# Generation of Bimodal Solitons in a Sapphire Whispering Gallery Mode Maser
at Millikelvin Temperatures

Catriona A. Thomson  ARC Centre of Excellence for Engineered Quantum Systems
and ARC Centre of Excellence for Dark Matter Particle Physics, Department of
Physics, University of Western Australia, 35 Stirling Highway, Crawley WA
6009, Australia  Michael E. Tobar  ARC Centre of Excellence for Engineered
Quantum Systems and ARC Centre of Excellence for Dark Matter Particle Physics,
Department of Physics, University of Western Australia, 35 Stirling Highway,
Crawley WA 6009, Australia  Maxim Goryachev<EMAIL_ADDRESS>ARC
Centre of Excellence for Engineered Quantum Systems and ARC Centre of
Excellence for Dark Matter Particle Physics, Department of Physics, University
of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia

###### Abstract

We present experimental observations of bimodal solitons in a solid state
three-level maser cooled to millikelvin temperatures. The maser is built on a
highly dilute $\textrm{Fe}^{3+}$ spin ensemble hosted by a high purity
$\textrm{Al}_{2}\textrm{O}_{3}$ crystal constituting a high quality factor
whispering-gallery-mode resonator. The maser is pumped through one of these
modes, near 31 GHz, generating two signals near 12.04 GHz from two distinct
modes, 8 MHz apart. The system demonstrates three regimes, namely, a
continuous wave regime, a dense soliton regime and a sparse soliton regime.
These results open new avenues for studying nonlinear wave phenomena using
microwave systems as well as new applications of solitons in this part of the
electromagnetic spectrum.

##  I Introduction

Optical frequency combs have become key in myriad applications from metrology
and precision spectroscopy to tests of beyond the standard model physics, with
their remarkable ability to compare clocks within 10-19 Hz [1, 2]. Initially
produced via mode-locked lasers [3, 4, 5], which revolutionized the field of
frequency metrology [6, 7], a particularly active area of current research is
in developing alternative production mechanisms for such combs, in engineering
systems balanced between nonlinear gain and dissipation.

Figure 1: Experimental setup used to generate maser solitons at mK
temperatures. A WGM crystal resonator is pumped from a room temperature signal
source (around 31 GHz) through an input filter ensuring minimal excitation of
the readout mode. Generated maser signals (around 12 GHz) are collected with a
single antenna and fed through a low temperature isolator and a low noise
cryogenic amplifier. These signals are analysed via a spectrum analyser and
down converted via two mixers to be observed via an oscilloscope. Inset:
Fitted quality factors of modes A and B at 4 K and mK, at a range of powers
incident on the modes, without any applied pump mode. These whispering gallery
modes require a certain threshold of incident power to be applied to the mode
for high quality factor resonance to be possible.

A frequency comb may occur in conjunction with solitons within a resonator,
equivalent to ultrashort phase-locked pulses circulating the cavity in the
time domain. Such pulses may be infinitely supported within the resonator,
given a continuous-wave pump source and a resonant medium which displays a
balance between dispersion and non-linear gain [8]. To this end, the nonlinear
Kerr effect has been much exploited as the dissipative structure in various
material systems [8, 9, 10] such as sillica optical fibres [11], and high-Q
crystalline and silicon nitride-based micro-resonators [8, 12].

Despite such plethora of systems demonstrating combs and solitons [13], they
are quite unique to the optical frequency domain. This article reports the
first, to the best of authors’ knowledge, observation of microwave solitons in
a maser system. The reported phenomena was produced via the iron paramagnetic
impurities (order of parts per billion) within the most pure HEMEX grade
synthetic sapphire crystal, when cooled to millikelvin temperatures and with
zero applied magnetic field [14]. The paramagnetic impurities have been shown
to offer non-linear gain in form of a $\chi^{3}$ non-linearity at 4K in
temperature, which besides masing [14], has produced four wave mixing [15],
frequency conversion [16] and time reversal symmetry breaking [17, 18].
Dispersion arises from frequency dependence of the effective permeability and
thus refractive index due to these paramagnetic impurities [18] which can be
altered by pumping a mode near the ion pump transition. Furthermore, at
millikelvin temperature, effects of paramagnetic impurities on the properties
of cooled sapphire resonators have been shown to be enhanced when compared
with the effects at 4K as the majority of the spins condense to the ground
state [18, 19, 20]. This work reports the first operation of such a maser at
millikelvin temperatures, with the observation of new effects not seen in the
4K system, providing a new avenue for implementing optical metrological
methods in the microwave and radio- frequency domains.

##  II Experimental Setup

The experimental setup is based on a cylindrical HEMEX grade sapphire (undoped
monocrystalline $\alpha$-Al2O3) resonator supporting extremely high quality
factor whispering gallery modes (WGMs) in the microwave frequency range. The
crystalline lattice of sapphire is host to a number of naturally occurring
impurities including Fe3+, Cr3+, and Ti3+ with typical concentrations of
parts-per-billion in HEMEX-grade crystals [20]. Our crystal has been annealed
to increase the Fe3+ population by conversion of Fe2+ to Fe3+, with a 100-fold
improvement achieved in the Fe3+ to Fe2+ ratio [21]. Paramagnetic Fe3+ ions in
sapphire, each substituting an Al3+ cation, was proposed [22] and implemented
as a gain medium for cryogenic masers, which do not require any external
magnetic field [23, 21] . Profitably, at zero applied field, this ion
represents a three level system with the spin-$|1/2\rangle$ (ground state),
$|3/2\rangle$ (intermediate state), and $|5/2\rangle$ (higher excited) states.
These three levels are ideally suited to be used as a $\Lambda$-scheme:
pumping the $|1/2\rangle\rightarrow|5/2\rangle$ transition via a high-
frequency pump WGM within the ion’s electron spin resonance (ESR) linewidth,
inducing a fast non-radiative transition to the intermediate level with a
short life time followed by a radiative maser transition
$|3/2\rangle\rightarrow|1/2\rangle$, stimulating photon emission at the ESR
frequencies around $\sim 12.04$ GHz, which happens to encompass the linewidth
of two signal WGMs in the crystal (see Fig 1). These signal WGMs have
sufficient quality factors to self-sustain resonance for sufficient pump
power. Fig. 1 (inset) shows the quality factors of modes A and B, displaying a
very strong nonlinearity in the form of quality factor dependence on power
incident on the mode.

The sapphire WGM crystal was housed in a silver-plated copper cavity and
interrogated by two microwave probes inserted through the cavity walls. The
output probe is an antenna oriented to couple to two modes in the region of
$12.04$ GHz (specifically, we investigate the responses of mode A at 12.03812
GHz and mode B at 12.02979 GHz). The input probe is an orthogonally oriented
loop which, ideally, strongly couples to the azimuthal magnetic component of
one of the (dominantly transverse-magnetic) modes in the vicinity of the
$31.9$ GHz transition and does not load quality factors of the signal modes,
which are quasi-transverse-electric modes.

The cavity is attached to the millikelvin stage of a dilution refrigerator.
The pump signal is generated by a microwave synthesizer locked to a hydrogen
maser and delivered to the cavity via a series of copper and phosphor-bronze
cables. The signal is cleaned by a narrow band-pass filter tuned to near
$31.5$ GHz and directly bolted to the cavity. The total attenuation of the
cables and the filter from the output of the synthesizer to the input of the
cavity is $31$ dB at the pump frequency. The output signals from the crystal
pass through a millikelvin isolator and a 4 K low noise amplifier. At room
temperature the target signal is split between a spectrum analyzer, used to
collect frequency data around the target mode, and two downconversion
channels, used to mix down the signal for collection of time-series data. To
achieve downconversion, two synthesizers generate frequencies of $f_{A}$ and
$f_{B}$ to bring the signal carriers to the kHz level whereby they are
individually digitized with a two-channel high-speed real-time oscilloscope. A
schematic of the experimental setup used to observe maser solitons is shown in
Fig. 1.

##  III Observations

Figure 2: Top: Spectra of masing on mode A ($f_{s}$) for different pump
frequencies around $31.3377$ GHz ($\Delta f_{p}$). A/B: snapshots of masing on
mode A/B at various $\Delta f_{p}$, illustrating the three masing regimes.
Note that $f_{s}$ is arbitrarily chosen, serving only to indicate scale, as
the signal frequency has no unambiguously natural centre. Figure 3: Top:
Spectra of masing on mode A ($f_{s}$) for different pump powers at $31.3377$
GHz. A/B: snapshots of masing on mode A/B at various pump powers, illustrating
the three masing regimes. Note that $f_{s}$ is arbitrarily chosen, serving
only to indicate scale, as the signal frequency has no unambiguously natural
centre.

Masing was observed on two signal modes ($f_{A}=12.03813$ GHz and
$f_{B}=12.02979$ GHz) for a range of pump frequencies, $\Delta f_{p}$, swept
around the pump mode, $f_{p}=31.33771$ GHz. Full spectra of the masing signal
from $f_{A}$ is given in Fig 2, along with snapshots from both $f_{A}$ and
$f_{B}$.

Table 1: *

Target Whispering Gallery Modes  Pump Mode Frequency (GHz) Readout Mode
Frequency (GHz) 1 31.33771 A 12.03812 2 31.33974 B 12.02979

Figure 4: Time-series output of mode A (blue) and B (pink) of the crystal, in
regime I, being pumped at $31.337707136$ GHz, after being mixed down to
baseband, representing modulation of the carrier mode. The soliton pulse train
is the time-series equivalent of the comb structure. The fitted sech envelope
of one pulse is highlighted. Figure 5: Maser peak emission ($P_{peak}$) for
various pump frequencies about $31.3377$ GHz ($\Delta f_{p}$) for mode A, with
$P_{pump}=0$ dBm. $P_{peak}$ is defined as the amplitude of the centre of the
dominant peak in the maser spectrum. Three points are highlighted, one for
each regime. The inset panels show maser spectra and associated soliton
outputs in the various regimes; (III) single spectral line regime (blue), (II)
the dense soliton regime (red), and (I) the sparse soliton regime (purple).
Discrete jumps between the regimes can be observed on the spectrum of peak
maser output. Top-right inset illustrates the relationship between the comb
frequencies, and the intracavity soliton whose existence implies synchronised
phases between the comb frequencies.

###  III.1 Comb Structure

At millikelvin temperatures (within a dilution fridge cooled to $\sim$16 mK,
but transiently heated to $\sim$20-50 mK), a comb-like structure was obtained,
the first such structure observed being derived from a maser system to the
knowledge of the authors (Figs. 2 and 3, inset). Comb lines only persist down
to approximately -40 dBm incident pump power, below which the comb collapses.
Data at 4 K were also obtained, but results are deferred to the appendices. It
has to be noted that the 4K and mK regimes for the maser system are
sufficiently different. The occupation number of the excited state for the
system at 4K is ${n}=\frac{1}{e^{\frac{h\nu}{k_{B}T}}+1}\approx 0.41$ whereas
for 50mK it is $8.643\times 10^{-14}$.

