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Over-the-Air Computing under Adaptive Channel
State Estimation
Nikos G. Evgenidis∗, Vasilis K. Papanikolaou∗, Panagiotis D. Diamantoulakis∗, and George K. Karagiannidis∗
∗Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
e-mails: nevgenid@ece.auth.gr, vpapanikk@auth.gr, padiaman@auth.gr, geokarag@auth.gr
Abstract—Over-the-air Computation (AirComp) has attracted
significant attention as an efficient way of data fusion by inte-
grating uncoded communication transmissions with computation
thanks to the superposition offered by the multiple access
channels. However, proper pre-processing and post-processing
is required to neutralize the wireless channel effect, in order
for AirComp to function successfully. Since, internet-of-things
(IoT) type of devices with limited capabilities are the target de-
mographic of AirComp, having perfect channel state information
(CSI) available is not always a practical assumption. In this work,
we examine the effect of imperfect CSI on the AirComp system
and we design a general optimization framework that takes into
account both magnitude and phase errors in CSI. On top of that,
a pilot retransmission policy is designed that offers a trade-off
between cost of retransmissions and gain in the accuracy of the
computations. Simulation results show the deterioration caused
by the imperfect CSI and also the value of the proposed policy
under various system conditions.
Index Terms—over-the-air computing, AirComp, imperfect
CSI, optimization framework, MSE minimization
I. INTRODUCTION
In 5G and beyond networks, a paradigm shift has been
noticed from human type communications to machine type
communications. To deal with the massive amount of dis-
tributed data, particularly from internet-of-things type sensors,
data aggregation through over-the-air computing (AirComp)
has recently emerged as a very attractive technology [1], [2].
AirComp can accommodate various objectives in the network
by taking advantage of the superposition property of trans-
mitted waveforms over the multiple access channel (MAC),
and proper pre- and post-processing to facilitate a family of
functions, called nomographic such as mean, geometric mean,
etc [3]. This exploitation can lead to significant increases in
latency and computation load at the central processing node,
especially when the number of sensors becomes excessively
large. As proven in [4], the uncoded AirComp is optimal in
the presence of a Gaussian multiple access channel (MAC)
with independent data sources. This optimality is in terms of
the mean squared error (MSE), which is a basic measure of
performance for the discussed system.
AirComp systems face a particular set of challenges that
need to be resolved to guarantee their operation. The time
synchronization of the devices is one such challenge, where
each device has to account for its message’s propagation delay
to ensure service. Moreover, carrier frequency offset (CFO)
issues that are usually satisfied through the use of high quality
oscillators need to be addressed with different means, since
AirComp devices are usually low-end [5]. On top of that, the
transmission through the wireless channel is prone to channel
fading and noise. To account for that, a power allocation
strategy is required, which is presented in [1], [6], i.e., the
optimal power allocation for the transmitting devices in order
to achieve a minimum MSE.
The integration of AirComp with popular wireless tech-
nologies such as MIMO has gathered a lot of attention from
the research community [7]. Also, recently, AirComp has
been discussed as an appealing technology to be implemented
with federated learning in the network edge [5], [8]–[14].
On top of that, AirComp has shown promise in different
scenarios such as in combination with aerial networks with
unmanned aerial vehicles (UAVs). More specifically, in [15],
the authors optimize the trajectory of the UAV to minimize
the time-average MSE, while in [16] they take into account
CSI imperfections when data are aggregated into a UAV fusion
center. Finally, promising new technologies such as intelligent
reconfigurable surfaces (IRS) has been examined as a way to
facilitate AirComp with higher performance gains [17], [18].
