# Unsourced Random Access Using Multiple Stages of Orthogonal Pilots: MIMO and
Single-Antenna Structures

Mohammad Javad Ahmadi, Mohammad Kazemi, and Tolga M. Duman This research is
accepted for publications in IEEE Transactions on Wireless Communications
[31], with DOI:10.1109/TWC.2023.3288376. The authors are with the Department
of Electrical and Electronics Engineering, Bilkent University, 06800 Ankara,
Turkey (e-mails: {ahmadi, kazemi, duman}@ee.bilkent.edu.tr).

###### Abstract

We study the problem of unsourced random access (URA) over Rayleigh block-
fading channels with a receiver equipped with multiple antennas. We propose a
slotted structure with multiple stages of orthogonal pilots, each of which is
randomly picked from a codebook. In the proposed signaling structure, each
user encodes its message using a polar code and appends it to the selected
pilot sequences to construct its transmitted signal. Accordingly, the
transmitted signal is composed of multiple orthogonal pilot parts and a polar-
coded part, which is sent through a randomly selected slot. The performance of
the proposed scheme is further improved by randomly dividing users into
different groups each having a unique interleaver-power pair. We also apply
the idea of multiple stages of orthogonal pilots to the case of a single
receive antenna. In all the set-ups, we use an iterative approach for decoding
the transmitted messages along with a suitable successive interference
cancellation technique. The use of orthogonal pilots and the slotted structure
lead to improved accuracy and reduced computational complexity in the proposed
set-ups, and make the implementation with short blocklengths more viable.
Performance of the proposed set-ups is illustrated via extensive simulation
results which show that the proposed set-ups with multiple antennas perform
better than the existing MIMO URA solutions for both short and large
blocklengths, and that the proposed single-antenna set-ups are superior to the
existing single-antenna URA schemes.

###### Index Terms:

Unsourced random access (URA), internet of things (IoT), orthogonal pilots,
massive MIMO, pilot detection, power diversity, CRC check, performance
analysis, fading channel.

##  I Introduction

In contrast to the conventional grant-based multiple access, where the base
station (BS) waits for the preamble from devices to allocate resources to
them, in grant-free random access, users transmit their data without any
coordination. Removing the need for scheduling results in some benefits, such
as reducing the latency and signaling overhead, which makes the grant-free
set-up interesting for serving many users. Sourced and unsourced random access
schemes are the main categories of grant-free random access. In the former,
both the messages and identities of the users are important to the BS, so each
user is assigned a unique pilot. However, this is inefficient, especially
considering the next-generation wireless networks with a massive number of
connected devices [1, 2]. In the so-called _unsourced random access_ (URA),
which was introduced by Polyanskiy in [3], the BS cares only about the
transmitted messages, i.e., the identity of the users is not a concern. The BS
is connected to millions of cheap devices, a small fraction of which are
active at a given time. In this set-up, the users employ a common codebook,
and they share a short frame for transmitting their messages. In URA, the per-
user probability of error (PUPE) is adopted as the performance criterion.

Many low-complexity coding schemes are devised for URA over a Gaussian
multiple-access channel (GMAC) including T-fold slotted ALOHA (SA) [4, 5, 6,
7], sparse codes [8, 9, 10, 11], and random spreading [12, 13, 14]. However,
GMAC is not a fully realistic channel model for wireless communications.
Therefore, in [15, 16, 17, 18, 19], the synchronous Rayleigh quasi-static
fading MAC is investigated, and the asynchronous set-up is considered in [20,
21]. Recently, several studies have investigated Rayleigh block-fading
channels in a massive MIMO setting [22, 23, 24, 25]. In [22], a covariance-
based activity detection (AD) algorithm is used to detect the active messages.
A pilot-based scheme is introduced in [23] where non-orthogonal pilots are
employed for detection and channel estimation, and a polar list decoder is
used for decoding messages. Furthermore, in a scheme called FASURA [24], each
user transmits a signal containing a non-orthogonal pilot and a randomly
spread polar code.
The coherence blocklength is defined as the period over which the channel
coefficients stay constant. As discussed in [23], the coherence time can be
approximated as $T_{c}\approx 1/4D_{s}$, where $D_{s}$ is the maximal Doppler
spread. For a typical carrier frequency of $2$ GHz, the coherence time may
vary in the range of $1$ ms–$45$ ms (corresponding to the transmitter speeds
between $3$ km/h–$120$ km/h). Moreover, the sampling frequency should be
chosen in the order of coherence bandwidth, whose typical value is between
$100$ kHz–$500$ kHz in outdoor environments. Consequently, the coherence
blocklength $L_{c}$ can range from $100$ to $20000$ samples. Although the AD
algorithm in [22] performs well in the fast fading scenario (e.g., when
$L_{c}\leq 320$), it is not implementable with larger blocklengths due to run-
time complexity scaling with $L_{c}^{2}$. In contrast, the schemes in [23, 24]
work well in the large-blocklength regimes (e.g., for $L_{c}=3200$); that is,
in a slow fading environment where large blocklengths can be employed, their
decoding performance is better than that of [22].

Most coding schemes in URA employ non-orthogonal pilots/sequences for
identification and estimation purposes [22, 13, 14, 23, 12, 24]. Performance
of detectors and channel estimators may be improved in terms of accuracy and
computational complexity by employing a codebook of orthogonal pilots;
however, this significantly increases the amount of collisions due to the
limited number of available orthogonal pilot sequences. To address this
problem, the proposed schemes in this paper employ multiple stages of
orthogonal pilots combined with an iterative detector.

In the proposed scheme, the transmitted signal of each user is composed of
$J+1$ stages: a polar codeword appended to $J$ independently generated
orthogonal pilots. Thus, the scheme is called multi-stage set-up with multiple
receive antennas (MS-MRA). At each iteration of MS-MRA at the receiver side,
only one of the pilot parts is employed for pilot detection and channel
estimation, and the polar codeword is decoded using a polar list decoder.
Therefore, the transmitted pilots in the remaining $J-1$ pilot parts are still
unknown. To determine the active pilots in these, we adopt two approaches. In
the first one, all the pilot bits are coded jointly with the data bits and
cyclic redundancy check (CRC) bits (therefore, the transmitted bits of all the
pilot parts are detected after successful polar decoding). As a second
approach, to avoid waste of resources, we propose an enhanced version of the
MS-MRA, where only data and CRC bits are fed to the polar encoder. At the
receiver side, the decoder iteratively moves through different $J+1$ parts of
the signal to detect all the parts of an active user’s message. Since it does
not encode the pilot bits, this is called MS-MRA without pilot bits encoding
(MS-MRA-WOPBE). We further improve the performance of the MS-MRA by randomly
dividing users into different groups. In this scheme (called multi-stage set-
up with user grouping for multiple receive antennas (MSUG-MRA)), each group is
assigned a unique interleaver-power pair. Transmission with different power
levels increases the decoding probability of the users with the highest power
(because they are perturbed by interfering users with low power levels). Since
successfully decoded signals are removed using successive interference
cancellation (SIC), users with lower power levels have increased chance of
being decoded in the subsequent steps. By repeating each user’s signal
multiple times, we further extend the idea in MS-MRA and MSUG-MRA to the case
of a single receive antenna. These extensions are called multi-stage set-up
with a single receive antenna (MS-SRA) and multi-stage set-up with user
grouping for a single receive antenna (MSUG-SRA).
We demonstrate that, while the covariance-based AD algorithm in [22] suffers
from performance degradation with large blocklengths, and the algorithms in
[23, 24] do not work well in the short blocklength regime (hence not suitable
for fast fading scenarios), the MS-MRA and MSUG-MRA have a superior
performance in both regimes. Furthermore, the MS-SRA and MSUG-SRA show better
performance compared to similar solutions with a single receive antenna over
fading channels [17, 20, 19].
Our contributions are as follows:

  * •

We propose a URA set-up with multiple receive antennas, namely MS-MRA. The
proposed set-up offers comparable performance with the existing schemes with
large blocklengths, while having lower computational complexity. Moreover, for
the short-blocklength scenario, it significantly improves the state-of-the-
art.

  * •

We provide a theoretical analysis to predict the error probability of the MS-
MRA, taking into account all the sources of error, namely, errors resulting
from pilot detection, channel estimation, channel decoding, SIC, and
collisions.

  * •

We extend the MS-MRA set-up by randomly dividing the users into groups, i.e.,
MSUG-MRA, which is more energy-efficient than MS-MRA and other MIMO URA
schemes.

  * •

Two URA set-ups with a single receive antenna, called MS-SRA and MSUG-SRA, are
provided by adopting the ideas of the MS-MRA and MSUG-MRA to the case of a
single receive antenna. They perform better than the alternative solutions
over fading channels.

The rest of the paper is organized as follows. Section II presents the system
model for the proposed framework. The encoding and decoding procedures of the
proposed schemes are introduced in Section III. In Section IV, extensive
numerical results and examples are provided. Finally, Section V provides our
conclusions.
The following notation is adopted throughout the paper. We denote the sets of
real and imaginary numbers by $\mathbb{R}$ and $\mathbb{C}$, respectively.
$\left[\mathbf{T}\right]_{(l,:)}$ and $\left[\mathbf{T}\right]_{(:,l)}$
represent the $l$th row and the $l$th column of $\mathbf{T}$, respectively;
the $\mathrm{Re}\left(\mathbf{t}\right)$ and
$\mathrm{Im}\left(\mathbf{t}\right)$ give the real and imaginary parts of
$\mathbf{t}$, respectively; the transpose and Hermitian of matrix $\mathbf{T}$
are denoted by $\mathbf{T}^{T}$ and $\mathbf{T}^{H}$, respectively; $|.|$
denotes the cardinality of a set, $\mathbf{I}_{M}$ and $\mathbf{1}_{s}$ denote
the $M\times M$ identity matrix and $1\times s$ all-ones vector, respectively;
we use $[a_{1}:a_{2}]$ to denote $\\{i\in\mathbb{Z}:a_{1}\leq i\leq a_{2}\\}$,
and $\delta_{i,j}$ is the Kronecker delta.

##  II System Model

We consider an unsourced random access model over a block-fading wireless
channel. The BS is equipped with $M$ receive antennas connected to $K_{T}$
potential users, for which $K_{a}$ of them are active in a given frame.
Assuming that the channel coherence time is larger than $L$, we divide the
length-$n$ time-frame into $S$ slots of length $L$ each ($n=SL$). Each active
user randomly selects a single slot to transmit $B$ bits of information. In
the absence of synchronization errors, the received signal vector
corresponding to the $s$th slot at the $m$th antenna is written as

$\displaystyle\mathbf{y}_{m,s}=\sum_{i\in\mathcal{K}_{s}}{h_{m,i}\mathbf{x}\left(\mathbf{w}(i)\right)+{\mathbf{z}_{m,s}}},$
(1)

where $\mathbf{y}_{m,s}\in\mathbb{C}^{1\times L}$, $\mathcal{K}_{s}$ denotes
the set of active user indices available in the $s$th slot,
$K_{s}:=|\mathcal{K}_{s}|$,
$\mathbf{x}\left(\mathbf{w}(i)\right)\in\mathbb{C}^{1\times L}$ is the encoded
and modulated signal corresponding to the message bit sequence
$\mathbf{w}(i)\in\\{0,1\\}^{B}$ of the $i$th user,
$h_{m,i}\sim\mathcal{CN}(0,1)$ is the channel coefficient between the $i$th
user and the $m$th receive antenna, and
${\mathbf{z}_{m,s}}\sim\mathcal{CN}(\mathbf{0},\sigma_{z}^{2}\mathbf{I}_{L})$
is the circularly symmetric complex white Gaussian noise vector. Letting
$\mathcal{K}_{a}$ and $\mathcal{L}_{d}$ be the set of active user indices and
the list of decoded messages, respectively, the PUPE of the system is defined
in terms of the probability of false-alarm, $p_{fa}$, and the probability of
missed-detection, $p_{md}$, as

$\displaystyle P_{e}=p_{fa}+p_{md},$ (2)

where
$p_{md}=\dfrac{1}{K_{a}}\sum_{i\in\mathcal{K}_{a}}{\mathrm{Pr}(\mathbf{w}(i)\notin\mathcal{L}_{d})}$
and $p_{fa}=\mathbb{E}\left\\{\dfrac{n_{fa}}{|\mathcal{L}_{d}|}\right\\}$,
with $n_{fa}$ being the number of decoded messages that were indeed not sent.
The energy-per-bit of the set-up can be written as
$\dfrac{E_{b}}{N_{0}}=\dfrac{LP}{\sigma_{z}^{2}B}$, where $P$ denotes the
average power of each user per channel use. The objective is to minimize the
required energy-per-bit for a target PUPE.

