The Magic of Superposition: A Survey on the Simultaneous Transmission Based Wireless Systems
11 The Magic of Superposition: A Survey on theSimultaneous Transmission Based Wireless Systems
Ufuk Altun,
Graduate Student Member, IEEE , Gunes Kurt,
Senior Member, IEEE andEnver Ozdemir,
Member, IEEE
Abstract —In conventional communication systems, any in-terference between two communicating points is regarded asunwanted noise since it distorts the received signals. On theother hand, allowing simultaneous transmission and intentionallyaccepting the interference of signals and even benefiting from ithave been considered for a range of wireless applications. Asprominent examples, non-orthogonal multiple access (NOMA),joint source-channel coding, and the computation codes are de-signed to exploit this scenario. They also inspired many other fun-damental works from network coding to consensus algorithms.Especially, federated learning is an emerging technology that canbe applied to distributed machine learning networks by allow-ing simultaneous transmission. Although various simultaneoustransmission 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 emphasizethe connections between these studies and provide a guidefor the readers on the wireless communication techniques thatbenefit from the superposition of signals. We classify the existingliterature depending on their purpose and application area andpresent their contributions. The survey shows that simultaneoustransmission can bring scalability, security, low-latency, low-complexity and energy efficiency for certain distributed wirelessscenarios which are inevitable with the emerging Internet ofthings (IoT) applications.
Index Terms —Simultaneous transmission, superposition, wire-less networks. A BBREVIATIONS
A&F
Amplify-and-Forward
AFC
Analog Function Computation
AWGN
Additive White Gaussian Noise
BER
Bit Error Rate
BICM-ID
Bit-Interleaved Coded Modulation with IterativeDecoding
C-RAN
Cloud-Radio Access Network
C&F
Compress-and-Forward
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
CoMAC
Computation over Multiple Access Channels
CP&F
Compute-and-Forward
CPC&F
Compute-Compress-and-Forward
CSI
Channel State Information
D&F
Decode-and-Forward
DFT
Discrete Fourier Transform
DSGD
Distributed Stochastic Gradient Descent FC Fusion Fenter i.i.d. independent and identically distributed IA Interference Alignment
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 ForcingI. I
NTRODUCTION T HE theory of communication, as presented in [1] byClaude Shannon, considers the mathematical relationbetween a source and a destination. The theory identifies anysignal between the source and the destination as noise. Thisassumption is still valid in advanced communication systems,especially in wireless communication networks that introducedestructive, as well as a limited medium to its users. Althoughits mobility feature attracts more and more users with eachwireless network generation, limited resources present extremedesign challenges since the wireless medium is shared by allof the participants. a r X i v : . [ c s . I T ] F e b Some of the most critical design challenges are providingsecurity, low latency and high communication rates and sup-porting a large number of users. From any of those aspects,the corresponding solution is sought, designed and tested forShannon’s pairwise communication model. Hence, the firststep of the network design generally starts with virtuallydividing the limited channel resources to each user andconsidering the crowded wireless channel as a combinationof multiple subchannels between the pairs of nodes. Forexample, assigning orthogonal frequencies to different usersis a practical and efficient approach to solve these challenges.It is needless to say that all aspects we listed above areimportant for a wireless network, however their importanceand the priority is closely related to the purpose of the network.With the emergence of Internet of things (IoT) networks, wecan add other design considerations to our list such as energyconsumption or the system complexity. More importantly, anIoT network enables an immense range of wireless applica-tions such that these aspects (and probably many more) arerequired in different and particular orders according to theapplication. For instance, security can be of high importanceand the complexity is less for finance applications, while themass sensor networks can require a low energy consumptionand a high bandwidth efficiency.We believe that this diversity of wireless applications at-tracts the researchers’ attention to the outside of the classicalperspective. The physical layer based studies have alreadygained attention on this matter in order to help the wirelesscommunication problems. One peculiar idea is related to thephysical layer and the interference of signals. In the last twodecades, the researchers have asked the question if the inter-ference of signals can be beneficial. The question is closelyinvestigated in several studies and positive answers werepresented for a limited number of scenarios. These studies aremostly classified in the literature according to their applicationarea, however their interactions with other studies regardingtheir unique channel model is usually overlooked. In this study,we consider the channel model aspect and survey the existingliterature for the studies that exploit the superposition propertyof the wireless channel. The contributions of this survey canbe listed as follows. • The studies in the literature that exploit the superpositionproperty of the wireless channel are reviewed:
A varietyof wireless communication techniques exist in the liter-ature (Fig. 1) with this apparent and unique connection(superposition property). Yet the literature lacks a com-prehensive guide that highlights this connection to thereaders. This survey presents a guide for both beginnerand experienced readers that are interested in the benefitsof simultaneous transmission. • The studies are presented along with their system model,contributions and performance metrics:
The simultaneoustransmission methods are grouped depending on theirapplication areas. The methods are introduced with briefexplanations and studies are investigated for their contri-butions and performance metrics.
SimultaneousTransmission BasedWireless Systems II. Multiple Access A. Code DivisionMultiple Access (CDMA)B. Non-orthogonalMultiple Access (NOMA)C. Type-Based MultipleAccess (TBMA)D. Compute-and-Forward Multiple Access(CFMA)III. Multiple Antenna A. Integer ForcingIV. Network Coding A. Physical LayerNetwork Coding (PLNC)B. Compute-and-Forward (CP&F)V. Interference,Computetion, FunctionAlignmentVI. FunctionComputation A. Digital FunctionComputationB. Analog FunctionComputation (AFC)VII. Federated LearningVIII. Spectrum SensingIX. Detection andEstimation A. DetectionB. EstimationX. Gossip andConsensusXI. Security
Fig. 1. Structure of the survey as the main application areas of thesimultaneous transmission based wireless communication techniques.
A. Boundaries of Our Scope
Since we consider redinterfering signals, we are naturallyinterested in the multiple users and the techniques for access-ing the wireless channel. It should be pointed out that theanswers to all our questions are not included within Shannon’scommunication model regarding the interference policy. Hereour boundaries may be perplexing since there are certaintechniques in the literature that accept the interference of othersignals, however still interested in the pairwise communicationwithout benefiting from interference.The general resources of a wireless network is use ofpower in the time and the frequency plane that is suitable for distribution among the users. However, communicationtechnologies make use of other resources such as code orenergy to divide among the users. The users in these studiesoccupy the same time and frequency blocks, yet they aredistributed with another resource. The code division multi-ple access (CDMA) is an example that enables controlledinterference 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 thatdoes not prohibit interference. However, MIMO technologyshould be considered under pairwise communication sinceits interference comes from the signals that are known andmanaged by a single user.These examples raise the question of whether our scope islimited to any technique that enables the interference of signals(i.e. uses the same time and frequency slots) or limited to amore narrow scenario. We could not answer this question witha single border since one of them would draw an incompletemap on the main idea and the other one would be too extensiveto 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 directlygot benefit from it. However, we briefly illustrate their relationto the interference and we refer the readers to more detailedresources on that matter.Our main focus is on the techniques that intentionallyexploit the interference of signals and get a direct benefit fromit. Multiple users in the techniques that we present here use thesame time and frequency blocks for communication and do notaim to distinguish between the individual information. Instead,these methods are only interested in the superimposed formof the signals. There are various techniques in the literatureand before giving a general map of these techniques, first wewould like to clarify some of the concepts that we use to avoidany ambiguity.We define the channel model as the wireless multipleaccess channel (W-MAC). The multiple access term statesthe fact that the simultaneously transmitted electromagneticwaves combine with each other. In general, this statementonly covers the time dimension, i.e. the waves from otherfrequencies also merge over the W-MAC. However, it isstraightforward to separate the superimposed waves in thefrequency dimension at the receiver. For this reason, weextend our definition of simultaneous transmission and includethe frequency dimension. Hereafter, we refer to simultaneous(concurrent) transmission to cover both time and frequencydomains, i.e. signals are transmitted at the same time andfrequency block. Similarly, we refer to the simultaneouslytransmitted signals as the superpositioned or superimposed signals. Furthermore, we use superposition or simultaneoustransmission to indicate the methods that we are interested in. B. A Map of the Simultaneous Transmission Based Commu-nication Techniques
There are several application areas of simultaneous trans-mission -based techniques for different purposes. Each applica-tion has its own design parameters and performance metrics,
Fig. 2. An illustration of the featured methods and their applications areas. for instance distributed detection problems usually requirea high detection probability while security applications areinterested in secrecy rates. As a result, this survey assumesa unique point of view that connects various problems andapplication areas of the wireless communication networks.However, the superposition property of the W-MAC impliesseveral common grounds as follows: • Its distributed character: All applications include decen-tralized multiple nodes. As a result, scalability, energyefficiency and complexity are often a design concern. • Its wireless character: The users communicate over thewireless channel. The amount of channel state infor-mation (CSI) knowledge at the users and the channelcharacteristics (e.g. fading) are important concerns.The structure of this survey (the application areas of thesimultaneous transmission based techniques) is illustrated inFig. 1. The multiple access and the multiple antenna ap-plications are the early examples that the superposition ofthe simultaneously transmitted signals is observed at thereceiver. CDMA and non-orthogonal multiple access (NOMA)techniques are essentially based on pairwise communications.For this reason, transmitters use a third resource (other thantime and frequency) to distinguish between the individualmessages, e.g. the CDMA or NOMA users uniquely encodetheir messages as a function of a code block or a powerlevel respectively. On the other hand, the traditional MIMOapplications benefit from the superimposed signals to gaindiversity although the network is not distributed.Joint source-channel coding proposed in [2] has createda paradigm shift on the multi-user communication modelsby considering the joint optimization of communication andcomputation aspects. Over the years, the paradigm has beenshaped into several core models such as type-based multipleaccess (TBMA) [3], computation over multiple access channel(CoMAC) [4], compute-and-forward (CP&F) [5] and analog
Fig. 3. An illustration of the relationship between the applications and their network model. function computation (AFC) [6]. These studies influencednovel techniques in the past decade and appeared at variousapplication areas from network coding to federated learning.An illustration of these techniques and their general applica-tion areas are given in Fig. 2.CP&F is mainly investigated for relaying and networkcoding purposes to improve the network capacity (i.e. com-putation rate), however it has also been extended to multipleaccess applications (CFMA [7]) to reduce complexity andsecurity applications (as in [8]) to improve secrecy rate. The computation alignment and the function alignment studieswhich are influenced by CP&F and CoMAC are proposed toreach the computing capacity in multiple antenna networks.CoMAC inspired various digital and analog functioncomputation methods that aims to allow low-latency, low-bandwidth computations in distributed wireless networks. Themain idea behind the CoMAC is later implemented in spec-trum sensing, federated learning and consensus algorithmsto improve network efficiency. Especially, federated learningalgorithms are one of the latest and most promising application area of the simultaneous transmission. With the help of Co-MAC, federated learning algorithms can decentralize machinelearning systems and improve energy or time efficiency of thesystem. TBMA is primarily used for distributed detection andestimation applications to design scalable and energy-efficientmodels.From an intriguing perspective, all these models and appli-cations can be viewed as a particular function that manipulatesthe wireless channel to perform a given task. For example,multiple access algorithms aim to transfer the transmitteddata to a destination. Eventually, simultaneous transmissionmanipulates the channel to perform a function which bothinputs and the outputs are the transferred data. However in thedetection algorithms, outputs are not data, instead they are thetest statistics. Hence, it can be said that a detection algorithmthat uses simultaneous transmission manipulates the wirelesschannel to perform a function that inputs are the sample setand the output is the test statistics. The corresponding input-output relationships are illustrated in Fig. 3.The security applications consider transferring data to thereceiver without leaking information to the eavesdroppers.This approach manipulates the W-MAC to transfer the initialdata to the receiver such that the received signal is onlymeaningful at the destination. Network coding applicationsconsider transferring data in layered network structures. Thesimultaneous transmission models propose a cascade of func-tions (composite functions) (relays) that both the initial inputsand 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 purposeof the application. Moreover, gossip and consensus algorithmsalso consider the topology of the network by using subgroupsor subfunctions in the network, i.e. composite functions.In this survey, we consider wireless communication tech-niques that benefit from simultaneous transmission and weclassify the existing literature according to their applicationpurposes. We draw a map of the existing literature and provideintroductory information on each application (Fig. 2). We alsogive details of these studies and investigate their contributionsas well as their performance metrics.The paper is organized as follows. In Section II, multipleaccess methods that benefit from the simultaneous transmis-sion are presented. Section III is dedicated to the multipleantenna algorithms. Network coding studies are investigated inSection IV. Interference alignment, computation alignment andfunction alignment studies are examined in Section V. Digitaland analog function computation applications are presented inSection VI, and the federated learning studies are presentedin Section VII. In Section VIII, spectrum sensing modelsare presented. Section IX is devoted to the detection andestimation studies, and Section X includes the gossip andconsensus studies. Lastly, the security applications are givenin Section XI. The paper is concluded in Section XII.II. M
ULTIPLE A CCESS
Wireless channel has limited resources (signaling dimen-sions) such as frequency, time, or space and accessing to (a) (b)Fig. 4. Distribution of resources in multiple access methods of our interest.(a) CDMA, (b) NOMA. the channel requires a portion from each resource. Multipleaccess methods aim to allocate these resources to multipleusers efficiently [9]. Our interest, simultaneous transmission,enables the dedication of all frequency and time domains to allusers. As a result, simultaneous transmission for the purposeof multiple access falls under the category of our interest.However, known multiple access methods such as CDMAand NOMA divide another dimension to its users rather thanexploiting the superposition of the signals. For this reason,we find it more adequate to give elementary information onthese methods and refer the readers to proper references. Inthis section, multiple access methods that enable users tosimultaneously transmit their messages are presented.
