∎

11institutetext: M. Mandloi (Corresponding author) 22institutetext: Department
of Electronics and Telecommunication Engineering, SVKM’s NMIMS (Deemed to be
University) Shirpur campus, Maharashtra 425405, India
Tel.: +91-9584858734
22email<EMAIL_ADDRESS>33institutetext: D. S. Gurjar
44institutetext: Department of Electronics and Communication Engineering,
National Institute of Technology Silchar, Assam 788010, India
44email<EMAIL_ADDRESS>
# Low-Complexity Interference Cancellation Algorithms for Detection in Media-
based Modulated Uplink Massive-MIMO Systems

Manish Mandloi  Devendra Singh Gurjar

(Received: date / Accepted: date)

###### Abstract

Media-based modulation (MBM) is a novel modulation technique that can improve
the spectral efficiency of the existing wireless systems. In MBM, multiple
radio frequency (RF) mirrors are placed near the transmit antenna(s) and are
switched ON/OFF to create different channel fade realizations. In such
systems, additional information is conveyed through the ON/OFF status of RF
mirrors along with conventional modulation symbols. A challenging task at the
receiver is to detect the transmitted information symbols and extract the
additional information from the channel fade realization used for
transmission. In this paper, we consider a massive MIMO (mMIMO) system where
each user relies on MBM for transmitting information to the base station, and
investigate the problem of symbol detection at the base station. First, we
propose a mirror activation pattern (MAP) selection based modified iterative
sequential detection algorithm. With the proposed algorithm, the most
favorable MAP is selected, followed by the detection of symbol corresponding
to the selected MAP. Each solution is subjected to the reliability check
before getting the update. Next, we introduce a $K$ favorable MAP search based
iterative interference cancellation (KMAP-IIC) algorithm. In particular, a
selection rule is introduced in KMAP-IIC for deciding the set of favorable
MAPs over which iterative interference cancellation is performed, followed by
a greedy update scheme for detecting the MBM symbols corresponding to each
user. Simulation results show that the proposed detection algorithms exhibit
superior performance-complexity trade-off over the existing detection
techniques in MBM-mMIMO systems.

###### Keywords:

Media-based modulation RF mirrors massive MIMO iterative interference
cancellation structured sparsity channel hardening

##  1 Introduction

The explosive increase in the number of subscribers, the use of data thirsty
applications, and ubiquitous computing for low latency applications pose a
serious challenge towards the design of advanced communication techniques for
5G and beyond wireless systems. Over the last decade, different wireless
techniques, such as non-orthogonal multiple access (NOMA), millimeter-wave
(mm-Wave) communications, vehicle-to-everything (V2X) communications, massive
MIMO (mMIMO), and machine learning in wireless networks, have been proposed in
the literature to enhance the performance of the existing wireless systems r1b
; r1c ; r1a . Amongst these, mMIMO has been considered as a promising
technique to satisfy the requirement of high data rate for 5G and beyond
wireless systems r1 ; r2 ; r3 . In mMIMO, a large number of base station (BS)
antennas are used to serve comparatively small number of single/multiple
antenna users r1 ; r2 . One of the key advantages of such systems is that
simple linear precoders and decoders can achieve near-optimal bit error rate
(BER) performance which includes zero-forcing and minimum mean squared error
detectors r1 ; r4 . Some of the key challenges in practical implementation of
multiple antenna systems include inter channel interference, requirement of
dedicated RF chains at each transmit antenna, and constraints on the total
number of receive antennas. Index modulation is one such digital modulation
technique proposed to overcome these challenging issues in MIMO systems r5 ;
r5a . Through index modulation, extra information bits are embedded in the
switching pattern of the building blocks along with the transmission of
symbols selected from the conventional constellation set such as M-ary PSK or
M-QAM r6 . This results in relaxing the need of all the available resources at
the transmitter such as transmit antennas and RF chains r7 ; r8 . In
conventional index modulation (IM) schemes, such as spatial modulation (SM)
and generalised SM (GSM), only one or fewer antenna elements and RF chains are
active which reduces the energy consumption as well as the inter channel
interference thereby providing a higher energy efficiency. IM together with
mMIMO r9 ; r10  has been evolved as an emerging technique for fulfilling the
higher spectral as well as the higher energy efficiency requirements in beyond
fifth-generation (B5G) wireless systems r6 . One of the key challenges in IM-
mMIMO systems is the detection of the transmitted information which requires
detection of the selected switching pattern and the detection of transmitted
symbol.
Recently, media-based modulation (MBM) has been introduced as a potential IM
scheme to enhance the spectral efficiency and the energy efficiency of the
existing wireless systems by embedding information in different channel fade
realizations r11 , r12 . These different end-to-end channel fade realizations
are created by modifying the radio frequency (RF) properties of the
propagation medium near to the transmitter r11 . To achieve these variations,
multiple RF mirrors (parasitic elements), which are digitally controlled by
the input data bits, are placed near the transmit antenna(s) r12 –Choc  . The
ON/OFF status of RF mirrors also referred to as mirror activation pattern
(MAP) convey additional information bits along with the conventional
modulation symbol. Through $n_{rf}$ RF mirrors, $2^{n_{rf}}$ MAPs can be
generated. Each MAP results in an independent channel fade realization which
is used by the transmitting antenna to transmit information symbol selected
from a constellation set. The practical advantages of MBM systems have been
demonstrated in r12  with 14 RF mirrors placed near a dipole transmit antenna.
Moreover, the antenna design, radiation pattern, and potential advantages of
MBM are discussed in r12 , r1111 . The use of MBM in mMIMO systems, which is
referred as MBM-mMIMO, has been manifested to achieve significant improvements
in terms of BER performance, and spectral efficiency r13 ; r14  over the
existing modulation schemes such as SM and GSM r15 ; r16 ; r17 ; r18 ; r19 ;
r20 . It is worth noting that, in MBM the throughput enhancement is linear
with respect to the number of RF mirrors, whereas, in SM and GSM, the increase
in throughput is logarithmic r13 . Therefore, MBM is being considered as an
emerging spectral efficient IM scheme in B5G wireless systems. However, one of
the major challenges in MBM-mMIMO systems is to detect the transmitted
information symbol, and to extract the additional information conveyed through
the selection of channel fade realization by each user, reliably with low
computational complexity r13 ; r21 .
Optimal detection rule suggests an exhaustive search over the set of all the
possible MBM symbol vector r13 ; r14 ; r21 , which is termed as maximum-
likelihood (ML) detection. However, due to exponentially high computational
complexity, ML detection is impractical in MBM-mMIMO systems r13 . It is due
to the fact that the set of all the possible symbol vectors for each user is
generated by using all the possible combinations of MAPs and constellation
points which ultimately results in an exponential increase in the size of the
search space for ML detection with respect to the number of transmit antennas
and $n_{rf}$. Though, ZF/MMSE achieves near-optimal performance in mMIMO
systems r1 , r22 ; r23 ; r24 ; r25 , their performance in MBM-mMIMO systems is
sub-optimal which makes ZF/MMSE less selective for MBM-mMIMO systems r7 ; r8 .
The sub-optimal BER performance of ZF/MMSE and exponentially high
computational complexity of ML detection motivates for the design of low-
complexity detection algorithms capable of achieving better BER performance in
MBM-mMIMO systems.
Recently, inclusion-exclusion subspace pursuit (IESP) r13  and iterative
interference cancellation (IIC) r14  algorithms have been proposed to achieve
better BER performance over the conventional detection techniques such as MMSE
and successive interference cancellation (SIC) r12 ; r14 ; r26  in MBM-mMIMO
systems. However, the IESP algorithm r13  requires computation of pseudo
inverse at multiple stages which is computationally expensive in MBM-mMIMO
system due to large dimensional channel matrix. On the other hand, IIC
algorithm r14  outperforms MMSE, SIC and IESP algorithms in terms of both the
BER performance and the computational complexity. In IIC r14 , symbol vector
transmitted by each user is detected by canceling interference from all the
other users followed by an exhaustive search over the set of all the possible
MAPs in an iterative manner. Although the computational complexity of IIC is
less compared with MMSE, SIC and IESP algorithms, there is scope to reduce the
computational complexity of IIC algorithm further by selecting a list of
favorable MAPs for performing the search. In order to solve the detection
problem in MBM-mMIMO systems, we propose two low-complexity detection schemes
in this paper. First, we propose a MAP selection based iterative sequential
detection (MAP-ISD) algorithm where the most favorable MAP is selected using a
selection metric. The symbol corresponding to the selected MAP is detected
sequentially for each user while nullifying the interference from all the
other users. Next, we propose a $K$ favorable MAP search based iterative
interference cancellation (KMAP-IIC) algorithm. In particular, $K$ most
favorable MAPs are selected over which the search is performed for detecting
the symbol corresponding to each user. The selection metric is obtained by
exploiting the channel hardening which occurs in mMIMO systems r27 .
Our key contributions in this article are;

  * •

A technique for selection of favorable mirror activation pattern (MAP) for
each user is proposed by utilizing channel hardening in mMIMO.

