1 A Survey on Over-the-Air Computation Alphan SahinMember IEEE and Rui YangyMember IEEE

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A Survey on Over-the-Air Computation

Alphan ¸Sahin∗,Member, IEEE and Rui Yang†,Member, IEEE

Abstract—Communication and computation are often viewed

as separate tasks. This approach is very effective from the

perspective of engineering as isolated optimizations can be per-

formed. However, for many computation-oriented applications,

the main interest is a function of the local information at

the devices, rather than the local information itself. In such

scenarios, information theoretical results show that harnessing

the interference in a multiple access channel for computation,

i.e., over-the-air computation (OAC), can provide a signiﬁcantly

higher achievable computation rate than separating communi-

cation and computation tasks. Moreover, the gap between OAC

and separation in terms of computation rate increases with more

participating nodes. Given this motivation, in this study, we

provide a comprehensive survey on practical OAC methods. After

outlining fundamentals related to OAC, we discuss the available

OAC schemes with their pros and cons. We provide an overview

of the enabling mechanisms for achieving reliable computation

in the wireless channel. Finally, we summarize the potential

applications of OAC and point out some future directions.

Index Terms—Over-the-air computation

I. INTRODUCTION

Over-the-air computation (OAC) refers to the computation

of mathematical functions by exploiting the signal superposi-

tion property of wireless multiple access channels. The distinct

feature of OAC is that the local data at the edge devices

(EDs), such as smartphones, laptops, tablets, vehicles, or

sensors, are not acquired over orthogonal channels to perform

a computation task at a fusion node, e.g., an edge server (ES)

at a base station or an access point. Instead, the computation

is handled by harnessing the interference via simultaneous

transmissions. For example, suppose that the goal is to evaluate

a function f(s1,..., sK)at an ES, where skis the symbol

at the kth ED. With the separation of communication and

computation tasks, the function is computed at the fusion node

after each symbol is received via orthogonal or non-orthogonal

resources (i.e., orthogonal multiple access (OMA) and non-

orthogonal multiple access (NOMA)), as illustrated for time-

domain multiple access (TDMA) in Fig. 1(a). On the other

hand, with OAC, the function is intended to be computed

through signal superposition in the channel as shown in

Fig. 1(b). In this example, the key observation is that if the ES

is not interested in the local information but only in a function

of them, OAC paves the way for reducing resource usage,

which otherwise scales with the number of EDs. Hence, it is

a fundamental and disruptive concept to the traditional way of

handling computation and communication tasks independently.

The idea of function computation over a multiple access

channel was ﬁrst thoroughly analyzed in Bobak’s pioneering

work in [1] and the theoretical limits of computation over

Alphan ¸Sahin and Rui Yang are afﬁliated with the University of South

Carolina, Columbia, SC and InterDigital, New York, NY, USA, respectively.

E-mails: asahin@mailbox.sc.edu∗, rui.yang@interdigital.com†.

Simultaneous

transmissions

(1, … )

1



Compute

…

Decoder

12

…

Bandwidth





Occupied time

2

Bandwidth

Occupied time

Simultaneous

transmissions

1





ED 1

ED 2

ED 

…

1

Encoder

2

Encoder



Encoder

ED 1

ED 2

ED 

…

1

Encoder

2

Encoder



Encoder

Decoder

CommunicationComputation

Over-the-air computation

(a) Separation of communication and computation. Computation occurs after

receiving the arguments of the function at the ES.

Signal

superposition

(1, … )

1



Compute

…

Decoder

12

…

Bandwidth





Occupied time

2

Bandwidth

Occupied time

Simultaneous

transmissions

1





ED 1

ED 2

ED 

…

1

Encoder

2

Encoder



Encoder

ED 1

ED 2

ED 

…

1

Encoder

2

Encoder



Encoder

Decoder

CommunicationComputation

Over-the-air computation

(b) Computing function by using the signal superposition property of multiple

access channels via simultaneous transmissions.