A. Code Division Multiple Access (CDMA)
In CDMA, each user is assigned with a spreading code todistinguish users from each other. Therefore, the channel canbe used by all users in the same time period and bandwidth.Especially in the uplink scenario, in which multiple userstransmit simultaneously, base station receives a combination ofsignals from all users. If the codes are orthogonal, despreadingthe received signal with the corresponding code outputs theinformation signal of the corresponding user. Fig. 4a illustratesthe resource distribution of CDMA technique. As seen fromthe figure, each user occupies a large and equal bandwidth andusually, a power balance is required. Non-orthogonal codesare also used in CDMA to flex the synchronization problemand support more users, however removing orthogonality addsinterference to the system.In [10], extensive information is given about the workingprinciples of the CDMA as well as an introduction to thevarious versions of the CDMA. An extended CDMA method,complementary code based MIMO CDMA, which aims torevive CDMA in the next-generation systems are investigatedin [11].
B. Non-Orthogonal Multiple Access (NOMA)
NOMA is an emerging multiple access method that is basedon the distribution of power or code domain . NOMA can beviewed as a complementary method for the existing multiple NOMA is usually referred for power or code domain distribution. How-ever, the name also suggests a category that includes any non-orthogonalmethod such as spatial division or pattern division. In this survey, we focuson power domain NOMA. access techniques since the power or code domain is notsuitable to support large number of users. As a result, thedistribution of another resource is required. However, NOMAbased systems still remain in our scope since NOMA enablesthe transmission of two users simultaneously. As an example,resource distribution of power domain NOMA is given inFig. 4b. The power is intentionally distributed to users witha level difference. The model depends on the successiveinterference cancellation (SIC) such that the receiver is ableto detect the higher level or lower level signal and then cancelit. NOMA is an emerging multiple access method and attractsattention for the next generation communication systems. Werefer the readers to [12] and [13] for comprehensive informa-tion on NOMA. In [14] an overview on the current challengesof NOMA (also the rejection of NOMA in 5G standards) andpossible solutions are analyzed. Additionally, performance ofNOMA is investigated with computer simulations.
C. Type-Based Multiple Access (TBMA)
TBMA is a unique method by means of channel-user rela-tionship since the users in TBMA does not aim to communi-cate with the destination individually. Instead, a data statistics(type) of all the users is transferred to the destination [15]. Itshould be noted that the individual data is not reconstructed atthe receiver and TBMA is not suitable for classical wirelesscommunications. However, it reduces the network latency andshows huge potential in sensor networks that only the totaldata statistics is desired. Specifically, TBMA is widely usedfor detection and estimation purposes.TBMA is not widespread in the literature and requires abroader explanation of its mechanism. For this reason, wededicate this section solely to the explanation of the TBMA.The literature review of TBMA is given in the detection andestimation section.Consider a network that consists of n users that aims totransfer messages (e.g. sensor readings) X i , X , ..., X n toa fusion center (FC). The i th user in the network assignsits message X i to a waveform s x i that is chosen from anorthonormal waveform set { s , s , ...s k } . When each user inthe network simultaneously transmit its data with energy E ,the FC obtains the following expression [16] z = n (cid:88) i =1 h i √ Es x i + w, (1)where h i and w are the channel and fading coefficientsrespectively. Assuming the channel gains are inverted, thesignal at the FC becomes z = k (cid:88) j =1 √ EN j s j + w, (2)where N j is the number of sensor readings that is mappedinto waveform s j . Note that each member of the messageset is mapped into a waveform and the transmitted signalsare superimposed over the channel. As a result, for eachwaveform, the FC receives the number of users that transmitted the corresponding waveform (i.e. the FC obtains histogram(type) of the sensor readings). D. Compute-and-Forward Multiple Access (CFMA)
CP&F is a relaying and network coding method that is basedon the simultaneous transmission of signals. The objective ofthe CP&F is to efficiently transfer the messages of multiplesources to a receiver with the help of multiple relays. Inessence, CP&F transfers a function of the source messagesto each relay by exploiting the superposition property. Therelays are unable to decode the individual messages sinceeach of them obtains a single function that contains multiplemessages (unknown parameters). The relays forward theirfunctions to the receiver and the receiver can reconstruct theindividual messages by solving the functions for the unknownparameters. Contrary to other network coding algorithms, thecommunication phase between the sources and the relays takesplace at the same time slot and the bandwidth in CP&F, henceit offers efficiency on the spectrum and latency.Compute-and-forward multiple access (CFMA) is inspiredby the CP&F and instead of relaying the messages, it enablesdirect access to the channel by exploiting the superpositionproperty. The CFMA is proposed in [17] by Zhu and Gastparfor the networks that two users aim to access to a singlereceiver. The main idea behind CFMA is based on the samecoding and decoding structure as the CP&F which yields afunction of the messages at the receiver. However, two usersdirectly communicate towards a receiver without multiplerelays. Two functions are required at the receiver to solvethe messages. For this reason, CFMA also uses successivecancellation decoding to obtain the coefficients of the secondfunction. The authors also examine the more than two usersscenario in [17] and low density parity check (LDPC) codedCFMA is considered in [18] and [7]. In the following section,multiple antenna techniques that benefit from the simultaneoustransmission are presented.III. M
ULTIPLE A NTENNA
Multiple antenna techniques became beneficial in the com-munication networks as a result of the developments in boththe antenna technology and the processing capacity. MIMOstructure is proven to improve the multiplexing capabilityof the single antenna networks as well as increasing theirdiversity gain [19]. MIMO presents a unique case for thescope of this survey since the communication is betweenpairwise nodes, e.g. the antennas are controlled by the samesource. The fundamental advantage of MIMO is the diversitygain which results from the superposition of the signals thatare transmitted from multiple antennas. As a result, MIMOnetworks draw our attention in the sense that the superpositionproperty is exploited to reduce the error rates or improve thebit rates. On the other hand, the conventional MIMO is anextensive topic and the superposition property is just a tool inMIMO studies which leads to numerous results [20]. Also, itis already well presented in the literature and we believe thatthe relation between MIMO and the superposition propertycan be better observed from the existing studies such as [21], [22]. Further reading on MIMO can be found in [23] for itssecurity applications and in [24] for its challenges and future.In addition to its conventional perspective, there also existsstudies that exploit the superposition property in a unique waywith multiple antennas. The integer-forcing receivers [25], [26]are an example of these studies and we would like to mentionits architecture and its difference from the traditional studies.
A. Integer-Forcing Architecture
An effective channel matrix can be defined and used forthe analysis of linear MIMO receivers. In the traditionalsense, the effective channel matrix should be matched withan identity matrix in order to recover the messages of eachantenna. However, integer-forcing receivers match the effec-tive channel matrix with integer value matrices as proposedin [25]. Inspired by CP&F [5], nested lattice codes are usedfor communication in order to obtain functions with integercoefficients. After matching with integer values, the receivercan solve the effective channel matrix and obtain messages ifthe matrix is full rank.The integer-forcing receiver is extended to mitigate theexternal signal interference in [27]. In addition to the resultsgiven in [25] that integer-forcing receivers obtain the inputswith integer coefficients, the authors later observe in [27]that the integer values can be also controlled to mitigate theexternal interference by considering the interference space.The results show that the proposed receiver presents significantgain over the traditional linear MIMO receivers.Successive interference cancellation (SIC) technique isadapted to the integer-forcing receivers in [28] as successiveinteger-forcing. The results indicate that the successive integer-forcing receiver can achieve the channel’s sum capacity andoutperform the traditional linear receivers with SIC in certainscenarios. IV. N
ETWORK C ODING
Source and channel coding are essentially concerned withthe communication between two nodes. Specifically, theyimprove the capacity and error performance of pairwise com-munication respectively. On the other hand, network codingbenefits from the architecture of the network in order toimprove its capacity, efficiency and security [29]. Networkcoding is interested in the information flow between nodes.The fundamental idea behind the network coding can be seenin Fig. 5a. The nodes n and n aims to exchange informationthrough an in-between node n in the given network. Withoutnetwork coding, the messages S and S of the nodes n and n would require a total number of four pairwise hops,hence four-time slots. A simple network coding algorithmcan be applied, as given in Fig. 5a, to reduce the requiredtime slots. After obtaining S and S sequentially, the relaynode n computes S = S ⊕ S and broadcasts it. Since thenodes know their initial messages, n and n can extract theunknown message from S . As a result, the network gains atime slot with the broadcast of the relay node.Network coding is one of the most effective and intriguingapplications of the simultaneous transmission since the multi-ple access nature of the channel presents unique opportunities (a) (b)Fig. 5. Information flow comparison between traditional network coding andPLNC [30]. (a) Traditional network coding, (b) PLNC. for the code design. These studies are specifically calledphysical layer network coding (PLNC) which outperformsthe traditional network coding in certain scenarios [31]. Weconsider the PLNC in two subsections since one study, theCP&F [5], made a name for itself and requires special atten-tion.The network architecture is an important parameter inthe investigation of these studies and some commonly usedarchitectures are illustrated in Fig. 6. Also, the studies areclassified according to their network architectures in Table Ialong with their performance metrics. In the coming part,we briefly explain the main ideas behind these methods andinvestigate the studies that exist in the literature. 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 thesignals to reduce the required time slots. A simple three-node example of the PLNC is given in Fig. 5b. Without theconventional network coding, it is obvious that the uncodedscheme requires a time slot for each transmission and theconventional network coding reduces the required number oftransmission by enabling broadcast at the relay.PLNC further improves this situation by accepting thesuperposition of the messages from n and n to the relaynode as seen in 5b. In PLNC scenario, relay node receivesthe superimposed message and can not extract S and S .However PLNC allows n and n to decode the superimposedmessage after n broadcasts it. Later, this basic example isextended to larger networks and different coding schemes asin [32] that a packet-based PLNC architecture is proposedand implemented. The study uses a testbed of computers andestablishes a proof of concept for the PLNC.Analog version of the PLNC [30] is proposed in [33]. Theauthors simply consider the superposition of the signals insteadof packets. Also, the relay uses amplify-and-forward (A&F) totransfer the superimposed signal. After presenting their results,the study also verifies them by implementing the proposedmethod with software defined radio (SDR) modules. (a) (b) (c) (d)Fig. 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. 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 througha single relay (equipped with multiple antennas) in stationaryand non-stationary channels. The STANC is proposed forthe stationary case and an alternative solution, the repetitiontransmission, is suggested for the non-stationary channels.Achievable sum rates of these two models are calculated andverified with simulations. The simulations also included acomparison with the zero-forcing (ZF) and maximization ofSNR beamforming models and it is shown that the STANCoutperforms other models regarding the sum rates. The net-work model in [35] is customized to the networks wheremultiple nodes transfer information to a single receiver bothdirectly and with the help of a relay node (equipped withmultiple antennae). The sum-rate performances and error rateperformances are evaluated with simulations for the cases ofdirect transmission, analog network coded transmission andthe STANC transmission.In [36], a network that consists of a single relay, twotransmitters and two receivers (compound multiple accesschannel with a relay (cMACr)) is considered. The relay nodeis assumed to have cognitive capabilities and able to include itsown message to the received message before forwarding. Thestudy investigates the achievable rate regions of three relaymethods; decode-and-forward (D&F), compress-and-forward(C&F) and lattice coded (CP&F) schemes. Also, a specialcase is examined where the cMACr network does not allowcross-reception (i.e. one of the sources always connects to thereceiver via the relay, not directly). In this scenario, the modulosum of the source messages is computed with the lattice codesand compared with the D&F and C&F schemes.A multi-hop network includes multiple layers of relaysbetween the sources and the destiny as illustrated in Fig 6a.In [37], a cross-layer strategy is followed for a multi-hopPLNC 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 witherror correction codes are investigated in [39]. Turbo codes,LDPC codes and bit-interleaved coded modulation with iter-ative decoding (BICM-ID) are simulated in a PLNC basednetwork. The results showed that the PLNC reduces the biterror rate (BER) performance of all three channel coding
Fig. 7. The network model of the CP&F method. E and D symbolize encoderand decoder, respectively. schemes.In [40], a secure PLNC scheme is designed. The methodis inspired by the forwarding algorithm given in [41] (basedon CP&F). The study analyzes two networks with a butterflytopology and a three source topology. Also, it is shown thatthe secure PLNC outperforms the secure network coding suchas given in [42]. B. Compute-and-Forward (CP&F)