Fig. 2 and 3 demonstrate three distinct generation regimes: (III) the
continuous wave regime at the centre (near zero $\Delta f_{p}$) with a narrow
masing peak, (II) the few peak regime for larger detuning, corresponding to
high density soliton emission, followed by (I), a sparse soliton regime just
surpassing the masing threshold on both sides of the masing regime, with few
solitons. The latter regime demonstrates a second broad feature that can be
attributed to the doublet phenomenon in WGM systems where the solution has two
degenerate components. Various symmetry breaking features in the sapphire
resonator lift this degeneracy and our readout mode thus exists as a double
peaked structure. It is worth noting that for the continuous wave regime with
$\Delta f_{p}\sim 0$, one peak supercedes the other within the doublet. The
dependence of the masing regime upon explicitly pump power, at a fixed pump
frequency, was also clearly observed (see Fig. 3), displaying analogous
regimes to the detuned frequency experiment. Due to the lineshape of the WGM
pump mode, pumping off resonance delivers less power to the pump transition;
hence, either parameter, pump power or pump frequency, can be used to control
power delivered to the pump transition. It should be noted that the
“bimodality” of the system refers, in fact, to the simultaneous masing on
modes A and B, which are separated by $\sim$ 8 MHz, and not to the doublet
phenomenon caused by lifted degeneracy.

###  III.2 Pulse Train

Examining the time-series data, this masing can be observed to be expelled in
discrete millisecond-scale solitons, the time-domain phenomenon associated
with the observed comb structure (see Fig. 4). The emission of discernible
pulses can be observed up to a saturation point, corresponding to single-
frequency maser emission (cyan in Fig. 3). The relationship between comb shape
and soliton emission period is illustrated in Fig. 5, which maps out the three
regimes, depeneant upon pump frequency/power.

##  IV Theoretical Description of the Maser Soliton System

In a host lattice of sapphire, Fe3+ ions effectively act as a three level
system with $|1/2\rangle$ being the ground state, $|5/2\rangle$ the excited
state and $|3/2\rangle$ the intermediate state. A diagram of this ion is shown
in Fig. 6. The maser works by pumping ions from the ground state to the exited
state from where they then decay into the intermediate state. The stimulated
emission between $|3/2\rangle$ and $|1/2\rangle$ leads to masing.

The full description of the system discussed in this work includes a cavity
pump mode (whispering gallery modes (WGM) at 31.33771 GHz or 31.33974 GHz),
two signal modes (WGMs at 12.03812 GHz and 12.02979 GHz), an ensemble of three
level systems, and associated dissipation baths. Generation of solitons in
such $\Lambda$ systems have been considered before in the context of
electromagnetically induced transparency (EIT) [24]. In this work, the system
is different to EIT as it represents a $\Lambda$ scheme maser.

###  IV.1 Simplified System Hamiltonian

For the Fe3+ ions in sapphire, the $|5/2\rangle$ state rapidly decays into the
$|3/2\rangle$ state and the lifetime of the latter is considerably longer. So,
observationally, the $N_{|5/2\rangle}$ population is near zero with total
population $N\approx N_{|1/2\rangle}+N_{|3/2\rangle}$ at any given time.
Furthermore, the dynamics of the pump mode can be adiabatically eliminated as
it is dominated by the spin ensemble. Both signal modes are very high $Q$
relative to losses in the Fe3+ ensemble, therefore the latter is the dominant
source of losses. Additionally, we note that the signal modes are sufficiently
far away from each other in frequency space, so that their direct coupling can
be neglected. These facts lead to the significantly simplified description of
the system as a dual mode laser system with an externally pumped TLS, with the
crystal and the ensemble providing dispersion. The corresponding Hamiltonian
(in units $\hbar=1$) is

$\begin{split}H=\omega_{a}\sum_{n}\sigma^{+}_{n}\sigma^{-}_{n}+\sum_{m=1}^{2}&\omega_{m}a_{m}^{\dagger}a_{m}\\\
+&\sum_{n,m}g_{m}\left(\sigma^{-}_{n}a^{\dagger}_{m}+\sigma^{+}_{n}a_{m}\right)\end{split}$
(1)

where $m$ is a summation index over the photonic signal modes with resonance
frequencies $\omega_{m}$ and creation (annihilation) operators
$a^{\dagger}_{m}$ ($a_{m}$); $n$ is an index over the spin ensemble;
$\sigma^{+}_{n}$, $\sigma^{-}_{n}$ and $\sigma^{z}_{n}$ are the usual Pauli
matrices and $g_{m}$ are the spin-photon coupling constants. The system is
completed by introducing photonic mode loss rates $\gamma_{m}$ and TLS
emission, decoherence and pump rates: $\gamma_{E}$, $\gamma_{D}$ and
$\gamma_{P}$ respectively.

Figure 6: Schematic of the three-level spin system of the Fe3+ ion (left)
which leads to masing. Right shows a simplification of the system to a two
level system which is justified by theoretical and experimental observations.

###  IV.2 Relationship to the Lugiato-Lefever Equation

In the limit of a large TLS ensemble, the corresponding Maxwell-Bloch
equations (see Chap. 11 in [25]) could be written as follows:

$\displaystyle\begin{array}[]{c}\displaystyle\frac{\partial F_{m}}{\partial
t}+\widetilde{c}\frac{\partial F_{m}}{\partial
z}=\left(i\omega_{m}-\gamma_{m}/2\right)F_{m}-ig_{m}J\\\
\displaystyle\frac{\partial{J}}{\partial
t}=\left(i\omega_{a}-\gamma_{a}/2\right)J+i\sum_{m=1}^{2}g_{m}F_{m}D,\\\
\displaystyle\frac{\partial{D}}{\partial
t}=-\gamma_{I}\left(D-D_{0}\right)+2i\sum_{m=1}^{2}g_{m}\left(F^{*}_{m}J-J^{\ast}F_{m}\right)\end{array}$
(2)

where $F$, $J$ and $D$ are proportional to the electric field $E_{m}$, the
collective atomic polarization and the population difference respectively;
$\gamma_{I}=\gamma_{P}+\gamma_{D}$;
$\gamma_{a}=\gamma_{P}+\gamma_{D}+\gamma_{E}$;
$D_{0}=\frac{\gamma_{P}-\gamma_{D}}{\gamma_{P}+\gamma_{D}}$ is the pump
parameter and $\widetilde{c}$ is the effective light velocity, and
$\gamma_{m}$ is the photonic loss of the $m$th mode.

Derivation of the Lugiato–Lefever equation from the Maxwell–Bloch equations
(2) for a case of a single photonic mode is given in [26] and discussed in
other sources, e.g [27]. In this work, we show the relationship between the
maser soliton system and the LLE as based on derivation in [26]. For
simplicity we consider only one photonic mode, so that the Maxwell-Bloch
equations in the frame associated with angular frequency $\omega_{0}$ can, by
substituting $J\rightarrow iJ$, be written as:

$\displaystyle\begin{array}[]{c}\displaystyle\frac{\partial F}{\partial
t}+\widetilde{c}\frac{\partial F}{\partial
z}=-\frac{\gamma}{2}\Big{[}\left(1+i\frac{2\Omega}{\gamma}\right)F-\frac{2g}{\gamma}J\Big{]}\\\
\displaystyle\frac{\partial{J}}{\partial
t}=-\frac{\gamma_{a}}{2}\Big{[}\left(1+i\frac{2\Omega_{a}}{\gamma_{a}}\right)J-\frac{2g}{\gamma_{a}}FD\Big{]},\\\
\displaystyle\frac{\partial{D}}{\partial
t}=-\gamma_{I}\Big{[}D-D_{0}+\frac{2g}{\gamma_{I}}\left(F^{*}J+J^{\ast}F\right)\Big{]}\end{array}$
(3)

where $\Omega=\omega_{0}-\omega$ and $\Omega_{a}=\omega_{0}-\omega_{a}$ are
mode and atom detunings. By substituting $D\rightarrow D\chi$, $J\rightarrow
J\xi$, $F\rightarrow F\zeta$:

$\displaystyle\begin{array}[]{c}\displaystyle\frac{\partial F}{\partial
t}+\widetilde{c}\frac{\partial F}{\partial
z}=-\frac{\gamma}{2}\Big{[}\left(1+i\frac{2\Omega}{\gamma}\right)F-\frac{2g\xi}{\gamma\zeta}J\Big{]}\\\
\displaystyle\frac{\partial{J}}{\partial
t}=-\frac{\gamma_{a}}{2}\Big{[}\left(1+i\frac{2\Omega_{a}}{\gamma_{a}}\right)J-\frac{2g\chi\zeta}{\gamma_{a}\xi}FD\Big{]},\\\
\displaystyle\frac{\partial{D}}{\partial
t}=-\gamma_{I}\Big{[}D-D_{0}/\chi+\frac{2g\xi\zeta}{\gamma_{I}\chi}\left(F^{*}J+J^{\ast}F\right)\Big{]}\end{array}$
(4)

So, to match the notation in [26], we have to choose
$\frac{2g\xi}{\gamma\zeta}=-2C$, $\frac{2g\chi\zeta}{\gamma_{a}\xi}=1$ and
$\frac{2g\xi\zeta}{\gamma_{I}\chi}=\frac{1}{2}$ giving us
$\zeta=-\frac{1}{2g}\sqrt{\frac{\gamma_{I}\gamma_{a}}{2}}$,
$\xi=\frac{C\gamma}{2g^{2}}\sqrt{\frac{\gamma_{I}\gamma_{a}}{2}}$ and
$\chi=-\frac{C\gamma\gamma_{a}}{2g^{2}}$, giving the following system:

$\displaystyle\begin{array}[]{c}\displaystyle\frac{\partial F}{\partial
t}+\widetilde{c}\frac{\partial F}{\partial
z}=-\frac{\gamma}{2}\Big{[}\left(1+i\theta\right)F+2CJ\Big{]}\\\
\displaystyle\frac{\partial{J}}{\partial
t}=-\frac{\gamma_{a}}{2}\Big{[}\left(1+i\Delta\right)J-FD\Big{]},\\\
\displaystyle\frac{\partial{D}}{\partial
t}=-\gamma_{I}\Big{[}D-D_{0}/\chi+\frac{1}{2}\left(F^{*}J+J^{\ast}F\right)\Big{]}\end{array}$
(5)

where $\theta=\frac{2\Omega}{\gamma}$ and
$\Delta=\frac{2\Omega_{a}}{\gamma_{a}}$ which is exactly the starting system
for derivation of the LLE in [26]. The only additional parameter is the scaled
population inversion parameter $D_{0}/\chi$ which is controlled in our
experiment via the high frequency pump. For the case $D_{0}=\chi$ and a
circular cavity of radius $R$, the corresponding LLE is written as:

$\begin{split}\frac{\partial}{\partial\bar{t}}\tilde{F}(\bar{t},\varphi)=-\tilde{F}(\bar{t},&\varphi)-i\theta_{0}\tilde{F}(\bar{t},\varphi)+i\eta\frac{\beta}{2}\frac{\partial^{2}}{\partial\varphi^{2}}\tilde{F}(\bar{t},\varphi)\\\
&+i\eta|\tilde{F}(\bar{t},\varphi)|^{2}\tilde{F}(\bar{t},\varphi)\end{split}$
(6)

where $\varphi$ is an angular coordinate, $\eta=-|\Delta|/\Delta$,
$\widetilde{F}=\sqrt{\frac{2C}{|\Delta|^{3}}}F$,
$\overline{t}=\frac{\gamma}{2}t$, $\theta_{0}=\theta-\frac{2C}{\Delta}$ and
the dispersion coefficient is

$\beta=\frac{2C\gamma_{a}\widetilde{c}^{2}}{|\Omega_{a}|^{3}R^{2}}$ (7)

Although we have demonstrated that the present system is related to the LLE
through its connection to the Maxwell-Bloch equations, its dynamics are far
more complicated. This additional dynamics makes the system deviate from
expected behaviour of typical LLE systems.