Fig. 1. Separation of communication and computation versus OAC.

multiple access channels were investigated for a ﬁxed many-

to-one function. In [2], Goldenbaum made the ﬁrst connection

between nomographic functions and OAC. In [3] and [4],

it was shown that OAC can provide a signiﬁcantly higher

achievable computation rate than separating communication

and computation. Given the promising information theoretical

results, OAC has been drawing more and more attention in the

literature. Initially, it has been applied to communication prob-

lems in the interference channel, e.g., physical layer network

coding [5], [6], compute-and-forward relaying strategy [7], and

wireless sensor networks (WSNs) to address the issues like

acceleration in gossip networks [8] and several computation

tasks [9], [10]. With the increased interest in applications

that require heavy computation, it has recently been utilized

in multi-disciplinary ﬁelds such as machine learning over

wireless networks [11], wireless control systems [12], and

computing frameworks like wireless data centers [13] and

wireless intra-chip computations [14].

The exciting applications have led to the investigation of

OAC from various perspectives and resulted in a wide-variety

of computation strategies. This paper aims to discuss these

OAC schemes without losing the mathematical rigor and how

these methods hamd;e the challenges such as the detrimental

impact of wireless channels on computation, synchronization

errors, maintaining accurate and fresh channel state informa-

tion (CSI) at the radios, security, and hardware impairments

such as power ampliﬁer non-linearity.

arXiv:2210.11350v5 [cs.IT] 2 Apr 2023

A. Relation to other surveys and our contributions

The reader can ﬁnd relevant discussions on distributed

inference over sensor networks in [15]. The methods relying

on compute-and-forward relaying scheme and uncoded strate-

gies for physical layer network coding are comprehensively

discussed and compared in [16], [17]. To reduce the per-round

communication latency for the implementation of distributed

learning over a wireless network, OAC has been used in many

recent works as an enabler. We refer the readers interested

in wireless systems for machine learning in general to the

excellent survey papers in [18]–[25] and the references therein.

In [24], federated edge learning (FEEL), i.e., implementation

of federated learning (FL) [26] over a wireless network, and

the resource management for FEEL are surveyed. In [11],

several exciting applications of OAC and research directions in

this area are discussed without mathematical details. In [27],

semantic communication is thoroughly surveyed and OAC is

mentioned as one of the task-oriented semantic communica-

tion paradigms. In [28], [29], OAC is particularly analyzed

from the perspective of integrated sensing, communication,

and computation. In [30], over-the-air distributed computing

for artiﬁcial intelligence applications is envisioned for 6G

wireless networks. In [31], the particular interest is in the

applications that enjoy signal superposition in general. Besides

OAC, the topics such as NOMA, interference alignment,

multiple antenna systems, security, and spectrum sensing are

investigated. In [32], the design of aeronautical networks

with computation paradigms such as edge computing and off-

loading are surveyed. We also acknowledge the reference [33]

which discusses the OAC from the perspective of various

network architectures and provides an excellent survey on the

OAC based on multiple antennas at the devices.1

The main focus of this study is to investigate how to com-

pute a function over a wireless network reliably and efﬁciently.

Our priority is to form a composition that can provide a relative

comparison of the state-of-the-art OAC techniques with pros

and cons, particularly from the perspective of the physical

layer of communication systems. Since a wide variety of

applications can beneﬁt from the OAC, in this study, we focus

on the computation itself, rather than a particular application.

We seek answers to three main questions:

1) What functions can potentially be calculated with OAC?

To answer this question, we review the nomographic

functions that appear in both mathematics and commu-

nication literature.

2) What are the OAC schemes in the state-of-the-art and

their trade-offs to deal with the distortion in wireless

channels? To address this question, we ﬁrst give a

general system model along with fundamental metrics

on OAC. Under this framework, we evaluate the methods

based on how they achieve computation under the fading

channel and the encoding strategies.

1Our paper and [33] are independently developed and compensate each

other from the perspective of classiﬁcations of available OAC approaches.

The corresponding pre-prints were listed on arXiv.org one day apart (October

19, 2022).

