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# Waveform Design for Over-the-Air Computing

Nikos G. Evgenidis, Nikos A. Mitsiou, , Sotiris A. Tegos, , Panagiotis D.
Diamantoulakis, , Panagiotis Sarigiannidis, ,
Ioannis T. Rekanos, , and George K. Karagiannidis N. G. Evgenidis, N. A.
Mitsiou, P. D. Diamantoulakis, and I. T. Rekanos are with the Department of
Electrical and Computer Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>rekanos@auth.gr).S. A. Tegos is with the Department of
Electrical and Computer Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece and also with the Department of Informatics and
Telecommunication Engineering, University of Western Macedonia, 50100 Kozani,
Greece (e-mail: sotiristegos@ieee.org).P. Sarigiannidis is with the Department
of Informatics and Telecommunication Engineering, University of Western
Macedonia, 50100 Kozani, Greece (e-mail: psarigiannidis@uowm.gr).G. K.
Karagiannidis is with the Department of Electrical and Computer Engineering,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece, and also
with the Artificial Intelligence & Cyber Systems Research Center, Lebanese
American University (LAU), Lebanon (e-mail: geokarag@auth.gr).This work was
funded from the Smart Networks and Services Joint Undertaking (SNS JU) under
European Union’s Horizon Europe research and innovation programme (Grant
Agreement No. 101096456 - NANCY).

###### Abstract

In response to the increasing number of devices anticipated in next-generation
networks, a shift toward over-the-air (OTA) computing has been proposed.
Leveraging the superposition of multiple access channels, OTA computing
enables efficient resource management by supporting simultaneous uncoded
transmission in the time and the frequency domain. Thus, to advance the
integration of OTA computing, our study presents a theoretical analysis
addressing practical issues encountered in current digital communication
transceivers, such as time sampling error and intersymbol interference (ISI).
To this end, we examine the theoretical mean squared error (MSE) for OTA
transmission under time sampling error and ISI, while also exploring methods
for minimizing the MSE in the OTA transmission. Utilizing alternating
optimization, we also derive optimal power policies for both the devices and
the base station. Additionally, we propose a novel deep neural network
(DNN)-based approach to design waveforms enhancing OTA transmission
performance under time sampling error and ISI. To ensure fair comparison with
existing waveforms like the raised cosine (RC) and the better-than-raised-
cosine (BRTC), we incorporate a custom loss function integrating energy and
bandwidth constraints, along with practical design considerations such as
waveform symmetry. Simulation results validate our theoretical analysis and
demonstrate performance gains of the designed pulse over RC and BTRC
waveforms. To facilitate testing of our results without necessitating the DNN
structure recreation, we provide curve fitting parameters for select DNN-based
waveforms as well.

###### Index Terms:

over-the-air computing, resource allocation, waveform design, deep neural
networks

##  I Introduction

The next generation of wireless networks is expected to enable many
applications for devices in the physical layer to reduce delay. One of these
is considered to be computing, a functionality currently performed in higher
layers, as the latter is extremely important for many real-world scenarios,
such as autonomous driving, etc. This shift from traditional communication
systems to more goal-oriented tasks requires the introduction of more
sophisticated techniques to manage the available resources according to the
task to be performed. In this direction, and with respect to (w.r.t.)
computing, over-the-air (OTA) computing has been proposed as an interesting
alternative to the conventional receive-then-compute paradigm [1]. OTA
computing takes advantage of the multiple access channel (MAC) superposition
principle to allow simultaneous transmission of multiple devices to compute a
desired target function through wireless data aggregation [2]. As a result,
OTA computing can achieve better resource management as well as improved
computation efficiency due to the distributed nature of the method.

On the other hand, as shown in the pioneering work [3], the use of uncoded
analog transmission is the optimal transmission method for OTA computing.
Analog transmission implies that any value can be transmitted and no
conversion to bits is performed on the received signal, while the latter
contains the information of the desired computation. Nevertheless, most
current communication systems are based on digital transmission of symbols
using appropriate waveforms [4, 5] to mitigate various phenomena such as
synchronization problems at sampling instances and intersymbol interference
(ISI), among others. The presence of such phenomena affects the performance of
the systems at the receiver and consequently can affect the accuracy of
computing even in the conventional computing paradigm. Therefore, for OTA
computing to be integrated into modern communication systems, its
compatibility with digital communication techniques and hardware used in
current generations of wireless networks must be facilitated, which is an open
research topic to be explored.

###  I-A Literature Review

Due to its ability to support a large number of devices while using the
available resources in an intelligent and computationally efficient manner,
OTA computing has attracted a lot of study in the last decade. Specifically,
the pioneering works [6, 7, 8, 9] investigated the use of OTA computing to
approximate various target functions that can be written in nomographic form,
while in [10], a deep neural network (DNN) framework to approximate target
functions whose nomographic decomposition is non-trivial was proposed.
Furthermore, optimal power policies for OTA computing were studied in [11,
12], while optimal power allocation schemes and techniques to improve accuracy
performance under imperfect channel state information (CSI) scenarios were
proposed in [13].

In addition to pure OTA computing, much research has been conducted to
facilitate the cooperation of OTA computing with other technologies to improve
its performance. For example, in [14, 15, 16], reconfigurable intelligent
surfaces (RISs) were proposed to provide better accuracy through optimal
handling of the RIS for improved channel conditions and power allocation. The
integration of RISs with OTA computing has also been further investigated by
considering unmanned aerial vehicle (UAV) trajectory optimization problems to
create optimal channel conditions [17]. Similarly, and with the aim of
improving the performance of OTA computing, MIMO systems have also been
proposed as an interesting way to achieve this goal. As such, many different
directions have been explored in [18, 19, 20, 21, 22], including joint hybrid
beamforming and zero-force beamforming under a general pool of OTA computing
related constraints associated with mean squared error (MSE) threshold and
outage probability. Additionally, multiple target function computation has
been investigated for MIMO systems, where different target functions, such as
for example arithmetic and geometric mean, can be computed simultaneously over
separate channels with appropriate pre-coding [23].

Another important practical application of OTA computing that has attracted
attention is the enabling of federated learning (FL). FL enables distributed
training of DNN frameworks by using individual devices to update isolated
parts of a DNN and then combining all the updates for a global update at a
fusion center (FC). In this context, OTA computing has been studied as an
effective way to provide the update from the distributed devices to the FC
[24, 25, 26, 27, 28].

Related to the applicability of OTA computing in FL, a convergence analysis of
distributed optimization based on OTA computing was performed in [29]. In the
latter, it was proven that even if all devices do not participate with updates
for each training round, the training will still converge, thus proving the
effectiveness of OTA computing when used to enable distributed optimization
techniques. Moreover, a reduction in the overall convergence time is achieved
as a result of the parallel nature of the distributed optimization techniques
enabled by OTA computing.

###  I-B Motivation & Contribution

Although OTA computing has been extensively investigated, either alone or in
conjunction with other technologies, all related works are based on the
assumption of analog transmission without considering the existing components
of current communication systems. Such components include waveform
transmission and filters at both ends, both of which face practical problems
such as sampling error and ISI insertion. In [30], a DNN framework was
proposed to enable OTA computing for digital systems, i.e., to perform to the
conversion to bits, without discussing the fundamental principles related to
the transmitted waveforms and the impact of the aforementioned issues such as
sampling error and ISI, while also losing the optimality of uncoded analog
transmission as shown in [3]. In order to facilitate the use of OTA computing
in modern devices, it is crucial to investigate the performance of OTA
computing under these phenomena, while at the same time aiming to establish
its compatibility with modern communication components, which motivates the
present work. Furthermore, in relation to the effects of these phenomena, the
established waveforms have been proposed for pure communication between
devices, where bit error rate (BER) is of interest, meaning that they neglect
any differences between the performed tasks, such as computing and
conventional data transfer, that are enabled by physical layer communication.
Therefore, since computing is of interest, which is more accurately evaluated
by MSE, it is reasonable to investigate more appropriate waveforms to mitigate
sampling error and ISI, without assuming that the accuracy of computing is
invariant to the implemented waveforms.

With this in mind, our work aims to fill the gap regarding the performance of
OTA computing when used in modern communication systems. At the same time, we
aim to improve the accuracy of OTA computing by proposing a method to generate
waveforms that can better handle phenomena that affect currently deployed
systems. Therefore, the contribution of our work is summarized in the
following points:

  * •

We focus on the theoretical analysis of the effect of sampling error and ISI
when the transmitted waveforms are incorporated into the system model of OTA
computing. For this purpose, we focus on the raised cosine (RC) and better-
than-raised-cosine (BTRC) waveforms which are commonly used in modern
communication systems. The statistical properties of the established waveforms
are also studied, and approximations are proposed to study the behavior of the
waveforms in order to study the average MSE of the OTA computing system.

  * •

In order to mitigate the effect of sampling error and ISI on the performance
of OTA computing, we provide a thorough analysis, formulating optimization
problems to address different conditions. Specifically, sampling error is
studied separately and also in combination with ISI when OTA computing is used
in frequency-selective channels. In both scenarios, optimal power allocation
policies are extracted by checking all critical points of the arising MSE
expressions.

  * •

For the first time, we propose a novel DNN framework that aims to generate
appropriate waveforms to improve the performance of OTA computing.
Specifically, a DNN architecture is proposed along with a custom loss function
that is introduced to ensure that the DNN-generated waveforms satisfy energy
and bandwidth constraints similar to those of the currently implemented
waveforms. Furthermore, the training phase of the DNN includes channel fading
and noise as well as sampling error and the presence of ISI, allowing the DNN
to mitigate their combined effects. Moreover, the proposed framework also
takes into account the division of the used waveforms between the transmitter
and the receiver side, thus correctly encompassing the design structure of
modern transceivers.