However, most of these works are focused on the perfect
channel state information (CSI) scenario, which cannot easily
be guaranteed in practice, when IoT-type devices are con-
cerned. Imperfect CSI can highly deteriorate the AirComp
system’s performance, as channel information is vital in ob-
taining the correct message at the receiver. In literature, there
are some attempts studying the effect of imperfect CSI and
its performance [16], [19]. Nevertheless, none of these look
into the general case of imperfect CSI both in magnitude and
phase, which is the main motivation of this work. In our
paper, we will investigate the problem of the minimization
of the MSE and the power allocation that must be used in
the transmitting and receiving sides, so that the optimum
can be achieved. First, we provide some theoretical analysis
based on order statistics in order to get more insight on how
the CSI imperfections affect the magnitude and the phase of
every channel. The derived results can help us identify the
necessary communication conditions that are needed for an
AirComp system to work without suffering extensively from
a noise-infused CSI. Then, we propose an algorithm that uses
alternating optimization for the minimization problem, based
on an approximation of the MSE at a worst case scenario.
This approach differs from the one in [19] and also takes
into consideration the phase difference between the real and
the estimated channel, which is obviously the general case to
be considered. Furthermore, based on our theoretical analysis
we propose a second retransmission round for the weaker
estimated channels in an attempt to improve the system’s
performance.
II. SYSTEM MODEL
Let Kbe the number of transmitting devices in the AirComp
system that are such that all of them are independent from
one another. For our work we will assume that the receiver
and all transmitters have a single antenna. Also let bk∈C
symbolize the transmitting power at the k-th device and a∈
C∗symbolize the receiver gain factor. For practical reasons we
will assume that all devices have a common maximum power
magnitude P,sothat|bk|≤√Pfor all k∈{1,···,K}.
We wish to find the computation distortion of the ideal
signal K
k=1 xk, which will be given by
MSE = E|r−
K
k=1
xk|2,(1)
where ris the received signal given by r=
aK
k=1 bkhkxk+n,sowehave
MSE = E
aK
k=1
bkhkxk+n−
K
k=1
xk
2.(2)
All signals and noise are independent with one another and
we assume that all signals xk∈[−u, u]have normalized
variance, thus E(|xk|2)=1,∀k. We will also assume that
the noise nfollows a complex Gaussian distribution with
E(n2)=σ2. Taking the expectation with respect to the signals
xkand noise ngives
MSE = E
K
k=1
(abkhk−1)xk+an
2(3)
and equivalently from the above assumptions
MSE =
K
k=1
abkhk−1
2
+σ2|a|2.(4)
In order to study possible imperfections in the CSI, we
will assume that the error in the estimation of the channel is
modeled as additive random variable nk. Hence, we assume
the following:
•nk∼CN(0,σ
2)=N(0,σ2
2)+jN(0,σ2
2),arethe
noise samples in the kth CSI estimation, uncorrelated and
independent between them.
•hk∼CN(0,σ
2
h)=N(0,σ2
h
2)+jN(0,σ2
h
2)because of
Rayleigh fading conditions in the channel of transmitter
k. Here we assume that σ2
h=2for all |hk|to follow a
Rayleigh distribution of mode σRayleigh =1as used in
the simulations of the paper.
For the estimation of hkwe use symbols of max power
√Phence the receiver gets yk=√Ph
k+nkand assumes
that yk=√Ph
k. Consequently, the channel estimation will
be given by
√Ph
k=√Ph
k+nk⇔h
k=hk+nk
√P=hk+ek,(5)
where ek∼CN(0,σ2
P)=N(0,σ2
2P)+jN(0,σ2
2P). Conse-
quently, we can assume that the channel estimation h
kis
distributed as h
k∼CN(0,σ2+Pσ
2
h
P)=N(0,σ2+Pσ
2
h
2P)+
jN(0,σ2+Pσ
2
h
2P).
