# The Magic of Superposition: A Survey on the Simultaneous Transmission Based
Wireless Systems

Ufuk Altun, , Gunes Kurt, and Enver Ozdemir

###### Abstract

In conventional communication systems, any interference between two
communicating points is regarded as unwanted noise since it distorts the
received signals. On the other hand, allowing simultaneous transmission and
intentionally accepting the interference of signals and even benefiting from
it have been considered for a range of wireless applications. As prominent
examples, non-orthogonal multiple access (NOMA), joint source-channel coding,
and the computation codes are designed to exploit this scenario. They also
inspired many other fundamental works from network coding to consensus
algorithms. Especially, federated learning is an emerging technology that can
be applied to distributed machine learning networks by allowing simultaneous
transmission. Although various simultaneous transmission applications exist
independently in the literature, their main contributions are all based on the
same principle; the superposition property. In this survey, we aim to
emphasize the connections between these studies and provide a guide for the
readers on the wireless communication techniques that benefit from the
superposition of signals. We classify the existing literature depending on
their purpose and application area and present their contributions. The survey
shows that simultaneous transmission can bring scalability, security, low-
latency, low-complexity and energy efficiency for certain distributed wireless
scenarios which are inevitable with the emerging Internet of things (IoT)
applications.

###### Index Terms:

Simultaneous transmission, superposition, wireless networks.

## Acronyms

AFC

    Analog Function Computation
A&F

    Amplify-and-Forward
AWGN

    Additive White Gaussian Noise
BER

    Bit Error Rate
BICM-ID

    Bit-Interleaved Coded Modulation with Iterative Decoding
CDMA

    Code Division Multiple Access
CEE

    Channel Estimation Error
CFMA

    Compute-and-Forward Multiple Access
CFO

    Carrier Frequency Offset
cMACr

    Compound Multiple Access Channel with a relay
C&F

    Compress-and-Forward
CoMAC

    Computation over Multiple Access Channels
CPC&F

    Compute-Compress-and-Forward
CP&F

    Compute-and-Forward
C-RAN

    Cloud-Radio Access Network
CSI

    Channel State Information
DFT

    Discrete Fourier Transform
D&F

    Decode-and-Forward
DSGD

    Distributed Stochastic Gradient Descent
FC

    Fusion Fenter
IA

    Interference Alignment
i.i.d.

    independent and identically distributed
IoT

    Internet of Things
LDLC

    Low Density Lattice Codes
LDPC

    Low Density Parity Check
LLR

    Log-Likelihood Ratio
MAC

    Multiple Access Channel
MAF

    Modified Amplify-and-Forward
MAP

    Maximum A Posteriori
MDF

    Modified Detect-and-Forward
MIMO

    Multiple Input Multiple Output
ML

    Machine Learning
MMSE

    Minimum Mean Square Error
MSE

    Mean Square Error
MWRC

    Multi-Way Relay Channel
NOMA

    Non-Orthogonal Multiple Access
OFDM

    Orthogonal Frequency Division Multiplexing
PLNC

    Physical Layer Network Coding
SDR

    Software Defined Radio
SER

    Symbol Error Rate
SIC

    Successive Interference Cancellation
SIMO

    Single Input Multiple Output
SNR

    Signal to Noise Ratio
STAC

    Simultaneous Transmitting and Air Computing
STANC

    Space-Time Analog Network Coding
SVP

    Shortest Vector Problem
TBMA

    Type-based Multiple Access
TDMA

    Time Division Multiple Access
UAV

    Unmanned Aerial Vehicles
W-MAC

    Wireless Multiple Access Channel
WSN

    Wireless Sensor Network
ZF

    Zero Forcing

##  I Introduction

The theory of communication, as presented in [1] by Claude Shannon, considers
the mathematical relation between a source and a destination. The theory
identifies any signal between the source and the destination as noise. This
assumption is still valid in advanced communication systems, especially in
wireless communication networks that introduce destructive, as well as a
limited medium to its users. Although its mobility feature attracts more and
more users with each wireless network generation, limited resources present
extreme design challenges since the wireless medium is shared by all of the
participants.

Some of the most critical design challenges are providing security, low
latency and high communication rates and supporting a large number of users.
From any of those aspects, the corresponding solution is sought, designed and
tested for Shannon’s pairwise communication model. Hence, the first step of
the network design generally starts with virtually dividing the limited
channel resources to each user and considering the crowded wireless channel as
a combination of multiple subchannels between the pairs of nodes. For example,
assigning orthogonal frequencies to different users is a practical and
efficient approach to solve these challenges.

It is needless to say that all aspects we listed above are important for a
wireless network, however their importance and the priority is closely related
to the purpose of the network. With the emergence of Internet of things (IoT)
networks, we can add other design considerations to our list such as energy
consumption or the system complexity. More importantly, an IoT network enables
an immense range of wireless applications such that these aspects (and
probably many more) are required in different and particular orders according
to the application. For instance, security can be of high importance and the
complexity is less for finance applications, while the mass sensor networks
can require a low energy consumption and a high bandwidth efficiency.

We believe that this diversity of wireless applications attracts the
researchers’ attention to the outside of the classical perspective. The
physical layer based studies have already gained attention on this matter in
order to help the wireless communication problems. One peculiar idea is
related to the physical layer and the interference of signals. In the last two
decades, the researchers have asked the question if the interference of
signals can be beneficial. The question is closely investigated in several
studies and positive answers were presented for a limited number of scenarios.
These studies are mostly classified in the literature according to their
application area, however their interactions with other studies regarding
their unique channel model is usually overlooked. In this study, we consider
the channel model aspect and survey the existing literature for the studies
that exploit the superposition property of the wireless channel. The
contributions of this survey can be listed as follows.

  * •

The studies in the literature that exploit the superposition property of the
wireless channel are reviewed: A variety of wireless communication techniques
exist in the literature (Fig. 1) with this apparent and unique connection
(superposition property). Yet the literature lacks a comprehensive guide that
highlights this connection to the readers. This survey presents a guide for
both beginner and experienced readers that are interested in the benefits of
simultaneous transmission.

  * •

The studies are presented along with their system model, contributions and
performance metrics: The simultaneous transmission methods are grouped
depending on their application areas. The methods are introduced with brief
explanations and studies are investigated for their contributions and
performance metrics.

Figure 1: Structure of the survey as the main application areas of the
simultaneous transmission based wireless communication techniques.

###  I-A Boundaries of Our Scope

Since we consider redinterfering signals, we are naturally interested in the
multiple users and the techniques for accessing the wireless channel. It
should be pointed out that the answers to all our questions are not included
within Shannon’s communication model regarding the interference policy. Here
our boundaries may be perplexing since there are certain techniques in the
literature that accept the interference of other signals, however still
interested in the pairwise communication without benefiting from interference.

The general resources of a wireless network is use of power in the time and
the frequency plane that is suitable for distribution among the users.
However, communication technologies make use of other resources such as code
or energy to divide among the users. The users in these studies occupy the
same time and frequency blocks, yet they are distributed with another
resource. The code division multiple access (CDMA) is an example that enables
controlled interference and uses the code dimension for multiple access. The
spatial diversity of the multiple input multiple output (MIMO) networks can be
given as another example that does not prohibit interference. However, MIMO
technology should be considered under pairwise communication since its
interference comes from the signals that are known and managed by a single
user.

These examples raise the question of whether our scope is limited to any
technique that enables the interference of signals (i.e. uses the same time
and frequency slots) or limited to a more narrow scenario. We could not answer
this question with a single border since one of them would draw an incomplete
map on the main idea and the other one would be too extensive to cover and
also include unrelated parts. For this reason, our scope also includes the
major techniques (e.g. CDMA, MIMO) that enable the interference of signals but
not directly got benefit from it. However, we briefly illustrate their
relation to the interference and we refer the readers to more detailed
resources on that matter.

Our main focus is on the techniques that intentionally exploit the
interference of signals and get a direct benefit from it. Multiple users in
the techniques that we present here use the same time and frequency blocks for
communication and do not aim to distinguish between the individual
information. Instead, these methods are only interested in the superimposed
form of the signals. There are various techniques in the literature and before
giving a general map of these techniques, first we would like to clarify some
of the concepts that we use to avoid any ambiguity.

We define the channel model as the wireless multiple access channel (W-MAC).
The multiple access term states the fact that the simultaneously transmitted
electromagnetic waves combine with each other. In general, this statement only
covers the time dimension, i.e. the waves from other frequencies also merge
over the W-MAC. However, it is straightforward to separate the superimposed
waves in the frequency dimension at the receiver. For this reason, we extend
our definition of simultaneous transmission and include the frequency
dimension. Hereafter, we refer to simultaneous (concurrent) transmission to
cover both time and frequency domains, i.e. signals are transmitted at the
same time and frequency block. Similarly, we refer to the simultaneously
transmitted signals as the superpositioned or superimposed signals.
Furthermore, we use superposition or simultaneous transmission to indicate the
methods that we are interested in.

###  I-B A Map of the Simultaneous Transmission Based Communication Techniques

There are several application areas of simultaneous transmission-based
techniques for different purposes. Each application has its own design
parameters and performance metrics, for instance distributed detection
problems usually require a high detection probability while security
applications are interested in secrecy rates. As a result, this survey assumes
a unique point of view that connects various problems and application areas of
the wireless communication networks. However, the superposition property of
the W-MAC implies several common grounds as follows:

  * •

Its distributed character: All applications include decentralized multiple
nodes. As a result, scalability, energy efficiency and complexity are often a
design concern.

  * •

Its wireless character: The users communicate over the wireless channel. The
amount of channel state information (CSI) knowledge at the users and the
channel characteristics (e.g. fading) are important concerns.

The structure of this survey (the application areas of the simultaneous
transmission based techniques) is illustrated in Fig. 1. The multiple access
and the multiple antenna applications are the early examples that the
superposition of the simultaneously transmitted signals is observed at the
receiver. CDMA and non-orthogonal multiple access (NOMA) techniques are
essentially based on pairwise communications. For this reason, transmitters
use a third resource (other than time and frequency) to distinguish between
the individual messages, e.g. the CDMA or NOMA users uniquely encode their
messages as a function of a code block or a power level respectively. On the
other hand, the traditional MIMO applications benefit from the superimposed
signals to gain diversity although the network is not distributed.

Figure 2: An illustration of the featured methods and their applications
areas. Figure 3: An illustration of the relationship between the applications
and their network model.

Joint source-channel coding proposed in [2] has created a paradigm shift on
the multi-user communication models by considering the joint optimization of
communication and computation aspects. Over the years, the paradigm has been
shaped into several core models such as type-based multiple access (TBMA) [3],
computation over multiple access channel (CoMAC) [4], compute-and-forward
(CP&F) [5] and analog function computation (AFC) [6]. These studies influenced
novel techniques in the past decade and appeared at various application areas
from network coding to federated learning. An illustration of these techniques
and their general application areas are given in Fig. 2.

CP&F is mainly investigated for relaying and network coding purposes to
improve the network capacity (i.e. computation rate), however it has also been
extended to multiple access applications (CFMA [7]) to reduce complexity and
security applications (as in [8]) to improve secrecy rate. The computation
alignment and the function alignment studies which are influenced by CP&F and
CoMAC are proposed to reach the computing capacity in multiple antenna
networks.

CoMAC inspired various digital and analog function computation methods that
aims to allow low-latency, low-bandwidth computations in distributed wireless
networks. The main idea behind the CoMAC is later implemented in spectrum
sensing, federated learning and consensus algorithms to improve network
efficiency. Especially, federated learning algorithms are one of the latest
and most promising application area of the simultaneous transmission. With the
help of CoMAC, federated learning algorithms can decentralize machine learning
systems and improve energy or time efficiency of the system. TBMA is primarily
used for distributed detection and estimation applications to design scalable
and energy-efficient models.

From an intriguing perspective, all these models and applications can be
viewed as a particular function that manipulates the wireless channel to
perform a given task. For example, multiple access algorithms aim to transfer
the transmitted data to a destination. Eventually, simultaneous transmission
manipulates the channel to perform a function which both inputs and the
outputs are the transferred data. However in the detection algorithms, outputs
are not data, instead they are the test statistics. Hence, it can be said that
a detection algorithm that uses simultaneous transmission manipulates the
wireless channel to perform a function that inputs are the sample set and the
output is the test statistics. The corresponding input-output relationships
are illustrated in Fig. 3.

The security applications consider transferring data to the receiver without
leaking information to the eavesdroppers. This approach manipulates the W-MAC
to transfer the initial data to the receiver such that the received signal is
only meaningful at the destination. Network coding applications consider
transferring data in layered network structures. The simultaneous transmission
models propose a cascade of functions (composite functions) (relays) that both
the initial inputs and the final outputs are the data. Function computation,
federated learning, spectrum sensing algorithms match the W-MAC with
particular functions that is unique to the purpose of the application.
Moreover, gossip and consensus algorithms also consider the topology of the
network by using subgroups or subfunctions in the network, i.e. composite
functions.