##  III URA with Multiple Stages of Orthogonal Pilots

###  III-A MS-MRA Encoder

Figure 1: Illustration of the encoding process in the proposed MS-MRA schemes.

In this part, we introduce a multi-stage signal structure which is used in
both of the proposed URA set-ups. As shown in Fig. 1, we divide the message of
the $i$th user into $J+1$ parts (one coded part and $J$ pilot parts) denoted
by $\mathbf{w}_{c}(i)$ and $\mathbf{w}_{p_{j}}(i),j=1,2,...,J$ with lengths
$B_{c}$ and $B_{p}$, respectively, where $B_{c}+JB_{p}=B$. The $i$th user
obtains its $j$th pilot sequence, $\mathbf{b}_{ji}$, with length
$n_{p}=2^{B_{p}}$ by mapping $\mathbf{w}_{p_{j}}(i)$ to the orthogonal rows of
an $n_{p}\times n_{p}$ Hadamard matrix $\mathbf{B}_{n_{p}}$, which is
generated as

$\displaystyle\mathbf{B}_{2}=\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix},\ \ \ \
\mathbf{B}_{2^{i}}=\mathbf{B}_{2}\otimes\mathbf{B}_{2^{i-1}}\ \ \forall\ \
i=2,3,\ldots,$

where $\otimes$ represents the Kronecker product. Since the number of possible
pilots in the orthogonal Hadamard codebook is limited, it is likely that the
users will be in collision in certain pilot segments, that is, they share the
same pilots with the other users. However, the parameters are chosen such that
two different users are in a complete collision in all the pilot parts with a
very low probability. To construct the coded sequence of the $i$th user, we
accumulate all the message bits in a row vector as

$\displaystyle\mathbf{w}(i)=\left[\mathbf{w}_{p_{1}}(i),\mathbf{w}_{p_{2}}(i),\ldots,\mathbf{w}_{p_{J}}(i),\mathbf{w}_{c}(i)\right],$
(3)

and pass it to an $\left(2n_{c},\ B+r\right)$ polar code, where $r$ is the
number of CRC bits. Note that contrary to the existing schemes in URA, we feed
not only data bits but the pilot bit sequences to the encoder. Hence, in the
case of successful decoding, all the pilot sequences for the user can be
retrieved. The polar codeword is then modulated using quadrature phase shift
keying (QPSK), resulting in $\mathbf{v}_{i}\in\\{\sqrt{{P_{c}}/{2}}(\pm 1\pm
j)\\}^{1\times n_{c}}$, where $P_{c}$ is the average power of the polar coded
part, and Gray mapping is used. The overall transmitted signal for the $i$th
user consists of $J$ pilot parts and one coded part, i.e.,

$\displaystyle\mathbf{x}_{i}=\left[\sqrt{P_{p}}\mathbf{b}_{1i},\sqrt{P_{p}}\mathbf{b}_{2i},\ldots,\sqrt{P_{p}}\mathbf{b}_{Ji},\mathbf{v}_{i}\right]\in\mathbb{C}^{1\times
L},$ (4)

where $L=n_{c}+Jn_{p}$ and $P_{p}$ denotes the average power of the pilot
sequence. Accordingly, the received signal in a slot is composed of $J+1$
parts, for which, at each iteration, the decoding is done by employing one of
the $J$ pilot parts (sequentially) and the coded part of the received signal.
Generally, only the non-colliding users can be decoded. Some non-colliding
users in the current pilot stage may experience collisions in the other pilot
parts. Therefore, by successfully decoding and removing them using SIC, the
collision density is reduced, and with further decoding iterations, the
effects of such collisions are ameliorated.

###  III-B MS-MRA Decoder

We now introduce the decoding steps of MS-MRA where the transmitted signal in
(4) is received by $M$ antennas through a fading channel. The $j$th pilot part
and the polar coded part of the received signal in the $s$th slot of the MS-
MRA can be modeled using (1) as

$\displaystyle\mathbf{Y}_{p_{j}}$
$\displaystyle=\sqrt{P_{p}}\mathbf{H}\mathbf{B}_{j}+\mathbf{Z}_{p_{j}}\in\mathbb{C}^{M\times
n_{p}},\ j=1,2,\ldots,J,$ (5) $\displaystyle\mathbf{Y}_{c}$
$\displaystyle=\mathbf{H}\mathbf{V}+\mathbf{Z}_{c}\in\mathbb{C}^{M\times
n_{c}},$ (6)

where $\mathbf{H}\in\mathbb{C}^{M\times K_{s}}$ is the channel coefficient
matrix with $h_{m,i}$ in its $m$th row and $i$th column, $\mathbf{Z}_{p_{j}}$
and $\mathbf{Z}_{c}$ consist of independent and identically distributed
(i.i.d.) noise samples drawn from $\mathcal{CN}(0,\sigma_{z}^{2})$ (i.e., a
circularly symmetric complex Gaussian distribution), and $\mathbf{b}_{ji}$ and
$\mathbf{v}_{i}$ determine the rows of $\mathbf{B}_{j}\in\\{\pm
1\\}^{K_{s}\times n_{p}}$ and $\mathbf{V}\in\\{\sqrt{{P_{c}}/{2}}(\pm 1\pm
j)\\}^{K_{s}\times n_{c}}$, respectively, with $i\in\mathcal{K}_{s}$. Note
that we have removed the slot indices from the above matrices to simplify the
notation.
The decoding process is comprised of five different steps that work in tandem.
A pilot detector based on a Neyman-Pearson (NP) test identifies the active
pilots in the current pilot part; channel coefficients corresponding to the
detected pilots are estimated using a channel estimator; maximum-ratio
combining (MRC) is used to produce a soft estimate of the modulated signal;
after demodulation, the signal is passed to a polar list decoder; and, the
successfully decoded codewords are added to the list of successfully decoded
signals before being subtracted from the received signal via SIC. The process
is repeated until there are no successfully decoded users in $J$ consecutive
SIC iterations. In the following, $\mathbf{Y}^{\prime}_{p_{j}}$ and
$\mathbf{Y}^{\prime}_{c}$ denote the received signals in (5) and (6) after
removing the list of messages successfully decoded in the current slot up to
the current iteration.

####  III-B1 Pilot Detection Based on NP Hypothesis Testing

At the $j$th pilot part, we can write the following binary hypothesis testing
problem:

$\displaystyle\mathbf{u}_{ji}|\mathcal{H}_{0}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{z}^{2}\mathbf{I}_{M}\right),$
$\displaystyle\mathbf{u}_{ji}|\mathcal{H}_{1}\sim\mathcal{CN}\left(\mathbf{0},\sigma^{2}_{1}\mathbf{I}_{M}\right),$
(7)

where $\sigma_{1}=\sqrt{\sigma_{z}^{2}+m_{ij}n_{p}P_{p}}$,
$\mathbf{u}_{ji}:=\mathbf{Y^{\prime}}_{p_{j}}\mathbf{\bar{b}}_{i}^{H}/\sqrt{n_{p}}$,
with $\mathbf{\bar{b}}_{i}=\left[\mathbf{B}_{n_{p}}\right]_{(i,:)}$,
$\mathcal{H}_{1}$ and $\mathcal{H}_{0}$ are alternative and null hypotheses
that show the existence and absence of the pilot $\mathbf{\bar{b}}_{i}$ at the
$j$th pilot part, respectively, and $m_{ij}$ is the number of users that pick
the pilot $\mathbf{\bar{b}}_{i}$ as their $j$th pilots.

######  Lemma 1.

([25, Appendix A]): Let $\hat{\mathcal{D}}_{j}$ be the estimate of the set of
active rows of $\mathbf{B}_{n_{p}}$ in the $j$th pilot part. Using a
$\gamma-$level Neyman-Pearson hypothesis testing (where $\gamma$ is the bound
on the false-alarm probability), $\hat{\mathcal{D}}_{j}$ can be obtained as

$\displaystyle\hat{\mathcal{D}}_{j}=\left\\{l:\mathbf{u}_{jl}^{H}\mathbf{u}_{jl}\geq\tau_{0}^{\prime}\right\\},$
(8)

where $\tau_{0}^{\prime}=0.5\sigma_{z}^{2}\Gamma^{-1}_{2M}(1-\gamma)$,
$\Gamma_{k}(.)$ denotes the cumulative distribution function of the chi-
squared distribution with $k$ degrees of freedom $\chi^{2}_{k}$, and
$\Gamma^{-1}_{k}(.)$ is its inverse.

The detection probability of a non-colliding user ($m_{ij}=1$) is then
obtained as

$\displaystyle P_{D}(\delta_{NP})=$
$\displaystyle\mathbb{P}\left(\mathbf{u}_{ji}^{H}\mathbf{u}_{ji}\geq\tau_{0}^{\prime}|\mathcal{H}_{1}\right)$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mua\mkern-1.5mu)}}{{=}}$
$\displaystyle
1-\Gamma_{2M}\left(\dfrac{2\tau_{0}^{\prime}}{\sigma_{z}^{2}+n_{p}P_{p}}\right)$
$\displaystyle=$ $\displaystyle
1-\Gamma_{2M}\left(\dfrac{\sigma_{z}^{2}\Gamma_{2M}^{-1}(1-\gamma)}{\sigma_{z}^{2}+n_{p}P_{p}}\right),$
(9)

where in (a), we use the fact that
$\dfrac{2}{\sigma_{1}^{2}}\mathbf{u}_{ji}^{H}\mathbf{u}_{ji}|\mathcal{H}_{1}\sim\chi^{2}_{2M}$.
Note that a higher probability of detection is obtained in the general case of
$m_{ij}>1$. It is clear that the probability of detection is increased by
increasing the parameters $\gamma$, $n_{p}$, $P_{p}$, and $M$.

####  III-B2 Channel Estimation

Let $\mathbf{B}_{\hat{\mathcal{D}}_{j}}\in\\{\pm
1\\}^{|\hat{\mathcal{D}}_{j}|\times n_{p}}$ be a sub-matrix of
$\mathbf{B}_{n_{p}}$ consisting of the detected pilots in (8), and suppose
that
$\tilde{\mathbf{b}}_{jk}=\left[\mathbf{B}_{\hat{\mathcal{D}}_{j}}\right]_{(k,:)}$
is the corresponding pilot of the $i$th user. Since the rows of the codebook
are orthogonal to each other, the channel coefficient vector of the $i$th user
can be estimated as

$\displaystyle\hat{\mathbf{h}}_{i}=\dfrac{1}{n_{p}\sqrt{P_{p}}}\mathbf{Y^{\prime}}_{p_{j}}\tilde{\mathbf{b}}_{jk}^{T}.$
(10)

If the $i$th user is in a collision ( $m_{ij}>1$), (10) gives an unreliable
estimate of the channel coefficient vector. However, this is not important
since a CRC check is employed after decoding and such errors do not propagate.