III. PROPOSED POLICY AND ANALYSIS
A. Theoretical Study on AirComp with Imperfect CSI
Assuming the receiver is unaware of the CSI imperfections,
the algorithm obtained for an AirComp system with perfect
CSI in [1], [6] would lead to a combination of full power and
channel inversion methods. Using this algorithm in (4), would
result in
MSE =
i∗
k=1 a√Ph
k
H
|h
k|hk−1
2
+σ2|a|2
+
K
k=i∗+1 hk
h
k−1
2
,(6)
where i∗is the critical number of transmitters that utilize their
full power as given in [1]. Moreover, the channel ordering has
been used |h
1|≤|h
2|≤···≤|h
K|. Since the order has been
taken with respect to the estimation of hk, it is clear that the
true order could be different, e.g., |h2|≤|h1|≤···≤|hK|,
which means that the proposed optimum solution can lead to
a different optimum number i∗
est of transmitters that use full
power, since
i∗=argmax
1≤i≤K{gi},(7)
where
gi=√Pi
k=1 |hk|
σ2+Pi
k=1 |hk|2(8)
and the magnitudes affect the result. Though i∗can be
incorrectly estimated the main problem is that the incorrect
values of giwill also affect a, which is given as a=gi∗.
Lemma 1: The expected value of the magnitude of the
ordered channel gains is given by
E(Ur)=π
4σhK
rr
r−1
k=0 r−1
k(−1)k
(K−r+1+k)3
2
.
(9)
Proof: The proof is presented in Appendix A.
Corollary 1: The expected value of the magnitude of the
error given by
E(|ek|)=∞
0
|ek|2
σ2
ek
exp −|ek|2
2σ2
ekd|ek|
=σekπ
2=σ2
2Pπ
2=1
√ρπ
4,(10)
where ρis the transmit SNR.
Proof: Following the proof of Lemma 1, (10) is easily
produced in the same way.
Using the above results, we can find statistically how many
channels are mostly affected by the CSI imperfection, which
is of great value when designing the AirComp system. The
main resulting issue is that the order of the estimated channels
is very likely to be different from the correct order, which
will result in an additional error in the minimization of MSE.
Ultimately, the most affected channels will also be more
susceptible to greater phase difference during CSI.
B. Optimization Framework
Due to AirComp’s operation, imperfections in CSI can
greatly hinder performance, even for small levels of noise.
Solutions that ignore the possible imperfections that result
in optimum controls for the perfect CSI case, by assuming
|h
k|=|hk|,∀k∈{1,···,K}, diverge remarkably from the
optimal strategy, which is defined by i∗and a, when imperfect
CSI is available.
In order to improve the overall performance, we propose
a new optimization framework that will take into account all
the extra error terms that occur. In order to do this we assume
that the statistical mean of the noise is known to the receiver.
First of all, since a∈C∗we define ∠a=apto symbolize its
complex phase. By setting Δhk=∠(hk,h
k), using the fact
that h
k=hk+ek, and (4), the optimization problem can be
expressed as
min
a,bMSE = K
k=1 |a||bk||hk|2
+1
−2K
k=1 |a||bk||hk|cos (ap+Δhk)+σ2|a|2,
s.t. C1:|bk|≤√Pk,∀k.
(11)
Theorem 1: The optimal power distribution is given by
|bk|=min√P, cos Δhk
a|hk|.(12)
Proof: In order to use the phase factor for minimiza-
tion in (11) we will need to approximate its term and find
its extrema. However, from trigonometry, cos (ap+Δhk)=
cos apcos Δhk−sin apsin Δhkand any approach to approx-
imate this quantity without knowledge about the sign of the
phase difference Δhkcannot be made because the sign of
sin Δhkwill be affected. Since the sign of Δhkis affected by
the phase of the noise in the CSI estimation, we cannot make
any assumptions about it and thus we cannot further use ap
in minimization. Hence, from now on we will consider ato
be a real number and we rewrite the mean squared error as
MSE =
K
k=1 a|bk||hk|2
+1
−2
K
k=1 a|bk||hk|cos Δhk+σ2a2.
(13)
O
hkB
h
k
|ek|
Fig. 1: Worst case approximation of phase difference Δhk.
In order to find the extrema for every variable |bk|we take
the first order partial derivative to be equal to zero, thus:
∂MSE
∂|bk|=0⇔|bk|=cos Δhk
a|hk|and since |bk|is the power
magnitude at device k,itmustalsobe0≤|bk|≤√P.
