimum beamforming gain of $N^{3}$ with $N$ nodes in transmitter and
receiver groups ($M=N$). Figure 3 shows the results. Note that there is a
threshold on the inter-group distance such that if the inter-group distance is
larger than this threshold, coherent communications are better than point-to-
point transmissions. Point-to-point transmissions can have larger data rate
than coherent communications if the inter-group distance is less than this
threshold, since the coherent communication gain is not large enough to
compensate the coherent communication overhead, namely the time to exchange
data in the transmitter/receiver group plus the time to collect more channel
information.

Figure 3: Coherent communications performance.

###  V-C Coherent Beamforming Protocols

The optimization problem for coherent beamforming is

$\displaystyle\max$ $\displaystyle G(\bm{\theta})$ (2) subject to
$\displaystyle(\ref{eq:gainnm})$ variables:
$\displaystyle\bm{\theta}=\\{\theta_{n},0\leq\theta_{n}\leq 2\pi,1\leq n\leq
N\\}\;.$

This optimization problem can be easily solved if $M=1$. The optimal solution
is not unique and one such solution is $\theta_{1}=0$ and
$\theta_{n}=\theta_{1}+\theta_{11}-\theta_{n1}$ for any $n\geq 2$, which
achieves phase coherence at receiver $1$. This case is called as multiple-
input-single-output (MISO) and was widely studied (see e.g., [7, 9, 10, 11,
12, 14, 15, 16, 25]). However, if $M>1$, there is no solution to achieve phase
coherence at all receivers in general. The optimization problem is not convex
and thus an optimal solution may not be found in polynomial time.

We design two coherent beamforming protocols for this optimization problem to
maximize the coherent communications gain $G(\bm{\theta})$. A _sequential
fixing_ (SF) protocol to adjust the phase angle $\theta_{n}$ values is given
in Algorithm 1. Note that in Step 2, we can set $\theta_{1}=0$ because if
there is an optimal solution with $\theta_{1}\neq 0$, we can always construct
another solution by letting $\hat{\theta}_{n}=\theta_{n}-\theta_{1}$ for
$n=1,\cdots,N$. It is easy to verify that the constructed solution has the
same coherent communication gain and $\hat{\theta}_{1}=0$. Thus, setting
$\theta_{1}=0$ can reduce the algorithm complexity without losing the
optimality.

Algorithm 1 The SF protocol.

1: Sort transmitters based on $\sum_{m=1}^{M}h_{nm}^{2}$, i.e., the expected
value for received power from a transmitter to all receivers. To simplify
discussion, we assume
$\sum_{m=1}^{M}h_{1m}^{2}\geq\cdots\geq\sum_{m=1}^{M}h_{Nm}^{2}\;.$

2: Set $\theta_{1}=0$.

3: for $n=2,\cdots,N$ do

4: Choose $\theta_{n}$ to maximize the coherent communication gain
$G(\bm{\theta})$ by $n$ transmitters
$\sum_{m=1}^{M}((\sum_{i=1}^{n}h_{im}\cos(\theta_{i}+\theta_{im}))^{2}+(\sum_{i=1}^{n}h_{im}\sin(\theta_{i}+\theta_{im}))^{2})/h_{sd}^{2}$,
where $\theta_{1},\cdots,\theta_{n-1}$ are already determined and $h_{sd}^{2}$
is the channel gain between the source and the destination.

5: end for

Algorithm 2 The IO protocol.

1: Sort transmitters based on $\sum_{m=1}^{M}h_{nm}^{2}$. To simplify
discussion, we assume
$\sum_{m=1}^{M}h_{1m}^{2}\geq\cdots\geq\sum_{m=1}^{M}h_{Nm}^{2}\;.$

2: Set $\theta_{1}=0$.

3: for $n=2,\cdots,N$ do

4: Choose $\theta_{n}$ to maximize coherent communication gain
$G(\bm{\theta})$ by $n$ transmitters
$\sum_{m=1}^{M}((\sum_{i=1}^{n}h_{im}\cos(\theta_{i}+\theta_{im}))^{2}+(\sum_{i=1}^{n}h_{im}\sin(\theta_{i}+\theta_{im}))^{2})/h_{sd}^{2}$.

5: end for

6: Set Improved = 1.

7: while Improved == 1 do

8: Set Improved = 0.