In this survey, we consider wireless communication techniques that benefit
from simultaneous transmission and we classify the existing literature
according to their application purposes. We draw a map of the existing
literature and provide introductory information on each application (Fig. 2).
We also give details of these studies and investigate their contributions as
well as their performance metrics.

The paper is organized as follows. In Section II, multiple access methods that
benefit from the simultaneous transmission are presented. Section III is
dedicated to the multiple antenna algorithms. Network coding studies are
investigated in Section IV. Interference alignment, computation alignment and
function alignment studies are examined in Section V. Digital and analog
function computation applications are presented in Section VI, and the
federated learning studies are presented in Section VII. In Section VIII,
spectrum sensing models are presented. Section IX is devoted to the detection
and estimation studies, and Section X includes the gossip and consensus
studies. Lastly, the security applications are given in Section XI. The paper
is concluded in Section XII.

##  II Multiple Access

Wireless channel has limited resources (signaling dimensions) such as
frequency, time, or space and accessing to the channel requires a portion from
each resource. Multiple access methods aim to allocate these resources to
multiple users efficiently [9]. Our interest, simultaneous transmission,
enables the dedication of all frequency and time domains to all users. As a
result, simultaneous transmission for the purpose of multiple access falls
under the category of our interest. However, known multiple access methods
such as CDMA and NOMA divide another dimension to its users rather than
exploiting the superposition of the signals. For this reason, we find it more
adequate to give elementary information on these methods and refer the readers
to proper references. In this section, multiple access methods that enable
users to simultaneously transmit their messages are presented.

###  II-A Code Division Multiple Access (CDMA)

In CDMA, each user is assigned with a spreading code to distinguish users from
each other. Therefore, the channel can be used by all users in the same time
period and bandwidth. Especially in the uplink scenario, in which multiple
users transmit simultaneously, base station receives a combination of signals
from all users. If the codes are orthogonal, despreading the received signal
with the corresponding code outputs the information signal of the
corresponding user. Fig. 4a illustrates the resource distribution of CDMA
technique. As seen from the figure, each user occupies a large and equal
bandwidth and usually, a power balance is required. Non-orthogonal codes are
also used in CDMA to flex the synchronization problem and support more users,
however removing orthogonality adds interference to the system.

In [10], extensive information is given about the working principles of the
CDMA as well as an introduction to the various versions of the CDMA. An
extended CDMA method, complementary code based MIMO CDMA, which aims to revive
CDMA in the next-generation systems are investigated in [11].

(a)

(b)

Figure 4: Distribution of resources in multiple access methods of our
interest. (a) CDMA, (b) NOMA.

###  II-B Non-Orthogonal Multiple Access (NOMA)

NOMA is an emerging multiple access method that is based on the distribution
of power or code domain111NOMA is usually referred for power or code domain
distribution. However, the name also suggests a category that includes any
non-orthogonal method such as spatial division or pattern division. In this
survey, we focus on power domain NOMA.. NOMA can be viewed as a complementary
method for the existing multiple access techniques since the power or code
domain is not suitable to support large number of users. As a result, the
distribution of another resource is required. However, NOMA based systems
still remain in our scope since NOMA enables the transmission of two users
simultaneously. As an example, resource distribution of power domain NOMA is
given in Fig. 4b. The power is intentionally distributed to users with a level
difference. The model depends on the successive interference cancellation
(SIC) such that the receiver is able to detect the higher level or lower level
signal and then cancel it.

NOMA is an emerging multiple access method and attracts attention for the next
generation communication systems. We refer the readers to [12] and [13] for
comprehensive information on NOMA. In [14] an overview on the current
challenges of NOMA (also the rejection of NOMA in 5G standards) and possible
solutions are analyzed. Additionally, performance of NOMA is investigated with
computer simulations.

###  II-C Type-Based Multiple Access (TBMA)

TBMA is a unique method by means of channel-user relationship since the users
in TBMA does not aim to communicate with the destination individually.
Instead, a data statistics (type) of all the users is transferred to the
destination [15]. It should be noted that the individual data is not
reconstructed at the receiver and TBMA is not suitable for classical wireless
communications. However, it reduces the network latency and shows huge
potential in sensor networks that only the total data statistics is desired.
Specifically, TBMA is widely used for detection and estimation purposes.

TBMA is not widespread in the literature and requires a broader explanation of
its mechanism. For this reason, we dedicate this section solely to the
explanation of the TBMA. The literature review of TBMA is given in the
detection and estimation section.

Consider a network that consists of $n$ users that aims to transfer messages
(e.g. sensor readings) $X_{i},X_{2},...,X_{n}$ to a fusion center (FC). The
$i^{th}$ user in the network assigns its message $X_{i}$ to a waveform
$s_{x_{i}}$ that is chosen from an orthonormal waveform set
$\\{s_{1},s_{2},...s_{k}\\}$. When each user in the network simultaneously
transmit its data with energy $E$, the FC obtains the following expression
[16]

$z=\sum_{i=1}^{n}h_{i}\sqrt{E}s_{x_{i}}+w,$ (1)

where $h_{i}$ and $w$ are the channel and fading coefficients respectively.
Assuming the channel gains are inverted, the signal at the FC becomes

$z=\sum_{j=1}^{k}\sqrt{E}N_{j}s_{j}+w,$ (2)

where $N_{j}$ is the number of sensor readings that is mapped into waveform
$s_{j}$. Note that each member of the message set is mapped into a waveform
and the transmitted signals are superimposed over the channel. As a result,
for each waveform, the FC receives the number of users that transmitted the
corresponding waveform (i.e. the FC obtains histogram (type) of the sensor
readings).

###  II-D Compute-and-Forward Multiple Access (CFMA)

CP&F is a relaying and network coding method that is based on the simultaneous
transmission of signals. The objective of the CP&F is to efficiently transfer
the messages of multiple sources to a receiver with the help of multiple
relays. In essence, CP&F transfers a function of the source messages to each
relay by exploiting the superposition property. The relays are unable to
decode the individual messages since each of them obtains a single function
that contains multiple messages (unknown parameters). The relays forward their
functions to the receiver and the receiver can reconstruct the individual
messages by solving the functions for the unknown parameters. Contrary to
other network coding algorithms, the communication phase between the sources
and the relays takes place at the same time slot and the bandwidth in CP&F,
hence it offers efficiency on the spectrum and latency.

Compute-and-forward multiple access (CFMA) is inspired by the CP&F and instead
of relaying the messages, it enables direct access to the channel by
exploiting the superposition property. The CFMA is proposed in [17] by Zhu and
Gastpar for the networks that two users aim to access to a single receiver.
The main idea behind CFMA is based on the same coding and decoding structure
as the CP&F which yields a function of the messages at the receiver. However,
two users directly communicate towards a receiver without multiple relays. Two
functions are required at the receiver to solve the messages. For this reason,
CFMA also uses successive cancellation decoding to obtain the coefficients of
the second function. The authors also examine the more than two users scenario
in [17] and low density parity check (LDPC) coded CFMA is considered in [18]
and [7]. In the following section, multiple antenna techniques that benefit
from the simultaneous transmission are presented.

##  III Multiple Antenna

Multiple antenna techniques became beneficial in the communication networks as
a result of the developments in both the antenna technology and the processing
capacity. MIMO structure is proven to improve the multiplexing capability of
the single antenna networks as well as increasing their diversity gain [19].
MIMO presents a unique case for the scope of this survey since the
communication is between pairwise nodes, e.g. the antennas are controlled by
the same source. The fundamental advantage of MIMO is the diversity gain which
results from the superposition of the signals that are transmitted from
multiple antennas. As a result, MIMO networks draw our attention in the sense
that the superposition property is exploited to reduce the error rates or
improve the bit rates. On the other hand, the conventional MIMO is an
extensive topic and the superposition property is just a tool in MIMO studies
which leads to numerous results [20]. Also, it is already well presented in
the literature and we believe that the relation between MIMO and the
superposition property can be better observed from the existing studies such
as [21, 22]. Further reading on MIMO can be found in [23] for its security
applications and in [24] for its challenges and future. In addition to its
conventional perspective, there also exists studies that exploit the
superposition property in a unique way with multiple antennas. The integer-
forcing receivers [25, 26] are an example of these studies and we would like
to mention its architecture and its difference from the traditional studies.

###  III-A Integer-Forcing Architecture

An effective channel matrix can be defined and used for the analysis of linear
MIMO receivers. In the traditional sense, the effective channel matrix should
be matched with an identity matrix in order to recover the messages of each
antenna. However, integer-forcing receivers match the effective channel matrix
with integer value matrices as proposed in [25]. Inspired by CP&F [5], nested
lattice codes are used for communication in order to obtain functions with
integer coefficients. After matching with integer values, the receiver can
solve the effective channel matrix and obtain messages if the matrix is full
rank.

The integer-forcing receiver is extended to mitigate the external signal
interference in [27]. In addition to the results given in [25] that integer-
forcing receivers obtain the inputs with integer coefficients, the authors
later observe in [27] that the integer values can be also controlled to
mitigate the external interference by considering the interference space. The
results show that the proposed receiver presents significant gain over the
traditional linear MIMO receivers.

Successive interference cancellation (SIC) technique is adapted to the
integer-forcing receivers in [28] as successive integer-forcing. The results
indicate that the successive integer-forcing receiver can achieve the
channel’s sum capacity and outperform the traditional linear receivers with
SIC in certain scenarios.

##  IV Network Coding

Source and channel coding are essentially concerned with the communication
between two nodes. Specifically, they improve the capacity and error
performance of pairwise communication respectively. On the other hand, network
coding benefits from the architecture of the network in order to improve its
capacity, efficiency and security [29]. Network coding is interested in the
information flow between nodes. The fundamental idea behind the network coding
can be seen in Fig. 5a. The nodes $n_{1}$ and $n_{3}$ aims to exchange
information through an in-between node $n_{2}$ in the given network. Without
network coding, the messages $S_{1}$ and $S_{3}$ of the nodes $n_{1}$ and
$n_{3}$ would require a total number of four pairwise hops, hence four-time
slots. A simple network coding algorithm can be applied, as given in Fig. 5a,
to reduce the required time slots. After obtaining $S_{1}$ and $S_{3}$
sequentially, the relay node $n_{2}$ computes $S_{2}=S_{1}\oplus S_{3}$ and
broadcasts it. Since the nodes know their initial messages, $n_{1}$ and
$n_{3}$ can extract the unknown message from $S_{2}$. As a result, the network
gains a time slot with the broadcast of the relay node.

(a)

(b)

Figure 5: Information flow comparison between traditional network coding and
PLNC [30]. (a) Traditional network coding, (b) PLNC.

Network coding is one of the most effective and intriguing applications of the
simultaneous transmission since the multiple access nature of the channel
presents unique opportunities for the code design. These studies are
specifically called physical layer network coding (PLNC) which outperforms the
traditional network coding in certain scenarios [31]. We consider the PLNC in
two subsections since one study, the CP&F [5], made a name for itself and
requires special attention.

The network architecture is an important parameter in the investigation of
these studies and some commonly used architectures are illustrated in Fig. 6.
Also, the studies are classified according to their network architectures in
Table I along with their performance metrics. In the coming part, we briefly
explain the main ideas behind these methods and investigate the studies that
exist in the literature.

(a)

(b)

(c)

(d)

Figure 6: Several network topologies that are commonly used in simultaneous
transmission based techniques. (a) Multi-hop, (b) Multi ($K$) transmitter
multi ($L$) relay, (c) Multi-way relay, (d) Cooperative.

###  IV-A Physical Layer Network Coding (PLNC)

The PLNC is proposed in 2006 by Zhang et al. [30]. The proposed network is
based on the superposition of the signals to reduce the required time slots. A
simple three-node example of the PLNC is given in Fig. 5b. Without the
conventional network coding, it is obvious that the uncoded scheme requires a
time slot for each transmission and the conventional network coding reduces
the required number of transmission by enabling broadcast at the relay.

PLNC further improves this situation by accepting the superposition of the
messages from $n_{1}$ and $n_{3}$ to the relay node as seen in 5b. In PLNC
scenario, relay node receives the superimposed message and can not extract
$S_{1}$ and $S_{3}$. However PLNC allows $n_{1}$ and $n_{3}$ to decode the
superimposed message after $n_{2}$ broadcasts it. Later, this basic example is
extended to larger networks and different coding schemes as in [32] that a
packet-based PLNC architecture is proposed and implemented. The study uses a
testbed of computers and establishes a proof of concept for the PLNC.

Analog version of the PLNC [30] is proposed in [33]. The authors simply
consider the superposition of the signals instead of packets. Also, the relay
uses amplify-and-forward (A&F) to transfer the superimposed signal. After
presenting their results, the study also verifies them by implementing the
proposed method with software defined radio (SDR) modules.

