# OFDM-Standard Compatible SC-NOFS Waveforms for Low-Latency and Jitter-
Tolerance
Industrial IoT Communications

Tongyang Xu, , Shuangyang Li, , Jinhong Yuan  This work was supported by the
UK Engineering and Physical Sciences Research Council (EPSRC) under Grant
EP/Y000315/1. T. Xu is with the School of Engineering, Newcastle University,
Newcastle upon Tyne, NE1 7RU, U.K. (e-mail: tongyang.xu@newcastle.ac.uk). S.
Li is with the Department of Electrical Engineering and Computer Science,
Technical University of Berlin, Berlin 10587, Germany (e-mail:
<EMAIL_ADDRESS>J. Yuan is with the School of Electrical
Engineering and Telecommunications, University of New South Wales, Sydney, NSW
2052, Australia (e-mail: j.yuan@unsw.edu.au).

###### Abstract

Traditional communications focus on regular and orthogonal signal waveforms
for simplified signal processing and improved spectral efficiency. In
contrast, the next-generation communications would aim for irregular and non-
orthogonal signal waveforms to introduce new capabilities. This work proposes
a spectrally efficient irregular Sinc (irSinc) shaping technique, revisiting
the traditional Sinc back to 1924, with the aim of enhancing performance in
industrial Internet of things (IIoT). In time-critical IIoT applications, low-
latency and time-jitter tolerance are two critical factors that significantly
impact the performance and reliability. Recognizing the inevitability of
latency and jitter in practice, this work aims to propose a waveform technique
to mitigate these effects via reducing latency and enhancing the system
robustness under time jitter effects. The utilization of irSinc yields a
signal with increased spectral efficiency without sacrificing error
performance. Integrating the irSinc in a two-stage framework, a single-carrier
non-orthogonal frequency shaping (SC-NOFS) waveform is developed, showcasing
perfect compatibility with 5G standards, enabling the direct integration of
irSinc in existing industrial IoT setups. Through 5G standard signal
configuration, our signal achieves faster data transmission within the same
spectral bandwidth. Hardware experiments validate an 18% saving in timing
resources, leading to either reduced latency or enhanced jitter tolerance.

###### Index Terms:

Waveform, non-orthogonal, 6G, industrial IoT, spectral efficiency, energy
efficiency, NOFS, OFDM, machine learning, low latency, jitter tolerance.

##  I Introduction

The advancement of industrial Internet of things (IIoT) [1, 2, 3] has enabled
various time-critical applications in factory automation, power system
control, and transportation. However, the practical implementation of IIoT
presents challenges [4, 5], particularly concerning latency and jitter.
Latency, or the delay in data transmission, is a critical concern in
applications where real-time responsiveness is critical, such as robotic
control or automated manufacturing. Jitter, the variability in data packet
delivery times, can disrupt the synchronization of processes, impacting the
reliability of industrial systems. The need for jitter-tolerance, low-latency
networks, adherence to standards, and careful consideration of the specific
requirements of industrial applications are essential for successful IIoT
implementations.

The predominant approaches currently focus on optimizing media access control
(MAC) layer protocols through methods like time-sensitive networking (TSN)
[6], deterministic networking (DetNet) [7], and time synchronous channel
hopping (TSCH) [8]. However, the options for enhancing physical layer (PHY)
techniques are restricted, given that incorporating a new PHY technique
requires adjustments to the existing PHY layer standards, posing a challenge
that is impractical to overcome. Consequently, existing PHY layer techniques
are mainly focused on adjusting communication system configurations, such as
incorporating mmWave frequencies and multiple input multiple output (MIMO)
technology. Nevertheless, the utilization of mmWave frequencies introduces
significant challenges, including signal attenuation, high power consumption,
and increased costs. The use of MIMO architecture adds complexity to antenna
design, increased power consumption, and extra latency stemming from channel
state information (CSI) feedback processing. Hence, there is a need to
innovate in PHY layer techniques to address latency and jitter challenges.

Figure 1: Time and frequency characteristics for Sinc, raised cosine (RC), and
irSinc patterns.

Waveform design is an important research topic in communications dated back to
1924 [9] where Nyquist discovered a signal shape to be used in band limited
telegraph systems. Furthermore, in 1928 [10], Nyquist defined the well-known
Nyquist rate that provides an interference-free signal design principle being
used in all communication systems. Building upon the communication principles
established by Nyquist, in 1949 [11], Shannon’s capacity theory was proposed,
shaping the communication research from that point onward. The orthogonal
frequency division multiplexing (OFDM) was developed in 1958 [12] and became
the foundation for modern communication systems. In 4G long term evolution
(LTE), OFDM was standardized due to its enhanced spectral efficiency and
simple signal processing. To further enhance spectral efficiency and lead 5G,
many waveform candidates have been proposed with non-orthogonal signal
features. Representative signals are spectrally efficient frequency division
multiplexing (SEFDM) [13], faster than Nyquist (FTN) [14], filter bank
multicarrier (FBMC) [15], and generalized frequency division multiplexing
(GFDM) [16]. They all improve spectral efficiency while intentionally
introduce inter carrier interference (ICI) and inter symbol interference
(ISI), leading to complicated signal generation, channel estimation, and
signal detection [17, 18, 19, 20]. The sophisticated signal processing
prevents the inclusion of the aforementioned non-orthogonal signals in 5G,
thus maintaining OFDM as the standardized waveform format.

For next-generation 6G [21, 22, 23, 24], diverse services are expected to
appear and new functional signal waveforms are needed. The next generation
waveform (NGW) can be roughly characterized by orthogonal and non-orthogonal
signal formats. The orthogonal waveform representatives may be the recently
proposed multicarrier waveform based on delay-Doppler processing, e.g.,
orthogonal time frequency space (OTFS) [25] and orthogonal delay-Doppler
division multiplexing (ODDM) [26]. These waveforms are designed carefully to
deal with delay-Doppler effect in doubly-selective wireless communications.
The non-orthogonal representative, on the other hand, may be the non-
orthogonal frequency spacing (NOFS) [27], whose main objective is to enhance
spectral efficiency beyond the Mazo limit [28] via compressing spectral or
timing utilization. Both orthogonal and non-orthogonal signal waveforms bring
apparent benefits, but they still have challenges in increased signal
processing complexity due to the break of traditional signal transmission
framework. It is expected that future 6G might still use existing hardware
infrastructure, driven by substantial prior investments [29, 30] and a
commitment to sustainability. In this case, opting for a fundamentally new
signal waveform may not be the primary choice, as it would require the
investment of new hardware into the existing system. Therefore, a more
suitable approach would maintain the current over-the-air signal waveform
format, OFDM, and implement minor modifications from other dimensions.

To simplify signal processing and have a better integration of non-orthogonal
signal waveforms in existing systems, this work aims to develop a new signal
framework, termed single-carrier non-orthogonal frequency shaping (SC-NOFS),
which reserves its increased spectral efficiency advantage and provides
compatibility with existing 3GPP standard. Similar to traditional single-
carrier orthogonal frequency division multiplexing (SC-OFDM), our proposed SC-
NOFS requires two-stage processing. The first stage is our proposed non-
orthogonal signal processing and the second stage employs traditional inverse
fast Fourier transform (IFFT) signal generation. This system architecture not
only enhances spectral efficiency but also reserves the benefit of OFDM signal
characteristics over the air leading to simplified signal detection and
channel estimation. In addition, this system architecture provides flexibility
in the first stage since any user defined signal patterns can be integrated
here.

Figure 2: irSinc pattern at each sub-carrier.

The contributions of this work are as follow:

  * •

An OFDM-standard compatible SC-NOFS signal is proposed. The two-stage
bandwidth compressed waveform framework offers flexibility in designing user-
defined signal patterns in the first stage, while maintaining compatibility
with the OFDM waveform standard in the second stage. The first stage enhances
spectral efficiency, while the second stage ensures signal orthogonality,
simplifying channel estimation and equalization.

  * •

Reduced signal processing complexity in SC-NOFS compared to SC-OFDM due to
dimension reduction with NOFS and inverse non-orthogonal frequency shaping
(INOFS) transforms. Neural networks are used for these transforms, with
tunable connections allowing for pruning of unimportant connections, resulting
in lower computational complexity, particularly in multiplication operations,
compared to SC-OFDM.

  * •

The waveform framework utilizes shallow neural networks to train optimal
signal sub-carriers, ensuring rapid training and providing explainability. The
trained model maintains robustness in wireless channels, eliminating the need
for re-training with changing channel conditions. This approach is suitable
for integrated artificial intelligence (AI) and communication in 6G usage
scenarios [31] defined by the International Telecommunication Union (ITU).

  * •

A proof-of-concept hardware implementation validates the proposed signal
waveform regarding bit error rate (BER), error vector magnitude (EVM),
spectrum, and modulation constellation performance. The experiment showcases
two practical scenarios. The first demonstrates data transmission using fewer
spectral resources, while the second shows increased data transmission rate in
a given time within the same spectral resources.

The rest of the paper is organized as follows. Section II briefly describes
the principle of irregular Sinc (irSinc). The principle of the multi-carrier
NOFS is explained in Section III, in which system fundamentals and challenges
are described. Section IV describes the proposed SC-NOFS signal and shows the
neural network training methodology to realize the signal framework.
Simulation results on BER, peak-to-average power ratio (PAPR), channel
impacts, and complexity analysis are presented in Section V. To validate the
proposed signal format, hardware implementation is evaluated in Section VI.
Finally, Section VII concludes the work.

Notations: Unless otherwise specified, matrices are denoted by bold uppercase
letters (i.e., $\mathbf{F}$), vectors are represented by bold lowercase
letters (i.e., $\mathbf{x}$, $\mathbf{s}$), and scalars are denoted by normal
font (i.e., $\rho$). Subscripts indicate the location of the entry in the
matrices or vectors (i.e., $c_{i,j}$ and $s_{n}$ are the ($i,j$)-th and the
$n$-th element in $\mathbf{C}$ and $\mathbf{s}$, respectively)

##  II Irregular Sinc (irSinc)

In practical OFDM communication systems, each subcarrier is shaped by a Sinc
as illustrated in Fig. 1(a). An ideal Sinc has an infinite length and is not
implementable in practical systems. Therefore the commonly adopted solution is
to truncate the frequency domain signal via bandlimited filters that are
chosen carefully to mitigate the power of sidelobes in the time domain.
Consequently, this can result in minor out-of-band leakage as demonstrated by
the corresponding time pulse in Fig. 1(b).

Commonly used non-Sinc signals are shaped in frequency-domain by the raised
cosine (RC) response. The RC with a roll-of-factor of 0.3 has a smooth
sidelobe as shown in Fig. 1(c), where we shall notice that the frequency
truncation has negligible effect on the corresponding time domain signal as
shown in Fig. 1(d). Specifically, the time domain pulse in Fig. 1(d) has zero
out-of-band leakage but at the cost of a wider time occupancy resulting in
potential ISI.