Consequently, the best power distribution will be given by
(12) and, thus, the proof is completed.
We can observe that the real MSE is at this point related to
the estimations h
konly through the terms cos Δhk. Assuming
the imperfections of CSI due to the noise have a constant
magnitude of |ek|, then from Figure 1, the worst case scenario
for these terms arise when the phase of the imperfection
is such that h
kbecomes tangent to the circle of constant
radius |ek|. This approximation will be quite accurate when
the conditions are such that the mean estimated error can be
assumed less than the mean of the estimated channels, i.e.
E(|hk|)>E(|ek|), and using it we obtain
cos Δhmax
k=|h2
k|−|e2
k|
|hk|=|hworst
k|
|hk|.(14)
Hence, if we consider the worst case scenario we can make
the following approximations |h
k|2+|ek|2≈|hk|2and
cos Δhk≈|h
k|
|hk|. We can also assume that |ek|2≈E(|ek|2)=
E(n2)
P=σ2
P=cfor all k. While this approach contains its own
errors, mostly for large scale fading or great levels of noise, it
provides a tight approximation of the sinusoidal term for the
greater majority of the involved channels, except for those that
are greatly affected by noise. Also, since it covers the worst
case in an almost optimal way it will partly compensate for
its near optimum behavior in more favorable scenarios.
Using the above mentioned approximation combined with
the attainable values of |bk|given by Theorem 1, we can now
express (13) as
MSEi=
i
k=1 a√P|h
k|−1
2
+a2(σ2+iP c)
+c
K
k=i+1
1
|h
k|2+c.
(15)
Lemma 2: The optimal aito minimize (15) is given by
ai=√Pi
k=1 |h
k|
(σ2+iP c)+Pi
k=1 |h
k|2.(16)
Proof: Considering (15) as a quadratic polynomial in
terms of aand using a well known property for the global
extremum of this function, we obtain that the global minimum
for every iis given by differentiating (15) in terms of aito
finally obtain (16), and thus, the proof is completed.
At this point, it is observed that (16) will be precise
mostly at conditions like relatively high SNR or good channel
conditions. This is so, because, due to our approximation
the power magnitudes, we will obtain |bk|opt =cos Δhk
a|hk|≈
|h
k|
a|hk|2≈|h
k|
a(|h
k|2+c), are points of the function f(x)= x
a(x2+c)
which is an increasing and then decreasing function in terms
of xthat achieves its maximum at x=√c. Hence, for these
to be in descending order we need |h
1|≥√c.Otherwise,we
observe that the receiver coefficient athat will be calculated
can be such that some of the weaker estimated channels will
use the inverse channel method. Finally, expression (16) is
quite similar to the optimum value of afor the perfect CSI
case, but will always be less than gi. In other words, the
uncertainty caused by the imperfect CSI will force the system
to use more transmitting power in an attempt to counter these
imperfections.
Corollary 2: At least one device must use its full power
during transmission.
Proof: The proof is given in Appendix B.
So, it suffices to solve Ksubproblems for i∈1, ..., K and,
then, compare the minimum values MSEmin
i=MSE
i(ai).
In order to get a feasible solution for a,itmustbeai>
|h
i+1|
(|h
i+1|2+c)√P.Ifai>|h
i+1|
(|h
i+1|2+c)√Pwe can always find a
better solution to minimize the overall MSE. Thus, we only
need to check the values MSEi(ai)when aiis feasible. This
way we can calculate i∗andthenestimateaiand bkfor all k.
It is important to note that this approach differs from the
one followed in [19] in its mathematical derivation, but also
in the fact that our approximation covers the more general
case of phase misalignment as opposed to the phase alignment
considered in [19].