  * •

Based on the theoretical part of our work and the DNN-generated waveforms,
simulation results are presented for varying conditions, including increasing
number of transmitting devices, transmit signal-to-noise ratio (SNR), and
sampling error variance. The results are performed for both RC and BTRC under
the extracted optimal power allocation to compare the performance of the
currently utilized waveforms. Furthermore, the DNN-generated waveform is
tested with the optimal policy showcasing a considerable performance gain over
RC and BTRC, which emphasizes the significance of the proposed framework.

###  I-C Structure

The remainder of this paper is organized as follows. Section II describes the
system model, presenting the basic concepts of OTA computing, the currently
used waveforms, and their combination. In Section III, we formulate the
optimization problems of minimizing the average MSE in the presence of
sampling error and ISI, and propose corresponding solutions. In Section IV, we
discuss the proposed DNN framework and how it incorporates energy and
bandwidth constraints. In Section V, we present simulation results and
discussion on the performance of OTA computing and the utilization of the
proposed DNN-generated waveforms, while Section VI concludes the work.

###  I-D Notation

From henceforth, vectors are denoted by bold lowercase letters. Sets and
sequences are denoted by $\\{\cdot\\}$. The expectation of a random
multivariate expression w.r.t. the random variable $X$ is denoted by
$\mathbb{E}_{X}[\cdot]$, while the discrete Fourier transform (DFT) operation
is denoted as $\mathcal{F}(\cdot)$.

##  II System Model

###  II-A OTA Computing Preliminaries

In our work, we consider an OTA computing system consisting of a receiver,
which acts as an FC, and multiple transmitting devices. Let $K$ be the number
of transmitting devices in the OTA computing system, while the measurements of
all devices are independent. Assume that we want to calculate a function
$f:\mathbb{R}^{K}\rightarrow\mathbb{R}$ of all transmitted data, denoted as
$f(x_{1},x_{2},\cdots,x_{K})$. When $f$ is a nomographic function, it is known
that there is an appropriate pre-processing function
$\varphi_{k}:\mathbb{R}\rightarrow\mathbb{R},\forall k\in\\{1,\cdots,K\\}$ and
a post-processing function $\psi:\mathbb{R}\rightarrow\mathbb{R}$ such that
the target function $f$ can be expressed as

$f(x_{1,t},x_{2,t},\cdots,x_{K,t})=\psi\left(\sum_{k=1}^{K}\varphi_{k}(x_{k,t})\right),$
(1)

where $x_{k,t}$ is the data sample of the $k$-th device at the $t$-th time
instance. Due to the stochastic nature of the wireless medium, all transmitted
data are subject to channel fading and noise at the receiver, resulting in the
following

$\hat{f}=\psi\left(\sum_{k=1}^{K}h_{k}\varphi_{k}(x_{k,t})+n\right),$ (2)

where $h_{k}$ denotes the channel fading of the $k$-th device, that is assumed
to follow a Rayleigh distribution, and $n$ denotes the additive white Gaussian
noise (AWGN) with $\mathbb{E}_{n}[n]=0$ and
$\mathbb{E}_{n}[n^{2}]=\sigma^{2}$, where $\sigma^{2}$ is the noise power. We
define the set of all devices as $\mathcal{K}=\\{1,\cdots,K\\}$, where the
devices are ordered in ascending order of their channel gains. For the
transmitted data of each device it is assumed that both
$\mathbb{E}_{x_{k}}[x_{k,t}]=0,\,\,\forall t$ and
$\mathbb{E}_{x_{k}}[x_{k,t}^{2}]=1,\,\,\forall t$ hold. In the context of this
paper, and without loss of generality, we assume that the receiver and all
transmitting devices are equipped with a single antenna. We assume that
perfect CSI is available at both the transmitter and the receiver.

###  II-B Overview of Basic Waveforms

Modern communication systems rely on the use of appropriate waveforms to deal
with the effects of phenomena such as ISI, that are provoked by the limited
bandwidth availability and the channels’ frequency selectivity. In general,
ISI occurs when the currently transmitted symbols are interfered with past and
future symbols. As a result, the base station (BS) may not be able to
correctly reconstruct the original symbol. Let $T$ be the symbol period of all
devices and $z_{k}(t)$ be the waveform associated with the $k$-th device and
its data $x_{k,t}$. Because of its ability to mitigate ISI, one of the most
commonly implemented waveforms for modern digital systems is the RC waveform,
expressed in the time domain as

$z_{\mathrm{RC}}(t)=\frac{1}{T}\mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\pi\alpha\frac{t}{T}\right)}{1-\left(2\alpha\frac{t}{T}\right)^{2}},$
(3)

where $0\leq\alpha\leq 1$ is the roll-off factor. Another widely used waveform
that is known to provide better performance than the raised cosine in single-
user scenarios is the BTRC waveform. The BTRC waveform satisfies the Nyquist
criterion and is given in the time domain as

$z_{\mathrm{BTRC}}(t)=\frac{1}{T}\mathrm{sinc}\left(\frac{t}{T}\right)\frac{4\beta\pi
t\sin{\left(\frac{\pi\alpha t}{T}\right)}+2\beta^{2}\cos\left(\frac{\pi\alpha
t}{T}\right)-\beta^{2}}{(2\pi t)^{2}+\beta^{2}},$ (4)

where $\beta=(2T\ln{2})/\alpha$. The bandwidth of both waveforms, $W_{SC}$, is
dependent on the selected roll-off factor of the system, as follows

$W_{\mathrm{SC}}=(1+\alpha)W,$ (5)

where $W$ is the utilized bandwidth. In general, the larger the roll-off
factor, the better the system’s ability to mitigate ISI. However, as observed
by (5), a larger roll-off factor also indicates a larger spectrum allocation,
thus creating a critical trade-off between communication performance and
resource efficiency.

It should also be noted that while the above waveforms are capable of
completely eliminating ISI, this is only possible if perfect time sampling is
performed at the receiver. Therefore, in practical wireless communication
systems, the use of these waveforms does not necessarily result in zero ISI.
Furthermore, it is important to emphasize that modern communication systems
mostly rely on a two-part split of the used waveform, i.e., for practical
reasons, both the transmitting device and the BS are equipped with the square-
root filter of the waveform [31] as shown in Fig. 1, which leads to optimal
SNR at the receiver side [32]. Note that this technique is the only filter
realization that can achieve maximum SNR at the receiver, facilitating it as
the optimal filter realization in modern communication systems.

Figure 1: OTA computing model with sampling error.

###  II-C OTA Transmission under Sampling Error and ISI

Without loss of generality, we assume that the target function is the sum of
all transmitted data. In this case, no pre- or post-processing functions are
needed, except for appropriate power allocation of the participating devices.
Let $b_{k}\in\mathbb{C}$ be the transmit equalization factor at the $k$-th
device such that $|b_{k}|^{2}$ denotes the transmit power, $\arg\\{b_{k}\\}$
denotes the phase of the transmit signal and the transmitted signal of the
$k$-th device in time is given as $x_{k}b_{k}z(t)$. Similarly,
$a\in\mathbb{C^{*}}$ denotes the receiver gain factor. All transmitting
devices are assumed to have a common maximum power magnitude
$P_{\mathrm{max}}$, so that $|b_{k}|^{2}E\leq P_{\mathrm{max}}$ for all
$k\in\mathcal{K}$, where $E=\int_{-\infty}^{\infty}|z(t)|^{2}\mathrm{d}t$ is
the waveform energy. This constraint can be equivalently written as
$|b_{k}|\leq\sqrt{P}$, where $P=P_{\mathrm{max}}/E$. Due to the perfect CSI
availability, the phase of $b_{k}$ can always be chosen in such a way that the
phase shift introduced by the fading is always eliminated. Thus, we consider
that the receiver gain, the transmit power, and the channel coefficients are
all real numbers, i.e., $a,b_{k},h_{k}\in\mathbb{R},\,\,\forall
k\in\mathcal{K}$.

For our analysis, we consider that the received signal is subject to two major
errors, namely the time sampling error and ISI. For the time sampling error,
$\epsilon=t/T$, we assume that it follows a Gaussian distribution centered
around the ideal sampling time [33] and it has variance
$\sigma_{\epsilon}^{2}$, hence
$\epsilon\sim\mathcal{N}(0,\sigma_{\epsilon}^{2})$. If the sampling error is
the only imperfection at the receiver, the received signal can be described as

$\hat{y}=a\left(\sum_{k=1}^{K}x_{k,0}z_{k}(\epsilon T)b_{k}h_{k}+n\right).$
(6)

It should be highlighted that (6) corresponds to the case of flat-frequency
channel fading. On the other hand, if the time sampling error coexists with
ISI, the received signal is given as

$\displaystyle\hat{y}_{\mathrm{ISI}}$
$\displaystyle=a\left(\sum_{k=1}^{K}x_{k,0}z_{k}(\epsilon
T)b_{k}h_{k,0}+n\right)$ (7)
$\displaystyle\quad+a\underbrace{\left(\sum_{\begin{subarray}{c}q=-\frac{\mu}{2}\\\
q\neq
0\end{subarray}}^{\frac{\mu}{2}}\sum_{k=1}^{K}x_{k,q}z_{k}(T(q+\epsilon))b_{k}h_{k,q}\right)}_{\text{ISI
terms}},$

where $\mu$ is assumed to be even and denotes the number of ISI symbols
affecting the currently transmitted symbol, while $h_{k,q}$ is the channel
coefficient of the symbol $x_{k,q}$, which is transmitted at the $q$-th time
instance relative to the current time, denoted by $t=0$ for convenience. It is
also assumed that $t=0$ corresponds to the ideal sampling time. Note that in
both cases the ideally received signal is given by

$r=\sum_{k=1}^{K}x_{k,0}.$ (8)

Figure 2: Signal amplitude of the target signal (8) and received signal (7)
with sampling error with $\sigma_{\epsilon}=0.1$ for RC waveform of
$\alpha=0.3$ in the absence of noise. The sampling instances for each case are
depicted as bullet points. For the rest of the selected parameters, see
Section V.