3) What are the mechanisms that play a role in achieving

a reliable OAC? To answer this question, we review the

impacts of synchronization impairments, power manage-

ment, and channel estimation on OAC and elaborate on

security aspects and computation architectures for OAC.

Finally, we provide an overview of the applications of OAC

in the literature and point out the potential areas that can be

improved for OAC.

Organization: The rest of the study is organized as follows.

In Section II, we provide an overview of the fundamentals

and discuss the functions that can potentially be computed

via OAC. In Section III, we discuss the state-of-the-art OAC

schemes, comprehensively. In Section IV, we discuss the

enabling mechanisms to achieve a reliable computation. We

summarize the potential applications of OAC in various ﬁelds

in Section V. We ﬁnalize our discussions with various topics

that need to be investigated further in Section VI.

Notation: The complex and real numbers are denoted by

Cand R, respectively. The K-times Cartesian product of

space Ais shown as AK.F(A)represents the space of every

function that maps Ato R.Edenotes the unit interval [0,1].

E{·} denotes the expectation over all random variables. The

function sign (·)results in 1,−1, or 0for a positive, a negative,

or a zero-valued argument, respectively. The symbol ~denotes

linear convolution. The function I[·]results in 1if its argument

holds, otherwise it is 0. Pr (·)is the probability of an event. The

zero-mean multivariate complex Gaussian distribution with the

covariance matrix CMof an M-dimensional random column

vector x∈CMis denoted by x∼ CN(0M,CM).N(µ, σ2)

is the normal distribution with the mean µand the variance

σ2. The trace of a matrix is denoted by tr{·}. The continuous

uniform distribution is denoted by U[a,b], where aand bare the

minimum and the maximum values, respectively. The function

log+

2(x)is deﬁned as max(log2(x),0). Kronecker delta is

expressed as δij .

II. WHAT CAN BE CALCULATED WITH OAC?

OAC aims to compute a multivariate function by relying on

its representation that can structurally match with the underly-

ing operation that multiple access channel naturally performs.

In wireless communications, multiple access channels are

modeled with additive property, i.e., the signal superposition.

With this property, the OAC problem boils down to the repre-

sentation of a target function with a special function, called a

nomographic function, or a set of nomographic functions over

multiple wireless resources. These functions are called nomo-

graphic because they are inline with the nomographs that solve

certain equations through some graphs, i.e., analog computing.

A well-known example of a nomograph is the Smith chart

which assists in solving problems related to transmission lines.

While the nomographs allow quick and accurate computations,

the use cases of nomographs diminished historically due

to the effectiveness of digital computers. Nevertheless, the

fundamental theories about nomography are intricate, arguably

connected to the neural networks, and pave the way for

addressing the scenarios where digital computation suffers

from latency, power consumption, and limited-communication

bandwidth. In this section, we discuss the preliminaries on

nomographic functions to reveal what can be calculated with

OAC.

A. Preliminaries

Deﬁnition 1 (Nomographic function [2], [34]–[36]).Let SK,

K≥2, be a compact metric space. A function f:SK→R

for which there exist functions ψk∈ F(S),k∈ {1,..., K},

and ϕ∈ F(R)such that fcan be represented as

f(s1, s2, . . . , sK) = ϕ K

k=1

ψk(sk)!,(1)

is called nomographic function and N(SK)is the space of

nomographic functions with the domain SK.

The functions ψk,∀k, and the function ϕare further

called pre-processing functions (or inner functions) and post-

processing function (or outer function), respectively. Equation

(1) reveals why a nomographic function is relevant to OAC:

Equation (1) can be interpreted as an evaluation of the function

fin an ideal uplink (UL) channel (i.e., no noise, no multi-

path channel distortion), where skand ψkare the symbols

and the pre-processing functions at kth data-generating node,

respectively, the sum of the signals from Knodes corresponds

to the superposition that naturally occurs in the channel, and

ϕis the post-processing function at the fusion center. To

the best of our knowledge, this connection is ﬁrst made in

Goldenbaum’s work in [2], [34]–[36] while the non-linear

function examples in the form of (1) appear in [37]–[39]

without discussing the family of nomographic functions.