CP&F is proposed by Nazer and Gastpar in [5] as a relay-ing method and draw the attention of numerous researchersthroughout the years. CP&F is inspired from the lattice codes(structured codes) that is previously used in [4], [43] and [44].A lattice Λ is a group in R such that for any t , t ∈ Λ , theirsummation is also t + t ∈ Λ . This property makes the latticecodes the building blocks of CP&F as for many simultaneoustransmission techniques.In a fundamental CP&F network as given in Fig. 7, N nodesencode their message w n ∈ { , ..., N } to the lattices as x n = E ( w n ) , (3)where E is the encoding function. Then, N nodes simultane-ously transmit their message x n to the channel. A relay obtainsthe superimposed signal y = N (cid:88) n =1 h n x n + z, (4)where h n is the channel coefficient and z is the AWGN. TheCP&F decoder scales the received message as αy = N (cid:88) n =1 αh n x n + αz, (5) TABLE IA
N OVERVIEW OF
PLNC
STUDIES .Author Year Networkmodel Contribution Performance metricZhang et al. [30] 2006 Two-way(multi-hop) Traditional PLNC is introduced. BERKatti et al. [32] 2008 Multi-way Throughtput improved with PLNC. Throughtput gainKatti et al. [33] 2007 Two-way Analog PLNC is introduced. Network throughputAmah and Klein [34] 2011 Multi-way Space-Time Analog Network Coding (STANC) introduced. Sum rateWie and Chen [35] 2013 Cooperative STANC adapted to multi-way cooperative networks. Sum rateG¨und¨uz et al. [36] 2010 Cooperative(cMACr) Lattice codes used for PLNC in cooperative networks. Achievable rate regionXu et al. [37] 2012 Multi-hop PLNC implemented to multi-hop networks with a cross-layerdesign. Network throughputBurr and Fang [38] 2014 Multi-hop Algebraic constructs are used in the network design. Network throughputAl-Rubaie et al. [39] 2013 Two-way LDPC and Turbo codes are compared. BERHayashi [40] 2019 Cooperative(Butterfly) PLNC and NC are compared. Number of time spans and then quantizes the scaled signal to the closest lattice asfollows αy = N (cid:88) n =1 β n x n (cid:124) (cid:123)(cid:122) (cid:125) approximation of αy + N (cid:88) n =1 ( αh n − β n ) x n βz (cid:124) (cid:123)(cid:122) (cid:125) effective noise . (6)Here, the receiver uses αy to approximate the appropriatecoefficients β n and obtains the following function of thecodewords N (cid:88) n =1 β n x n . (7)A single relay in the given scenario only obtains an integerfunction of the messages which gives no information aboutthe individual messages. However, the receiver can detect themessages if it obtains independent functions as many as thenumber of unknown messages. This idea inspired many studiesin the literature and several aspects of the CP&F have beeninvestigated. Additionally, the CP&F is extended to variouschannel and network models and several design challenges(e.g. CSI estimation) are addressed. We collect the relatedportion of these studies in Table II and classify them for theirnetwork topology, objective and performance metrics.In [46], Nazer and Gastpar propose a PLNC scheme basedon the nested lattice codes and CP&F. It has been shown thatthe lattice codes can be exploited to transfer a function of theinputs to a sink node and the receiver can recover the messagesif it obtains enough functions. The study considers a two-way relay channel and provides an introduction to the existingPLNC approaches. Later, the study proposes the lattice codebased PLNC and compares with other schemes consideringtransfer rates.In [47], a complementary scenario to the CP&F scheme, in-verse compute-and-forward, is considered. The CP&F schemecomputes a function of transmitted messages at relay nodes.The proposed network aims to recover back the computedfunctions of the CP&F at the receiver. In the proposed networkmodel, two CP&F relays send their computed functions toa sink node. The inverse CP&F is considered as a cascadeto the traditional scheme and the rate region of the cascade network is investigated. It is shown that the proposed cascadenetwork outperforms the traditional pairwise communicationbased relay networks on the rate region. In [48], the authorsextend their previous inverse CP&F study to three transmittersscenario and investigate the rate region. Their results showthat transmitting equations with a correlation between themprovides superior performance than transmitting independentequations.Challenges of CP&F on lattice decoding is consideredin [49]. The study analyzes the lattice decoders that aresuitable to practical scenarios. Specifically, the performanceof the maximum likelihood decoder, Diophantine approxima-tion and the sphere decoder is investigated. The computersimulations are used to compare the decoders and to verifythe previous theoretical results. Also, the performance of oneand two dimensional lattice codes are investigated and itis shown via simulations that the performance degrades forlarger constellations. This study is later extended to providean overall base on the decoding of the CP&F in [50]. Inaddition to the previous study, [50] includes a novel maximuma posteriori (MAP) decoder and a Diophantine approximationbased maximum likelihood decoder.A two-way relay channel with CP&F is considered in [51]and [52]. The study [51] proposes a Fincke-Pohst strategybased code search algorithm to find the appropriate coeffi-cients. 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 networkcoefficients separately for each relay, the proposed designjointly optimizes the coefficient matrix for all relays. Themethod utilizes a candidate set search algorithm based on theFincke-Pohst strategy (as in [51]) to select the coefficients.The performance of the proposed method is investigated withsimulations. In [52], a channel inversion precoding is pro-posed. The achievable rates and the symbol error rate (SER) ofthe channel inversion precoded CP&F are calculated and it isstated that the proposed precoding improves the performanceof the CP&F.The communication between the nodes of a hexagonal lat-tice network is analyzed in [54]. Four communication modelsare derived depending on the broadcast and superposition TABLE IIA
N OVERVIEW OF
CP&F
STUDIES .Study Year Network Contribution Performance metricNazer and Gastpar [5] 2011 K × K × CP&F is proposed. Achievable ratesNazer and Gastpar [46] 2011 Two-way Considered for PLNC.Huang et al. [68] 2013 Multi-way Rate optimization is investigated. Sum ratesTan and Yuan [58], [59] 2015 K × L × ,multi-hop CPC&F is proposed.Nokleby and Nazer [70] 2013 × × Amplify-and-Compute model proposed.Ntranos et al. [71], Tan etal. [45] 2013,4 K × , K × L × Asymmetric power allocation case is considered.Pappi et al. [93] 2015 C-RAN Investigated from coalition game perspective.El Soussi et al. [61] 2014 × × coop-erative Rate optimization is investigated. Symmetric ratesOrdentlich et al. [90] 2014 K × 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 probabilitySong et al. [47], [48] 2011,3 × × , × × Inverse CP&F model proposed. Rate regionHuang et al. [72] 2013 Two-way Capacity bounds investigated.Zhu and Gastpar [108] 2013 K × Considered for cognitive radios.Nazer and Gastpar [91] 2014 K × Adapted to DMC.Lim et al. [100]–[103] 2016-9 K × L × Joint typicality decoder proposed.Pappi et al. [84] 2013 K × L × CEE investigated. Computation rateOrdentlich et al. [89] 2015 × Feedback included.Hong and Caire [74]–[76] 2011-3 K × L A low complexity design presented.Liu [52] 2014 Two-way Channel inversion precoding considered. Achievable rates, SERTunali et al. [55], [56] 2012,5 K × L × Eisenstein integer lattices considered. Outage prob., SERWang and Burr [95] 2014 × × Coding gain improved with LDLC. SERMejri et al. [49], [50] 2012,5 K × × , K × Decoding schemes compared. Error probabilityWei and Chen [51] 2012 Two-way Fincke-Pohst code search implemented Average rate, zero entry prob.Niesen and Whiting [57] 2012 × × Degrees of freedom investigated. Degrees of freedomFeng et al. [85], [86] 2013 K × Blind CP&F (without CSI) presented. Throughput, complexityNajafi et al. [87] 2013 K × L × Synchronisation problems investigated. Outage rate, average rateSakzad et al. [88] 2014 K × L × Phase precoding included. Equation error rateWen et al. [96] 2015 K × L × SVP considered with sphere decoding. Average computation rateNokleby and AAzhang [98] 2016 K × L × Node cooperation case investigated. Computation rate, outage prob.Goldenbaum et al. [94] 2016 × × Designed for OFDM and 5G. Message rateZhu and Gastpar [97] 2016 × Typical sumsets of lattices investigated. Density of the setsHuang and Burr [80]–[82] 2016,7 K × L × , K × × A low complexity coefficient selection designsuggested. Cumulative distribution func.Goseling et al. [104], [105] 2013,4 K × L Random access included. Throughput communications between the nodes. Two models that areapplicable with CP&F are investigated on the subject of net-work capacity. The study achieves an improved lower boundcompared to the previous studies. The results reveal that theminimum transport capacity of the broadcast or superpositionenabled case ( / ) is larger than the maximum capacity of thedisabled cases ( / ).In [55] and [56], the alphabet of the lattice codes thatis used in CP&F is restricted to the Eisenstein integers .The traditional CP&F (as in [5]) uses integer-based latticecodes and the study exploits the Eisenstein integers to obtaina better pair of nested lattice structure (the coarse and thefine lattice). It is shown that the outage performance and theerror-correction performance of the proposed codebooks aresuperior to the integer-based lattice codebooks.In traditional CP&F, received signals at the relays are scaledup in order to ensure that the coefficients are close to aninteger. This is a result of the lattice codes that involve onlyinteger codebooks and scaling the signals up also amplifiesthe noise at the receiver [5]. Eventually, scaling the signals Eisenstein integers are the complex numbers in the form of z = a + b exp[2 πi/ where a, b ∈ Z up establishes a Diophantine trade-off between the amplifiednoise level and the approximation performance. The Diophan-tine trade-off of the CP&F scheme is investigated in [57] and itis stated that the asymptotic rate of the scheme in [5] is belowthe 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 samedegrees 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 isthe fact that relay forwarding rates can exceed the informationrate of the sources. For this reason, CPC&F includes acompressing phase to improve network efficiency (e.g powergain). The compressing and the recovering algorithms of theCPC&F is designed and verified with numerical results. Later,the authors generalize the CPC&F method in which the com-pression algorithm includes more operations and shows bettercompression performance as demonstrated with simulations.The same problem that results from the redundant forward rateis also considered in [60]. However, in [60], the compressionis 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, wheretwo nodes are able to send their messages to a receiver bothdirectly and over a relay. The study investigates two codingmethods (CP&F and C&F) that are based on lattice codesand aims to optimize the symmetric rate. The authors proposean iterative coordinate descent method that focuses on thepower allocation and integer coefficient selection processesfor the optimization problem. The results reveal that CP&Fshows better performance than lattice-based C&F. This work islater extended to a multi-user multi-relay cooperative scenarioin [62]. Cooperative networks are also considered in [63]–[67].The authors in [64], [65], [67] propose an CP&F method forthe two transmitter single relay two receiver networks. TheCP&F is considered for the same scenario with single receiverin [63], [66].The pairwise CP&F model is applied to multi-way relaychannels (MWRCs) in [68]. The pairwise structure of thenetwork is accomplished with two phases: the broadcasttransmission phase and the multiple access transmission phase.The sum rates of the pairwise CP&F case and the pairwisesuccessive transmission case are derived for the MWRC andcompared with each other.The outage probability of the CP&F scheme is derivedin [69] for MWRCs. Also the CP&F is compared withthe non-network coding scheme with respect to the outageprobabilities. The results show that in a canonical two-wayrelay channel, CP&F achieves dB gain against the non-network coding at the outage probability of − .In [70], amplify-and-compute method is proposed, whichcombines CP&F and A&F methods. The relays in the networkreceive the superpositioned lattice codes from the sources asin CP&F and transmit to the next network layer as in A&F.A CP&F scheme that allows asymmetric power allocationto the nodes is proposed in [71] and [45]. The method in [71]is based on the lattice codes in which a fine lattice and acoarse lattice provide codebooks that are decodable and underthe power limit respectively. The method maps the messagesto the codebooks depending on the power and noise tolerance.Specifically, the top of the message vector is set to zerodepending on the power of the codebook and the bottom of themessage vector is set to zero depending on the noise tolerance.A bi-directional relay network in which two nodes exchangeinformation through a relay node under an inter-symbol inter-ference channel is considered in [72]. The proposed model isseparated into a multiple access phase and a broadcast phaseand the capacity region is derived. The inner bound of thecapacity region is computed with the help of the CP&F methodand the outer bound is computed with the cut-set argumentgiven in [73]. The numerical results revealed that the proposedCP&F scheme has a higher exchange rate than the D&F.The complexity reduced version of the CP&F is proposedin [74], which only involves scaling, offset and scalar quan-tization at the receivers. The method aims to reach the samecapacity of the CP&F with the low-complexity, low-powerdecentralized antenna networks. For this purpose, the methodconsiders quantization at the receivers as a part of the wirelesschannel. The numerical results show that the computation rate of the quantized CP&F is within the shaping error of 0.25bits per symbol compared to the traditional CP&F of [5].The authors extended their work to downlink scenario of thequantized CP&F in [75]. The study derives the computationrate of the proposed scheme and compares it with the downlinkCP&F via simulations. The reverse CP&F is generalized andextended in [76] to cover additional scenarios and to includecomprehensive simulation results and comparisons.Another low-complexity CP&F scheme is given in [77].The outage probability of the proposed model is derivedand compared with the standard CP&F scheme. Additionally,channel estimation error (CEE) is introduced to the system andits effect is investigated. The results show that the proposedsimple method is also more resistant to the CEE than thetraditional CP&F. In [78], the authors extend their studyand propose two CP&F based methods. The first methodis designed to reduce the computational complexity and thesecond method is proved to have better performance comparedto the traditional CP&F, which is verified via simulations.The complexity of the coefficient selection of the CP&Fis considered in [79]–[82]. The authors propose a low-complexity algorithm to optimize the integer coefficients thatis essential for the CP&F performance in [79]. This work islater improved and generalized to cover CP&F and integer-forcing algorithms