9: for $n=1,\cdots,N$ do

10: Choose $\theta_{n}$ to maximize the coherent communication gain
$G(\bm{\theta})=\sum_{m=1}^{M}((\sum_{i=1}^{N}h_{im}\cos(\theta_{i}+\theta_{im}))^{2}+(\sum_{i=1}^{N}h_{im}\sin(\theta_{i}+\theta_{im}))^{2})/h_{sd}^{2}$,
where $\theta_{i}$ values, $i\neq n$, are unchanged.

11: If $\theta_{n}$ is updated, set Improved = 1.

12: end for

13: end while

Next, we introduce the _iterative optimization_ (IO) protocol to iteratively
optimize the phase angle $\theta_{n}$ values. The algorithmic steps of the IO
protocol are given in Algorithm 2. The first three steps in the IO protocol
are the same as in the SF protocol and the next two steps try to improve the
coherent communication gain $G(\bm{\theta})$. Thus, the IO protocol always
achieves better performance than the SF protocol but it has higher complexity.

The designed distributed protocols can also be implemented as centralized
algorithms (by assuming all information available at one node). The
performance and computational complexity are the same for the distributed
protocol and its centralized version. However, the distributed protocol incurs
additional communication overhead (exchanging intermediate results) that is
not present in its centralized version.

For comparison purposes, we also design other protocols.

  1. 1.

_Random beamforming_ (RB) protocol. Randomly choose $\theta_{n}$ values
corresponding to no coherent communication.

  2. 2.

_Random target_ (RT) protocol. Choose $\theta_{n}$ values to achieve phase
coherence at a random receiver. If the target receiver is $m$, this can be
done by setting $\theta_{1}=0,\theta_{n}=\theta_{1m}-\theta_{nm}$ for
$n=2,\cdots,N$.

  3. 3.

_Best target_ (BT) protocol. Instead of considering $M$ receivers jointly, we
study the coherent communication gain achieved by $N$ transmitters and each
receiver, and identify the the best receiver with the largest coherent
communication gain. Then we choose $\theta_{n}$ values to achieve this gain
and let all receivers participate in coherent communications.

  4. 4.

_Exhaustive search_ (ES) protocol. Exhaustively search for the best
$\theta_{n}$ values to maximize the coherent communication gain
$G(\bm{\theta})$. Given the high complexity of the ES protocol, we implement
this protocol for $N\leq 3$.

The performance achieved by ES protocol, as well as the upper bound
$G_{\text{UB}}$ in Corollary 1, are used as benchmarks.

TABLE I: Coherent communication gain under Channel Model 1 with $D=1000$ and $r=10$. $(N,M)$ | RB | RT | BT | SF | IO | ES | $G_{\text{UB}}$
---|---|---|---|---|---|---|---
$(1,1)$ | 1 | 1 | 1 | 1 | 1 | 1 | 1
$(1,10)$ | 9.97 | 9.97 | 9.97 | 9.97 | 9.97 | 9.97 | 10
$(3,1)$ | 3.04 | 8.98 | 8.98 | 8.98 | 8.98 | 8.98 | 9
$(3,10)$ | 26.82 | 87.27 | 88.71 | 88.73 | 88.73 | 88.73 | 90
$(10,10)$ | 99.82 | 840.17 | 909.13 | 909.71 | 910.47 | - | 1000
TABLE II: Coherent beamforming protocols. RB | Random beamforming protocol
---|---
RT | Random target protocol
BT | Best target protocol
SF | Sequential fixing protocol
IO | Iterative optimization protocol
ES | Exhaustive search protocol

To evaluate the proposed protocols, we consider the following setting. There
is one data stream ($K=1$) from transmitter $1$ to receiver $1$ and the
distance between them is $D$. There are $N-1$ additional transmitters and
$M-1$ additional receivers that can participate in coherent communications.
The distance between transmitters $i$ and $1$ is no more than $r$ and the
distance between receivers $i$ and $1$ is also no more than $r$. We set $D\gg
r$ such that transmitters (and receivers) can collaborate to transmit over a
long distance. Channel gain is modeled by $h^{2}=d^{-2}$ (Channel Model 1) and
phase shift is modeled by $\theta=2\pi(d/\lambda-\lfloor d/\lambda\rfloor)$,
where $d$ is the distance between a transmitter and a receiver, and $\lambda$
is the wavelength.

Table I shows the coherent communication gain results obtained for $D=1000$m,
$r=10$m, and $\lambda=0.125$m (corresponding to $2.4$GHz), where full names of
protocols are listed in Table II. The RB protocol uses a random algorithm and
thus we show the average performance in $100$ runs.