  * •

A low-complexity MAP-ISD algorithm is proposed for symbol detection in the
MBM-mMIMO systems which utilizes the concept of favorable MAP and low-
complexity ML search in sparse vectors.

  * •

The concept of reliability check of the solution in each iteration and
stopping rule are utilized in the MAP-ISD algorithm to obtain better BER with
low computational complexity.

  * •

Selection of $K$ favorable MAPs and low-complexity ML search are also
integrated with IIC and proposed KMAP-IIC algorithm.

  * •

Simulation results on BER and computational complexity of the proposed
algorithms are shown to validate superiority of the proposed algorithms over
the existing algorithms.

Simulation results show that the proposed detection algorithms exhibit
superior performance-complexity trade-off over the existing detection
techniques such as MMSE, IESP, and IIC r13 ; r14  in MBM-mMIMO systems. It is
also observed through simulations that the selection of a detection scheme
between MAP-ISD and KMAP-IIC depends on the number of RF mirrors and the
number of users in the system.
Notation: Boldface upper-case and lower-case letters denote matrices and
column vectors, respectively. $(\cdot)^{H}$ and $(\cdot)^{-1}$ represent
matrix Hermitian and matrix-inversion, respectively. $x_{i}$ is the $i^{th}$
element of x. $\textbf{I}_{U}$ refers to $U\times U$ identity matrix,
$a_{i,j}$ denotes the element in $i^{th}$ row and $j^{th}$ column of A, and
$\textbf{a}_{j}$ denotes $j^{th}$ column of A. $\mathcal{Q}[\cdot]$ denotes
the quantization operation which maps the soft values to the nearest
constellation point.

##  2 Background

In this section, we discuss the mathematical model of MBM-mMIMO system, and
introduce the ML detection rule for optimal detection of symbols in such
systems. We also shed light on the prohibitive computational complexity of ML
detection in MBM-mMIMO systems. Finally, we provide the detailed description
of ISD and IIC algorithms for detection in mMIMO systems and MBM-mMIMO
systems, respectively.

Table 1: List of Notations $N_{r}$ | Number of BS antennas
---|---
$U$ | Number of users
$n_{rf}$ | Number of RF mirrors
$M=2^{n_{rf}}$ | Number of mirror activation pattern (MAP)
$\mathbb{A}$ | Constellation set
$\mathbb{S}_{MBM}$ | MBM signal set for a single user
$\mathbb{S}_{MBM}^{U}$ | MBM signal set for $U$ user system
$L$ | Number of iterations

###  2.1 System model

Figure 1: System model for MBM-mMIMO system.

We consider an $N_{r}\times U$ mMIMO system with $N_{r}$ BS antennas and $U$
single antenna users ($N_{r}>>U$ for e.g., $N_{r}=128,U=16$ ). Each user
employs MBM for transmission of information to the BS as depicted in Fig. 1.
We also consider that each user is having $n_{rf}$ RF mirrors placed near the
antenna. Using MBM, the information is transmitted in two parts: 1. the mirror
ON/OFF status, and 2. the conventional modulation symbol using the
constellation set $\mathbb{A}$ (e.g., 4-QAM, QPSK, 16-QAM). To switch mirrors
ON/OFF, each user requires $n_{rf}$ bits of incoming information, and
therefore, there are $M=2^{n_{rf}}$ ON/OFF combinations possible which are
termed as MAPs. In a rich scattering environment, each MAP corresponds to an
independent channel fade realization r11 ; r12 . After selection of an MAP,
$\log_{2}{|{\mathbb{A}}|}$ additional bits are transmitted through the antenna
by selecting one symbol from the modulation alphabet $\mathbb{A}$. Therefore,
the spectral efficiency of a multi-user MBM system in terms of bits per
channel use (bpcu) can be mathematically given by

$\eta_{MBM}=U(n_{rf}+\log_{2}{|{\mathbb{A}}|}).$ (1)

Let, the channel state vector between the $j$th MAP selected by $k$-th user
and the base station is represented by $\mathbf{h}^{j}_{k}$. Each
$\mathbf{h}^{j}_{k}=[h^{j}_{1,k},h^{j}_{2,k},\cdots,h^{j}_{N_{r},k}]_{N_{r}\times
1}^{T}$, for all $j=1,2,\cdots,M$ and $k=1,2,\cdots,U$, where $h^{j}_{i,k}$ is
assumed to be independent and identically distributed (i.i.d.) complex
Gaussian random variable with zero mean and unit variance i.e.
$\sim\mathcal{CN}(0,1)$. Therefore, the channel matrix comprising of channel
vectors corresponding to all the possible MAPs for the $k$-th user is
$\mathbf{H}_{k}=[\mathbf{h}^{1}_{k},\mathbf{h}^{2}_{k},\cdots,\mathbf{h}^{M}_{k}]$.
The MBM signal set for a single user can be defined as the set of all the
possible transmit symbol vectors as

$\displaystyle\mathbb{S}_{MBM}=\\{\mathbf{s}_{j,i}:j=1,2,\cdots,M,i=1,2,\cdots,|\mathbb{A}|\\},$
$\displaystyle\text{s.t.}\ \
\mathbf{s}_{j,i}=[0,\cdots,0,\underbrace{q_{i}}_{j\text{th
coordinate}},0,\cdots,0]^{T},q_{i}\in\mathbb{A},$ (2)

where $\mathbf{s}_{j,i}$ is an $M\times 1$ vector with only one non-zero entry
$q_{i}\in\mathbb{A}$ corresponding to the $j$th channel fade realization. For
the $k$-th user, let us denote the transmit symbol vector is denoted by
$\mathbf{x}_{k}\in\mathbb{S}_{MBM}$. Thus, the received symbol vector at the
BS after performing matched filtering and sampling operations can be written
as

$\mathbf{y}=\sum_{k=1}^{U}\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{n},$ (3)

where $\mathbf{n}$ is an $N_{r}\times 1$ additive white Gaussian noise (AWGN)
vector with $\mathbf{n}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N_{r}})$.
Further, for simplicity, we can rewrite the received symbol vector as

$\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n},$ (4)

where $\mathbf{H}=[\mathbf{H}_{1}\ \ \mathbf{H}_{2}\ \ \cdots\ \
\mathbf{H}_{U}]$ is an $N_{r}\times UM$ MBM-mMIMO channel matrix and
$\mathbf{x}=[\mathbf{x}_{1}^{T}\ \ \mathbf{x}_{2}^{T}\ \ \cdots\ \
\mathbf{x}_{U}^{T}]^{T}$ is $UM\times 1$ MBM-mMIMO transmit symbol vector
comprising of transmit symbol vectors of all the users.