in [83]. The same problem is studiedin [80], [81]. An exhaustive search algorithm and a latticereduction algorithm is proposed in [80], [81] to reduce thehardness of the coefficient selection problem. Also in [82],a low-complexity coefficient method is proposed for massiveMIMO enabled CP&F networks.The effect of the channel estimation error to the perfor-mance of CP&F scheme is analyzed in [84]. The computationrate region for the imperfect channel estimation case is derivedand the expression is closely approximated for the Gaussiandistributed CEE. Additionally, the distribution of the rate lossis given in the closed-form. The simulations are used todemonstrate the vulnerability of the CP&F to the CEE.The traditional CP&F requires CSI to decide the appropriatescale factors which are essential in the decoding of theinteger lattices. Otherwise, the non-integer channel coefficientsincrease symbol error. In [85], [86], a practical CP&F schemeis proposed that does not require CSI to compute the mostsuitable scale factors. Instead, the proposed method choosessub-optimal however sufficient scaling factors to gain fromthe system complexity. Simulations show that in some cases,the computation complexity can be reduced ten times whencompared to the CP&F of [5].The synchronization problem of the CP&F is consideredin [87]. CP&F networks can exhibit asynchronization of thenodes as a result of the decentralized node structure of the net-work. The study solves the symbol asynchronization problemwith an equalizer by converting the nature of the network fromasynchronous to synchronous. The frame asynchronization issolved by eliminating delays with multiple antennas at therelay node. Also, it is shown that the achievable rate can bemaximized for all SNR regions by applying a linear filter.A phase precoding method for the CP&F scheme is pro-posed in [88] for multi-user multi-relay networks. The objec- tive of the method is to reach higher computation rates thanthe traditional CP&F. However, it also requires an optimalprecoding matrix and an optimal network equation matrix tofulfill that objective. For this reason, the study introduces apartial feedback channel between the relays and the nodessince the precoding matrix is needed at the nodes and thenetwork equation matrix has to be computed at the relays.The relays compute the optimal precoder and the networkequations, then forward the precoder information to the nodesthrough the feedback channel. With the simulations, the studyshows that the proposed phase precoding can improve theequation error rate.Another feedback enabled CP&F method is given in [89].The method aims to design the optimal CP&F model toachieve the maximum computation rates for the scenario thattransmitters have access to an ideal feedback channel towardsthe relay. The method is designed for two users (and a relay)networks and it is demonstrated that the proposed schemeobtains better computation rates than the CP&F without feed-back.CP&F is considered for the Gaussian multi-user interferencechannels in [90]. A comprehensive study is given on theapproximate 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 moduloarithmetics in [92]. It is shown that only five lattice codefamilies are capable of complex modulo arithmetics over theEuclidean geometry and their coding gains are calculated.In [93], CP&F scheme is considered for the cloud-radioaccess networks (C-RANs). The study aims to maximize theinformation flow from the nodes to the FC of the network.For this reason, a coalition game is designed that maximizesthe defined profits. Exploiting the interference of signals isconsidered for the fifth generation ( G) networks in [94]. Theobjective is to provide channel access to a massive numberof nodes that are required by the IoT. For this purpose, thestudy combines the PLNC with the pulse shaped orthogonalfrequency division multiplexing (OFDM).In [95], a low density lattice codes (LDLC) based CP&Fmethod is proposed in order to reach high coding gains.In [96], the authors propose a sphere decoding method tomaximize the computation rate by considering the problemin hand as a shortest vector problem (SVP). The sumsetscan be defined in simple terms as the set of received latticepoints which is the sum of the transmitted lattice points.In [97], typical sumsets are defined and analyzed accordingto their sizes, distributions and densities. The study aims toobtain results that can improve the performance of the latticedecoding in CP&F.Cooperation between the transmitters is considered in [98].This is different than the cooperative networks as in Fig. 6d. Inthis study, cooperation indicates that the nodes can partiallyhear the messages of the other nodes which resembles thediversity improvement of a multiple antenna network. Theresults reveal that the partial cooperation between the nodescan 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 theCP&F method in [100]–[103]. The CP&F is also consideredfor random access channels in [104]–[106]. The impact ofthe input distribution to the computation rate of the CP&Fis considered in [107] for the Gaussian W-MAC. It is shownthat the Gaussian input distribution is not optimal and thecomputation rates can be improved if the input distributions arechosen wisely. In [108], CP&F is extended to cognitive radionetworks. In the following section, we present the computationand function alignment methods that are inspired by thefunction alignment and CP&F methods.V. I
NTERFERENCE / C
OMPUTATION / F
UNCTION A LIGNMENT
The interference alignment (IA) is an interference man-agement technique and can be compared with the multipleaccess methods for their application purpose. The conventionalmultiple access methods divide the time and frequency re-sources among the users. In the IA, all users share the sameresources, however, the IA algorithm affects the transmittedsignals (precoding) such that the received signals are alignedinto two subspaces. As the algorithm aims, the unintendedsignals (the interference from the other users) fall under onesubspace and the intended signal can be extracted from theother subspace. We are partially interested in the IA since,on one hand, the network model enables the interference; onthe other hand, it aims to cancel the interference instead ofbenefiting from it.We are much more interested in the computation alignmentmethod that is inspired by the IA and the CP&F. Similar tothe IA, the computation alignment divides the signals intosubspaces, however, the aligned signals are not discarded,instead, the interference is exploited for the computation. Inthis section, we present an elementary description of the IAand refer the readers to [109] and [110] for detailed informa-tion. Later, we continue with the computation alignment andpresent the current studies. The fundamental IA, computationand function alignment studies are exhibited in Table III.The IA studies are mainly centered upon the space, fre-quency, or time dimensions to align the interference [109]. Wefocus on a simple space dimension example, which is basedon multiple antenna techniques. Consider a MIMO networkthat consists of three transmitters and three receivers all ofwhich equipped with two antennas as shown in Fig. 8a. Afterthe simultaneous transmission, the first receiver obtains thefollowing signal. y = H v s + H v s + H v s + n . (8)The subscripts indicate the users, i.e. y j ∈ C × is thereceived signal vector of the j th receiver. The first and secondrow of the vector are the received signal at the first and secondantenna, respectively. H ij ∈ C × is the channel matrix vectorfrom the i th transmitter to the j th receiver. v i ∈ C × isthe diagonal precoding matrix of the i th transmitter, s i isthe message of the i th transmitter and the n j ∈ C × is theGaussian noise vector at the j th receiver, where its rows arethe noise of the first and second antenna, respectively. TABLE IIIA
N OVERVIEW OF
IA,
COMPUTATION ALIGNMENT AND FUNCTION ALIGNMENT STUDIES .Author Year Networkmodel Contribution PerformancemetricNiesen et al. [111], [112] 2011,2013 K × K Provides a capacity approximation for multi-layer networksthat is independent of network depth. Capacity ap-proximationGoela et al. [113] 2012 Investigates coding schemes that reach computation capacitywith network decomposition. CodingcapacitySuh et al. [114], [115] 2012, 2016 × Derives a new upper bound on the computing capacity andpropose a network decomposition theorem. ComputingcapacitySuh and Gastpar [116] 2013 Considers feedback for function alignment.Suh and Gastpar [117] 2013 Investigates the scenarios where network decomposition isoptimal. Symmetriccapacity(a)(b)Fig. 8. Vector illustration of the interference alignment and computationalignment methods. (a) Interference alignment, (b) Computation alignment.
Assuming perfect CSI at the receivers, the MIMO algo-rithms require three antennas since each of the three users’inputs is an unknown variable. In IA schemes, the precodingvector ( v i ) elegantly aligns the two unknown vectors (theinterference from the other users) such that (8) can be writtenwith two unknown vectors. After that, the two unknownvariables can be solved with the two equations. The vectorrepresentation of how the IA works is illustrated in Fig. 8.Each node is represented with a square and the transmittedand received signals are given in the inside of these squares.The precoding at the transmitter divides the three vectors intotwo vectors; one of them is the intended vector and the otheris the aligned (unintended) signals.The perspective given above exploits the physical layer tosuppress the interference. The computation alignment method is inspired from this perspective, however, it focuses onthe unintended aligned vector for the function computation,which genuinely exploits the signal interference. A simple twotransmitter two receiver network example of the computationalignment is given in Fig. 8b. In this example, the timedimension is used to create the subspaces. Assume that thefirst transmitter aims to send s and s while the secondtransmitter aims to send s . In this scenario, transmit vectorsadjust the messages as follows (cid:20) x ( t ) x ( t ) (cid:21) = v s + v s (cid:20) x ( t ) x ( t ) (cid:21) = v s , (9)where t , t denote the time slots and x , x denote the trans-mitted signals of the first and second transmitters, respectively.Proper selection of the transmit vectors lead to the followingreceived signal vectors y = (cid:20) (cid:21) ( s + s ) + h (cid:20) − (cid:21) s + n y = h (cid:20) (cid:21) ( s + s ) + (cid:20) − (cid:21) s + n , (10)where the subscripts indicate the users, i.e. y j ∈ C × is thereceived signal vector of the j th receiver. However, subspacesare obtained with multiple transmissions rather than multipleantennas in this example, hence the first and second row ofthe vector are the received signal at the first and second timeslots, respectively. After obtaining signals from two time slots,the first receiver can obtain s + s by using y ( t ) + y ( t ) and the second user can obtain s by using y ( t ) − y ( t ) .The design of the transmit vector enables computationalignment. A vectorial illustration of (10) is given in Fig. 8b. Itshould be noted that the illustration omits the representationof the transmit vectors and the channel gain vectors in thefigure for better appearance. As a result of the computationalignment, the receivers obtain the summation of the messagesin the aligned subspace. Specifically, the first receiver obtains s + s aligned and s in the other subspace while thereceiver two obtains s + s and s .In a relay network, C&F method adds additional noise tothe network in each layer. As a result, the approximation gapof the network capacity widens for the increasing number ofnetwork layers. The computation alignment scheme in [111] and [112] presents a relay network with an approximationgap that is independent of the layer depth. The computationalignment technique depends on lattice codes and the CP&F tobe able to recover the integer-valued messages. However, theCP&F scheme also produce errors as a result of the non-integerchannel gains. This problem is solved by IA by dividing thechannel into multiple subchannels and aligning. The resultsreveal that the approximation gap is not constant as opposedto C&F results, it depends on fading characteristics.In [113], multiple transmitter multiple receiver summationnetworks are considered. The scalar and vectorial linear codesare investigated for these networks and it is stated that thecomputation alignment is essential to reach the computationcapacity. For this purpose, the network is decomposed intosub-networks with the network equivalence theorems. Also,the linear coding capacity of the computation is derived forvarious channel parameters.In [114] and [115], × modulo-2 sum networks areconsidered. The study is inspired by the IA and similar tothe computation alignment given in [112], and named as thefunction alignment. A new upper bound is derived for thecomputing capacity of the two receiver CP&F networks withlinear codes in [114], [115]. Also, the studies define a networkdecomposition theorem to divide the network into elementarysubnetworks. Using the theorem, the computing capacity isgeneralized for the N -transmitter N -receiver CP&F networks.In [116], the authors extend their previous work, [114], toinclude feedback. The study derives the feedback includedcomputing capacity and compare it with the no-feedbackscheme (as in [114]). It should be stated that network de-composition is crucial to create subnetworks. The authors alsoinvestigate the network decomposition and its importance inCP&F networks thoroughly in [117].In [118], memoryless bivariate Gaussian sources are con-sidered for a two-source one receiver network. The receiveraims to obtain the information of the two sources with theminimum distortion. In the paper, perfect causal feedback isassumed and the power-distortion relationship is investigated.The results are given as a function of the source correlationand SNR. Also, the necessary and sufficient conditions toreach the minimum distortion levels are derived. The studyis later extended to two transmitter two receiver networksin [119]. Simultaneous transmission is also the basis of func-tion computation techniques that are inspired by lattice codesand CP&F. Moreover, the function computation methods areconsidered for analog signals and the resulting analog functioncomputation studies gained popularity in the literature.VI. F UNCTION C OMPUTATION
Function computation is one of the most striking applica-tions of simultaneous transmission. The joint source-channelcoding paradigm by Gastpar and Vetterli [2] and later Nazerand Gastpar [4] establish the basis for the function computa-tion. This approach inspires a wide range of studies that oftentargets one of the two main aspects; computation or multipleaccess. The motivation behind the studies that focus on the In particular Avestimehr-Diggavi-Tse (ADT) network is considered. multiple access aspect is generally to improve the networkthroughput as in CP&F and considered in Section IV. Here,we present the studies that focus on the computation aspectas the digital function computation. The computation aspectlater inspires the analog transmission based studies that purelyfocus on function computation. The motivation behind thesestudies is strictly computation related and presented as theanalog function computation [6].