####  III-B3 MRC, Demodulation, and Channel Decoding

Let $\mathbf{h}_{i}$ be the channel coefficient vector of the $i$th user,
where $i\in\tilde{\mathcal{S}}_{s}$ with $\tilde{\mathcal{S}}_{s}$ denoting
the set of remaining users in the $s$th slot. Using $\hat{\mathbf{h}}_{i}$ in
(10), the modulated signal of the $i$th user can be estimated employing the
MRC technique as

$\displaystyle\hat{\mathbf{v}}_{i}=\hat{\mathbf{h}}_{i}^{H}\mathbf{Y^{\prime}}_{c}.$
(11)

Plugging (6) into (11), $\hat{\mathbf{v}}_{i}$ is written as

$\displaystyle\hat{\mathbf{v}}_{i}=\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}\mathbf{v}_{i}+\mathbf{n}_{i},$
(12)

where $\mathbf{n}_{i}=\sum_{k\in\tilde{\mathcal{S}}_{s},k\neq
i}\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{k}\mathbf{v}_{k}+\hat{\mathbf{h}}_{i}^{H}\mathbf{Z}_{c}$.
The first and second terms on the right-hand side of (12) are the signal and
interference-plus-noise terms, respectively. We can approximate
$\mathbf{n}_{i}$ to be Gaussian distributed, i.e.,
$\mathbf{n}_{i}\sim\mathcal{CN}(\mathbf{0},\sigma_{oi}^{2}\mathbf{I}_{n_{c}})$,
where
$\sigma_{oi}^{2}=\dfrac{1}{n_{c}}\mathbb{E}\\{\mathbf{n}_{i}\mathbf{n}_{i}^{H}\\}=P_{c}\sum_{k\in\hat{\mathcal{D}}_{j},k\neq
i}|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{k}|^{2}+\sigma_{z}^{2}\|\hat{\mathbf{h}}_{i}\|^{2}$,
which is obtained by treating the coded data sequences of different users to
be uncorrelated. The demodulated signal can be obtained as

$\displaystyle\mathbf{g}_{i}=\left[\mathrm{Im}\left(\vartheta_{1i}\right),\mathrm{Re}\left(\vartheta_{1i}\right),\ldots,\mathrm{Im}\left(\vartheta_{n_{c}i}\right),\mathrm{Re}\left(\vartheta_{n_{c}i}\right)\right],$
(13)

where $\vartheta_{ti}=\left[\hat{\mathbf{v}}_{i}\right]_{(:,t)}$. From (12)
and (13), and using
$\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}\approx\|\hat{\mathbf{h}}_{i}\|^{2}$,
each sample of $\mathbf{g}_{i}$ can be approximated as
$\pm\sqrt{{P_{c}}/{2}}\|\hat{\mathbf{h}}_{i}\|^{2}+n^{\prime}$, where
$n^{\prime}\sim\mathcal{CN}\left(0,\dfrac{\sigma_{oi}^{2}}{2}\right)$. The
following log-likelihood ratio (LLR) is obtained as the input to the polar
list decoder

$\displaystyle\mathbf{f}_{i}=\dfrac{2\sqrt{2P_{c}}\|\hat{\mathbf{h}}_{i}\|^{2}}{\hat{\sigma}_{oi}^{2}}\mathbf{g}_{i},$
(14)

where $\hat{\sigma}_{oi}^{2}$ is an approximation of $\sigma_{oi}^{2}$ which
is obtained by replacing $\mathbf{h}_{k}$’s by their estimates. At the $j$th
pilot part, the $i$th user is declared as successfully decoded if 1) its
decoder output satisfies the CRC check, and 2) by mapping the $j$th pilot part
of its decoded message to the Hadamard codebook, $\tilde{\mathbf{b}}_{jk}$ is
obtained. Then, all the successfully decoded messages (in the current and
previous iterations) are accumulated in the set $\mathcal{S}_{s}$, where
$|\mathcal{S}_{s}|+|\tilde{\mathcal{S}}_{s}|=K_{s}$.

####  III-B4 SIC

we can see in (3) that the successfully decoded messages contain bit sequences
of pilot parts and the coded part ($\mathbf{w}_{p_{j}}(i),j=1,2,...,J$ and
$\mathbf{w}_{c}(i)$). Having the bit sequences of successfully decoded
messages, we can construct the corresponding transmitted signals using (4).
The received signal matrix can be written as

$\displaystyle\mathbf{Y}=\mathbf{H}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}+\mathbf{H}_{\tilde{\mathcal{S}}_{s}}\mathbf{X}_{\tilde{\mathcal{S}}_{s}}+\mathbf{Z}_{s},$
(15)

where $\mathbf{Y}$ is obtained by merging received signal matrices of
different parts, i.e.,

$\mathbf{Y}=\left[\mathbf{Y}_{p_{1}},\ldots,\mathbf{Y}_{p_{J}},\mathbf{Y}_{c}\right]\in\mathbb{C}^{M\times
L}$

with $\mathbf{X}_{\mathcal{S}_{s}}\in\mathbb{C}^{|\mathcal{S}_{s}|\times L}$
and
$\mathbf{X}_{\tilde{\mathcal{S}}_{s}}\in\mathbb{C}^{|\tilde{\mathcal{S}}_{s}|\times
L}$ including the signals in the sets $\mathcal{S}_{s}$ and
$\tilde{\mathcal{S}}_{s}$, and
$\mathbf{H}_{\mathcal{S}_{s}}\in\mathbb{C}^{M\times|\mathcal{S}_{s}|}$ and
$\mathbf{H}_{\tilde{\mathcal{S}}_{s}}\in\mathbb{C}^{M\times|\tilde{\mathcal{S}_{s}}|}$
comprising the channel coefficients corresponding to the users in the sets
$\mathcal{S}_{s}$ and $\tilde{\mathcal{S}}_{s}$, respectively. Employing the
least squares (LS) technique, $\mathbf{H}_{\mathcal{S}_{s}}$ is estimated as

$\displaystyle\hat{\mathbf{H}}_{\mathcal{S}_{s}}=\mathbf{Y}\mathbf{X}_{\mathcal{S}_{s}}^{H}(\mathbf{X}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}^{H})^{-1}.$
(16)

Note that $\mathbf{X}_{\mathcal{S}_{s}}$ consists of all the successfully
decoded signals in the $s$th slot so far, and $\mathbf{Y}$ is the initially
received signal matrix (not the output of the latest SIC iteration). The SIC
procedure is performed as follows

$\displaystyle\mathbf{Y^{\prime}}=\left[\mathbf{Y^{\prime}}_{p_{1}},\mathbf{Y^{\prime}}_{p_{2}},\ldots,\mathbf{Y^{\prime}}_{p_{J}},\mathbf{Y^{\prime}}_{c}\right]=\mathbf{Y}-\hat{\mathbf{H}}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}.$
(17)

Finally, $\mathbf{Y}^{\prime}$ is fed back to the pilot detection algorithm
for the next iteration, where the next pilot part is employed. We note that if
no user is successfully decoded in $J$ consecutive iterations (corresponding
to $J$ different pilot parts), the algorithm is stopped. The details of the
decoding stages of MS-MRA are shown in Fig. 2 and Algorithm 1. Note that we
will discuss MS-MRA-WOPBE, which deviates from the above model, in Section
III-D.

######  Theorem 1.

The signal-to-interference-plus-noise ratio (SINR) at the output of MRC for a
non-colliding user in the $s$th slot can be approximated as

$\displaystyle\beta_{s}\approx\dfrac{\omega_{c_{s}}P_{c}\left(\omega_{p_{s}}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{4}\\}+\dfrac{\sigma_{z}^{2}}{n_{p}P_{p}}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}\right)}{\left(P_{c}(|\tilde{\mathcal{S}}_{s}|-1)+\sigma_{z}^{2}\right)\left(\omega_{p_{s}}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}+\dfrac{M\sigma_{z}^{2}}{n_{p}P_{p}}\right)},$
(18)

where $\omega_{p_{s}}=\omega_{c_{s}}=1-\dfrac{|\mathcal{S}_{s}|}{L}$ if the
transmitted signals are randomly interleaved, and
$\omega_{p_{s}}=1-\dfrac{1}{E_{x}}P_{p}|\mathcal{S}_{s}|$,
$\omega_{c_{s}}=1-\dfrac{1}{E_{x}}P_{c}|\mathcal{S}_{s}|$, otherwise, with
$E_{x}=Jn_{p}P_{p}+n_{c}P_{c}$.

###### Proof.

See Appendix A. ∎

We employ the above approximate SINR expression 1) to estimate the error
probability of MS-MRA analytically, and 2) to determine the optimal power
allocation for each group in MSUG-MRA. We further note that using this SINR
approximation, the performance of the MS-MRA is well predicted in the low and
medium $K_{a}$ regimes (see Fig. 6). The reason why the SINR approximation
does not work well in the high $K_{a}$ regime is the employed approximations
in Lemma 4 (see the Appendix A for details).

Figure 2: The decoding process of MS-MRA at the $j$th pilot part and the $s$th
slot.

for  _$l=0,1,\ldots,S$_ do Different slots

$\mathcal{S}_{s}$ $\ \ =\emptyset$.

$\mathrm{flag}\ =1$.

$t^{\prime}=0$ ($t^{\prime}$ denotes the iteration index).

while  _$\mathrm{flag}=1$_ do

$t^{\prime}\ \ =t^{\prime}+1.$

for  _$j=1,2,...,J$_ do different pilot parts

Pilot detection: estimate $\hat{\mathcal{D}}_{j}$ using (8).

Ch. estimation: estimate channel coefficient using (10).

if  _MS-MRA_ then

for  _$i\in\hat{\mathcal{D}}_{j}$_ do different detected pilots

MRC estimation: obtain $\hat{\mathbf{v}}_{i}$ using (11).

Demodulation: obtain $\mathbf{g}_{i}$ using (13).

Decoding: pass $\mathbf{f}_{i}$ in (14) to list decoder.

end for

end if

if  _MS-MRA-WOPBE_ then

Perform IISD in Section III-D2.

end if

$\mathcal{S}_{t^{\prime}j}$: set of successfully decoded users in the current
iteration.

$\mathcal{S}_{s}=\mathcal{S}_{s}\bigcup\mathcal{S}_{t^{\prime}j}$.

LS-based ch. estimation: estimate $\hat{\mathbf{H}}_{\mathcal{S}_{s}}$ using
(16).

SIC: update $\mathbf{Y^{\prime}}_{p_{j}}$ and $\mathbf{Y^{\prime}}_{c}$ using
(17).

end for

if  _$\bigcup_{j=1}^{J}\mathcal{S}_{t^{\prime}j}=\emptyset$_ then

$\mathrm{flag}=0$.

end if

end while

end for

Algorithm 1 The proposed MS-MRA decoder.

###  III-C Analysis of MS-MRA

In this part, the PUPE of the MS-MRA is analytically calculated, where errors
resulting from the collision, pilot detection, and polar decoder are
considered. For our analyses, we assume that after successfully decoding and
removing a user using a pilot part, the decoder moves to the next pilot part.
Hence, in the $t$th iteration of the $s$th slot, we have

$\displaystyle|\mathcal{S}_{s}|$ $\displaystyle=t-1,$ (19a)
$\displaystyle|\tilde{\mathcal{S}}_{s}|$ $\displaystyle=K_{s}-t+1.$ (19b)

######  Lemma 2.

Let $\xi_{k}$ be the event that $k$ out of $K_{s}$ users remain in the $s$th
slot, and define $\eta_{i}:=\|\mathbf{h}_{i}\|^{2}$, where
$\mathbf{h}_{i}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})$. Assuming that the
strongest users with highest $\eta_{i}$ values are decoded first, we have

$\displaystyle\mathbb{E}\\{\eta_{i}^{m}|\xi_{k}\\}=\mu_{(k,m)},$ (20)

where
$\mu_{(k,m)}:=\dfrac{\int_{-\infty}^{\bar{x}_{k}}\eta^{m}f_{2M}^{\chi^{2}}(2\eta)d\eta}{\int_{-\infty}^{\bar{x}_{k}}f_{2M}^{\chi^{2}}(2\eta)d\eta}$,
with $f_{k}^{\chi^{2}}(.)$ denoting the PDF of the chi-squared distribution
with $k$ degrees of freedom and $\bar{x}_{k}=0.5\Gamma_{2M}^{-1}(k/K_{s})$.