###  2.2 Maximum Likelihood Detection for MBM-mMIMO systems

The objective at the receiver is to extract the information embedded in the
selection of MAP as well as detect the transmitted information symbol from the
constellation set for realizing the potential benefits of MBM. An optimal
detection rule termed as ML detection suggests an exhaustive search over all
the possible combinations of the transmit symbol vector $\mathbf{x}$ which
minimize the ML cost, where ML cost associated with the symbol vector
$\mathbf{x}$ is given by

$\displaystyle C(\mathbf{x})$ $\displaystyle=$
$\displaystyle\|\mathbf{y}-\mathbf{H}\mathbf{x}\|^{2},$ (5) $\displaystyle=$
$\displaystyle\|\mathbf{y}-\sum_{k=1}^{U}\mathbf{H}_{k}\mathbf{x}_{k}\|^{2}.$
(6)

Therefore, the ML detection problem given the received vector $\mathbf{y}$ and
the channel state information $\mathbf{H}$ can be formulated as

$\hat{\mathbf{x}}=\arg\min_{\mathbf{x}\in\mathbb{S}^{U}_{MBM}}\|\mathbf{y}-\mathbf{H}\mathbf{x}\|^{2}.$
(7)

where $\mathbb{S}^{U}_{MBM}$ is the set of all the possible combinations of
$\mathbf{x}$. In MBM-mMIMO systems, for a given value of $U$, $M$ and
$|{\mathbb{A}}|$ the dimensions of $\mathbf{x}$ equals $M\times U$ and the set
$\mathbb{S}^{U}_{MBM}$ contains $\left(M\times|{\mathbb{A}}|\right)^{U}$
possible combinations of $\mathbf{x}$ for e.g., if $U=16$ and $M=2^{n_{rf}}=8$
then for a 4-QAM modulated MBM-mMIMO systems the dimensions of $\mathbf{x}$ is
128 ($M\times U=16\times 8=128$) and the size of
$|\mathbb{S}^{U}_{MBM}|=32^{16}\approx 1.2\times 10^{24}$. Clearly, ML
detection is unreasonable in such systems. Therefore, the design of low-
complexity detection techniques is a challenging and a crucial problem for
practical realization of MBM-mMIMO in the B5G wireless systems.

###  2.3 Iterative Sequential Detection for mMIMO Systems

In this section, we discuss the ISD algorithm proposed in r24  for detecting
the transmitted information symbols in mMIMO systems. In ISD, symbols
corresponding to each user are detected in a sequential manner. In each
iteration of ISD, symbols corresponding to all the users are detected which
are refined in next iterations. An initial solution is used to initialize the
algorithm, which could be an all zero solution, i.e.,
$\hat{\mathbf{x}}_{j}^{0}=0$ for all $j=1,2,\cdots,U$. Next, for detection of
symbol corresponding to the $k$-th user in the $t$-th iteration, interference
from all the other users is cancelled as

$\mathbf{r}_{k}^{(t)}=\mathbf{y}-\sum_{i=1}^{k-1}\mathbf{h}_{i}\hat{\mathbf{x}}_{i}^{(t)}-\sum_{i=k+1}^{U}\mathbf{h}_{i}\hat{\mathbf{x}}_{i}^{(t-1)},$
(8)

where $\hat{\mathbf{x}}_{i}^{t-1}$ and $\hat{\mathbf{x}}_{i}^{t}$ is the
symbol corresponding to the $i$-th user in the $(t-1)$-th and the $(t)$-th
iterations, respectively. The residual error vector $\mathbf{r}_{k}^{(t)}$ is
then passed through a matched filter for detecting $\hat{\mathbf{x}}_{i}^{t}$
as

$\hat{\mathbf{x}}_{i}^{t}=\mathcal{Q}\left[\frac{\mathbf{h}_{i}^{H}}{\|\mathbf{h}_{i}\|^{2}}\mathbf{r}_{k}^{(t)}\right],$
(9)

where $\mathcal{Q}[\cdot]$ is the quantization operation which maps the soft
values to the nearest constellation points. These steps are repeated for
multiple iterations to obtain a better solution. It is worth noting that the
detection of MBM symbols using the ISD algorithm directly is an inefficient
way of detection due to the presence of MAPs. Moreover, selecting a reliable
solution for each user and a proper stopping rule are required to avoid error
propagation and terminate the algorithm early, respectively. Therefore, the
selection of a reliable MAP and solution for each user is crucial for
detecting the transmitted symbol in MBM-mMIMO systems. Also, the stopping rule
will help in terminating the algorithm early, thereby saving the required
computations. This motivates further modifications in ISD to detect symbols in
MBM-mMIMO systems with low-complexity. Algorithm 1 presents the pseudo code of
ISD algorithm for mMIMO detection.

Algorithm 1 ISD Algorithm

1:Input: ${\mathbf{y}}$, ${\mathbf{H}}$, $N_{r},U,L$

2:Initialization: $\mathbf{x}^{(0)}=\mathbf{0},\ \
\mathbf{r}^{(0)}=\mathbf{y},\ \ t=0\ \ (\text{Iteration index})$

3:repeat

4: $\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)},\ \
\mathbf{r}^{(t)}=\mathbf{r}^{(t-1)},\ \ t=t+1$

5: for $(j=1,++j,j\leq U)$ do

6: Compute :
$\mathbf{r}_{k}^{(t)}=\mathbf{y}-\sum_{i=1}^{k-1}\mathbf{h}_{i}\hat{\mathbf{x}}_{i}^{(t)}-\sum_{i=k+1}^{U}\mathbf{h}_{i}\hat{\mathbf{x}}_{i}^{(t-1)}$

7: Compute :
$z_{i}=\frac{\mathbf{h}_{i}^{H}}{\|\mathbf{h}_{i}\|^{2}}\mathbf{r}_{k}^{(t)}$

8: Update the symbol for $i$th user as :
$\hat{\mathbf{x}}_{i}^{t}=\mathcal{Q}\left[\mathbf{z}_{i}\right]$

9: end for

10:until $t=L$ (terminate the iterative processing)

11:Output:$\hat{{\mathbf{x}}}_{L}$

###  2.4 Iterative Interference Cancellation Algorithm for MBM-mMIMO Systems

In this section, we discuss the IIC algorithm proposed in r14 . In IIC, the
symbol detection corresponding to each user is a two step process. The
algorithm starts with an all zero solution, i.e.,
$\hat{\mathbf{x}}_{i}^{(0)}=0$ $\forall i=1,2,\cdots,U$. In the first step,
the residual signal is obtained as

$\displaystyle\mathbf{r}^{(t)}$ $\displaystyle=$
$\displaystyle\mathbf{y}-\sum_{i=1}^{U}\mathbf{H}_{i}\hat{\mathbf{x}}_{i}^{(t)},$
(10) $\displaystyle\mathbf{y}_{j}^{(t)}$ $\displaystyle=$
$\displaystyle\mathbf{r}^{(t)}+\mathbf{H}_{j}\hat{\mathbf{x}}_{j}^{(t)}$ (11)

where $\hat{\mathbf{x}}_{i}^{(t-1)}$ is the estimated MBM symbol vector of the
$i$-th user in $(t-1)$-th iteration, and $\mathbf{r}^{(t)}$ is the residual
signal. $\mathbf{y}_{j}^{(t)}$ is the interference cancellation vector for the
$j$-th user in the $t$-th iteration. Next, a search is performed over the set
of all the possible MBM symbol vectors for detecting the symbol vector
corresponding to the $j$th user as

$\hat{\mathbf{s}}_{j}^{(t)}=\arg\min_{\mathbf{s}_{j}\in\mathbb{S}_{MBM}}\|{\mathbf{y}_{j}^{(t)}-\mathbf{H}_{j}\mathbf{s}_{j}}\|^{2}.$
(12)

IIC algorithm is initialised with an all-zero solution. After performing these
two steps for all the users, a greedy search is performed multiple times to
update the solution for each user by using the selection metric given as

$u=\arg\min_{k=1,\cdots,U}\|{\mathbf{r}^{(t)}+\mathbf{H}_{j}(\hat{\mathbf{s}}_{j}^{(t)}-\hat{\mathbf{x}}_{i}^{(t)})}\|^{2}.$
(13)

These steps are performed for multiple iterations so that the algorithm
converge to a better solution. The pseudo code of IIC algorithm is described
in Algorithm 2.