A. Digital Function Computation
A list of the digital function computation studies is pre-sented in Table IV. Nazer and Gastpar proposed the com-putation codes in [4] that is based on lattices. The objectiveof the study is to send a linear function of multiple usersto a receiver with the simultaneous transmission. The studyinvestigates the achievable rates with the proposed compu-tation codes and compares them with the separation basedmethods. Computation of linear functions is also consideredin [120] for a wireless network that consists of two correlatedGaussian sources. The work aims to find the optimum codingscheme that upper bounds the distortion at the received signal.The numerical results are given for the subtraction ( a − b )and weighted addition ( a + 2 b ) functions as a function of thecorrelation coefficient of the two sources.Goldenbaum et al. generalize the computation of nomo-graphic functions with nested lattice codes in [121]. The modelis based on the fact that the nomographic functions can bewritten in the form of pre and post processing functions asstudied in the analog function computation research. However,the authors implement a digital model to reduce the destructiveeffects of the noise. The study also examines the requirednumber of channel use and the accuracy performance of thesystem. Lattice codes are used in [122] for the computationof nomographic functions. The study thoroughly analyzesthe relation between the lattice codes and the nomographicfunctions and derive the computation rate performances. Oneof their observations reveals that any continuous function canbe computed over the channel.In [123] and [124], the authors include orthogonal com-ponents to their coding scheme in a similar manner that theTBMA benefits from the orthogonal signals. The method isinterested in calculating the arithmetic summation and thetype functions. The type function computes the histogramof the transmitted signals as explained in TBMA. Then theresulting statistics can be used to obtain the mean, variance,maximum, minimum and median functions. The W-MAC isfirstly decomposed into multiple modulo sum subchannelswith nested lattice codes and linear network codes. Then thelinear Slepian–Wolf source coding is used to calculate thedesired functions. It is shown that the joint source-channelcoding provides better performance than the separate codingschemes in certain cases.The simultaneous transmitting and air computing (STAC)method is proposed in [125] and [126] to improve the functioncomputation capability and the network efficiency of datacenter networks. The proposed method is based on a traditionalfunction computation model that combines communication TABLE IVA
N OVERVIEW OF DIGITAL FUNCTION COMPUTATION STUDIES .Author Year Networkmodel Contribution PerformancemetricJeon et al. [123], [124] 2013, 2014 K × L × Extends the lattice code computable function set by consider-ing orthogonal components and derives an approximation ofcomputation capacity. ComputationrateWu et al. [125], [126] 2015, 2016 Proposes a low-bandwidth, low-energy SDR network archi-tecture, STAC. SER, sessionrateNazer and Gastper [4] 2007 K × Proposes the computation codes and the CoMAC whichenhance the communication performance by utilizing latticecodes and the joint source-channel coding. Computation rateGoldenbaum et al. [121], [122] 2013, 2015 Improves the reliability of nomographic function computationby adapting lattice codes to AFC based consensus methods.Jeon and Jung [127], [128] 2015, 2016 Provides non-vanishing computation rates (asymptoticallypositive) by allowing only a subset of nodes with high gainsto transmit.Wu et al. [132]–[135] 2019 Considers the wide-band implementation of CoMAC by allo-cating sub-functions to subcarriers.Soundararajan and Vishwanath [120] 2012 × Derives a lower bound on the distortion of CoMAC andconsiders correlated source scenario. DistortionrateZhan et al. [129] 2011 Butterfly Investigates the duality between computation andcommunication aspect of simultaneous transmissionschemes. DistortionlevelZhu et al. [130], [131] 2017, 2019 × CapacityregionChen et al. [136] 2020 K × A low-complexity transceiver model is designed to maximizethe achievable function rate. Achievablefunction rate and computation; however, the model is also developed uponan enhanced software-defined network structure that providesside information to the nodes. Computer simulations demon-strate the spectrum and energy efficiency of STAC in datacenter networks.Function computation problem is considered for the fadingMACs in [127] and [128]. The main idea behind the studyis that only the nodes with high channel gains participate inthe in-network computation rather than the whole network.The study investigates the computation rates of the proposedmodel for the fading channels.A duality between the function computation problem andthe multicast problem is considered in [129]. The functioncomputation problem is interested in receiving a function(e.g. summation) of the transmitted messages. In the multicastproblem, the objective is to receive individual messages. Afterdefining the duality relation for the deterministic networks,Gaussian MACs are considered. The achievable distortionlevels are derived for the summation of two Gaussian sourcesin these networks. The results revealed that there is a constantgap between the cut-set bound and the distortion of thesummation function of the independent Gaussian sources.It is obvious that the superposition of the signals destroysthe individual information of the transmitted messages. Thiscan be viewed as renouncing the capability of the nodesto (multiple) access to the receiver [130], [131]. However,computation codes benefit from this idea to improve the effi-ciency of the computations in the network. Here, access to theindividual information is traded with improved computationefficiency and this can be highly beneficial if the networkis only interested in a function of the transmitted data. Inother words, computation codes present a duality between themultiple access and computation. This duality is investigatedin [130] and [131] to check the existence of the computation codes that also allow the individual access to the receiver. Theinvestigation results indicate that efficient computation codesprevent the individual access to the receiver.The wideband computation over multiple access channel(CoMAC) schemes that are adopted for the frequency se-lective channels is proposed in [132]–[135]. These studiesrely on the NOMA and OFDM models to use widebandfrequencies. A NOMA assisted function computation networkis proposed in [132] and [133]. As given in the previoussections, NOMA is a multiple access method that is based onthe superposition of signals. In [132], functions to be computedare intentionally divided into sub-functions and these sub-functions are computed simultaneously under different NOMAaccess slots. As a result, the computations can be made at thewideband frequencies where the fading is more challenging.The results reveal that the proposed NOMA assisted approachachieves higher computation rates and prevents vanishingcomputation. Additionally, expressions for the diversity orderof the computation rate is derived in [133]. In [134] and [135],the sub-functions are allocated to the OFDM carriers and anoptimization problem is considered for the power allocation.In [136], a transceiver model is designed for digital functioncomputation that reduces the time-complexity. The authorsderive the achievable function rates of the proposed modelby considering the number of nodes, the maximum value ofmessages and the quantization error threshold.
B. Analog Function Computation (AFC)
The main idea behin the AFC is to match the W-MAC withthe desired function. In its base form, the W-MAC constitutesa natural summation operation with its superposition property.The AFC adjusts the channel with proper signal processingat the transmitter and receiver ends such that the W-MAC Fig. 9. The network model of the AFC studies. ψ and ϕ denote pre -processingand post -processing functions, respectively. can compute other mathematical operations. For this purpose,transmitters use pre -processing functions, ϕ n ( · ) : R → R ,before transmitting their signals and the receiver applies a post -processing function, ψ ( · ) : R → R , after receivingthe superpositioned signal as depicted in Fig. 9. The userssimultaneously transmit their pre -processed signals and thereceiver obtains the following function output after the post -process, f ( x , x , ..., x N ) = ψ (cid:32) N (cid:88) n =1 ϕ n ( x n ) (cid:33) . (11)The functions that can be expressed as in (11) with the sum-mation operation are called nomographic functions. The AFCis firstly shown to be applicable to nomographic functions andlater it is proven that any function can be computed with theAFC [6]. The AFC studies are listed in Table V with theircontributions and performance metrics.Goldenbaum and Sta´nczak pioneered the studies that com-pute mathematical functions at the communication process.In [137], the communication system carries the information atthe transmit powers which relieves the synchronization burden.Their method manages to compute a variety of functionswhich involve nonlinear functions. The authors lastly exhibitthe error analysis of the arithmetic mean function. In [139],Goldenbaum and Sta´nczak extend their previous work byanalyzing the geometric mean function. They also comparetheir method with TDMA via simulations and present theirresults that AFC outperforms the TDMA on the functioncomputation time.In their following works, Goldenbaum and Sta´nczak providean extensive theory of AFC in [140] and [6]. They analyzethe function sets and network topologies that are compatiblewith AFC. Their results show that every function can becomputed with AFC as the pre -processing is independent ofthe computed function.The error performance of the estimators that are used inthe receivers of AFC is analyzed in [141] for arithmetic andgeometric mean functions. Additionally, function computationsimulations are performed for AFC, TDMA and CDMAmodels. The results showed that the AFC brings out bettercomputation accuracy in less time than time or code dividednetworks.Dependency of AFC to CSI is investigated in [142]. Theletter firstly shows that the transmitters only need the mag- nitude of the CSI rather than the full CSI. Then it is proventhat any CSI knowledge requirement on the transmitters canbe removed if the receiver is equipped with multiple antennas.The computation of the l p -norms with the AFC is inves-tigated in [143]. The authors later propose an algorithm toapproximate certain continuous multivariate functions by thenomographic function in [144]. In [80], the max functionwhich aims to obtain the maximum value at the FC isimplemented with the AFC and CDMA.In [145], Goldenbaum and Sta´nczak investigate the relationbetween the reliability and efficiency of an AFC model froman information-theoretic point of view. In order to evaluatethe relation in question, the computation rate metric is definedas the number of functions that can be computed per channeluse. Then, achievable computation rates are given for linearcombination and special polynomial functions. The letter statesthat the computation rate is dependant to the required accuracylevel and the number of transmitters as well as the functiontype.An AFC method based on the free deconvolution theorem(see [146] for the further information on the free deconvolu-tion) is proposed in [147]. The contribution of the study isthat the mean function can be computed with fewer channeluses and without the channel estimation.In [148], a MIMO AFC model is proposed that the FC andthe multiples transmitting nodes are equipped with multipleantennas. The method allows channel estimation errors in thesystem model to consider realistic scenarios and design a non-convex optimization problem to achieve the optimum solutionin this scenario. The optimization problem that minimizes theworst-case mean square error (MSE) is converted to a simplerversion and solved.An adaptive AFC method is given in [149]. Different fromthe traditional AFC methods, this model is based on thecausal CSI at the transmitters. Simulation results of the studystate that the proposed adaptive model shows better outageprobability performance than the traditional AFC.In [150], a function computation method that can computefunctions for multiple variables is proposed. For example,multi-model sensor measurements such as temperature, hu-midity and pollution can be computed over the air simulta-neously. This is possible with the utilization of beamformingtechnology over MIMO enabled nodes. The study designs anoptimization problem that minimizes the sum mean-squarederror at the receiver. The optimization problem is shown tobe an NP-hard problem and an approximate version is solvedwith differential geometry. Also, the result of the optimizationproblem (based on Grassmann Manifold) is validated withcomputer simulations. This work is later extended to high-mobility sensing networks where an environment is moni-tored by unmanned aerial vehicles (UAVs) [151]. In additionto [150], the study includes enhanced equalization and channelfeedback methods to provide reliable information exchangebetween the sensors and the receiver.Non-uniform fading in different nodes of an AFC networkis a performance degrading problem. In order to mitigate theeffects of non-uniform fading, a uniform-forcing transceiverdesign is proposed in [138]. The authors formulate an opti- TABLE VA
N OVERVIEW OF ANALOG FUNCTION COMPUTATION STUDIES .Author Year Contribution PerformancemetricGoldenbaum et al. [137] 2009 Computes functions (e.g. arithmetic mean) over the channel with low-complexity and low-energy consumption. OutageprobabilityGoldenbaum and Sta´nczak [139] 2010 Extends [137] to investigate the geometric mean.Sta´nczak et al. [140], [141] 2012, 2013 Provides a complete theory of AFC.Goldenbaum and Sta´nczak [142] 2014 Investigates the AFC for different CSI assumptions.Jeon and Jung [149] 2018 Improves the traditional AFC against fading environment by utilizingcausal CSI.Chen et al. [152] 2018 Extends [138], considers the computation of multiple functions andinvestigates the method’s performance.Goldenbaum et al. [6] 2013 Investigates and generalizes the functions that is computable withAFC. MSELimmer and Sta´nczak [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 beam-forming 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 acqui-sition.Li et al. [156], [157] 2018, 2019 Considers an AirComp method that transfers power to the distributednodes over the air to relax energy constraints.Cao et al. [158] 2019 Considers joint optimization of the transmit powers and the denoisingfactor to improve AirComp.Basaran et al. [159] 2020 Provides an energy efficient AirComp method by exploiting thecorrelation between the measurements.Chen et al. [138] 2018 Considers non-uniform fading scenario in the proposed AirCompmethod and brings robustness with the uniform-forcing transceiverdesign.Limmer et al. [144] 2015 Provides a method that computes certain multivariate functions withnomographic function approximation. ApproximationerrorGoldenbaum et al. [145] 2015 Adapts computation codes to AFC to provide reliability and investigateits rates. ComputationrateWang et al. [147] 2015 Without channel estimation, reduces the required number of channeluses on the computation of mean function with free deconvolution. Relative errorDong et al. [160] 2020 Uses Wirtinger flow to provide a low-complexity, low-latency Air-Comp method that requires no CSI).Chen et al. [162] 2019 Investigates and compares the computation based and communicationbased methods. Function rateJakimovski et al. [165] 2011 Provides a testbed implementation. Average errorSigg et al. [166] 2012 Mean errorKortke et al. [163] 2014 Relative errorAbari et al. [167] 2015 CFOAltun et al. [164] 2017 MSE mization problem that minimizes the mean square error ofthe computed function output at the receiver. Their resultsshow that a semidefinite relaxation is necessary to solve theproblem and successive convex approximation can furtherincrease the accuracy of the solution. The proposed transceivermodel in [138] is extended in [152] to compute multiplefunctions simultaneously. The nodes in the new networkmodel are equipped with antenna arrays and zero-forcingbeamforming technology. Beamforming technology is adoptedto remove the interference of other functions that are computedsimultaneously and multiple antenna arrays relieve the massiveCSI knowledge requirement. Also as in [138], the method isresilient to the non-uniform fading problem. The performanceof the proposed network is analyzed with both simulations andanalytical expressions.Another beamforming and MIMO enabled function compu-tation method is given in [153]. The study proposes a receiverbeamforming model design that reduces channel dimensionsand equalizes channel covariances and small scale fading components. Additionally, the method uses a feedback schemeto accurately provide massive CSI knowledge to the network.The proposed approach mainly aims to reduce the computationerror resulting from the discrepancy of the received functionoutput. The simulations demonstrate the positive effect of theproposed model on error reduction.A training model for faster acquisition of the CSI isproposed in [154] and [155]. The method is based on theeffective CSI definition that is obtained with the simultaneouspilot transmission of the nodes. The method requires iterativebroadcasts from FC to estimate the effective CSI and onelast simultaneous pilot transmission yields the CSI vector atthe FC while the traditional AFC methods individually