Both Sinc and non-Sinc shaped sub-carriers are ‘Symmetric’ while in non-
orthogonal signals the symmetric feature might not be the optimal considering
ICI when the transmission rate is higher than the Nyquist rate [32].
Therefore, instead of optimizing traditional ‘Symmetric’ frequency response in
Fig. 1(a)(c), we proposed in [27] an ‘Asymmetric’ frequency response pattern,
termed as irregular Sinc (irSinc). One example of the irSinc is illustrated in
Fig. 1(e) where its irregular shaping can adaptively mitigate multicarrier
ICI. Its time domain pulse is illustrated in Fig. 1(f) with a rectangular
shape but with the rippled features at the top. It should be noted that the
irSinc in Fig. 1(e) is just one example. For multicarrier transmissions, each
sub-carrier should utilize a unique irSinc signal as illustrated in Fig. 2 in
order to improve the communication performance. The shape of each sub-carrier
in NOFS signals varies, and it is adjustable based on the signal’s specific
characteristics. To optimize error performance or in other words to maximize
the signal-to-interference power ratio, the irregular Sinc waveform must be
tailored for each particular signal configuration. As the number of sub-
carriers and sub-carrier spacing compression change, the signal interference
characteristics also change accordingly. Thus, it becomes necessary to re-
optimize the shape of all irregular Sinc for each sub-carrier to maximize
error performance.

##  III Multi-Carrier NOFS

Figure 3: Block diagram of (a) multi-carrier OFDM communication link with
complex-value signal processing, (b) multi-carrier NOFS communication link
with real-value signal processing. C/R indicates complex to real conversion.
R/C indicates real to complex conversion.

###  III-A System Fundamentals

In a traditional OFDM system, IFFT [33] and fast Fourier transform (FFT) are
used for signal modulation and demodulation in Fig. 3(a). To realize an irSinc
enabled multicarrier system, a new pair of signal transforms, i.e., INOFS and
NOFS, has to be applied as shown in Fig. 3(b). To show the effect of the
irSinc transforms, we demonstrate the NOFS signal with the sub-carrier packing
scheme in Fig. 4(b). Compared to the OFDM sub-carrier packing in Fig. 4(a),
the spectral efficiency of NOFS is improved via non-orthogonally packing sub-
carriers closer, where each sub-carrier is shaped by an independent and unique
irSinc.

Unlike the complex-value signal processing in Fig. 3(a), the INOFS and NOFS
transforms require real-value signal processing in order to be compatible with
neural network architectures. Therefore, considering a complex signal with $N$
sub-carriers, its real-value signal generation model, with a dimension of
$2N$, is defined in a neural network format as

$\displaystyle\mathbf{x}=\sigma(\mathbf{G_{\gamma}}\mathbf{s}+\Psi_{{}_{tx}}),$
(1)

where the vector $\mathbf{x}\in\mathbb{R}^{2N{\times}1}$ represents the real-
value NOFS signal. The matrix
$\mathbf{G_{\gamma}}\in\mathbb{R}^{2N{\times}2N}$ represents the INOFS
transform that is similar to the function of IFFT in OFDM. The subscript
$\gamma$ indicates the matrix is in a neural network format. The vector
$\mathbf{s}\in\mathbb{R}^{2N{\times}1}$ represents the real-value modulation
quadrature amplitude modulation (QAM) symbols by packing real and imaginary
parts together. The vector $\Psi_{{}_{tx}}\in\mathbb{R}^{2N{\times}1}$ is the
transmitter side bias that is used to fine tune the output from the
multiplication of $\mathbf{G_{\gamma}}\mathbf{s}$, and $\sigma(\cdotp)$ is the
activation function used to convert the input into a target output.

Before passing through a channel, the transmitter side NOFS signal
$\mathbf{x}\in\mathbb{R}^{2N{\times}1}$ is reconstructed into a complex signal
as

$\displaystyle\mathbf{\bar{x}}=\mathbf{x}_{(1:N)}+i\cdotp\mathbf{x}_{(N+1:2N)}.$
(2)

Figure 4: Sub-carrier shaping and packing schemes for Sinc shaped OFDM signal
and irSinc shaped NOFS signal. Both signals have the same sub-carrier
bandwidth while the NOFS signal has compressed sub-carrier spacing. Figure 5:
BER for multi-carrier NOFS signals. $\alpha$=0.7,0.5,0.4 indicate
43%,100%,150% spectral efficiency improvement, respectively.

The complex signal $\mathbf{\bar{x}}\in\mathbb{C}^{N{\times}1}$ passes through
a multipath channel $\mathbf{H}\in\mathbb{C}^{N{\times}N}$ and additive white
Gaussian noise (AWGN) $\mathbf{z}\in\mathbb{C}^{N{\times}1}$ leading to the
received signal $\mathbf{\bar{y}}\in\mathbb{C}^{N{\times}1}$ given by

$\displaystyle\mathbf{\bar{y}}=\mathbf{H}\mathbf{\bar{x}}+\mathbf{z}=\mathbf{H}(\mathbf{x}_{(1:N)}+i\cdotp\mathbf{x}_{(N+1:2N)})+\mathbf{z}.$
(3)

At the receiver, assuming $\mathbf{H}$ is well structured, the received signal
$\mathbf{\bar{y}}\in\mathbb{C}^{N{\times}1}$ is equalized to
$\mathbf{\hat{y}}\in\mathbb{C}^{N{\times}1}$ in time-domain by removing the
effect of $\mathbf{H}\in\mathbb{C}^{N{\times}N}$ as

$\displaystyle\mathbf{\hat{y}}=\mathbf{H}^{-1}\mathbf{\bar{y}}=\mathbf{x}_{(1:N)}+i\cdotp\mathbf{x}_{(N+1:2N)}+\mathbf{H}^{-1}\mathbf{z}.$
(4)

The complex signal $\mathbf{\hat{y}}\in\mathbb{C}^{N{\times}1}$ is converted
back to its real value format $\mathbf{{y}}\in\mathbb{R}^{2N{\times}1}$ as

$\displaystyle\mathbf{{y}}=\mathbf{x}+[\Re(\mathbf{H}^{-1}\mathbf{z});\Im(\mathbf{H}^{-1}\mathbf{z})].$
(5)

Then a matched operation of (1), aiming to demultiplex the signals to baseband
QAM symbols, is given by

$\displaystyle\begin{split}\mathbf{\hat{s}}&=\sigma(\mathbf{G^{*}_{\gamma}}\mathbf{{y}}+\Psi_{{}_{rx}})\\\
&=\sigma(\mathbf{G^{*}_{\gamma}}(\sigma(\mathbf{G_{\gamma}}\mathbf{s}+\Psi_{{}_{tx}})+\mathbf{\epsilon})+\Psi_{{}_{rx}}),\end{split}$
(6)

where the term
$\mathbf{\epsilon}\in\mathbb{R}^{2N{\times}1}=[\Re(\mathbf{H}^{-1}\mathbf{z});\Im(\mathbf{H}^{-1}\mathbf{z})]$,
and $\Psi_{{}_{rx}}\in\mathbb{R}^{2N{\times}1}$ indicates the receiver side
bias.

Utilizing the adaptive irSinc shaping for each sub-carrier, we examine the BER
performance of the multi-carrier NOFS system, as illustrated in Fig. 5 (reused
from [27]). The results indicate that, with the bandwidth compression factor
$\alpha$=0.4, the NOFS system achieves a significant 60% saving of
communication resources, leading to an impressive 150% enhancement in spectral
efficiency. Notably, it approaches the performance of the traditional OFDM
system.

###  III-B Challenges

The existing 5G standard employs the OFDM waveform format where sub-carriers
are orthogonally packed leading to simple signal generation, signal detection,
and channel estimation. The non-orthogonal multi-carrier NOFS achieves
significant spectral efficiency improvement, which is potential to enable new
capabilities in future communications. However, regardless of advantageous
features, multi-carrier NOFS poses challenges, especially the increased power
leakage, high complexity channel estimation/equalization, and incompatibility
with existing standards.

####  III-B1 Power Leakage

Out-of-band power leakage refers to the emission of signal power outside of
the designated frequency band of a communication system. In co-existence
communication scenarios, where different signal bands are positioned adjacent
to each other in the frequency domain, out-of-band power leakage can have
significant implications. It can lead to interference with neighbouring
communication systems operating in adjacent frequency bands, thereby affecting
their performance and reliability. For the bandwidth compressed NOFS, its sub-
carrier packing pattern is illustrated in Fig. 4 with irregular out-of-band
patterns. Therefore, it is obvious that the NOFS signal waveform may introduce
higher out-of-band power leakage than other waveforms. Such a high out-of-band
power leakage signal design is not suitable for co-existence communication
scenarios because the leakage would cause interference to adjacent signals.
Some non-orthogonal signals cope with this issue relying on pulse shaping at
the cost of increased signal processing complexity. Therefore, aiming for
wider application scenarios, optimization of the out-of-band power is of
importance.

Figure 6: Block diagram of (a) single-carrier OFDM communication link with
complex-value signal processing, (b) single-carrier NOFS communication link
with real-value signal processing. C/R indicates complex to real conversion.
R/C indicates real to complex conversion.

####  III-B2 Channel Estimation and Equalization

Channel estimation and equalization for non-orthogonal signals are very
challenging due to the self-created ICI such that the commonly used frequency-
domain one-tap channel estimation and equalization are no longer applicable.
In NOFS, due to the irSinc frequency response on each sub-carrier, the
conventional Fourier transform is not applicable. Therefore, time-domain
channel estimation and equalization may perform better than frequency domain
counterparts. However, time-domain operations require de-convolution
processing, which is more complicated than the frequency-domain approach and
maybe impractical in real world communication systems.

For ease of explanation, we consider multipath frequency selective channel
models and ignore Doppler effects. To estimate the channel from the distorted
NOFS signal in (3), pilot symbols are needed prior to data symbols. For each
pilot symbol, cyclic prefix (CP) is added to mitigate ISI and simplify channel
estimation. The mathematical model of pilot symbols going through the channel
is given by

$\displaystyle\mathbf{\bar{y}_{{}_{cp}}}=\mathbf{H}\mathbf{\bar{x}_{{}_{cp}}}+\mathbf{z_{{}_{cp}}},$
(7)

where $\mathbf{\bar{x}_{{}_{cp}}}\in\mathbb{C}^{N_{cp}{\times}1}$ is the
transmitted pilot symbol including both CP and pilot data. $N_{cp}=N+L$ where
$L$ indicates the length of CP. The channel matrix
$\mathbf{H}\in\mathbb{C}^{N_{cp}{\times}N_{cp}}$ becomes larger due to the
addition of CP, and $\mathbf{z_{{}_{cp}}}\in\mathbb{C}^{N_{cp}{\times}1}$
indicates the AWGN vector. At the receiver, after removing CP, the Toeplitz
channel matrix $\mathbf{H}\in\mathbb{C}^{N_{cp}{\times}N_{cp}}$ becomes a
circulant channel matrix $\mathbf{H_{c}}\in\mathbb{C}^{N{\times}N}$ with the
following new format

$\displaystyle\mathbf{\bar{y}_{p}}$
$\displaystyle=\mathbf{H_{c}}\mathbf{\bar{x}_{p}}+\mathbf{z},$ (8)

where $\mathbf{\bar{x}_{p}}\in\mathbb{C}^{N{\times}1}$ is the transmitted
pilot vector, $\mathbf{\bar{y}_{p}}\in\mathbb{C}^{N{\times}1}$ is the received
and distorted pilot vector.