Algorithm 2 IIC Algorithm

1:Input: ${\mathbf{y}}$, ${\mathbf{H}}$, $N_{r},U,n_{rf},L$

2:Initialization: $\mathbf{x}^{(0)}=\mathbf{0}\ \ (\text{Symbol
estimation}),\mathbf{r}^{(0)}=\mathbf{y}\ \ (\text{Residual
signal}),\mathbf{W}_{i},\forall i=1,2,\cdots,U,\ t=0\ \ (\text{Iteration
index})$

3:repeat

4:
$\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)},\mathbf{r}^{(t)}=\mathbf{r}^{(t-1)},\
\ t=t+1$

5: for $(j=1,++j,j\leq U)$ do

6: Apply interference cancellation for the $j$-th user as:
$\mathbf{y}_{j}^{(t)}=\mathbf{r}^{(t)}+\mathbf{H}_{j}\hat{\mathbf{x}}_{j}^{(t)}$

7: Perform ML search for the $j$-th user as:
$\hat{\mathbf{s}}_{j}^{(t)}=\arg\min_{\mathbf{s}_{j}\in\mathbb{S}_{MBM}}\|{\mathbf{y}_{j}^{(t)}-\mathbf{H}_{j}\mathbf{s}_{j}}\|^{2}$

8: end for

9: for $(j=1,++j,j\leq U)$ do

10: Select the first element of
$u=\arg\min_{k=1,\cdots,U}\|{\mathbf{r}^{(t)}+\mathbf{H}_{j}(\hat{\mathbf{s}}_{j}^{(t)}-\hat{\mathbf{x}}_{i}^{(t)})}\|^{2}$

11: if
$\|\mathbf{r}^{(t)}\|^{2}>\|\mathbf{r}^{(t)}+\mathbf{H}_{u}(\hat{\mathbf{x}}_{u}^{(t)}-\hat{\mathbf{s}}_{u}^{(t)})\|^{2}$
then

12: Update the residual error vector as
$\mathbf{r}^{(t)}=\mathbf{r}^{(t)}+\mathbf{H}_{u}(\hat{\mathbf{x}}_{u}^{(t)}-\hat{\mathbf{s}}_{u}^{(t)})$

13: Update the symbol vector for $v$th user as
$\hat{\mathbf{x}}_{u}^{(t)}=\hat{\mathbf{s}}_{u}^{(t)}$

14: else

15: break

16: end if

17: end for

18:until $\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)}$ or $t=L$ (terminate
the iterative processing or if no update in the solution)

19:$\hat{\mathbf{x}}=\hat{\mathbf{x}}^{(t)}$

20:Output:$\hat{{\mathbf{x}}}$

The size of search space $\mathbb{S}_{MBM}$ for each user in MBM-mMIMO is
$\left(M\times|\mathbb{A}|\right)$ which grows exponentially with the number
of RF mirrors and linearly with the size of constellation set used. To perform
ML search for each user in a single iteration, it requires IIC to search over
$M\times|\mathbb{A}|$ possible solutions. Therefore, the total search
operations in performing ML search is $\left(M\times|\mathbb{A}|\right)UL$.
This makes the algorithm computationally expensive and motivates for further
research in the algorithm to improve the practical feasibility of IIC in MBM-
mMIMO systems.

##  3 Proposed Detection Algorithms

In this section, we discuss the proposed algorithms, namely, MAP-ISD algorithm
and KMAP-IIC algorithm for MBM symbol detection in mMIMO systems.

###  3.1 MAP Selection based ISD

In this algorithm, we propose to modify the ISD algorithm r24 , which was
originally proposed to detect the symbols in mMIMO systems. Here, we improvise
the ISD algorithm for detecting MBM symbols corresponding to each user in MBM-
mMIMO systems. In particular, we introduce a rule for selecting the most
favorable MAP corresponding to each user from the list of all the possible
MAPs in MBM-mMIMO system. For this, first, we utilize the concept of channel
hardening, which occurs in mMIMO systems, and then find the pseudo-inverse of
the individual channel matrices $\mathbf{H}_{k}$. The key idea for computing
the pseudo-inverse is to find the highly erroneous locations by applying low-
complexity zero-forcing over the residual vector (as discussed later in this
section). The MAP corresponding to these erroneous locations are then explored
for detecting the symbol transmitted by a particular user. Upon detection, the
residual vector is updated accordingly.

####  3.1.1 Channel Hardening

In mMIMO systems, due to a large number of receive antennas as compared to the
number of users, the Gram matrix, $\mathbf{G}=\mathbf{H}^{H}\mathbf{H}$,
becomes diagonal dominant which is a consequence of channel hardening r1 ; r27
. Mathematically, it can be expressed as

$\frac{\mathbf{h}_{i}^{T}\mathbf{h}_{j}}{N_{r}}\rightarrow 0,\quad\forall
i\neq j,~{}~{}i,j\in\left\\{{1,2,\ldots,UM}\right\\},$ (14)

where $\mathbf{h}_{j}$ is the $j$th column of channel gain matrix
$\mathbf{H}$.

####  3.1.2 Low-Complexity Pseudo-Inverse

Due to diagonal dominance, we can approximate the inverse of matrix
$\mathbf{G}$ by using matrix $\mathbf{D}$ r16  as

$\mathbf{G}^{-1}\approx\mathbf{D}^{-1},$ (15)

where $\mathbf{D}$ is the diagonal matrix containing only the diagonal entries
of $\mathbf{G}$. Therefore, the problem of finding pseudo-inverse in mMIMO
systems r16 ; r17 ; r18  can be approximated by using a low-complexity
approximation as

$(\mathbf{H}^{H}\mathbf{H})^{-1}\mathbf{H}^{H}=\mathbf{G}^{-1}\mathbf{H}^{H}\approx\mathbf{D}^{-1}\mathbf{H}^{H}.$
(16)

It is worth noting that, the approximate inverse of matrix $\mathbf{G}$ is
computed only once for each user and requires the computational complexity of
$\mathcal{O}(MN_{r})$.

####  3.1.3 MAP Selection Rule

Next, we use the approximate pseudo-inverse obtained in Eq. (16) over the
residual vector defined in Eq. 10 to obtain the favorable MAP as

$\mathbf{e}_{j}^{(t)}=\mathbf{W}_{j}\left(\mathbf{y}-\sum_{i=1,i\neq
j}^{U}\mathbf{H}_{i}\hat{\mathbf{x}}_{i}^{(t-1)}\right).$ (17)

where $\mathbf{W}_{j}=\mathbf{D}^{-1}_{j}\mathbf{H}_{j}^{H}$ and
$\mathbf{D}_{j}$ is the diagonal of the Gram matrix
$\mathbf{H}_{j}^{H}\mathbf{H}_{j}$. From $\mathbf{e}_{j}^{(t)}$, we select the
element having the largest magnitude value, i.e., $|{e}_{j,i}^{(t)}|$ for all
$i=1,2,\cdots,M$ and select $k$ as the index of the largest value which is
nothing but the index of the most favorable MAP. The magnitude value of
$|{e}_{j,i}^{(t)}|$ resembles the non-zero location in the transmitted symbol
vector $\mathbf{x}_{j}$ which can be mathematically analysed using Eq. (17).
The vector $\mathbf{e}_{j}^{(t)}$ can be simplified as

$\displaystyle\mathbf{e}_{j}^{(t)}$ $\displaystyle=$
$\displaystyle\mathbf{W}_{j}\left(\mathbf{H}_{j}\mathbf{x}_{j}+\sum_{i=1,i\neq
j}^{U}\mathbf{H}_{i}\mathbf{x}_{i}+\mathbf{n}-\sum_{i=1,i\neq
j}^{U}\mathbf{H}_{i}\hat{\mathbf{x}}_{i}^{(t-1)}\right)$ (18) $\displaystyle=$
$\displaystyle\mathbf{W}_{j}\mathbf{H}_{j}\mathbf{x}_{j}+\sum_{i=1,i\neq
j}^{U}\mathbf{W}_{j}\mathbf{H}_{i}\left(\mathbf{x}_{i}-\hat{\mathbf{x}}_{i}^{(t-1)}\right)+\mathbf{W}_{j}\mathbf{n}.$

After incorporating the value of $\mathbf{W}_{j}$ in Eq. (18) and further
solving we get