Two scenarios with $N=1$ are used to verify the correctness of protocol
implementation. For $N=1$ and $M=10$, we have $G_{\text{UB}}=10$ under the
assumption that all channel gains are the same. Since this assumption does not
hold in simulation, the ES protocol finds that the maximum coherent
communication gain $G(\bm{\theta})$ is $9.97$, which is slightly less than
$G_{\text{UB}}=10$. Due to the same reason, the maximum gain $G(\bm{\theta})$
obtained by the ES protocol for scenario $(N,M)=(3,1)$ or $(3,10)$ is slightly
less than $G_{\text{UB}}=9$ or $90$.

For scenario $(N,M)=(3,1)$, most protocols (except the RB protocol) find the
optimal solution with the maximum gain $G(\bm{\theta})=8.98$ while the RB
protocol cannot. In fact, the RB protocol is the case of non-coherent
communication and thus has the expected gain $G(\bm{\theta})=NM$. For scenario
$(N,M)=(3,10)$, the SF and IO protocols can find the optimal solution with the
maximum gain $G(\bm{\theta})=88.73$. The RT and BT protocols can find near-
optimal solutions, while the RB protocol finds a solution with gain
$G(\bm{\theta})=26.82$, which is far from the optimum.

For scenario $(N,M)=(10,10)$, the ES protocol cannot find a solution due to
high complexity and thus the maximum gain is unknown. Since the IO protocol
finds the best solution (with gain $G(\bm{\theta})=910.47$) among all
protocols and $G_{\text{UB}}=1000$, the maximum gain is within
$[910.47,1000]$. The BT and SF protocols can find solutions with a gain close
to the best known gain $G(\bm{\theta})=910.47$. The RB protocol again cannot
find a good solution.

The state-of-the-art methods focus on the MISO communications, which
corresponds to the case of $M=1$ in Table I. For the MISO case, the optimal
solution can be achieved. Most schemes developed in this paper, such as
schemes RT to IO in Table I, can achieve the optimal gain. For the case of
$M>1$, there is no previous method and thus we develop the ES scheme for small
$N$ and establish an upper bound $G_{UB}$ for large $N$ as the performance
benchmarks.

In summary, the IO protocol always has the best performance among all
protocols and achieves the maximum gain $G(\bm{\theta})$ when the ES protocol
is applicable. The BT and SF protocols are also viable in terms of providing
near-optimal solutions, while the RB protocol should not be used due to its
poor performance.

We obtain more results under different parameters and channel models, when the
radius $r$ of transmitter/receiver group is $100$m and the distance $D$
between these two groups is $1$km or $10$km. We use realistic channel models:

  1. 1.

_Free space model_ (Channel Model 2):
$h^{2}=\frac{G_{t}G_{r}\lambda^{2}}{(4\pi d)^{2}}$, where $G_{t}$ and $G_{r}$
are the transmitter and receiver antenna gains, respectively.

  2. 2.

_Two-ray model_ (Channel Model 3): If $d$ is less than crossover distance
$d_{c}$, $h^{2}$ is calculated by the free space model, else
$h^{2}=\frac{G_{t}G_{r}(h_{t}h_{r})^{2}}{d^{4}}$, where $d_{c}=\frac{4\pi
h_{t}h_{r}}{\lambda}$, $h_{t}$ and $h_{r}$ are the transmitter and receiver
antenna heights, respectively.

We set $G_{t}=G_{r}=1,\lambda=0.125$ and $h_{t}=h_{r}=0.5$ in simulation and
obtain results in Table III–Table VI for $r=100$m and $D=1$km or $10$km. The
column of $G_{\text{UB}}$ shows the upper bound $N^{2}M$. The ES protocol can
only find the optimal solution for small $N$, i.e., it cannot find a solution
for $N=10$. Due to larger $r$ ($r=100$m), the gap from the condition
$h_{nm}^{2}\approx h_{11}^{2}$ for $m=1,\cdots,M$ and $n=1,\cdots,N$ in
Corollary 1 is larger than the case when $r=10$m. As a result, the upper bound
$G_{\text{UB}}$ for $r=100$m obtained by Corollary 1 is not as tight (not as
close to the optimal value by the ES protocol) as in the case when $r=10$m.
The IO protocol has the largest power gain $G(\bm{\theta})$ but this gain is
not close to $G_{\text{UB}}$. For $N=3$, the IO protocol finds a solution with
similar performance as that achieved by the ES protocol, which indicates that
this solution is near optimal.