##  III Theoretical Analysis

In this section, we theoretically investigate the MSE of OTA computing between
the ideal and actual received signal for both the RC and the BTRC waveforms.
To this end, both cases of time sampling error and ISI are considered, while
the optimal power allocation for any waveform during the OTA transmission is
also extracted.

###  III-A OTA Computing under Time Sampling Error

By definition, ideal sampling of any waveform occurs exactly at intervals of
symbol period $T$. Since we assume that ideal sampling occurs at $t=0$, it is
clear that introducing the sampling error $\epsilon$ will lead to (6) and (8).
Consequently, the MSE in this scenario can be expressed as

$\mathrm{MSE}(a,\bm{b})=\mathbb{E}_{\epsilon}\left[\sum_{k=1}^{K}\big{(}az_{k}(\epsilon
T)b_{k}h_{k}-1\big{)}^{2}+\sigma^{2}a^{2}\right],$ (9)

where the expectation w.r.t. $x_{k},\,\forall k\in\mathcal{K}$, and $n$ have
been calculated and $\bm{b}=[b_{1},\cdots,b_{K}]$ is the transmission power
vector of all users. To study the performance of the MSE as described by (9),
we have to first calculate the statistics of the waveform w.r.t. the sampling
error. In order to simplify this calculation we consider a geometric series
based approximation of the fraction term of the expression for RC in (3),
which allows us to write

$\frac{1}{1-4\alpha^{2}\epsilon^{2}}=\sum_{n=0}^{\infty}(4\alpha^{2}\epsilon^{2})^{n}=1+4\alpha^{2}\epsilon^{2}+\mathcal{O}(16\alpha^{4}\epsilon^{4}),$
(10)

where $\mathcal{O}(\cdot)$ denotes the summary of the least-significant terms
which can be considered negligible. The first equality holds due to the
convergence of the geometric series whenever $\epsilon<0.5$ and the
approximation is strong due to the small values of the roll-off factor and the
sampling error leading to negligible values as the powers of the sampling
error $\epsilon$ increase.

Thus, the mean amplitude, $\bar{\epsilon}_{1}$, and the mean squared
amplitude, $\bar{\epsilon}_{2}$, at the moment of sampling can be strongly
approximated as described by the following lemma.

###### Lemma 1

The mean amplitude value and the mean squared amplitude value of the
transmitted OTA signal at $t=0$ are strongly approximated by

$\bar{\epsilon}_{1}=1+\sum_{m=1}^{\infty}\kappa_{m}\sigma_{\epsilon}^{2m}(2m-1)!!+4\alpha^{2}\sum_{m=0}^{\infty}\kappa_{m}\sigma_{\epsilon}^{2m+2}(2m+1)!!,$
(11)

where $\kappa_{m}$ denotes the coefficients given as

$\kappa_{m}=\sum_{n=0}^{m}\frac{(-1)^{m}\pi^{2m}\alpha^{2(m-n)}}{(2n+1)!(2m-2n)!}$
(12)

and

$\displaystyle\bar{\epsilon}_{2}$
$\displaystyle=1+\sum_{p=1}^{\infty}\lambda_{p}\sigma_{\epsilon}^{2p}(2p-1)!!+8\alpha^{2}\sum_{p=0}^{\infty}\lambda_{p}\sigma_{\epsilon}^{2p+2}(2p+1)!!$
(13)
$\displaystyle\qquad+16\alpha^{4}\sum_{p=0}^{\infty}\lambda_{p}\sigma_{\epsilon}^{2p+4}(2p+3)!!,$

where $\lambda_{p}$ denotes the coefficients given as

$\lambda_{p}=\sum_{n=0}^{p}\sum_{l=0}^{p-n}\sum_{k=0}^{p-n-l}\frac{(-1)^{p}\pi^{2p}\alpha^{2(p-l-n)}}{(2n+1)!(2l+1)!(2k)!(2(p-n-
l-k))!}\\!.$ (14)

###### Proof:

The proof is presented in Appendix A. ∎

Then, taking the expected value of (9) w.r.t. the sampling error $\epsilon$
occurring in the OTA computing signal, it is straightforward to prove that the
MSE under time sampling error is given as

$\mathrm{MSE}(a,\bm{b})=\sum_{k=1}^{K}\left(\left(ab_{k}h_{k}\right)^{2}\bar{\epsilon}_{2}+1\right)-2\sum_{k=1}^{K}ab_{k}h_{k}\bar{\epsilon}_{1}+\sigma^{2}a^{2}.$
(15)

It is important to emphasize that the following power policy allocation
technique is not restricted by the use of the approximations given in (11) and
(13) and can be used by numerically calculating
$\bar{\epsilon}_{1}=\mathbb{E}_{\epsilon}[z_{k}(\epsilon T)]$ and
$\bar{\epsilon}_{2}=\mathbb{E}_{\epsilon}[z_{k}^{2}(\epsilon T)]$. However,
the strong approximation in (15) shows interest because it allows a direct
calculation of the roll-off factors effect on the MSE in contrast to the
numerical approach, meaning that it can also be used to solve optimization
problems that would include the roll-off factor as part of the optimization
variables, for instance in joint MSE-bandwidth problems

The MSE in (15) can now be minimized by finding the optimal power allocation
of the OTA transmission. This problem is formulated as

$\begin{array}[]{ll}\mathop{\mathrm{min}}\limits_{a,\bm{b}}&\mathrm{MSE}(a,\bm{b})\\\
\,\,\textbf{s.t.}&\mathrm{C}_{1}:b_{k}\leq\sqrt{P},\ \forall
k\in\mathcal{K}.\end{array}$ (P1)

The Lagrangian can be written as

$\mathcal{L}=\mathrm{MSE}(a,\bm{b})+\sum_{k=1}^{K}\lambda_{k}(b_{k}-\sqrt{P})$
(16)

where $\lambda_{k}$ are the Lagrange multipliers.

By using the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution must
satisfy the following:

$\frac{\partial\mathcal{L}}{\partial a}=0,\frac{\partial\mathcal{L}}{\partial
b_{k}}=0,\forall k\in\mathcal{K}$ (17)

and

$\lambda_{k}(b_{k}-\sqrt{P})=0,\forall k\in\mathcal{K},$ (18)

which equivalently leads to

$\frac{\partial\mathrm{MSE}(a,\bm{b})}{\partial b_{k}}=0\Rightarrow
b_{k}=\bar{\epsilon}_{1}/(ah_{k}\bar{\epsilon}_{2})\text{ }\mathrm{or}\text{
}b_{k}=\sqrt{P},\,\,\forall k\in\mathcal{K}.$ (19)

Also, since $\mathrm{MSE}(a,\bm{b})$ is a convex function with respect of
$b_{k}$, it holds that

$b_{k}=\min\left\\{\sqrt{P},\frac{\bar{\epsilon}_{1}}{ah_{k}\bar{\epsilon}_{2}}\right\\},\forall
k\in\mathcal{K}.$ (20)

Based on (20), to further reduce the size of the set of the potential optimal
points, it is sufficient to consider $K$ different cases for the values of
$\lambda_{k}$. More specifically, it is noted that if there exists a device
$(i+1)\in\mathcal{K}$ such that
$b_{i+1}=\bar{\epsilon}_{1}/(ah_{i+1}\bar{\epsilon}_{2})$, any device
$j\in\\{\mathcal{K}|j>i\\}$ can select the same inverse channel-like transmit
power due to the ascending channel order. Thus, it holds that if
$\lambda_{i+1}=0$, then $\lambda_{j}=0,\forall j>i$. Regarding the $i$-th
case, by using (20), the Lagrangian simplifies to the MSE for the
corresponding values of $\lambda_{k}$ and $b_{k}$, which can be written as

$\displaystyle\mathcal{L}_{i}=\mathrm{MSE}_{i}(a)$
$\displaystyle=a^{2}\left(\sum_{k=1}^{i}Ph^{2}_{k}\bar{\epsilon}_{2}+\sigma^{2}\right)-2a\sum_{k=1}^{i}\sqrt{P}h_{k}\bar{\epsilon}_{1}$
(21)
$\displaystyle\quad+\sum_{k=i+1}^{K}\left(1-\frac{\bar{\epsilon}_{1}^{2}}{\bar{\epsilon}_{2}}\right)+i.$

It should be noted that for the primal feasibility conditions of (P1), i.e.,
$\mathrm{C}_{1}$, to be satisfied, since the channel gains have been ordered,
the following must hold for the receiver gain

$a\geq\frac{\bar{\epsilon}_{1}}{\sqrt{P}h_{i+1}\bar{\epsilon}_{2}}.$ (22)

If (22) is satisfied, the minimum value of (21) is reached when
$\partial{\mathcal{L}_{i}}/\partial a=0$, i.e.,

$a=a_{i}=\frac{\sqrt{P}\bar{\epsilon}_{1}\sum_{k=1}^{i}h_{k}}{P\bar{\epsilon}_{2}\sum_{k=1}^{i}h^{2}_{k}+\sigma^{2}}.$
(23)

Then, we can define the set of optimal solutions of $a$ as follows

$\mathcal{A}=\left\\{a=a_{i}\bigg{|}a_{i}\geq\frac{\bar{\epsilon}_{1}}{\sqrt{P}h_{i+1}\bar{\epsilon}_{2}},\forall
i\in\mathcal{K}\right\\}.$ (24)

Observe that if $a\not\in\mathcal{A}$, it cannot be optimal since the KKT
conditions described by (17) are not satisfied. Comparing the values of the
sequence $\mathrm{MSE}_{i}(a_{i}),\ \forall i\in\mathcal{K},\ \forall
a_{i}\in\mathcal{A}$, described by (21), we can identify the number of devices
$i^{*}$ that must transmit with maximum power and is equal to

$i^{*}=\underset{\begin{subarray}{c}1\leq i\leq K\\\
a_{i}\in\mathcal{A}\end{subarray}}{\mathrm{argmin}}\,\,\\{\mathrm{MSE}_{i}(a_{i})\\}.$
(25)

Then, the optimal power allocation at the devices and the receiver can be
calculated by combining (25), (23) and (20) in this specific order.