Another analog PLNC scheme, space-time network coding (STANC), is proposed in
[34]. The study considers a non-regenerative multi-way relay network that
multiple nodes (equipped with single antenna) exchange information through a
single relay (equipped with multiple antennas) in stationary and non-
stationary channels. The STANC is proposed for the stationary case and an
alternative solution, the repetition transmission, is suggested for the non-
stationary channels. Achievable sum rates of these two models are calculated
and verified with simulations. The simulations also included a comparison with
the zero-forcing (ZF) and maximization of SNR beamforming models and it is
shown that the STANC outperforms other models regarding the sum rates. The
network model in [35] is customized to the networks where multiple nodes
transfer information to a single receiver both directly and with the help of a
relay node (equipped with multiple antennae). The sum-rate performances and
error rate performances are evaluated with simulations for the cases of direct
transmission, analog network coded transmission and the STANC transmission.

In [36], a network that consists of a single relay, two transmitters and two
receivers (compound multiple access channel with a relay (cMACr)) is
considered. The relay node is assumed to have cognitive capabilities and able
to include its own message to the received message before forwarding. The
study investigates the achievable rate regions of three relay methods; decode-
and-forward (D&F), compress-and-forward (C&F) and lattice coded (CP&F)
schemes. Also, a special case is examined where the cMACr network does not
allow cross-reception (i.e. one of the sources always connects to the receiver
via the relay, not directly). In this scenario, the modulo sum of the source
messages is computed with the lattice codes and compared with the D&F and C&F
schemes.

A multi-hop network includes multiple layers of relays between the sources and
the destiny as illustrated in Fig 6a. In [37], a cross-layer strategy is
followed for a multi-hop PLNC network to design the efficient routing paths.
In [38], algebraic frameworks are considered for the design of multi-hop PLNC
networks.

The compatibility and the performance of the PLNC with error correction codes
are investigated in [39]. Turbo codes, LDPC codes and bit-interleaved coded
modulation with iterative decoding (BICM-ID) are simulated in a PLNC based
network. The results showed that the PLNC reduces the bit error rate (BER)
performance of all three channel coding schemes.

In [40], a secure PLNC scheme is designed. The method is inspired by the
forwarding algorithm given in [41] (based on CP&F). The study analyzes two
networks with a butterfly topology and a three source topology. Also, it is
shown that the secure PLNC outperforms the secure network coding such as given
in [42].

TABLE I: An overview of PLNC studies. Author |  Year |  Network model |  Contribution |  Performance metric
---|---|---|---|---
Zhang et al. [30] |  2006 |  Two-way (multi-hop) |  Traditional PLNC is introduced. |  BER
Katti et al. [32] |  2008 |  Multi-way |  Throughtput improved with PLNC. |  Throughtput gain
Katti et al. [33] |  2007 |  Two-way |  Analog PLNC is introduced. |  Network throughput
Amah and Klein [34] |  2011 |  Multi-way |  Space-Time Analog Network Coding (STANC)introduced. |  Sum rate
Wie and Chen [35] |  2013 |  Cooperative |  STANC adapted to multi-way cooperative networks. |  Sum rate
Gündüz et al. [36] |  2010 |  Cooperative (cMACr) |  Lattice codes used for PLNC in cooperative networks. |  Achievable rate region
Xu et al. [37] |  2012 |  Multi-hop |  PLNC implemented to multi-hop networks with a cross-layer design. |  Network throughput
Burr and Fang [38] |  2014 |  Multi-hop |  Algebraic constructs are used in the network design. |  Network throughput
Al-Rubaie et al. [39] |  2013 |  Two-way |  LDPC and Turbo codes are compared. |  BER
Hayashi [40] |  2019 |  Cooperative (Butterfly) |  PLNC and NC are compared. |  Number of time spans
Figure 7: The network model of the CP&F method. $\mathcal{E}$ and
$\mathcal{D}$ symbolize encoder and decoder, respectively.

###  IV-B Compute-and-Forward (CP&F)

CP&F is proposed by Nazer and Gastpar in [5] as a relaying method and draw the
attention of numerous researchers throughout the years. CP&F is inspired from
the lattice codes (structured codes) that is previously used in [4, 43] and
[44]. A lattice $\Lambda$ is a group in $\mathbb{R}^{2}$ such that for any
$t_{1},t_{2}\in\Lambda$, their summation is also $t_{1}+t_{2}\in\Lambda$. This
property makes the lattice codes the building blocks of CP&F as for many
simultaneous transmission techniques.

In a fundamental CP&F network as given in Fig. 7, $N$ nodes encode their
message $w_{n}\in\\{1,...,N\\}$ to the lattices as

$x_{n}=\mathcal{E}(w_{n}),$ (3)

where $\mathcal{E}$ is the encoding function. Then, $N$ nodes simultaneously
transmit their message $x_{n}$ to the channel. A relay obtains the
superimposed signal

$y=\sum_{n=1}^{N}h_{n}x_{n}+z,$ (4)

where $h_{n}$ is the channel coefficient and $z$ is the AWGN. The CP&F decoder
scales the received message as

$\alpha y=\sum_{n=1}^{N}\alpha h_{n}x_{n}+\alpha z,$ (5)

and then quantizes the scaled signal to the closest lattice as follows

$\alpha y=\underbrace{\sum_{n=1}^{N}\beta_{n}x_{n}}_{\text{approximation of
$\alpha y$}}+\underbrace{\sum_{n=1}^{N}(\alpha h_{n}-\beta_{n})x_{n}\beta
z}_{\text{effective noise}}.$ (6)

Here, the receiver uses $\alpha y$ to approximate the appropriate coefficients
$\beta_{n}$ and obtains the following function of the codewords

$\sum_{n=1}^{N}\beta_{n}x_{n}.$ (7)

A single relay in the given scenario only obtains an integer function of the
messages which gives no information about the individual messages. However,
the receiver can detect the messages if it obtains independent functions as
many as the number of unknown messages. This idea inspired many studies in the
literature and several aspects of the CP&F have been investigated.
Additionally, the CP&F is extended to various channel and network models and
several design challenges (e.g. CSI estimation) are addressed. We collect the
related portion of these studies in Table II and classify them for their
network topology, objective and performance metrics.

TABLE II: An overview of CP&F studies. Study |  Year |  Network |  Contribution |  Performance metric
---|---|---|---|---
Nazer and Gastpar [5] |  2011 |  $K\times K\times 1$ |  CP&F is proposed. |  Achievable rates
Nazer and Gastpar [46] |  2011 |  Two-way |  Considered for PLNC.
Huang et al. [68] |  2013 |  Multi-way |  Rate optimization is investigated. |  Sum rates
Tan and Yuan [58, 59] |  2015 |  $K\times L\times 1$, multi-hop |  CPC&F is proposed.
Nokleby and Nazer [70] |  2013 |  $2\times 2\times 1$ |  Amplify-and-Compute model proposed.
Ntranos et al. [71], Tan et al. [45] |  2013,4 |  $K\times 1$, $K\times L\times 1$ |  Asymmetric power allocation case is considered.
Pappi et al. [93] |  2015 |  C-RAN |  Investigated from coalition game perspective.
El Soussi et al. [61] |  2014 |  $2\times 1\times 1$ cooperative |  Rate optimization is investigated. |  Symmetric rates
Ordentlich et al. [90] |  2014 |  $K\times K$ |  Interference channel considered.
Zhu and Gastpar [107] |  2015 |  Two-way |  Input distributions are investigated.
Wang et al. [69] |  2012 |  Multi-way |  Outage prob. investigated. |  Outage probability
Song et al. [47, 48] |  2011,3 |  $2\times 2\times 1$, $3\times 3\times 1$ |  Inverse CP&F model proposed. |  Rate region
Huang et al. [72] |  2013 |  Two-way |  Capacity bounds investigated.
Zhu and Gastpar [108] |  2013 |  $K\times 1$ |  Considered for cognitive radios.
Nazer and Gastpar [91] |  2014 |  $K\times 1$ |  Adapted to DMC.
Lim et al. [100, 101, 102, 103] |  2016-9 |  $K\times L\times 1$ |  Joint typicality decoder proposed.
Pappi et al. [84] |  2013 |  $K\times L\times 1$ |  CEE investigated. |  Computation rate
Ordentlich et al. [89] |  2015 |  $2\times 1$ |  Feedback included.
Hong and Caire [74, 75, 76] |  2011-3 |  $K\times L$ |  A low complexity design presented.
Liu [52] |  2014 |  Two-way |  Channel inversion precoding considered. |  Achievable rates, SER
Tunali et al. [55, 56] |  2012,5 |  $K\times L\times 1$ |  Eisenstein integer lattices considered. |  Outage prob., SER
Wang and Burr [95] |  2014 |  $2\times 1\times 2$ |  Coding gain improved with LDLC. |  SER
Mejri et al. [49, 50] |  2012,5 |  $K\times 1\times 1$, $K\times 1$ |  Decoding schemes compared. |  Error probability
Wei and Chen [51] |  2012 |  Two-way |  Fincke-Pohst code search implemented |  Average rate, zero entry prob.
Niesen and Whiting [57] |  2012 |  $2\times 2\times 1$ |  Degrees of freedom investigated. |  Degrees of freedom
Feng et al. [85, 86] |  2013 |  $K\times 1$ |  Blind CP&F (without CSI) presented. |  Throughput, complexity
Najafi et al. [87] |  2013 |  $K\times L\times 1$ |  Synchronisation problems investigated. |  Outage rate, average rate
Sakzad et al. [88] |  2014 |  $K\times L\times 1$ |  Phase precoding included. |  Equation error rate
Wen et al. [96] |  2015 |  $K\times L\times 1$ |  SVP considered with sphere decoding. |  Average computation rate
Nokleby and AAzhang [98] |  2016 |  $K\times L\times 1$ |  Node cooperation case investigated. |  Computation rate, outage prob.
Goldenbaum et al. [94] |  2016 |  $2\times 2\times 2$ |  Designed for OFDM and 5G. |  Message rate
Zhu and Gastpar [97] |  2016 |  $2\times 1$ |  Typical sumsets of lattices investigated. |  Density of the sets
Huang and Burr [80, 81, 82] |  2016,7 |  $K\times L\times 1$, $K\times 1\times 1$ |  A low complexity coefficient selection design suggested. |  Cumulative distribution func.
Goseling et al. [104, 105] |  2013,4 |  $K\times L$ |  Random access included. |  Throughput

In [46], Nazer and Gastpar propose a PLNC scheme based on the nested lattice
codes and CP&F. It has been shown that the lattice codes can be exploited to
transfer a function of the inputs to a sink node and the receiver can recover
the messages if it obtains enough functions. The study considers a two-way
relay channel and provides an introduction to the existing PLNC approaches.
Later, the study proposes the lattice code based PLNC and compares with other
schemes considering transfer rates.

In [47], a complementary scenario to the CP&F scheme, inverse compute-and-
forward, is considered. The CP&F scheme computes a function of transmitted
messages at relay nodes. The proposed network aims to recover back the
computed functions of the CP&F at the receiver. In the proposed network model,
two CP&F relays send their computed functions to a sink node. The inverse CP&F
is considered as a cascade to the traditional scheme and the rate region of
the cascade network is investigated. It is shown that the proposed cascade
network outperforms the traditional pairwise communication based relay
networks on the rate region. In [48], the authors extend their previous
inverse CP&F study to three transmitters scenario and investigate the rate
region. Their results show that transmitting equations with a correlation
between them provides superior performance than transmitting independent
equations.

Challenges of CP&F on lattice decoding is considered in [49]. The study
analyzes the lattice decoders that are suitable to practical scenarios.
Specifically, the performance of the maximum likelihood decoder, Diophantine
approximation and the sphere decoder is investigated. The computer simulations
are used to compare the decoders and to verify the previous theoretical
results. Also, the performance of one and two dimensional lattice codes are
investigated and it is shown via simulations that the performance degrades for
larger constellations. This study is later extended to provide an overall base
on the decoding of the CP&F in [50]. In addition to the previous study, [50]
includes a novel maximum a posteriori (MAP) decoder and a Diophantine
approximation based maximum likelihood decoder.

A two-way relay channel with CP&F is considered in [51] and [52]. The study
[51] proposes a Fincke-Pohst strategy based code search algorithm to find the
appropriate coefficients. In [53], the authors consider a multi-source multi-
relay network to maximize the network flow with CP&F. Contrary to the
traditional CP&F which optimizes the network coefficients separately for each
relay, the proposed design jointly optimizes the coefficient matrix for all
relays. The method utilizes a candidate set search algorithm based on the
Fincke-Pohst strategy (as in [51]) to select the coefficients. The performance
of the proposed method is investigated with simulations. In [52], a channel
inversion precoding is proposed. The achievable rates and the symbol error
rate (SER) of the channel inversion precoded CP&F are calculated and it is
stated that the proposed precoding improves the performance of the CP&F.

The communication between the nodes of a hexagonal lattice network is analyzed
in [54]. Four communication models are derived depending on the broadcast and
superposition communications between the nodes. Two models that are applicable
with CP&F are investigated on the subject of network capacity. The study
achieves an improved lower bound compared to the previous studies. The results
reveal that the minimum transport capacity of the broadcast or superposition
enabled case ($3/7$) is larger than the maximum capacity of the disabled cases
($2/5$).