traineach CSI. The study analytically obtains the computationcomplexity of both traditional and proposed methods. Also,an error improvement method is proposed to compensate forthe estimation error of the novel approach since the proposedmodel causes larger estimation errors.Function computation is integrated with wireless power transfer in [156] for IoT networks. The framework aims tominimize the computation error that is seen at the aggregateddata by jointly optimizing power transfer and function com-putation tasks. The framework is designed for networks thatconsist of MIMO beamforming capable nodes. The joint opti-mization problem is divided into two sections; wireless powercontrol optimization and function computation optimization.As applied earlier in [138], the semidefinite relaxation methodis implemented to solve the function computation section ofthe optimization problem while the wireless power controlsection is solved in closed form. The study also states thatthe integration of the wireless power transfer into the AFCnetwork brings an additional design dimension which canincrease the computation accuracy. A combination of functioncomputation and wireless power transfer is also implementedfor high-mobility sensing in smart cities where UAVs are usedto collect sensor readings [157].Power control problem of the AFC systems are consideredin [158]. Poor distribution of transmit powers in an AFCnetwork can cause high computation errors as a result ofchannel distortion. The proposed method aims to find the opti-mum transmit power level design by solving the optimizationproblem that minimizes the computation error at the FC. Thedefinition of the problem involves the optimization of bothtransmit powers and the denoising factor of the FC. Moreover,the problem is solved for additional scenarios such that onlyone transmitter has power constraints instead of all devices.Lastly, the simulations reveal that the proposed power controlscheme has a notable mitigating effect on the computationerror of the AFC network.The problem of energy consumption in function compu-tation networks is considered in [159]. The method mainlybenefits from the spatial correlations between the sensor read-ings in order to reduce energy consumption. For this purpose, aminimum mean square error (MMSE) estimator is designed toobtain estimations with less number of samples. The proposedestimator requires significantly less energy consumption tooperate, hence nearly doubles the lifetime of the network. Inaddition to the improved energy efficiency, the estimator alsoyields better MSE performance compared to the traditionalmethods as illustrated via simulations.Wirtinger flow is an algorithm that is usually used inthe solution of non-convex optimization problems such asphase retrieval from the received signal magnitudes. Dong etal. use the Wirtinger flow method to solve the non-convexoptimization problem of the function computation withoutthe knowledge of CSI in [160]. The proposed algorithmonly requires data samples and randomly initialized Wirtingerflow iterations, i.e. pilot transmission for CSI acquisition isnot needed. As a result, the study can reduce the latencyof the function computation applications by removing thedependency on the pilot transmission process of the CSIestimation. Also, the study reveals that the estimation errorof the Wirtinger flow-based function computation model issufficiently small.In [161], the power alignment problem of the functioncomputation networks is considered. The power alignment isusually performed with pre -processing or pre-coding in tradi- tional function computation networks. In [161], this problemis addressed with a backscatter framework instead of pre-coding. The proposed method is designed for mass densitysensor networks where multiple UAVs are responsible for thecollection of sensor measurements. The model consists oftwo phases; channel gain acquisition and data aggregation.In the first phase, the UAVs are the power emitters and thesensor nodes act as the backscatter object in which the nodesbackscatter the ambient signal that comes from the UAV. Asa result, the UAV collects the sum channel gain. In the secondphase, the UAVs are the readers that receive the aggregatedsensor data. The results show that using the sum channel gainfor the aggregation of the sensor readings can improve MSEup to 10 dB.The function computation capability of the AFC is com-pared with the separate communication schemes in [162]. Theachievable function rates are derived for the two cases and it isshown and verified via simulations that the computation overthe channel (AFC) is not always the optimum scenario for thefunction computation.
1) Test-bed Implementations:
The feasibility of the AFC isalso investigated with testbed implementations in the literature.In [163], the authors deploy the model of [141] with SDRs. Anetwork with 11 nodes and an FC is implemented to computethe arithmetic and geometric mean of the sensor readingsof the nodes. In another implementation study [164], thesummation of sensor readings over the channel is tested viaSDR modules. The network that includes three transmittersand an FC is used to analyze the effect of distance andsignal power level. Also, the error performance of the AFCnetwork is compared with the TDMA scheme via computersimulations. The TDMA scheme requires communication slotsfor each node while the AFC completes the transmission inone slot. As a result, the simulation results of AFC displaysless error since TDMA introduces additional thermal noise tothe system with each communication.In [165], a simultaneous transmission method is proposedand implemented with SDR modules. The method is based onthe hamming distance of the superimposed received signalsthat are affected by each transmit vector. The calculation ofcorrupted goods in a pallet via temperature readings is sug-gested as an application example. The testbed implementationdemonstrates the feasibility of the method and the success ofcomputations over the channel. Another implementation studyis given by Sigg et al. in [166]. They suggested a computationmodel that carries information in the mean value of the Poissondistribution. The nodes of the system transmit burst sequencesthat are Poisson distributed and the density of the bursts ina time interval represents the mean of the distribution. Themodel then extracts the mean value of the superimposed signalat the receiver. The proposed model is implemented with 15sensor nodes and a receiver that is driven by microcontrollers.The realized scenario successfully recovered the average tem-perature of the simultaneously transmitted sensor readings.AirShare method is presented in [167] to improve the re-sistance of the distributed wireless applications against carrierfrequency offset (CFO). The method uses a broadcast clocksignal as a reference to other nodes that aim to transmit simultaneously. The paper firstly investigates the feasibilityof the AirShare method and then implements the network viaSDR modules. VII. F EDERATED L EARNING
An emerging application area of the AFC is the federatedlearning algorithms which enable the computation of learningdata over the air. In this section, the federated learning studiesthat are based on the simultaneous transmission of signals areexamined.Machine learning (ML) is the rising technology of the lastdecade as a result of the increasing computational capabilitiesof the electronic devices (machines) (comprehensive informa-tion on the main contents of this section can be found in [168]).ML is based on the usage of massive data or data sets tolet those machines make the classification or prediction op-erations. Moreover, many aspects of a communication systemsuch as modulation, demodulation, channel estimation etc. canbe considered as a classification or prediction problem [169].This observation brought out the relationship between the MLand wireless communication. Although traditional communi-cation systems use model-based solutions for their problems,ML based data-driven techniques have already started to showpromising results in the wireless communication area [170].In the following years, this relationship proved to be two wayssuch that the communication networks can also be beneficialin the learning area.The data-driven nature of the ML requires the collection ofmassive data in centralized points before the learning process.However, data sources of today’s technology are often at thewireless edges. As a result, collecting massive data from wire-less devices to a center can cost a high amount of energy andbandwidth [171]. Collaborative machine learning or federatedlearning is a solution to the data collection problem whichproposes the process of the data at the edge users or distributedcenters instead of a local center [172], [173]. The superpositionproperty of the wireless channel can further relieve some ofthe costs in the federated learning schemes by performingcomputations over the wireless channel. A list of federatedlearning studies can be found in Table VI.One fundamental example of this paradigm is given byAmiri and G¨und¨uz [171], [174], [175]. The learning algorithmis based on the minimization of a loss function which issolved with the distributed stochastic gradient descent (DSGD)method. In this method, the learning parameters are updatedwith multiple iterations. Different from the traditional collab-orative ML that uses DSGD, this study exploits the wirelesschannel for the parameter update. In other words, the functionthat updates the learning parameters are calculated over the airas a result of the superposition property. Two models, digitaland analog, are proposed to reduce the required number ofiterations to reach an accuracy level. Simulations show thatthe analog model requires fewer iterations, hence saves bothenergy and bandwidth of the system.In [176], Amiri and G¨und¨uz extend their previous model tofading 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 thedimension of their transmit vector. The proposed model isalso compared with the model of another study that doesnot reduce the transmit vector (without scheduling) and theresults show that the compressed worker-wise scheduled ana-log DSGD model increases accuracy. In [177], the authorsremove the CSI knowledge assumption on the transmitters.Instead, the receiver is equipped with multiple antennas. Theresults show that the increasing number of antennas alleviatethe destructiveness of the wireless channel such that the infiniteantennas result in totally removed fading and noise.Zhu et al. take the same approach on federated learningin [178]; error function is minimized with each iterationand the iterations are calculated over the air. In addition toother studies, tree performance metrics are defined to analyzethe network performance and the relations between them areobtained in closed forms. Also, the network is implemented tocompare the proposed model with OFDM. The results confirmthe relationships between the defined metrics and show the lowlatency contribution of the proposed method.Yang et al. consider the same model which is based onthe over-the-air computation in [179]. The study includes userselection as used in [176] which schedules the users accordingto their channel state. As an addition, [179] considers the usageof beamforming to improve learning performance. Also, theydefine and solve the optimization problems that govern theperformance of the device selection model. Computation ofthe input data of a decentralized network is also consideredfor spectrum sensing algorithms. Different from the federatedlearning methods, the channel is matched to compute thesensing algorithm functions rather than the learning algorithmfunctions. In the following section, the studies that benefitfrom the superposition of signals for the purpose of spectrumsensing are investigated.VIII. S
PECTRUM S ENSING
Supporting a high amount of users is one of the most impor-tant challenges of wireless communication since the frequencyspectrum is limited. However, the distribution of the spectrumto the users is usually more challenging than the physicalscarcity of the channel itself [180]. The cognitive radio isa delicate approach that aims to handle the spectrum accessproblem efficiently. Cognitive radio is designed to be aware ofits surroundings with SDR capabilities, e.g. spectrum analysisor CSI estimation. The main objective of cognitive radio isto find the idle channels, that is dedicated to primary users,and utilize them to secondary users. The spectrum sensingproblem is the first aspect of this objective and attracteddistributed network solutions in the literature [181]. Some ofthe simultaneous transmission based spectrum sensing studiesare listed in Table VII with their contribution and performancemetrics.The spectrum sensing is a particular case of the detectionproblem (Section IX) and involves certain network dynamics.In a distributed cognitive radio network, multiple secondaryusers sense the spectrum and report back to an FC. A coopera-tive method that utilizes the simultaneous transmission for this TABLE VIA
N OVERVIEW OF FEDERATED LEARNING STUDIES .Author Year Contribution PerformancemetricTran et al. [172] 2019 The method provides a duality between learning time and energyefficiency. Time vs. en-ergy costYang et al. [173] Overviews the existing federated learning studies and promotes its dataaggregation aspect. ClassificationAmiri and G¨und¨uz [176] Improve the error performance of [171] by compressing the gradientestimate. AccuracyAmiri 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 aswell as promoting a trade-off between communication and learningperformance.Yang et al. [179] Reduces the convergence rate of the learning algorithm by consideringdevice selection and beamforming.Amiri and G¨und¨uz [171], [174], [175] 2019, 2020 Analog and digital DSGD reduce the learning time in bandwidth andenergy limited networks.TABLE VIIA
N OVERVIEW OF SPECTRUM SENSING STUDIES .Author Year Contribution Performance metricZheng et al. [182] 2015 The method manages to transfer the sensing data in one time slot,analyze its detection and throughput performance. Approximation error,throughputZheng et al. [183] 2017 Define an energy optimization problem and find the expression ofoptimal threshold. Energy efficiency,sensing timeChen et al. [184] 2018 Considers the effect of CFO in the sensing algorithm. Signal to aliasing andnoise ratio problem is proposed in [182]. Instead of individual sensing, themethod takes help from the decentralized nodes to detect thefree frequency bands. Spectrum sensing mechanism is basedon observed energy from the frequency channels. This energyinformation is carried simultaneously from the distributednodes to the FC as encoded to the signal energies and the finaldecision is made at the FC. The detection probability and thefalse alarm probability of the proposed method are derivedand used in order to optimize the detection performance.Also, the proposed scheme is compared with the traditionalspectrum sensing methods with respect to the approximationerror and network throughput via simulations. The authorslater consider the energy efficiency of the spectrum sensingproblem in [183]. For this reason, an optimization problem isderived that considers the sensing time, the detection thresholdand the symbol sequence length. The problem is simplified,solved and the results are verified with simulations.In [184], the proposed cognitive radio network focuseson the wideband spectrum. The objective of the method isto find the occupied wideband channels with low latencyand high accuracy. The method utilizes a distributed networkmodel to improve the sensing accuracy and benefits fromthe superposition of the signals to reduce the delay time.Specifically, the discrete Fourier transform (DFT) of the data iscomputed over the nodes and the air. The network is examinedfor the synchronization errors (e.g. synchronization phaseoffset) that occur between the nodes. Furthermore, a robustestimation and equalization technique is proposed to mitigatethe effects of the imperfect synchronization. The performanceof the proposed method is analyzed theoretically and verifiedwith both simulation and SDR based implementation. Thecomputation of functions over the air is also extended for detection and estimation algorithms to reduce the latency andimprove the error performance. In these studies, the channelis adopted to collect the data from distributed sources andprocess them simultaneously over the air for the detector orestimator. IX. D
ETECTION AND E STIMATION
Detection and estimation are essential parts of many engi-neering applications and their performance profoundly affectsthe performance of the system. While the detection andestimation theory is concerned with the optimal design of thedecision mechanisms, the acquisition of the samples is anotherimportant parameter for the whole system. In general, largersample sizes produce better decision performance, however,it requires more energy consumption and long computationtimes. For this reason, the response of the decision perfor-mance to the sample size is an important aspect to consider.The error exponent is a measure that indicates how fast theerror changes with the increasing sample size and is often usedin the analysis of asymptotic behavior of the sample size. Thisaspect is particularly important in the IoT applications that caninvolve massive sensor networks in which the observationsfrom multiple sensors together create the sample set. Forthis reason, distributed detection or estimation strategies aredeveloped to improve the performance of this process. In thissection, we present the distributed detection and estimationstudies that exploit simultaneous transmission.
