# Unfolding for Joint Channel Estimation and Symbol Detection in MIMO
Communication Systems

Swati Bhattacharya1, K.V.S. Hari1, Yonina C. Eldar2

1 Statistical Signal Processing Lab, Department of Electrical Communication
Engineering,
Indian Institute of Science, Bangalore, India.
2 Faculty of Math and Computer Science, Weizmann Institute of Science, Israel.
Email: 1{swatibh<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper proposes a Joint Channel Estimation and Symbol Detection (JED)
scheme for Multiple-Input Multiple-Output (MIMO) wireless communication
systems. Our proposed method for JED using Alternating Direction Method of
Multipliers (JED-ADMM) and its model-based neural network version JED using
Unfolded ADMM (JED-U-ADMM) markedly improve the symbol detection performance
over JED using Alternating Minimization (JED-AM) for a range of MIMO antenna
configurations. Both proposed algorithms exploit the non-smooth constraint,
that occurs as a result of the Quadrature Amplitude Modulation (QAM) data
symbols, to effectively improve the performance using the ADMM iterations. The
proposed unfolded network JED-U-ADMM consists of a few trainable parameters
and requires a small training set. We show the efficacy of the proposed
methods for both uncorrelated and correlated MIMO channels. For certain
configurations, the gain in SNR for a desired BER of $10^{-2}$ for the
proposed JED-ADMM and JED-U-ADMM is upto $4$ dB and is also accompanied by a
significant reduction in computational complexity of upto $75\%$, depending on
the MIMO configuration, as compared to the complexity of JED-AM.

###### Index Terms:

MIMO, wireless, communication, detection, channel estimation, ADMM, unfolding

##  I Introduction

Massive multiple-input multiple-output (MIMO) systems with tens or hundreds of
antennas at both the transmitter and receiver have emerged as a promising
technique for achieving high spectral efficiency [1]. The conventional MIMO
configuration has a larger number of receive antennas at the base station (BS)
than the number of transmit antennas at the users (UEs). With the rise of
fifth-generation wireless communication, an emerging scenario is where many
users connect to a single BS with limited antennas i.e., $N\leq K$, referred
to as overloaded MIMO [2, 3, 4, 5, 6, 7, 8].

One of the critical aspects in utilizing the full potential of massive MIMO is
accurate symbol detection at the BS for which Maximum Likelihood (ML)
detection is optimal but computationally prohibitive. Linear detectors such as
Zero Forcing (ZF), Minimum Mean Square Error (MMSE) or more advanced nonlinear
detectors [9, 10] and deep learning based detectors [11, 12] perform well when
$N$ is much larger than $K$ but quickly deteriorate in performance as the
ratio $N/K$ approaches $1$ which is the overloaded MIMO regime. Symbol
detection for overloaded MIMO has been studied in the literature using
sparsity [2, 3, 4], convex optimization [5] and deep learning [6, 7, 8, 13].

Traditionally, the tasks of channel estimation and symbol detection are
decoupled and performed separately using two independent modules in cascade,
in which Channel State Information (CSI) is first estimated followed by symbol
detection. Joint Channel Estimation and Symbol Detection (JED) methods [14,
15, 16, 17, 18] consider the transmitted data as additional pilots, and
exchange information iteratively between the two aforesaid modules as depicted
in Fig. 1, thereby resulting in better channel and symbol estimates. Message
passing based techniques for JED were proposed in [14, 15, 16, 17], however,
they depend heavily on the channel sparsity and data statistics and require
extensive probability calculations. In contrast, a simple iterative technique
for JED presented in [18], which is based on Alternating Minimization (AM),
updates the channel and symbols using MMSE-based and ZF estimates,
respectively. We refer to the technique proposed in [18] as JED-AM and provide
further details in Section III-B.