$\displaystyle=$
$\displaystyle\mathbf{D}^{-1}_{j}\mathbf{G}_{j}\mathbf{x}_{j}+\sum_{i=1,i\neq
j}^{U}\mathbf{D}^{-1}_{j}\mathbf{G}{ji}\left(\mathbf{x}_{i}-\hat{\mathbf{x}}_{i}^{(t-1)}\right)+\widetilde{\mathbf{n}},$
(19)

where $\widetilde{\mathbf{n}}=\mathbf{D}^{-1}_{j}\mathbf{H}_{j}^{H}\mathbf{n}$
and $\mathbf{G}_{ji}=\mathbf{H}_{j}^{H}\mathbf{H}_{i}$. Due to channel
hardening $\mathbf{D}^{-1}_{j}\mathbf{G}_{j}\approx\mathbf{I}$ and
$\mathbf{D}^{-1}_{j}\mathbf{G}{ji}\approx\mathbf{O}$. Clearly, the vector
$\mathbf{e}_{j}^{(t)}$ reduces to a combination the transmitted symbol
$\mathbf{x}_{j}$ and the noise vector $\widetilde{\mathbf{n}}$, i.e.,
$\mathbf{e}_{j}^{(t)}=\mathbf{x}_{j}+\widetilde{\mathbf{n}}$. Obviously, the
vector $\mathbf{x}_{j}$ has only one non-zero location, and therefore, the
magnitude $|{e}_{j,i}^{(t)}|$ for all $i=1,2,\cdots,M$ have the maximum value
corresponding to that non-zero location.
Finally, we decide the MBM symbol corresponding to the $j$th user as

$\hat{\mathbf{s}}_{j}^{(t)}=[0,\cdots,0,\underbrace{\beta}_{k\text{th
coordinate}},0,\cdots,0]^{T}$ (20)

where $\beta=\mathcal{Q}[{e}_{j,k}]$ such that $\beta\in\mathbb{A}$. The
symbol for the $j$th user is updated only if it results in the reduction of
the norm of residual vector, i.e.,
$\hat{\mathbf{x}}_{j}^{(t)}=\hat{\mathbf{s}}_{j}^{(t)}$ if
$\|\mathbf{r}^{(t)}\|^{2}>\|\mathbf{r}^{(t)}+\mathbf{H}_{j}(\hat{\mathbf{x}}_{j}^{(t-1)}-\hat{\mathbf{s}}_{j}^{(t)})\|^{2}$.
Similarly, the symbol corresponding to each user is detected in a single
iteration. The algorithm is run for multiple iterations say $L$ for refining
the detected symbols and obtaining the minimum norm of residual vector.
Algorithm 3 presents the pseudo code of the MAP-ISD algorithm for symbol
detection in MBM-mMIMO systems.

Algorithm 3 MAP-ISD Algorithm

1:Input: ${\mathbf{y}}$, ${\mathbf{H}}$, $N_{r},U,n_{rf},L$

2:Initialization:
$\mathbf{x}^{(0)}=\mathbf{0},\mathbf{r}^{(0)}=\mathbf{y},\mathbf{W}_{i},\forall
i=1,2,\cdots,U,\ \ t=0\ \ (\text{Iteration index})$

3:repeat

4:
$\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)},\mathbf{r}^{(t)}=\mathbf{r}^{(t-1)},\
\ t=t+1$

5: for $(j=1,++j,j\leq U)$ do

6: Compute :
$\mathbf{e}_{j}=\mathbf{W}_{j}(\mathbf{r}^{(t)}+\mathbf{H}_{j}\hat{\mathbf{x}}_{j}^{(t-1)})$

7: Find the element having largest $|{e}_{j,i}^{(t)}|$ for all
$i=1,2,\cdots,M$

8: Find the MBM symbol for the $j$th user as:
$\hat{\mathbf{s}}_{j}^{(t)}=[0,\cdots,0,\underbrace{\beta}_{k\text{th
coordinate}},0,\cdots,0]^{T}$ where $\beta=\mathcal{Q}[{e}_{j,k}]$

9: if
$\|\mathbf{r}^{(t)}\|^{2}>\|\mathbf{r}^{(t)}+\mathbf{H}_{j}(\hat{\mathbf{x}}_{j}^{(t-1)}-\hat{\mathbf{s}}_{j}^{(t)})\|^{2}$
then

10: Update the residual vector as
$\mathbf{r}^{(t)}=\mathbf{r}^{(t)}+\mathbf{H}_{j}(\hat{\mathbf{x}}_{j}^{(t-1)}-\hat{\mathbf{s}}_{j}^{(t)})$

11: Update the symbol vector for $j$th user as
$\hat{\mathbf{x}}_{j}^{(t)}=\hat{\mathbf{s}}_{j}^{(t)}$

12: end if

13: end for

14:until $\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)}$ or $t=L$ (terminate
the iterative processing or if no update in the solution)

15:Output:$\hat{{\mathbf{x}}}_{t}$

###  3.2 $K$-favorable MAP based IIC

In this section, we discuss the proposed KMAP-IIC algorithm for detecting
symbols in MBM-mMIMO systems. Through selection of multiple favorable MAPs,
the computational complexity of the existing IIC can be reduced significantly
without compromising the BER performance (as discussed in Section 4). In the
proposed algorithm, multiple favorable MAPs are selected using Eq. (17) and a
list $\mathbb{L}_{j}^{(t)},\ \forall j=1,2,\cdots,U$, of $K$-favorable MAPs is
generated. From $\mathbf{e}_{j}^{(t)}$, we select $K$ elements having largest
magnitude value to update $\mathbb{L}_{j}^{(t)}$ i.e. we sort
$|{e}_{j,i}^{(t)}|$ for all $i=1,2,\cdots,M$ in descending order, and select
the indices of first $K$ element.
Next, we define the reduced size set $\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ of
favorable MBM symbol vectors for the $j$th user in the $t$th iteration as

$\displaystyle\tilde{\mathbb{S}}_{MBM}^{j,(t)}=\\{\mathbf{s}_{k,i}\in\mathbb{A}_{0}:k=\mathbb{L}_{j}^{(t)}(1),\mathbb{L}_{j}^{(t)}(2),\cdots,\mathbb{L}_{j}^{(t)}(K),$
$\displaystyle i=1,2,\cdots,|\mathbb{A}|\\}$ $\displaystyle\text{s.t.}\ \
\mathbf{s}_{k,i}=[0,\cdots,0,\underbrace{\alpha_{i}}_{k\text{th
coordinate}},0,\cdots,0]^{T},\alpha_{i}\in\mathbb{A}.$ (21)

After finding the list of favorable MAPs and the reduced size set of MBM
symbol vectors, a search is performed over the set
$\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ for all the users in order to determine the
best candidate solution corresponding to each user. The solution in the set
$\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ which minimizes the norm of residual vector
is selected as the best candidate solution for the $j$th user defined as

$\hat{\mathbf{s}}_{j}^{(t)}=\arg\min_{\tilde{\mathbf{s}}_{q}\in\tilde{\mathbb{S}}_{MBM}^{j,(t)}}\|{\mathbf{r}_{j}^{(t)}-\mathbf{H}_{j}\tilde{\mathbf{s}}_{q}}\|^{2}.$
(22)

Moreover, for a square- and rectangular-QAM constellation, a low-complexity
search, proposed in r22 , is utilized in the proposed algorithm to reduce the
computational complexity further. The real and imaginary parts of the
estimated solution for square- and rectangular-QAM constellations can be
expressed mathematically as

$\displaystyle\Re(\hat{\mathbf{s}}_{l})=\min\left[\max\left(2\lfloor\frac{m_{1}+1}{2}\rceil-1,-N_{1}+1\right),N_{1}-1\right],$
$\displaystyle\Im(\hat{\mathbf{s}}_{l})=\min\left[\max\left(2\lfloor\frac{m_{2}+1}{2}\rceil-1,-N_{2}+1\right),N_{2}-1\right],$
(23)

where $m_{1}=\Re({z}_{l})$, $m_{2}=\Im({z}_{l})$ and
$z_{l}=\frac{\mathbf{h}_{j,l}^{H}\mathbf{r}_{j}^{(t)}}{\|\mathbf{h}_{j,l}\|^{2}}$,
and $N_{1},N_{2}$ are the sizes two PAMs on the real and imaginary axis of the
QAM constellation, respectively. Thus the best possible solution corresponding
to the $l$th MAP of the $j$th user is
$\hat{\mathbf{s}}_{l}=\Re(\hat{\mathbf{s}}_{l})+i\Im(\hat{\mathbf{s}}_{l})$
r22 . This makes the search independent of the number of points in the
constellation set, and therefore, reduces the computational complexity
incurred during the detection.
The search for the best possible solution is followed by symbol update stage
where symbol vectors corresponding to the selected user is updated. A greedy
update strategy proposed in r8  is used which requires $U$ iterations to
update the solution corresponding to several users in order to minimize the ML
cost. In each iteration, the user for which the ML cost is selected by using

$v=\arg\min_{i=1,2,\cdots,U}\|\mathbf{r}^{(t)}+\mathbf{H}_{i}(\hat{\mathbf{x}}_{i}^{(t)}-\hat{\mathbf{s}}_{i}^{(t)})\|^{2},$
(24)

and the solution corresponding to the selected user is updated. This update
strategy is initiated only after completion of search over reduced size set
for all the users. Algorithm 4 presents the pseudo code of the KMAP-IIC
algorithm.