###### Proof.

In the first iteration of the $s$th slot for which no user is decoded yet (all
the $K_{s}$ active users are available), since
$\mathbf{h}_{i}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})$, we have
$2\eta_{i}|\xi_{K_{s}}\sim\chi^{2}_{2M}$. We assume that the users with higher
values of $\eta_{i}$ are decoded first. Hence, if in an iteration, $k$ out of
$K_{s}$ users remain in the slot, the distribution of $\eta_{i}$ is obtained
by $2\eta_{i}|\xi_{k}\sim\left\\{\chi^{2}_{2M}\right\\}_{k/{K_{s}}},$ where
$\left\\{.\right\\}_{\beta}$ removes $1-\beta$ portion of the samples with
higher values from the distribution and normalizes the distribution of the
remaining samples, i.e.,

$\mathbb{P}(\eta_{i}=y|\xi_{k})=\dfrac{f_{2M}^{\chi^{2}}(2y)}{\int_{-\infty}^{\bar{x}_{k}}f_{2M}^{\chi^{2}}(2y)dy},y<\bar{x}_{k},$

where $\bar{x}_{k}$ is obtained by solving the following equation
$\mathbb{P}\left(\eta_{i}<\bar{x}_{k}|\xi_{K_{s}}\right)=k/{K_{s}}$, which
results in $\bar{x}_{k}=0.5\Gamma_{2M}^{-1}(k/K_{s})$. Therefore, we obtain

$\displaystyle\mathbb{E}\\{\eta_{i}^{m}|\xi_{k}\\}=\dfrac{\int_{-\infty}^{\bar{x}_{k}}\eta^{m}f_{2M}^{\chi^{2}}(2\eta)d\eta}{\int_{-\infty}^{\bar{x}_{k}}f_{2M}^{\chi^{2}}(2\eta)d\eta}.$
(21)

∎

We can see from (13) and (14) that the input of the polar decoder is a
$1\times 2n_{c}$ real codeword. Thus, the average decoding error probability
of a non-colliding user in the $t$th iteration of a slot with $K_{s}$ users
can be approximated as (see [27])

$\displaystyle P_{K_{s},t}^{dec}\approx
Q\left(\dfrac{0.5\log\left(1+\alpha_{K_{s},t}\right)-\dfrac{B+r}{2n_{c}}}{\sqrt{\dfrac{1}{2n_{c}}\dfrac{\alpha_{K_{s},t}(\alpha_{K_{s},t}+2)\log^{2}e}{2(\alpha_{K_{s},t}+1)^{2}}}}\right),$
(22)

where $Q(.)$ denotes the standard $Q$-function, and $\alpha_{K_{s},t}$ is the
SINR of a non-colliding user in the $t$th iteration of a slot with $K_{s}$
users, which is calculated using Theorem 1, Lemma 2, and (19) as

$\displaystyle\alpha_{K_{s},t}\approx\dfrac{s_{c_{t}}P_{c}\left(s_{p_{t}}\mu_{(K_{s}-t+1,2)}+\dfrac{\sigma_{z}^{2}}{n_{p}P_{p}}\mu_{(K_{s}-t+1,1)}\right)}{\left(P_{c}(K_{s}-t)+\sigma_{z}^{2}\right)\left(s_{p_{t}}\mu_{(K_{s}-t+1,1)}+\dfrac{M\sigma_{z}^{2}}{n_{p}P_{p}}\right)},$
(23)

where $s_{p_{t}}=1-P_{p}\dfrac{t-1}{E_{x}}$ and
$s_{c_{t}}=1-P_{c}\dfrac{t-1}{E_{x}}$. Note that since the powers of signal
and interference-plus-noise terms of $\hat{\mathbf{v}}_{i}$ are equal in their
real and imaginary parts, the SINRs of $\mathbf{f}_{i}$ in (14) and
$\hat{\mathbf{v}}_{i}$ are the same. Therefore, in (22), we employ the SINR
calculated in Theorem 1 for the input of the polar list decoder.
Since decoding in the initial iterations well represents the overall decoding
performance of the MS-MRA, we approximate the SINR of the first iteration by
setting $t=1$ in (23) as

$\displaystyle\alpha_{K_{s},1}\approx\dfrac{P_{c}M}{\left(\sigma_{z}^{2}+P_{c}K_{s}\right)\left(1+\dfrac{\sigma_{z}^{2}}{n_{p}P_{p}}\right)}.$
(24)

Concentrating on (22), we notice that $P_{K_{s},1}^{dec}$ is a decreasing
function of $n_{c}$ and $\alpha_{K_{s},1}$. Besides, (24) shows that
$\alpha_{K_{s},1}$ increases by decreasing $n_{c}$ and $J$ (considering
$K_{s}\approx K_{a}(Jn_{p}+n_{c})/n$), and increasing $M$, $P_{c}$, and
$P_{p}$, however, it is not a strictly monotonic function of $n_{p}$. Since
our goal is to achieve the lowest $P_{K_{s},t}^{dec}$ by spending the minimum
${E_{b}}/{N_{0}}=(n_{c}P_{c}+Jn_{p}P_{p})/B$, we can optimize the parameters
$n_{c}$, $n_{p}$, $P_{c}$, and $P_{p}$.

######  Theorem 2.

In the $t$th iteration of the $s$th slot, the probability of collision for a
remaining user $i\in\tilde{\mathcal{S}}_{s}$ can be approximated as

$\displaystyle P_{K_{s},t}^{col}\approx
1-\dfrac{N_{1}^{\left(t\right)}}{K_{s}-t+1},$ (25)

where $N_{i}^{(k)}$ denotes the average number of pilots that are in
$i$-collision (selected by $i$ different users) in the $k$th iteration, which
is calculated as

$\displaystyle N_{i}^{(k+1)}\approx
N_{i}^{(k)}+\begin{cases}\kappa_{k}\left((i+1)N_{i+1}^{(k)}-iN_{i}^{(k)}\right)&i\geq
2\\\
\kappa_{k}\left(2N_{2}^{(k)}-N_{1}^{(k)}\right)-\dfrac{1}{J}&i=1\end{cases},$
(26)

where $\kappa_{k}=\dfrac{J-1}{J(K_{s}-k+1)}$, and $N_{i}^{(1)}\approx
n_{p}f_{p}(i;K_{s}/n_{p})$ with $f_{p}(i;a)$ denoting the probability mass
function (PMF) of the Poisson distribution with the parameter $a$.

###### Proof.

See Appendix B. ∎

Note that to extend the result in Theorem 2 to an SIC-based system with only
one pilot sequence (orthogonal or non-orthogonal), we only need to set $J=1$
in the above expressions. From (25), the collision probability in the first
iteration can be calculated as $P_{K_{s},1}^{col}\approx 1-e^{-K_{s}/n_{p}}$,
which is a decreasing function of $n_{p}$. Since the overall decoding
performance of the system depends dramatically on the collision probability in
the first iteration, we can increase $n_{p}$, however, this results in
additional overhead.

######  Corollary 1.

Assuming a relatively large CRC length (hence negligible $p_{fa}$), the PUPE
of the MS-MRA with $S$ slots and $K_{a}$ active users can be approximated as

$\displaystyle P_{e}\approx
1-\sum_{r=1}^{K_{a}}(1-\epsilon_{r})\binom{K_{a}-1}{r-1}\left(\dfrac{1}{S}\right)^{r-1}\left(1-\dfrac{1}{S}\right)^{K_{a}-r},$
(27)

where $\epsilon_{r}$ denotes the PUPE of a slot with $r$ users, which is
obtained as

$\displaystyle\epsilon_{r}\approx\sum_{j=1}^{r}\dfrac{r-j+1}{r}p_{j,r},$ (28)

with
$p_{j,r}=(e_{j,r})^{r-j+1}\prod_{f=1}^{j-1}\left(1-(e_{f,r})^{{r}-f+1}\right)$,
and

$\displaystyle
e_{t,r}=1-P_{D}(\delta_{NP})\left(1-P_{r,t}^{dec}\right)\left(1-P_{r,t}^{col}\right),$
(29)

where $P_{r,t}^{dec}$, $P_{r,t}^{col}$, and $P_{D}(\delta_{NP})$ are computed
in (22), Theorem 2, and (9), respectively.

Note that the result in Corollary 1 can also be used in any other slotted
system with SIC by replacing appropriate $e_{j,r}$.

###  III-D MS-MRA-WOPBE

As discussed in Section III-A, in the MS-MRA scheme, the pilot bits are fed to
the polar encoder along with the data and CRC bits. To improve the performance
by decreasing the coding rate, the MS-MRA-WOPBE scheme passes only the data
and CRC bits to the encoder. To detect the bit sequences of different parts of
the message, it employs an extra iterative decoding block called iterative
inter-symbol decoder (IISD) (described in Section III-D2). At each step of
IISD, it detects one part of a user’s signal (polar or pilot part), appends
the detected part to the current pilot (which was used for channel estimation
in the previous step) to have an extended pilot, and re-estimates the channel
coefficients accordingly. The encoding and decoding procedures of MS-MRA-WOPBE
are described below.

####  III-D1 Encoder

The $i$th user encodes its bits using the following steps (the general
construction is shown in Fig. 1). Similar to the MS-MRA encoder in Section
III-A, $B$ information bits are divided into $J+1$ parts as in (3), and the
transmitted signal is generated as in (4). The only difference is in the
construction of the QPSK signal. The encoder in MS-MRA-WOPBE defines two CRC
bit sequences as $\mathbf{c}_{2}(i)=\mathbf{w}(i)\mathbf{G}_{2}$ and
$\mathbf{c}_{1}(i)=[\mathbf{w}_{c}(i),\mathbf{c}_{2}(i)]\mathbf{G}_{1}$, where
$\mathbf{G}_{2}\in\\{0,1\\}^{B\times r_{2}}$ and
$\mathbf{G}_{1}\in\\{0,1\\}^{(B_{c}+r_{2})\times r_{1}}$ are generator
matrices known by the BS and users. Then, it passes
$[\mathbf{w}_{c}(i),\mathbf{c}_{2}(i),\mathbf{c}_{1}(i)]$ to an
$\left(2n_{c},\ B_{c}+r_{1}+r_{2}\right)$ polar encoder, and modulates the
output by QPSK to obtain $\mathbf{v}_{i}\in\\{\sqrt{{P_{c}}/{2}}(\pm 1\pm
j)\\}^{1\times n_{c}}$.

####  III-D2 Decoder

As shown in Algorithm 1, MS-MRA-WOPBE exploits the same decoding steps as the
MS-MRA scheme, except for the IISD step. We can see in Algorithm 1 that the
$j$th pilot of the $i$th user is detected before employing the IISD. Then,
IISD must detect the data (polar) sequence and the $f$th pilot of the $i$th
user, where $f=1,...,J,f\neq j$. In the following, IISD is described in
detail.
Step 1 [Detecting $\mathbf{w}_{c}(i)\forall i\in\hat{\mathcal{D}}_{j}$]: We
first obtain $\mathbf{g}_{i}$ using (13), where
$\hat{\mathbf{v}}_{i}=\hat{\mathbf{h}}_{i}^{H}\mathbf{R}_{h}^{-1}\mathbf{Y^{\prime}}_{c}$,
and
$\mathbf{R}_{h}=\sigma_{z}^{2}\mathbf{I}_{M}+P_{c}\sum_{l\in\hat{\mathcal{D}}_{j}}\hat{\mathbf{h}}_{l}\hat{\mathbf{h}}_{l}^{H}$.
Then, we pass
$\mathbf{f}_{i}=\dfrac{2\sqrt{2P_{c}}}{1-P_{c}\hat{\mathbf{h}}_{i}^{H}\mathbf{R}_{h}^{-1}\hat{\mathbf{h}}_{i}}\mathbf{g}_{i}$
to the list decoder. A CRC check $\mathrm{flag_{CRC1}}(i)\in\\{0,1\\}$ and an
estimate of $[\mathbf{w}_{c}(i),\mathbf{c}_{2}(i),\mathbf{c}_{1}(i)]$ 111In
the output of the polar list decoder, there is a list of possible messages. If
more than one messages satisfy the CRC check
($\mathbf{c}_{1}(i)=[\mathbf{w}_{c}(i),\mathbf{c}_{2}(i)]\mathbf{G}_{1}$), the
most likely of them is returned as the detected message and the CRC flag is
set to one. Otherwise, the most likely message is returned as the detected
message and the CRC flag is set to zero. are obtained by the polar list
decoder.
Step 2 [Updating $\hat{\mathbf{h}}_{i}$]: Since the $j$th pilot and polar
codeword of the $i$th user are detected so far, we append them to construct a
longer signal as
$\mathbf{q}_{i}=[\mathbf{b}_{ji},\mathbf{v}_{i}]\in\mathbb{C}^{1\times(n_{p}+n_{c})}$.
Then, we update $\hat{\mathbf{h}}_{i}$ by MMSE estimation as
$\hat{\mathbf{h}}_{i}=\mathbf{Y^{\prime}}_{q}\mathbf{R}_{q}^{-1}\mathbf{q}_{i}^{H}$,
where
$\mathbf{R}_{q}=\sigma_{z}^{2}\mathbf{I}_{(n_{p}+n_{c})}+\sum_{l\in\hat{\mathcal{D}}_{j}}\mathbf{q}_{l}^{H}\mathbf{q}_{l}$,
and
$\mathbf{Y^{\prime}}_{q}=[\mathbf{Y^{\prime}}_{p_{j}},\mathbf{Y^{\prime}}_{c}]$.
Step 3 [Detecting $\mathbf{w}_{p_{f}}(i)\forall
i\in\hat{\mathcal{D}}_{j},f\neq j$]: Assuming that the $t$th row of the
Hadamard matrix is active in the $f$th pilot part ($f\neq j$), we estimate the
corresponding channel coefficient as
$\mathbf{s}_{ft}=\dfrac{1}{n_{p}\sqrt{P_{p}}}\mathbf{Y^{\prime}}_{p_{f}}\tilde{\mathbf{b}}_{ft}^{T}$
(see (10)). To find the $f$th pilot sequence of the $i$th user, we find the
pilot whose corresponding channel coefficient vector is most similar to
$\hat{\mathbf{h}}_{i}$, i.e., we maximize the correlation between
$\hat{\mathbf{h}}_{i}$ and $\mathbf{s}_{ft}$ as

$\displaystyle\hat{t}_{fi}=\max_{t}{\dfrac{|\hat{\mathbf{h}}_{i}^{H}\mathbf{s}_{ft}|^{2}}{\mathbf{s}_{ft}^{H}\mathbf{s}_{ft}}},f=1,...,J,f\neq
j.$ (30)

Step4 [Updating $\hat{\mathbf{h}}_{i}$]: Since the bit sequences of all $J+1$
parts are detected, we can construct $\mathbf{x}_{i}$ using (4). The channel
coefficient vector can be updated by MMSE as
$\hat{\mathbf{h}}_{i}=\mathbf{Y^{\prime}}\mathbf{R}^{-1}\mathbf{x}_{i}^{H}$,
where
$\mathbf{R}=\sigma_{z}^{2}\mathbf{I}_{L}+\sum_{l\in\hat{\mathcal{D}}_{j}}\mathbf{x}_{l}^{H}\mathbf{x}_{l}$.
If the number of users that satisfy $\mathrm{flag_{CRC1}}(i)=1$ is not changed
in an iteration, the iteration is stopped, otherwise, the algorithm goes to
Step 1 for another iteration with updated $\hat{\mathbf{h}}_{i}$. Users whose
bit sequences satisfy $\mathbf{c}_{2}(i)=\mathbf{w}(i)\mathbf{G}_{2}$ and
$\mathbf{c}_{1}(i)=[\mathbf{w}_{c}(i),\mathbf{c}_{2}(i)]\mathbf{G}_{1}$ are
added to the set $\mathcal{S}_{t^{\prime}j}$ as successfully decoded users of
the current iteration.