Deep Unfolding, a model-based neural network providing interpretability with
reduced number of tunable parameters and training, was introduced in the
seminal work of Gregor and LeCun in [19] and has gained attention over the
last few years. These neural networks which are derived from a parent
iterative algorithm are shown to improve the performance over fully-connected
black-box detectors along with reduced computational complexity [20, 21, 22,
23, 12, 7, 6, 8, 11, 24, 25, 26, 13, 27, 28, 29, 30, 31, 32] and have been
widely used in the domain of image processing [20, 21, 22, 23].

In the context of wireless communications, unfolding has been used for symbol
detection assuming knowledge of CSI, [12, 7, 6, 8, 11, 25, 26, 27] and for
jointly obtaining the channel and symbol estimates [13, 24, 33, 34]. One of
the first works presented a deep unfolded version of Orthogonal AMP (OAMP),
named OAMP-Net2, in [13]. An unfolded soft-output based symbol detection
method is outlined in [24]. A more recent work in [34] base their unfolded
network on the Expectation-Maximization (EM) algorithm. A related work on JED
is presented in [33] which uses a convolutional neural network instead of a
model-based unfolded network.

Figure 1: A generic JED architecture with channel estimation and symbol
detection blocks that iteratively exchange information until convergence.

Our main contributions are summarized as follows:

  * •

The first algorithm, JED-ADMM, exploits the property of the data symbols being
in the Quadrature Amplitude Modulation (QAM) constellation to impose a non-
smooth constraint and propose an equivalent ADMM formulation for JED. We see
from the simulation studies of Section VI-B1 that JED-ADMM results in an
improved bit error rate (BER) performance over the existing JED-AM [18]. The
performance of ADMM-based algorithms greatly depends on the penalty parameter
[35]. We see that the BER yielded by JED-ADMM is also affected by the ADMM
penalty parameter, which is decided empirically. To mitigate the loss in
performance that could possibly be caused by an improper choice of the ADMM
penalty parameter, we let it be trained from the data leading to the next
algorithm.

  * •

The second algorithm unfolds the ADMM iterations of JED-ADMM to build a model-
based network (JED-U-ADMM). We also introduce some additional trainable
parameters, otherwise not included in JED-ADMM, to construct a flexible
unfolded network. Simulation studies in Section VI-B2 demonstrate the
improvement in performance as well as reduction in computational complexity.

  * •

We note that the previous works on JED, that use unfolding, demonstrate the
performance of their algorithms for either $N>K$ in [24], $N=K$ in [13] or
$N\geq K$ as in [34]. In contrast, we provide simulation studies for JED-ADMM
and JED-U-ADMM in all possible MIMO configurations $N>K$ and $N\leq K$.

We describe the system model and existing JED in Sections II and III-B
respectively. The reformulation via ADMM is in Section IV-A and its deep
unfolded version in Section V-A. Simulation parameters for our experiments are
described in Section VI-A and the experimental results for our proposed
approach are detailed in Section VI-B.

Symbol | Definition | Symbol | Definition
---|---|---|---
$L$ | Number of iterations/layers in the unfolded network | $\mathbb{S}_{\beta}$ | rectangular $\beta$-QAM constellation
$N$ | Number of BS antennas | $K$ | Number of single-antenna UEs
$T_{t}$ | Number of pilot symbols | $T_{d}$ | Number of data symbols
$\mathbf{H}$ | channel between all the $K$ UEs and the BS with $N$ antennas | $\mathbf{X}_{d}$ | data matrix transmitted from $K$ UEs over $T_{d}$ time instants
$\mathbf{X}_{t}$ | pilot matrix transmitted from $K$ UEs over $T_{t}$ time instants | $\mathbf{V}$ | noise matrix at the receiver
$\mathbf{Y}_{t}$ | pilot matrix received at BS over $T_{t}$ time instants | $\mathbf{Y}_{d}$ | data matrix received at BS over $T_{d}$ time instants
$\mathcal{C}$ | convex hull of $\mathbb{S}_{\beta}$ | $\mathbf{Z}_{d}$ | auxiliary variable for $\mathbf{X}_{d}$
$\rho_{c}$ | Spatial correlation coefficient between antennas. | $\mathbf{\Lambda}$ | dual variable for ADMM updates
$\rho$ / $\rho_{l}$ | ADMM penalty parameter for MMSE-like update of $\tilde{\mathbf{X}}^{(l)}_{d}$ |  $\theta$ / $\theta_{l}$ | relaxation parameter in $\tanh(.)$
$\alpha$ / $\alpha_{l}$ | relaxation parameter for dual variable $\mathbf{\Lambda}$ |  $\gamma$ / $\gamma_{l}$ | relaxation parameter for MMSE-like update of $\tilde{\mathbf{H}}^{(l)}$