Algorithm 4 KMAP-IIC Algorithm

1:Input: ${\mathbf{y}}$, ${\mathbf{H}}$, $N_{r},U,n_{rf},L,K$

2:Initialization:
$\mathbf{x}^{(0)}=\mathbf{0},\mathbf{r}^{(0)}=\mathbf{y},\mathbf{W}_{i},\forall
i=1,2,\cdots,U,\ \ t=0\ \ (\text{Iteration index})$

3:repeat

4:
$\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)},\mathbf{r}^{(t)}=\mathbf{r}^{(t-1)},\
\ t=t+1$

5: for $(j=1,++j,j\leq U)$ do

6: Compute :
$\mathbf{e}_{j}=\mathbf{W}_{j}(\mathbf{r}^{(t)}+\mathbf{H}_{j}\hat{\mathbf{x}}_{j}^{(t-1)})$

7: Sort the elements in Descending Order and update $\mathbb{L}_{j}^{(t)}$
i.e. select indices of first $K$ elements from $\arg
sort(|{e}_{j,1}^{(t)}|,|{e}_{j,2}^{(t)}|,\ldots,|{e}_{j,M}^{(t)}|,descend)$

8: Generate the Reduced Search Space $\tilde{\mathbb{S}}_{MBM}^{j,(t)}$

9: Perform search for the $j$th user as:
$\hat{\mathbf{s}}_{j}^{(t)}=\arg\min_{\tilde{\mathbf{s}}_{q}\in\tilde{\mathbb{S}}_{MBM}^{j,(t)}}\|{\mathbf{r}_{j}^{(t)}-\mathbf{H}_{j}\tilde{\mathbf{s}}_{q}}\|^{2}$

10: end for

11: for $(j=1,++j,j\leq U)$ do

12: Select the first element of
$v=\arg\min_{i=1,2,\cdots,U}\|\mathbf{r}^{(t)}+\mathbf{H}_{i}(\hat{\mathbf{x}}_{i}^{(t)}-\hat{\mathbf{s}}_{i}^{(t)})\|^{2}$

13: if
$\|\mathbf{r}^{(t)}\|^{2}>\|\mathbf{r}^{(t)}+\mathbf{H}_{v}(\hat{\mathbf{x}}_{v}^{(t)}-\hat{\mathbf{s}}_{v}^{(t)})\|^{2}$
then

14: Update the residual vector as
$\mathbf{r}^{(t)}=\mathbf{r}^{(t)}+\mathbf{H}_{v}(\hat{\mathbf{x}}_{v}^{(t)}-\hat{\mathbf{s}}_{v}^{(t)})$

15: Update the symbol vector for $v$th user as
$\hat{\mathbf{x}}_{v}^{(t)}=\hat{\mathbf{s}}_{v}^{(t)}$

16: else

17: break

18: end if

19: if $\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)}$ then

20: break

21: end if

22: end for

23:until $\hat{\mathbf{x}}^{(t)}=\hat{\mathbf{x}}^{(t-1)}$ or $t=L$ (terminate
the iterative processing or if no update in the solution)

24:Output:$\hat{{\mathbf{x}}}_{t}$

##  4 Simulation results

In this section, we present the simulation results of the proposed detection
algorithms in terms of BER performance and the computational complexity for
MBM-mMIMO systems. We also compare the BER performance of the proposed
algorithms with the performance of MMSE detector r7 , IESP algorithm r7  and
IIC algorithm r8 . We consider $128\times 16$ and $128\times 20$ MBM-mMIMO
systems with $n_{rf}=3$, $n_{rf}=4$, $n_{rf}=5$ and $n_{rf}=6$, respectively.
In summary, Table 2 shows list of the parameters used in simulation of BER
performance and computational complexity.

Table 2: List of parameters used in simulation. Figure | MBM-mMIMO System | Parameters Used
---|---|---
Fig. 2 |  $N_{r}=128$, $U=20$, $n_{rf}=3$ | 4-QAM, $L=1,2,4,6,8$
Fig. 3 |  $N_{r}=128$, $U=16$, $n_{rf}=4$ | 4-QAM, $L=1,2,4,6,8$
Fig. 4 |  $N_{r}=128$, $U=20$, $n_{rf}=3$ | 4-QAM, $L=1,2,4,6$, $K=M/2$
Fig. 5 |  $N_{r}=128$, $U=20$, $n_{rf}=3$ | 4-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 6 |  $N_{r}=128$, $U=20$, $n_{rf}=4$ | 4-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 7 |  $N_{r}=128$, $U=16$, $n_{rf}=6$ | 4-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 8 |  $N_{r}=128$, $U=20$, $n_{rf}=3$ | 16-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 9 |  $N_{r}=128$, $U=20$, $n_{rf}=5$ | 16-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 10 |  $U=20$, $n_{rf}=4$ | 4-QAM, $L=6$, $K=1,M/4,M/2$
Fig. 11 |  $N_{r}=128$, $U=16,20$, $n_{rf}=3,4$ | 4-QAM, $L=6$, $K=1,M/4,M/2$, SNR = 5 dB
Fig. 12 |  $U=16$, $n_{rf}=4$ | 4-QAM, $L=6$, $K=1,M/4,M/2$, SNR = 5 dB

###  4.1 Bit Error Rate Performance

In Figs. 2 and 3, we present the BER performance of MAP-ISD with different
iterations $L=1,2,4,6$ and $8$ for $128\times 20$ mMIMO system, $n_{rf}=3$ and
$128\times 16$ mMIMO system, $n_{rf}=4$, respectively. It is observed that the
BER performance of MAP-ISD improves with increase in the number of iterations
and converges after $L=6$. Significant improvement in BER performance is
observed for increase in the value of $L$ from 1 to 2 and from 2 to 4 whereas
marginal improvement is observed for further increase in the value of $L$.

Figure 2: BER performance of MAP-ISD for $128\times 20$, $n_{rf}=3$ MBM-mMIMO
system with 4-QAM modulation. Figure 3: BER performance of MAP-ISD for
$128\times 16$, $n_{rf}=4$ MBM-mMIMO system with 4-QAM modulation. Figure 4:
BER performance of KMAP-IIC with $K=M/2$ for $128\times 20$, $n_{rf}=4$ MBM-
mMIMO system with 4-QAM modulation.

In Fig. 4, we simulate the BER performance of KMAP-IIC algorithm with
different iterations $L=1,2,4$ and $6$ for $128\times 20$ mMIMO systems with
4-QAM, $K=M/2$ and $n_{rf}=3$ MBM. Similar observations can be drawn which
suggests that the BER performance of the KMAP-IIC algorithm converges for
$L=6$. Therefore, for comparison of BER of the proposed algorithms, i.e. MAP-
ISD and KMAP-IIC, with other detection techniques (discussed later in the
section) we use $L=6$ iterations.

Figure 5: BER performance comparison for $128\times 20$, $n_{rf}=3$ MBM-mMIMO
system with 4-QAM modulation.