In [55] and [56], the alphabet of the lattice codes that is used in CP&F is
restricted to the Eisenstein integers222Eisenstein integers are the complex
numbers in the form of $z=a+b\exp[2\pi i/3]$ where $a,b\in\mathcal{Z}$. The
traditional CP&F (as in [5]) uses integer-based lattice codes and the study
exploits the Eisenstein integers to obtain a better pair of nested lattice
structure (the coarse and the fine lattice). It is shown that the outage
performance and the error-correction performance of the proposed codebooks are
superior to the integer-based lattice codebooks.

In traditional CP&F, received signals at the relays are scaled up in order to
ensure that the coefficients are close to an integer. This is a result of the
lattice codes that involve only integer codebooks and scaling the signals up
also amplifies the noise at the receiver [5]. Eventually, scaling the signals
up establishes a Diophantine trade-off between the amplified noise level and
the approximation performance. The Diophantine trade-off of the CP&F scheme is
investigated in [57] and it is stated that the asymptotic rate of the scheme
in [5] is below the MIMO schemes. The authors design a novel compute-and-
forward model that benefits from the interference alignment (IA) in [57]. The
proposed model is shown to reach the same degrees of freedom with the MIMO
scheme.

In [58] and [59], the compute-compress-and-forward (CPC&F) method is proposed
to establish an efficient multi-hop design of the CP&F. The main idea behind
the CPC&F is the fact that relay forwarding rates can exceed the information
rate of the sources. For this reason, CPC&F includes a compressing phase to
improve network efficiency (e.g power gain). The compressing and the
recovering algorithms of the CPC&F is designed and verified with numerical
results. Later, the authors generalize the CPC&F method in which the
compression algorithm includes more operations and shows better compression
performance as demonstrated with simulations. The same problem that results
from the redundant forward rate is also considered in [60]. However, in [60],
the compression is applied at the symbol level rather than the message level
and introduces a mapping to the system.

In [61], a cooperative relay network is considered, where two nodes are able
to send their messages to a receiver both directly and over a relay. The study
investigates two coding methods (CP&F and C&F) that are based on lattice codes
and aims to optimize the symmetric rate. The authors propose an iterative
coordinate descent method that focuses on the power allocation and integer
coefficient selection processes for the optimization problem. The results
reveal that CP&F shows better performance than lattice-based C&F. This work is
later extended to a multi-user multi-relay cooperative scenario in [62].
Cooperative networks are also considered in [63, 64, 65, 66, 67]. The authors
in [64, 65, 67] propose an CP&F method for the two transmitter single relay
two receiver networks. The CP&F is considered for the same scenario with
single receiver in [63, 66].

The pairwise CP&F model is applied to multi-way relay channels (MWRC s) in
[68]. The pairwise structure of the network is accomplished with two phases:
the broadcast transmission phase and the multiple access transmission phase.
The sum rates of the pairwise CP&F case and the pairwise successive
transmission case are derived for the MWRC and compared with each other.

The outage probability of the CP&F scheme is derived in [69] for MWRC s. Also
the CP&F is compared with the non-network coding scheme with respect to the
outage probabilities. The results show that in a canonical two-way relay
channel, CP&F achieves $7$ dB gain against the non-network coding at the
outage probability of $10^{-2}$.

In [70], amplify-and-compute method is proposed, which combines CP&F and A&F
methods. The relays in the network receive the superpositioned lattice codes
from the sources as in CP&F and transmit to the next network layer as in A&F.

A CP&F scheme that allows asymmetric power allocation to the nodes is proposed
in [71] and [45]. The method in [71] is based on the lattice codes in which a
fine lattice and a coarse lattice provide codebooks that are decodable and
under the power limit respectively. The method maps the messages to the
codebooks depending on the power and noise tolerance. Specifically, the top of
the message vector is set to zero depending on the power of the codebook and
the bottom of the message vector is set to zero depending on the noise
tolerance.

A bi-directional relay network in which two nodes exchange information through
a relay node under an inter-symbol interference channel is considered in [72].
The proposed model is separated into a multiple access phase and a broadcast
phase and the capacity region is derived. The inner bound of the capacity
region is computed with the help of the CP&F method and the outer bound is
computed with the cut-set argument given in [73]. The numerical results
revealed that the proposed CP&F scheme has a higher exchange rate than the
D&F.

The complexity reduced version of the CP&F is proposed in [74], which only
involves scaling, offset and scalar quantization at the receivers. The method
aims to reach the same capacity of the CP&F with the low-complexity, low-power
decentralized antenna networks. For this purpose, the method considers
quantization at the receivers as a part of the wireless channel. The numerical
results show that the computation rate of the quantized CP&F is within the
shaping error of 0.25 bits per symbol compared to the traditional CP&F of [5].
The authors extended their work to downlink scenario of the quantized CP&F in
[75]. The study derives the computation rate of the proposed scheme and
compares it with the downlink CP&F via simulations. The reverse CP&F is
generalized and extended in [76] to cover additional scenarios and to include
comprehensive simulation results and comparisons.

Another low-complexity CP&F scheme is given in [77]. The outage probability of
the proposed model is derived and compared with the standard CP&F scheme.
Additionally, channel estimation error (CEE) is introduced to the system and
its effect is investigated. The results show that the proposed simple method
is also more resistant to the CEE than the traditional CP&F. In [78], the
authors extend their study and propose two CP&F based methods. The first
method is designed to reduce the computational complexity and the second
method is proved to have better performance compared to the traditional CP&F,
which is verified via simulations.

The complexity of the coefficient selection of the CP&F is considered in [79,
80, 81, 82]. The authors propose a low-complexity algorithm to optimize the
integer coefficients that is essential for the CP&F performance in [79]. This
work is later improved and generalized to cover CP&F and integer-forcing
algorithms in [83]. The same problem is studied in [80, 81]. An exhaustive
search algorithm and a lattice reduction algorithm is proposed in [80, 81] to
reduce the hardness of the coefficient selection problem. Also in [82], a low-
complexity coefficient method is proposed for massive MIMO enabled CP&F
networks.

The effect of the channel estimation error to the performance of CP&F scheme
is analyzed in [84]. The computation rate region for the imperfect channel
estimation case is derived and the expression is closely approximated for the
Gaussian distributed CEE. Additionally, the distribution of the rate loss is
given in the closed-form. The simulations are used to demonstrate the
vulnerability of the CP&F to the CEE.

The traditional CP&F requires CSI to decide the appropriate scale factors
which are essential in the decoding of the integer lattices. Otherwise, the
non-integer channel coefficients increase symbol error. In [85, 86], a
practical CP&F scheme is proposed that does not require CSI to compute the
most suitable scale factors. Instead, the proposed method chooses sub-optimal
however sufficient scaling factors to gain from the system complexity.
Simulations show that in some cases, the computation complexity can be reduced
ten times when compared to the CP&F of [5].

The synchronization problem of the CP&F is considered in [87]. CP&F networks
can exhibit asynchronization of the nodes as a result of the decentralized
node structure of the network. The study solves the symbol asynchronization
problem with an equalizer by converting the nature of the network from
asynchronous to synchronous. The frame asynchronization is solved by
eliminating delays with multiple antennas at the relay node. Also, it is shown
that the achievable rate can be maximized for all SNR regions by applying a
linear filter.

A phase precoding method for the CP&F scheme is proposed in [88] for multi-
user multi-relay networks. The objective of the method is to reach higher
computation rates than the traditional CP&F. However, it also requires an
optimal precoding matrix and an optimal network equation matrix to fulfill
that objective. For this reason, the study introduces a partial feedback
channel between the relays and the nodes since the precoding matrix is needed
at the nodes and the network equation matrix has to be computed at the relays.
The relays compute the optimal precoder and the network equations, then
forward the precoder information to the nodes through the feedback channel.
With the simulations, the study shows that the proposed phase precoding can
improve the equation error rate.

Another feedback enabled CP&F method is given in [89]. The method aims to
design the optimal CP&F model to achieve the maximum computation rates for the
scenario that transmitters have access to an ideal feedback channel towards
the relay. The method is designed for two users (and a relay) networks and it
is demonstrated that the proposed scheme obtains better computation rates than
the CP&F without feedback.

CP&F is considered for the Gaussian multi-user interference channels in [90].
A comprehensive study is given on the approximate sum capacity and the
capacity bounds. In [91], CP&F is investigated for the discrete memoryless
channels. The lattice codes are considered for the complex modulo arithmetics
in [92]. It is shown that only five lattice code families are capable of
complex modulo arithmetics over the Euclidean geometry and their coding gains
are calculated. In [93], CP&F scheme is considered for the cloud-radio access
networks (C-RAN s). The study aims to maximize the information flow from the
nodes to the FC of the network. For this reason, a coalition game is designed
that maximizes the defined profits. Exploiting the interference of signals is
considered for the fifth generation ($5$G) networks in [94]. The objective is
to provide channel access to a massive number of nodes that are required by
the IoT. For this purpose, the study combines the PLNC with the pulse shaped
orthogonal frequency division multiplexing (OFDM).

In [95], a low density lattice codes (LDLC) based CP&F method is proposed in
order to reach high coding gains. In [96], the authors propose a sphere
decoding method to maximize the computation rate by considering the problem in
hand as a shortest vector problem (SVP). The sumsets can be defined in simple
terms as the set of received lattice points which is the sum of the
transmitted lattice points. In [97], typical sumsets are defined and analyzed
according to their sizes, distributions and densities. The study aims to
obtain results that can improve the performance of the lattice decoding in
CP&F.

Cooperation between the transmitters is considered in [98]. This is different
than the cooperative networks as in Fig. 6d. In this study, cooperation
indicates that the nodes can partially hear the messages of the other nodes
which resembles the diversity improvement of a multiple antenna network. The
results reveal that the partial cooperation between the nodes can increase the
computation rate almost to the capacity.

In [99], a CP&F transform is proposed in which the W-MAC is transformed to a
modulo-lattice MIMO channel with the help of SIC. Joint typicality decoders
are adopted to the CP&F method in [100, 101, 102, 103]. The CP&F is also
considered for random access channels in [104, 105, 106]. The impact of the
input distribution to the computation rate of the CP&F is considered in [107]
for the Gaussian W-MAC. It is shown that the Gaussian input distribution is
not optimal and the computation rates can be improved if the input
distributions are chosen wisely. In [108], CP&F is extended to cognitive radio
networks. In the following section, we present the computation and function
alignment methods that are inspired by the function alignment and CP&F
methods.

##  V Interference / Computation / Function Alignment

The interference alignment (IA) is an interference management technique and
can be compared with the multiple access methods for their application
purpose. The conventional multiple access methods divide the time and
frequency resources among the users. In the IA, all users share the same
resources, however, the IA algorithm affects the transmitted signals
(precoding) such that the received signals are aligned into two subspaces. As
the algorithm aims, the unintended signals (the interference from the other
users) fall under one subspace and the intended signal can be extracted from
the other subspace. We are partially interested in the IA since, on one hand,
the network model enables the interference; on the other hand, it aims to
cancel the interference instead of benefiting from it.

We are much more interested in the computation alignment method that is
inspired by the IA and the CP&F. Similar to the IA, the computation alignment
divides the signals into subspaces, however, the aligned signals are not
discarded, instead, the interference is exploited for the computation. In this
section, we present an elementary description of the IA and refer the readers
to [109] and [110] for detailed information. Later, we continue with the
computation alignment and present the current studies. The fundamental IA,
computation and function alignment studies are exhibited in Table III.

TABLE III: An overview of IA, computation alignment and function alignment studies. Author |  Year |  Network model |  Contribution |  Performance metric
---|---|---|---|---
Niesen et al. [112, 111] |  2011,2013 |  $K\times K$ |  Provides a capacity approximation for multi-layer networks that is independent of network depth. |  Capacity approximation
Goela et al. [113] |  2012 |  Investigates coding schemes that reach computation capacity with network decomposition. |  Coding capacity
Suh et al. [114, 115] |  2012, 2016 |  $2\times 2$ |  Derives a new upper bound on the computing capacity and propose a network decomposition theorem. |  Computing capacity
Suh and Gastpar [116] |  2013 |  Considers feedback for function alignment.
Suh and Gastpar [117] |  2013 |  Investigates the scenarios where network decomposition is optimal. |  Symmetric capacity

The IA studies are mainly centered upon the space, frequency, or time
dimensions to align the interference [109]. We focus on a simple space
dimension example, which is based on multiple antenna techniques. Consider a
MIMO network that consists of three transmitters and three receivers all of
which equipped with two antennas as shown in Fig. 8a. After the simultaneous
transmission, the first receiver obtains the following signal.

$\mathbf{y}_{1}=\mathbf{H}_{11}\mathbf{v}_{1}s_{1}+\mathbf{H}_{21}\mathbf{v}_{2}s_{2}+\mathbf{H}_{31}\mathbf{v}_{3}s_{3}+\mathbf{n}_{1}.$
(8)

(a)

(b)

Figure 8: Vector illustration of the interference alignment and computation
alignment methods. (a) Interference alignment, (b) Computation alignment.