The relationship between a real-value matrix and a complex-value matrix is
given in the following

$\mathbf{G_{\gamma}}=\begin{bmatrix}\Re(\mathbf{\bar{G}_{\gamma}})&-\Im(\mathbf{\bar{G}_{\gamma}})\\\
\Im(\mathbf{\bar{G}_{\gamma}})&\Re(\mathbf{\bar{G}_{\gamma}})\\\
\end{bmatrix}\in\mathbb{R}^{2N{\times}2N},$ (9)

where $\mathbf{\bar{G}_{\gamma}}\in\mathbb{C}^{N{\times}N}$ is the complex-
value neural network based NOFS generation matrix with the following format

$\mathbf{\bar{G}_{\gamma}}=\begin{bmatrix}\mathbf{g}_{1}\\\ \mathbf{g}_{2}\\\
\vdots\\\ \mathbf{g}_{N}\\\ \end{bmatrix}\in\mathbb{C}^{N{\times}N},$ (10)

where $\mathbf{g}_{i}$ indicates the $i^{th}$ row in
$\mathbf{\bar{G}_{\gamma}}$. To simplify the analysis, considering
$\mathbf{\bar{x}_{p}}$=$\mathbf{x_{p}}_{(1:N)}+1i\cdotp\mathbf{x_{p}}_{(N+1:2N)}$
and
$\mathbf{x_{p}}$=$\sigma(\mathbf{G_{\gamma}}\mathbf{s_{p}}+{\Psi_{{}_{tx}}})$,
(8) is converted to

$\displaystyle\mathbf{\bar{y}_{p}}=\mathbf{H_{c}}(\sigma_{c}(\mathbf{\bar{G}_{\gamma}}\mathbf{\bar{s}_{p}}+{\bar{\Psi}_{{}_{tx}}}))+\mathbf{z},$
(11)

where $\mathbf{\bar{s}_{p}}\in\mathbb{C}^{N{\times}1}$ represents complex-
value QAM symbols, ${\bar{\Psi}_{{}_{tx}}}\in\mathbb{C}^{N{\times}1}$
indicates the transmitter side bias in a complex format. In practice, the
signal processing inside an activation function should be in real-value
formats due to the characteristics of neural network operations. However, to
simplify the expression here, we use a unique activation function
$\sigma_{c}(\cdotp)$ to indicate that the values in the bracket could be
complex. But in practical operations, the computation is still in real-value
formats. Due to the circulant channel matrix feature, $\mathbf{{H}_{c}}$ can
be reconstructed via copying and shifting
$\mathbf{h}\in\mathbb{C}^{N{\times}1}$ at each column. Therefore, (11) is
further converted to

$\mathbf{\bar{y}_{p}}=\sigma_{c}(\mathbf{P})\mathbf{h}+\mathbf{z},$ (12)

where $\mathbf{P}=(\mathbf{P}_{1}+\mathbf{P}_{2})\in\mathbb{C}^{N{\times}N}$
is a circulant matrix defined by

$\mathbf{P}_{1}=\begin{bmatrix}\mathbf{g}_{1}&\mathbf{g}_{Q}&\cdots&\mathbf{g}_{2}\\\
\mathbf{g}_{2}&\mathbf{g}_{1}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\mathbf{g}_{Q}\\\
\mathbf{g}_{Q}&\cdots&\mathbf{g}_{2}&\mathbf{g}_{1}\end{bmatrix}\cdotp\begin{bmatrix}\mathbf{\bar{s}_{p}}&0&\cdots&0\\\
0&\mathbf{\bar{s}_{p}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\mathbf{\bar{s}_{p}}\end{bmatrix}$ (13)

$\mathbf{P}_{2}=\begin{bmatrix}\bar{\Psi}_{{}_{tx,1}}&\bar{\Psi}_{{}_{tx,Q}}&\cdots&\bar{\Psi}_{{}_{tx,2}}\\\
\bar{\Psi}_{{}_{tx,2}}&\bar{\Psi}_{{}_{tx,1}}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\bar{\Psi}_{{}_{tx,Q}}\\\
\bar{\Psi}_{{}_{tx,Q}}&\cdots&\bar{\Psi}_{{}_{tx,2}}&\bar{\Psi}_{{}_{tx,1}}\end{bmatrix}$
(14)

where $\bar{\Psi}_{{}_{tx,i}}$ indicates the $i^{th}$ element in
$\bar{\Psi}_{{}_{tx}}$. The channel elements are therefore computed as

$\mathbf{\hat{h}}=\mathbf{\sigma_{c}(P)^{*}(\sigma_{c}(P)\sigma_{c}(P)^{*})^{-1}}\mathbf{\bar{y}_{p}}.$
(15)

It is apparent from the above signal processing that time-domain channel
estimation is far more complicated than its frequency-domain operations
because matrix multiplications and matrix inverse calculations are needed for
the time-domain computations. In addition, channel equalization relies on the
reconstruction of the channel matrix resulting in extra matrix multiplications
and matrix inverse operations. Hence, there is a pressing demand for a novel
signal design that maintains compatibility with frequency-domain channel
estimation and equalization operations.

####  III-B3 Compatibility

Compatibility with available standards is a big challenge for all advanced
physical layer signal waveform design. Operators have invested huge [29, 30]
on hardware infrastructure where OFDM is integrated as the physical layer
waveform. A fundamentally new physical layer design will require replacement
of existing hardware, which is not a wise investment when previous investment
has not been repaid. In available standards, sub-carriers are shaped by
conventional Sinc where it can be implemented directly by using IFFT. For
NOFS, its irregular Sinc shaped sub-carriers change the physical layer
characteristics fundamentally and will require new hardware to replace the
OFDM supported hardware. In addition, the irregular Sinc sub-carriers would
also cause interference to already deployed standards. Therefore, there is a
critical need for a physical layer signal design that inherits the advantages
of NOFS signals while ensuring compatibility with existing standards.

##  IV Single-Carrier NOFS

###  IV-A Proposed SC-NOFS Signal Framework

Due to the simple signal processing and robustness to multipath channels, OFDM
has been standardized in 4G, 5G, WiFi, and other relevant systems. Therefore,
to enable straightforward integration in existing standards, new physical
layer waveforms could adopt advantages from OFDM through a two-stage waveform
generation framework.

A conventional two-stage signal framework for SC-OFDM is illustrated in Fig.
6(a) where the first stage is an $M$-point discrete Fourier transform (DFT)
[34] on an $M$-point QAM symbol sequence and the second stage follows the
traditional $N$-point OFDM signal generation. However, the original motivation
of SC-OFDM is to reduce PAPR but without any gains in spectral efficiency.

Therefore, we propose and demonstrate a two-stage SC-NOFS signal framework in
Fig. 6(b) where the user-defined signal generation is integrated at the first
stage and OFDM signal generation is applied at the second stage such that
over-the-air signal format still follows the existing OFDM standard. To
enhance spectral efficiency, we intentionally use a $2Q\times{2M}$ NOFS
transform such that the $2Q$-point output is shorter than its $2M$-point input
leading to a compression factor defined as $\alpha$=$Q/M$. Therefore, a
reduced number of data sub-carriers are required for the second stage, leading
to a reduction in occupied spectral resources, as illustrated in Fig. 6,
ultimately leading to improved spectral efficiency. It is noted that C/R and
R/C modules are required to convert between complex-value symbols and real-
value symbols because the NOFS and INOFS transforms are operated based on
real-value neural network architectures.

The benefits of the two-stage SC-NOFS waveform processing are

  * •

SC-NOFS achieves higher spectral efficiency than SC-OFDM because a spectrally
efficient waveform pattern is integrated at the first stage.

  * •

It is compatible with existing standards since over-the-air interface is in
the OFDM format.

  * •

Its out-of-band power leakage is reduced and is within the standard
requirement.

  * •

Its PAPR is reduced compared to its multi-carrier version.

  * •

Channel estimation and equalization follow the traditional frequency-domain
one-tap signal processing.

Figure 7: Training mechanism for SC-NOFS.

In the first stage, the $2Q\times{2M}$ real-value NOFS transform aims to
realize the following

$\displaystyle\mathbf{x_{a}}=\sigma(\mathbf{W^{*}_{\gamma}}\mathbf{s}+\Psi_{{}_{tx}}),$
(16)

where the matrix $\mathbf{W^{*}_{\gamma}}\in\mathbb{R}^{2Q{\times}2M}$
indicates the first-stage neural network based NOFS transform where $M>Q$. The
aim of the non-square neural network matrix $\mathbf{W^{*}_{\gamma}}$ is to
output a shorter symbol vector than its input symbol vector for the purpose of
spectral efficiency enhancement. The vector
$\mathbf{s}\in\mathbb{R}^{2M{\times}1}$ represents the real-value QAM symbols.
Due to the dimension reduction of $\mathbf{W^{*}_{\gamma}}$, the bias
$\Psi_{{}_{tx}}\in\mathbb{R}^{2Q{\times}1}$ and the output
$\mathbf{x_{a}}\in\mathbb{R}^{2Q{\times}1}$ become shorter.

The complex signal $\mathbf{\bar{x}_{a}}\in\mathbb{C}^{Q{\times}1}$ is
reconstructed from $\mathbf{x_{a}}\in\mathbb{R}^{2Q{\times}1}$ and is
symmetrically zero padded to obtain
$\mathbf{\bar{\bar{x}}_{a}}\in\mathbb{C}^{N{\times}1}$ for the second stage
$N$-point IFFT processing as

$\displaystyle\mathbf{\bar{x}_{b}}=\mathbf{F}\mathbf{\bar{\bar{x}}_{a}},$ (17)

where the matrix $\mathbf{F}\in\mathbb{C}^{N{\times}N}$ represents the IFFT
matrix.