In Fig. 5, we compare the BER performance of the proposed algorithms with
other algorithms such as IIC, IESP and MMSE for $128\times 20$ mMIMO with
4-QAM, $n_{rf}=3$ MBM system. For comparison of KMAP-IIC, we consider three
different values for $K$, i.e. $K=1$, $K=M/4$ and $K=M/2$. It is observed
that, MAP-ISD and KMAP-IIC algorithms outperform the MMSE and the IESP
algorithms. Moreover, the KMAP-IIC achieves BER performance close to within
$0.1$ dB to that of the IIC algorithm with $K=M/2$. For a target BER of
$10^{-4}$, MAP-ISD achieves performance within 0.5 dB of the IIC algorithm.
Therefore, due to significant low-computational complexity of MAP-ISD
(discussed later in Section 4.2), it would be a better choice over other
detection techniques with marginal loss in performance. Fig. 6 shows the
comparison of BER for $128\times 20$ mMIMO with 4-QAM, $n_{rf}=4$ MBM systems.
It is observed that in contrast to Fig. 5, the BER performance gap between
MAP-ISD and KMAP-IIC increases with increase in $n_{rf}$ from 3 to 4. The key
reason behind such performance degradation is the error propagation in
interference cancellation of MAP-ISD which increases with increase in
$n_{rf}$. It is interesting to note that, KMAP-IIC with $K=1$ achieves BER
performance close to within 0.1 dB to that of IIC, KMAP-IIC with $K=M/4$ and
KMAP-IIC with $K=M/2$. Therefore, in MBM-mMIMO systems with moderate values of
$n_{rf}$, KMAP-IIC with $K=1$ could be a better choice.

Figure 6: BER performance comparison for $128\times 20$, $n_{rf}=4$ MBM-mMIMO
system with 4-QAM modulation.

Fig. 7 presents the BER performance comparison for $128\times 20$ mMIMO with
4-QAM, $n_{rf}=6$ MBM system. It is observed that the performance of KMAP-IIC
with $K=M/2$ is same as that of the IIC algorithm. The performance of KMAP-IIC
with $K=M/4$ and $K=1$ is close to within $0.2$ dB and $0.5$ dB, respectively,
of the performance of IIC algorithm. However, the performance of MAP-ISD is
far inferior as compared to the KMAP-IIC, IIC and IESP algorithms. Clearly,
from Figs. 5, 6 and 7, the performance of MAP-ISD degrades as compared to the
IIC algorithm which makes MAP-ISD less selective for systems having high
values of $n_{rf}$. However, in such scenarios KMAP-IIC still performs up to
mark. Therefore, from the aforementioned BER performance results, it can be
concluded that the choice of low-complexity (discussed in next subsection)
reasonable detection technique between MAP-ISD and KMAP-IIC depends
significantly on the value of $n_{rf}$ RF mirrors and the number of uplink
users.

Figure 7: BER performance comparison for $128\times 16$, $n_{rf}=6$ MBM-mMIMO
system with 4-QAM modulation. Figure 8: BER performance comparison for
$128\times 20$, $n_{rf}=3$ MBM-mMIMO system with 16-QAM modulation. Figure 9:
BER performance comparison for $128\times 20$, $n_{rf}=5$ MBM-mMIMO system
with 16-QAM modulation.

In Figs. 8 and 9, the BER performance comparison is performed for $128\times
20$ mMIMO system with $n_{rf}=3$ and $n_{rf}=5$ with 16-QAM modulation.
Observations reveal that the proposed algorithms perform superior over the
IESP algorithm. However, the performance of MAP-ISD and KMAP-IIC with $K=1$
degrades with an increase in $n_{rf}$ as compared to KMAP-IIC with $K=M/4$,
KMAP-IIC with $K=M/2$ and IIC. It is also observed that KMAP-IIC with $K=M/4$
achieves performance close to KMAP-IIC with $K=M/2$ and IIC algorithms.
In Fig. 10, we present the BER performance with variation in the number of BS
antennas from $N_{r}=80$ to $N_{r}=140$ for $U=20$, $n_{rf}=4$ and 4-QAM MBM-
mMIMO system. It is observed that the BER performance improves with an
increase in the BS antennas. Also, the performance of KMAP-IIC with different
values of $K$, i.e., $K=1,M/4,M/2$, is close to that of the IIC algorithm. On
the other hand, the performance of MAP-ISD is far from IIC. For example, for a
target BER of $5\times 10^{-4}$, required BS antennas for IIC and KMAP-IIC are
around $125$, whereas MAP-ISD requires around $140$ BS antennas. Moreover,
channel-hardening is not observed to degrade the performance compared to the
IIC algorithm.

Figure 10: BER performance comparison for $U=20$ mMIMO with $n_{rf}=4$ and
4-QAM modulation for different values of $N_{r}$.

###  4.2 Computational Complexity

This section discusses the computational complexity of the proposed algorithms
and compares them with the IIC algorithm r8 . First, we discuss the
approximate computations involved in different steps of the IIC algorithm in
Table 3 in terms of floating-point operations (FLOPs) r23 ,r24 . It is to note
that the algorithms’ initialization does not require any computation due to
all zero initial solutions.

Table 3: Approximate number of computations in each iteration of IIC algorithm Steps involved in IIC | Computational complexity
---|---
Interference cancellation | $2N_{r}$
Performing ML search | $(4N_{r}-1)MU|\mathbb{A}|$
Finding the solution using greedy search | $(6N_{r}-1)U^{2}$

In Table 4 and 5, we present the computations required in different steps of
the MAP-ISD and KMAP-IIC algorithms, respectively.

Table 4: Approximate number of computations in each iteration of MAP-ISD Steps involved in MAP-ISD | Computational complexity
---|---
Finding low-complexity matrix inverse (required only once) | $(2N_{r}-1)MU$
Finding the most favorable MAP | $2MN_{r}U$
Finding reliability of the solution | $(4N_{r}-1)U$
Table 5: Approximate number of computations in each iteration of KMAP-IIC Steps involved in KMAP-IIC | Computational complexity
---|---
Finding low-complexity matrix inverse (required only once) | $(2N_{r}-1)MU$
Finding favorable MAPs | $2MN_{r}U$
Performing low-complexity search | $(2N_{r}+4)KU$
Finding the solution using greedy search | $(6N_{r}-1)U^{2}$

It is observed from Tables 3, 4, and 5 that the computations in MAP-ISD are
significantly less than that of IIC and KMAP-IIC algorithm, which is due to
the complex search mechanism in IIC and KMAP-IIC algorithm. The computation of
finding a low-complexity matrix inversion in MAP-ISD and KMAP-IIC is required
only once. Therefore, the reduction in search space due to favorable MAP
selection and low-complexity search are the key factors for the overall
reduction in the proposed algorithms’ computational complexity. Fig. 11
presents the comparison of average number of FLOPs for different MBM-mMIMO
systems at SNR= 5 dB. In Fig. 11, System 1 refers to $128\times 20$ mMIMO
systems with $n_{rf}=3$, system 2 refers to $128\times 20$ mMIMO systems with
$n_{rf}=4$ and system 3 refers to $128\times 16$ mMIMO systems with
$n_{rf}=6$. Observations reveal that KMAP-IIC with $K=M/2$ achieves
significant computational gain of upto $70\%$ and $63\%$ for $128\times 16$,
$n_{rf}=6$ and $128\times 20$, $n_{rf}=4$ systems, respectively. This gain
increases further to $76\%$ and $80\%$ when MAP-ISD algorithm is used for
$128\times 16$, $n_{rf}=6$ and $128\times 20$, $n_{rf}=4$ systems,
respectively. It is clear from the figure that the computational complexity of
MAP-ISD is significantly less as compared to the KMAP-IIC with $K=1$, $K=M/4$
and $K=M/2$. Therefore, in case of MBM-mMIMO systems with less value of
$n_{rf}$, MAP-ISD is suitable over KMAP-IIC with $K=1$ and other algorithms.
On the other hand for moderate values of $n={rf}$ KMAP-IIC with $K=1$ and
$K=M/4$ can be used to achieve a reasonable BER performance.

Figure 11: Computational complexity comparison of different detection
techniques for different MBM-mMIMO systems. Figure 12: Computational
complexity comparison of different detection techniques versus the number of
receive antennas.

In Fig. 12, we present the comparison on the average number of floating point
operations (FLOPs) of the IIC algorithm with MAP-ISD and KMAP-IIC algorithms,
respectively, with respect to the number of receive antennas for MBM-mMIMO
system having $16$ users and $n_{rf}=4$ at each user. Observation reveals that
the computational complexity increase linearly with increase in the number of
receive antennas at the station. It is also observed that the computational
complexity increases with increase in $K$. Furthermore, the complexity of MAP-
ISD is significantly less compared with the KMAP-IIC algorithm and the IIC
algorithm r8 .