TABLE III: Coherent communication gain under Channel Model 2 with $D=1000$ and $r=100$. $(N,M)$ | RB | RT | BT | SF | IO | ES | $G_{\text{UB}}$
---|---|---|---|---|---|---|---
$(3,10)$ | 34.91 | 36.84 | 41.56 | 42.87 | 43.59 | 43.59 | 90
$(10,10)$ | 106.88 | 177.68 | 195.74 | 223.24 | 257.74 | - | 1000
TABLE IV: Coherent communication gain under Channel Model 3 with $D=1000$ and $r=100$. $(N,M)$ | RB | RT | BT | SF | IO | ES | $G_{\text{UB}}$
---|---|---|---|---|---|---|---
$(3,10)$ | 42.84 | 42.84 | 48.81 | 49.91 | 50.85 | 50.85 | 90
$(10,10)$ | 110.27 | 186.30 | 202.40 | 229.01 | 266.46 | - | 1000
TABLE V: Coherent communication gain under Channel Model 2 with $D=10000$ and $r=100$. $(N,M)$ | RB | RT | BT | SF | IO | ES | $G_{\text{UB}}$
---|---|---|---|---|---|---|---
$(3,10)$ | 29.40 | 38.36 | 49.47 | 50.51 | 50.63 | 50.63 | 90
$(10,10)$ | 101.38 | 228.66 | 295.63 | 295.63 | 321.06 | - | 1000
TABLE VI: Coherent communication gain under Channel Model 3 with $D=10000$ and $r=100$. $(N,M)$ | RB | RT | BT | SF | IO | ES | $G_{\text{UB}}$
---|---|---|---|---|---|---|---
$(3,10)$ | 29.66 | 38.69 | 49.89 | 50.94 | 51.06 | 51.06 | 90
$(10,10)$ | 98.62 | 226.54 | 292.12 | 291.76 | 317.08 | - | 1000
Figure 4: Coherent communications for multiple data streams.

##  VI Network Formation and Beamforming in Self-organizing Wireless Networks

When there are $K>1$ data streams (see Fig. 4), each with its own source-
destination pair, we need to assign each transmitter and each receiver to a
source and a destination, respectively. There are three _network formation_
approaches.

  * •

Approach 1: Use all nodes in the transmitter group together for beamforming.
The problem is that the cost/overhead for synchronization and coordination for
beamforming will be very high (increasing with the number of nodes).

  * •

Approach 2: Remove some nodes (potentially distant ones) from the
transmitter/receiver group. The problem is that sets of nodes to be removed
for different data streams (source-destination pairs) may be different.

  * •

Approach 3: Dynamically group transmitters and receivers to different
transmitter and receiver groups, respectively, depending on the data stream
requirements.

We follow Approach 3 since it selects suitable transmitters and receivers for
different data streams and accounts for factors such as traffic backlogs at
transmitters and priorities and QoS requirements of data streams.

###  VI-A Coherent Communication Gain with Network Formation

Network formation affects the coherent communication gain $G(\bm{\theta})$.
Note that so far we used the power gain as the performance measure for $K=1$.
However, when $K>1$, the power gain cannot capture the effect of interference
from other groups. Thus, we use the SIR gain as the measure. There are $K$
data streams, where data stream $k$ is from transmitter $s_{k}$ to receiver
$d_{k}$. There are $N$ potential transmitters and $M$ potential receivers for
each of these $K$ data streams. The signal amplitude $A$ and the signal period
$T$ are the same for all transmitters. Denote the channel gain and the phase
shift from transmitter $n$ to receiver $m$ as $h_{nm}^{2}$ and $\theta_{nm}$,
respectively. The signal from transmitter $n$ to receiver $m$ is
$Ah_{nm}\sin{\left(\frac{2\pi}{T}t+\theta_{n}+\theta_{nm}\right)}$, where
transmitter $n$ can tune $\theta_{n}$. Denote $\mathcal{T}_{k}$ and
$\mathcal{R}_{k}$ as the set of transmitters (including $s_{k}$) and receivers
(including $d_{k}$) assigned to data stream $k$, respectively. The total
received energy for data stream $k$ is
$\sum_{m\in\mathcal{R}_{k}}\int_{t=0}^{T}\left(\sum_{n\in\mathcal{T}_{k}}Ah_{nm}\sin{\left(\frac{2\pi}{T}t+\theta_{n}+\theta_{nm}\right)}\right)^{2}dt$.

Theorem 3 presents the SIR gain result. The proof of Theorem 3 is given in
Appendix D.