A. Detection
Several detection studies based on simultaneous transmis-sion are presented in Table VIII. The distributed detectionproblem of wireless networks is considered in [185]. The network consists of spatially distributed sensors that aim totransfer 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 theirstatistics. The study examines the detection performance ofthe proposed method from the perspective of the number ofmeasurements and power consumption. Two sensor types areconsidered for the review; intelligent and dumb. The intelligentsensors are aware of the source statistics and transfer the log-likelihood ratio (LLR) to the FC. It is shown that the LLRbased detection is asymptotically optimal, i.e. it can reach theperformance of the centralized detection. The dumb sensorsare unaware of the source statistics and transfer the histogramof their measurements to the FC, however, source statisticsare required at the FC. The results show that the histogramtransfer is also asymptotically optimal. The results are alsoverified with simulations that show the detection error as afunction of the number of sensors.Sensor measurements of a wireless sensor network are usedfor target detection in [16]. The sensor measurements aretransferred to the test center via type-based multiple accesswhere the same frequency and time resources are allocated toall users. In TBMA, the receiver only accesses to the histogramof the transmitted data (we refer the reader to Section II fordetailed information) and the detection mechanism is basedon the histogram of the observations. The study focuses onthe performance analyses of TBMA based detection schemes(includes a large deviations approach). An asymptoticallyoptimal detection mechanism is proposed and its detectionerror exponents are derived. Channels with i.i.d. and non-i.i.d. gains are considered for the evaluations. Also, the errorprobabilities of TBMA scheme is compared with TDMA viasimulations.In [187], the existence of a signal is checked with a binaryhypothesis test where the observation samples are obtainedfrom a decentralized sensor network. The sensor readings arecollected in the FC via two-channel models; each sensor hasits own dedicated channel or a single channel is dedicatedto all sensors. The second model that dedicates a singlechannel to the sensors attracts our interest since it allows thesuperposition of the signals. The study focuses on the detectionperformance of these two-channel models and their compar-ison with the centralized detection model. In the analyses,two Gaussian noise cases (correlated and independent) andtwo power constraint cases (average and total) are considered.Bayesian error exponents of the detection probabilities are de-rived for these scenarios (includes large deviations approach).Several results are presented in the study and verified viasimulations that examine the effect of the number of sensors onthe detection error probability and error exponent. The resultfor the average power constraint case is that the superimposedsignals with the correlated Gaussian noise can reach the errorperformance of the centralized detection scenario. However,dedicating channels to each sensor always leads to errorperformance reductions. For the total power constraint case, anincreasing number of sensors exponentially reduces the errorexponent 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 thesignal detection at each distributed node and transfers the testresults to the FC. The second scheme, modified amplify-and-forward MAF, transfers the observations to the FC before thehypothesis testing. The study examines the detection perfor-mance of these schemes under individual power constraintand total power constraint. The results show that the MAF isasymptotically (i.e. infinite number of sensors) optimal underindividual power constraint, however MDF is not optimal.In [189], multiple antennas at the FC is considered for thedistributed detection problem. The upper and lower boundsof the error exponents are derived for the scenarios: AWGNchannels, Rayleigh channels, full CSI enabled, phase-only CSIenabled. It is shown that equipping the FC with multipleantennas presents a gain of π/ for the scenario of Rayleighchannel and full CSI. Four algorithms are proposed for thedesign of the sensors’ power allocation. Lastly, the errorexponents of these algorithms are compared with the derivedbounds via simulations.An anomaly detection algorithm is proposed in [186] whichis based on the AFC given in [141]. A supervised learningalgorithm is used for the hypothesis testing and the classifierof the algorithm is generated at the FC with the sensorreadings. Since the FC is only interested in a function of thesensor readings, AFC is proposed for the classifier generationover the channel. The main contribution of the study is itsenergy efficiency and it is shown that the proposed schemecan significantly reduce the consumed energy compared tothe TDMA scheme. Later, in [190], the study on anomalydetection is extended to include the upper bounds on theprobability of mislabeling. B. Estimation
A list of the estimation studies that benefit from the super-position of signals can be found in Table IX. An estimationmethod that is based on the TBMA scheme is proposed in [3]. The method benefits from the superposition of the sensorreadings over the channel to gain from the bandwidth and thedelay time. The study aims to design the optimum estimationprocess that includes the transfer of the samples to the FC andthe estimator. The study derives the Cramer-Rao bound of theestimation problem and designs the estimation process as thecombination of TBMA and an ML detector. The results showthat the TBMA based estimation is asymptotically optimal ifthe channel gains of all sensors are the same. Additionally,the proposed model is analyzed for the fading channels andcompared with the TDMA based schemes via simulations.In [191], power and bandwidth limited networks are con-sidered for the distributed estimation of unknown signals. Theauthors especially propose a power scheduling model andinvestigate the cases where the observations are scalar andvectorial. The optimization problem for power scheduling isshown to be convex and solved for the scalar observations.Additionally, it is shown via simulations that the proposedscheduling generates better MSE performance than the uni-form scheduling. The optimal scenario to estimate the vectorial TABLE VIIIA
N OVERVIEW OF DETECTION STUDIES .Author Year Contribution Performance metricLiu and Sayeed [185] 2007 A low-complexity distributed detection mechanism (based on type function) isproposed and investigated for the error exponent. Error probabilityMergen et al. [16] 2007 Provides an asymptotically optimal detector and analyzes its error exponentperformance. Error exponentLi and Dai [187] 2007 Proposes a bandwidth and delay efficient method and compares with theseparation based methods.Li et al. [188] 2011 Analyzes and compares the MDF and the MAF methods for error exponentsunder 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, energyconsumptionRaceala-Motoc et al. [190] 2018 Gives the upper-bounds for the probability of mislabeling for [186]. Probability of mis-labelingTABLE IXA
N OVERVIEW OF ESTIMATION STUDIES .Author Year Contribution Performance metricMergen and Tong [3] 2006 Analyzes the asymptotic behaviour of TBMA based estimation. MSEXiao et al. [191] 2008 Propose a bandwidth and power efficient method and defines and solves anoptimization problem for power scheduling. MSEBajwa et al. [192] 2007 Proposes a energy efficient joint source-channel based method and defines therelationship between its power consumption, error rate and latency. Power, distortion,scaling exponentsWang and Yang [193] 2010 The method removes the CSI requirement by using more bandwidth. MSE, bandwidthBanavar et al. [194] 2010 Considers and investigates different fading models and their effect on theperformance. Asymptoticvariance observations is investigated for both noiseless and noisy chan-nel models. The results for the noiseless channel model aregiven in closed form; however, the solution of the optimizationproblem for the noisy channel required semidefinite relaxationmethods.A joint source-channel communication structure is consid-ered for decentralized wireless sensor networks in [192]. Theobjective of the study is to build a communication infrastruc-ture that reduces bandwidth and power consumption. For thispurpose, optimization of the acquisition, communication andprocessing of the measured data is considered collectively.The relationships between the power consumption, error rateand the latency of the network are derived for the increasingnumber of sensors and verified with simulations. It is shownthat the efficient and healthy estimation of the unknown signalsis possible for a large number of sensors if prior knowledge isallowed in the proposed approach. Moreover, the method stillproduces healthy estimation results when partial or no priorknowledge is given if the power and latency constraints areincreased sublinearly as a function of the number of sensors.A TBMA based estimation method is proposed in [193] fordistributed WSNs. In traditional TBMA schemes; the fadingof the wireless channel notably degrades the communicationperformance and providing CSI to each node via pilot trans-mission or feedback channel brings excessive burden for alarge number of sensors. The proposed method aims to providerobustness against these problems by partitioning additionalbandwidth to the network. The main idea behind the modelis to use multiple orthogonal waveforms for each message(type) where the traditional TBMA uses only one. The authorsalso designed a maximum likelihood estimator for the systemand derived the Cramer-Rao lower bound expression. The simulations compare the MSE performance of the method withtraditional TBMA and illustrate the relationship between thebandwidth, SNR and the number of nodes.Fading characteristics of the channel and the amount ofallowed CSI at the network is highly important for the per-formance of the distributed sensor networks that exploit thesuperposition of the signals. In [194], various fading modelsand different CSI amounts are tested for distributed estimationnetworks. The variance expression for the network’s estimateis derived for the perfect and partial CSI cases. Also, differentchannel characteristics are considered and the correspondingvariance expressions are derived for a large number of sensors.The authors consider the impact of the errors that occur atthe feedback process in their evaluations. The convergencerates are calculated for several scenarios. It is shown thatthe asymptotic results can be nearly reached with a practicalnumber of sensor nodes. The results are verified with computersimulations. Simultaneous transmission is an effective tool forthe consensus algorithms that reduce the convergence time ofthe consensus process. In the following section, the studiesthat exploit the superposition property of the W-MAC arepresented. X. G
OSSIP AND C ONSENSUS
The physical layer is generally exploited for many-to-onenetworks where an FC requires the message of multipleusers ( K × ). However, in consensus problems, all usersin 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 consensusproblem that provides fast and reliable convergence, whilerequiring low-energy, is a challenging task [195]. Gossiping is a distributed form of consensus that the nodes locallycommunicate with their neighbors instead of following anetwork-wide protocol. It has been shown that simultaneoustransmission presents opportunities for the improvement ofgossip and consensus algorithms. A list of existing consensusstudies is given in Table X. In this section, we examine theconsensus studies that exploit the wireless channel to provideenergy and time efficiency.The average consensus problem of wireless networks isconsidered in [196]. The method exploits the physical layerwith the simultaneous transmission in order to update theconsensus algorithm. The results show that each node in theproposed network can obtain the average of the data undera sufficient MSE level. Since the channel is accessed by thenodes simultaneously, the method presents better convergenceperformance compared to the conventional consensus algo-rithms as the number of nodes increase.In [197], Nazer et al. use the computation coding in a gossipalgorithm that averages the sensor readings. The method isbased on the simultaneous transmission of the nodes byexploiting the superposition property of the physical layer.The network is arbitrarily divided into local neighborhoodssuch that the average consensus is initially provided for eachof neighborhood, later the global consensus is achieved forthe network. The results reveal that the energy efficiency ofthe network increases exponentially and the time efficiency in-creases polynomial as the number of nodes increase. In [198],the previous studies are extended and compared with thenearest neighborhood gossip algorithms via simulations. Also,it is shown that the proposed method converges in O ( n /m ) rounds while the nearest neighborhood algorithm convergesin O ( n ) rounds where n and m are the network and localneighborhood sizes respectively.The averaging gossip problem of a wireless networkis extended to nomographic functions in [199]. The mainidea behind the study is the analog computation of thenomographic consensus functions over the wireless channelwith the concurrent transmission. The proposed model iscluster/neighborhood-based as in [197] and [198] such thatthe consensus is initially provided for local clusters rather thanthe whole network. Different from [197] and [198], the globalconsensus is obtained with the existence of the common nodeswhich connect the clusters with each other. Two algorithms,deterministic and randomized, are proposed and it is shownthat both of them significantly increase the convergence rate.Another gossip algorithm that exploits the physical layeris proposed in [200]. The novelty of the study lies in thefull-duplex communication model of the network which al-lows simultaneous transmission and reception of signals withspecial self-interference cancellation techniques. Each nodein the network broadcasts to its neighbors and receives fromthem simultaneously and only requires synchronization. Thetheoretical results show that the proposed gossip algorithm hasa three times faster convergence rate than the randomized gos-sip algorithms. Also, the study includes computer simulationssupporting the presented theoretical results.In [201], the average consensus of a wireless network isconsidered. Although many of the simultaneous transmission based consensus algorithms are neighborhood/cluster-based(as in [197]–[199]), this study arbitrarily divides the networkinto two subgroups rather than many. In each iteration, all thenodes of a subgroup simultaneously broadcast their data whilethe nodes in the other subgroup receive the superimposedsignal. After an iteration, the subgroups change roles such thatthe receiver subgroup becomes the transmitting subgroup andvice versa. The nodes in the transmitting subgroup create theirnew data as a weighted sum of the received signal and theoriginal data. The proposed method offers energy efficiencyand significant convergence performance as the simulationsillustrate the comparison of the proposed method with therandomized and the broadcast gossip algorithms.The superposition of the transmitted signals is also consid-ered in [202]. The study uses similar grounds that are usedin [199]. [202] also includes unknown channel coefficients inthe 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 tocontrol the convergence rate. The results show that the smallervalues of the tuning parameter (stubborn systems) reduce theeffect of the time-variant channel coefficients and increasethe convergence time. However, the tuning parameter onlyaffects the convergence rate in the time-invariant systems. Thestudy also verifies the theoretical results via simulations. Amax consensus scheme ( ScalableMax ) is proposed in [203].The method focuses on the dense networks where a largenumber of nodes aims to find the maximum function. Themain contribution of the proposed method is its scalability.Moreover, an error correction mechanism is added to thesystem in order to increase the system’s resilience to thelow SNR region. The CP&F and AFC also inspired securityapplication in the literature. In the following section, wepresent the security applications that benefit from simultaneoustransmission. XI. S