##  5 Conclusion

We proposed low complexity interference cancellation based algorithms for
symbol detection in MBM-mMIMO systems. First, we proposed a MAP selection
based ISD algorithm which detects symbols in a sequential manner while
nullifying interference from all other users. Then, we presented an approach
to reduce the search space in the IIC algorithm and devised the KMAP-IIC
algorithm. The key idea was to introduce a metric and a selection rule for
selecting the list of the favorable MAPs in MBM corresponding to each user.
The proposed algorithms achieve superior performance-complexity trade-off over
the existing detection techniques in MBM-mMIMO systems. Observations reveal
that when the number of users and the RF mirrors are comparatively less MAP-
ISD algorithm achieves similar performance with almost $75\%$ and $87\%$
savings in computational complexity as compared to KMAP-IIC and IIC algorithm,
respectively. However, with an increase in the number of users and the RF
mirrors, KMAP-IIC delivers better BER performance over MAP-ISD with almost
$65\%$ to $70\%$ savings in computational complexity for different value of
$K$ over the IIC algorithm.

###### Acknowledgements.

This work is supported by the Start-up Research Grant (file no.
SRG/2019/000654) scheme of Science and Engineering Research Board, Department
of Science and Technology, Government of India.

## Conflict of interest

The authors declare that they have no conflict of interest.

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Maria Cristina Volpe

# Neutrino flavor evolution in dense environments and the r-process

###### Abstract

In dense environments, standard and non-standard neutrino interactions with
the background particles trigger a variety of flavor mechanisms, which can
impact r-process nucleosynthetic abundances. Future observations of a(n)
(extra)galactic supernova will tell us about properties of neutrinos and of
the astrophysical source that produce them. The upcoming measurement of the
diffuse supernova neutrino background constitute a unique source of
information. We highlight some recent developments.

##  1 Introduction

Confirmation that core-collapse supernovae emit 99 $\%$ of a few $10^{53}$ erg
as neutrinos was given by the measurement of the 24 SN1987A $\nu$ events in
Kamiokande [1], IMB [2] and Baksan [3]. Core-collapse supernovae (CC SNe) and
binary neutron star mergers (BNS) are candidate sites for the synthesis of
elements heavier than iron through the rapid neutron capture process
($r$-process). Weak processes involving neutrinos play an important role in
these sites, for the explosion dynamics, for future observations and for
r-process nucleosynthesis. Evidence for the production of r-process elements
in BNS was provided by the kilonova observed with GW170817 [4]. Gravitational
wave measurements will furnish, among others, a precise measurement of the BNS
rate, crucial for unravelling the sites for the $r$-process, a longstanding
open issue.

Numerous theoretical investigations clearly show that flavor evolution impacts
the $r$-process. Challenging self-consistent multidimensional calculations of
the matter composition in neutrino driven winds and neutrino flavor evolution
are still missing, A reference phenomenon, for flavor evolution studies, is
the Mikheev-Smirnov-Wolfenstein (MSW) mechanism [5, 6], due to neutrino-matter
interactions, a phenomenon responsible for high energy solar 8B neutrinos to
1/3. In dense environments, neutrinos change flavor due to neutrino self-
interactions (see e.g. [7, 8] for a review), shock waves and turbulence.
Pinning down the conditions, their impact and their description has triggered
intense investigations since more than a decade. Moreover, efforts have been
made to extend the current theoretical description of neutrino flavor
evolution in dense media and questioned the (widely employed) mean-field
approximation (see e.g. [9, 10]). Here we highlight some MSW-like mechanisms
found in BNS based on the mean-field neutrino evolution equations.

##  2 Flavor evolution from standard and non-standard interactions in BNS

BNS produce large amount of low energy neutrinos with luminosities and average
energies comparable to CC SNe, but with an excess of $\bar{\nu}_{e}$ over
$\nu_{e}$, due to the matter neutron richness and the geometry of the neutrino
emission. Moreover the $\nu_{\mu},\nu_{\tau}$ fluxes can be much smaller than
$\nu_{e}$ ones. The flux predictions present sizeable variations depending on
simulations (see Table VII of Ref.[11]).

The $\bar{\nu}_{e}$ excess in BNS can produce a cancellation between the
matter and the neutrino self-interactions contributions to the neutrino
propagation Hamiltonian. This cancellation can trigger the matter-neutrino
resonance (MNR) and adiabatic conversion of $\nu_{e}$ (standard) and
$\bar{\nu}_{e}$ (symmetric MNR), influencing $r$-process nucleosynthetic
abundances [12]. The flavor content depends on the initial conditions for the
neutrino emission and the trajectories followed by the neutrinos, modifying
$\nu_{e}$ and $\bar{\nu}_{e}$ rates on neutrons and protons by several tens of
percent depending on the neutrino trajectories [11].

The MNR underlying mechanisms is a multiple MSW effect, as shown by Ref.[13]
with a schematic model, and by Ref.[14] based on detailed BNS simulations. By
using a perturbative argument, one can show that the $\nu$ self-interaction
adjusts itself to the matter term multiple times, due to the non-linear
feedback. On the contrary, non-linear feedback does not operate for helicity
coherence that arises because of wrong helicity contributions to the evolution
equations, due to the neutrino mass [14].

Constraints on non-standard interactions (NSI) come from solar, oscillation
experiments, scattering measurements [15] and coherent neutrino-nucleus
scattering [16]. NSI can trigger new flavor conversion mechanisms in dense
environments, such as the MNR in core-collapse supernovae [17] or the
I-resonance, triggered by a cancellation of standard and non-standard matter
terms [18].

The first investigation of NSI effects in BNS has uncovered that the
I-resonance can also be triggered in presence of sizeable $\nu$ self-
interactions as a synchronized MSW effect. The study has also shown that, as
in the supernova context, a complex pattern of flavor conversion mechanisms
takes place along different trajectories, even for very small values of the
NSI couplings (well below current bounds). This can influence significantly
the electron fraction and nucleosynthetic abundances [19].

Moreover, in dense environments, very short scale conversions, termed fast
modes, can occur when more realistic emission at the neutrinosphere is
considered [20], with crossings between the $\nu_{e}$ and $\bar{\nu}_{e}$
angular distributions (see [21] for a review). Such modes, that do not appear
to produce flavor equilibration (in a two neutrino beam schematic model) [22],
can take place in detailed multidimensional simulations (when the ratio of
$\nu_{e}$ and $\bar{\nu}_{e}$ number densities is close to one) [23]. This
result was confirmed by analysis of multidimensional simulations from
different groups.

##  3 Future supernova neutrino observations

The Supernova Early Warning System (SNEWS) is awaiting for the next
(extra)galactic supernova [24] that will bring invaluable information of the
supernova explosion mechanism, the supernova location, on neutrino properties,
non-standard particles, NSI and on the neutron star.

The gravitational binding energy can be determined with $11\%$ precision in
Super-Kamiokande and $3\%$ in Hyper-Kamiokande, with likelihood analysis of
the neutrino fluences (MSW effect included) and combining inverse beta-decay
and elastic scattering. The $\bar{\nu}_{e},\nu_{\mu},\nu_{\tau}$ flux
parameters can be precisely reconstructed (except the pinching parameters).
Based on the EOS of neutron stars, the compactness or mass-radius relation can
be inferred with rather good precision [25, 26].

Complementary to future supernova neutrino observations is the measurement of
the diffuse supernova background from past supernovae (DSNB). The DSNB is
sensitive to redshifted neutrino spectra and flavor mechanisms, the core-
collapse supernova rate, the debated fraction of failed supernovae [27, 28]and
$\nu$ properties, such as neutrino decay [29]. As for flavor phenomena, beyond
the MSW, shock waves (including phase effects) and $\nu$ self-interactions
impact the DSNB fluxes and rates. Shock wave effects can change the latter by
about 10-20 $\%$ (depending on the $\nu$ mass ordering) and can reduce the
sensitivity to neutrino self-interactions [30]. The DSNB rates are also
sensitive to the shock revival time [31].

The sensitivity of the combined analysis (90 $\%$ C.L.) of SK-IV and previous
phases, is on par with DSNB rates from four models, whereas more conservative
predictions are lower about a factor of 4 [32]. The DSNB measurement is
expected with the running Super-Kamiokande (SK)+Gd and the upcoming JUNO and
Hyper-Kamiokande experiments. Exciting observations are ahead of us.

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