######  Theorem 3.

The SIR gain of coherent communications for data stream $k$ with transmitter
set $\mathcal{T}_{k}$ and receiver set $\mathcal{R}_{k}$ is

$\displaystyle G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta})$
$\displaystyle=$ $\displaystyle\frac{\sum_{1\leq n\leq K}^{n\neq
k}h_{s_{n}d_{k}}^{2}\rho_{k}(\bm{\theta})}{h_{s_{k}d_{k}}^{2}\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}}\;,$ (3)

where $\bm{\mathcal{T}}$ and $\bm{\mathcal{R}}$ are sets of all
$\mathcal{T}_{k}$ and $\mathcal{R}_{k}$, respectively, and
$\rho_{k}(\bm{\theta})$ is given by

$\displaystyle\rho_{k}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\sum_{m\in\mathcal{R}_{k}}\left(\left(\sum_{n\in\mathcal{T}_{k}}h_{nm}\cos{(\theta_{n}+\theta_{nm})}\right)^{2}\right.$
(4)
$\displaystyle\left.+\left(\sum_{n\in\mathcal{T}_{k}}h_{nm}\sin{(\theta_{n}+\theta_{nm})}\right)^{2}\right)\;.$

For the case of different transmit powers, we define $A_{n}$ as the signal
amplitude of transmitter $n$ and replace each $Ah_{nm}$ by $A_{n}h_{nm}$. The
result in Theorem 3 is changed to

$\displaystyle G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta})$
$\displaystyle=$ $\displaystyle\frac{\sum_{1\leq n\leq K}^{n\neq
k}A_{s_{n}}^{2}h_{s_{n}d_{k}}^{2}\rho_{k}(\bm{\theta})}{A_{s_{k}}^{2}h_{s_{k}d_{k}}^{2}\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}A_{n}^{2}h_{nm}^{2}}\;,$ $\displaystyle\rho_{k}(\bm{\theta})$
$\displaystyle=$
$\displaystyle\sum_{m\in\mathcal{R}_{k}}\left(\left(\sum_{n\in\mathcal{T}_{k}}A_{n}h_{nm}\cos{(\theta_{n}+\theta_{nm})}\right)^{2}\right.$
$\displaystyle\left.+\left(\sum_{n\in\mathcal{T}_{k}}A_{n}h_{nm}\sin{(\theta_{n}+\theta_{nm})}\right)^{2}\right)\;.$

TABLE VII: Network formation and coherent beamforming protocols. RRB | Random network formation and random beamforming protocol
---|---
RRT | Random network formation and random target protocol
RBT | Random network formation and best target protocol
DRB | Distance-based network formation and random beamforming protocol
DRT | Distance-based network formation and random target protocol
DBT | Distance-based network formation and best target protocol
TABLE VIII: Coherent communication gain under Channel Model 2 with $D=1000$ and $r=100$. $(N,M)$ | RRB | RRT | RBT | DRB | DRT | DBT
---|---|---|---|---|---|---
$(3,10)$ | 2.34 | 2.39 | 2.39 | 2.50 | 2.50 | 2.50
$(10,10)$ | 2.98 | 5.74 | 6.74 | 1.31 | 7.94 | 8.31

###  VI-B Joint Beamforming and Network Formation Protocols

The joint coherent beamforming and network formation problem can be formulated
as

$\displaystyle\max$ $\displaystyle
f\left(\\{G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta})\\}_{k=1,2,\cdots}\right)$
(5) subject to $\displaystyle(\ref{eq:gaink}),(\ref{eq:rho})$ $\displaystyle
s_{k}\in\mathcal{T}_{k},d_{k}\in\mathcal{R}_{k}$ $\displaystyle(1\leq k\leq
K)$
$\displaystyle\mathcal{T}_{i}\bigcap\mathcal{T}_{j}=\emptyset,\mathcal{R}_{i}\bigcap\mathcal{R}_{j}=\emptyset$
$\displaystyle(1\leq i\neq j\leq K)$ $\displaystyle 0\leq\theta_{n}\leq 2\pi$
$\displaystyle(1\leq n\leq N)$ variables
$\displaystyle\mathcal{T}_{k},\mathcal{R}_{k},\theta_{n}\;.$

In (5), $f(\cdot)$ captures the optimization objective and the trade-off among
all data streams. Different functions, e.g., minimum or average, can be used
for the function $f(\cdot)$. In simulation results, we use

$f\left(\\{G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta}))\\}_{k=1,2,\cdots}\right)=\min_{k}\\{G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta}))\\}\;.$

The problem is to choose the optimal values of
$\mathcal{T}_{k},\mathcal{R}_{k}$ and $\theta_{n}$ according to the
optimization problem (5). The joint optimization protocol includes network
formation policy and beamforming policy. The network formation policy has a
solution with integer variables to describe the assignment a transmitter
(receiver) to one of the transmitter (receiver) set. Such a mixed-integer
optimization problem is NP-hard in general. The complexity to obtain optimal
solution is high while a tight upper bound is not obvious. Thus, to solve this
problem, we consider two policies for network formation:

  1. 1.

random network formation,

  2. 2.

distance-based network formation that assigns a transmitter (receiver) to the
closest source (destination).