###  III-E MSUG-MRA

Different from MS-MRA where the power of every user is the same and signals
are not interleaved, MSUG-MRA defines $G$ groups, each being assigned unique
interleaver and power pair ($\pi_{g}(.),P_{p_{g}},P_{c_{g}}$), $g=1,2,...,G$.
We assume that $\phi=\dfrac{P_{p_{g}}}{P_{c_{g}}}$ is constant in all groups,
hence each group can be identified with a unique interleaver-power pair
$(\pi_{g}(.),P_{c_{g}})$, which is known at both transmitter and receiver
sides. The details of encoding and decoding procedures as well as the power
selection strategy are explained below. Note that we assume without loss of
generality that $P_{c_{1}}<P_{c_{2}}...<P_{c_{G}}$.

####  III-E1 Encoder

The encoding is adopted as follows:

  * •

Every user randomly selects a group, e.g., with index $g$.

  * •

Each user employs $P_{c_{g}}$ and $\phi P_{c_{g}}$ as the powers of the coded
and pilot parts, with which it generates its multi-stage signal
$\mathbf{x}_{i}$ similar to MS-MRA (according to (4)).

  * •

The transmitted signal is created as
$\tilde{\mathbf{x}}_{i}=\pi_{g}(\mathbf{x}_{i})$.

####  III-E2 Decoder

In each iteration, the decoder tends to decode the messages belonging to the
users of the dominant group (the $G$th group with the highest power level).
After decoding and removing users in the $G$th group, users in the $(G-1)$st
group become the dominant ones. Using the same trend, all the groups have the
chance to be the dominant group at some point. Since users in different groups
are interleaved differently, signals of users in other groups are uncorrelated
from the signals in the dominant group. Thus, letting the $g_{0}$th group to
be dominant, we approximately model the $f$th signal in the the $g$th group
($g\neq g_{0}$) as

$\displaystyle\tilde{\mathbf{x}}_{f}\sim\mathcal{CN}(\mathbf{0},\zeta
P_{c_{g}}\mathbf{I}_{L}),$ (31)

where $\zeta=\dfrac{J\phi n_{p}+n_{c}}{L}$. Therefore, when the $g_{0}$th
group is dominant (the users in the groups with indices greater than $g_{0}$
are already removed using SIC), users in the $g_{0}$th group are perturbed by
i.i.d. noise samples drawn from $\mathcal{CN}(0,\delta_{g_{0}})$, with
$\delta_{g_{0}}\approx\zeta
K_{0}\sum_{g=1}^{g_{0}-1}P_{c_{g}}+\sigma_{z}^{2}$, where
$K_{0}=\dfrac{K_{a}}{SG}$ is the average number of users in each group of the
current slot. Consequently, by replacing $\sigma_{z}^{2}$, $P_{p}$, and
$P_{c}$ with $\delta_{g_{0}}$, $\phi P_{c_{g_{0}}}$, and $P_{c_{g_{0}}}$ in
the decoding steps of MS-MRA (in Section III-B), the decoding procedure of
MSUG-MRA is obtained as:

  * •

Deinterleave the rows of the received
signals:$\mathbf{\tilde{Y}^{\prime}}_{p_{j}}=\pi_{g_{0}}^{-1}(\mathbf{Y^{\prime}}_{p_{j}})$
and
$\mathbf{\tilde{Y}^{\prime}}_{c}=\pi_{g_{0}}^{-1}(\mathbf{Y^{\prime}}_{c})$.

  * •

Find active pilots as

$\hat{\mathcal{D}}_{j}=\left\\{l:\mathbf{\tilde{u}}_{jl}^{H}\mathbf{\tilde{u}}_{jl}\geq
0.5\delta_{g_{0}}\Gamma^{-1}_{2M}(1-\gamma)\right\\},$

where
$\mathbf{\tilde{u}}_{ji}:=\mathbf{\tilde{Y}^{\prime}}_{p_{j}}\mathbf{\bar{b}}_{i}^{H}/\sqrt{n_{p}}$.

  * •

Channel estimation and MRC:
$\hat{\mathbf{v}}_{i}=\hat{\mathbf{h}}_{i}^{H}\mathbf{\tilde{Y}^{\prime}}_{c}$,
where $\hat{\mathbf{h}}_{i}=\dfrac{1}{n_{p}\sqrt{\phi
P_{c_{g_{0}}}}}\mathbf{\tilde{Y}^{\prime}}_{p_{j}}\tilde{\mathbf{b}}_{jk}^{T}$,
and $\tilde{\mathbf{b}}_{jk}$ is one of the detected pilots.

  * •

Pass
$\mathbf{f}_{i}=\dfrac{2\sqrt{2P_{c_{g_{0}}}}\|\hat{\mathbf{h}}_{i}\|^{2}}{\hat{\sigma}_{oi}^{2}}\mathbf{g}_{i}$
to the polar decoder, where
$\hat{\sigma}_{oi}^{2}=P_{c_{g_{0}}}\sum_{k\in\hat{\mathcal{D}}_{j},k\neq
i}|\hat{\mathbf{h}}_{i}^{H}\hat{\mathbf{h}}_{k}|^{2}+\delta_{g_{0}}\|\hat{\mathbf{h}}_{i}\|^{2}$,
and $\mathbf{g}_{i}$ is defined in (13).

  * •

Regenerate signals of successfully decoded users according to Section III-E1
(using ($\pi_{g_{0}}(.)$,$P_{c_{g_{0}}}$) pair), and collect them in the rows
of $\mathbf{\tilde{X}}_{\mathcal{S}_{s}}$.

  * •

Apply LS-based SIC similar to (17), i.e.,
$\mathbf{Y}^{\prime}=\mathbf{Y}(\mathbf{I}_{L}-\mathbf{\tilde{X}}_{\mathcal{S}_{s}}^{H}(\mathbf{\tilde{X}}_{\mathcal{S}_{s}}\mathbf{\tilde{X}}_{\mathcal{S}_{s}}^{H})^{-1}\mathbf{\tilde{X}}_{\mathcal{S}_{s}})$.

Note that this loop is repeated for $G$ different group indices and $J$
different pilot parts, and the iteration is stopped if there is no
successfully decoded users in $GJ$ consecutive iterations.

####  III-E3 Power Calculation

When MSUG-MRA starts the decoding in the $g_{0}$th group, there are
$|\mathcal{S}_{s}|\approx K_{0}(G-g_{0})$ successfully decoded users from
previous groups (with higher power levels), $|\tilde{\mathcal{S}}_{s}|=K_{0}$
users remain in the $g_{0}$th group, and users in the current group are
perturbed with a complex Gaussian noise with covariance matrix
$\delta_{g_{0}}\mathbf{I}_{M}$. Therefore, the SINR of a non-colliding user in
the current group can be calculated by replacing
$|\tilde{\mathcal{S}}_{s}|\approx K_{0}$, $|\mathcal{S}_{s}|=K_{0}(G-g_{0})$,
$\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}=M$,
$\mathbb{E}\\{\|\mathbf{h}_{i}\|^{4}\\}=M^{2}$, $P_{c}=P_{c_{g_{0}}}$,
$P_{p}=\phi P_{c_{g_{0}}}$, $\sigma_{z}^{2}\approx\delta_{g_{0}}$, and
$\omega_{p_{s}}=\omega_{c_{s}}=1-\dfrac{|\mathcal{S}_{s}|}{L}$ in (18) as

$\displaystyle\beta_{g_{0}}^{\prime}\approx\dfrac{\rho_{g_{0}}MP_{c_{g_{0}}}^{2}+\dfrac{\delta_{g_{0}}}{n_{p}\phi}P_{c_{g_{0}}}}{\left(P_{c_{g_{0}}}(K_{0}-1)+\delta_{g_{0}}\right)\left(P_{c_{g_{0}}}+\dfrac{\delta_{g_{0}}}{\rho_{g_{0}}n_{p}\phi}\right)},$
(32)

where $\rho_{g_{0}}=1-\dfrac{K_{0}(G-g_{0})}{L}$. To impose similar
performance on different groups, we set
$\beta_{1}^{\prime}=\beta_{2}^{\prime}=\ldots=\beta_{G}^{\prime}$. Solving
this equation, the power of the $g$th group satisfies
$c_{1}P_{g}^{2}+c_{2}P_{g}+c_{3}=0$, where
$c_{1}=(K_{0}-1)-\dfrac{\rho_{g}M}{\beta_{g-1}^{\prime}}$,
$c_{2}=\delta_{g}\left(1+\dfrac{(K_{0}-1)}{\phi n_{p}\rho_{g}}-\dfrac{1}{\phi
n_{p}\beta_{g-1}^{\prime}}\right)$, $c_{3}=\dfrac{\delta_{g}^{2}}{\phi
n_{p}\rho_{g}}$. Solving this equation, we have

$\displaystyle P_{t}=\dfrac{-c_{2}+\sqrt{c_{2}^{2}-4c_{1}c_{3}}}{2c_{1}},$
(33) $\displaystyle\mathrm{s.t.\ }\dfrac{1}{G}\sum_{f=1}^{G}P_{f}=P\
\mathrm{and}\ \ P_{t}\in\mathbb{R}^{+}.$

Note that the MS-MRA scheme is a special case of the MSUG-MRA with $G=1$.

###  III-F MS-SRA and MSUG-SRA

In this part, we apply the proposed MIMO coding schemes to the case of a
single receive antenna. To accomplish this, we repeat each user’s length-$L$
signal multiple times to create temporal diversity in MS-SRA and MSUG-SRA.
Accordingly, we divide the whole frame into $V$ sub-frames of length
$n^{\prime}=n/V$, then divide each sub-frame into $S$ slots of length
$L=n^{\prime}/S$. Each user randomly selects a slot index, namely $s$, and
transmits its signal, through the $s$th slot of each sub-frame. Assuming the
coherence time to be $L$, each sub-frame is analogous to a receive antenna.
Therefore, the transmitted messages in MS-SRA and MSUG-SRA can be decoded
using MS-MRA and MSUG-MRA decoders in Sections III-B and III-E2, respectively,
considering $V$ receive antennas. Since each user repeats its signal $V$
times, for this case, we have ${E_{b}}/{N_{0}}=\dfrac{VLP}{\sigma_{z}^{2}B}$.