###  III-B OTA Computing under Time Sampling Error and ISI

In limited bandwidth systems, in addition to the time sampling error, ISI is
also present. Although waveforms such as RC and BTRC have relatively small
amplitudes around their roots at multiples of $T$, the effect of ISI cannot be
neglected, especially in multiple access schemes such as OTA computing where a
large number of connected devices are present in the system. As a result, the
study of ISI is important for understanding the performance of OTA computing
in practice and for extracting optimal power allocation policies to mitigate
its effect.

In this case the received signal is affected by the currently transmitted
signal as well as earlier and later transmissions captured at time instances
$t=T(q+\epsilon)$ where $q\in\\{-\mu/2,\cdots,\mu/2\\}$ and the received
signal is given by (7). Therefore, the MSE now contains additional terms due
to ISI terms and is equal to

$\displaystyle\mathrm{MSE}^{\mathrm{ISI}}(a,\bm{b})$
$\displaystyle=\sum_{k=1}^{K}\left(az_{k}(\epsilon
T)b_{k}h_{k}-1\right)^{2}+\sigma^{2}a^{2}$ (26)
$\displaystyle\quad+\sum_{\begin{subarray}{c}q=-\frac{\mu}{2}\\\ q\neq
0\end{subarray}}^{\frac{\mu}{2}}\sum_{k=1}^{K}\left(az_{k}(T(q+\epsilon))b_{k}h_{k,q}\right)^{2},$

where $h_{k,q}$ denotes the channel of the $k$-th device at the $q$-th time
instance affected by the delay spread. Since the samples at different times
are independent of each other, $\mathbb{E}_{x_{k}}[x_{k,q_{1}}x_{k,q_{2}}]=0$,
whenever $q_{1}\neq q_{2}$. Then, taking the expectation w.r.t. $\epsilon$ we
derive an expression similar to the one in (P1) described by

$\displaystyle\mathrm{MSE}^{\mathrm{ISI}}(a,\bm{b})$
$\displaystyle\\!=\\!\sum_{k=1}^{K}\left(\left(ab_{k}h_{k}\right)^{2}\tilde{\epsilon}_{0}+1\right)-2\sum_{k=1}^{K}ab_{k}h_{k}\check{\epsilon}$
(27) $\displaystyle\quad+\sum_{\begin{subarray}{c}q=-\frac{\mu}{2}\\\ q\neq
0\end{subarray}}^{\frac{\mu}{2}}\sum_{k=1}^{K}\left(\left(ab_{k}h_{k,q}\tilde{\epsilon}_{q}\right)^{2}\right)+\sigma^{2}a^{2},$

where $\tilde{\epsilon}_{q}=\mathbb{E}_{\epsilon}[z_{k}^{2}(T(q+\epsilon))]$
and $\check{\epsilon}=\mathbb{E}_{\epsilon}[z_{k}(\epsilon T)]$. In
particular, for the strongest symbol interference caused by the previous and
the next symbol of the currently sampled symbol, the mean values can be
rigorously calculated in closed form as described by the following lemma.

###### Lemma 2

The mean squared amplitude of the raised cosine waveform at the time instances
corresponding to $q=\pm 1$ with roll-off factor $\alpha=0$ is equal to

$\tilde{\epsilon}_{1}=1+\sum\limits_{m=1}^{\infty}\left(\sum\limits_{n=0}^{2m}A_{n}A_{2m-n}+B_{n}B_{2m-n}\right)\sigma_{\epsilon}^{2m}(2m-1)!!,$
(28)

and $\tilde{\epsilon}_{-1}=\tilde{\epsilon}_{1}$, where the coefficients
$A_{n},B_{n}$ are given as

$\begin{split}A_{n}&=\sum\limits_{l=\frac{n+1}{2}}^{\infty}\frac{(-1)^{l}\pi^{2l}}{(2l+1)!}\binom{2l}{n}\text{
if }n\text{ odd, }A_{n}=0\text{ if }n\text{ even }\\\
B_{n}&=\sum\limits_{l=\frac{n}{2}}^{\infty}\frac{(-1)^{l}\pi^{2l}}{(2l+1)!}\binom{2l}{n}\text{
if }n\text{ even, }B_{n}=0\text{ if }n\text{ odd }.\end{split}$ (29)

###### Proof:

The proof is presented in Appendix B. ∎

Moreover, the transmitting devices have no knowledge of the dynamically
changing characteristics of the channel, thus they assume that the same
channel is present at all time instances, i.e., each device assumes that
$h_{k,q}=h_{k},\forall q$.

Considering ISI, the new optimization problem to solve can be formulated as

$\begin{array}[]{ll}\mathop{\mathrm{min}}\limits_{a,\bm{b}}&\mathrm{MSE}^{\mathrm{ISI}}(a,\bm{b})\\\
\,\,\textbf{s.t.}&\mathrm{C}_{1}:b_{k}\leq\sqrt{P},\forall
k\in\mathcal{K}.\end{array}$ (P2)

The Lagrangian of the latter can be written as

$\mathcal{L}^{\mathrm{ISI}}=\mathrm{MSE}^{\mathrm{ISI}}(a,\bm{b})+\sum_{k=1}^{K}\lambda_{k}^{\mathrm{ISI}}(b_{k}-\sqrt{P})$
(30)

where $\lambda^{\mathrm{ISI}}_{k}$ are the Lagrange multipliers.

By using the KKT conditions, the optimal solution must satisfy the following
conditions:

$\frac{\partial\mathcal{L}^{\mathrm{ISI}}}{\partial
a}=0,\frac{\partial\mathcal{L}^{\mathrm{ISI}}}{\partial b_{k}}=0,\forall
k\in\mathcal{K}$ (31)

and

$\lambda^{\mathrm{ISI}}_{k}\mathcal{L}^{\mathrm{ISI}}(b_{k}-\sqrt{P})=0,\forall
k\in\mathcal{K}.$ (32)

which equivalently leads to

$\frac{\partial\mathrm{MSE}^{\mathrm{ISI}}(a,\bm{b})}{\partial
b_{k}}=0\Rightarrow b_{k}=\check{\epsilon}/(ah_{k}\hat{\epsilon})\text{
}\mathrm{or}\text{ }b_{k}=\sqrt{P},\,\forall k\in\mathcal{K}.$ (33)

Also, since $\mathrm{MSE}(a,\bm{b})$ is a convex function with respect of
$b_{k}$, it holds that

$b_{k}=\min\left\\{\sqrt{P},\frac{\check{\epsilon}}{ah_{k}\hat{\epsilon}}\right\\},\forall
k\in\mathcal{K}.$ (34)

For (P2), similar to (P1), based on (20), to further reduce the size of the
set of the potential optimal points, it is sufficient to consider $K$
different cases for the values of $\lambda^{\mathrm{ISI}}_{k}$. More
specifically, it is noted that if there exists a device $(i+1)\in\mathcal{K}$
such that $b_{i+1}=\check{\epsilon}/(ah_{i+1}\hat{\epsilon})$, any device
$j\in\\{\mathcal{K}|j>i\\}$ can select the same inverse channel-like transmit
power due to the ascending channel order. Thus, it holds that if
$\lambda^{\mathrm{ISI}}_{i+1}=0$, then $\lambda^{\mathrm{ISI}}_{j}=0,\forall
j>i$. Regarding the $i$-th case, by using (34), the Lagrangian simplifies to
the MSE for the corresponding values of $\lambda^{\mathrm{ISI}}_{k}$ and
$b_{k}$, which can be written as

$\displaystyle\mathcal{L}^{\mathrm{ISI}}_{i}=\mathrm{MSE}_{i}^{\mathrm{ISI}}(a)$
$\displaystyle=a^{2}\left(\sum_{k=1}^{i}Ph^{2}_{k}\hat{\epsilon}+\sigma^{2}\right)-2a\sum_{k=1}^{i}\sqrt{P}h_{k}\check{\epsilon}$
(35)
$\displaystyle\quad+\sum_{k=i+1}^{K}\left(1-\frac{\check{\epsilon}^{2}}{\hat{\epsilon}}\right)+i.$

It should be noted that for the primal feasibility conditions of (P2), i.e.,
$\mathrm{C}_{1}$, to be satisfied, since the channel gains have been ordered,
the following must hold for the receiver gain factor

$\frac{\check{\epsilon}}{\sqrt{P}h_{i+1}\hat{\epsilon}}\leq a.$ (36)

If (36) is satisfied, the minimum value of (35) is reached when
$\partial{\mathcal{L}^{\mathrm{ISI}}_{i}}/\partial a=0$, i.e.,

$a=a_{i}=\frac{\sqrt{P}\check{\epsilon}\sum_{k=1}^{i}h_{k}}{P\hat{\epsilon}\sum_{k=1}^{i}h^{2}_{k}+\sigma^{2}}.$
(37)

Then, similar to the non-ISI case, we can define the set of optimal solutions
of $a$ as follows

$\mathcal{A}^{\mathrm{ISI}}=\left\\{a=a_{i}\bigg{|}a_{i}\geq\frac{\check{\epsilon}}{\sqrt{P}h_{i+1}\hat{\epsilon}},\forall
i\in\mathcal{K}\right\\}.$ (38)

Observe that if $a\not\in\mathcal{A}^{\mathrm{ISI}}$, it cannot be optimal
since the KKT conditions described by (31) are not satisfied. Comparing the
values of the sequence $\mathrm{MSE}^{\mathrm{ISI}}_{i}(a_{i}),\ \forall
i\in\mathcal{K},\ \forall a_{i}\in\mathcal{A}^{\mathrm{ISI}}$, described by
(35), we can identify the number of devices $i^{*}$ that must transmit with
maximum power and is equal to

$i^{*}=\underset{\begin{subarray}{c}1\leq i\leq K\\\
a_{i}\in\mathcal{A}^{\mathrm{ISI}}\end{subarray}}{\mathrm{argmin}}\,\,\\{\mathrm{MSE}^{\mathrm{ISI}}_{i}(a_{i})\\}.$
(39)

Then, the optimal power allocation at the devices and the receiver can be
calculated by combining (39), (37) and (34) in this specific order.