We consider three policies for beamforming within a group:

  1. 1.

random beamforming,

  2. 2.

random target (perform coherent beamforming towards a random receiver),

  3. 3.

best target (perform coherent beamforming towards the best receiver).

The combination of the above policies results in $2\times 3=6$ protocols,
namely “Random network formation and Random Beamforming” (RRB), “Random
network formation and Random Target” (RRT), “Random network formation and Best
Target” (RBT), “Distance-based network formation and Random Beamforming”
(DRB), “Distance-based network formation and Random Target” (DRT), and
“Distance-based network formation and Best Target” (DBT), respectively. We
list these protocol names in Table VII. Simulation results are shown in Table
VIII–Table XI. Table VIII shows results for channel model 2 with distance
between transmitter and receiver groups $D=1000$m and group radius $r=100$m.
There are $K=2$ data streams. If the number of transmitters and receivers
$(N,M)$ is $(3,10)$, policies DRB, DRT, and DBT can all achieve the best
performance. But if the number of transmitters $N$ is increased to $10$, only
DBT can achieve the best performance among all policies. We obtain the same
conclusion from the other three tables that DBT achieves the best performance
among all policies, highlighting the importance of the distance for network
formation and the optimal selection of target receiver for beamforming.

TABLE IX: Coherent communication gain under Channel Model 3 with $D=1000$ and $r=100$. $(N,M)$ | RRB | RRT | RBT | DRB | DRT | DBT
---|---|---|---|---|---|---
$(3,10)$ | 2.26 | 2.32 | 2.32 | 2.52 | 2.52 | 2.52
$(10,10)$ | 3.21 | 6.13 | 7.21 | 1.23 | 7.67 | 7.67
TABLE X: Coherent communication gain under Channel Model 2 with $D=10000$ and $r=100$. $(N,M)$ | RRB | RRT | RBT | DRB | DRT | DBT
---|---|---|---|---|---|---
$(3,10)$ | 2.45 | 2.51 | 2.51 | 2.87 | 4.04 | 4.04
$(10,10)$ | 2.57 | 7.52 | 8.69 | 2.62 | 11.19 | 12.35
TABLE XI: Coherent communication gain under Channel Model 3 with $D=10000$ and $r=100$. $(N,M)$ | RRB | RRT | RBT | DRB | DRT | DBT
---|---|---|---|---|---|---
$(3,10)$ | 2.47 | 2.37 | 2.37 | 2.84 | 4.07 | 4.07
$(10,10)$ | 2.60 | 7.24 | 8.31 | 2.89 | 11.20 | 12.38

In the above study, we focus on network formation and coherent beamforming. We
can also consider power control to further improve system performance. That
is, noting that the signal power of one data streams is interference to other
data streams, maximum power may not be an optimal solution for optimization
problem (5). Power control problem has been well studied for SISO links and
there are many existing solutions based on nonlinear optimization [35, 36,
37], game theory [38, 39], metaheuristic [40, 41], etc. Given the complexity
of power control problem, we leave the extension of existing SISO power
control algorithms as a future work.

##  VII Conclusion

We presented distributed coherent group communication protocols that consist
of coherent beamforming and network formation protocols. We started with an
optimization problem for a single data stream that formulates coherent
beamforming for a given pair of transmitter group and receiver group, and
obtained a number of policies with near optimal performance in terms of the
achieved power gain at the receiver group. We then considered the joint
optimization of coherent beamforming and network formation for multiple data
streams to maximize the SIR gain at the receiver group for each coherent
communication group. The network formation protocol forms multiple coherent
communication groups for multiple source-destination pairs, where each
transmitter group includes a source and each receiver group includes a
destination. Then the beamforming protocol enables transmissions of signals
from each transmitter in a group to each receiver in the corresponding
receiver group. The power and SIR gains achieved by our design are higher than
those in point-to-point transmissions and bring various advantages such as
improvement in communication range, power efficiency, reliability, and
throughput.

## Acknowledgement

We would like to thank Dr. Satya Ponnaluri and Dr. Eric van Doorn for
discussions on beamforming and synchronization.