The second stage signal $\mathbf{\bar{x}_{b}}$ goes through the multipath
channel $\mathbf{H}\in\mathbb{C}^{N{\times}N}$ and AWGN
$\mathbf{z}\in\mathbb{C}^{N{\times}1}$ giving an expression as

$\displaystyle\mathbf{\bar{y}}=\mathbf{H}\mathbf{\bar{x}_{b}}+\mathbf{z}=\mathbf{H}\mathbf{F}\mathbf{\bar{\bar{x}}_{a}}+\mathbf{z}.$
(18)

At the receiver, unlike the time-domain channel equalization processing in
(4), the proposed SC-NOFS signal operates FFT first to convert the time-domain
signal to its frequency-domain version as

$\displaystyle\begin{split}\mathbf{r}=\mathbf{F^{*}}\mathbf{\bar{y}}&=\mathbf{F^{*}}\mathbf{H}\mathbf{F}\mathbf{\bar{\bar{x}}_{a}}+\mathbf{F^{*}}\mathbf{z}\\\
&=\mathbf{D}\mathbf{\bar{\bar{x}}_{a}}+\mathbf{F^{*}}\mathbf{z},\end{split}$
(19)

where $\mathbf{F^{*}}\in\mathbb{C}^{N{\times}N}$ indicates the FFT matrix. The
multiplication term $\mathbf{F^{*}}\mathbf{H}\mathbf{F}$ leads to a diagonal
matrix $\mathbf{D}$ which simplifies channel estimation and equalization into
one-tap processing.

The channel frequency response at each sub-carrier, represented by the
diagonal elements $diag(\mathbf{D})=[{d}_{1},{d}_{2},...,{d}_{N}]$ in
$\mathbf{D}$, is extracted to equalize signals as

$\displaystyle\begin{split}\mathbf{\hat{y}}&=diag(\mathbf{D})^{-1}\mathbf{r}\\\
&=\mathbf{\bar{\bar{x}}_{a}}+diag(\mathbf{D})^{-1}\mathbf{F^{*}}\mathbf{z}\\\
&=\mathbf{\bar{\bar{x}}_{a}}+\mathbf{\xi},\end{split}$ (20)

where $\mathbf{\xi}\in\mathbb{C}^{N{\times}1}$. The complex signal
$\mathbf{\hat{y}}\in\mathbb{C}^{N{\times}1}$ is truncated to remove the
symmetrically padded zeros and is converted back to its real value format
$\mathbf{{y}}\in\mathbb{R}^{2Q{\times}1}$ as

$\displaystyle\begin{split}\mathbf{{y}}&=\mathbf{{{x}}_{a}}+[\Re(\mathbf{\xi}_{((N-Q)/2+1:N/2)});\Im(\mathbf{\xi}_{(N/2+1:N/2+(N-Q)/2)})]\\\
&=\sigma(\mathbf{W^{*}_{\gamma}}\mathbf{s}+\Psi_{{}_{tx}})+\mathbf{\varrho},\end{split}$
(21)

where $\mathbf{\varrho}\in\mathbb{R}^{2Q{\times}1}$. Then the INOFS transform
on (21) is given by

$\displaystyle\begin{split}\mathbf{\hat{s}}&=\sigma(\mathbf{W_{\gamma}}\mathbf{{y}}+\Psi_{{}_{rx}})\\\
&=\sigma(\mathbf{W_{\gamma}}(\sigma(\mathbf{W^{*}_{\gamma}}\mathbf{s}+\Psi_{{}_{tx}})+\mathbf{\varrho})+\Psi_{{}_{rx}}),\end{split}$
(22)

where $\Psi_{{}_{rx}}\in\mathbb{R}^{2M{\times}1}$ indicates the receiver side
bias.

###  IV-B Neural Network Training Framework

The NOFS/INOFS transform blocks in Fig. 6(b) take advantages of machine
learning to provide new features and achieve better signal performance. The
aim of the NOFS transform block is to partially remove some samples such that
the total transmission samples are reduced leading to increased spectral
efficiency. The removal of samples causes information loss and therefore
difficulties in accurate signal recovery. Therefore, the neural network has to
be specifically trained to generate optimal output sequence such that original
signal can be recovered easily by using its INOFS transform at the receiver.

The training mechanism of the NOFS/INOFS transform neural networks
$\mathbf{W^{*}_{\gamma}}$/$\mathbf{W_{\gamma}}$ are detailed in Fig. 7. The
first stage training aims to learn legacy signals and emulate them using
neural network architectures, which is an efficient mechanism to transform
sophisticated signal model into simple neural network architectures with fine-
tuning capabilities for each neural connection. In this work, the legacy
signal is in an OFDM format but it could be any signals when considering
different application scenarios. It is noted that conventional mathematical
models have solid theoretical analysis but further improvement or optimization
is challenging. Neural networks include an array of internal connections and
each connection is associated with a weight to determine the importance of
this connection. The obvious advantage of neural networks is its tuning
capability where changing internal neural connections can bring new features.
This is the reason why in the first stage the framework needs to learn and
construct signal models in a neural network format.

The second stage of training aims to fine-tune the neural network obtained
from the first stage by optimizing neural connection topology and their
weights through sample dimension truncation and neural connection truncation.
The objective is to obtain a fine-tuned neural network matrix
$\mathbf{W_{\gamma}}$ that can enhance spectral efficiency and deal with non-
orthogonal signal interference. In the second stage, the trained neural
network matrix $\mathbf{W^{*}_{\gamma}}$ is used to modulate $\mathbf{s}$ in
the Mod block with the manipulations of activation function $\sigma(\cdotp)$
and bias $\Psi$. The SC-NOFS signal is obtained after the R/C and IFFT blocks,
and is distorted by the AWGN channel. It is noted that the framework
intentionally removes frequency selective channel effects in the training
process to avoid variation effects on neural networks training. The channel
effect will be removed using traditional channel estimation and equalization
algorithms. At the receiver, the signal is demultiplexed at the FFT block and
is demodulated in the De-Mod block using the INOFS transform. At this step,
original symbols are recovered but with interference. The optimal signal
detection scheme is based on the maximum likelihood principle, which is not
the preferred solution for communication sustainability and net-zero
objectives. Moreover, the maximum likelihood method is not scalable and its
processing complexity is exponentially increased with the signal size.
Therefore, a simple signal detection approach, termed iterative detection (ID)
[17], based on successive interference cancellation, is preferred in the Dec
block.

The training objective in Fig. 7 is to achieve the best BER performance and
therefore minimizing the BER gap between the conventional SC-OFDM and the
target SC-NOFS is the focus. The gap is calculated based on mean squared error
(MSE), and once a smaller MSE is obtained, the training iteration continues.
Otherwise, the training stops with the current neural network matrix
$\mathbf{W_{\gamma}}$ as the final solution. We use 600,000 4QAM symbols to
train the neural network and $10^{7}$ 4QAM symbols to test the system. The
learning rate is 0.01 and the training loss is calculated based on MSE. To
maintain signal features during training, we use linear activation functions
$\sigma(\cdot)$. Random values are initially given to the elements in the
neural network and the optimal values will be obtained after 1000 training
iterations via back propagation using the stochastic gradient descent (SGD)
optimization algorithm. The training framework is flexible and it can set up
other training objectives such as to achieve low PAPR or to cut out-of-band
power leakage.

##  V Performance Evaluation

To show the advantages of our proposed SC-NOFS signal framework, we include
two related works in high compaction multi-carrier communication (HC-MCM) [35]
and truncated-OFDM [36], where only one-stage signal generation with very
limited number of sub-carriers were investigated. To have fair comparisons, we
have to integrate the one-stage signal techniques from [35, 36] into our
proposed two-stage single-carrier signal framework using the structure in Fig.
6. In addition, it is noted that the works in [35, 36] utilize a small number
of sub-carriers because they have to rely on maximum likelihood or sphere
decoding signal detection algorithms. In contrast, our approach allows for the
utilization of 1024 sub-carriers while employing simpler signal detection
methods. The key novelty of SC-NOFS is the NOFS/INOFS transform aiming to
generate a shorter sequence relying on irSinc shaped sub-carrier and
artificial intelligence. Conventional signal truncation solutions in HC-MCM
[35] and truncated-OFDM [36] have limitations because their signals are Sinc
shaped, which is inflexible and is not the optimal solution for non-orthogonal
signals [27]. To simplify the notations, we use ‘Sinc-Traditional([35],[36])’
to indicate the benchmark in the comparison of the following results.

###  V-A BER Performance

Figure 8: BER performance in AWGN for SC-NOFS signals with $Q$=64, $M$=76,
$N$=128, $\alpha$=$Q/M$=0.8421, spectral efficiency improvement is
$(M/Q-1)\times$100%=18.75%.  Figure 9: BER performance in AWGN for SC-NOFS
signals with $Q$=125, $M$=150, $N$=256, $\alpha$=$Q/M$=0.8333, spectral
efficiency improvement is $(M/Q-1)\times$100%=20%.  Figure 10: BER performance
in AWGN for SC-NOFS signals with $Q$=492, $M$=600, $N$=1024,
$\alpha$=$Q/M$=0.82, spectral efficiency improvement is
$(M/Q-1)\times$100%=22%.

We firstly evaluate the signal performance in Fig. 8 with $M$=76 input 4QAM
symbols and $N$=128. Based on the training methodology in Fig. 7, the proposed
SC-NOFS signal can reach the compression factor $\alpha$=64/76=0.8421, where
12 output samples are removed out of 76 samples. Its performance curve is
overlapping with the SC-OFDM/OFDM indicating the performance reliability and
18.75%=(76/64-1)$\times$100% spectral efficiency improvement. On the other
hand, under the same compression ratio, the Sinc-Traditional([35],[36])
signals are not recoverable and their performance obviously moves away from
SC-OFDM/OFDM.

It is noted that the size of an NOFS/INOFS transform matrix is determined by
the number of sub-carriers. The larger size of the NOFS/INOFS transform
matrix, the more flexibility of the neural network architecture because more
neural connections and weights are available to be tuned. As the number of
sub-carriers decreases, the benefits of signal compression gradually diminish
owing to the reduced flexibility in neural network tuning. Therefore, it is of
great importance to integrate more neural connections in the transform matrix.
One efficient solution is to increase the number of input samples $M$ such
that the size of the generator matrix is increased and the diversity of matrix
elements is enhanced.

In Fig. 9, we increase the number of input samples to $M$=150. Results show
clearly that the higher number of input samples enables better spectral
efficiency because the compression factor can be further reduced to
$\alpha$=0.8333. On the other hand, for those Sinc shaped solutions in
[35][36], the performance curve is away from SC-OFDM/OFDM. When we further
increase the number of input samples to $M$=600 in Fig. 10, the spectral
efficiency is further improved with the compression factor reduced to
$\alpha$=0.82.

###  V-B PAPR Performance

Figure 11: PAPR performance for SC-NOFS signals.

In Fig. 11, we evaluate the PAPR of SC-NOFS signals with different numbers of
sub-carriers. Firstly, both single-carrier signals (SC-NOFS and SC-OFDM)
exhibit a noticeable improvement in PAPR compared to traditional multi-carrier
OFDM signals. This improvement is reasonable since the overlap of multiple
sub-carriers in OFDM leads to a high peak power. Secondly, SC-NOFS signals
demonstrate nearly identical PAPR when compared to SC-OFDM, with only minor
differences observed. It should be emphasized that the primary goal of
training or optimizing irSinc in SC-NOFS is to mitigate intercarrier
interference, rather than focusing on improving PAPR. Therefore, any
improvement in PAPR achieved through the use of irSinc is considered an added
benefit.