###  IV.3 Distinguishing Features of the Maser Soliton System

While in optics, a system may be represented by a pure 1D circular cavity, our
sapphire resonator is a large 3D cavity with a more complicated wave structure
involving some dynamics of two additional degrees of freedom. Its large
spatial footprint leads to significant spin ensemble broadening. A more
complicated level structure of the $\textrm{Fe}^{3+}$ ensemble induces
additional decoherence effects through interaction with higher energy levels.
The same is true for the high frequency pump mode exhibiting additional
dynamics potentially involving decoherence effects; in optical counterparts
involving only the Kerr nonlinearity this additional decoherence is avoided
all together. Furthermore, apart from the target $\textrm{Fe}^{3+}$ ensemble,
the crystal hosts additional spin ensembles (e.g. chromium) involving spin-
spin interaction as well as interactions with nuclei spin which is the main
source of local decoherence in our target spin system. Clearly, the system is
highly susceptibility to external magnetic fluctuations due to the fact that
it is based on a spin ensemble. Finally, there are two cavity modes involved
which are both coupled to the same spin ensemble and thus can interact.
Overall, all these additional effects give observed deviations from the
picture expected from the LLE equation.

##  V Conclusion

These observations provide a window into the spontaneous self-organization
behaviours within this nonlinear many-body spin system, worthy of further
study. The system may be related to the Lugiato-Lefever equation (a damped,
driven, non-linear wave equation) for which temporal solitons on top of a
continuous wave pump are a solution and which likewise describes the dynamics
of Kerr frequency combs [27]. Due to various inhomogeneities of the spin
ensemble, and imperfections in the crystal itself, inhomogeneity and non-
periodicity can be observed in the soliton trains. These temporal results are
more thoroughly presented in the appendices. The production of this maser-
based comb structure, and associated soliton pulse train, in a simple sapphire
system has great potential for further investigation, both theoretically, into
the behaviour of the dissipative masing system, and into the potential
applications of this comb, for example in precision spectroscopy [6], tunable
narrowband microwave signal generation [28] or reconfigurable RF filtering
[29]. Detailed characterization of the soliton evolution and comb structure
will be undertaken in the future.

## Acknowledgements

This research was supported by the ARC Centre of Excellence for Engineered
Quantum Systems (EQUS, CE170100009) and the ARC Centre of Excellence for Dark
Matter Particle Physics (CDM, CE200100008).

##  Appendix A Results at 4 K

Fig. 7 presents the power-dependence of masing, for modes A and B. Unlike the
masing results at mK, shown in the main paper, there is no two peak structure
present, and although there are modulation sidebands, a fine frequency comb is
not produced as in the mK case. Fig 8 presents the response of masing on modes
A and B when the pump frequency is swept. As in the mK case, the doublet
structure of the pump mode is evident.

(a)  (b)

Figure 7: Pumping on the first pump mode at an array of pump powers for modes
A (a) and B (b). Measurements taken at 4 K. Power at signal output is gauged
by the color bar and the masing peak can be tracked relative to the centre of
the measurement spectrum ($\Delta f_{s}$). The bottom plots isolate the
frequency shift ($\Delta f_{s}$, red) and maximum power ($P_{s}$, purple) of
the maser peaks, depending on incident pump power.

(a)

(b)  (c)

(d)

(e)  (f)

Figure 8: Pumping on the first pump mode at an array of pump frequencies about
$f_{p}$ for modes A (a,b,c) and B (d,e,f) with pump power set to -33 dBm, -20
dBm and -13 dBm respectively. Measurements taken at 4 K. Power at signal
output is gauged by the color bar and the masing peak can be tracked relative
to the centre of the measurement spectrum ($\Delta f_{s}$). The bottom plots
isolate the frequency shift ($\Delta f_{s}$, red) and maximum power ($P_{s}$,
purple) of the maser peaks, depending on incident pump frequency.

##  Appendix B Further Results at mK

Fig. 9 and 10 present masing observed at mK, that is, between 16 mK and 50 mK
after transient heating, over a range of pump frequencies and at different
incident powers. Fig. 9 presents results when pump mode 1 was pumped and Fig.
10 presents results when pump mode 2 was pumped. Fig. 11 and Fig. 12 display
the time series soliton output.

(a)  (b)

(c)

(d)

(e)  (f)

Figure 9: Pumping on the first pump mode at an array of pump frequencies about
$f_{p}$ for modes A (a,b) and B (d,e,f) with pump power set to -31 dBm, -41
dBm and -37 dBm (mode B only) respectively. Measurements taken at mK. Power at
signal output is gauged by the color bar and the masing peak can be tracked
relative to the centre of the measurement spectrum ($\Delta f_{s}$). The
bottom plots isolate the frequency shift ($\Delta f_{s}$, red) and maximum
power ($P_{s}$, purple) of the maser peaks, depending on input pump frequency.
The dependence of mode B upon pump power is represented in (c), but neglected
for mode A, as this is shown in the main paper.

(a)  (b)

(c)

(d)

(e)  (f)

Figure 10: Pumping on the second pump mode at an array of pump frequencies
about $f_{p}$ for modes A (a,b) and B (d,e) with pump power set to -31 dBm and
-34.5 dBm respectively. Measurements taken at mK. Power at signal output is
gauged by the color bar and the masing peak can be tracked relative to the
centre of the measurement spectrum ($\Delta f_{s}$). The bottom plots isolate
the frequency shift ($\Delta f_{s}$, red) and maximum power ($P_{s}$, purple)
of the maser peaks, depending on input pump frequency. The dependence of modes
A and B upon pump power is also represented ((c) and (f)).  (a)

(b)  (c)

Figure 11: At incident powers of -41 dBm, -36 dBm and -31 dBm ((a),(b), and
(c)), the pump frequency was swept around pump mode 1 ($\Delta f_{p}$), and
solitons were observed at the mixed down mode outputs A and B. The top plots
record the standard deviation (red) and mean output voltage (black) of the
soliton train (or continuous wave) emanating from the crystal at the signal
modes. Depending on the input pump frequency, solitons may be emitted
discretely in pulses, or the output may evolve into a continuous wave. The
bottom plot (right) presents snapshots of the soliton output produced at -31
dBm incident power (corresponding to (c), output A), for various $\Delta
f_{p}$. In the color density plot (left), traces are aligned to the first
pulse to illustrate intersoliton-period dependence upon pump frequency.

(a)  (b)

(c)  (d)

Figure 12: At incident powers of -34.5 dBm ((a) and (b)) and -32 dBm ((c) and
(d)), the pump frequency was swept around pump mode 2, and solitons were
observed at the mixed down mode outputs A and B. The left plots record the
standard deviation (red) and mean output voltage (black) of the soliton train
(or continuous wave) emanating from the crystal at the signal modes. Plots (b)
and (d) present snapshots of the soliton outputs produced at -34.5 dBm and -32
dBm incident power, from mode A and B respectively, for various $\Delta
f_{p}$. In the latter, we can observe the pump twice breaching the power level
at which discrete soliton emission can be sustained, caused by the doublet
shape of the pump mode. Highlighted in (a), the mean output voltage shows
clear jumps at certain $\Delta f_{p}$ steps, which suggests the elimination of
one intracavity soliton at each step. The yellow region (corresponding to the
yellow time-trace) is therefore representative of the single soliton state.

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# Data-oriented Coordinated Uplink Transmission for Massive IoT System

Jyri Hämäläinen, Rui Dinis, , Mehmet C. Ilter

###### Abstract

Recently, the paradigm of massive ultra-reliable low-latency IoT
communications (URLLC-IoT) has gained growing interest. Reliable delay-
critical uplink transmission in IoT is a challenging task since low-complex
devices typically do not support multiple antennas or demanding signal
processing tasks. However, in many IoT services the data volumes are small and
deployments may include massive number of devices. We consider on a clustered
uplink transmission with two cooperation approaches: First, we focus on
scenario where location-based channel knowledge map (CKM) is applied to enable
cooperation. Second, we consider a scenario where scarce channel side-
information is applied in transmission. In both scenarios we also model and
analyse the impact of erroneous information. In the performance evaluation we
apply the recently introduced data-oriented approach that has gathered
significant attention in the context of short-packet transmissions.
Specifically, it introduces a transient performance metric for small data
transmissions, where the amount of data and available bandwidth play crucial
roles. Results show that cooperation between clustered IoT devices may provide
notable benefits in terms of increased range. It is noticed that the
performance is heavily depending on the strength of the static channel
component in the CKM based cooperation. The channel side-information based
cooperation is robust against changes in the radio environment but sensitive
to possible errors in the channel side-information. Even with large IoT device
clusters, side-information errors may set a limit for the use of services
assuming high-reliability and low-latency. Analytic results are verified
against simulations, showing only minor differences at low probability levels.

###### Index Terms:

Data-oriented approach, coordinated transmission, channel knowledge map,
channel side-information, erroneous side-information, 6G, Rice fading.

##  I Introduction

The Internet of Things (IoT) is an umbrella term for concepts connecting
various physical objects and devices to the internet and allowing them to
collect and exchange data. While IoT is an critically important field for
future research, several comprehensive surveys with different focus areas and
aspects of IoT have been published including 5G IoT [1] and 6G IoT [2], as
well as many special topics such as ultra low-power communications [3],
machine-type communications [4], IoT system autonomy [5], to name few.

As noticed in the cellular communication research community, the IoT equipment
can be divided into ’high-end’ and ’low-end’ devices with different kind of
quality of service (QoS) requirements. The wireless collection of
environmental data, for example, is typically carried out by sensors that on
one hand admit low transmission power on narrow bandwidth but, on other hand,
the collected data is not aging in milliseconds and communication is delay
tolerant. In contrary, if, e.g., sensor data from a large machinery is used
for wireless control purposes, then demanding QoS requirements in terms of
reception reliability and low communication latency are faced. While delay
tolerant IoT communications has been extensively studied, the paradigm of
massive ultra-reliable low-latency IoT communications (URLLC-IoT) has also
gained growing interest recently. In 5G domain the discussion on URLLC already
covered some IoT use cases [6] but topic is more integral part of the 6G
development [2],[24].

Executing a reliable delay-critical transmission in IoT is a challenging task.
The low-complex devices are typically not supporting multiple antennas and
demanding signal processing tasks. Yet, there are two aspects that can help
supporting URLLC in the IoT domain. First, in many IoT services the data
volumes are small. Second, IoT deployments may include massive number of
devices. Due to small data volumes the maximisation of link capacity is not as
important as in e.g. massive MIMO applications but emphasis is merely in the
link reliability. Furthermore, the large number of IoT devices potentially
support cooperative transmission approaches even though it must be taken into
account that cooperation cannot assume very sophisticated methods or radio
resource intensive information exchange between receiver and transmitting
devices. Accordingly, we focus on an approach where a reliable low-latency
uplink transmission is executed by a group of clustered and cooperating IoT
devices. We expect that successful cooperative transmission requires only
scarce location or channel side-information. Moreover, an important aspect of
our study is that we assume incomplete and erroneous side-information in the
transmitting cluster.