From (34) and (37), it is evident that the MSE of an OTA transmission is
affected by the statistics of the selected waveform. Therefore, it is
important to emphasize that the theoretical analysis presented in this section
is general and can be used to minimize the MSE for any chosen waveform,
including RC and BTRC. This will be also used later to perform optimal
waveform design for the OTA transmission.

##  IV DNN-generated Waveforms

Let $\bm{w}$ denote the weights of the DNN and $z_{\bm{w}}$ denote the output
of the DNN model. To approximate a continuous waveform with the DNN, time
discretization is used, i.e., each element of the output vector $z_{\bm{w}}$
corresponds to a specific time instance of the desired continuous waveform.
Moreover, in practice, a windowed version of a waveform is transmitted, thus
we assume that the waveform of the DNN, and the RC, BTRC waveforms expand only
to a limited time window. The larger the time window, the more ISI symbols,
$\mu$, are considered. To design the waveform, we utilize supervised learning
by feeding as input to the DNN the vector
$\bm{u}=[a,\bm{b},\bm{h},\bm{x}]\in\mathbb{R}^{3K+1}$, where
$\bm{h}=[h_{1},h_{2},\cdots,h_{K}]$ denotes the channels of the devices and
$\bm{x}=[x_{1,0},x_{2,0},\cdots,x_{K,0}]$ denotes the transmitted data at the
current transmission time. It should be noted that we initialize the power
allocation $a,\bm{b}$ with the values that correspond to the same OTA
transmission without sampling error and ISI as found in [11]. The rationale
behind this choice is that, as shown in Section III, finding the optimal
$a,\bm{b}$ requires the statistics of the waveform, which are unknown a
priori. Nevertheless, we note that to test the extracted waveform, after the
DNN has converged, we obtain the optimal power allocation for the waveform
through the analysis of Section III \- specifically equations (34),(37),(39) -
and then calculate its MSE. It should be noted that this iterative procedure
is performed only once. An alternating optimization inspired method was also
tested, where the optimal values of $a,\bm{b}$ are given as input to the DNN
which is retrained, and new values for $a,\bm{b}$ are calculated. However,
this iterative alternating optimization approach offered trivial performance
gains, thus it was not used.

The architecture of the designed DNN consists of three fully connected hidden
layers, each consisting of multiple neurons and implementing the rectified
linear unit (ReLU) activation function. The output layer consists of $N_{s}$
output nodes which are equal to the number of discretized time instances
considered. Assuming that the $N_{s}$ output nodes of the DNN correspond to
$\mu+1$ time periods of the generated waveform, the waveform expands in the
time domain from $-(\mu+1)T/2$ to $(\mu+1)T/2$. Then, the time resolution of
the waveform is $\Delta t=(\mu+1)T/(N_{s}-1)$ and the time instances
correspond to $t=m\Delta t$, with $-(N_{s}-1)/2\leq m\leq(N_{s}-1)/2$ and
$m\in\mathbb{Z}$. To train the DNN, a training dataset is generated, denoted
as $\\{\bm{u}_{d},r_{d}\\}_{d\in\mathcal{D}}$, where $\mathcal{D}$ is the set
of all training samples and $\bm{u}_{d}$ is the $d$-th input sample and
$r_{d}=\left\lVert\bm{x}_{d}\right\rVert_{0}$ is the target value. The
received signal from the generated waveform is given similarly to (7) as

$\displaystyle y_{\mathrm{DNN}}[\bm{w},$
$\displaystyle\bm{u}_{d},m]=a\left(\sum_{k=1}^{K}x_{k,0}z_{k_{\bm{w}}}[m]b_{k}h_{k,0}+n\right)$
(40)
$\displaystyle+a\underbrace{\left(\sum_{\begin{subarray}{c}q=-\frac{\mu}{2}\\\
q\neq
0\end{subarray}}^{\frac{\mu}{2}}\sum_{k=1}^{K}x_{k,q}z_{k_{\bm{w}}}[m+qT]b_{k}h_{k,q}\right)}_{\text{ISI
terms}}.$

It should be noted that since the channel response at different time instances
is unknown, $x_{k,q},h_{k,q}$ are passed as arguments to the loss function,
but cannot be considered as inputs to the DNN framework itself. However, since
both draw values from known distributions, the training phase will allow the
DNN to capture their general statistics.

For the DNN to generate a valid and feasible waveform, the DNN-based waveform
must meet the same criteria as the waveforms studied in the literature.
Therefore, it is crucial to ensure that the DNN-based waveform has the same
energy as the other waveforms and that it uses the same bandwidth to ensure
that the same resources in terms of energy and frequency are utilized for
fairness. Both of these constraints must be included in the loss function of
the DNN model to be considered. Regarding the MSE of the OTA transmission, a
challenge arises due to the stochastic nature of the time sampling error. In
particular, since the sampling error is random, it is not known a priori which
value of the waveform should be used in (40). In essence, considering the time
discretization of the DNN output, during each forward pass a random index of
the DNN output should be chosen and the waveform value corresponding to this
index should be used in (40). However, this approach causes the gradient of
the output to be lost during the forward propagation and the DNN cannot be
trained. To overcome this problem, the sampling error distribution is
approximated by an equivalent probability mass function, denoted as
$f_{P}[m]$, with points at the discrete time instances considered by the DNN.
For practical reasons, we assume that no sampling error greater than half a
period can occur, i.e., $f_{P}[m]$ is non-zero only for $-T/2\leq t\leq T/2$
or equivalently $-(N_{s}-1)/(2(\mu+1))\leq m\leq(N_{s}-1)/(2(\mu+1))$. Then,
the loss function of the DNN is given by

$\mathcal{L}_{\mathrm{MSE}}=\frac{1}{|\mathcal{D}|}\sum_{d\in\mathcal{D}}\sum_{m=-\frac{N_{s}-1}{2(\mu+1)}}^{\frac{N_{s}-1}{2(\mu+1)}}f_{P}[m]\big{(}y_{\mathrm{DNN}}[\bm{w},\bm{u}_{d},m]-r_{d}\big{)}^{2}.$
(41)

The utilized bandwidth, as expressed by the discrete Fourier transform (DFT)
of the waveform, must first be considered for the bandwidth constraint.
Considering the time resolution $\Delta t$, the frequency resolution can be
found equal to $\Delta f=1/((\mu+1)T)-1/((\mu+1)TN_{s})\approx 1/((\mu+1)T)$,
since the number of output nodes as well as the overall studied time duration
are in general large. As known, for a baseband signal like the waveforms
described by (3) and (4) the utilized bandwidth is given as $W=1/(2T)$. Thus,
the effect of the roll-off factor on the total bandwidth spread can be studied
for a resolution step of $1/((\mu+1)T)$, with $\alpha=0$ corresponding to the
$\mu/2$-th sample, since then we obtain bandwidth utilization $W=1/(2T)$, and
$\alpha=1$ corresponding to the $\mu$-th sample of the DFT, since then the
previous utilized bandwidth is doubled. Obviously, this has a restrictive
effect on the roll-off factors that can be investigated, thus a way to
increase the frequency resolution is needed. This can be achieved by applying
zero-padding, which expands a signal in time to achieve the desired frequency
resolution. Denoting the additional duration of the waveforms due to zero-
padding as $T^{\mathrm{ZP}}$, the frequency resolution is correspondingly
increased in $\Delta f^{\mathrm{ZP}}\approx 1/((\mu+1)T+T^{\mathrm{ZP}})$
which can be derived as before but using the extended time duration of the
signal. In this way, the number of roll-off factors to be examined can be
chosen so that the frequency resolution corresponds to the desired roll-off
factor step. For example, if the examined roll-off factors have a step of
$\Delta\alpha=0.1$ corresponding to a desired frequency resolution equal to
$\Delta W=\Delta\alpha/(2T)$, we can choose $T^{\mathrm{ZP}}$ to satisfy