It is worth noting that the compactness mentioned in

Deﬁnition 1 is an important assumption, especially in the

analysis of continuous functions. For example, the range of

a continuous function f(s1, s2, . . . , sK)on a compact space

SKis compact. Since the function is bounded, one can ensure

that the limits exist, or that suprema and inﬁma are taken by

the function. If the space is not compact, it can be harder to

analyze the behavior of a given function and more structural

properties related to the function need to be known. From the

perspective of OAC, compactness is inherited due to practical

limitations. For instance, the measure space of a sensor is

typically compact because a sensor can quantify values in a

ﬁnite closed interval, e.g., 0◦C≤sk≤100◦C, ∀k. Hence, to

make general statements about entire function spaces and not

only about speciﬁc examples, the space SKin Deﬁnition 1 is

considered to be compact.

Now, let us denote the space of nomographic functions,

the space of nomographic functions with the restriction of

continuous pre- and post-processing functions, and the space

of continuous functions with the domain EKas N(EK),

N0(EK), and C0(EK), respectively.2Sprecher and Buck pro-

vide insights into the representation of a function f∈ C0(EK)

as a nomographic function as follows:

2Nomographic functions in mathematics are often investigated by deﬁning

the compact space Sas E.

Theorem 1 (Sprecher’65 [40]).Every function f∈ C0(EK)

can be represented with real, monotonic increasing pre-

processing functions and possibly a discontinuous post-

processing function.

Theorem 2 (Buck’79 [41]).Every function f∈ F(EK)is

nomographic (i.e. N(EK) = F(EK)).

The key idea for the proof of Theorem 2 is to show there

exists a one-to-one mapping from EKto a space Γ⊂Rin the

form of g(s1,..., sK) = PK

k=1 ψk(sk). Given the existence

of such g(therefore, the pre-processing functions exist), the

post-processing function can then be expressed as ϕ(x) =

f(g−1(x)), where g−1is the inverse function that maps x∈Γ

to (s1,..., sK). Without any restriction on the pre-functions

and the post-processing function, such a map can be obtained

by choosing Γ = Eand constructing the binary representation

of x∈Γby uniformly interleaving the digits of the binary

representations of the symbol sk,∀k(see [41, p. 287] and

[42, p. 2]). For this speciﬁc constructive proof, ψkrelies on

reading the binary representation of skin base 2K, which

implicitly causes discontinuity in its range. The proof also

shows the existence of special nomographic functions with an

interesting property:

Deﬁnition 2 (Universality).The pre-processing functions are

universal if they are ﬁxed and can be used to calculate every

function in F(EK).

The universality is a desirable property for OAC because

the pre-processing functions do not need to be re-designed

(i.e., less communication overhead) if the target function

changes over time. This property is exploited in [2], [34]

for multi-cluster computation as discussed in Section IV-C.

It is also mentioned that universality provides robustness

against changes in network topology (via dropping and joining

devices) in the sense that transmitting nodes do not need to

adapt their pre-processing functions.

If one desires the pre- and post-processing functions to be

continuous for an arbitrary continuous function f, Theorem 2

is unfortunately not valid:

Theorem 3 (Buck’82 [43]).N0(EK)is nowhere dense in

C0(EK).

A canonical example of Theorem 3 is geometric mean,

i.e., f(s1, s2, . . . , sK) = (Qksk)1

K. This function cannot be

represented as ϕ(Pk=1 ψk(sk)) with the continuous functions

ψ1,..., ψK, ϕ on Eas demonstrated for K= 2 by Arnold

[44] and for an arbitrary Kby Goldenbaum [2]. Theorem 3

implies that there exist inﬁnite number of continuous functions

in C0(EK)that cannot be approximated with a nomographic

function in N0(EK)for a given arbitrary precision. Kol-

mogorov remarkably addresses the issue of representing a

continuous function with a set of nomographic functions in

N0(EK):

Theorem 4 (Kolmogorov’57 [45]).Every function f∈

C0(EK)can be represented as the superposition of at most

2K+ 1 nomographic functions in N0(EK), i.e.,

f(s1, s2, . . . , sK) =

2K+1

`=1

ϕ` K

k=1

ψk`(sk)!,(2)

where the post-processing functions ϕ`depend on fand the

functions ψk` are independent of f.