The subscripts indicate the users, i.e. $\mathbf{y}_{j}\in\mathbb{C}^{2\times
1}$ is the received signal vector of the $j^{th}$ receiver. The first and
second row of the vector are the received signal at the first and second
antenna, respectively. $\mathbf{H}_{ij}\in\mathbb{C}^{1\times 2}$ is the
channel matrix vector from the $i^{th}$ transmitter to the $j^{th}$ receiver.
$\mathbf{v}_{i}\in\mathbb{C}^{2\times 2}$ is the diagonal precoding matrix of
the $i^{th}$ transmitter, $s_{i}$ is the message of the $i^{th}$ transmitter
and the $\mathbf{n}_{j}\in\mathbb{C}^{2\times 1}$ is the Gaussian noise vector
at the $j^{th}$ receiver, where its rows are the noise of the first and second
antenna, respectively.

Assuming perfect CSI at the receivers, the MIMO algorithms require three
antennas since each of the three users’ inputs is an unknown variable. In IA
schemes, the precoding vector ($\mathbf{v}_{i}$) elegantly aligns the two
unknown vectors (the interference from the other users) such that (8) can be
written with two unknown vectors. After that, the two unknown variables can be
solved with the two equations. The vector representation of how the IA works
is illustrated in Fig. 8. Each node is represented with a square and the
transmitted and received signals are given in the inside of these squares. The
precoding at the transmitter divides the three vectors into two vectors; one
of them is the intended vector and the other is the aligned (unintended)
signals.

The perspective given above exploits the physical layer to suppress the
interference. The computation alignment method is inspired from this
perspective, however, it focuses on the unintended aligned vector for the
function computation, which genuinely exploits the signal interference. A
simple two transmitter two receiver network example of the computation
alignment is given in Fig. 8b. In this example, the time dimension is used to
create the subspaces. Assume that the first transmitter aims to send $s_{11}$
and $s_{12}$ while the second transmitter aims to send $s_{21}$. In this
scenario, transmit vectors adjust the messages as follows

$\begin{split}\begin{bmatrix}x_{1}(t_{1})\\\
x_{1}(t_{2})\end{bmatrix}&=\mathbf{v}_{11}s_{11}+\mathbf{v}_{12}s_{12}\\\
\begin{bmatrix}x_{2}(t_{1})\\\
x_{2}(t_{2})\end{bmatrix}&=\mathbf{v}_{21}s_{21},\end{split}$ (9)

where $t_{1}$, $t_{2}$ denote the time slots and $x_{1}$, $x_{2}$ denote the
transmitted signals of the first and second transmitters, respectively. Proper
selection of the transmit vectors lead to the following received signal
vectors

$\begin{split}\mathbf{y}_{1}&=\begin{bmatrix}1\\\
1\end{bmatrix}(s_{11}+s_{21})+h\begin{bmatrix}1\\\
-1\end{bmatrix}s_{21}+\mathbf{n}_{1}\\\ \mathbf{y}_{2}&=h\begin{bmatrix}1\\\
1\end{bmatrix}(s_{12}+s_{21})+\begin{bmatrix}1\\\
-1\end{bmatrix}s_{11}+\mathbf{n}_{1},\end{split}$ (10)

where the subscripts indicate the users, i.e.
$\mathbf{y}_{j}\in\mathbb{C}^{2\times 1}$ is the received signal vector of the
$j^{th}$ receiver. However, subspaces are obtained with multiple transmissions
rather than multiple antennas in this example, hence the first and second row
of the vector are the received signal at the first and second time slots,
respectively. After obtaining signals from two time slots, the first receiver
can obtain $s_{11}+s_{21}$ by using $y_{1}(t_{1})+y_{1}(t_{2})$ and the second
user can obtain $s_{11}$ by using $y_{2}(t_{1})-y_{2}(t_{2})$.

The design of the transmit vector enables computation alignment. A vectorial
illustration of (10) is given in Fig. 8b. It should be noted that the
illustration omits the representation of the transmit vectors and the channel
gain vectors in the figure for better appearance. As a result of the
computation alignment, the receivers obtain the summation of the messages in
the aligned subspace. Specifically, the first receiver obtains $s_{11}+s_{21}$
aligned and $s_{12}$ in the other subspace while the receiver two obtains
$s_{12}+s_{21}$ and $s_{11}$.

In a relay network, C&F method adds additional noise to the network in each
layer. As a result, the approximation gap of the network capacity widens for
the increasing number of network layers. The computation alignment scheme in
[111] and [112] presents a relay network with an approximation gap that is
independent of the layer depth. The computation alignment technique depends on
lattice codes and the CP&F to be able to recover the integer-valued messages.
However, the CP&F scheme also produce errors as a result of the non-integer
channel gains. This problem is solved by IA by dividing the channel into
multiple subchannels and aligning. The results reveal that the approximation
gap is not constant as opposed to C&F results, it depends on fading
characteristics.

In [113], multiple transmitter multiple receiver summation networks are
considered. The scalar and vectorial linear codes are investigated for these
networks and it is stated that the computation alignment is essential to reach
the computation capacity. For this purpose, the network is decomposed into
sub-networks with the network equivalence theorems. Also, the linear coding
capacity of the computation is derived for various channel parameters.

In [114] and [115], $2\times 2$ modulo-2 sum networks333In particular
Avestimehr-Diggavi-Tse (ADT) network is considered. are considered. The study
is inspired by the IA and similar to the computation alignment given in [112],
and named as the function alignment. A new upper bound is derived for the
computing capacity of the two receiver CP&F networks with linear codes in
[114, 115]. Also, the studies define a network decomposition theorem to divide
the network into elementary subnetworks. Using the theorem, the computing
capacity is generalized for the $N$-transmitter $N$-receiver CP&F networks. In
[116], the authors extend their previous work, [114], to include feedback. The
study derives the feedback included computing capacity and compare it with the
no-feedback scheme (as in [114]). It should be stated that network
decomposition is crucial to create subnetworks. The authors also investigate
the network decomposition and its importance in CP&F networks thoroughly in
[117].

In [118], memoryless bivariate Gaussian sources are considered for a two-
source one receiver network. The receiver aims to obtain the information of
the two sources with the minimum distortion. In the paper, perfect causal
feedback is assumed and the power-distortion relationship is investigated. The
results are given as a function of the source correlation and SNR. Also, the
necessary and sufficient conditions to reach the minimum distortion levels are
derived. The study is later extended to two transmitter two receiver networks
in [119]. Simultaneous transmission is also the basis of function computation
techniques that are inspired by lattice codes and CP&F. Moreover, the function
computation methods are considered for analog signals and the resulting analog
function computation studies gained popularity in the literature.

##  VI Function Computation

Function computation is one of the most striking applications of simultaneous
transmission. The joint source-channel coding paradigm by Gastpar and Vetterli
[2] and later Nazer and Gastpar [4] establish the basis for the function
computation. This approach inspires a wide range of studies that often targets
one of the two main aspects; computation or multiple access. The motivation
behind the studies that focus on the multiple access aspect is generally to
improve the network throughput as in CP&F and considered in Section IV. Here,
we present the studies that focus on the computation aspect as the digital
function computation. The computation aspect later inspires the analog
transmission based studies that purely focus on function computation. The
motivation behind these studies is strictly computation related and presented
as the analog function computation [6].

###  VI-A Digital Function Computation

A list of the digital function computation studies is presented in Table IV.
Nazer and Gastpar proposed the computation codes in [4] that is based on
lattices. The objective of the study is to send a linear function of multiple
users to a receiver with the simultaneous transmission. The study investigates
the achievable rates with the proposed computation codes and compares them
with the separation based methods. Computation of linear functions is also
considered in [120] for a wireless network that consists of two correlated
Gaussian sources. The work aims to find the optimum coding scheme that upper
bounds the distortion at the received signal. The numerical results are given
for the subtraction ($a-b$) and weighted addition ($a+2b$) functions as a
function of the correlation coefficient of the two sources.

TABLE IV: An overview of digital function computation studies. Author |  Year |  Network model |  Contribution |  Performance metric
---|---|---|---|---
Jeon et al. [123, 124] |  2013, 2014 |  $K\times L\times 1$ |  Extends the lattice code computable function set by considering orthogonal components and derives an approximation of computation capacity. |  Computation rate
Wu et al. [125, 126] |  2015, 2016 |  Proposes a low-bandwidth, low-energy SDR network architecture, STAC. |  SER, session rate
Nazer and Gastper [4] |  2007 |  $K\times 1$ |  Proposes the computation codes and the CoMAC which enhance the communication performance by utilizing lattice codes and the joint source-channel coding. |  Computation rate
Goldenbaum et al. [121, 122] |  2013, 2015 |  Improves the reliability of nomographic function computation by adapting lattice codes to AFC based consensus methods.
Jeon and Jung [127, 128] |  2015, 2016 |  Provides non-vanishing computation rates (asymptotically positive) by allowing only a subset of nodes with high gains to transmit.
Wu et al. [132, 133, 134, 135] |  2019 |  Considers the wide-band implementation of CoMAC by allocating sub-functions to subcarriers.
Soundararajan and Vishwanath [120] |  2012 |  $2\times 1$ |  Derives a lower bound on the distortion of CoMAC and considers correlated source scenario. |  Distortion rate
Zhan et al. [129] |  2011 |  Butterfly |  Investigates the duality between computation and communication aspect of simultaneous transmission schemes. |  Distortion level
Zhu et al. [130, 131] |  2017, 2019 |  $2\times 2$ |  Capacity region
Chen et al. [136] |  2020 |  $K\times 1$ |  A low-complexity transceiver model is designed to maximize the achievable function rate. |  Achievable function rate

Goldenbaum et al. generalize the computation of nomographic functions with
nested lattice codes in [121]. The model is based on the fact that the
nomographic functions can be written in the form of pre and post processing
functions as studied in the analog function computation research. However, the
authors implement a digital model to reduce the destructive effects of the
noise. The study also examines the required number of channel use and the
accuracy performance of the system. Lattice codes are used in [122] for the
computation of nomographic functions. The study thoroughly analyzes the
relation between the lattice codes and the nomographic functions and derive
the computation rate performances. One of their observations reveals that any
continuous function can be computed over the channel.

In [123] and [124], the authors include orthogonal components to their coding
scheme in a similar manner that the TBMA benefits from the orthogonal signals.
The method is interested in calculating the arithmetic summation and the type
functions. The type function computes the histogram of the transmitted signals
as explained in TBMA. Then the resulting statistics can be used to obtain the
mean, variance, maximum, minimum and median functions. The W-MAC is firstly
decomposed into multiple modulo sum subchannels with nested lattice codes and
linear network codes. Then the linear Slepian–Wolf source coding is used to
calculate the desired functions. It is shown that the joint source-channel
coding provides better performance than the separate coding schemes in certain
cases.

The simultaneous transmitting and air computing (STAC) method is proposed in
[125] and [126] to improve the function computation capability and the network
efficiency of data center networks. The proposed method is based on a
traditional function computation model that combines communication and
computation; however, the model is also developed upon an enhanced software-
defined network structure that provides side information to the nodes.
Computer simulations demonstrate the spectrum and energy efficiency of STAC in
data center networks.

Function computation problem is considered for the fading MAC s in [127] and
[128]. The main idea behind the study is that only the nodes with high channel
gains participate in the in-network computation rather than the whole network.
The study investigates the computation rates of the proposed model for the
fading channels.

A duality between the function computation problem and the multicast problem
is considered in [129]. The function computation problem is interested in
receiving a function (e.g. summation) of the transmitted messages. In the
multicast problem, the objective is to receive individual messages. After
defining the duality relation for the deterministic networks, Gaussian MAC s
are considered. The achievable distortion levels are derived for the summation
of two Gaussian sources in these networks. The results revealed that there is
a constant gap between the cut-set bound and the distortion of the summation
function of the independent Gaussian sources.

It is obvious that the superposition of the signals destroys the individual
information of the transmitted messages. This can be viewed as renouncing the
capability of the nodes to (multiple) access to the receiver [130, 131].
However, computation codes benefit from this idea to improve the efficiency of
the computations in the network. Here, access to the individual information is
traded with improved computation efficiency and this can be highly beneficial
if the network is only interested in a function of the transmitted data. In
other words, computation codes present a duality between the multiple access
and computation. This duality is investigated in [130] and [131] to check the
existence of the computation codes that also allow the individual access to
the receiver. The investigation results indicate that efficient computation
codes prevent the individual access to the receiver.

The wideband computation over multiple access channel (CoMAC) schemes that are
adopted for the frequency selective channels is proposed in [132, 133, 134,
135]. These studies rely on the NOMA and OFDM models to use wideband
frequencies. A NOMA assisted function computation network is proposed in [132]
and [133]. As given in the previous sections, NOMA is a multiple access method
that is based on the superposition of signals. In [132], functions to be
computed are intentionally divided into sub-functions and these sub-functions
are computed simultaneously under different NOMA access slots. As a result,
the computations can be made at the wideband frequencies where the fading is
more challenging. The results reveal that the proposed NOMA assisted approach
achieves higher computation rates and prevents vanishing computation.
Additionally, expressions for the diversity order of the computation rate is
derived in [133]. In [134] and [135], the sub-functions are allocated to the
OFDM carriers and an optimization problem is considered for the power
allocation. In [136], a transceiver model is designed for digital function
computation that reduces the time-complexity. The authors derive the
achievable function rates of the proposed model by considering the number of
nodes, the maximum value of messages and the quantization error threshold.