$\frac{1}{(\mu+1)T+T^{\mathrm{ZP}}}=\frac{\Delta\alpha}{2T}\Leftrightarrow
T^{\mathrm{ZP}}=\left(\frac{2}{\Delta\alpha}-\mu-1\right)T,$ (42)

and since the symbol period $T$ corresponds to a number of $(N_{s}-1)/(\mu+1)$
samples, it is straightforward to calculate the desired number of samples
needed to perform zero-padding which are denoted as $N_{s}^{\mathrm{ZP}}$ and
for which $T^{\mathrm{ZP}}=N_{s}^{\mathrm{ZP}}\Delta t$ holds. Therefore, to
ensure that the generated waveform has the same bandwidth utilization as the
other waveforms, a constraint is applied to keep the frequency response
outside the desired spectrum as close to zero as possible. Note that if the
term of the parenthesis in (42) is negative then the duration of the signal is
enough to capture the desired frequency resolution and no zero-padding is
needed, resulting in $N_{s}^{\mathrm{ZP}}=0$. With these in mind, we use the
following as part of the loss function

$\mathcal{L}_{f}=\frac{1}{|\mathcal{D}|}\sum_{d\in\mathcal{D}}\sum_{n>N_{t}(\alpha)}^{N_{s}+N_{s}^{\mathrm{ZP}}}\big{(}\mathcal{F}(z_{\bm{w}}[m])[n]-\Gamma_{\mathrm{thr}}\big{)}^{2},$
(43)

where $N_{t}(\alpha)$ is the DFT sample after which no bandwidth is used for a
given frequency resolution, $\Gamma_{\mathrm{thr}}$ is a frequency response
threshold near zero, and $\mathcal{F}(\cdot)$ denotes the DFT. For example,
assuming that $\Delta\alpha=0.1$ we want to generate a waveform for
$\alpha=0.3$, it would be $N_{t}(0.3)=14$ as a result of the chosen frequency
resolution and the studied roll-off factor. Since the frequency resolution is
$\Delta W=0.1/(2T)$, the first $11$ samples expand in a bandwidth equal to
that for $\alpha=0$, and $3$ more samples are needed to get to the utilized
bandwidth when $\alpha=0.3$. Therefore, for different values of the roll-off
factor, $\mathcal{L}_{f}$ must be adjusted accordingly by $N_{t}(\alpha)$ to
capture the used bandwidth of the waveform found by the DFT.

In addition to equal bandwidth utilization, it is important to ensure that the
generated waveform has similar energy to the others for a fair comparison. The
energy of the generated waveform can be calculated using Parseval’s theorem in
its DFT form and is equal to

$E_{\mathrm{DNN}}=\frac{1}{N}\sum_{n=0}^{N-1}|\mathcal{F}(z_{\bm{w}}[m])[n]|^{2},$
(44)

where $N=N_{s}+N_{s}^{\mathrm{ZP}}$ is the total number of samples in the
zero-padded version of the waveform that is extracted through (42). Then,
assuming that $E$ is the target energy of another waveform to compare to and
is calculated in the same way, we arrive at the following constraint, which is
passed as part of the loss function

$\mathcal{L}_{e}=\frac{1}{|\mathcal{D}|}\sum_{d\in\mathcal{D}}\big{(}E_{\mathrm{DNN}}-E\big{)}^{2}.$
(45)

Note that, as explained in Section II, current communication systems employ a
two-part waveform break between the transmitter and receiver with each part
having the following frequency response $\sqrt{|\mathcal{F}(z_{\bm{w}}[m])|}$.
While this does not affect (43), it does affect the form of (44), which can
now be written equivalently as

$E_{\mathrm{DNN}}=\frac{1}{N}\sum_{n=0}^{N-1}|\mathcal{F}(z_{\bm{w}}[m])[n]|.$
(46)

Obviously, constraint (45) changes accordingly, while the MSE term in the loss
function remains unchanged.

Finally, the generated waveform requires symmetry for its samples w.r.t. the
ideal sampling time. To achieve this, a constraint is introduced that aims to
minimize the distance between the negative and positive parts of the waveform,
as follows

$\mathcal{L}_{s}=\frac{1}{|\mathcal{D}|}\sum_{d\in\mathcal{D}}\sum_{m=1}^{\frac{N_{s}-1}{2}}\big{(}z_{\bm{w}}[m]-z_{\bm{w}}[-m]\big{)}^{2}.$
(47)

Combining all of the above loss function components described by (41), (43),
(45), and (47) yields a total training loss function that satisfies all of the
required constraints and aims to minimize the weighted MSE of the received
signal for more accurate OTA computing

$\mathcal{L}_{tot}=\mathcal{L}_{\mathrm{MSE}}+M_{f}\mathcal{L}_{f}+M_{e}\mathcal{L}_{e}+M_{s}\mathcal{L}_{s}.$
(48)

Note that $M_{f},M_{e}$ and $M_{s}$ act as penalty factors, meaning that if
their corresponding constraint is violated, the training loss will increase,
so that the training will take the constraint into account and learn to stay
within small violation limits. The same reasoning applies to all considered
constraints in order to ensure that bandwidth, energy, and symmetry violations
are kept at extremely low levels, which in turn allows the generation of a
waveform that has the same characteristics as RC and BTRC, but outperforms
both due to the consideration of MSE in the training phase.

However, during training, the DNN will generate a different waveform for each
input sample of the training dataset. Therefore, in order to generate only one
waveform that shows good performance for each input sample of the dataset,
during testing only, we generate a waveform by averaging all the generated
output waveforms for all input samples. Since the imposed constraints in the
loss function have the distributive property, this final waveform also
satisfies all constraints. This final waveform is the one we plot in Section V
and on which the MSE is calculated.

##  V Simulation Results

In this section, we present the simulation results. For all simulations, we
assume that the channel fading follows the circularly symmetric complex
Gaussian distribution, i.e., $h_{k}\sim\mathcal{CN}(0,1),\ \forall
k\in\mathcal{K}$, and without loss of generality, the transmitted data are
generated from a uniform distribution in the interval $[-\sqrt{3},\sqrt{3}]$.
Unless otherwise stated, we assume that $K=20$ devices participate in the OTA
transmission and that the maximum transmit power at each device is such that
the transmit SNR is $P_{\mathrm{max}}/\sigma^{2}=10$ dB. We study the problem
of MSE minimization for $\mu=6$ ISI symbols, which means that the time
duration of the studied waveforms extends from $-3.5T$ to $3.5T$, and we are
interested in generating waveforms that perform better for a roll-off factor
step equal to $\Delta\alpha=0.1$, as an indicator for the effectiveness of the
proposed framework while it also provides a satisfactory performance
estimation for the whole range of roll-off factors.

Figure 3: DNN generated waveforms for OTA computing and various roll-off
factors.

###  V-A DNN Setup

Regarding the DNN setup, the DNN has $3$ hidden layers, each consisting of
$256$ neurons, while the ReLU activation function was used. For each simulated
roll-off factor of interest, a training dataset of $|\mathcal{D}|=5\times
10^{4}$ channel and data realizations is generated, while the corresponding
optimal power allocation vectors were calculated according to (37) and (34).
The validation and test datasets are generated in a similar manner and consist
of $5\times 10^{3}$ and $5\times 10^{4}$ realizations, respectively. The Adam
optimizer was used, while the DNN loss function was defined in (48). A batch
size of 100 was selected, while the DNN was trained for 10 epochs.

Regarding the parameters associated with (48), the number of outputs of the
proposed DNN is selected equal to $N_{s}=3501$ samples, which allows a time
resolution of $\Delta t=T/500$. To achieve the desired spectral resolution,
the zero-padded version of the waveform requires an additional
$N_{s}^{\mathrm{ZP}}=6500$ time instances as indicated by (42). For the
penalty factors, we selected $M_{f}=22.4,M_{e}=9000,M_{s}=4300$ and for the
bandwidth constraint we selected $\Gamma_{\mathrm{thr}}=0.2$ as a reasonably
small frequency response. Other parameter values can also be selected in the
spirit of experimentation.

TABLE I: Waveform Approximation Approximation Coefficients
---
Coef. $\alpha$ | $\alpha=0.2$ | $\alpha=0.5$ | $\alpha=0.8$
$a_{0}$ | $0.0939$ | $0.1313$ | $0.1360$
$a_{1}$ | $0.2168$ | $0.2638$ | $0.2507$
$a_{2}$ | $0.1841$ | $0.2371$ | $0.2046$
$a_{3}$ | $0.2092$ | $0.1676$ | $0.1712$
$a_{4}$ | $0.1647$ | $0.1406$ | $0.1315$
$a_{5}$ | $0.0950$ | $0.0764$ | $0.0994$
$a_{6}$ | $0.0121$ | $0.0053$ | $0.0405$
$p$ | $0.6481$ | $0.8378$ | $0.8739$
RMSE | $3.041\times 10^{-3}$ | $3.094\times 10^{-3}$ | $3.132\times 10^{-3}$

###  V-B Generated Waveforms

In Fig. 3, waveforms for various roll-off factors are depicted, each
normalized in time. These waveforms exhibit behavior reminiscent of both RC
and BTRC waveforms, with larger sidelobes for smaller $\alpha$ values and
steeper decay for larger $\alpha$ values. This behavior has significant
implications for system performance, particularly in terms of mitigating
sampling error and ISI. For small values of the roll-off factor $\alpha$, the
generated waveforms illustrate slower amplitude decay. This characteristic
helps mitigate the effects of sampling error, which occurs when a signal is
not sampled frequently enough, resulting in inaccuracies in its representation
at discrete time points. The slower decay allows the system to better
accommodate such errors, thereby enhancing overall performance. Conversely,
larger values of $\alpha$ increase the attenuation of ISI, which occurs when
symbols in a digital communication system interfere with each other due to
channel characteristics. In this scenario, the smaller amplitudes of the
sidelobes in the waveforms generated for large $\alpha$ values contribute to
the reduction of ISI, thereby improving the system robustness. It is noted
that although the maximum values of the generated waveforms are close, they
are different. This is a direct consequence of the fact that channel fading is
considered during the training phase.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$00.020.040.06DFT SampleFrequency
ResponseBTRCDNN-Gen$a=0.2$$a=0.5$$a=0.8$ Figure 4: Comparison of bandwidth
utilization through DFT.