###  III-G Computational Complexity

We focus on the number of multiplications as a measure of the computational
complexity, and make a complexity comparison among the proposed and existing
URA solutions. The per-iteration computational complexity of the MS-MRA in a
slot is calculated as follows: The pilot detection in (8) has a complexity of
$\mathcal{O}(n_{p}^{2}MJS)$ corresponding to $J$ different pilot parts and $S$
different slots, where $\mathcal{O}(.)$ is the standard big-O notation,
denoting the order of complexity. The channel estimator in (10) does not
require any extra computation, because $\hat{\mathbf{h}}_{i}$ corresponds to
$\mathbf{u}_{ji}$ which is calculated before for pilot detection; the MRC in
(11) has a complexity of
$\mathcal{O}(\sum_{j=1}^{J}|\mathcal{D}_{j}|Mn_{c}S)$; to compute the LLR in
(14), the required computational complexity is
$\mathcal{O}(\sum_{j=1}^{J}|\mathcal{D}_{j}|^{2}MS)$; the computational
complexity of the polar list decoder is [26]
$\mathcal{O}(\sum_{j=1}^{J}|\mathcal{D}_{j}|n_{c}\log n_{c}S)$; and, the SIC
has a complexity of
$\mathcal{O}(ML|\mathcal{S}_{s}|S+|\mathcal{S}_{s}|^{2}LS)$. We know from (44)
that in the first iteration, we have $|\mathcal{D}_{j}|\approx
n_{p}-n_{p}e^{{-K_{a}}/{(n_{p}S)}}<{K_{a}}/{S}$, and $|\mathcal{S}_{s}|=0$; in
the last iterations, we have $|\mathcal{S}_{s}|\approx K_{a}/S$ and
$|\mathcal{D}_{j}|\approx 0$. Hence, considering $M\gg\log n_{c}$ and
$n_{c}|\mathcal{D}_{j}|\gg n_{p}$, we can compute the computational complexity
of the MS-MRA in the first and last iterations as
$\mathcal{O}\left(K_{a}MJ(n_{c}+K_{a}/S)\right)$ and
$\mathcal{O}\left(LK_{a}(M+K_{a}/S)\right)$, respectively. Considering the
computational complexity in the intermediate iterations to be in the same
order, the per-iteration computational complexity of the MS-MRA can be bounded
by
$\mathcal{O}\Bigl{(}n_{p}^{2}MJS+\max\left(K_{a}MJ(n_{c}+K_{a}/S),LK_{a}(M+K_{a}/S)\right)\Bigl{)}$.
Note that the computational complexity of MSUG-MRA is in the same order as MS-
MRA, and for MS-SRA and MSUG-SRA schemes, the computational complexity is
obtained by replacing $M$ by $V$ in the above figures. Note that by employing
a low-complexity adaptive filter [28, 29, 30], we can considerably reduce the
computational complexity of the LS-based channel estimator in (16) (hence the
total computational complexity of the proposed schemes).

Looking at Algorithm 1, we can infer that MS-MRA-WOPBE is obtained by
employing the same pilot detector (with complexity
$\mathcal{O}(n_{p}^{2}MJS)$), channel estimator (does not incur any extra
computational complexity), and SIC (with complexity
$\mathcal{O}(ML|\mathcal{S}_{s}|S+|\mathcal{S}_{s}|^{2}LS)$) as in the MS-MRA
case, except for employing the IISD block. In Step 1 of IISD, the complexity
for computing $\mathbf{f}_{i}$ and implementing polar decoder are
$\mathcal{O}((Mn_{c}+M^{2})T_{I}S\sum_{j=1}^{J}|\mathcal{D}_{j}|)$ and
$\mathcal{O}(T_{I}n_{c}\log n_{c}S\sum_{j=1}^{J}|\mathcal{D}_{j}|)$,
respectively, where $T_{I}$ denotes the number of iterations of IISD. In the
Step 2 of IISD, computing $\hat{\mathbf{h}}_{i}$ and $e_{k}$ has the
complexity of
$\mathcal{O}(T_{I}(n_{c}+n_{p})^{2}S\sum_{j=1}^{J}|\mathcal{D}_{j}|+T_{I}(n_{c}+n_{p})MS\sum_{j=1}^{J}|\mathcal{D}_{j}|)$
and $\mathcal{O}(T_{I}(J-1)n_{p}MS\sum_{j=1}^{J}|\mathcal{D}_{j}|)$,
respectively. The computational complexity of obtaining ${\hat{\bf h}}_{i}$ in
Step 3 of IISD is
$\mathcal{O}(T_{I}(L^{2}+LM)S\sum_{j=1}^{J}|\mathcal{D}_{j}|)$. Then,
replacing $|\mathcal{D}_{j}|$ and $|\mathcal{S}_{s}|$ with their approximate
values (discussed in the previous paragraph), the overall computational
complexity of the MS-MRA-WOPBE is bounded by
$\mathcal{O}\Bigl{(}n_{p}^{2}MJS+\max\left(\left((L^{2}+M^{2})+ML\right)T_{I}JK_{a},LK_{a}(M+K_{a}/S)\right)\Bigl{)}$.

For comparison purposes, the dominant per-iteration computational complexity
of the FASURA in [24] (which is due to energy detector and SIC operation) can
also be computed as
$\mathcal{O}\left(M(n_{p}+L^{\prime}n_{c})2^{B_{f}}+K_{a}(nM+n^{2})\right)$,
where $B_{f}$ denotes the number of pilot bits, $n$ is the frame length, and
$L^{\prime}$ is the length of the spreading sequence.

##  IV Numerical Results

We provide a set of numerical results to assess the performance of the
proposed URA set-ups. In all the results, we set $B=100$, the number of CRC
bits $r=11$, the Neyman-Pearson threshold $\gamma=0.1$, and the list size of
the decoder to $64$. For MS-MRA and MSUG-MRA, we set the frame length
$n\approx 3200$, and $P_{e}=0.05$. The corresponding values for the MS-SRA and
MSUG-SRA are $n\approx 30000$, and $P_{e}=0.1$.
In Fig. 3, the performance of the proposed MS-MRA and MSUG-MRA is compared
with the short blocklength scheme of [22] with the number of antennas $M=100$
and slot length $L=200$. (In this scenario, we consider a fast-fading
environment, where the coherence blocklength is considered as $L_{c}=200$). To
facilitate a fair comparison, we consider $(J,n_{p},n_{c})=(2,32,128)$
($L=192$) and $P_{p}/P_{c}=1$ ($\phi=1$ for MSUG-MRA) for all the proposed
schemes. For MSUG-MRA, the value of $G$ is set as $G=1$ for $K_{a}\leq 400$,
$G=3$ for $K_{a}=500$, $G=6$ for $600\leq K_{a}\leq 800$, $G=8$ for $900\leq
K_{a}\leq 1000$, and $G=10$ for $K_{a}>1000$. The superiority of the proposed
schemes over the one in [22] is mostly due to the more powerful performance of
the polar code compared to the simple coding scheme adopted in [22] and the
use of the SIC block, which significantly diminishes the effect of
interference. We also observe that MS-MRA-WOPBE outperforms MS-MRA, which is
due to 1) employing IISD, which iteratively improves the accuracy of the
channel estimation, and 2) lower coding rate by not encoding the pilot bits.
Besides, the range of the number of active users that are detected by the
MSUG-MRA is higher than those of MS-MRA and MS-MRA-WOPBE schemes. This
improvement results from randomly dividing users into different groups, which
provides each group with a lower number of active users (hence a lower
effective interference level).

Figure 3: The required $E_{b}/N_{0}$ in the proposed MIMO set-ups and the
scheme in [22] for $L\approx 200$, $M=100$, and $P_{e}=0.05$.

In Fig. 4, we compare the proposed MS-MRA and MSUG-MRA with the ones in [23,
24], considering the slow-fading channel with coherence blocklength
$L_{c}=3200$. We set $(J,n_{p},n_{c})=(2,256,512)$, $M=50$, $P_{p}/P_{c}=0.66$
for MS-MRA. We choose $(J,n_{p},n_{c},G)=(2,256,512,1)$ for $K_{a}\leq 700$,
$(J,n_{p},n_{c},G)=(2,64,512,6)$ for $K_{a}=900$, and
$(J,n_{p},n_{c},G)=(2,64,512,18)$ for $K_{a}>900$ with $\phi=0.66$. Thanks to
employing the slotted structure, SIC, and orthogonal pilots, all the proposed
schemes have superior performance compared to [23]. Due to employing random
spreading and an efficient block called NOPICE, FASURA in [24] performs better
than the proposed MS-MRA and MSUG-MRA in the low $K_{a}$ regimes; however, its
performance is worse than the MSUG-MRA in higher values of $K_{a}$ (thanks to
the random user grouping employed in MSUG-MRA). The proposed MS-MRA-WOPBE also
shows a similar performance as FASURA. To achieve the result in Fig. 4, FASURA
sets $n_{p}=896$, $L^{\prime}=9$, $n_{c}=256$, $n=3200$, $B_{f}=16$, and
$M=50$. The order of computational complexity for these schemes is given in
the performance-complexity plot in Fig. 5. It can be interpreted from this
figure that the proposed MS-MRA-WOPBE has comparable accuracy to FASURA while
offering a lower computational complexity. Note also that despite the higher
required $E_{b}/N_{0}$ compared to FASURA, MS-MRA offers very large savings in
terms of computational complexity, which is attributed to employing orthogonal
pilots, slotted structure, and simpler decoding blocks.

Figure 4: The required $E_{b}/N_{0}$ in the proposed MIMO set-ups and the
results in [23, 24] for $M=50$. Figure 5: Performance-complexity curve for the
proposed MIMO schemes and FASURA in [24].

As a further note, FASURA considers $2^{B_{p}}$ possible spreading sequences
of length $L^{\prime}$ for each symbol of the polar codeword; hence every
transceiver should store $n_{c}2^{B_{p}}$ vectors of length $L^{\prime}$, as
well as a pilot codebook of size $2^{J}\times n_{p}$. For typical values
reported in [24], the BS and every user must store $1.6\times 10^{7}$ vectors
of length $9$ and a matrix of size $5.8\times 10^{7}$. For the proposed
schemes in this paper, every transceiver must store only an orthogonal
codebook of size $n_{p}\times n_{p}$, where $n_{p}=256$. Thus, FASURA requires
about 3000 times larger memory than our proposed schemes, which may be
restrictive for some target URA applications such as sensor networks, where a
massive number of cheap sensors are deployed. Moreover, unlike FASURA, the
proposed solutions are implementable with short blocklengths (see Fig. 3),
which makes them appropriate for fast fading scenarios as well.

In Fig. 6, we compare the theoretical PUPE in (27) with the simulation results
of the MS-MRA for three different scenarios ($M=50,100,200$) with
$P_{p}/P_{c}=0.66$ and $(J,n_{p},n_{c})=(2,256,512)$. It is shown that the
approximate theoretical analysis well predicts the performance of the MS-MRA
for $K_{a}\leq 700$, however, the results are not consistent for higher values
of $K_{a}$. The reason for the mismatch for the $K_{a}>800$ regime is the
approximations employed while analyzing SIC in Lemma 4 (e.g., $n_{c},n_{p}\gg
1$, $|\mathcal{S}_{s}|\gg 1$, uncorrelated QPSK codewords of two different
users, and uncorrelated samples of $\mathbf{x}_{i}$).

Figure 6: Comparison of the simulation and analytical performance of the MS-
MRA for different values of $M$.

Fig. 7 compares the MS-SRA and MSUG-SRA with the existing single-antenna
solutions [17, 20, 19]. For both set-ups, we set $(J,n_{p},n_{c})=(2,64,512)$,
$P_{p}/P_{c}=1$ ($\phi=1$ for MSUG-SRA), $(S,V)=(6,8)$ for $K_{a}\leq 200$,
and $(S,V)=(12,4)$ for $K_{a}\geq 300$. For MSUG-SRA, we also choose $G=1$ for
$K_{a}\leq 300$, $G=3$ for $500\leq K_{a}\leq 700$, and $G=6$ for $K_{a}\geq
900$. It is observed that the proposed MS-SRA has a superior performance
compared to the existing URA approaches for the low number of active users,
However, it performs worse than the scheme in [19] for higher values of
$K_{a}$. Furthermore, the proposed MSUG-SRA outperforms existing solutions,
and its effective range of $K_{a}$ is up to $1500$ users.

##  V Conclusions

We propose a family of unsourced random access solutions for MIMO Rayleigh
block fading channels. The proposed approaches employ a slotted structure with
multiple stages of orthogonal pilots. The use of a slotted structure along
with the orthogonal pilots leads to the lower computational complexity at the
receiver, and also makes the proposed designs implementable for fast fading
scenarios. We further improve the performance of the proposed solutions when
the number of active users is very large by randomly dividing the users into
different interleaver-power groups. The results show that the proposed MIMO
URA designs are superior for both short and large blocklengths, while offering
a lower computational complexity.