####  I-1 Channel knowledge map based approach

While joint transmission is typically based on either direction information or
channel feedback from the receiver, recently a new concept of CKM has been
introduced [7]. The CKM represents a development of the radio and/or built
environment aware communications, see e.g. [9, 8], providing from a database
certain channel-related information for transmitter-receiver pairs. It is
envisioned that CKM-enabled communications could even replace the need for
real-time channel state information (CSI) when e.g. channel path information
for a specific transmitter-receiver pair is known, including the number of
significant channel paths and their powers, phases and delays [10]. From CKM
perspective a favourable scenario takes place when there is a strong LoS
between transmitter and receiver. In such a situation the path loss between
transmitter and receiver can be easily computed if locations are known.
However, since IoT devices are typically power limited, the path loss
information as such may provide only understanding on the power gap that
transmitter needs to overcome for a successful reception. To tackle this kind
of challenge cooperative transmission methods can be used especially if IoT
devices occur in clusters.

####  I-2 Feedback based cooperation

Since CKM based approach has its limitations, we will also consider a more
conventional feedback based cooperation in cluster of IoT devices. Even though
multiple-input single-output (MISO) with co-located antennas is a well studied
topic [11], [12], the use of explicitly defined simple methods with quantised
channel information has not been extensively studied. Such methods are
typically suboptimal but they provide a good insight on the impact of feedback
quantisation and erroneous channel information [13, 14, 15, 16]. Low
resolution feedback is an especially attractive choice for IoT communications
because therein a limited reverse channel capacity may set strict constraints
for feedback data volume. When compared to CKM-based approach the use of
explicit feedback from receiver has a clear advantage if LoS is weak and
channel contains a fast fading component. On the other hand, the drawback is
that obtaining channel information in the receiver may require training or
sounding signals from IoT devices consuming both radio and power resources of
IoT devices. Finally, feedback information is vulnerable to errors in the
reverse channel and, in addition to power consumption, the channel sounding
signal from an individual IoT device might be too weak for a reliable
reception in the receiver.

####  I-3 Delay Outage Rate

To assess the proposed system feasible key performance indicators (KPIs) are
needed. In this respect, a new concept called as data-oriented approach was
introduced for the design and analysis of advanced wireless systems [17], [23]
where data transmission sessions complete within one channel coherence time
over short packet transmission sessions. Basically, this concept is simply
representing a new way to look at the link performance from the transmission
delay outage point of view and authors in [17] named it as the Delay Outage
Rate (DOR) which refers the probability that the required information delivery
time for a specific transmission exceeds the predefined threshold. The DOR
metric for the short-packet transmission over fading channels was presented in
[18] and inspired by it, [19] showed how the DOR expression can be utilized
for searching optimum constellation points over a coded scenario. Furthermore,
the DOR metric has been recently utilized in the coverage analysis of
reflective intelligent surfaces (RIS)-aided communication systems [20] and in
visible light communication scenarios [21]. In [22], the data-oriented
approach was adopted into downlink RSMA systems where two different precoder
designs were investigated based on the proposed DOR metric.

The mentioned DOR studies provide a new perspective on the wireless link
performance and it can be used to evaluate simultaneously reliability and
latency. In simplest setting the DOR study can be carried out
straightforwardly by using the cumulative distribution function (CDF) of the
signal-to-noise-ratio (SNR). In all mentioned studies, obtaining the DOR
expression basically falls back to finding the CDF of received SNR. While, in
this respect, it can be considered as a simple extension of the outage
probability analysis, there is a notable technical difference. Namely, study
of high reliability and low latency require accurate knowledge of the tail of
the CDF of SNR. If the exact distribution is known, then this is not a problem
but in case when the CDF is obtained as an approximation it is typical that
approximation works well in 0.1 - 0.9 probability levels but when looking at
the tail’s fit in the logarithmic scale notable errors may occur as
illustrated in Fig.1. Therefore, obtaining DOR requires an accurate
approximation of the tail of the CDF of SNR.

Figure 1: The illustration of DOR and outage probability.

####  I-4 Contributions

We focus on a URLLC-IoT system, where a cluster of IoT devices collaborates to
perform reliable low-latency uplink transmission towards an Information
Collecting Unit (ICU), which can be either a fixed node or, for example, a
drone. The primary emphasis is on scenarios involving Rician fading, where a
LoS link is established between the transmitting devices and the receiver in
the ICU. We consider two scenarios: In the first scenario, information from a
CKM is used to roughly adjust the phases of the transmitting IoT devices. In
the second scenario, IoT devices employ quantized channel side-information
provided by the ICU receiver. The main contributions are:

  * •

We propose a cooperative transmission model designed to address scenarios with
either limited receiver locations or inaccurate channel side-information in
the transmitting devices.

  * •

The calculations of the DOR and outage probability are derived under the
assumption of Rician fading, enabling a comprehensive performance evaluation
ranging from static channels with a clear Line of Sight (LoS) to Rayleigh
fading cases where a dominant channel direction is absent.

  * •

The study explicitly demonstrates the impact of reduced side-information
quality in the transmission cluster on the cooperative IoT transmission
performance. Our findings suggest that performance degradation due to
erroneous side-information may be tolerable in terms of outage probability but
can be critical for DOR performance.

  * •

We develop analytical expressions that establish a clear relationship between
the DOR, the required number of cooperative IoT devices for successful
transmission, and the influence of side-information errors.

The rest of the paper is organized as follows: The IoT system model and its
protocol are described in Section II, including two different scenarios which
are CKM-based coordination and feedback-based coordination, respectively.
Then, the DOR expression and outage probability have been derived in Section
III. The numerical results are presented and analyzed in Section IV. Finally,
Section V concludes the paper.

##  II IoT System model and protocol

We examine a system where IoT devices are organized into a cluster,
communicating with a transceiver unit that gathers information from the
cluster, as depicted in Fig.2. This transceiver, referred to as the
Information Collecting Unit (ICU), can be either a drone or a stationary radio
access point with the capability to utilize, store, or forward the collected
information. The primary focus of the study is on the synchronized data
communication from the IoT device cluster to the ICU. The coordination of IoT
devices within the cluster is managed by a cluster head (CH), highlighted in
red in Fig. 2. The CH engages in control information exchange with the ICU,
while the IoT devices handle data transmission. The transmitted data packet
remains consistent across all IoT devices.

Figure 2: Illustration of IoT system: IoT devices in a cluster communicate
with the ICU, coordinated by a CH.

The communication procedure is depicted in Fig. 3. In this illustration, we
consider a scenario where the CH predominantly handles or entirely manages the
control messaging with the ICU. Such messaging encompasses various aspects,
including admission control, transmission timing information, and other
relevant details. These messages serve the purpose of preparing for a
communication session or facilitating maintenance for a more prolonged
communication between the IoT device cluster and the ICU. During the
coordination phase, the CH locally readies specific IoT devices for
transmission, identifying those that will participate in the coordinated
transmission. The selection criteria for IoT devices may be based on factors
such as available power resources in different devices.

After completing the initial steps, we consider two scenarios for the
coordinated transmission:

Scenario 1: CKM-based coordination. In this scenario we assume that CKM is
available, with mapping $(Z_{\text{ICU}},Z_{\text{IoTD}})\rightarrow\theta$,
where $Z_{\text{ICU}}$ and $Z_{\text{IoTD}}$ include the ICU and IoT device
coordinates, respectively, and $\theta$ is a phase to be used in the
transmission. That is, IoT device apply the weight $e^{j\theta}$ in the
transmission to ensure in ICU receiver a coherent sum of signals from
different IoT devices. The CKM can be available either in the CH or in IoT
devices. In the first option CH carries the burden of storing and maintaining
the CKM but, on the other hand, it only needs to inform IoT devices on the
applied transmission phases. In the second option IoT device needs to store
the CKM of its own but it can choose the phase based on the ICU location. In
either of the cases there is no need for channel sounding or CSI feedback from
ICU, so at least two stages of the protocol in Fig.3 become redundant. Those
stages are denoted by dotted arrows.

Figure 3: Illustration of the communication protocol depicting the
coordination phase, where the CH prepares selected IoT devices for
transmission and engages in control messaging with the ICU.

Scenario 2: Feedback-based coordination. The CKM-based coordination requires
only the location of ICU due to the availability of CKM either in IoT devices
(individual CKM) or in the cluster head (joint CKM). As noted, location based
phasing may work only in case of strong LoS. In case of weak LoS coherent
signal combining over IoT devices may take place only if there is timely
phasing information available in the transmitting devices. Such information
can be based on the quantized phase feedback from the ICU. Accordingly, we
assume in this scenario that the ICU sends a quantized phase information per
IoT device. This phase information is either sent directly to each IoT device
or to the CH which shares the information with IoT devices and organizes
locally the cooperative transmission later on. In both options the phase
information should contain only very few bits since the feedback channel
capacity is limited and amount of bits to fed back is increasing with number
of IoT devices.

##  III Signal, channel and side-information models

This section introduces models for signals, channels, and side-information
which will be used in the analysis later on.

###  III-A Signal and channel models

Consider a signal model $r=\langle{\bf h},{\bf w}\rangle s+n$, where ${\bf
h}=(h_{k})_{k=1}^{K}\in\mathbb{C}^{K}$ contains complex channel coefficients.
Each IoT device admits a fixed transmission power $P$ and ${\bf
w}=(\sqrt{P}\delta_{k}e^{j\phi_{k}})_{k=1}^{K}$, where $\delta_{k}=1$ when
$k^{th}$ device is active and $\delta_{k}=0$ otherwise. Thus, the total
transmission power $\sum_{k=1}^{K}P\delta_{k}$ applied in the cluster of
devices increases with the number of transmitting devices unless we set
$P=P_{0}/|\delta|$. By $\phi_{k}$ we denote the phase applied in the $k^{th}$
device based on the side-information. Finally, $s$ refers to the transmitted
symbol and $n$ represents a complex AWGN term.

The following assumptions regarding the channels are considered:

  * •

The average propagation loss (including distance dependent loss, shadowing and
antenna gains) is the same for all IoT devices belonged by the same cluster
such that $\mathop{\mathbb{E}}\left[|h_{k}|^{2}\right]=\bar{\gamma}$ where
$\mathop{\mathbb{E}}\left[\cdot\right]$ is the expectation operator.

  * •

Rician fading is assumed so the channel coefficients are of the form
$h_{k}=g_{k}+\sqrt{\gamma_{d}}e^{j\varphi_{k}}$, where $g_{k}$ represents the
scattered and reflected signal part that is a complex zero-mean Gaussian with
mean power $\mathop{\mathbb{E}}\left[|g_{k}|^{2}\right]=\bar{\gamma}_{s}$.
Furthermore, $\gamma_{d}\geq 0$ is the power of the static signal part and the
amplitudes $|h_{k}|=\sqrt{\gamma_{k}}$ are independent and identically
distributed (i.i.d.) Rician random variables with the same Rice factor
$\nu=\gamma_{d}/\bar{\gamma}_{s}$.

  * •

The phase $\varphi_{k}$ of the static signal part is a sample from a uniform
distribution. Yet, we assume that phase $\varphi_{k}$ is fixed or phase drift
is so small that it can be neglected.

Based on these assumptions, the IoT devices are organized in clusters, where
the mean path loss is uniform for all devices in relation to the ICU. This
assumption is particularly valid under LoS conditions, as the distance between
the ICU and the IoT device cluster is typically much greater than the size of
the cluster. Consequently, the path loss and ICU antenna gain remain nearly
constant for all devices within the cluster. The choice of Rice fading is
deliberate, as it offers versatility in modeling the fading channel. In
environments characterized by rich scattering around the ICU or device
cluster, the Rice factor may be small. Conversely, in the presence of a strong
LoS component, the Rice factor is large, resulting in an almost static
channel.