## Appendix A – Proof of Theorem 1

We first consider a receiver $j$. Suppose that the local time $t$ at
transmitter $i$ corresponds to the local time
$\hat{t}$̂atreceiver$j$andthedifferencebetweenlocaltimes$t_ij=t- ^t$̂ is a
fixed number. If transmitter $i$ sends a signal $A\sin\frac{2\pi t}{T}$, where
$A$ is the amplitude and $T$ is the carrier period, the received signal at
receiver $j$ is $\hat{A}\sin\left(\frac{2\pi t}{T}+\frac{2\pi
t_{ij}}{T}+\theta_{ij}\right)$, where $\theta_{ij}$ is the phase shift on link
$(i,j)$, and the phase shift is observed as $s_{ij}=\frac{2\pi
t_{ij}}{T}+\theta_{ij}$. In this case, receiver $j$ can simply ask transmitter
$i$ to tune its initial phase value from $0$ to $-s_{ij}$, i.e., transmitter
$i$ sends a signal $A\sin\left(\frac{2\pi t}{T}-s_{ij}\right)$. Then the
received signal becomes $\hat{A}\sin\frac{2\pi t}{T}$, i.e., with the initial
phase value $0$ under the local time at receiver $j$. If receiver $j$ performs
the same operation for each transmitter and each transmitter can tune its
signal as requested, phase coherence can be achieved.

## Appendix B – Proof of Theorem 2

We first analyze the received power at a receiver $m$ under the coherent
communication scenario. We obtain

$\displaystyle
E_{NM}(m)=\int_{t=0}^{T}\left(\sum_{n=1}^{N}Ah_{nm}\sin\left(\frac{2\pi}{T}t+\theta_{n}+\theta_{nm}\right)\right)^{2}dt$
$\displaystyle=$ $\displaystyle A^{2}\int_{t=0}^{T}\left(\sin\frac{2\pi
t}{T}\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})+\right.$
$\displaystyle\left.\cos\frac{2\pi
t}{T}\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})\right)^{2}dt$
$\displaystyle=$ $\displaystyle
A^{2}\left(\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})\right)^{2}\int_{t=0}^{T}\sin^{2}\frac{2\pi
t}{T}dt+$ $\displaystyle
A^{2}\left(\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})\right)^{2}\int_{t=0}^{T}\cos^{2}\frac{2\pi
t}{T}dt+$ $\displaystyle
A^{2}\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})$
$\displaystyle\cdot\int_{t=0}^{T}2\sin\frac{2\pi t}{T}\cos\frac{2\pi t}{T}dt$
$\displaystyle=$ $\displaystyle
A^{2}\left(\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})\right)^{2}\frac{T}{2}+$
$\displaystyle
A^{2}\left(\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})\right)^{2}\frac{T}{2}+$
$\displaystyle
A^{2}\sum_{n=1}^{N}h_{nm}\cos(\theta_{n}+\theta_{nm})\sum_{n=1}^{N}h_{nm}\sin(\theta_{n}+\theta_{nm})\cdot
0$ $\displaystyle=$ $\displaystyle\frac{A^{2}T}{2}\beta_{m}(\bm{\theta})$

Thus, the total received power at all receivers is

$\displaystyle
E_{NM}=\sum_{m=1}^{M}E_{NM}(m)=\frac{A^{2}T}{2}\sum_{m=1}^{M}\beta_{m}(\bm{\theta})\;.$

Then we divide it by $\frac{A^{2}h_{11}^{2}T}{2}$ to obtain $G(\bm{\theta})$.

## Appendix C – Proof of Corollary 1

We obtain an upper bound for $G(\bm{\theta})$ as follows. We have

$\displaystyle\beta_{m}(\bm{\theta})$ $\displaystyle\leq$ $\displaystyle
N\sum_{n=1}^{N}h_{nm}^{2}\cos^{2}(\theta_{n}+\theta_{nm})+N\sum_{n=1}^{N}h_{nm}^{2}\sin^{2}(\theta_{n}+\theta_{nm})$
$\displaystyle=$ $\displaystyle
N\sum_{n=1}^{N}h_{nm}^{2}(\cos^{2}(\theta_{n}+\theta_{nm})+\sin^{2}(\theta_{n}+\theta_{nm}))\approx
N^{2}h_{11}^{2},$

where the equality in $\leq$ holds if

$\displaystyle\cos(\theta_{1}+\theta_{1m})=\cos(\theta_{2}+\theta_{2m})=\cdots=\cos(\theta_{N}+\theta_{Nm})$
$\displaystyle\sin(\theta_{1}+\theta_{1m})=\sin(\theta_{2}+\theta_{2m})=\cdots=\sin(\theta_{N}+\theta_{Nm})\;,$