Geometrically, the 2K+ 1 inner sums in (2)

ensure the existence of a continuous and bijective

correspondence between (s1, . . . , sK)∈EKand

(ϕ1(Pk=1 ψk`(sk)),..., ϕ2K+1(Pk=1 ψk`(sk))) ∈R2K+1.

Hence, the inner sums describe a homeomorphism that

continuously embeds EKinto R2K+1. In [46], Sternfeld

enhances the statement of Theorem 4 by showing that the

2K+ 1 nomographic functions in (2) cannot be reduced to

represent every f∈ C0(EK). Hence, from the perspective

of OAC, Theorem 4 implies that at least 2K+ 1 wireless

resources need to be allocated where each resource is

dedicated to a nomographic function in N0(EK)to calculate

every function in C0(EK).

In mathematics, Theorem 4, also known as Kolmogorov’s

superposition or Kolmogorov-Arnold representation theorem,

is notable because it solves a more constrained (i.e., the

function fneeds to be continuous), but a more general

form (i.e., the superposition of only one variable functions)

of Hilbert’s the thirteenth problem in [47]. There are also

other variants of Kolmogorov’s superposition and constructive

proofs that show how to obtain the pre- and post-processing

functions. For a comprehensive discussion on the variants and

constructions, we refer the reader to [48, Chapter 2]. A variant

that is mentioned in the OAC literature [2] is as follows:

Theorem 5 (Braun’09 [49]).For every function f∈ C0(EK),

there exist 2K+ 1 nomographic functions in N0(EK)such

that

f(s1, s2, . . . , sK) =

2K+1

`=1

ϕ` K

k=1

αkψ(sk+ (`−1)β)!,

(3)

where the pre-processing function ψis a well-deﬁned, con-

tinuous, monotone, and independent of f, the coefﬁcients αk,

∀k, and βare appropriate non-negative real constants.

The key observation made in [2] based on Theorem 5 is that

to calculate every function in C0(EK)with continuous nomo-

graphic functions over 2K+ 1 resources, the pre-processing

functions can be designed to be universal. Note that the

superposition in (3) involves 2K+ 1 post-processing function

and one single pre-processing function. In the literature, it

is shown that the superposition can also be expressed with

a single pre-processing function and a single post-processing

function as discussed in [48, Theorem 1] and [50, Theorem

2.14] by introducing a shift to the arguments of the post-

processing functions in (3). Also, Kolmogorov’s superposition

can be interpreted as a special feed-forward neural network

and is useful to predict the complexity of neural networks

(see the discussions in [51]–[53]).

In some cases, it may be desirable not to consume 2K+ 1

wireless resources to calculate a speciﬁc continuous function

with 2K+ 1 continuous nomographic functions. In this case,

one may follow one of two different directions: Manipulating

the domain of the target function or constructing a nomo-

graphic function that approximates the target function. In the

ﬁrst approach, some part of the domain is cut out so that the

nomographic function can be calculated with continuous pre-

and post-processing functions. For instance, if Sis chosen as

[α, 1] for α∈(0,1), the geometric mean can be calculated

with a nomographic function with ψk(x) = ln(x),∀k, and

ϕ(x) = ex/K on S. In the second approach, a nomographic

approximation can be deﬁned as follows [2]:

Deﬁnition 3 (Nomographic approximation).Let  > 0be an

arbitrary constant. The space of approximable nomographic

functions with respect to the precision is deﬁned by

(EK),(f∈ F(EK)|∃(ψ1,..., ψK, ϕ)∈ C0(E)×...