###  VI-B Analog Function Computation (AFC)

The main idea behin the AFC is to match the W-MAC with the desired function.
In its base form, the W-MAC constitutes a natural summation operation with its
superposition property. The AFC adjusts the channel with proper signal
processing at the transmitter and receiver ends such that the W-MAC can
compute other mathematical operations. For this purpose, transmitters use pre-
processing functions, $\varphi_{n}(\cdot):\mathbb{R}\rightarrow\mathbb{R}$,
before transmitting their signals and the receiver applies a post-processing
function, $\psi(\cdot):\mathbb{R}\rightarrow\mathbb{R}$, after receiving the
superpositioned signal as depicted in Fig. 9. The users simultaneously
transmit their pre-processed signals and the receiver obtains the following
function output after the post-process,

$f(x_{1},x_{2},...,x_{N})=\psi\left(\sum_{\begin{subarray}{c}n=1\end{subarray}}^{N}\varphi_{n}(x_{n})\right).$
(11)

The functions that can be expressed as in (11) with the summation operation
are called nomographic functions. The AFC is firstly shown to be applicable to
nomographic functions and later it is proven that any function can be computed
with the AFC [6]. The AFC studies are listed in Table V with their
contributions and performance metrics.

Figure 9: The network model of the AFC studies. $\psi$ and $\varphi$ denote pre-processing and post-processing functions, respectively. TABLE V: An overview of analog function computation studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Goldenbaum et al. [137] |  2009 |  Computes functions (e.g. arithmetic mean) over the channel with low-complexity and low-energy consumption. |  Outage probability
Goldenbaum and Stańczak [139] |  2010 |  Extends [137] to investigate the geometric mean.
Stańczak et al. [140, 141] |  2012, 2013 |  Provides a complete theory of AFC.
Goldenbaum and Stańczak [142] |  2014 |  Investigates the AFC for different CSI assumptions.
Jeon and Jung [149] |  2018 |  Improves the traditional AFC against fading environment by utilizing causal CSI.
Chen et al. [152] |  2018 |  Extends [138], considers the computation of multiple functions and investigates the method’s performance.
Goldenbaum et al. [6] |  2013 |  Investigates and generalizes the functions that is computable with AFC. |  MSE
Limmer and Stańczak [143] |  2014 |  Investigates the computation of $l_{p}$-norm functions with AFC.
Huang et al. [148] |  2015 |  Extends AFC to MIMO networks and includes imperfect CSI.
Zhu et al. [150, 151] |  2018 |  Introduces a multi-modal AirComp technique with MIMO and beamforming that minimizes distortion.
Farajzadeh et al. [161] |  2020 |  Removes the CSI requirement of AirComp.
Wen et al. [153] |  2019 |  Provides low-complexity and low-latency with MIMO AirComp.
Ang et al. [154, 155] |  2019 |  Reduce the complexity of the training process in massive CSI acquisition.
Li et al. [156, 157] |  2018, 2019 |  Considers an AirComp method that transfers power to the distributed nodes over the air to relax energy constraints.
Cao et al. [158] |  2019 |  Considers joint optimization of the transmit powers and the denoising factor to improve AirComp.
Basaran et al. [159] |  2020 |  Provides an energy efficient AirComp method by exploiting the correlation between the measurements.
Chen et al. [138] |  2018 |  Considers non-uniform fading scenario in the proposed AirComp method and brings robustness with the uniform-forcing transceiver design.
Limmer et al. [144] |  2015 |  Provides a method that computes certain multivariate functions with nomographic function approximation. |  Approximation error
Goldenbaum et al. [145] |  2015 |  Adapts computation codes to AFC to provide reliability and investigate its rates. |  Computation rate
Wang et al. [147] |  2015 |  Without channel estimation, reduces the required number of channel uses on the computation of mean function with free deconvolution. |  Relative error
Dong et al. [160] |  2020 |  Uses Wirtinger flow to provide a low-complexity, low-latency AirComp method that requires no CSI).
Chen et al. [162] |  2019 |  Investigates and compares the computation based and communication based methods. |  Function rate
Jakimovski et al. [165] |  2011 |  Provides a testbed implementation. |  Average error
Sigg et al. [166] |  2012 |  Mean error
Kortke et al. [163] |  2014 |  Relative error
Abari et al. [167] |  2015 |  CFO
Altun et al. [164] |  2017 |  MSE

Goldenbaum and Stańczak pioneered the studies that compute mathematical
functions at the communication process. In [137], the communication system
carries the information at the transmit powers which relieves the
synchronization burden. Their method manages to compute a variety of functions
which involve nonlinear functions. The authors lastly exhibit the error
analysis of the arithmetic mean function. In [139], Goldenbaum and Stańczak
extend their previous work by analyzing the geometric mean function. They also
compare their method with TDMA via simulations and present their results that
AFC outperforms the TDMA on the function computation time.

In their following works, Goldenbaum and Stańczak provide an extensive theory
of AFC in [140] and [6]. They analyze the function sets and network topologies
that are compatible with AFC. Their results show that every function can be
computed with AFC as the pre-processing is independent of the computed
function.

The error performance of the estimators that are used in the receivers of AFC
is analyzed in [141] for arithmetic and geometric mean functions.
Additionally, function computation simulations are performed for AFC, TDMA and
CDMA models. The results showed that the AFC brings out better computation
accuracy in less time than time or code divided networks.

Dependency of AFC to CSI is investigated in [142]. The letter firstly shows
that the transmitters only need the magnitude of the CSI rather than the full
CSI. Then it is proven that any CSI knowledge requirement on the transmitters
can be removed if the receiver is equipped with multiple antennas.

The computation of the $l_{p}$-norms with the AFC is investigated in [143].
The authors later propose an algorithm to approximate certain continuous
multivariate functions by the nomographic function in [144]. In [80], the max
function which aims to obtain the maximum value at the FC is implemented with
the AFC and CDMA.

In [145], Goldenbaum and Stańczak investigate the relation between the
reliability and efficiency of an AFC model from an information-theoretic point
of view. In order to evaluate the relation in question, the computation rate
metric is defined as the number of functions that can be computed per channel
use. Then, achievable computation rates are given for linear combination and
special polynomial functions. The letter states that the computation rate is
dependant to the required accuracy level and the number of transmitters as
well as the function type.

An AFC method based on the free deconvolution theorem (see [146] for the
further information on the free deconvolution) is proposed in [147]. The
contribution of the study is that the mean function can be computed with fewer
channel uses and without the channel estimation.

In [148], a MIMO AFC model is proposed that the FC and the multiples
transmitting nodes are equipped with multiple antennas. The method allows
channel estimation errors in the system model to consider realistic scenarios
and design a non-convex optimization problem to achieve the optimum solution
in this scenario. The optimization problem that minimizes the worst-case mean
square error (MSE) is converted to a simpler version and solved.

An adaptive AFC method is given in [149]. Different from the traditional AFC
methods, this model is based on the causal CSI at the transmitters. Simulation
results of the study state that the proposed adaptive model shows better
outage probability performance than the traditional AFC.

In [150], a function computation method that can compute functions for
multiple variables is proposed. For example, multi-model sensor measurements
such as temperature, humidity and pollution can be computed over the air
simultaneously. This is possible with the utilization of beamforming
technology over MIMO enabled nodes. The study designs an optimization problem
that minimizes the sum mean-squared error at the receiver. The optimization
problem is shown to be an NP-hard problem and an approximate version is solved
with differential geometry. Also, the result of the optimization problem
(based on Grassmann Manifold) is validated with computer simulations. This
work is later extended to high-mobility sensing networks where an environment
is monitored by unmanned aerial vehicles (UAV s) [151]. In addition to [150],
the study includes enhanced equalization and channel feedback methods to
provide reliable information exchange between the sensors and the receiver.

Non-uniform fading in different nodes of an AFC network is a performance
degrading problem. In order to mitigate the effects of non-uniform fading, a
uniform-forcing transceiver design is proposed in [138]. The authors formulate
an optimization problem that minimizes the mean square error of the computed
function output at the receiver. Their results show that a semidefinite
relaxation is necessary to solve the problem and successive convex
approximation can further increase the accuracy of the solution. The proposed
transceiver model in [138] is extended in [152] to compute multiple functions
simultaneously. The nodes in the new network model are equipped with antenna
arrays and zero-forcing beamforming technology. Beamforming technology is
adopted to remove the interference of other functions that are computed
simultaneously and multiple antenna arrays relieve the massive CSI knowledge
requirement. Also as in [138], the method is resilient to the non-uniform
fading problem. The performance of the proposed network is analyzed with both
simulations and analytical expressions.

Another beamforming and MIMO enabled function computation method is given in
[153]. The study proposes a receiver beamforming model design that reduces
channel dimensions and equalizes channel covariances and small scale fading
components. Additionally, the method uses a feedback scheme to accurately
provide massive CSI knowledge to the network. The proposed approach mainly
aims to reduce the computation error resulting from the discrepancy of the
received function output. The simulations demonstrate the positive effect of
the proposed model on error reduction.

A training model for faster acquisition of the CSI is proposed in [154] and
[155]. The method is based on the effective CSI definition that is obtained
with the simultaneous pilot transmission of the nodes. The method requires
iterative broadcasts from FC to estimate the effective CSI and one last
simultaneous pilot transmission yields the CSI vector at the FC while the
traditional AFC methods individually train each CSI. The study analytically
obtains the computation complexity of both traditional and proposed methods.
Also, an error improvement method is proposed to compensate for the estimation
error of the novel approach since the proposed model causes larger estimation
errors.

Function computation is integrated with wireless power transfer in [156] for
IoT networks. The framework aims to minimize the computation error that is
seen at the aggregated data by jointly optimizing power transfer and function
computation tasks. The framework is designed for networks that consist of MIMO
beamforming capable nodes. The joint optimization problem is divided into two
sections; wireless power control optimization and function computation
optimization. As applied earlier in [138], the semidefinite relaxation method
is implemented to solve the function computation section of the optimization
problem while the wireless power control section is solved in closed form. The
study also states that the integration of the wireless power transfer into the
AFC network brings an additional design dimension which can increase the
computation accuracy. A combination of function computation and wireless power
transfer is also implemented for high-mobility sensing in smart cities where
UAV s are used to collect sensor readings [157].

Power control problem of the AFC systems are considered in [158]. Poor
distribution of transmit powers in an AFC network can cause high computation
errors as a result of channel distortion. The proposed method aims to find the
optimum transmit power level design by solving the optimization problem that
minimizes the computation error at the FC. The definition of the problem
involves the optimization of both transmit powers and the denoising factor of
the FC. Moreover, the problem is solved for additional scenarios such that
only one transmitter has power constraints instead of all devices. Lastly, the
simulations reveal that the proposed power control scheme has a notable
mitigating effect on the computation error of the AFC network.

The problem of energy consumption in function computation networks is
considered in [159]. The method mainly benefits from the spatial correlations
between the sensor readings in order to reduce energy consumption. For this
purpose, a minimum mean square error (MMSE) estimator is designed to obtain
estimations with less number of samples. The proposed estimator requires
significantly less energy consumption to operate, hence nearly doubles the
lifetime of the network. In addition to the improved energy efficiency, the
estimator also yields better MSE performance compared to the traditional
methods as illustrated via simulations.

Wirtinger flow is an algorithm that is usually used in the solution of non-
convex optimization problems such as phase retrieval from the received signal
magnitudes. Dong et al. use the Wirtinger flow method to solve the non-convex
optimization problem of the function computation without the knowledge of CSI
in [160]. The proposed algorithm only requires data samples and randomly
initialized Wirtinger flow iterations, i.e. pilot transmission for CSI
acquisition is not needed. As a result, the study can reduce the latency of
the function computation applications by removing the dependency on the pilot
transmission process of the CSI estimation. Also, the study reveals that the
estimation error of the Wirtinger flow-based function computation model is
sufficiently small.

In [161], the power alignment problem of the function computation networks is
considered. The power alignment is usually performed with pre-processing or
pre-coding in traditional function computation networks. In [161], this
problem is addressed with a backscatter framework instead of pre-coding. The
proposed method is designed for mass density sensor networks where multiple
UAV s are responsible for the collection of sensor measurements. The model
consists of two phases; channel gain acquisition and data aggregation. In the
first phase, the UAV s are the power emitters and the sensor nodes act as the
backscatter object in which the nodes backscatter the ambient signal that
comes from the UAV. As a result, the UAV collects the sum channel gain. In the
second phase, the UAV s are the readers that receive the aggregated sensor
data. The results show that using the sum channel gain for the aggregation of
the sensor readings can improve MSE up to 10 dB.