In Fig. 4 the bandwidth utilization of the BTRC waveform and the DNN-generated
waveforms are presented for some indicative roll-off factors. As can be
observed, all waveforms have the same spectrum spread in terms of DFT samples
thus, proving the efficiency of the bandwidth-related constraint introduced in
the proposed DNN framework through (43). This guarantees that the extracted
waveform can be useful in practice by not causing increased intercarrier
interference.

To facilitate the testing of our results without the need to create the DNN
architecture, we provide the curve fitting parameters for the DNN-based
waveforms of 3 in Table I. This may also allow practical deployment in real
communication systems, since curve fitting can generate waveforms for
different roll-off factors, $\alpha$. We use sinusoidal functions for fitting
due to the symmetric shape of the curves, thus the generated waveforms can be
represented as follows

$\displaystyle\hat{z}_{\bm{w}}(t)$
$\displaystyle=a_{0}+a_{1}\cos(pt)+a_{2}\cos(2pt)+a_{3}\cos(3pt)$ (49)
$\displaystyle\quad+a_{4}\cos(4pt)+a_{5}\cos(5pt)+a_{6}\cos(6pt),$

with $p$ and $a_{j},j\in\\{0,\cdots,6\\}$ being adjustable coefficients for
the approximation. The root MSE (RMSE) of the fitting approximation is given
in Table I as well.

###  V-C Performance Results

In this subsection, we evaluate the performance of the generated waveforms
along with the proposed optimal power policies discussed in Section III,
aiming to minimize the MSE.

####  V-C1 OTA under Sampling Error

In Fig. 5, we observe the closeness of the proposed approximation to the
statistics of the RC waveform, as outlined in Lemma 1. Both
$\bar{\epsilon}_{1}$ and $\bar{\epsilon}_{2}$, closely approximate the actual
statistics of the waveform although, as the roll-off factor increases, the gap
between the bounds and the actual statistics gets slightly bigger. This is
expected, since a higher roll-off factor leads to a slightly larger
discrepancy in the approximation of the term $1/(1-4\alpha^{2}\epsilon^{2})$.
Nevertheless, the proposed approximations remain extremely tight (around
$0.1\%$ and $0.3\%$ at most, respectively) and can effectively approximate the
average MSE, as calculated in (P1).

$0.2$$0.25$$0.3$$0.35$$0.4$$0.45$$0.5$$0.55$$0.6$$0.65$$0.7$$0.75$$0.8$$0.96$$0.96$$0.97$$0.97$$0.98$$0.98$$0.99$Roll-
off factor $\alpha$Waveform ResponseActual
$\mathbb{E}_{\epsilon}[z_{k}(\epsilon)]$$\bar{\epsilon}_{1}$Actual
$\mathbb{E}_{\epsilon}[z_{k}^{2}(\epsilon)]$$\bar{\epsilon}_{2}$ Figure 5:
Approximation of mean values for the RC waveform based on Lemma 1 when only
sampling error with $\sigma_{\epsilon}=0.1$ occurs.
$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$0.055$$0.056$$0.057$$0.058$$0.059$$0.060$$0.061$$0.062$$0.063$$0.064$$0.065$Roll-
off factor $\alpha$Average MSERCBTRCDNN-Gen Figure 6: Waveform MSE performance
for varying roll-off factor and sampling error variance
$\sigma_{\epsilon}=0.1$ without the presence of ISI when there are $K=20$
devices.

In Fig. 6, we compare the MSE performance of established and DNN-generated
waveforms under only the effect of sampling error. Notably, optimal
performance for all waveforms is achieved for mid-range roll-off factor
values, where the RC waveform performs best. This result is due to the fact
that the primary goal of both the BTRC and the DNN-generated waveforms is to
mitigate ISI, resulting in inferior performance when only sampling error is
present. Nevertheless, the DNN-generated waveform demonstrates competitive
performance across all roll-off factors, even outperforming RC and BTRC for
small to medium values of $\alpha$. It should be noted that the DNN-based
waveform in Fig. 5 was also trained by taking into account ISI.

####  V-C2 OTA under Sampling Error and ISI

$10$$15$$20$$25$$30$$35$$40$$45$$50$$0.04$$0.05$$0.06$$0.07$$0.08$$0.09$$0.10$$0.11$$0.12$$0.13$$0.14$Number
of devices $K$Average MSERCBTRCDNN-Gen$a=0.2$$a=0.5$$a=0.8$ Figure 7: Waveform
MSE performance for varying roll-off factor and sampling error variance
$\sigma_{\epsilon}=0.1$.

Figs. 7 and 8 illustrate the average MSE for varying numbers of devices in the
system. It’s evident that employing the DNN-generated waveform offers
satisfactory performance compared to the established RC and BTRC waveforms.
Notably, this advantage remains consistent across all selected roll-off
factors, particularly for mid-range values of $\alpha$, which are commonly
used in practice. Moreover, despite the decreased effect of ISI when
increasing the roll-off factor, the choice of waveform remains significant.
Remarkably, the DNN-generated waveform outperforms the RC waveform, despite
the latter utilizing a $30\%$ larger bandwidth. Moreover, for a larger
sampling error value of $\sigma_{\epsilon}=0.2$, the improvement provided by
the DNN-generated waveform becomes pronounced, aiding in the convergence of
the MSE curve. In addition, for small roll-off factor values, the strong
effect of ISI leads the MSE to increase regardless of the waveform. Thus,
achieving optimal performance requires considering both roll-off factor and
waveform selection to reduce the MSE.

$10$$15$$20$$25$$30$$35$$40$$45$$50$$8\cdot
10^{-2}$$0.1$$0.12$$0.14$$0.16$$0.18$$0.2$$0.22$$0.24$$0.26$$0.28$$0.3$Number
of devices $K$Average MSERCBTRCDNN-Gen$a=0.2$$a=0.5$$a=0.8$ Figure 8: Waveform
MSE performance for varying roll-off factor and sampling error variance
$\sigma_{\epsilon}=0.2$. TABLE II: MSE Improvement Table MSE Performance Gain
($\%$)
---
| $\sigma_{\epsilon}=0.1$ | $\sigma_{\epsilon}=0.2$
Roll-off Factor Waveform | BTRC | RC | BTRC | RC
$\alpha=0.1$ | $2.16$ | $4.27$ | $4.09$ | $6$
$\alpha=0.2$ | $5.65$ | $12.39$ | $7.20$ | $13.38$
$\alpha=0.3$ | $3.38$ | $13.14$ | $9.45$ | $19.95$
$\alpha=0.4$ | $5.95$ | $15.96$ | $12$ | $25.31$
$\alpha=0.5$ | $8.82$ | $21.06$ | $14.54$ | $29.56$
$\alpha=0.6$ | $5.62$ | $19.68$ | $13.15$ | $30.97$
$\alpha=0.7$ | $5.78$ | $18.66$ | $12.88$ | $31.31$
$\alpha=0.8$ | $2.49$ | $15.73$ | $9.36$ | $29.08$
$\alpha=0.9$ | $1.76$ | $12.58$ | $7.59$ | $26.14$
$\alpha=1.0$ | $2.16$ | $8.29$ | $4.69$ | $20.84$

Table II presents the percentage improvement gain of DNN-generated waveforms
over their RC and BTRC counterparts for various roll-off factors and sampling
errors. The DNN-based waveforms demonstrate significant gains, particularly
for mid-range roll-off factors. This is reasonable as waveforms in this range
retain relatively large sidelobe amplitudes, resulting in a greater impact of
ISI on the average MSE. Conversely, for large roll-off factors, the ISI effect
diminishes due to the waveform having smaller sidelobes, while the DNN’s
ability to generate higher-gain waveforms is constrained by bandwidth
resolution limitations for small roll-off factors. Nonetheless, in practical
applications where spectrum allocation matters, mid-range roll-off factors are
common. In such cases, DNN-generated waveforms exhibit significant performance
gains, making them a good alternative to established waveforms in the
literature.

In Figs. 9 and 10, we examine the MSE performance as the transmit SNR varies.
Typically, lower SNRs result in larger MSE due to significant noise effect in
addition to ISI. However, we observe that as the SNR increases, the MSE
decreases, with the DNN-generated waveform consistently outperforming both RC
and BTRC across the entire SNR range of 0 to 20 dB. This demonstrates the
robustness of the proposed approach to changes in transmit power and noise
levels. Notably, despite increasing SNR, for small roll-off factors, the
impact of ISI remains dominant, outweighing the benefits of power increase.
This behavior arises from the fact that increased power can worsen ISI,
mitigating potential performance gains at the ideal time sampling instance.
Nevertheless, the DNN-generated waveform exhibits faster convergence behavior
compared to the other waveforms, which is crucial for practical OTA
applications.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0.02$$0.04$$0.06$$0.08$$0.10$$0.12$$0.14$$0.16$$0.18$$0.20$$0.22$Transmit
SNRAverage MSERCBTRCDNN-Gen$a=0.2$$a=0.5$$a=0.8$ Figure 9: Waveform MSE
performance for varying roll-off factor and sampling error variance
$\sigma_{\epsilon}=0.1$. $0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$5\cdot
10^{-2}$$0.1$$0.15$$0.2$$0.25$$0.3$$0.35$Transmit SNRAverage MSERCBTRCDNN-
Gen$a=0.2$$a=0.5$$a=0.8$ Figure 10: Waveform MSE performance for varying roll-
off factor and sampling error variance $\sigma_{\epsilon}=0.2$.