###  III-B Side-information models

The side-information is related to the phases $\\{\phi_{k}\\}_{k=1}^{K}$. We
consider two models for the use of side-information for the CKM-based
coordination and the feedback-based coordination, respectively:

CKM-based coordination. In this scenario IoT devices in the cluster have
predefined, location dependent, phase values that they apply once the position
of the ICU is known. That is, in each device there is a scarce CKM that
defines a mapping between the ICU location and a phase weight. Another
implementation option is to store a joint CKM in the cluster head. After
receiving/estimating the ICU location the cluster head informs each device
regarding the applied phase weight. We assume that CKM is computed or
otherwise defined based on pure LoS condition. That is, if the static signal
part has the phase $\varphi_{k}$, then the applied phasing is
$\phi_{k}=-\varphi_{k}+\epsilon_{k}$, where $\epsilon_{k}$ represent the error
emerging from the estimation and possible quantisation of the CKM information.
Now we have

$\begin{split}h_{k}w_{k}&=\sqrt{P}(g_{k}+\sqrt{\gamma_{d}}e^{j\varphi_{k}})e^{-j\phi_{k}}\\\
&=\sqrt{P}(g_{k}+\sqrt{\gamma_{d}}e^{j\varphi_{k}})e^{-j(\varphi_{k}-\epsilon_{k})}\\\
&=\sqrt{P}(\tilde{g}_{k}+\sqrt{\gamma_{d}}e^{j\epsilon_{k}}).\end{split}$ (1)

We note that phasing does not impact the properties of the complex Gaussian
part of the signal. That is, $\tilde{g}_{k}$ admit the same statistical
properties as $g_{k}$. Furthermore, if the Rice factor is small, then it
becomes apparent that the usefulness of CKM information will be small as well.
Since the phase error may include contribution from many sources of
unidealites, we assume $\epsilon_{k}\sim{\cal N}(0,\sigma_{\epsilon}^{2})$.
That is, error follows the zero-mean Gaussian distribution with variance
$\sigma_{\epsilon}^{2}$.

Feedback-based coordination. Here we assume that $N$-bit phase information is
available in IoT devices. If this phase information is send directly from ICU
to each device, then dedicated feedback channels are needed. Another option is
that ICU send all phase information ($|\delta|\cdot N$ bits) to the CH that
then forwards the $N$ bits of information to each IoT device. We apply a
uniform phase quantisation:
$\phi_{k}\in\\{n\pi/2^{N-1}:\,n=1,2,\dots,2^{N}\\}$. If the phase information
is error-free, then the received signal is given by

$\langle{\bf h},{\bf
w}\rangle=\sqrt{P}\sum_{k=1}^{K}\delta_{k}\sqrt{\gamma_{k}}e^{j\theta_{k}},\quad\theta_{k}\sim{\cal
U}\Big{(}-\frac{\pi}{2^{N}},\frac{\pi}{2^{N}}\Big{)},$ (2)

where ${\cal U}(-a,a)$ refers to the uniform distribution over the interval
$(-a,a)$. However, side-information errors may occur and, in that case, the
whole word of $N$ bits including the side-information to a certain device,
becomes random. The side-information word error probability is denoted by
$p_{w}$.

##  IV Analytic Performance Evaluation

###  IV-A Outage probability and DOR

The outage probability can be defined as a probability that link rate falls
below a certain predefined minimum level. Assume a transmission bandwidth $W$
and a single data stream transmission. Then we may use the classical Shannon
formula $R=W\cdot\log_{2}(1+\gamma)$ for an instantaneous rate $R$ and

$\begin{split}P_{\text{out}}&=P(R<R_{\min})\\\
&=P(\gamma<2^{R/R_{\min}}-1)=F_{\gamma}(2^{R/R_{\min}}-1),\end{split}$ (3)

where $F_{\gamma}(\cdot)$ is the CDF of SNR. Analyzing outage probability is
crucial for evaluating the coverage range of a wireless service with a
specified rate requirement $R_{\min}$. The computation of outage relies on the
distribution of SNR. While closed-form expressions for SNR distribution exist
in classical scenarios with single transmitting devices/antennas,
incorporating practical aspects such as errors, quantized side-information,
and multiple devices poses challenges, often requiring approximations due to
the complexity of the problem.

Conventionally, the outage probability in (3) is considered in the 5% - 95%
range. For classical coverage studies 5% outage probability is typical while
in case of delay tolerant IoT services even 95% packet loss maybe tolerable if
numerous re-transmissions can be used, as in narrow band (NB) IoT system [26,
27]. If the communication latency is limited, then it is favorable to avoid
re-transmissions by using coordinated transmission and apply DOR to measure
the link latency. While the CDF of SNR can be used in the computation of DOR
as in case of classical coverage studies, there is a an important practical
difference. Namely, the left tail of the SNR distribution is in a key role and
if approximation is used for the CDF, then the approximation error easily
becomes the limiting factor in the DOR assessment.

The DOR is defined as the probability that the required information delivery
time for a specific transmission session exceeds the predefined threshold
[23]. Assume an ideal continuous-rate adaptive transmission, where transmitter
always transmits at the maximum transmission rate that the channel can
reliably support. In such optimal rate adaptation transmission the
instantaneous rate is given by $R=W\cdot\log_{2}(1+\gamma)$. Furthermore, if
$D$ bits are transmitted, then the data delivery time $T_{d}$ is given by
$T_{d}=D/R$. Following the discussion in [23] we define the DOR as the
probability

$P_{\text{dor}}=P(T_{d}>T_{th})=P(R<D/T_{th}),$ (4)

where $T_{th}$ is time delay threshold for the transmission. Thus, we obtain

$P_{\text{dor}}=P\big{(}\gamma<2^{D/WT_{th}}-1\big{)}=F_{\gamma}\big{(}2^{D/WT_{th}}-1\big{)}.$
(5)

As evident from (5), $P_{\text{dor}}$ approaches one when $T_{th}$ is very
small. However, as $T_{th}$ increases, the fraction $D/WT_{th}$ becomes small,
and at a certain point, $P_{\text{dor}}$ decays rapidly. This behavior is
indicative, especially in a logarithmic scale, of the threshold $T_{th}$ at
which reliable transmission, such as a five-nine transmission, becomes
possible. At the end, to compute both the outage probability and delay outage
rate, the distribution of the SNR seen by the receiver is required. For that
purpose we deduce in the following accurate analytic approximations for the
SNR distribution in both Scenario 1 and Scenario 2 introduced in Section II.

###  IV-B SNR distribution for CKM scenario

In the CKM there is a mapping defined between the location of the ICU receiver
and the favorable phase in each device. Since the mapping is defined in case
of pure LoS, it will impact only the static channel part while the phase of
the reflected/scattered part of the signal is random. Using (1) we write

$\begin{split}\langle{\bf h},{\bf
w}\rangle_{1}&=\sqrt{P}\Big{(}\sum_{k=1}^{K}\delta_{k}\sqrt{\gamma_{s,k}}e^{j\psi_{k}}+\sqrt{\gamma_{d}}\sum_{k=1}^{K}\delta_{k}e^{j\epsilon_{k}}\Big{)}\\\
&=\sqrt{P}\left(g_{s}+\sqrt{\gamma_{d,\Sigma}}\cdot
e^{j\epsilon_{\Sigma}}\right),\end{split}$ (6)

where subscript in $\langle{\bf h},{\bf w}\rangle_{1}$, ’1’ ,refers to the
Scenario 1. We note that here $g_{s}$ contains the reflected/scattered signal
part and $\gamma_{d,\Sigma}$ is the static part of the sum signal after the
phasing defined by CKM is used, and notation $\epsilon_{\Sigma}$ refers to the
phase of the static part. Since $g_{s}$ is composed by complex zero-mean
Gaussian variables, it is also itself a complex zero-mean Gaussian with power
$\mathop{\mathbb{E}}\left[|g_{s}|^{2}\right]=\bar{\gamma}_{s}|\delta|$. Thus,
$|\langle{\bf h},{\bf w}\rangle_{1}|$ is a Rice variable with Rice factor
depending on the phasing errors $\epsilon=(\epsilon_{k})_{k}$.

In what follows, we approximate the distribution of $|\langle{\bf h},{\bf
w}\rangle_{1}|$ by the Rice distribution with a Rice factor
$\nu_{\Sigma}=\mathop{\mathbb{E}}\left[\gamma_{d,\Sigma}\right]/E\mathop{\mathbb{E}}\left[|g_{s}|^{2}\right]$.
We have

$\begin{split}\bar{\gamma}_{d,\Sigma}&=\mathop{\mathbb{E}}\left[\gamma_{d,\Sigma}\right]=\gamma_{d}\mathop{\mathbb{E}}\left[\Big{|}\sum_{k=1}^{K}\delta_{k}e^{j\epsilon_{k}}\Big{|}^{2}\right]\\\
&=\gamma_{d}\Big{(}\mathop{\mathbb{E}}\left[\langle\delta,\cos\epsilon\rangle^{2}\right]+\mathop{\mathbb{E}}\left[\langle\delta,\sin\epsilon\rangle^{2}\right]\Big{)}\end{split}$
(7)

In order to obtain expectations of $\langle\delta,\cos\epsilon\rangle^{2}$ and
$\langle\delta,\sin\epsilon\rangle^{2}$ we need to compute
$\mathop{\mathbb{E}}\left[\cos^{n}\epsilon_{k}\right]$ and
$\mathop{\mathbb{E}}\left[\sin^{n}\epsilon_{k}\right]$for $n=1,2$. Since
$\epsilon_{k}\sim{\cal N}(0,\sigma_{\epsilon}^{2})$ we have

$\begin{split}\mathop{\mathbb{E}}\left[\cos\epsilon_{k}\right]&=\frac{1}{\sqrt{2\pi}\sigma_{\epsilon}}\int_{-\infty}^{\infty}e^{-\frac{\epsilon_{k}^{2}}{2\sigma_{\epsilon}^{2}}}\cos\epsilon_{k}\,d\epsilon_{k}=e^{-\frac{\sigma_{\epsilon}^{2}}{2}},\end{split}$
(8)

where the second equality follows by using formula (3.896.2) of [31]. Function
$\sin\epsilon_{k}$ is odd and phase error is symmetric around zero, implying
$\mathop{\mathbb{E}}\left[\sin\epsilon_{k}\right]=0$. It remains to compute
$\mathop{\mathbb{E}}\left[\cos^{2}\epsilon_{k}\right]$ and
$\mathop{\mathbb{E}}\left[\sin^{2}\epsilon_{k}\right]$. By using the
identities [(3.898.1-2), [31]], we get

$\mathop{\mathbb{E}}\left[\cos^{2}\epsilon_{k}\right]=\tfrac{1}{2}(1+e^{-\frac{\sigma_{\epsilon}^{2}}{2}}),\
\mathop{\mathbb{E}}\left[\sin^{2}\epsilon_{k}\right]=\tfrac{1}{2}(1-e^{-\frac{\sigma_{\epsilon}^{2}}{2}}).$
(9)