The function computation capability of the AFC is compared with the separate
communication schemes in [162]. The achievable function rates are derived for
the two cases and it is shown and verified via simulations that the
computation over the channel (AFC) is not always the optimum scenario for the
function computation.

####  VI-B1 Test-bed Implementations

The feasibility of the AFC is also investigated with testbed implementations
in the literature. In [163], the authors deploy the model of [141] with SDR s.
A network with 11 nodes and an FC is implemented to compute the arithmetic and
geometric mean of the sensor readings of the nodes. In another implementation
study [164], the summation of sensor readings over the channel is tested via
SDR modules. The network that includes three transmitters and an FC is used to
analyze the effect of distance and signal power level. Also, the error
performance of the AFC network is compared with the TDMA scheme via computer
simulations. The TDMA scheme requires communication slots for each node while
the AFC completes the transmission in one slot. As a result, the simulation
results of AFC displays less error since TDMA introduces additional thermal
noise to the system with each communication.

In [165], a simultaneous transmission method is proposed and implemented with
SDR modules. The method is based on the hamming distance of the superimposed
received signals that are affected by each transmit vector. The calculation of
corrupted goods in a pallet via temperature readings is suggested as an
application example. The testbed implementation demonstrates the feasibility
of the method and the success of computations over the channel. Another
implementation study is given by Sigg et al. in [166]. They suggested a
computation model that carries information in the mean value of the Poisson
distribution. The nodes of the system transmit burst sequences that are
Poisson distributed and the density of the bursts in a time interval
represents the mean of the distribution. The model then extracts the mean
value of the superimposed signal at the receiver. The proposed model is
implemented with 15 sensor nodes and a receiver that is driven by
microcontrollers. The realized scenario successfully recovered the average
temperature of the simultaneously transmitted sensor readings.

AirShare method is presented in [167] to improve the resistance of the
distributed wireless applications against carrier frequency offset (CFO). The
method uses a broadcast clock signal as a reference to other nodes that aim to
transmit simultaneously. The paper firstly investigates the feasibility of the
AirShare method and then implements the network via SDR modules.

##  VII Federated Learning

An emerging application area of the AFC is the federated learning algorithms
which enable the computation of learning data over the air. In this section,
the federated learning studies that are based on the simultaneous transmission
of signals are examined.

Machine learning (ML) is the rising technology of the last decade as a result
of the increasing computational capabilities of the electronic devices
(machines) (comprehensive information on the main contents of this section can
be found in [168]). ML is based on the usage of massive data or data sets to
let those machines make the classification or prediction operations. Moreover,
many aspects of a communication system such as modulation, demodulation,
channel estimation etc. can be considered as a classification or prediction
problem [169]. This observation brought out the relationship between the ML
and wireless communication. Although traditional communication systems use
model-based solutions for their problems, ML based data-driven techniques have
already started to show promising results in the wireless communication area
[170]. In the following years, this relationship proved to be two ways such
that the communication networks can also be beneficial in the learning area.

The data-driven nature of the ML requires the collection of massive data in
centralized points before the learning process. However, data sources of
today’s technology are often at the wireless edges. As a result, collecting
massive data from wireless devices to a center can cost a high amount of
energy and bandwidth [171]. Collaborative machine learning or federated
learning is a solution to the data collection problem which proposes the
process of the data at the edge users or distributed centers instead of a
local center [172, 173]. The superposition property of the wireless channel
can further relieve some of the costs in the federated learning schemes by
performing computations over the wireless channel. A list of federated
learning studies can be found in Table VI.

TABLE VI: An overview of federated learning studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Tran et al. [172] |  2019 |  The method provides a duality between learning time and energy efficiency. |  Time vs. energy cost
Yang et al. [173] |  Overviews the existing federated learning studies and promotes its data aggregation aspect. |  Classification
Amiri and Gündüz [176] |  Improve the error performance of [171] by compressing the gradient estimate. |  Accuracy
Amiri et al. [177] |  Removes the requirement on CSI for the distributed ML method.
Zhu et al. [178] |  The method is for broadband communications and reduces latency as well as promoting a trade-off between communication and learning performance.
Yang et al. [179] |  Reduces the convergence rate of the learning algorithm by considering device selection and beamforming.
Amiri and Gündüz [171, 174, 175] |  2019, 2020 |  Analog and digital DSGD reduce the learning time in bandwidth and energy limited networks.

One fundamental example of this paradigm is given by Amiri and Gündüz [171,
174, 175]. The learning algorithm is based on the minimization of a loss
function which is solved with the distributed stochastic gradient descent
(DSGD) method. In this method, the learning parameters are updated with
multiple iterations. Different from the traditional collaborative ML that uses
DSGD, this study exploits the wireless channel for the parameter update. In
other words, the function that updates the learning parameters are calculated
over the air as a result of the superposition property. Two models, digital
and analog, are proposed to reduce the required number of iterations to reach
an accuracy level. Simulations show that the analog model requires fewer
iterations, hence saves both energy and bandwidth of the system.

In [176], Amiri and Gündüz extend their previous model to fading channels. The
authors propose the compressed worker-wise scheduled analog DSGD model in
which the edge users accumulate the error from previous iterations and reduce
the dimension of their transmit vector. The proposed model is also compared
with the model of another study that does not reduce the transmit vector
(without scheduling) and the results show that the compressed worker-wise
scheduled analog DSGD model increases accuracy. In [177], the authors remove
the CSI knowledge assumption on the transmitters. Instead, the receiver is
equipped with multiple antennas. The results show that the increasing number
of antennas alleviate the destructiveness of the wireless channel such that
the infinite antennas result in totally removed fading and noise.

Zhu et al. take the same approach on federated learning in [178]; error
function is minimized with each iteration and the iterations are calculated
over the air. In addition to other studies, tree performance metrics are
defined to analyze the network performance and the relations between them are
obtained in closed forms. Also, the network is implemented to compare the
proposed model with OFDM. The results confirm the relationships between the
defined metrics and show the low latency contribution of the proposed method.

Yang et al. consider the same model which is based on the over-the-air
computation in [179]. The study includes user selection as used in [176] which
schedules the users according to their channel state. As an addition, [179]
considers the usage of beamforming to improve learning performance. Also, they
define and solve the optimization problems that govern the performance of the
device selection model. Computation of the input data of a decentralized
network is also considered for spectrum sensing algorithms. Different from the
federated learning methods, the channel is matched to compute the sensing
algorithm functions rather than the learning algorithm functions. In the
following section, the studies that benefit from the superposition of signals
for the purpose of spectrum sensing are investigated.

##  VIII Spectrum Sensing

Supporting a high amount of users is one of the most important challenges of
wireless communication since the frequency spectrum is limited. However, the
distribution of the spectrum to the users is usually more challenging than the
physical scarcity of the channel itself [180]. The cognitive radio is a
delicate approach that aims to handle the spectrum access problem efficiently.
Cognitive radio is designed to be aware of its surroundings with SDR
capabilities, e.g. spectrum analysis or CSI estimation. The main objective of
cognitive radio is to find the idle channels, that is dedicated to primary
users, and utilize them to secondary users. The spectrum sensing problem is
the first aspect of this objective and attracted distributed network solutions
in the literature [181]. Some of the simultaneous transmission based spectrum
sensing studies are listed in Table VII with their contribution and
performance metrics.

TABLE VII: An overview of spectrum sensing studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Zheng et al. [182] |  2015 |  The method manages to transfer the sensing data in one time slot, analyze its detection and throughput performance. |  Approximation error, throughput
Zheng et al. [183] |  2017 |  Define an energy optimization problem and find the expression of optimal threshold. |  Energy efficiency, sensing time
Chen et al. [184] |  2018 |  Considers the effect of CFO in the sensing algorithm. |  Signal to aliasing and noise ratio

The spectrum sensing is a particular case of the detection problem (Section
IX) and involves certain network dynamics. In a distributed cognitive radio
network, multiple secondary users sense the spectrum and report back to an FC.
A cooperative method that utilizes the simultaneous transmission for this
problem is proposed in [182]. Instead of individual sensing, the method takes
help from the decentralized nodes to detect the free frequency bands. Spectrum
sensing mechanism is based on observed energy from the frequency channels.
This energy information is carried simultaneously from the distributed nodes
to the FC as encoded to the signal energies and the final decision is made at
the FC. The detection probability and the false alarm probability of the
proposed method are derived and used in order to optimize the detection
performance. Also, the proposed scheme is compared with the traditional
spectrum sensing methods with respect to the approximation error and network
throughput via simulations. The authors later consider the energy efficiency
of the spectrum sensing problem in [183]. For this reason, an optimization
problem is derived that considers the sensing time, the detection threshold
and the symbol sequence length. The problem is simplified, solved and the
results are verified with simulations.

In [184], the proposed cognitive radio network focuses on the wideband
spectrum. The objective of the method is to find the occupied wideband
channels with low latency and high accuracy. The method utilizes a distributed
network model to improve the sensing accuracy and benefits from the
superposition of the signals to reduce the delay time. Specifically, the
discrete Fourier transform (DFT) of the data is computed over the nodes and
the air. The network is examined for the synchronization errors (e.g.
synchronization phase offset) that occur between the nodes. Furthermore, a
robust estimation and equalization technique is proposed to mitigate the
effects of the imperfect synchronization. The performance of the proposed
method is analyzed theoretically and verified with both simulation and SDR
based implementation. The computation of functions over the air is also
extended for detection and estimation algorithms to reduce the latency and
improve the error performance. In these studies, the channel is adopted to
collect the data from distributed sources and process them simultaneously over
the air for the detector or estimator.

##  IX Detection and Estimation

Detection and estimation are essential parts of many engineering applications
and their performance profoundly affects the performance of the system. While
the detection and estimation theory is concerned with the optimal design of
the decision mechanisms, the acquisition of the samples is another important
parameter for the whole system. In general, larger sample sizes produce better
decision performance, however, it requires more energy consumption and long
computation times. For this reason, the response of the decision performance
to the sample size is an important aspect to consider. The error exponent is a
measure that indicates how fast the error changes with the increasing sample
size and is often used in the analysis of asymptotic behavior of the sample
size. This aspect is particularly important in the IoT applications that can
involve massive sensor networks in which the observations from multiple
sensors together create the sample set. For this reason, distributed detection
or estimation strategies are developed to improve the performance of this
process. In this section, we present the distributed detection and estimation
studies that exploit simultaneous transmission.

###  IX-A Detection

Several detection studies based on simultaneous transmission are presented in
Table VIII. The distributed detection problem of wireless networks is
considered in [185]. The network consists of spatially distributed sensors
that aim to transfer the statistics of the sensor measurements to the FC. The
method exploits the superposition property of the W-MAC for this purpose by
simultaneously transmitting their statistics. The study examines the detection
performance of the proposed method from the perspective of the number of
measurements and power consumption. Two sensor types are considered for the
review; intelligent and dumb. The intelligent sensors are aware of the source
statistics and transfer the log-likelihood ratio (LLR) to the FC. It is shown
that the LLR based detection is asymptotically optimal, i.e. it can reach the
performance of the centralized detection. The dumb sensors are unaware of the
source statistics and transfer the histogram of their measurements to the FC,
however, source statistics are required at the FC. The results show that the
histogram transfer is also asymptotically optimal. The results are also
verified with simulations that show the detection error as a function of the
number of sensors.

TABLE VIII: An overview of detection studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Liu and Sayeed [185] |  2007 |  A low-complexity distributed detection mechanism (based on type function) is proposed and investigated for the error exponent. |  Error probability
Mergen et al. [16] |  2007 |  Provides an asymptotically optimal detector and analyzes its error exponent performance. |  Error exponent
Li and Dai [187] |  2007 |  Proposes a bandwidth and delay efficient method and compares with the separation based methods.
Li et al. [188] |  2011 |  Analyzes and compares the MDF and the MAF methods for error exponents under power constraints.
Banavar et al. [189] |  2012 |  Includes multiple antenna receivers and investigates different CSI scenarios.
Ralinovski et al. [186] |  2016 |  The anomaly detection method reduces the communication costs (energy, bandwidth) and investigate relation between accuracy and communication costs. |  Reliability, energy consumption
Raceala-Motoc et al. [190] |  2018 |  Gives the upper-bounds for the probability of mislabeling for [186]. |  Probability of mislabeling

Sensor measurements of a wireless sensor network are used for target detection
in [16]. The sensor measurements are transferred to the test center via type-
based multiple access where the same frequency and time resources are
allocated to all users. In TBMA, the receiver only accesses to the histogram
of the transmitted data (we refer the reader to Section II for detailed
information) and the detection mechanism is based on the histogram of the
observations. The study focuses on the performance analyses of TBMA based
detection schemes (includes a large deviations approach). An asymptotically
optimal detection mechanism is proposed and its detection error exponents are
derived. Channels with i.i.d. and non-i.i.d. gains are considered for the
evaluations. Also, the error probabilities of TBMA scheme is compared with
TDMA via simulations.