##  Appendix A Proof of Theorem 1

######  Lemma 3.

Assuming that the transmitted data part contains uncorrelated and equally
likely QPSK symbols, for $i,j\in\mathcal{S}_{s}$ and $n_{p},n_{c}\to\infty$,
the transmitted signals satisfy

$\displaystyle\dfrac{1}{E_{x}}\mathbf{x}_{i}\mathbf{x}_{j}^{H}$
$\displaystyle\overset{p}{\to}0,$ (34)

where $E_{x}=Jn_{p}P_{p}+n_{c}P_{c}$.

###### Proof.

Let $\mathbf{b}_{ji}$ and $\mathbf{b}_{jr}$ be the $j$th pilots of the $i$th
and $r$th users, and $\mathbf{v}_{i}$ and $\mathbf{v}_{r}$ be the
corresponding polar-coded and QPSK-modulated signals. Since $\mathbf{b}_{ji}$
and $\mathbf{b}_{jr}$ are randomly chosen rows of the Hadamard matrix,
$\mathbf{b}_{ji}\mathbf{b}_{ji}^{T}=n_{p}$ with probability
$\dfrac{1}{n_{p}}$, and it is zero with probability $1-\dfrac{1}{n_{p}}$.
Besides, for $n_{c}\to\infty$, $v_{it}$ and $v_{rt}$ are zero-mean and
uncorrelated, where $v_{it}=[\mathbf{v}_{i}]_{(:,t)}$. Therefore,

$\displaystyle\lim_{n_{p},n_{c}\to\infty}$
$\displaystyle\mathbb{P}\left({\dfrac{1}{E_{x}}}|\mathbf{x}_{r}\mathbf{x}_{i}^{H}|>0\right)$
$\displaystyle=$
$\displaystyle\lim_{n_{p},n_{c}\to\infty}\mathbb{P}\left({\dfrac{1}{E_{x}}}\left|P_{p}\sum_{j=1}^{J}\mathbf{b}_{jr}\mathbf{b}_{ji}^{H}+\mathbf{v}_{r}\mathbf{v}_{i}^{H}\right|>0\right)$
$\displaystyle\leq$
$\displaystyle\lim_{n_{p},n_{c}\to\infty}\mathbb{P}\left(\dfrac{P_{p}}{{E_{x}}}\sum_{j=1}^{J}\left|\mathbf{b}_{jr}\mathbf{b}_{ji}^{H}\right|+{\dfrac{1}{E_{x}}}\left|\mathbf{v}_{r}\mathbf{v}_{i}^{H}\right|>0\right)$
$\displaystyle\leq$
$\displaystyle\lim_{n_{p},n_{c}\to\infty}\sum_{j=1}^{J}\mathbb{P}\left(\dfrac{P_{p}}{{E_{x}}}\left|\mathbf{b}_{jr}\mathbf{b}_{ji}^{H}\right|>0\right)$
$\displaystyle\indent\indent\indent\indent\indent\indent\indent\indent\indent+\mathbb{P}\left({\dfrac{1}{E_{x}}}\left|\sum_{t=1}^{n_{c}}v_{rt}v_{it}^{H}\right|>0\right)$
$\displaystyle\approx$
$\displaystyle\lim_{n_{p},n_{c}\to\infty}\dfrac{JP_{p}}{n_{p}{E_{x}}}+\mathbb{P}\left(\dfrac{n_{c}}{{E_{x}}}\left|\mathbb{E}\left\\{v_{rt}v_{it}^{H}\right\\}\right|>0\right)$
$\displaystyle\approx$ $\displaystyle\ 0.$

Note that, strictly speaking, the uncorrelated QPSK symbol assumption is not
accurate for coded systems. Nevertheless, it is useful to obtain a good
approximation of SINR, as we will show later. ∎

Figure 7: The required $E_{b}/N_{0}$ of the proposed MS-SRA and MSUG-SRA for
the case of single-antenna receiver.

######  Lemma 4.

By applying LS-based SIC, the residual received signal matrices of pilot and
coded parts can be written based on the signal and interference-plus-noise
terms as

$\displaystyle\mathbf{Y}^{\prime}_{p_{j}}\approx$
$\displaystyle\sqrt{P_{p}}\mathbf{h}_{i}\mathbf{b}_{ji}\mathbf{L}_{p_{j}}+\sqrt{P_{p}}\sum_{k\in\tilde{\mathcal{S}}_{s},k\neq
i}\mathbf{h}_{k}\mathbf{b}_{jk}\mathbf{L}_{p_{j}}+\mathbf{Z}_{n,p_{j}},$ (35)
$\displaystyle\mathbf{Y}^{\prime}_{c}\approx\mathbf{h}_{i}\mathbf{v}_{i}\mathbf{L}_{c}+\sum_{k\in\tilde{\mathcal{S}}_{s},k\neq
i}\mathbf{h}_{k}\mathbf{v}_{k}\mathbf{L}_{c}+\mathbf{Z}_{n,c},$ (36)

where $\mathbf{h}_{i}\in\mathbb{C}^{M\times 1}$ is the channel coefficient
vector of the $i$th user,
$\mathbf{L}_{p_{j}}=\omega_{p_{s}}\mathbf{I}_{n_{p}}$,
$\mathbf{L}_{c}=\omega_{c_{s}}\mathbf{I}_{n_{c}}$, and the elements of
$\mathbf{Z}_{n,p_{j}}$ and $\mathbf{Z}_{n,c}$ are drawn from
$\mathcal{CN}\left(0,\omega_{c_{s}}\sigma_{z}^{2}\right)$ and
$\mathcal{CN}\left(0,\omega_{p_{s}}\sigma_{z}^{2}\right)$, respectively, with
$\omega_{p_{s}}$ and $\omega_{c_{s}}$ are as defined in the statement of the
Theorem 1.

###### Proof.

Plugging (15) and (16) into (17), we obtain

$\displaystyle\mathbf{Y^{\prime}}$
$\displaystyle=\mathbf{H}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}\mathbf{L}+\mathbf{H}_{\tilde{\mathcal{S}}_{s}}\mathbf{X}_{\tilde{\mathcal{S}}_{s}}\mathbf{L}+\mathbf{Z}_{s}\mathbf{L}$
$\displaystyle=\mathbf{H}_{\tilde{\mathcal{S}}_{s}}\mathbf{X}_{\tilde{\mathcal{S}}_{s}}\mathbf{L}+\mathbf{Z}_{s}\mathbf{L}$
$\displaystyle=\mathbf{h}_{i}\mathbf{x}_{i}\mathbf{L}+\sum_{k\in\tilde{\mathcal{S}}_{s},k\neq
i}\mathbf{h}_{k}\mathbf{x}_{k}\mathbf{L}+\mathbf{Z}_{n},$ (37)

where
$\mathbf{L}=\mathbf{I}_{L}-\mathbf{X}_{\mathcal{S}_{s}}^{H}(\mathbf{X}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}^{H})^{-1}\mathbf{X}_{\mathcal{S}_{s}}$,
and $\mathbf{Z}_{n}=\mathbf{Z}_{s}\mathbf{L}$. Since
$\mathbf{L}^{H}\mathbf{L}=\mathbf{L}$ and
$\mathbf{Z}_{s}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{z}^{2}\mathbf{I}_{L}\right)$,
we have

$\displaystyle\mathbf{Z}_{n}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{z}^{2}\mathbb{E}\\{\mathbf{L}\\}\right).$
(38)

Since the values of $n_{p}$ and $n_{c}$ are large, and using (34), we have
$\dfrac{1}{{E_{x}}}\mathbf{X}_{\mathcal{S}_{s}}\mathbf{X}_{\mathcal{S}_{s}}^{H}\approx\mathbf{I}_{|\mathcal{S}_{s}|}$,
where $E_{x}=Jn_{p}P_{p}+n_{c}P_{c}$. In other words, we can approximate
$\mathbf{L}$ as

$\displaystyle\mathbf{L}\approx\mathbf{I}_{L}-\dfrac{1}{E_{x}}\sum_{r\in\mathcal{S}_{s}}\mathbf{x}_{r}^{H}\mathbf{x}_{r}.$
(39)

Using the weak law of large numbers, and assuming samples of $\mathbf{x}_{r}$
to be uncorrelated and $|\mathcal{S}_{l}|\gg 1$, we can rewrite $\mathbf{L}$
in (39) as

$\displaystyle\mathbf{L}\approx\begin{bmatrix}\mathbf{L}_{p_{1}}&...&\mathbf{0}&\mathbf{0}\\\
\vdots&\ddots&\vdots&\vdots\\\ \mathbf{0}&...&\mathbf{L}_{p_{J}}&\mathbf{0}\\\
\mathbf{0}&...&\mathbf{0}&\mathbf{L}_{c}\end{bmatrix},$ (40)

where $\mathbf{L}_{p_{j}}=\omega_{p_{s}}\mathbf{I}_{n_{p}}$ and
$\mathbf{L}_{c}=\omega_{c_{s}}\mathbf{I}_{n_{c}}$ with
$\omega_{p_{s}}=\omega_{c_{s}}=1-\dfrac{|\mathcal{S}_{s}|}{L}$ if the
transmitted signals are randomly interleaved, and
$\omega_{p_{s}}=1-\dfrac{1}{E_{x}}P_{p}|\mathcal{S}_{s}|$,
$\omega_{c_{s}}=1-\dfrac{1}{E_{x}}P_{c}|\mathcal{S}_{s}|$, otherwise.

Letting
$\mathbf{Z}_{n}=\left[\mathbf{Z}_{n,p_{1}},\ldots,\mathbf{Z}_{n,p_{J}},\mathbf{Z}_{n,c}\right]$,
we can infer from (38) and (40) that the elements of $\mathbf{Z}_{n,p_{j}}$
and $\mathbf{Z}_{n,c}$ approximately follow
$\mathcal{CN}\left(0,\omega_{p_{s}}\sigma_{z}^{2}\right)$ and
$\mathcal{CN}\left(0,\omega_{c_{s}}\sigma_{z}^{2}\right)$, respectively.
Besides, using (40) and the signal structure in (4), we can divide (37) into
pilot and coded parts as in (35) and (36). ∎

######  Lemma 5.

The estimated channel coefficients of a non-colliding user approximately
satisfy the following expressions:

$\displaystyle\mathbb{E}\\{\|\hat{\mathbf{h}}_{i}\|^{2}\\}$
$\displaystyle\approx\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}+\dfrac{M\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}},$
(41a)
$\displaystyle\mathbb{E}\\{|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{k}|^{2}\\}$
$\displaystyle\approx\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}+\dfrac{M\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}},$
(41b)
$\displaystyle\mathbb{E}\\{|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}|^{2}\\}$
$\displaystyle\approx\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{4}\\}+\dfrac{\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}.$
(41c)

###### Proof.

Using the approximation of $\mathbf{Y}^{\prime}_{p_{j}}$ in (35) in (10), the
channel coefficient vector of the $i$th user can be estimated as

$\displaystyle\hat{\mathbf{h}}_{i}\approx$ $\displaystyle\
\dfrac{\omega_{p_{s}}}{n_{p}}\mathbf{h}_{i}\mathbf{b}_{ji}\tilde{\mathbf{b}}_{jk}^{H}+\dfrac{\omega_{p_{s}}}{n_{p}}\sum_{f\in\tilde{\mathcal{S}}_{s},f\neq
i}\mathbf{h}_{f}\mathbf{b}_{jf}\tilde{\mathbf{b}}_{jk}^{H}+\mathbf{z}_{p_{j},n}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mua\mkern-1.5mu)}}{{=}}$
$\displaystyle\ \omega_{p_{s}}\mathbf{h}_{i}+\mathbf{z}_{p_{j},n},$ (42)

where
$\mathbf{z}_{p_{j},n}=\dfrac{1}{n_{p}\sqrt{P_{p}}}\mathbf{Z}_{n,p_{j}}\tilde{\mathbf{b}}_{jk}^{H}$,
and in (a), we use the assumption that the $i$th user is non-colliding, hence
$\tilde{\mathbf{b}}_{jk}$ is only selected by the $i$th user
($\mathbf{b}_{ji}=\tilde{\mathbf{b}}_{jk}$ and
$\mathbf{b}_{jf}\neq\tilde{\mathbf{b}}_{jk}^{H}$ for
$f\in\tilde{\mathcal{S}}_{s},f\neq i$). We can argue the following
approximation
$\mathbf{z}_{p_{j},n}\sim\mathcal{CN}\left(0,\dfrac{\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}}\right)$.
Using (42), we can show that
$\mathbb{E}\\{\|\hat{\mathbf{h}}_{i}\|^{2}\\}\approx\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}+\dfrac{M\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}}$,
$\mathbb{E}\\{|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}|^{2}\\}\approx\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{4}\\}+\dfrac{\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}$,
and
$\mathbb{E}\\{|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{k}|^{2}\\}=\mathbb{E}\\{\|\hat{\mathbf{h}}_{i}\|^{2}\\}$.
∎