By expanding $\langle\delta,\cos\epsilon\rangle^{2}$ we obtain

$\begin{split}\mathop{\mathbb{E}}\left[\langle\delta,\cos\epsilon\rangle^{2}\right]&=\sum_{k=1}^{K}\delta_{k}\mathop{\mathbb{E}}\left[\cos^{2}\epsilon_{k}\right]+\dots\\\
\dots+\sum_{k=1}^{K}&\sum_{l\neq
k}\delta_{k}\delta_{l}\mathop{\mathbb{E}}\left[\cos\epsilon_{k}\right]\mathop{\mathbb{E}}\left[\cos\epsilon_{l}\right]\\\
=\tfrac{1}{2}|\delta|\big{(}1&+e^{-\frac{\sigma_{\epsilon}^{2}}{2}}\big{)}+|\delta|(|\delta|-1)e^{-\sigma_{\epsilon}^{2}}.\end{split}$
(10)

Similarly, we get

$\begin{split}\mathop{\mathbb{E}}\left[\langle\delta,\sin\epsilon\rangle^{2}\right]&=\sum_{k=1}^{K}\delta_{k}\mathop{\mathbb{E}}\left[\sin^{2}\epsilon_{k}\right]=\tfrac{1}{2}|\delta|\big{(}1-e^{-\frac{\sigma_{\epsilon}^{2}}{2}}\big{)}.\end{split}$
(11)

The Rice factor of the approximation attains the form

$\begin{split}&\nu_{\Sigma}=\frac{\bar{\gamma}_{d,\Sigma}}{\bar{\gamma}_{s}|\delta|}=\nu\big{(}1+(|\delta|-1)e^{-\sigma_{\epsilon}^{2}}\big{)}.\\\
\end{split}$ (12)

We note that Rice factor increases with the number of devices cooperating but
phase errors may crucially impact the resulting performance gain. The
approximate CDF of SNR obtained from Rice distribution is given in terms of
the Marqum Q-function:

$F_{\hat{\gamma}}(\gamma)=1-Q_{1}\Big{(}\tfrac{\sqrt{\bar{\gamma}_{d,\Sigma}}}{\sigma_{\Sigma}},\tfrac{\sqrt{\gamma}}{\sigma_{\Sigma}}\Big{)},\
\sigma_{\Sigma}=\sqrt{\tfrac{1}{2}|\delta|\bar{\gamma}_{s}},$ (13)

where $\hat{\gamma}$ refers to the approximation of $|\langle{\bf h},{\bf
w}\rangle_{1}|^{2}$. We further note that the terms $\bar{\gamma}_{d,\Sigma}$
and $\sigma_{\Sigma}$ in (13) are functions of $\nu$, $|\delta|$,
$\sigma_{\epsilon}$ and $\bar{\gamma}$. Using (13) we obtain for the DOR

$P_{\text{dor}}=1-Q_{1}\Big{(}\tfrac{\sqrt{\bar{\gamma}_{d,\Sigma}}}{\sigma_{\Sigma}},\tfrac{1}{\sigma_{\Sigma}}\small{\sqrt{2^{D/WT_{th}}-1}}\Big{)}.$
(14)

That is, for a given data volume $D$ and bandwidth $W$, we can compute the DOR
as a function of delay threshold $T_{th}$.

Now we can also consider the following question: If, for a certain service,
the DOR and delay threshold are given, how many transmitting devices are
needed to ensure that service requirements are fulfilled?

It was shown in the Appendix that

$|\delta|>-\tfrac{e^{\sigma_{\epsilon}^{2}}}{4\nu}\log(2P_{\text{dor}})\left(1+\sqrt{1-4e^{-\frac{\sigma_{\epsilon}^{2}}{2}}\tfrac{\sqrt{\nu(1+\nu)\gamma/\bar{\gamma}}}{\log(2P_{\text{dor}})}}\right)^{2},$
(15)

where the required SNR is $\gamma=2^{D/WT_{th}}-1$ and (15) is valid only if

$T_{th}>D\big{(}W\log_{2}(1+\tfrac{\bar{\gamma}\nu}{1+\nu}|\delta|^{2}e^{-\sigma_{\epsilon}^{2}})\big{)}^{-1}.$
(16)

The bound (15) provides means to assess the number of required devices in
cooperative transmission but it assumes that the mean power $\bar{\gamma}$ and
Rice factor $\nu$ of the channel are known and standard deviation
$\sigma_{\epsilon}$ of the errors can be assessed. The first two are related
to locations of transmitter and receiver, and can be obtained from CKM. The
standard deviation of error can be also predefined when CKM is formed.

If the total transmission power in cluster is kept constant, i.e. the
transmission power in IoT devices is scaled by number of devices, then we have
shown in Appendix that the requirement for the number of devices attains the
form

$|\delta|>e^{\sigma_{\epsilon}^{2}}\Big{(}\sqrt{\tfrac{1}{\nu}\log(2P_{\text{dor}})^{-1}}+\sqrt{\tfrac{1+\nu}{\nu}\tfrac{\gamma}{\bar{\gamma}}}\Big{)}^{2}.$
(17)

###  IV-C SNR distribution when using channel side-information

In this case, the sum channel is composed of complex components that are
adjusted based on the side-information from the receiver. Therefore, we adopt
approximation that is based on the central limit theorem. Let us start by
writing the SNR of the received signal in the form

$\gamma=\big{|}\langle{\bf h},{\bf
w}\rangle_{2}\big{|}^{2}=(\text{Re}\langle{\bf h},{\bf
w}\rangle_{2})^{2}+(\text{Im}\langle{\bf h},{\bf w}\rangle_{2})^{2},$ (18)

where subscript ’2’ refers to the Scenario 2 and both $\text{Re}\langle{\bf
h},{\bf w}\rangle_{2}$ and $\text{Im}\langle{\bf h},{\bf w}\rangle_{2}$ are
composed of signals from $|\delta|$ devices. If the number of devices is
large, then we can use the central limit theorem to approximate both real and
imaginary parts of $\langle{\bf h},{\bf w}\rangle_{2}$ by Gaussian variables
[30]. More precisely, we approximate $\text{Re}\langle{\bf h},{\bf
w}\rangle_{2}\approx X$, $\text{Im}\langle{\bf h},{\bf w}\rangle_{2}\approx Y$
and $\gamma\approx\hat{\gamma}=X^{2}+Y^{2}$, where $X$ and $Y$ are Gaussian
with the following means and variances:

$\begin{cases}\mu_{r}=\mathop{\mathbb{E}}\left[\text{Re}\langle{\bf h},{\bf
w}\rangle_{2}\right],\
\sigma_{r}^{2}=\mathop{\mathbb{E}}\left[(\text{Re}\langle{\bf h},{\bf
w}\rangle_{2})^{2}\right]-\mu_{r}^{2},\\\
\mu_{i}=\mathop{\mathbb{E}}\left[\text{Im}\langle{\bf h},{\bf
w}\rangle_{2}\right],\
\sigma_{i}^{2}=\mathop{\mathbb{E}}\left[(\text{Im}\langle{\bf h},{\bf
w}\rangle_{2})^{2}\right]-\mu_{i}^{2}.\\\ \end{cases}$ (19)

Here we note that the distribution of variable $\theta_{k}$ in (2) is
symmetric around zero and $\sin\theta_{k}$ is an odd function. Therefore
$\mu_{i}=0$ and $Y^{2}$ follows the central $\chi^{2}$-distribution with one
degree of freedom. On the other hand, $\mu_{r}\neq 0$ and $X^{2}$ follow the
non-central $\chi^{2}$-distribution with one degree of freedom. The related
non-centrality parameter of this distribution is $\lambda=\mu_{r}^{2}$.

From the classical theory of statistics, it is known that the sum
$\hat{\gamma}=X^{2}+Y^{2}$ follows the $\chi^{2}$-distribution with two
degrees of freedom. Moreover, the related PDF and CDF can be given in the
following form:

$\begin{split}f_{\hat{\gamma}}(\gamma)&=\sum_{n=0}^{\infty}a_{n}f_{n}(\gamma),\
F_{\hat{\gamma}}(\gamma)=\sum_{n=0}^{\infty}a_{n}F_{n}(\gamma),\\\
\end{split}$ (20)

where the weights $a_{n}$ are defined as

$a_{n}=\tfrac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)\Gamma(\frac{1}{2})}\left(\tfrac{\sigma_{r}}{\sigma_{i}}\right)\left(\tfrac{\sigma_{i}^{2}-\sigma_{r}^{2}}{\sigma_{i}^{2}}\right)^{n}$
(21)

and the component functions $f_{n}$ and $F_{n}$ are of the form:

$\begin{split}f_{n}(\gamma)&=\tfrac{1}{2\sigma_{r}^{2}}\left(\tfrac{\gamma}{\lambda}\right)^{\frac{n}{2}}e^{-\frac{\lambda+\gamma}{2\sigma_{r}^{2}}}I_{n}\left(\tfrac{\sqrt{\lambda}}{\sigma_{r}^{2}}\sqrt{\gamma}\right),\\\
F_{n}(\gamma)&=1-Q_{n+1}\left(\tfrac{\sqrt{\lambda}}{\sigma_{r}},\tfrac{\sqrt{\gamma}}{\sigma_{r}}\right),\\\
\end{split}$ (22)

where $I_{n}(\cdot)$ is the modified Bessel function of order $n$ and
$Q_{m}(\cdot,\cdot)$ is the generalised Marcum Q-function of order $m$.

Before computing the parameters (19), we consider the impact of an erroneous
side-information. The side-information is composed by $|\delta|$ $N$-bit
words, each word dedicated to a certain device. If we assume that the word
error probability is $p_{w}$ and errors occur independently, then

$\begin{split}F_{\hat{\gamma}}(\gamma)=\sum_{m=0}^{|\delta|}\binom{|\delta|}{m}(1-p_{w})^{|\delta|-m}p_{w}^{m}F_{\hat{\gamma}}(\gamma|m),\\\
\end{split}$ (23)

where $F_{\hat{\gamma}}(\gamma|m)$ refers to the CDF of SNR in case there are
$m$ erroneous phasing words. We may use the formulas (20)-(22) to compute
$F_{\hat{\gamma}}(\gamma|m)$, but it is important to notice that parameters
defined in (19) will depend on the number $m$ of erroneous phasing words.
Therefore, the definitions in (19) for the error-free case and for the case
where the phasing word error exists should be computed.

Let us now deduce closed-form expressions for the parameters (19). We have

$\langle{\bf h},{\bf
w}\rangle_{2}=\sqrt{P}\cdot\sum_{k=1}^{K}\delta_{k}\sqrt{\gamma_{k}}e^{j\theta_{k}}.$
(24)

To find the required expressions, we need expectations of the first and second
power of real and imaginary parts of $\langle{\bf h},{\bf w}\rangle_{2}$.
First, to obtain $\mu_{r}$ we compute
$\mathop{\mathbb{E}}\left[\text{Re}\langle{\bf h},{\bf w}\rangle_{2}\right]$.
For that purpose we recall that the expected value for the amplitude of a
Rician variable is given by

$\begin{split}\mathop{\mathbb{E}}\left[\sqrt{\gamma_{k}}\right]&=\sqrt{\tfrac{\pi\bar{\gamma}}{4(1+\nu)}}e^{-\frac{\nu}{2}}\left((1+\nu)I_{0}(\tfrac{\nu}{2})+\nu
I_{1}(\tfrac{\nu}{2})\right).\\\ \end{split}$ (25)

Moreover, in the error-free case $\theta_{k}\sim{\cal
U}\left(-\frac{\pi}{2^{N}},\frac{\pi}{2^{N}}\right)$ and we obtain

$\mathop{\mathbb{E}}\left[\cos\theta_{k}\right]=\frac{2^{N-1}}{\pi}\int_{-\frac{\pi}{2^{N}}}^{\frac{\pi}{2^{N}}}\cos\theta\,d\theta=\text{sinc}(2^{-N}),$
(26)

where $\text{sinc}(x)=\sin(\pi x)/(\pi x)$. If error occurs, then
$\theta_{k}\sim{\cal U}\left(-\pi,\pi\right)$ and
$\mathop{\mathbb{E}}\left[\cos\theta_{k}\right]=0$.