###  V-C Frequency Selective Channel Performance

Figure 12: BER performance comparisons in a time-variant frequency selective
channel.

This work also tests the proposed signals in a time-variant frequency
selective channel. The channel’s power delay profile is defined by
${h(t)}=0.8765\delta(t)-0.2279\delta(t-T_{s})+0.1315\delta(t-4T_{s})-0.4032e^{\frac{j\pi}{2}}\delta(t-7T_{s})$,
with each element following a Rayleigh distribution. The frequency-domain one-
tap channel estimation/equalization method is applied with BER demonstrations
to show the quality of signals under time-variant multiplath channel effects.
It is noted that the channel estimation for the proposed SC-NOFS follows the
principle of conventional OFDM and is easily integrated in existing standards.
Other non-orthogonal signals need to face complicated and inaccurate channel
estimation and equalization challenges while our proposed solution perfectly
avoids this. The channel distorted BER results are shown in Fig. 12, in which
the Sinc-Traditional([35],[36]) signals exhibit the worst performance,
particularly at high Eb/N0 ranges due to the signal internal ICI using Sinc
shaped sub-carriers. At sufficiently high Eb/N0 levels, the curve tends
towards flatness, indicating unrecoverable performance degradation. In the
case of traditional OFDM signals, superior performance is observed within low
Eb/N0 ranges, showing its resilience when noise power dominates. Both SC-NOFS
and SC-OFDM outperform the traditional OFDM due to improved frequency
diversity against the multipath propagation impact. It is noted that the SC-
NOFS signal exhibits similar performance to SC-OFDM but with 22% higher
spectral efficiency. These results reveal the advantages of SC-NOFS signals
over other waveform candidates, even in a time-variant frequency-selective
channel.

###  V-D Complexity Comparison

Figure 13: Computational complexity in terms of real-value multiplication
operations under different numbers of sub-carriers ($N$).

In SC-NOFS, a significant innovation is the NOFS transform block and its
inverse counterpart, the INOFS transform. Unlike the conventional DFT and
inverse discrete Fourier transform (IDFT), illustrated in Fig. 6(a), where the
input and output have the same dimensions, the NOFS transform yields a reduced
output dimension, while the INOFS transform features a reduced input
dimension. Consequently, the computational complexity in SC-NOFS is lower
compared to that of SC-OFDM. Moreover, to fine tune the neural network matrix
$\mathbf{W_{\gamma}}$, unimportant neural connections are pruned, further
decreasing the computational load. In contrast, in SC-OFDM, its first stage
signal processing relies on the DFT where the traditional FFT is impossible,
leading to the use of direct matrix multiplication, resulting in increased
computational complexity. Based on this analysis, our proposed SC-NOFS offers
enhanced spectral efficiency while simultaneously reducing computational
complexity when compared to SC-OFDM.

In Fig. 13, we test three scenarios with varying numbers of sub-carriers
(N=128, 256, 1024). We evaluate the number of real-value multiplications as
the metric for computational complexity comparison. It is evident that as the
number of sub-carriers increases, so does the demand for multiplications.
However, the proposed SC-NOFS demands fewer operations due to the use of NOFS
and INOFS transforms.

##  VI Hardware Prototyping

Instead of using high-value equipment [37, 38, 39, 40] to validate our
proposed communication systems, this work employs the Analog Devices software-
defined radio (SDR) PLUTO [41] to test the functionality of our proposed
signals. Our experiment is designed to have the following advantages:

  * •

Operation on Low-Cost Hardware: The experiment is conducted using the low-cost
Analog Devices SDR PLUTO, enabling cost-efficient Internet of things (IoT)
testing and validation of our proposed signals in IoT preferred devices.

  * •

Integration with Standard OFDM: The designed signal is seamlessly integrated
into OFDM signals, ensuring compatibility with existing OFDM-supported
standards.

  * •

Avoidance of AI Model Retraining: Even with changes in channel conditions,
there is no need to retrain the AI model, simplifying the experimentation
process and demonstrating the feasibility of AI in communication systems with
variable channels.

Figure 14: Standard compatible SC-NOFS signal experiment block diagram
illustration.

The experiment block diagram is illustrated in Fig. 14. The signal processing
blocks within the upper dashed box are standard compatible because the
proposed signal framework applies the same signal modulation, demodulation,
channel estimation, and channel equalization methods with OFDM standards. The
aim is to showcase that our signal is able to be easily integrated into
existing standards where simple OFDM signal processing algorithms can be
reused.

In Fig. 14, at the transmitter side, symbols such as QAM, are firstly
modulated by the NOFS transform. Pilot symbols are inserted for the channel
estimation purpose. Guard band is added to reserve a protection gap in
frequency-domain. IFFT is applied for OFDM signal generation. CP is added to
avoid ISI and later is used to assist channel estimation/equalization. Timing
synchronization sequence is added at the beginning of the data to locate the
beginning of the data stream at the receiver. Analog signals are obtained
after the digital-to-analogue converter (DAC). It is noted that software-based
channel effect emulation is included at the Tx-DSP side. At the receiver,
after the software emulated channel effects and the wired cable transmission,
the channel distorted signal is received and converted to digital signals via
the analogue-to-digital converter (ADC). After timing synchronization, OFDM
demodulation, channel estimation, and channel equalization, the signals are
fed to the INOFS transform block for the second stage symbol recovery. With a
signal detector followed, the original symbols are recovered.

Figure 15: Experimental setup at 2.4 GHz carrier frequency using a low-cost
SDR device in an indoor lab environment. A wired cable is connected between
the Tx port and the Rx port to block over-the-air signal interference. Figure
16: The configuration of the PHY-layer time frame and the demonstration of
measured spectral and constellation performance. To provide a comprehensive
illustration of the entire framework and show standard compatibility, we
present a full PHY-layer time frame. In industrial applications, it is worth
noting that a shorter packet segment within the time frame could be utilized
to align with specific timing requirements.  Figure 17: Experimental BER
performance in the AWGN channel.  Figure 18: Measured frequency selective
distortions.  Figure 19: Experimental BER performance in a frequency selective
channel.

The real hardware setup is shown in Fig. 15 where a physical wired cable is
connected between the transmitter and the receiver on the SDR device. This
deliberate choice avoids the influence of over-the-air signal interference,
ensuring a controlled environment for experimentation and a convincing and
reliable basis for comparisons. To accurately emulate channel effects, the
experiment relies on a software emulation methodology, as depicted in Fig. 14.
This approach allows for the controlled channel conditions and evaluates their
impact on signal transmission, providing valuable insights into system
performance under various scenarios. A laptop serves as the central control
unit for the entire experimental setup. This setup ensures smooth operation
and reliable data collection throughout the experiment. Considering the
advantages of SC-NOFS, we design two application scenarios:

  * •

Scenario-A: Bandwidth saving. As noted in Fig. 6, the NOFS transform can
efficiently prune symbols, indicating that a signal can convey the same
information while occupying a reduced spectral bandwidth. This feature is
suitable for IoT applications because a large number of IoT devices could be
connected in future networks, where potential spectral interference among them
is a concern [42, 43]. With a narrower data bandwidth and therefore a wider
protection guard band, SC-NOFS signals can effectively mitigate such
interference. To validate the effectiveness and feasibility of the bandwidth
saving scenario, metrics such as spectral utilization, EVM, and data rate are
measured to provide valuable insights into the performance.

  * •

Scenario-B: Faster transmission. By manipulating the signal in Scenario-A via
increasing sampling rate, the signal bandwidth expands to that of the
conventional SC-OFDM but with a higher data rate. In this case, a system could
complete a transmission in a shorter time slot. This leads to a reduction in
latency or allowing for the reservation of a wider time protection gap to
mitigate severe time jitter effects. To validate the effectiveness and
feasibility of the faster transmission scenario, metrics such as spectral
utilization, EVM, data rate, latency, jitter, and BER are measured to quantify
the quality and reliability of signal transmission.

To follow standard specifications such as 5G [44], we set the number of data
sub-carriers $M$=600, the IFFT size $N$=1024, the sampling rate is 15.36
Mbit/s. The 5G standard defines different subcarrier mapping schemes, and this
experiment utilizes a 15 kHz configuration for demonstration purposes. With
such configurations, the PHY-layer time frame structure is depicted in Fig.
16(a) where $T_{{}_{frame}}$=10 ms and each frame contains 10 sub-frames. It
is noted that in the experiment we employ a full PHY-layer time frame for
illustrations that adhere to standard compatibility. In industrial
applications, a shorter packet segment within the time frame could be utilized
to meet specific timing requirements. In Fig. 16(b), it is evident that the
occupied bandwidth of the SC-OFDM signal is 9 MHz. Utilizing QPSK, the data
rate is 18 Mbit/s. With this configuration, the recovered signal constellation
is displayed in Fig. 16(c) with EVM at -22.3 dB.

In the Scenario-A for SC-NOFS, our proposed signal allows for symbol pruning
from 600 to 492, achieving a compression ratio of $\alpha$=0.82. As
illustrated in Fig. 16(d), the signal bandwidth is compressed by this ratio to
a narrower bandwidth of 7.38 MHz. Despite this reduction in bandwidth due to
symbol pruning in our SC-NOFS signal, the receiver can recover the complete
set of symbols using specifically designed algorithms. Consequently, the data
rate with SC-NOFS remains at 18 Mbit/s. The performance of SC-NOFS is
displayed in Fig. 16(e), where four distinct constellation points are clearly
observed, with an EVM of -22.1 dB, comparable to that of SC-OFDM.

For the Scenario-B using SC-NOFS, a higher sampling rate of 18.73 Mbit/s could
be employed. The higher data rate indicates that the system can transmit more
data in a given time slot. On the other hand, the system can transmit the same
amount of data faster indicating time saving. As illustrated in Fig. 16(f),
the duration of one frame is reduced to $T_{{}_{frame}}$=8.2 ms, with a 1.8 ms
protection gap introduced to mitigate the time jitter effect. The spectral
bandwidth under the higher sampling rate expands to 9 MHz, similar to that of
SC-OFDM, but with a higher data rate at 22 Mbit/s, as shown in Fig. 16(g). Its
performance is presented in Fig. 16(h) where four constellation points are
clearly observable with an EVM of -22.5 dB.

For a more comprehensive analysis of SC-NOFS, it is essential to investigate
the BER in relation to signal-to-noise ratio (SNR). We introduce noise to the
signal in the digital domain to simulate a wide range of SNR conditions. The
resulting BER performance is illustrated in Fig. 17. It is evident that both
signals exhibit similar performance across different SNR values. One
explanation for this similarity lies in the use of OFDM as the air interface
for both SC-OFDM and SC-NOFS. As a result, they demonstrate comparable channel
estimation and equalization performance. Furthermore, the NOFS transform,
having been verified to perform equivalently to DFT, contributes to the
comparable BER performance observed in both signals.