...C0(E)× C0(R) : 



f−ϕ K

k=1

ψk(sk)!



∞≤).(4)

If f∈ N0

(EK), we write f(s1, . . . , sK)≈ϕ(Pk=1 ψk(sk)).

For example, under Deﬁnition 3, the geometric mean on EK

is a function in N0

(EK)because it can be approximated with

ψk(x) = ln(x+ 1/p0()) and ϕ(x)=ex/K for p0()>0.

Nevertheless, a complete characterization of the approximate

nomographic functions is still an area that requires more inves-

tigation as it is possible to deﬁne the space of approximable

nomographic functions in different ways. For instance, in [54,

Eq. (5)], an approximate nomographic function is deﬁned in

a stochastic manner. For further theoretical investigations on

approximate nomography, the reader is also referred to [54]–

[57].

Another interesting function space is the class of symmetric

functions elaborated in [58]:

Deﬁnition 4 (Symmetric function).Let σ:SK→SKdenotes

a permutation. A function f:SK→Rthat is invariant with

respect to permutations of its arguments, i.e.,

f(s1, . . . , sK) = f(σ(s1, . . . , sK)),∀σ(5)

is called a symmetric function.

A distinct feature of the space of symmetric function is

that only the data itself is important, rather than its origin.

From an application standpoint, many functions such as mean,

maximum, minimum, median, and majority vote (MV) that

have either exact or approximate nomographic functions be-

long to this class. The second important feature is that the

functions in this space can be calculated through the type

function, i.e., frequency histogram [9], [10], [58], [59]. Type

function can be deﬁned as multiple weighted arithmetic sums

of indicator functions, i.e., counting the number of devices

based on a certain set, which is also investigated under type-

based multiple access (TBMA) in [9], [10].

B. Common nomographic functions

In TABLE I, we list several exact and approximate nomo-

graphic functions discussed in the literature. While arithmetic

mean, weighted sum, and MV are used in distributed learning

applications, modulo-2 sum often appears in physical layer

network coding. The product operation is used for key genera-

tion in [60]. The maximum, minimum, and counting functions

are used in WSN (e.g. generating an alert if the temperature

rises) or to calculate histogram (e.g., calculating measurement

statistics with TBMA [9], [10]). Geometric mean, p-norm,

and polynomial functions are often mentioned to provide

nomographic function examples that can be calculated over a

wireless network. In particular, p-norm is used for computing

average-pooling or max-pooling over the air in [61].

An interesting direction is to calculate an approximate

nomographic function with a continuous and monotone post-

processing function and continuous pre-processing functions

for a given continuous function. In [56], an approximation

is obtained by using a combination of a dimensionwise

function decomposition. In this approach, the target function

is skewed with a bijective function such that the resulting

function can be approximated well with a ﬁrst-order analysis

of variance (ANOVA) decomposition. To calculate the skew

function, Bernstein polynomials are used. It is worth noting

that Bernstein polynomials can be utilized to constructively

prove the Weierstrass approximation theorem that states every

continuous function can be approximated with an arbitrary pre-

cision over any ﬁnite interval by a polynomial of a sufﬁcient

order.

Another interesting direction is to calculate the target

function by expressing it as a solution to an optimization

problem and solving the problem through iterations that can be

expressed with some elementary nomographic functions. For

example, as discussed in Section IV-E1, the geometric median

can be calculated through iterations over-the-air by using the

Weiszfeld algorithm in [62]. In [63], [64], and [65], by using

the binary representations of the parameters, several non-linear

functions, e.g., maximum or minimum, are proposed to be

calculated through the communication rounds. For instance, to

calculate the maximum of the parameters, in the ﬁrst round,

the ES inquires to the EDs with bit 1in the most signiﬁcant

bit position of the binary representation of the parameter. If

there is any response to the inquiry, the ES detects that the

most signiﬁcant bit of the maximum of the parameters is 1;

otherwise, it is 0. In the second round, if the most signiﬁcant

bit is detected as 1, the ES inquires to the EDs with bit 1

in both the most and the second signiﬁcant bits. Otherwise,

the ES inquires to all EDs with bit 1in the second signiﬁcant

bit position. From the responses to the second inquiry, the ES

determines the second signiﬁcant bit. The procedure continues

until the least signiﬁcant bit is detected. The same procedure

can be used for computing minimum function by using the

reciprocal of the parameters. The reader is also referred to

[59] for successive partitions to compute functions.