Assume that there are $m$ erroneous phasing words. Then, using equations
(24)-(26) we obtain

$\begin{split}\mu_{r}(m)&=\sqrt{P}\cdot(|\delta|-m)\cdot\sqrt{\tfrac{\pi\bar{\gamma}}{4(1+\nu)}}e^{-\frac{\nu}{2}}\big{(}(1+\nu)I_{0}(\tfrac{\nu}{2})+\\\
&\dots+\nu I_{1}(\tfrac{\nu}{2})\big{)}\text{sinc}(2^{-N})\end{split}$ (27)

where $1\leq m\leq|\delta|$. Since the $\sin\theta_{k}$ is odd function,
$\mathop{\mathbb{E}}\left[\sin\theta_{k}\right]=0$ whether
$\theta_{k}\sim{\cal U}\left(-\frac{\pi}{2^{N}},\frac{\pi}{2^{N}}\right)$ or
$\theta_{k}\sim{\cal U}\left(-\pi,\pi\right)$. Therefore $\mu_{i}(m)=0$ for
all $m$.

Next we compute the expectations of $\left(\text{Re}\langle{\bf h},{\bf
w}\rangle_{2}\right)^{2}$ and $\left(\text{Im}\langle{\bf h},{\bf
w}\rangle_{2}\right)^{2}$. For the former we have

$\left(\text{Re}\langle{\bf h},{\bf
w}\rangle_{2}\right)^{2}=P\sum_{k=1}^{K}\sum_{k=1}^{K}\delta_{k}\delta_{l}\sqrt{\gamma_{k}}\sqrt{\gamma_{l}}\cos\theta_{k}\cos\theta_{l}$
(28)

and thus, we need the following expectations:
$\mathop{\mathbb{E}}\left[\sqrt{\gamma_{k}}\right]$,
$\mathop{\mathbb{E}}\left[\cos\theta_{k}\right]$,
$\mathop{\mathbb{E}}\left[\gamma_{k}\right]$ and
$\mathop{\mathbb{E}}\left[\cos^{2}\theta_{k}\right]$. The first two were
already obtained, so it remains to compute
$\mathop{\mathbb{E}}\left[\cos^{2}\theta_{k}\right]$. If phasing word is
correct, then $\theta_{k}\sim{\cal
U}\left(-\frac{\pi}{2^{N}},\frac{\pi}{2^{N}}\right)$, and we find that

$\begin{split}\mathop{\mathbb{E}}\left[\cos^{2}\theta_{k}\right]&=\tfrac{1}{2}(1+\text{sinc}(2^{-(N-1)})).\\\
\end{split}$ (29)

On the other hand, if phasing word is random, then $\theta_{k}\sim{\cal
U}\left(-\pi,\pi\right)$, and we obtain
$\mathop{\mathbb{E}}\left[\cos^{2}\theta_{k}\right]=1/2$. By taking the
expectation of (28) we obtain after some elementary manipulations:

$\small\begin{split}\mathop{\mathbb{E}}\left[\left(\text{Re}\langle{\bf
h},{\bf
w}\rangle_{2}\right)^{2}\right]&=P\big{(}\tfrac{1}{2}m\bar{\gamma}+\tfrac{1}{2}(|\delta|-m)\bar{\gamma}(1+\text{sinc}(\tfrac{1}{2^{N-1}}))\\\
+(|\delta|-m)&(|\delta|-m-1)\mathop{\mathbb{E}}\left[\sqrt{\gamma_{k}}\right]^{2}\text{sinc}(\tfrac{1}{2^{N}})^{2}\big{)}.\end{split}\normalsize$
(30)

As in (29), we have
$\mathop{\mathbb{E}}\left[\sin^{2}\theta_{k}\right]=\tfrac{1}{2}(1-\text{sinc}(2^{-(N-1)}))$
and

$\small\begin{split}\mathop{\mathbb{E}}\left[\left(\text{Im}\langle{\bf
h},{\bf
w}\rangle_{2}\right)^{2}\right]&=P\big{(}\tfrac{1}{2}m\bar{\gamma}+\tfrac{1}{2}(|\delta|-m)\bar{\gamma}(1-\text{sinc}(\tfrac{1}{2^{N-1}})).\\\
\end{split}\normalsize$ (31)

The parameters $\sigma_{r}(m)$ and $\sigma_{i}(m)$, $1\leq m\leq|\delta|$ are
obtained using (30), (31) and the previously computed $\mu_{r}$ and $\mu_{i}$.

##  V Performance results

In this section, we present performance results for both Scenario 1 and 2,
utilizing two performance indicators. Initially, the outage probability is
investigated for both scenarios. Subsequently, we delve into the results for
the delay outage rate, as discussed in Section IV-A. Although both the outage
probability and the delay outage rate employ the same CDF approximation, it’s
noteworthy that the latter typically focuses on the tail of the distribution,
necessitating a highly accurate approximation for the SNR distribution. In
contrast, the classical outage probability is plotted on a linear scale, where
values and errors of the approximation are discernible around the mean. To
assess the precision of our analytical results, we include simulated point
values in each figure.

###  V-A Simulation parameters

To begin with, the mean received SNR (denoted by $\bar{\gamma}$) of an
individual device transmission is expected to be low. While the value we apply
in performance evaluations is $-15$dB, it could be lower especially for the
CKM based phasing of Scenario 1. Yet, if $\bar{\gamma}$ is very small, then
even scarce estimation of channel phase in the receiver becomes difficult,
making the side-information based approach of Scenario 2 challenging.

We have scaled the transmission power in the cluster by the number of
transmitting devices. This makes easier assessing the gain from cooperation
when power gain from additional IoT devices is ignored. Yet, we recall that,
for example, with 20 IoT devices the power benefit would be 13dB when compare
to a single device transmission. Furthermore, since the focus of this work is
on the system that is composed of large number of cooperating IoT devices, we
have set the number of cooperating devices to 20 in most performance
illustrations.

In Scenario 1, we assume that phasing error is composed by factors that
jointly make the error $\epsilon$ follow the zero-mean Gaussian distribution
with standard deviation $\sigma_{\epsilon}$. Furthermore, we expect that only
only in very rare occasions phasing errors lead to destructive summation of
static signal components. This expectation is consistent with the assumption
$\sigma_{\epsilon}<30^{\circ}$ because then
$|\epsilon|<3\sigma_{\epsilon}<90^{\circ}$ in almost all cases. The number of
information bits in each per-device phasing word is either one or two. Using
three bits would improve results slightly but focus is more on the use of
scarce phase information, since accurate phase estimation of weak device
signal is challenging. Finally, phasing word error probability can be high if
channel coding is not used.

Finally, we note that in the DOR performance evaluations in Figs. 6 and 10 we
have set $W=200$kHz and $D=100$bits. These parameters are compatible with e.g.
NB-IoT system [26]. Even though NB-IoT is typically used for delay tolerant
services, it is of great interest to understand how to enable low-latency
services in similar IoT systems.

###  V-B CKM based phasing (Scenario 1)

In Fig. 4 we have the outage probability curves as a function of SNR when $20$
clustered IoT devices cooperate in transmission. The CKM-based phase
information is incomplete and phase error $\epsilon$ follows the Gaussian
distribution with standard deviation $\sigma_{\epsilon}=20$ degrees. As can be
expected, the strength of the Rice factor strongly impacts on the performance.
When there is no static channel component the channel becomes Rayleigh and
outage performance is very poor, as can be seen from part of the dashed curve
that is visible. However, it is somewhat surprising that cooperative
transmission is already providing some gain when $\nu=-6dB$. Furthermore, the
performance difference on 10% outage level between $\nu=0$dB and $\nu=9$dB is
4dB while the difference between $\nu=-6$dB and $\nu=9$dB is around 10dB. We
also note that the performance difference is especially large in low outage
levels. The outage probability performance of the selection diversity with 20
diversity branches is presented by the dotted curve in Fig. 4. The performance
of selection diversity is clearly inferior when compared to CKM based
approach. More importantly, the power gain that is obtained from multiple
devices transmitting simultaneously cannot be achieved in selection diversity,
since transmission power of an individual device cannot be scaled up.

Figure 4: Outage probability for CKM-based phasing when $|\delta|=20$ and
Gaussian phase error with standard deviation $\sigma_{\epsilon}=20^{\circ}$ is
assumed. Curves: $\nu=-6$dB (+), $\nu=-3$dB ($\nabla$), $\nu=0$dB (*),
$\nu=3$dB (o), $\nu=6$dB (x), $\nu=9$dB ($\Delta$). Ticks refer to simulated
values and solid curves are plotted using analytic formulae. The dotted curve
refers to the outage probability when selection diversity over 20 devices is
applied and dashed curve refers to the Rayleigh fading case.

The impact of Gaussian phasing error on the outage performance of cooperative
transmission is illustrated in Fig. 5 assuming 6dB Rice factor. Increasing the
standard deviation of the error from one degree to 10 degrees has only a minor
impact on the performance. However, if it increases to 30 degrees, then the
performance loss becomes well visible. We also note that our approximation
works best when standard deviation of the error is less than 30 degrees. For
larger standard deviation the number of occasions where static components of
some IoT devices admit destructive summation increases, degrading both the
system performance and the accuracy of the approximation. As can be seen in
Fig. 5, there is already some discrepancy between simulated and analytical
results when $\sigma_{\epsilon}=30$ degrees.

Figure 5: Outage probability for CKM-based phasing when $|\delta|=20$,
$\nu=6$dB and $\sigma_{\epsilon}=1^{\circ}$ ($\Delta$),
$\sigma_{\epsilon}=10^{\circ}$ (x), $\sigma_{\epsilon}=20^{\circ}$ (o) and
$\sigma_{\epsilon}=30^{\circ}$ (*). Ticks refer to simulated values and solid
curves are plotted using analytic formulae.

The DOR performance results has been presented in Fig. 6. While Fig. 4
indicated that a Rice factor decrease (e.g., from $\nu=6$dB to $\nu=0$dB)
leads to small SNR losses in probability levels between 10% and 90%, Fig. 6
shows that such decrease might be crucial for the delay threshold. If the DOR
requirement is e.g. $10^{-4}$ or lower, then even small change in the Rice
factor may seriously degrade the quality of service. Thus, low-latency
services are very sensitive for any changes in the radio environment. Finally,
we note that our approximation works well in the probability range of Fig. 6.
However, if the probability range is extended to very low probabilities, the
approximation errors start to become more visible.

Figure 6: Delay outage rate for CKM-based phasing when $|\delta|=20$ and
Gaussian phase error with standard deviation $\sigma_{\epsilon}=20^{\circ}$ is
assumed. Curves: $\nu=0$dB (*), $\nu=3$dB (o), $\nu=6$dB (x), $\nu=9$dB
($\Delta$). Ticks refer to simulated values and solid curves are plotted using
analytic formulae.