The use of a wired cable connection eliminates over-the-air signal
interference but comes at the cost of losing multipath effects, resulting in
flat spectra, as displayed in Fig. 16. To have a robust performance evaluation
under multipath effects and show the advantage of SC-NOFS, the experiment
specifically designed a frequency-selective channel model in a software
environment defined by
${h_{B}(t)}=0.6965\delta(t)-0.3279\delta(t-T_{s})+0.1765\delta(t-3T_{s})+0.5991\delta(t-5T_{s})+0.1315\delta(t-7T_{s})$.
By applying the same channel model to SC-OFDM and SC-NOFS, the signals with
frequency selective distortions are displayed in Fig. 18. It is noted that the
faster transmission of SC-NOFS signals helps avoid the power distortion
segment of the spectrum in SC-OFDM. This avoidance contributes to the superior
BER performance of SC-NOFS compared to SC-OFDM, even with a 22% higher data
transmission rate, as illustrated in Fig. 19.

Figure 20: Examples showing how SC-NOFS signals reduce transmission latency
and enhance time jitter tolerance.

Considering the timing frame and signal features in Fig. 16(f)(g), SC-NOFS
signals offer significant advantages in reducing time latency and addressing
challenges associated with time jitter. This makes SC-NOFS a promising
candidate for applications where precise timing and minimized variations in
signal arrival times are crucial. The advantages of SC-NOFS are displayed in
Fig. 20 where various time jitter conditions are considered. In Fig. 20(b),
the 1.8 ms time saving results in a shorter transmission time, creating a low-
latency scenario. Furthermore, the saved time resource can also be utilized to
improve time jitter robustness by reserving a wider time gap in Fig. 20(c),
thereby enhancing jitter tolerance.

##  VII Conclusions

This work proposed a spectrally efficient SC-NOFS waveform enabled by irSinc
to enhance performance in industrial IoT applications. Specifically, we
propose a two-stage signal waveform framework that aligns with existing 5G
standards. The framework leverages the dimension-reduced NOFS transform, which
has higher spectral efficiency and lower computational complexity compared to
DFT. Furthermore, the framework employs machine learning to determine the
optimal irSinc sub-carriers rather than the signal itself. As a result, the
AI-based communication framework is robust, eliminating the need for model re-
training even with changing channels. The signal achieves a spectral
compression of $\alpha$=0.82, indicating a 22%=(1/$\alpha$)$\times$100%
improvement in spectral efficiency. Simulation results demonstrate that the
SC-NOFS signal achieves a better PAPR and competitive BER in both AWGN and
frequency-selective channels. Experimental validation on a low-cost SDR device
with practical signal transmissions confirms two scenarios in a 5G setup: 1)
Bandwidth saving. Conventional SC-OFDM requires a 9 MHz bandwidth for an 18
Mbit/s data rate, while the proposed SC-NOFS only requires 7.38 MHz for the
same data rate. 2) Faster transmission. Utilizing the same spectral bandwidth
of 9 MHz, SC-OFDM provides an 18 Mbit/s data rate, while SC-NOFS achieves 22
Mbit/s. In a 5G setup, the faster transmission saves 1.8 ms of timing
resources, leading to either reduced latency or enhanced time jitter
tolerance.

## References

  * [1] L. D. Xu, W. He, and S. Li, “Internet of things in industries: A survey,” _IEEE Transactions on Industrial Informatics_ , vol. 10, no. 4, pp. 2233–2243, 2014.
  * [2] M. Wollschlaeger, T. Sauter, and J. Jasperneite, “The future of industrial communication: Automation networks in the era of the Internet of things and industry 4.0,” _IEEE Industrial Electronics Magazine_ , vol. 11, no. 1, pp. 17–27, 2017.
  * [3] A. Mahmood, L. Beltramelli, S. Fakhrul Abedin, S. Zeb, N. I. Mowla, S. A. Hassan, E. Sisinni, and M. Gidlund, “Industrial IoT in 5G-and-beyond networks: Vision, architecture, and design trends,” _IEEE Transactions on Industrial Informatics_ , vol. 18, no. 6, pp. 4122–4137, 2022.
  * [4] Y. Li, Y. Dong, P. Fan, and K. B. Letaief, “How far are wireless networks from being truly deterministic?” _IEEE Internet of Things Magazine_ , vol. 5, no. 4, pp. 64–71, 2022.
  * [5] E. Sisinni, A. Saifullah, S. Han, U. Jennehag, and M. Gidlund, “Industrial Internet of things: Challenges, opportunities, and directions,” _IEEE Transactions on Industrial Informatics_ , vol. 14, no. 11, pp. 4724–4734, 2018\.
  * [6] N. Finn, “Introduction to time-sensitive networking,” _IEEE Communications Standards Magazine_ , vol. 2, no. 2, pp. 22–28, 2018.
  * [7] A. Nasrallah, A. S. Thyagaturu, Z. Alharbi, C. Wang, X. Shao, M. Reisslein, and H. ElBakoury, “Ultra-low latency (ULL) networks: The IEEE TSN and IETF DetNet standards and related 5G ULL research,” _IEEE Communications Surveys and Tutorials_ , vol. 21, no. 1, pp. 88–145, 2019.
  * [8] M. R. Palattella, N. Accettura, L. A. Grieco, G. Boggia, M. Dohler, and T. Engel, “On optimal scheduling in duty-cycled industrial IoT applications using IEEE802.15.4e TSCH,” _IEEE Sensors Journal_ , vol. 13, no. 10, pp. 3655–3666, 2013.
  * [9] H. Nyquist, “Certain factors affecting telegraph speed,” _Journal of the A.I.E.E._ , vol. 43, no. 2, pp. 124–130, 1924.
  * [10] ——, “Certain topics in telegraph transmission theory,” _Transactions of the American Institute of Electrical Engineers_ , vol. 47, no. 2, pp. 617–644, 1928.
  * [11] C. E. Shannon and W. Weaver, _The Mathematical Theory of Communication_. University of Illinois Press, 1949.
  * [12] R. R. Mosier and R. G. Clabaugh, “Kineplex, a bandwidth-efficient binary transmission system,” _American Institute of Electrical Engineers, Part I: Communication and Electronics, Transactions of the_ , vol. 76, no. 6, pp. 723–728, Jan. 1958.
  * [13] T. Xu and I. Darwazeh, “Transmission experiment of bandwidth compressed carrier aggregation in a realistic fading channel,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 5, pp. 4087–4097, May 2017.
  * [14] T. Ishihara, S. Sugiura, and L. Hanzo, “The evolution of faster-than-Nyquist signaling,” _IEEE Access_ , vol. 9, pp. 86 535–86 564, 2021.
  * [15] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” _IEEE Signal Processing Magazine_ , vol. 28, no. 3, pp. 92–112, May 2011.
  * [16] N. Michailow, M. Matthe, I. Gaspar, A. Caldevilla, L. Mendes, A. Festag, and G. Fettweis, “Generalized frequency division multiplexing for 5th generation cellular networks,” _IEEE Transactions on Communications_ , vol. 62, no. 9, pp. 3045–3061, Sep. 2014.
  * [17] T. Xu, R. C. Grammenos, F. Marvasti, and I. Darwazeh, “An improved fixed sphere decoder employing soft decision for the detection of non-orthogonal signals,” _IEEE Communications Letters_ , vol. 17, no. 10, pp. 1964–1967, Oct. 2013.
  * [18] J. B. Anderson, F. Rusek, and V. Öwall, “Faster-than-nyquist signaling,” _Proceedings of the IEEE_ , vol. 101, no. 8, pp. 1817–1830, Aug. 2013.
  * [19] S. Li, B. Bai, J. Zhou, P. Chen, and Z. Yu, “Reduced-complexity equalization for faster-than-nyquist signaling: New methods based on ungerboeck observation model,” _IEEE Transactions on Communications_ , vol. 66, no. 3, pp. 1190–1204, Nov. 2018.
  * [20] S. Li, W. Yuan, J. Yuan, B. Bai, D. Wing Kwan Ng, and L. Hanzo, “Time-domain vs. frequency-domain equalization for ftn signaling,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 8, pp. 9174–9179, Jun. 2020\.
  * [21] F. Tariq, M. R. A. Khandaker, K. K. Wong, M. A. Imran, M. Bennis, and M. Debbah, “A speculative study on 6G,” _IEEE Wireless Communications_ , vol. 27, no. 4, pp. 118–125, 2020.
  * [22] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,” _IEEE Network_ , vol. 34, no. 3, pp. 134–142, 2020.
  * [23] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” _IEEE Communications Magazine_ , vol. 58, no. 3, pp. 55–61, 2020.
  * [24] W. Tong and P. Zhu, _6G: The Next Horizon_. Cambridge University Press, 2021.
  * [25] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” in _2017 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2017, pp. 1–6.
  * [26] H. Lin and J. Yuan, “Orthogonal delay-Doppler division multiplexing modulation,” _IEEE Transactions on Wireless Communications_ , vol. 21, no. 12, pp. 11 024–11 037, 2022.
  * [27] T. Xu and I. Darwazeh, “Identification and practical validation of spectrally efficient non-orthogonal frequency shaping waveform,” _Nature Communications Engineering_ , vol. 2, no. 58, 2023.
  * [28] J. Mazo, “Faster-than-Nyquist signaling,” _Bell Syst. Tech. J_ , vol. 54, no. 8, pp. 1451–1462, 1975.
  * [29] NGMN, “6G position statement: an operator view,” Sept. 2023.
  * [30] Keysight, “Next-generation wireless: A guide to the fundamentals of 6G,” Dec 2023.
  * [31] International Telecommunication Union, “IMT towards 2030 and beyond,” https://www.itu.int/en/ITU-R/study-groups/rsg5/rwp5d/imt-2030/Pages/default.aspx, June 2023.
  * [32] T. Xu and I. Darwazeh, “Nyquist-SEFDM: Pulse shaped multicarrier communication with sub-carrier spacing below the symbol rate,” in _2016 10th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP)_ , Jul. 2016, pp. 1–6.
  * [33] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” _Mathematics of Computation_ , vol. 19, no. 90, pp. 297–301, 1965.
  * [34] J. Fourier, _The Analytical Theory of Heat_. Chez Firmin Didot, père et fils, 1822.
  * [35] M. Hamamura and S. Tachikawa, “Bandwidth efficiency improvement for multi-carrier systems,” in _2004 IEEE 15th International Symposium on Personal, Indoor and Mobile Radio Communications (IEEE Cat. No.04TH8754)_ , vol. 1, 2004, pp. 48–52 Vol.1.
  * [36] W. Ozan, K. Jamieson, and I. Darwazeh, “Truncating and oversampling ofdm signals in white gaussian noise channels,” in _2016 10th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP)_ , 2016, pp. 1–6.
  * [37] T. Xu, C. Masouros, and I. Darwazeh, “Waveform and space precoding for next generation downlink narrowband IoT,” _IEEE Internet of Things Journal_ , vol. 6, no. 3, pp. 5097–5107, Jun. 2019.
  * [38] T. Xu, C. Masouros, and I. Darwazeh, “Design and prototyping of hybrid analogue digital multiuser MIMO beamforming for non-orthogonal signals,” _IEEE Internet of Things Journal_ , vol. 7, no. 3, pp. 1872–1883, Mar. 2020\.
  * [39] T. Xu and I. Darwazeh, “Non-orthogonal narrowband internet of things: A design for saving bandwidth and doubling the number of connected devices,” _IEEE Internet of Things Journal_ , vol. 5, no. 3, pp. 2120–2129, 2018.
  * [40] T. Xu, F. Liu, C. Masouros, and I. Darwazeh, “An experimental proof of concept for integrated sensing and communications waveform design,” _IEEE Open Journal of the Communications Society_ , vol. 3, pp. 1643–1655, 2022.
  * [41] Analog-Devices, “ADALM-PLUTO SDR active learning module,” https://www.analog.com/media/en/news-marketing-collateral/product-highlight/ADALM-PLUTO-Product-Highlight.pdf, technical document, Jul. 2019.
  * [42] M. Hampson, “Climate monitoring and 6G must learn to coexist,” https://spectrum.ieee.org/climate-monitoring-satellites, Jul. 2023.
  * [43] M. Polese, X. Cantos-Roman, A. Singh, M. J. Marcus, T. J. Maccarone, T. Melodia, and J. M. Jornet, “Coexistence and spectrum sharing above 100 GHz,” _Proceedings of the IEEE_ , vol. 111, no. 8, pp. 928–954, 2023.
  * [44] E. Dahlman, S. Parkvall, and J. Sköld, _5G NR: The Next Generation Wireless Access Technology_. Academic Press, 2018.