and the approximation holds if

$\displaystyle h_{nm}^{2}\approx h_{11}^{2}$ $\displaystyle m=1,\cdots,M\;.$

Thus, the gain at receiver $m$,
$E_{NM}(m)/\frac{A^{2}h_{11}^{2}T}{2}=\frac{A^{2}T}{2}\beta_{m}(\bm{\theta})/\frac{A^{2}h_{11}^{2}T}{2}$,
can be up to $\frac{1}{h_{11}^{2}}N^{2}h_{11}^{2}=N^{2}$ times if phase
coherence holds and channel gains are similar, i.e.,

$\displaystyle\theta_{1}+\theta_{1m}=\cdots=\theta_{N}+\theta_{Nm}$
$\displaystyle h_{nm}^{2}\approx h_{11}^{2},\>\>\>n=1,\cdots,N\;.$

Note that for coherent beamforming scenarios, the transmitters are close to
each other and receivers are close to each other while the distance between
transmitters and receivers is large. For such a scenario, we can assume
$h_{nm}^{2}\approx h_{11}^{2}$. If the above two conditions hold for all
receivers, the total gain by all receivers can be up to $G_{\text{UB}}=N^{2}M$
(such that $G(\bm{\theta})\leq G_{\text{UB}}$ for any $\bm{\theta}$).

## Appendix D – Proof of Theorem 3

The total received energy in one period is

$\displaystyle
E_{\mathcal{T}_{k},\mathcal{R}_{k}}=\frac{A^{2}T}{2}\rho_{k}(\bm{\theta})\;.$

At the same time, transmitters for other data streams interfere at receivers
for data stream $k$. The interference in one period is given by

$\displaystyle I_{\bm{\mathcal{T}},\mathcal{R}_{k}}$ $\displaystyle=$
$\displaystyle\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}\frac{A^{2}h_{nm}^{2}}{2}T=\frac{A^{2}T}{2}\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}\;,$

where $\bm{\mathcal{T}}$ is the set of all $\mathcal{T}_{k}$. Thus, the
achieved SIR is

$\displaystyle
R_{\bm{\mathcal{T}},\mathcal{R}_{k}}=\frac{\frac{A^{2}T}{2}\rho_{k}(\bm{\theta})}{\frac{A^{2}T}{2}\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}}=\frac{\rho_{k}(\bm{\theta})}{\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}}.$

If coherent communication is not applied, the received energy in one period is
given by $\frac{A^{2}h_{s_{k}d_{k}}^{2}}{2}T$, where $h_{s_{k}d_{k}}^{2}$ is
the channel gain from the source to the destination of data stream $k$. The
interference in one period is $\sum_{1\leq n\leq K}^{n\neq
k}\frac{A^{2}h_{s_{n}d_{k}}^{2}}{2}T$. Thus, the achieved SIR is

$\displaystyle R_{\\{\\{s_{1}\\},\cdots,\\{s_{K}\\}\\},\\{d_{k}\\}}$
$\displaystyle=$
$\displaystyle\frac{\frac{A^{2}h_{s_{k}d_{k}}^{2}}{2}T}{\sum_{1\leq n\leq
K}^{n\neq k}\frac{A^{2}h_{s_{n}d_{k}}^{2}}{2}T}$ $\displaystyle=$
$\displaystyle\frac{h_{s_{k}d_{k}}^{2}}{\sum_{1\leq n\leq K}^{n\neq
k}h_{s_{n}d_{k}}^{2}}\;.$

Based on the above two SIR values, the coherent communication gain is

$\displaystyle G_{k}(\bm{\mathcal{T}},\bm{\mathcal{R}},\bm{\theta})$
$\displaystyle=$
$\displaystyle\frac{\rho_{k}(\bm{\theta})}{\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}\frac{h_{s_{k}d_{k}}^{2}}{\sum_{1\leq n\leq K}^{n\neq
k}h_{s_{n}d_{k}}^{2}}}$ $\displaystyle=$ $\displaystyle\frac{\sum_{1\leq n\leq
K}^{n\neq
k}h_{s_{n}d_{k}}^{2}\rho_{k}(\bm{\theta})}{h_{s_{k}d_{k}}^{2}\sum_{m\in\mathcal{R}_{k}}\sum_{n\in\mathcal{T}_{l}}^{l\neq
k}h_{nm}^{2}}\;.$

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