The number of required IoT devices in cooperative transmission is plotted in
Fig. 7 using the analytic bound (17) and assuming that $P_{dor}=10^{-4}$. The
dashed curves have been obtained through simulations and they indicate that
analytic bounds are relatively tight. As can be expected, the strength of the
Rice factor has a notable impact on the number of required IoT devices in the
cooperative transmission. Namely, if Rice factor is 6dB, then the number of
required IoT devices is less than half when compared to the case with 0dB Rice
factor. We have shown results only for the case
$\sigma_{\epsilon}=20^{\circ}$, but we recall that the impact of phase error
is relatively small when $0<\sigma_{\epsilon}\leq 20^{\circ}$.

Figure 7: Numbers of required IoT devices in CKM-based phasing when
$P_{dor}=10^{-4}$ and Gaussian phase error is assumed with standard deviation
$\sigma_{\epsilon}=20^{\circ}$. Rice factors: $\nu=0$dB (x), $\nu=3$dB (o) and
$\nu=6$dB (*). Solid curves represent the bounds (17) and dashed curves are
obtained through simulations.

###  V-C Side-information based phasing (Scenario 2)

The outage probability results for channel side-information based approach are
presented in Fig. 8 assuming cooperative transmission of $20$ devices and 5%
error probability for each IoT device specific phasing word. In contrary to
CKM based cooperative transmission, phasing is now based on side-information
and, accordingly, the role of Rice factor is small. Therefore we have only
plotted curves for $\nu=-3$dB and $\nu=9$dB. In side-information based
approach the volume and quality of side-information is important. As can be
seen from Fig. 8, there is a notable performance difference between the case
where only a single phasing bit is send with respect to case where two bits
are provided. Yet, increasing further the number of bits in the phasing would
not provide much additional performance gain.

Figure 8: Outage probability for side-information based phasing when
$|\delta|=20$ and $p_{w}=0.05$. Solid curves: $N=1$, dashed curves: $N=2$.
Rice factors are $\nu=-3$dB (*), $\nu=9$dB (o). Ticks refer to simulated
values and solid curves are plotted using analytic formulae. The dotted curve
refers to the outage probability when selection diversity over 20 devices is
applied.

Fig. 9 shows the impact of the impact of word errors in the side-information
phasing for each device on the performance when Rice factor is 6dB and $N=2$.
Even though erroneous side-information degrades the performance, results
indicate that in the range of 10% or higher probabilities the impact of errors
is tolerable. Yet, performance losses increase towards lower probability
levels. This can be crucial in case of services that require low latency and
high reliability as depicted in Fig. 10. Therein the impact of errors on the
distribution tail is clearly visible. If the DOR level is low, then side-
information errors may have a huge impact on the performance in terms of very
high delay threshold. Fig. 10 also suggests that incomplete side-information
may set a lower bound for the DOR such that certain service requirements
cannot be fulfilled at all in case of corrupted side-information. Finally, we
note that there are small approximation errors seen in lower DOR levels in
Fig. 10. This is due the fact accurate approximation of the distribution tail
becomes increasingly difficult on lower probability levels.

Figure 9: Outage probability for side-information based phasing when
$|\delta|=20$, $\nu=6$dB and $N=2$. Phasing word error probabilities are
$p_{w}=0.01$ (*), $p_{w}=0.05$ (o), $p_{w}=0.10$ (x) and $p_{w}=0.20$
($\Delta$). Ticks refer to simulated values and solid curves are plotted using
analytic formulae.  Figure 10: Delay outage rate for side-information based
phasing when $|\delta|=20$, $\nu=0$dB and $N=2$. Curves: $p_{w}=0.01$ (*),
$p_{w}=0.05$ (o), $p_{w}=0.10$ (x), $p_{w}=0.20$ ($\Delta$). Ticks refer to
simulated values and solid curves are plotted using analytic formulae.

###  V-D Comparison of Scenario 1 and 2 approaches

Let us briefly compare the approaches used in Scenario 1 and 2. First, we note
that CKM based approach has clear advantage since no information exchange
between the ICU receiver and individual IoT devices is not required.
Basically, IoT devices need to coordinate transmission with cluster head that
may provide either the location or the phasing information, depending where
the CKM including the mapping from ICU position to phasing is stored. However,
the CKM based approach is sensitive to situations where the radio environment
includes a scattered/reflected channel part since CKM includes only phasing
information related to the static channel component. Thus, the approach of
Scenario 1 is feasible only in a radio environment with LoS between IoT
devices and ICU.

The approach used in Scenario 2 is more robust to the radio environment. If
timely side-information can be provided reliably to IoT devices, then the
performance benefit is notable. For that it is important to notice that ICU
should be able to estimate the phase of the channel between IoT device and ICU
with some tolerable accuracy. This would require a sounding mechanism or use
of pilot signals that will use the IoT device power resources. Also, if
received SNR in ICU from an individual IoT device is very low, then even rough
phase estimation of the channel between ICU and IoT device might be
challenging. It is worth noticing that even when side-information is
corrupted, the approach of Scenario 2 works well if delay-tolerant services
are considered. However, side-information errors may limit the provision of
services with high reliability and low latency.

##  VI Conclusions

We examined a transmission system where clustered IoT devices collaborated to
enhance communication range with the information collecting unit (ICU). Two
scenarios were presented: the first utilizing the Channel Knowledge Map (CKM)
for phasing information, assuming a channel with a strong static component,
and the second based on channel side-information estimated and provided to IoT
devices by the ICU. In both scenarios, we considered the presence of errors in
the information used for cooperation. Key performance indicators included
conventional outage probability and the recently introduced delay outage rate
(DOR).

Analytic approximations for SNR distributions in both scenarios, accounting
for errors, were derived and verified against simulation results,
demonstrating close agreement. The main findings include the significant
benefits of cooperation between clustered IoT devices, resulting in increased
range to the ICU. However, CKM-based cooperation is sensitive to disruptions
in the Line of Sight (LoS) component, diminishing potential gains when LoS is
disturbed. The cluster size required for reliable transmission becomes
impractically large. On the other hand, channel side-information-based
cooperation is robust against changes in the radio environment but susceptible
to errors in the channel side-information, potentially limiting its use in
scenarios with high-reliability and low-latency requirements.

## Appendix: Derivation of (15)

For Marqum Q-function there holds [33]:

$1-\tfrac{1}{2}\big{(}e^{-\frac{1}{2}(a-b)^{2}}-e^{-\frac{1}{2}(a+b)^{2}}\big{)}\leq
Q_{1}(a,b),\quad a>b$ (32)

and from the CDF expression in (13) we obtain

$F_{\hat{\gamma}}(\gamma)\leq\tfrac{1}{2}\Big{(}e^{-\frac{1}{2\sigma_{\tiny{\Sigma}}^{2}}(\sqrt{\bar{\gamma}_{d,\tiny{\Sigma}}}-\sqrt{\gamma})^{2}}-e^{-\frac{1}{2\sigma_{\tiny{\Sigma}}^{2}}(\sqrt{\bar{\gamma}_{d,\tiny{\Sigma}}}+\sqrt{\gamma})^{2}}\Big{)}.$
(33)

Here the latter term in the sum becomes small when either $\nu$ or
$\nu|\delta|$ is large since

$\tfrac{\bar{\gamma}_{d,\tiny{\Sigma}}}{2\sigma_{\tiny{\Sigma}}^{2}}=\nu(1+(|\delta|-1)e^{-\sigma_{\epsilon}^{2}}).$
(34)

That is, if the Rice factor is large or if e.g. $\nu>1$ and the number of
cooperative devices is large we can estimate

$F_{\hat{\gamma}}(\gamma)\leq\tfrac{1}{2}e^{-\frac{1}{2\sigma_{\tiny{\Sigma}}^{2}}(\sqrt{\bar{\gamma}_{d,\tiny{\Sigma}}}-\sqrt{\gamma})^{2}}<P_{\text{dor}},$
(35)

where the latter inequality represents the requirement that the upper bound
for CDF is smaller than the target DOR. From (35) we obtain

$\sqrt{\bar{\gamma}_{d,\tiny{\Sigma}}}-\sqrt{\gamma}>\sqrt{2\sigma_{\tiny{\Sigma}}^{2}\log(2P_{\text{dor}})^{-1}}.$
(36)

Following the notation in (13) we substitute here
$2\sigma_{\tiny{\Sigma}}^{2}=|\delta|\bar{\gamma}_{s}$ and approximate

$\bar{\gamma}_{d,\tiny{\Sigma}}=\gamma_{d}|\delta|(1+(|\delta|-1)e^{-\sigma_{\epsilon}^{2}})\approx\gamma_{d}|\delta|^{2}e^{-\sigma_{\epsilon}^{2}}.$
(37)

We note that approximation is feasible since
$1+(|\delta|-1)e^{-\sigma_{\epsilon}^{2}}=|\delta|e^{-\sigma_{\epsilon}^{2}}+(1-e^{-\sigma_{\epsilon}^{2}})$
and the latter term in the brackets is notably smaller than the first term in
cases we focus on. That is, if STD of error is less than 30 degrees and the
number of devices is larger than 10. We can now write (36) in the form

$\sqrt{\bar{\gamma}_{d}}|\delta|-\sqrt{\bar{\gamma}_{s}\log(2P_{\text{dor}})^{-1}}\sqrt{|\delta|}-\sqrt{\gamma}>0.$
(38)

Formula in the left is a second order polynomial with respect to
$\sqrt{|\delta|}$ and, after substituting
$\gamma_{d}=\bar{\gamma}\nu/(1+\nu)$, $\bar{\gamma}_{s}=\bar{\gamma}/(1+\nu)$
and applying some elementary math, we obtain

$|\delta|>-\tfrac{e^{\sigma_{\epsilon}^{2}}}{4\nu}\log(2P_{\text{dor}})\left(1+\sqrt{1-4e^{-\frac{\sigma_{\epsilon}^{2}}{2}}\tfrac{\sqrt{\nu(1+\nu)\gamma/\bar{\gamma}}}{\log(2P_{\text{dor}})}}\right)^{2}.$
(39)

We note that, if transmission power in IoT devices is scaled by number of
devices to keep the total transmitted power same regardless the number of
cooperating devices, then $\gamma_{d}=\bar{\gamma}\nu/|\delta|(1+\nu)$ and
$\bar{\gamma}_{s}=\bar{\gamma}/|\delta|(1+\nu)$ in (38) and we obtain
requirement

$|\delta|>e^{\sigma_{\epsilon}^{2}}\Big{(}\sqrt{\tfrac{1}{\nu}\log(2P_{\text{dor}})^{-1}}+\sqrt{\tfrac{1+\nu}{\nu}\tfrac{\gamma}{\bar{\gamma}}}\Big{)}^{2}.$
(40)

Finally, we note that requirement

$T_{th}>D\big{(}W\log_{2}(1+\tfrac{\bar{\gamma}\nu}{1+\nu}|\delta|^{2}e^{-\sigma_{\epsilon}^{2}})\big{)}^{-1}$
(41)

is obtained from the condition $a>b$ in (32) by setting
$a=\sqrt{\bar{\gamma}_{d,\tiny{\Sigma}}}/\sigma_{\tiny{\Sigma}}$ and
$b=\sqrt{\gamma}/\sigma_{\tiny{\Sigma}}$.

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