  *[IIoT]: industrial Internet of things
  *[MAC]: media access control
  *[MIMO]: multiple input multiple output
  *[CSI]: channel state information
  *[OFDM]: orthogonal frequency division multiplexing
  *[LTE]: long term evolution
  *[SEFDM]: spectrally efficient frequency division multiplexing
  *[FTN]: faster than Nyquist
  *[FBMC]: filter bank multicarrier
  *[GFDM]: generalized frequency division multiplexing
  *[ICI]: inter carrier interference
  *[ISI]: inter symbol interference
  *[OTFS]: orthogonal time frequency space
  *[ODDM]: orthogonal delay-Doppler division multiplexing
  *[NOFS]: non-orthogonal frequency spacing
  *[SC-NOFS]: single-carrier non-orthogonal frequency shaping
  *[SC-OFDM]: single-carrier orthogonal frequency division multiplexing
  *[IFFT]: inverse fast Fourier transform
  *[INOFS]: inverse non-orthogonal frequency shaping
  *[AI]: artificial intelligence
  *[BER]: bit error rate
  *[EVM]: error vector magnitude
  *[PAPR]: peak-to-average power ratio
  *[RC]: raised cosine
  *[FFT]: fast Fourier transform
  *[QAM]: quadrature amplitude modulation
  *[AWGN]: additive white Gaussian noise
  *[CP]: cyclic prefix
  *[DFT]: discrete Fourier transform
  *[ID]: iterative detection
  *[MSE]: mean squared error
  *[HC-MCM]: high compaction multi-carrier communication
  *[IDFT]: inverse discrete Fourier transform
  *[SDR]: software-defined radio
  *[IoT]: Internet of things
  *[DAC]: digital-to-analogue converter
  *[ADC]: analogue-to-digital converter
  *[SNR]: signal-to-noise ratio