Plugging (36) into the MRC expression in (11), $\hat{\mathbf{v}}_{i}$ can be
estimated as

$\displaystyle\hat{\mathbf{v}}_{i}\approx\omega_{c_{s}}\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}\mathbf{v}_{i}+\mathbf{z}_{in},$
(43)

where the first term on the right-hand side is the signal term, and
$\mathbf{z}_{in}=\sum_{k\in\tilde{\mathcal{S}}_{s},k\neq
i}\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{k}\mathbf{v}_{k}\mathbf{L}_{c}+\hat{\mathbf{h}}_{i}^{H}\mathbf{Z}_{n,c}$
is the interference-plus-noise term. Since
$\mathbf{L}^{H}\mathbf{L}=\mathbf{L}$, and using (40), we can show
$\mathbf{L}_{c}^{H}\mathbf{L}_{c}\approx\mathbf{L}_{c}$. Therefore, by
employing Lemma 5, we can approximate
$\mathbf{z}_{in}\sim\mathcal{CN}(\mathbf{0},\sigma_{in}^{2}\mathbf{I}_{n_{c}})$,
where

$\displaystyle\sigma_{in}^{2}=\omega_{c_{s}}\left(P_{c}(|\tilde{\mathcal{S}}_{s}|-1)+\sigma_{z}^{2}\right)\left(\omega_{p_{s}}^{2}\mathbb{E}\\{\|\mathbf{h}_{i}\|^{2}\\}+\dfrac{M\omega_{p_{s}}\sigma_{z}^{2}}{n_{p}P_{p}}\right).$

Besides, the per-symbol power of the signal term can be obtained as
$\sigma_{s}^{2}\approx\omega_{c_{s}}^{2}\mathbb{E}\\{|\hat{\mathbf{h}}_{i}^{H}\mathbf{h}_{i}|^{2}\\}P_{c}$.
Then, using Lemma 5, the SINR of $\hat{\mathbf{v}}_{i}$ can be calculated as
in (18).

##  Appendix B Proof of Theorem 2

In the first iteration of the $s$th slot, since $K_{s}$ users have selected
one out of $n_{p}$ pilots randomly, the number of users that select an
arbitrary pilot approximately follows a Poisson distribution with the
parameter $K_{s}/n_{p}$. In the $k$th iteration of the $s$th slot, let
$T_{j,i}^{(k)}$ be the average number of $i$-collision pilots (pilots selected
by $i$ different users) in the $j$th pilot part. We have

$\displaystyle T_{j,i}^{(1)}\approx n_{p}f_{p}(i;K_{s}/n_{p}),$ (44)

where $f_{p}(i;a)$ denotes the PMF of the Poisson distribution with the
parameter $a$. The average number of i-collision users in the $k$th iteration
of the $j$th pilot part is then calculated as $K_{j,i}^{(k)}\approx
iT_{j,i}^{(k)}$. Supposing that in the $k$th iteration (using the assumption
in (19)), the decoder employs the $j$th pilot part for channel estimation, the
removed user is non-colliding (1-collision) in its $j$th pilot part (we assume
that the decoder can only decode the non-colliding users), and it is in
$i$-collision in its $j^{\prime}$th ($j^{\prime}\neq j$) pilot part with
probability
$p_{i,j^{\prime}}^{(k)}=\dfrac{K_{j^{\prime},i}^{(k)}}{K_{s}-k+1}$. Therefore,
removing a user from the $j$th pilot part results in

  * •

In the $j$th pilot part, we have $T_{j,1}^{(k+1)}=T_{j,1}^{(k)}-1$, and
$T_{j,i}^{(k+1)}=T_{j,i}^{(k)}$ for $i>1$.

  * •

In the $j^{\prime}$th pilot part ($j^{\prime}\neq j$), we have
$T_{j^{\prime},i}^{(k+1)}=T_{j^{\prime},i}^{(k)}+p_{i+1,j^{\prime}}^{(k)}-p_{i,j^{\prime}}^{(k)}$.

The collision probability of the $j$th pilot part in the $t$th iteration is
then obtained as $P_{col}(j,t)=1-\dfrac{T_{j,1}^{(t)}}{K_{s}-t+1}$. Finally,
by approximating $T_{j,i}^{(t)}$ by its average over different pilot parts
(i.e., $T_{j,i}^{(t)}\approx
N_{i}^{(t)}:=\dfrac{1}{J}\sum_{j=1}^{J}T_{j,i}^{(t)}$) in above equations, the
results in Theorem 2 are obtained. Note that since all the pilot parts are
equally likely in the first iteration, we have $N_{i}^{(1)}\approx
T_{j,i}^{(1)}\approx n_{p}f_{p}(i;K_{s}/n_{p}),\forall j=1,...,J$.

## References

  * [1] X. Shao, L. Cheng, X. Chen, C. Huang and D. W. K. Ng, “Reconfigurable intelligent surface-Aided 6G massive access: coupled tensor modeling and sparse bayesian learning,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 12, pp. 10145–10161, Dec. 2022.
  * [2] X. Shao, X. Chen, D. W. K. Ng, C. Zhong and Z. Zhang, “Cooperative activity detection: sourced and unsourced massive random access paradigms,” _IEEE Trans. Signal Process._ , vol. 68, no. , pp. 6578–6593, Nov. 2020.
  * [3] Y. Polyanskiy, “A perspective on massive random-access,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Aachen, Germany, June 2017, pp. 2523–2527.
  * [4] O. Ordentlich and Y. Polyanskiy, “Low complexity schemes for the random access Gaussian channel,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Aachen, Germany, June 2017, pp. 2528–2532.
  * [5] G. Kasper Facenda and D. Silva, “Efficient scheduling for the massive random access Gaussian channel,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 11, pp. 7598–7609, Nov. 2020.
  * [6] A. Vem, K. R. Narayanan, J.-F. Chamberland, and J. Cheng, “A user-independent successive interference cancellation based coding scheme for the unsourced random access Gaussian channel,” _IEEE Trans. Commun._ , vol. 67, no. 12, pp. 8258–8272, Dec. 2019.
  * [7] A. Glebov, N. Matveev, K. Andreev, A. Frolov and A. Turlikov, “Achievability bounds for T-fold irregular repetition slotted ALOHA scheme in the Gaussian MAC,” in _Proc. IEEE Wireless Commun. Netw. Conf. (WCNC)_ , Marrakesh, Morocco, Apr. 2019, pp. 1–6.
  * [8] V. K. Amalladinne, A. Vem, D. K. Soma, K. R. Narayanan, and J.-F. Chamberland, “A coupled compressive sensing scheme for unsourced multiple access,” in _Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , Calgary, Canada, Sep. 2018, pp. 6628–6632.
  * [9] A. K. Tanc and T. M. Duman, “Massive random access with trellis based codes and random signatures,” _IEEE Commun. Lett._ , vol. 25, no. 5, pp. 1496–1499, May 2021.
  * [10] Z. Han, X. Yuan, C. Xu, S. Jiang and X. Wang, “Sparse Kronecker-product coding for unsourced multiple access,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 10, pp. 2274-2278, Oct. 2021.
  * [11] J. R. Ebert, V. K. Amalladinne, S. Rini, J. -F. Chamberland and K. R. Narayanan, “Stochastic binning and coded demixing for unsourced random access,” in _Proc. IEEE Workshop Signal Process. Adv. Wireless Commun. (SPAWC)_ , Lucca, Italy, Sep. 2021, pp. 351–355.
  * [12] A. K. Pradhan, V. K. Amalladinne, K. R. Narayanan, and J.-F. Chamberland, “Polar coding and random spreading for unsourced multiple access,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Dublin, Ireland, June 2020, pp. 1–6.
  * [13] M. J. Ahmadi and T. M. Duman, “Random spreading for unsourced MAC with power diversity,” in _IEEE Commun. Lett._ , vol. 25, no. 12, pp. 3995–3999, Dec. 2021.
  * [14] A. K. Pradhan, V. K. Amalladinne, K. R. Narayanan and J. -F. Chamberland, “LDPC codes with soft interference cancellation for uncoordinated unsourced multiple access,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Montreal, Canada, June 2021, pp. 1–6.
  * [15] S. S. Kowshik and Y. Polyanskiy, “Quasi-static fading MAC with many users and finite payload,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Paris, France, July 2019, pp. 440–444.
  * [16] S. S. Kowshik, K. Andreev, A. Frolov and Y. Polyanskiy, “Energy efficient coded random access for the wireless uplink,” _IEEE Trans. Commun._ , vol. 68, no. 8, pp. 4694–4708, Aug. 2020.
  * [17] S. S. Kowshik, K. Andreev, A. Frolov and Y. Polyanskiy, “Energy efficient random access for the quasi-static fading MAC,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Paris, France, July 2019, pp. 2768–2772.
  * [18] S. S. Kowshik and Y. Polyanskiy, “Fundamental limits of many-user MAC with finite payloads and fading,” _IEEE Trans. Inf. Theory_ , vol. 67, no. 9, pp. 5853–5884, Sep. 2021.
  * [19] K. Andreev, E. Marshakov and A. Frolov, “A polar code based TIN-SIC scheme for the unsourced random access in the quasi-static fading MAC,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Los Angeles, USA, June 2020, pp. 3019–3024.
  * [20] K. Andreev, S. S. Kowshik, A. Frolov and Y. Polyanskiy, “Low complexity energy efficient random access scheme for the asynchronous fading MAC,” in _Proc. IEEE Veh. Technol. Conf. (VTC)_ , Honolulu, USA, Sep. 2019, pp. 1–5.
  * [21] V. K. Amalladinne, K. R. Narayanan, J. -F. Chamberland and D. Guo, “Asynchronous neighbor discovery using coupled compressive sensing,” in _Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , Brighton, UK, May 2019, pp. 4569–4573.
  * [22] A. Fengler, S. Haghighatshoar, P. Jung and G. Caire, “Non-Bayesian activity detection, large-scale fading coefficient estimation, and unsourced random access with a massive MIMO receiver,” _IEEE Trans. Inf. Theory_ , vol. 67, no. 5, pp. 2925–2951, May 2021.
  * [23] A. Fengler, P. Jung and G. Caire, “Pilot-based unsourced random access with a massive MIMO receiver in the Quasi-static fading regime,” in _Proc. IEEE Workshop Signal Process. Adv. Wireless Commun. (SPAWC)_ , Lucca, Italy, Sep. 2021, pp. 356–360.
  * [24] M. Gkagkos, K. R. Narayanan, J. F. Chamberland and C. N. Georghiades, “FASURA: A scheme for quasi-static massive MIMO unsourced random access channels,” in _Proc. IEEE Workshop Signal Process. Adv. Wireless Commun. (SPAWC)_ , Oulu, Finland, July 2022, pp. 1–5.
  * [25] M. J. Ahmadi and T. M. Duman, “Unsourced random access with a massive MIMO receiver using multiple stages of orthogonal pilots,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Espoo, Finland, July 2022, pp. 2880-2885.
  * [26] E. Arikan, “Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 7, pp. 3051–3073, May 2009.
  * [27] Y. Polyanskiy, H. V. Poor and S. Verdu, “Channel coding rate in the finite blocklength regime,” _IEEE Trans. Inf. Theory_ , vol. 56, no. 5, pp. 2307–2359, May 2010.
  * [28] M. J. Ahmadi, R. Arablouei and R. Abdolee, “Efficient Estimation of Graph Signals With Adaptive Sampling,” _IEEE Trans. Signal Process._ , vol. 68, no. , pp. 3808-3823, June 2020.
  * [29] M. S. E. Abadi and M. J. Ahmadi, “Diffusion improved multiband-structured subband adaptive filter algorithm with dynamic selection of nodes over distributed networks,” _IEEE Trans. Circuits Syst. II, Exp. Briefs,_ , vol. 66, no. 3 , pp. 507-511, Mar. 2019.
  * [30] M. S. E. Abadi, J. H. Husøy, and M. J. Ahmadi, “Two improved multiband structured subband adaptive filter algorithms with reduced computational complexity,” _Signal Process._ , vol. 154, no. , pp. 15-29, Jan. 2019.
  * [31] M. J. Ahmadi, M. Kazemi, and T. M. Duman, “Unsourced random access with a massive MIMO receiver using multiple stages of orthogonal pilots: MIMO and single-antenna structures,” _IEEE Trans. Wireless Commun._ , June 2023.