C. Pilot Retransmission Policy
From our theoretical analysis it is clear that statistically
there will be a number of channels whose estimations will
be greatly affected by the noise-induced error during the CSI
procedure. In order to limit the effect of this in the MSE of the
system we need to have better estimations for the channels. As
a result we can consider the possibility of making a second
CSI round at least for some channels. Apparently, the most
obvious choice is for the weaker channels to retransmit pilot
symbols and then use the average of the two estimations as the
new correct estimated channel. For this approach we propose
the following heuristic algorithm in order to find the number of
channels that will need to re-estimate their channels. In order
to track the trade-off between the extra resources required for
the retransmissions and the resulting MSE, we propose and
define the following cost function, called the Retransmission
Policy Cost (RPC) as
RPC = (Power costk)b·E(MSEk)
Kd
,(17)
where Power costksymbolizes the power resources needed for
kretransmissions, E(MSEk)
Kdenotes the MSE for kretransmis-
sions and are our primary concern over the available resources
is taken to be their power. Moreover parameters band dare
considered to be weights for power and MSE, respectively.
Any combination of parameters can be used to give emphasis
to either the cost in terms of resources for the retransmissions
or its maximum error tolerance. Withous loss of generality, the
time penalty required for the retransmission can be included
in the RPC metric, but given that the decrease of E(MSE) is
vital for the correct interpretation of the superimposed signals
and the fact that the AirComp scheme needs less time than
other traditional schemes like NOMA due to the superposition
property of MAC, we suppose that the small time latency
created by a few retransmission rounds does not affect our
system.
IV. SIMULATION RESULTS
In this section we present the simulation results of our work
for an AirComp system. Fading channels have been simu-
lated with Circular-Symmetric Complex Normal distributed
variables CSCN(0,1) to simulate Rayleigh fading channel
conditions. Unless otherwise stated, the transmit SNR is set
as ρ= 10dB. We apply Monte Carlo analysis averaging over
104channel realizations (snapshots). Finally, we define the
average per user MSE as AMSE = E(MSE)/K .
In Fig. 2 we look at the performance of the perfect
CSI algorithm and the proposed optimization technique for
imperfect CSI under imperfect channel estimation. As we
can see the perfect CSI algorithm not only achieves worse
E(MSE)/K values, but also fails to converge as the number
of devices in the system increases. In contrast to this, the
proposed technique both achieves better performance and has
a diminishing behavior for increasing number of devices.
In Fig. 3 we look at the performance of the retransmission
policy discussed in section III. For comparison reasons, the
retransmissions have also been simulated for random selec-
tions of channels instead of the weaker ones as proposed. As
expected both policies achieve better MSE values than the no-
retransmission policy due to the better channel estimations for
their corresponding re-estimated channels. We can see that the
proposed policy achieves a much better improvement rate over
the random channel selection policy. It is important to point
out that this improvement is greater mainly for the first few
weaker channels, which confirms the idea that a re-estimation
of these channels can be used to decrease the MSE of the
system. We can also observe that though a bigger number of
retransmissions can achieve further performance improvement,
both policies tend to converge on the same MSE, because a
lot of channel re-estimations will now be common.
In order to evaluate in terms of RPC the proposed policy,
we design a simple example of an RPC function as follows.
To obtain the first, necessary round of channel estimates
exactly Kpower resources are needed. Assuming that each
power resource is equal to 1, then, for every retransmission,
1additional power resource is needed from the corresponding
device. Thus, for kretransmissions, Power costk=(K+k)P
and according to (17) the minimum of RPC is studied. RPC
in this case is expressed as
RPC = [(K+k)P]b·E(MSEk)
Kd
.(18)
Using Monte Carlo analysis we see that such a minimum
exists, it is global and it appears that for the statistically weaker
channels all retransmission choices achieve a better trade-off
between power and MSE than the original proposed scheme
without any retransmissions.
Fig. 2: MSE vs the number of devices in the AirComp system
accounting or not for imperfect CSI (Transmit SNR = 10dB).