Finally, another important observation drawn from the presented diagrams
relates to the magnitude of the average MSE caused by ISI. Comparing these
figures, specifically when there are 20 devices ($K=20$) and the transmit SNR
is 10 dB, it is visible that the average MSE is considerably greater (around
$60-70\%$) when ISI occurs. Also, under ISI the average MSE is heavily
dependent on the choice of the roll-off factor. In contrast, under only time
sampling error, the MSE is almost the same for all roll-off factor values.

##  VI Conclusions

Our study provides a practical examination of OTA computing performance
considering time sampling error and ISI. We derived the theoretical MSE for
the OTA transmission and established optimal power allocation strategies to
minimize the MSE. Additionally, we introduced a novel DNN-based method for
waveform design, integrating power, bandwidth, and design constraints into the
DNN loss function. Simulation results confirmed our theoretical findings and
demonstrated the superior performance of the designed waveform compared to the
traditional RC and BTRC waveforms. Future research extensions could explore
waveform design in a MIMO system, address imperfect CSI, or enhance the
current DNN-based approach. An area for improvement lies in our method of
obtaining a single waveform from the DNN output, which currently involves
averaging multiple output waveforms. Exploring more sophisticated techniques
for extracting a single waveform could be beneficial.

##  Appendix A Proof of Lemma 1

It is known that for the moments of a Gaussian random variable
$X\sim\mathcal{N}(0,\sigma^{2})$ it holds that

$\mathbb{E}_{X}[X^{m}]=\begin{cases}0,&\text{if }m\text{ is odd}\\\
\sigma^{m}(m-1)!!,&\text{if }m\text{ is even},\end{cases}$ (50)

where $(m-1)!!=1\cdots(m-3)(m-1)$ is the double factorial.

Let $g(\epsilon)=\mathrm{sinc}(\epsilon)\cos(\pi\alpha\epsilon)$ be the raised
cosine waveform without the denominator term, $1-(2a\epsilon)^{2}$, in (3)
which is strongly approximated by (10). Using the Taylor expansion of the
$\mathrm{sinc}$ and $\cos$ functions, we can write

$\displaystyle g(\epsilon)$
$\displaystyle=\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}(\pi\epsilon)^{2n}}{(2n+1)!}\right)\left(\sum_{k=0}^{\infty}\frac{(-1)^{k}(\pi\alpha\epsilon)^{2k}}{2k!}\right)$
(51)
$\displaystyle=\sum_{m=0}^{\infty}\underbrace{\sum_{n=0}^{m}\frac{(-1)^{m}\pi^{2m}\alpha^{2(m-n)}}{(2n+1)!(2m-2n)!}}_{\kappa_{m}}\epsilon^{2m},$

where $m=n+k$. With the same technique, we can also write

$\displaystyle g^{2}(\epsilon)$
$\displaystyle=\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}(\pi\epsilon)^{2n}}{(2n+1)!}\right)\left(\sum_{m=0}^{\infty}\frac{(-1)^{m}(\pi\epsilon)^{2m}}{(2m+1)!}\right)$
(52)
$\displaystyle\quad\times\left(\sum_{k=0}^{\infty}\frac{(-1)^{k}(\pi\alpha\epsilon)^{2k}}{2k!}\right)\left(\sum_{l=0}^{\infty}\frac{(-1)^{l}(\pi\alpha\epsilon)^{2l}}{2l!}\right)$
$\displaystyle=\sum_{p=0}^{\infty}\underbrace{\sum_{n=0}^{p}\sum_{l=0}^{p-n}\sum_{k=0}^{p-n-l}\tfrac{(-1)^{p}\pi^{2p}\alpha^{2(p-l-n)}}{(2n+1)!(2l+1)!(2k)!(2(p-n-
k-l))!}}_{\lambda_{p}}\epsilon^{2p}$

with $p=n+m+k+l$. Then, using (50), (51) and (10), for the mean value of the
waveform amplitude
$\bar{\epsilon}_{1}=\mathbb{E}_{\epsilon}[g(\epsilon)(1+4\alpha^{2}\epsilon^{2})]$,
it holds that

$\bar{\epsilon}_{1}=\sum_{m=0}^{\infty}\\!\kappa_{m}\\!\left(\\!\frac{1}{\sqrt{2\pi\sigma_{\epsilon}^{2}}}\int_{-\infty}^{\infty}\epsilon^{2m}(1+4\alpha^{2}\epsilon^{2})e^{-\frac{\epsilon^{2}}{2\sigma_{\epsilon}^{2}}}\mathrm{d}\epsilon\right)$
(53)

and (11) follows. Similarly, using (50), (52) and (10), for the mean squared
amplitude
$\bar{\epsilon}_{2}=\mathbb{E}_{\epsilon}[g^{2}(\epsilon)(1+4\alpha^{2}\epsilon^{2})^{2}]$,
we obtain

$\bar{\epsilon}_{2}=\sum_{p=0}^{\infty}\lambda_{p}\left(\frac{1}{\sqrt{2\pi\sigma_{\epsilon}^{2}}}\int_{-\infty}^{\infty}\epsilon^{2p}(1+4\alpha^{2}\epsilon^{2})^{2}e^{-\frac{\epsilon^{2}}{2\sigma_{\epsilon}^{2}}}\mathrm{d}\epsilon\right)$
(54)

and (13) follows, which completes the proof.

##  Appendix B Proof of lemma 2

To compute the mean squared amplitude at $t=T(1+\epsilon)$, the procedure is
similar to the one in Lemma 1 by expanding the $\mathrm{sinc}$ function and
combining it with binomial expansion as

$\displaystyle\mathrm{sinc}(1+\epsilon)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{(2n+1)!}{(1+\epsilon)}^{2n}$
(55)
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{(2n+1)!}\left(\sum_{k=0}^{2n}\binom{2n}{k}\epsilon^{k}\right)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}k=1,\\\ k\text{
odd}\end{subarray}}^{\infty}\underbrace{\left(\sum\limits_{n=\frac{k+1}{2}}^{\infty}\frac{(-1)^{n}\pi^{2n}}{(2n+1)!}\binom{2n}{k}\right)}_{A_{k}}\epsilon^{k}$
$\displaystyle\quad+\sum\limits_{\begin{subarray}{c}k=0,\\\ k\text{
even}\end{subarray}}^{\infty}\underbrace{\left(\sum\limits_{n=\frac{k}{2}}^{\infty}\frac{(-1)^{n}\pi^{2n}}{(2n+1)!}\binom{2n}{k}\right)}_{B_{k}}\epsilon^{k},$

where $A_{k}=0$ whenever $k$ is even and $B_{k}=0$ whenever $k$ is odd,
respectively. Then, according to (50), for the mean value of
$\mathrm{sinc}^{2}(1+\epsilon)$ to be non-zero there are two possible
combinations, i.e., odd with odd terms and even with even terms. Therefore, it
holds that

$\displaystyle\mathbb{E}_{\epsilon}[\mathrm{sinc}^{2}(1+\epsilon)]$
$\displaystyle=\mathbb{E}_{\epsilon}\left[\left(\sum\limits_{\begin{subarray}{c}k=1,\\\
k\text{
odd}\end{subarray}}^{\infty}A_{k}\epsilon^{k}\right)\left(\sum\limits_{\begin{subarray}{c}l=1,\\\
l\text{ odd}\end{subarray}}^{\infty}A_{l}\epsilon^{l}\right)\right]$ (56)
$\displaystyle\quad+\mathbb{E}_{\epsilon}\left[\left(\sum\limits_{\begin{subarray}{c}k=0,\\\
k\text{
even}\end{subarray}}^{\infty}B_{k}\epsilon^{k}\right)\left(\sum\limits_{\begin{subarray}{c}l=0,\\\
l\text{ even}\end{subarray}}^{\infty}B_{l}\epsilon^{l}\right)\right]$
$\displaystyle=\mathbb{E}_{\epsilon}\left[\sum\limits_{\begin{subarray}{c}m=2,\\\
m\text{
even}\end{subarray}}^{\infty}\left(\sum\limits_{\begin{subarray}{c}n=1,\\\
n\text{ odd}\end{subarray}}^{m}A_{n}A_{m-n}\right)\epsilon^{m}\right]$
$\displaystyle\quad+\mathbb{E}_{\epsilon}\left[\sum\limits_{\begin{subarray}{c}m=0,\\\
m\text{
even}\end{subarray}}^{\infty}\left(\sum\limits_{\begin{subarray}{c}n=0,\\\
n\text{ even}\end{subarray}}^{m}B_{n}B_{m-n}\right)\epsilon^{m}\right],$

which can be equivalently reduced to

$\tilde{\epsilon}_{1}=\mathbb{E}_{\epsilon}[\mathrm{sinc}^{2}(1+\epsilon)]=\mathbb{E}_{\epsilon}\left[\sum\limits_{m=0}^{\infty}\left(\sum\limits_{n=0}^{2m}A_{n}A_{2m-n}+B_{n}B_{2m-n}\right)\epsilon^{2m}\right],$
(57)

from which we can conclude (28) using (50) and $B_{0}^{2}=1$.

Then, due to the symmetry of the Gaussian random variable with zero mean, it
holds that

$\mathbb{E}_{\epsilon}[\mathrm{sinc}^{2}(-1+\epsilon)]=\mathbb{E}_{\epsilon}[\mathrm{sinc}^{2}(-1-\epsilon)]=\mathbb{E}_{\epsilon}[\mathrm{sinc}^{2}(1+\epsilon)],$
(58)

where the second equality holds due to the fact that
$\mathrm{sinc}^{2}(\cdot)$ is an even function. Thus,
$\tilde{\epsilon}_{1}=\tilde{\epsilon}_{-1}$, which completes the proof.

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