ECURITY
Superposition of signals is shown to be beneficial to providesecurity to wireless networks. Several security studies thatuse simultaneous transmission are listed in Table XI. In thissection, we present the security applications in detail.The CP&F method is utilized for the security purposesin [204]. The study considers a two-transmitter network thatcommunicates with a curious relay using CP&F over theAWGN channel. The security of the method is confirmed byinvestigating the mutual information between the individualmessages and the superimposed signal at the relay. It is shownthat the mutual information is significantly small for largeblock lengths and gives insufficient information to the relay.The authors also derive the achievable rates and the necessaryconditions for secure communication. Lastly, the network isgeneralized for the multi-hop networks.Another secure communication scheme based on the CP&Fis given in [205]. In the study, an asymmetric CP&F model(as in [71]) is considered which assumes asymmetric channelgains towards the eavesdropper and the secure sum rates arederived. TABLE XA
N OVERVIEW OF CONSENSUS STUDIES .Author Year Contribution Performance metricKirti et al. [196] 2007 Provides a scalable consensus algorithm that the convergence time isindependent of the network size. MSEGoldenbaum et al. [199] 2012 The consensus of multiple nodes with respect to nomographic functions isestablished 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 exponentialto the network size. Number ofgossip roundsNazer 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 networksize. Averaging timeMolinari et al. [202] 2018 Robust to fading and provides fast convergence. Convergence timeAgrawal et al. [203] 2019 Provides robustness to noise and low SNR for max-consensus networks. Error rate, numberof iterationsTABLE XIA
N OVERVIEW OF SECURITY STUDIES .Author Year Contribution Performance metricShashank and Kashyap [204] 2013 Achieves strong secrecy with lattice codes and randomized encoding. Achievablepower-rate pairVatedka 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 securesum rates. Achievable sum rateRichter et al. [206] 2015 Includes fading and provides weak secrecy for multi-way relay networksusing CP&F. Secrecy capacityKarpuk and Chorti [207] 2016 Derive the secrecy rate upper bounds, investigates the effect of synchroniza-tion error and extends the proposed model to MIMO sceheme.Ren et al. [41] 2017 Investigates internal and external eavesdropper scenarios, includes two-hopchannel, jammer and improves the secrecy rate.Goldenbaum et al. [208] 2016 Defines and derives the secrecy computation-capacity and shows that it canbe achieved without sacrificing capacity. Secrecycomputation-rateGoldenbaum et al. [209] 2016 Secure against the eavesdroppers that has good channel conditions andapplicable with low-complexity.Babaheidarian et al. [211] 2017 Considers the scenario of malicious receiver and provides secrecy in thisscenario with beamforming and jamming. Achievable securerate.Hynek Sykora [212] 2015 Uses game theory to provide secrecy. Pay-off matrixNegi and Goel [213] 2016 Provides secrecy by using artificial noise for networks with multiple antennaor helper nodes. Secrecy capacity,outage probabilityAltun [214] 2020 Obtains individual information at the FC and provides authentication to thetransferred messages. Probability of detec-tion and false alarm
A comprehensive study on secure communication is givenin [8] which is based on the CP&F over bi-directional relaychannels. Two nodes in the proposed model aim to commu-nicate with the help of a relay without leaking information tothe relay. Perfect and strong secrecy conditions are defined toevaluate the secrecy performance. It is shown that the proposedlattice coding and the CP&F provide both strong and perfectsecrecy even in a noiseless design. Also, the results reveal thatthe noise only affects the computation performance. Lastly, thecomputation rate under the Gaussian noise is derived. Anothercomprehensive secure communication study is given in [206].In addition to [8], this study considers single-input-multiple-output (SIMO) multi-user multi-way networks and includesfading to the channel model. The study derives the secrecyregion under the weak secrecy condition. The results showthat the securely achievable sum-rate is equal to the differencebetween the computation rate and the MAC capacity. Thesimulations verify the derived results. Also, the proposedmodel is compared with the traditional insecure CP&F andthe security schemes of [8] from the secrecy rate perspective.Secure communication of two nodes through a relay nodeis considered in [207] with the help of PLNC. The study con- siders perfect secrecy conditions and defines the upper boundson the achievable secrecy rate for noiseless channels. Differentcoding algorithms are designed for scenarios where the nodesare alone or they cooperate. Moreover, the achievable ratesare calculated for these scenarios and it is shown that thegiven algorithms reach close to the calculated upper bounds.Lastly, the study is extended to a scenario, where the nodesare equipped with multiple antennas.In [208], the secure communication of multiple nodes isaimed in a modulo- adder network. The objective is toleak no information to the eavesdropper while the legitimatereceiver obtains the modulo- sum of the transmitters. Thestudy defines the secrecy computation communication capacitywhich is the maximum achievable secure computation rateand it is shown that the secrecy computation communicationcapacity is the same as the computation capacity under certainconditions. In other words, the proposed scheme can achievethe security constraints without reducing the capacity.Secure function computation based on the AFC over awiretap channel is considered in [209]. The study aims to keepthe eavesdropper ignorant and contributes to the complexitysuch that the method can provide security without the need for additional stochastic encoding. Also, the study assumes noadvantage over the eavesdropper e.g. better channel quality.In [41], a modified CP&F given in [210] is exploited for thepurpose of secure communication. Two scenarios where theadversary is external and internal (the relay) are considered.In the internal case, the relay is assumed to be curious thateavesdrops the received signal and the destination node isassumed to be cooperative by jamming the relay to preventinformation leakage. The random binning and the lattice chaincodes are used in the proposed scheme. It is shown that theproposed methods can achieve the secrecy capacity at thehigh SNR regions. Moreover, the results revealed that theobtained secrecy capacity is close to the channel capacity.Specifically, the security constraints only lower the insecurechannel capacity by / bits per channel use. Cooperativejamming is also applied in [211]. Secure communication frommultiple sources to multiple receivers is aimed with the helpof multiple relays. Contrary to [41], the relays are assumed tobe trustworthy and cooperative in the way that the relays usebeamforming to jam the malicious receivers.[212] considers a scenario such that the relays of a PLNCnetwork can be malicious and intentionally use a deceptivemapping. The authors view this problem as an incompleteinformation game and derive an equilibrium. In [213], Negiand Goel propose a secure transmission model that uses arti-ficial noise. The proposed model intentionally introduces ANto the wireless channel in order to degrade the eavesdropper’schannel. Naturally, the added AN also degrades the legitimatereceiver’s channel. This problem is solved by separating theAN source from the signal source. The authors propose twomethods for the separation; using other antennas (multipleantenna) or other users (helpers). In both methods, AN andsignal source transmit simultaneously at the same time andfrequency slot. However, AN source chooses the noise vectorsuch that the noise and the channel coefficient of the legitimatereceiver cancel each other. In other words, AN lies in thenull space of the channel coefficient of the legitimate receiver.The paper makes two critical assumptions. Firstly, transmittersperfectly know the receiver’s channel. Secondly, legitimatereceiver’s and the eavesdropper’s channels are uncorrelated,which enables AN to degrade any channel other than thelegitimate receiver’s. Presented methods are also supported andcompared with simulations.An authentication method is proposed in [214]. The authorsconsider an uplink communication scenario where multipleusers simultaneously transmit their information to a receiver.Contrary to the conventional AFC studies, this method enablesthe reconstruction of individual data at the receiver. For thispurpose, unique pre and post processing functions that arebased on Gaussian primes are proposed. As a result, themethod can simultaneously transfer the individual data of mul-tiple users to the receiver. More importantly, the authors arguethat the post -processing function also enables the receiver toauthenticate the legitimate transmitters. The reason behind thisargument comes from the uniqueness of the fading coefficientsbetween any two points and the uniqueness of the Gaussianprimes. The study firstly proves the feasibility of the proposedapproach with numerical results and later investigates the security aspect of the method by simulating the probabilityof detection and false alarm metrics.XII. C ONCLUSION
The simultaneous transmission based communication tech-niques are the objective of this study. An extensive literatureoverview is presented on the wireless applications that exploitthe interference of signals. These methods are grouped de-pending on their application areas and presented with detailedinformation. The wireless channel presents a natural weightedsummation operation for the simultaneously transmitted in-puts. This summation is exploited in various applications toperform desired tasks. From this perspective, simultaneoustransmission techniques can be viewed as a channel manip-ulation that performs a specific function over the channel.The survey investigates these functions and their applicationssuch as multiple access, network coding, detection, securityetc. The studies in each application are listed along with theircontributions and performance metrics.
A. Challenges and Open Areas
There are various challenges in simultaneous transmissionnetworks that should be addressed and open to improvements.Most of the studies assume perfect CSI knowledge at bothtransmitters and receivers. Moreover, some of these studies arehighly vulnerable to channel estimation errors which preventimplementation of these methods. For this reason, efficientand accurate CSI acquisition is essential and an open issuefor simultaneous transmission methods. Various applicationsalso require time synchronization among transmitting nodes.The lack of perfect synchronization and CSI acquisitionmethods also effect existence of testbed implementation stud-ies. Although there are AFC implementations via SDR mod-ules, the literature lacks testbed implementation studies onvarious applications such as network coding, multiple accessor security.Scalability and power constraints are other challenges forwireless networks. Although simultaneous transmission basedmethods improve scalability compared to the conventionalpairwise methods, increasing network size still degrades thesystem performance in most applications. Moreover, the net-work size is also limited by the power constraints of thewireless channel.Despite the challenges of the wireless environment, thesimultaneous transmission techniques can bring scalability,security, bandwidth efficiency, low-complexity or low-latencyfor distributed networks. The future of communication isdirected at the distributed networks, hence the simultaneoustransmission can become a key component in future networks.The federated learning studies are already prominent examplesthat support this claim. Various application areas are still opento the improvements of simultaneous transmission. We believethat the hand-over process of mobile networks can be im-proved with simultaneous transmission. Moreover, implemen-tation of simultaneous transmission methods for heterogeneousnetworks is another open area that is worth investigating. R EFERENCES[1] C. E. Shannon, “A mathematical theory of communication,”
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