V. C ONCLUSIONS
In this paper, an AirComp system under imperfect CSI
assumptions was considered. The detrimental effect of im-
perfect CSI is presented for the MSE, especially when the
optimization framework does not account for the imperfections
in the channel estimations. We presented a comprehensive
analysis on how channel estimation errors affect the AirComp
system and a novel optimization framework to minimize MSE
under those conditions. In order to counter the effect of the
imperfections, an adaptive policy based on pilot retransmission
was presented, where the proposed policy shows the potential
to greatly improve the performance. Moreover, a new metric
was presented alongside the retransmission policy to showcase
Fig. 3: MSE vs the number of retransmission with various
policies (Transmit SNR = 10 dB).
Fig. 4: Retransmission Policy Cost vs the Number of Retrans-
missions for the proposed policy, the random policy, and the
original scheme without retransmission for b=d=1and
Pk=1,∀k.
the efficiency of the approach. Finally, simulation results were
presented to validate the effectiveness of the proposed analysis,
showcasing that it can offer great insights into the design of
AirComp systems.
ACKNOWLEDGMENT
This paper was supported by the European Union’s Horizon
2020 Research and Innovation Program under Agreement
957406. The implementation of the doctoral thesis of Vasilis
K. Papanikolaou was co-financed by Greece and the European
Union (European Social Fund-ESF) through the Operational
Programme “Human Resources Development, Education and
Lifelong Learning” in the context of the Act “Enhancing Hu-
man Resources Research Potential by undertaking a Doctoral
Research” Sub-action 2: IKY Scholarship Programme for PhD
candidates in the Greek Universities.
APPENDIX A
PROOF OF LEMMA 1
We start the proof by presenting an analysis based on order
statistics, in order to relate the mean estimation error with the
mean of the magnitude of the ordered channels. We denote the
correct channel gain ordering in the following way |hcor
1|≤
|hcor
2|≤···≤|hcor
K|. From the order statistics we can find that
every ordered sample Ur=|hcor
r|from a total of Ksamples
has the following probability distribution function (PDF)
fUr(ur)= K!
(K−r)!(r−1)!
2ur
σ2
h
×exp −u2
r
σ2
h
(K−r+1)
1−exp −u2
r
σ2
hr−1
.
(19)
Hence, the expected value E(Ur)can be calculated by
E(Ur)=
K
rr∞
0
2u2
r
σ2
h
exp −u2
r
σ2
h
(K−r+1)
×1−exp −u2
r
σ2
hr−1
dur.(20)
Setting t=ur
σhwe obtain
E(Ur)=2σhK
rr∞
0
t2exp −t2(K−r+1)
×1−exp (−t2)r−1
dt
=2σhK
rr∞
0
t2e−Kt2[et2−1]r−1dt. (21)
Using binomial expansion and that ∞
0exp (−mt2)=
π
4mand after some algebraic manipulations, after the in-
tegration, we get
∞
0
t2e−Kt2(et2−1)ndt =
=1
2π
4
n
k=0 n
k(−1)k1
(K−n+k)3
2
.(22)
Then, we can easily get (9) from (21) and (22), and, thus, the
proof is concluded.
APPENDIX B
PROOF OF COROLLARY 2
We begin with a proof by contradiction, assuming that a full
inverse channel method is utilized. Then, by (16) the overall
MSE of the system will be
MSE0(a0)=σ2a2
0+c
K
k=1
1
|h
k|2+c,(23)
where a0>|h
1|
√P(|h
1|2+c)for feasibility reasons. Then, taking
a1=|h
1|
√P(|h
1|2+c)observe that
MSE1(a1)=a1√P|h
1|−1
2
+a2
1(σ2+Pc)
+c
K
k=2
1
|h
k|2+c=a2
1σ2+c
K
k=1
1
|h
k|2+c
=MSE
0(a1)(24)
and since a0>a
1we see that MSE1(a1)<MSE0(a0)which
means that we get that the performance of the system is better
than the full inverse channel method, and, thus the proof is
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