Technology, Pasadena, USA A. Bornheim, O. Cerri, I. Dutta, J.M. Lawhorn, N.
Lu, J. Mao, H.B. Newman, J. Ngadiuba, T.Q. Nguyen, M. Spiropulu, J.R. Vlimant,
C. Wang, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon University, Pittsburgh,
USA J. Alison, M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, I.
Vorobiev University of Colorado Boulder, Boulder, USA J.P. Cumalat, W.T. Ford,
E. MacDonald, R. Patel, A. Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner
Cornell University, Ithaca, USA J. Alexander, Y. Cheng, J. Chu, D.J. Cranshaw,
A. Datta, A. Frankenthal, K. Mcdermott, J. Monroy, J.R. Patterson, D. Quach,
A. Ryd, W. Sun, S.M. Tan, Z. Tao, J. Thom, P. Wittich, M. Zientek Fermi
National Accelerator Laboratory, Batavia, USA M. Albrow, M. Alyari, G.
Apollinari, A. Apresyan, A. Apyan, S. Banerjee, L.A.T. Bauerdick, A. Beretvas,
D. Berry, J. Berryhill, P.C. Bhat, K. Burkett, J.N. Butler, A. Canepa, G.B.
Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi, K.F. Di Petrillo, V.D.
Elvira, J. Freeman, Z. Gecse, L. Gray, D. Green, S. Grünendahl, O. Gutsche,
R.M. Harris, S. Hasegawa, R. Heller, T.C. Herwig, J. Hirschauer, B.
Jayatilaka, S. Jindariani, M. Johnson, U. Joshi, P. Klabbers, T. Klijnsma, B.
Klima, M.J. Kortelainen, S. Lammel, D. Lincoln, R. Lipton, M. Liu, T. Liu, J.
Lykken, K. Maeshima, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V.
O’Dell, V. Papadimitriou, K. Pedro, C. Pena57, O. Prokofyev, F. Ravera, A.
Reinsvold Hall, L. Ristori, B. Schneider, E. Sexton-Kennedy, N. Smith, A.
Soha, W.J. Spalding, L. Spiegel, S. Stoynev, J. Strait, L. Taylor, S. Tkaczyk,
N.V. Tran, L. Uplegger, E.W. Vaandering, H.A. Weber University of Florida,
Gainesville, USA D. Acosta, P. Avery, D. Bourilkov, L. Cadamuro, V.
Cherepanov, F. Errico, R.D. Field, D. Guerrero, B.M. Joshi, M. Kim, J.
Konigsberg, A. Korytov, K.H. Lo, K. Matchev, N. Menendez, G. Mitselmakher, D.
Rosenzweig, K. Shi, J. Sturdy, J. Wang, X. Zuo Florida State University,
Tallahassee, USA T. Adams, A. Askew, D. Diaz, R. Habibullah, S. Hagopian, V.
Hagopian, K.F. Johnson, R. Khurana, T. Kolberg, G. Martinez, H. Prosper, C.
Schiber, R. Yohay, J. Zhang Florida Institute of Technology, Melbourne, USA
M.M. Baarmand, S. Butalla, T. Elkafrawy16, M. Hohlmann, D. Noonan, M. Rahmani,
M. Saunders, F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA
M.R. Adams, L. Apanasevich, H. Becerril Gonzalez, R. Cavanaugh, X. Chen, S.
Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, C. Mills, G. Oh,
T. Roy, M.B. Tonjes, N. Varelas, J. Viinikainen, X. Wang, Z. Wu, Z. Ye The
University of Iowa, Iowa City, USA M. Alhusseini, K. Dilsiz90, S. Durgut, R.P.
Gandrajula, M. Haytmyradov, V. Khristenko, O.K. Köseyan, J.-P. Merlo, A.
Mestvirishvili91, A. Moeller, J. Nachtman, H. Ogul92, Y. Onel, F. Ozok93, A.
Penzo, C. Snyder, E. Tiras94, J. Wetzel Johns Hopkins University, Baltimore,
USA O. Amram, B. Blumenfeld, L. Corcodilos, M. Eminizer, A.V. Gritsan, S.
Kyriacou, P. Maksimovic, C. Mantilla, J. Roskes, M. Swartz, T.Á. Vámi The
University of Kansas, Lawrence, USA C. Baldenegro Barrera, P. Baringer, A.
Bean, A. Bylinkin, T. Isidori, S. Khalil, J. King, G. Krintiras, A.
Kropivnitskaya, C. Lindsey, N. Minafra, M. Murray, C. Rogan, C. Royon, S.
Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang, J. Williams, G. Wilson Kansas
State University, Manhattan, USA S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y.
Maravin, T. Mitchell, A. Modak, A. Mohammadi Lawrence Livermore National
Laboratory, Livermore, USA F. Rebassoo, D. Wright University of Maryland,
College Park, USA E. Adams, A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng,
N.J. Hadley, S. Jabeen, G.Y. Jeng, R.G. Kellogg, T. Koeth, A.C. Mignerey, S.
Nabili, M. Seidel, A. Skuja, S.C. Tonwar, L. Wang, K. Wong Massachusetts
Institute of Technology, Cambridge, USA D. Abercrombie, B. Allen, R. Bi, S.
Brandt, W. Busza, I.A. Cali, Y. Chen, M. D’Alfonso, G. Gomez Ceballos, M.
Goncharov, P. Harris, D. Hsu, M. Hu, M. Klute, D. Kovalskyi, J. Krupa, Y.-J.
Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mironov, S. Narayanan, X. Niu, C.
Paus, D. Rankin, C. Roland, G. Roland, Z. Shi, G.S.F. Stephans, K. Sumorok, K.
Tatar, D. Velicanu, J. Wang, T.W. Wang, Z. Wang, B. Wyslouch University of
Minnesota, Minneapolis, USA R.M. Chatterjee, A. Evans, P. Hansen, J.
Hiltbrand, Sh. Jain, M. Krohn, Y. Kubota, Z. Lesko, J. Mans, M. Revering, R.
Rusack, R. Saradhy, N. Schroeder, N. Strobbe, M.A. Wadud University of
Mississippi, Oxford, USA J.G. Acosta, S. Oliveros University of Nebraska-
Lincoln, Lincoln, USA K. Bloom, S. Chauhan, D.R. Claes, C. Fangmeier, L.
Finco, F. Golf, J.R. González Fernández, C. Joo, I. Kravchenko, J.E. Siado,
G.R. Snow${}^{\textrm{\textdagger}}$, W. Tabb, F. Yan State University of New
York at Buffalo, Buffalo, USA G. Agarwal, H. Bandyopadhyay, L. Hay, I.
Iashvili, A. Kharchilava, C. McLean, D. Nguyen, J. Pekkanen, S. Rappoccio
Northeastern University, Boston, USA G. Alverson, E. Barberis, C. Freer, Y.
Haddad, A. Hortiangtham, J. Li, G. Madigan, B. Marzocchi, D.M. Morse, V.
Nguyen, T. Orimoto, A. Parker, L. Skinnari, A. Tishelman-Charny, T. Wamorkar,
B. Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA S.
Bhattacharya, J. Bueghly, Z. Chen, A. Gilbert, T. Gunter, K.A. Hahn, N. Odell,
M.H. Schmitt, K. Sung, M. Velasco University of Notre Dame, Notre Dame, USA R.
Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop, K.
Lannon, N. Loukas, N. Marinelli, I. Mcalister, F. Meng, K. Mohrman, Y.
Musienko50, R. Ruchti, P. Siddireddy, M. Wayne, A. Wightman, M. Wolf, L.
Zygala The Ohio State University, Columbus, USA J. Alimena, B. Bylsma, B.
Cardwell, L.S. Durkin, B. Francis, C. Hill, A. Lefeld, B.L. Winer, B.R. Yates
Princeton University, Princeton, USA B. Bonham, P. Das, G. Dezoort, P. Elmer,
B. Greenberg, N. Haubrich, S. Higginbotham, A. Kalogeropoulos, G. Kopp, S.
Kwan, D. Lange, M.T. Lucchini, J. Luo, D. Marlow, K. Mei, I. Ojalvo, J. Olsen,
C. Palmer, P. Piroué, D. Stickland, C. Tully University of Puerto Rico,
Mayaguez, USA S. Malik, S. Norberg Purdue University, West Lafayette, USA V.E.
Barnes, R. Chawla, S. Das, L. Gutay, M. Jones, A.W. Jung, G. Negro, N.
Neumeister, C.C. Peng, S. Piperov, A. Purohit, J.F. Schulte, M. Stojanovic18,
N. Trevisani, F. Wang, A. Wildridge, R. Xiao, W. Xie Purdue University
Northwest, Hammond, USA J. Dolen, N. Parashar Rice University, Houston, USA A.
Baty, S. Dildick, K.M. Ecklund, S. Freed, F.J.M. Geurts, A. Kumar, W. Li, B.P.
Padley, R. Redjimi, J. Roberts${}^{\textrm{\textdagger}}$, J. Rorie, W. Shi,
A.G. Stahl Leiton University of Rochester, Rochester, USA A. Bodek, P. de
Barbaro, R. Demina, J.L. Dulemba, C. Fallon, T. Ferbel, M. Galanti, A. Garcia-
Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R. Taus Rutgers, The
State University of New Jersey, Piscataway, USA B. Chiarito, J.P. Chou, A.
Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E. Hughes, S.
Kaplan, O. Karacheban26, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M.
Osherson, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S.A. Thayil, S.
Thomas, H. Wang University of Tennessee, Knoxville, USA H. Acharya, A.G.
Delannoy, S. Spanier Texas A&M University, College Station, USA O. Bouhali95,
M. Dalchenko, A. Delgado, R. Eusebi, J. Gilmore, T. Huang, T. Kamon96, H. Kim,
S. Luo, S. Malhotra, R. Mueller, D. Overton, L. Perniè, D. Rathjens, A.
Safonov Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, V. Hegde,
S. Kunori, K. Lamichhane, S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S.
Undleeb, I. Volobouev, Z. Wang, A. Whitbeck Vanderbilt University, Nashville,
USA E. Appelt, S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A.
Melo, H. Ni, K. Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska University
of Virginia, Charlottesville, USA M.W. Arenton, B. Cox, G. Cummings, J.
Hakala, R. Hirosky, M. Joyce, A. Ledovskoy, A. Li, C. Neu, B. Tannenwald, E.
Wolfe Wayne State University, Detroit, USA P.E. Karchin, N. Poudyal, P. Thapa
University of Wisconsin - Madison, Madison, WI, USA K. Black, T. Bose, J.
Buchanan, C. Caillol, S. Dasu, I. De Bruyn, P. Everaerts, C. Galloni, H. He,
M. Herndon, A. Hervé, U. Hussain, A. Lanaro, A. Loeliger, R. Loveless, J.
Madhusudanan Sreekala, A. Mallampalli, D. Pinna, A. Savin, V. Shang, V.
Sharma, W.H. Smith, D. Teague, S. Trembath-reichert, W. Vetens †: Deceased 1:
Also at Vienna University of Technology, Vienna, Austria 2: Also at Institute
of Basic and Applied Sciences, Faculty of Engineering, Arab Academy for
Science, Technology and Maritime Transport, Alexandria, Egypt, Alexandria,
Egypt 3: Also at Université Libre de Bruxelles, Bruxelles, Belgium 4: Also at
IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 5: Also at
Universidade Estadual de Campinas, Campinas, Brazil 6: Also at Federal
University of Rio Grande do Sul, Porto Alegre, Brazil 7: Also at UFMS, Nova
Andradina, Brazil 8: Also at Universidade Federal de Pelotas, Pelotas, Brazil
9: Also at Nanjing Normal University Department of Physics, Nanjing, China 10:
Now at The University of Iowa, Iowa City, USA 11: Also at University of
Chinese Academy of Sciences, Beijing, China 12: Also at Institute for
Theoretical and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov
Institute’, Moscow, Russia 13: Also at Joint Institute for Nuclear Research,
Dubna, Russia 14: Also at Helwan University, Cairo, Egypt 15: Now at Zewail
City of Science and Technology, Zewail, Egypt 16: Also at Ain Shams
University, Cairo, Egypt 17: Now at British University in Egypt, Cairo, Egypt
18: Also at Purdue University, West Lafayette, USA 19: Also at Université de
Haute Alsace, Mulhouse, France 20: Also at Tbilisi State University, Tbilisi,
Georgia 21: Also at Erzincan Binali Yildirim University, Erzincan, Turkey 22:
Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland
23: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany 24: Also at University of Hamburg, Hamburg, Germany 25: Also at
Department of Physics, Isfahan University of Technology, Isfahan, Iran,
Isfahan, Iran 26: Also at Brandenburg University of Technology, Cottbus,
Germany 27: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow
State University, Moscow, Russia 28: Also at Institute of Physics, University
of Debrecen, Debrecen, Hungary, Debrecen, Hungary 29: Also at Physics
Department, Faculty of Science, Assiut University, Assiut, Egypt 30: Also at
Eszterhazy Karoly University, Karoly Robert Campus, Gyongyos, Hungary 31: Also
at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 32: Also at MTA-
ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd
University, Budapest, Hungary, Budapest, Hungary 33: Also at Wigner Research
Centre for Physics, Budapest, Hungary 34: Also at IIT Bhubaneswar,
Bhubaneswar, India, Bhubaneswar, India 35: Also at Institute of Physics,
Bhubaneswar, India 36: Also at G.H.G. Khalsa College, Punjab, India 37: Also
at Shoolini University, Solan, India 38: Also at University of Hyderabad,
Hyderabad, India 39: Also at University of Visva-Bharati, Santiniketan, India
40: Also at Indian Institute of Technology (IIT), Mumbai, India 41: Also at
Deutsches Elektronen-Synchrotron, Hamburg, Germany 42: Also at Sharif
University of Technology, Tehran, Iran 43: Also at Department of Physics,
University of Science and Technology of Mazandaran, Behshahr, Iran 44: Now at
INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari c, Bari,
Italy 45: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy 46: Also at Centro Siciliano
di Fisica Nucleare e di Struttura Della Materia, Catania, Italy 47: Also at
Università di Napoli ’Federico II’, NAPOLI, Italy 48: Also at Riga Technical
University, Riga, Latvia, Riga, Latvia 49: Also at Consejo Nacional de Ciencia
y Tecnología, Mexico City, Mexico 50: Also at Institute for Nuclear Research,
Moscow, Russia 51: Now at National Research Nuclear University ’Moscow
Engineering Physics Institute’ (MEPhI), Moscow, Russia 52: Also at St.
Petersburg State Polytechnical University, St. Petersburg, Russia 53: Also at
University of Florida, Gainesville, USA 54: Also at Imperial College, London,
United Kingdom 55: Also at Moscow Institute of Physics and Technology, Moscow,
Russia, Moscow, Russia 56: Also at P.N. Lebedev Physical Institute, Moscow,
Russia 57: Also at California Institute of Technology, Pasadena, USA 58: Also
at Budker Institute of Nuclear Physics, Novosibirsk, Russia 59: Also at
Faculty of Physics, University of Belgrade, Belgrade, Serbia 60: Also at
Trincomalee Campus, Eastern University, Sri Lanka, Nilaveli, Sri Lanka 61:
Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy, Pavia,
Italy 62: Also at National and Kapodistrian University of Athens, Athens,
Greece 63: Also at Universität Zürich, Zurich, Switzerland 64: Also at Ecole
Polytechnique Fédérale Lausanne, Lausanne, Switzerland 65: Also at Stefan
Meyer Institute for Subatomic Physics, Vienna, Austria, Vienna, Austria 66:
Also at Laboratoire d’Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS,
Annecy-le-Vieux, France 67: Also at Şırnak University, Sirnak, Turkey 68: Also
at Department of Physics, Tsinghua University, Beijing, China, Beijing, China
69: Also at Near East University, Research Center of Experimental Health
Science, Nicosia, Turkey 70: Also at Beykent University, Istanbul, Turkey,
Istanbul, Turkey 71: Also at Istanbul Aydin University, Application and
Research Center for Advanced Studies (App. & Res. Cent. for Advanced Studies),
Istanbul, Turkey 72: Also at Mersin University, Mersin, Turkey 73: Also at
Piri Reis University, Istanbul, Turkey 74: Also at Adiyaman University,
Adiyaman, Turkey 75: Also at Ozyegin University, Istanbul, Turkey 76: Also at
Izmir Institute of Technology, Izmir, Turkey 77: Also at Necmettin Erbakan
University, Konya, Turkey 78: Also at Bozok Universitetesi Rektörlügü, Yozgat,
Turkey, Yozgat, Turkey 79: Also at Marmara University, Istanbul, Turkey 80:
Also at Milli Savunma University, Istanbul, Turkey 81: Also at Kafkas
University, Kars, Turkey 82: Also at Istanbul Bilgi University, Istanbul,
Turkey 83: Also at Hacettepe University, Ankara, Turkey 84: Also at Vrije
Universiteit Brussel, Brussel, Belgium 85: Also at School of Physics and
Astronomy, University of Southampton, Southampton, United Kingdom 86: Also at
IPPP Durham University, Durham, United Kingdom 87: Also at Monash University,
Faculty of Science, Clayton, Australia 88: Also at Bethel University, St.
Paul, Minneapolis, USA, St. Paul, USA 89: Also at Karamanoğlu Mehmetbey
University, Karaman, Turkey 90: Also at Bingol University, Bingol, Turkey 91:
Also at Georgian Technical University, Tbilisi, Georgia 92: Also at Sinop
University, Sinop, Turkey 93: Also at Mimar Sinan University, Istanbul,
Istanbul, Turkey 94: Also at Erciyes University, KAYSERI, Turkey 95: Also at
Texas A&M University at Qatar, Doha, Qatar 96: Also at Kyungpook National
University, Daegu, Korea, Daegu, Korea