In [187], the existence of a signal is checked with a binary hypothesis test
where the observation samples are obtained from a decentralized sensor
network. The sensor readings are collected in the FC via two-channel models;
each sensor has its own dedicated channel or a single channel is dedicated to
all sensors. The second model that dedicates a single channel to the sensors
attracts our interest since it allows the superposition of the signals. The
study focuses on the detection performance of these two-channel models and
their comparison with the centralized detection model. In the analyses, two
Gaussian noise cases (correlated and independent) and two power constraint
cases (average and total) are considered. Bayesian error exponents of the
detection probabilities are derived for these scenarios (includes large
deviations approach). Several results are presented in the study and verified
via simulations that examine the effect of the number of sensors on the
detection error probability and error exponent. The result for the average
power constraint case is that the superimposed signals with the correlated
Gaussian noise can reach the error performance of the centralized detection
scenario. However, dedicating channels to each sensor always leads to error
performance reductions. For the total power constraint case, an increasing
number of sensors exponentially reduces the error exponent for the
superimposed signals.

Two superposition based schemes are proposed in [188] for the distributed
signal detection via binary hypothesis testing. The modified detect-and-
forward (MDF) scheme considers the signal detection at each distributed node
and transfers the test results to the FC. The second scheme, modified amplify-
and-forward MAF, transfers the observations to the FC before the hypothesis
testing. The study examines the detection performance of these schemes under
individual power constraint and total power constraint. The results show that
the MAF is asymptotically (i.e. infinite number of sensors) optimal under
individual power constraint, however MDF is not optimal.

In [189], multiple antennas at the FC is considered for the distributed
detection problem. The upper and lower bounds of the error exponents are
derived for the scenarios: AWGN channels, Rayleigh channels, full CSI enabled,
phase-only CSI enabled. It is shown that equipping the FC with multiple
antennas presents a gain of $\pi/8$ for the scenario of Rayleigh channel and
full CSI. Four algorithms are proposed for the design of the sensors’ power
allocation. Lastly, the error exponents of these algorithms are compared with
the derived bounds via simulations.

An anomaly detection algorithm is proposed in [186] which is based on the AFC
given in [141]. A supervised learning algorithm is used for the hypothesis
testing and the classifier of the algorithm is generated at the FC with the
sensor readings. Since the FC is only interested in a function of the sensor
readings, AFC is proposed for the classifier generation over the channel. The
main contribution of the study is its energy efficiency and it is shown that
the proposed scheme can significantly reduce the consumed energy compared to
the TDMA scheme. Later, in [190], the study on anomaly detection is extended
to include the upper bounds on the probability of mislabeling.

###  IX-B Estimation

A list of the estimation studies that benefit from the superposition of
signals can be found in Table IX. An estimation method that is based on the
TBMA scheme is proposed in [3] . The method benefits from the superposition of
the sensor readings over the channel to gain from the bandwidth and the delay
time. The study aims to design the optimum estimation process that includes
the transfer of the samples to the FC and the estimator. The study derives the
Cramer-Rao bound of the estimation problem and designs the estimation process
as the combination of TBMA and an ML detector. The results show that the TBMA
based estimation is asymptotically optimal if the channel gains of all sensors
are the same. Additionally, the proposed model is analyzed for the fading
channels and compared with the TDMA based schemes via simulations.

TABLE IX: An overview of estimation studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Mergen and Tong [3] |  2006 |  Analyzes the asymptotic behaviour of TBMA based estimation. |  MSE
Xiao et al. [191] |  2008 |  Propose a bandwidth and power efficient method and defines and solves an optimization problem for power scheduling. |  MSE
Bajwa et al. [192] |  2007 |  Proposes a energy efficient joint source-channel based method and defines the relationship between its power consumption, error rate and latency. |  Power, distortion, scaling exponents
Wang and Yang [193] |  2010 |  The method removes the CSI requirement by using more bandwidth. |  MSE, bandwidth
Banavar et al. [194] |  2010 |  Considers and investigates different fading models and their effect on the performance. |  Asymptotic variance

In [191], power and bandwidth limited networks are considered for the
distributed estimation of unknown signals. The authors especially propose a
power scheduling model and investigate the cases where the observations are
scalar and vectorial. The optimization problem for power scheduling is shown
to be convex and solved for the scalar observations. Additionally, it is shown
via simulations that the proposed scheduling generates better MSE performance
than the uniform scheduling. The optimal scenario to estimate the vectorial
observations is investigated for both noiseless and noisy channel models. The
results for the noiseless channel model are given in closed form; however, the
solution of the optimization problem for the noisy channel required
semidefinite relaxation methods.

A joint source-channel communication structure is considered for decentralized
wireless sensor networks in [192]. The objective of the study is to build a
communication infrastructure that reduces bandwidth and power consumption. For
this purpose, optimization of the acquisition, communication and processing of
the measured data is considered collectively. The relationships between the
power consumption, error rate and the latency of the network are derived for
the increasing number of sensors and verified with simulations. It is shown
that the efficient and healthy estimation of the unknown signals is possible
for a large number of sensors if prior knowledge is allowed in the proposed
approach. Moreover, the method still produces healthy estimation results when
partial or no prior knowledge is given if the power and latency constraints
are increased sublinearly as a function of the number of sensors.

A TBMA based estimation method is proposed in [193] for distributed WSN s. In
traditional TBMA schemes; the fading of the wireless channel notably degrades
the communication performance and providing CSI to each node via pilot
transmission or feedback channel brings excessive burden for a large number of
sensors. The proposed method aims to provide robustness against these problems
by partitioning additional bandwidth to the network. The main idea behind the
model is to use multiple orthogonal waveforms for each message (type) where
the traditional TBMA uses only one. The authors also designed a maximum
likelihood estimator for the system and derived the Cramer-Rao lower bound
expression. The simulations compare the MSE performance of the method with
traditional TBMA and illustrate the relationship between the bandwidth, SNR
and the number of nodes.

Fading characteristics of the channel and the amount of allowed CSI at the
network is highly important for the performance of the distributed sensor
networks that exploit the superposition of the signals. In [194], various
fading models and different CSI amounts are tested for distributed estimation
networks. The variance expression for the network’s estimate is derived for
the perfect and partial CSI cases. Also, different channel characteristics are
considered and the corresponding variance expressions are derived for a large
number of sensors. The authors consider the impact of the errors that occur at
the feedback process in their evaluations. The convergence rates are
calculated for several scenarios. It is shown that the asymptotic results can
be nearly reached with a practical number of sensor nodes. The results are
verified with computer simulations. Simultaneous transmission is an effective
tool for the consensus algorithms that reduce the convergence time of the
consensus process. In the following section, the studies that exploit the
superposition property of the W-MAC are presented.

##  X Gossip and Consensus

The physical layer is generally exploited for many-to-one networks where an FC
requires the message of multiple users ($K\times 1$). However, in consensus
problems, all users in the network aim to agree upon a common value, e.g.
average, which carries a much higher communication burden. Finding an
efficient communication structure for the consensus problem that provides fast
and reliable convergence, while requiring low-energy, is a challenging task
[195]. Gossiping is a distributed form of consensus that the nodes locally
communicate with their neighbors instead of following a network-wide protocol.
It has been shown that simultaneous transmission presents opportunities for
the improvement of gossip and consensus algorithms. A list of existing
consensus studies is given in Table X. In this section, we examine the
consensus studies that exploit the wireless channel to provide energy and time
efficiency.

TABLE X: An overview of consensus studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Kirti et al. [196] |  2007 |  Provides a scalable consensus algorithm that the convergence time is independent of the network size. |  MSE
Goldenbaum et al. [199] |  2012 |  The consensus of multiple nodes with respect to nomographic functions is established and analyzed.
Steffens and Pesavento [201] |  2012 |  Provides a low-complexity and scalable consensus algorithm.
Nazer et al. [197] |  2009 |  Faster than the pairwise gossip and provides energy efficiency exponential to the network size. |  Number of gossip rounds
Nazer et al. [198] |  2011 |  Provides time and energy efficiency polynomial to the network size.
Nokleby et al. [200] |  2011 |  Based on full duplex communication and provides scalability to the network size. |  Averaging time
Molinari et al. [202] |  2018 |  Robust to fading and provides fast convergence. |  Convergence time
Agrawal et al. [203] |  2019 |  Provides robustness to noise and low SNR for max-consensus networks. |  Error rate, number of iterations

The average consensus problem of wireless networks is considered in [196]. The
method exploits the physical layer with the simultaneous transmission in order
to update the consensus algorithm. The results show that each node in the
proposed network can obtain the average of the data under a sufficient MSE
level. Since the channel is accessed by the nodes simultaneously, the method
presents better convergence performance compared to the conventional consensus
algorithms as the number of nodes increase.

In [197], Nazer et al. use the computation coding in a gossip algorithm that
averages the sensor readings. The method is based on the simultaneous
transmission of the nodes by exploiting the superposition property of the
physical layer. The network is arbitrarily divided into local neighborhoods
such that the average consensus is initially provided for each of
neighborhood, later the global consensus is achieved for the network. The
results reveal that the energy efficiency of the network increases
exponentially and the time efficiency increases polynomial as the number of
nodes increase. In [198], the previous studies are extended and compared with
the nearest neighborhood gossip algorithms via simulations. Also, it is shown
that the proposed method converges in $O(n^{2}/m^{2})$ rounds while the
nearest neighborhood algorithm converges in $O(n^{2})$ rounds where $n$ and
$m$ are the network and local neighborhood sizes respectively.

The averaging gossip problem of a wireless network is extended to nomographic
functions in [199]. The main idea behind the study is the analog computation
of the nomographic consensus functions over the wireless channel with the
concurrent transmission. The proposed model is cluster/neighborhood-based as
in [197] and [198] such that the consensus is initially provided for local
clusters rather than the whole network. Different from [197] and [198], the
global consensus is obtained with the existence of the common nodes which
connect the clusters with each other. Two algorithms, deterministic and
randomized, are proposed and it is shown that both of them significantly
increase the convergence rate.

Another gossip algorithm that exploits the physical layer is proposed in
[200]. The novelty of the study lies in the full-duplex communication model of
the network which allows simultaneous transmission and reception of signals
with special self-interference cancellation techniques. Each node in the
network broadcasts to its neighbors and receives from them simultaneously and
only requires synchronization. The theoretical results show that the proposed
gossip algorithm has a three times faster convergence rate than the randomized
gossip algorithms. Also, the study includes computer simulations supporting
the presented theoretical results.

In [201], the average consensus of a wireless network is considered. Although
many of the simultaneous transmission based consensus algorithms are
neighborhood/cluster-based (as in [197, 198, 199]), this study arbitrarily
divides the network into two subgroups rather than many. In each iteration,
all the nodes of a subgroup simultaneously broadcast their data while the
nodes in the other subgroup receive the superimposed signal. After an
iteration, the subgroups change roles such that the receiver subgroup becomes
the transmitting subgroup and vice versa. The nodes in the transmitting
subgroup create their new data as a weighted sum of the received signal and
the original data. The proposed method offers energy efficiency and
significant convergence performance as the simulations illustrate the
comparison of the proposed method with the randomized and the broadcast gossip
algorithms.

The superposition of the transmitted signals is also considered in [202]. The
study uses similar grounds that are used in [199]. [202] also includes unknown
channel coefficients in the design of the consensus method. Time-variant and
time-invariant cases are considered for the wireless channel model. A tuning
parameter (a stubbornness index) is proposed to control the convergence rate.
The results show that the smaller values of the tuning parameter (stubborn
systems) reduce the effect of the time-variant channel coefficients and
increase the convergence time. However, the tuning parameter only affects the
convergence rate in the time-invariant systems. The study also verifies the
theoretical results via simulations. A max consensus scheme (ScalableMax) is
proposed in [203]. The method focuses on the dense networks where a large
number of nodes aims to find the maximum function. The main contribution of
the proposed method is its scalability. Moreover, an error correction
mechanism is added to the system in order to increase the system’s resilience
to the low SNR region. The CP&F and AFC also inspired security application in
the literature. In the following section, we present the security applications
that benefit from simultaneous transmission.

##  XI Security

Superposition of signals is shown to be beneficial to provide security to
wireless networks. Several security studies that use simultaneous transmission
are listed in Table XI. In this section, we present the security applications
in detail.

TABLE XI: An overview of security studies. Author |  Year |  Contribution |  Performance metric
---|---|---|---
Shashank and Kashyap [204] |  2013 |  Achieves strong secrecy with lattice codes and randomized encoding. |  Achievable power-rate pair
Vatedka et al. [8] |  2015 |  Provides perfect or strong secrecy to two-way relay networks using CP&F.
Babaheidarian and Salimi [205] |  2015 |  Combines lattice alignment and asymmetric CP&F to improve the secure sum rates. |  Achievable sum rate
Richter et al. [206] |  2015 |  Includes fading and provides weak secrecy for multi-way relay networks using CP&F. |  Secrecy capacity
Karpuk and Chorti [207] |  2016 |  Derive the secrecy rate upper bounds, investigates the effect of synchronization error and extends the proposed model to MIMO sceheme.