III. WHAT ARE THE OAC SCHEMES?

An OAC scheme aims to realize (1) (or (2)) over a

wireless multiple-access channel (MAC) with a ﬁdelity cri-

TABLE I

EXAMPLE NOMOGRAPHIC FUNCTIONS.

Description f(s1, s2,...,sK)ψk(x)ϕ(x)

Arithmetic mean 1

k=1

skxx

Weighted sum

k=1

wksk,wk∈Rwkx x

Polynomial function

k=1

cksk−1

k,ck∈Rck−1xk−1x

Majority vote sign K

k=1

sign (sk)!sign (x)sign (x)

Counting number of EDs

with the class C

k=1

I[sk∈ C]xI[x∈ C]

p-norm K

k=1 |sk|p!1/p

|x|px1

Modulo-2 sum s1⊕s2...⊕sK,sk∈Z2x x mod 2

Approximation of

the product Y

sk, sk≥0 ln x+1

p0()ex

Approximation of

the geometric mean Y

sk!1/K

, sk≥0 ln x+1

p0()ex

Approximation of the

cosine of the product cos Y

sk!, sk≥0 ln x+1

p0()cos(ex)

Approximation of

the maximum max

k{sk}, sk≥0xp0()x

p0()

Approximation of

the minimum min

k{sk}, sk≥0x−p0()x−1

p0()

terion. For a rigorous classiﬁcation and a generalization of

the OAC schemes in the state-of-the-art, consider an OAC

scheme that targets to calculate the nomographic function

z[n] = f(s[n]) ,f(s1[n], . . . , sK[n]) for the symbol vector

s[n] = [s1[n], . . . , sK[n]]T,∀n∈ {1,..., Nf},Nf≥1. Let

sk= [sk[1],..., sk[Nf]]Tand pk= [pk[1],..., pk[Nf]]Tdenote

the symbol vector and the pre-processed symbol vector at the

kth ED for pk[n] = ϕk(sk[n]) ∈R,∀n. The kth ED calculates

the encoded vector ck∈CBas

ck= [c1,..., cB]T=k{pk},(6)

where k:RNf→CBis the encoder (e.g., source encoder,

channel encoder, constellation mapping, or a combination of

these operations) and Bis the number of modulation symbols

in a complex-valued codeword.

Let Mbe a resource mapper that maps Bmodulation

symbols to the Bavailable resources. Now, consider L

modulation symbols among Bsymbols, denoted by mk=

[mk,1,..., mk,L]T, that are processed with a linear precoder

Bk∈CNt×Las

xk= [xk,1,..., xk,Nt]T=Bkmk,(7)

where xk∈CNtis the transmitted symbols from the kth

ED over Ntdimensions. Hence, each ED applies Naccess ,

dB/Lelinear precoders to the modulation symbols in total.

The received vector at the ES, denoted by y∈CNr, can be

written as

y= [y1,..., yNr]T=

k=1 pPkHkxk+n

k=1 pPkHkBkmk+n,(8)

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1ASurveyonOver-the-AirComputationAlphan¸Sahin,Member,IEEEandRuiYangy,Member,IEEEAbstractCommunicationandcomputationareoftenviewedasseparatetasks.Thisapproachisveryeffectivefromtheperspectiveofengineeringasisolatedoptimizationscanbeper-formed.However,formanycomputation-orientedapplications,themaini...

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1 A Survey on Over-the-Air Computation Alphan SahinMember IEEE and Rui YangyMember IEEE

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