TABLE I: List of Symbols

Symbol | Definition | Symbol | Definition
---|---|---|---
JED | Joint Channel Estimation and Symbol Detection | AM | Alternating Minimization
ADMM | Alternating Direction Method of Multipliers | DL | Deep Learning based
CSI | Channel State Information | QAM | Quadrature Amplitude Modulation

TABLE II: List of Acronyms

Throughout the paper we use the following notation: Bold-faced uppercase and
lowercase letters $\mathbf{A},\mathbf{a}$ and plain faced letters $a$
represent a matrix, column vector and scalar respectively. The conjugate-
transpose and the Moore-Penrose pseudoinverse of $\mathbf{A}$ are given by
$\mathbf{A}^{*},\mathbf{A}^{\dagger}$ respectively. The symbol
$\mathbb{S}_{\beta}$ is the constellation for $\beta$-QAM and $\mathbf{I}_{n}$
is the $n\times n$ identity matrix. The normal distribution with mean $\mu$
and variance $\sigma^{2}$ is given by
$x\sim\mathcal{CN}\left(\mu,\sigma^{2}\right)$ and
$x\sim\mathcal{U}\left[a,b\right]$ is the continuous uniform distribution on
$\left[a,b\right]$. We use
$\mathcal{P}_{\infty,\sqrt{\beta}}\left(\mathbf{X}\right)$ to denote the
element-wise projection of $\mathbf{X}$ onto the infinity-norm ball of radius
$\sqrt{\beta}$ and $\mathbbm{1}$ is the indicator function. The list of
symbols and acronyms used in the paper are summarized in Tables I and II,
respectively.

##  II System Model and Problem Formulation

Consider the uplink transmission in a narrow-band MIMO system with $N$ BS
antennas and $K$ single-antenna users, as depicted in Fig. 2.

Figure 2: Uplink MIMO communication scenario with $N$ antennas at the BS and
$K$ single-antenna UEs.

The channel is considered to have a block fading structure and remains
constant over $T$ transmissions. The first $T_{t}$ symbols are pilot symbols
used to obtain an initial estimate of CSI and the remaining $T_{d}=T-T_{t}$
symbols are for data i.e., $\mathbf{X}_{t}\in\mathbb{C}^{K\times T_{t}}$ is
the transmitted pilot matrix and $\mathbf{X}_{d}\in\mathbb{C}^{K\times T_{d}}$
is the transmitted data matrix. At the $n^{th}$ instant, user $k$ transmits
data $\\{x_{k}\left(n\right)\in\mathbb{S}_{\beta}:n\in\\{1,2,...,T_{d}\\}\\}$.
Here, $\mathbb{S}_{\beta}$ is the rectangular $\beta$-QAM constellation with
real and imaginary parts in
$\\{-\sqrt{\beta}+1,-\sqrt{\beta}+3,...,\sqrt{\beta}-3,\sqrt{\beta}-1\\}$. We
consider two channel models:- (a) a Rayleigh fading channel
$\mathbf{H}\in\mathbb{C}^{N\times K}$ with elements drawn from the Gaussian
distribution
$\mathbf{H}\sim\mathcal{CN}\left(\boldsymbol{0},\sigma_{h}^{2}\mathbf{I}_{K}\right)$
and (b) a correlated channel
$\mathbf{H}=\mathbf{R}^{\frac{1}{2}}_{r}\mathbf{H}_{w}\mathbf{R}^{\frac{1}{2}}_{t}$,
with $\mathbf{R}_{r}$ and $\mathbf{R}_{t}$ being the correlation matrices at
the receiver and the transmitter respectively and $\mathbf{H}_{w}$ being a
Rayleigh fading channel. Let $\mathbf{V}\in\mathbb{C}^{N\times T}$ be
circularly symmetric complex white Gaussian noise at the receiver;
$\mathbf{V}\sim\mathcal{CN}\left(\boldsymbol{0},\sigma_{v}^{2}\mathbf{I}_{N}\right)$.
Denoting the received pilot and data matrix by
$\mathbf{Y}_{t}\in\mathbb{C}^{N\times T_{t}}$ and
$\mathbf{Y}_{d}\in\mathbb{C}^{N\times T_{d}}$ respectively, the system model
is as follows:

$\mathbf{Y}=\mathbf{H}\mathbf{X}+\mathbf{V}\,=\,\mathbf{H}\left[\mathbf{X}_{t}\hskip
5.69046pt\mathbf{X}_{d}\right]+\left[\mathbf{V}_{t}\hskip
5.69046pt\mathbf{V}_{d}\right]\,=\,\left[\mathbf{Y}_{t}\hskip
5.69046pt\mathbf{Y}_{d}\right]$ (1)

where $\mathbf{X}\in\mathbb{C}^{K\times T}$ is the complete transmitted signal
matrix and $\mathbf{Y}\in\mathbb{C}^{N\times T}$ is the complete received
signal matrix over $T$ transmissions.

##  III Background

###  III-A Symbol Detection using ADMM

Symbol detection, assuming perfect CSI, using ADMM has previously been studied
in [36] where the idea of introducing an auxiliary variable for solving a non-
smooth convex problem was shown to enhance the performance.

The task of estimating symbols $\hat{\mathbf{x}}$ from
$\mathbf{y}=\mathbf{Hx}+\mathbf{v}$, with
$\mathbf{y}\in\mathbb{C}^{N},\mathbf{H}\in\mathbb{C}^{N\times K}$ known and
$\mathbf{v}\in\mathbb{C}^{N}$ being the noise can be posed as (2).

$\hat{\mathbf{x}}=\underset{\mathbf{x}\in\mathbb{S}^{K}_{\beta}}{\arg\min}\frac{1}{2}\left\lVert\mathbf{y}-\mathbf{Hx}\right\rVert^{2}_{2}.$
(2)

The data symbol $x_{k}$, the $k^{th}$ element of $\mathbf{x}$, is discrete
valued with the probability mass function (pmf) :
$\mathbb{P}\\{x_{k}\left(n\right)=a\\}=\frac{\textstyle{1}}{\textstyle{\beta}},\hskip
8.5359pta\in\mathbb{S}_{\beta}$. A common relaxation is to assume that the
symbols are continuous variables that are uniformly distributed over
$\mathcal{C}$ which is the convex hull of $\mathbb{S}_{\beta}$ [18] i.e.,
$Re\\{x_{k}\left(n\right)\\},Im\\{x_{k}\left(n\right)\\}\sim\mathcal{U}\left[-\sqrt{\beta}+1,\sqrt{\beta}-1\right]$,

and

$\mathcal{C}=\biggl{\\{}\mathbf{x}\bigg{|}\left\lVert\begin{bmatrix}vec\left(Re\\{\mathbf{x}\\}\right)\\\
vec\left(Im\\{\mathbf{x}\\}\right)\end{bmatrix}\right\rVert_{\infty}\leq\sqrt{\beta}-1\biggr{\\}}.$
(3)

We can approximate the constraint $\mathbf{x}\in\mathbb{S}^{K}_{\beta}$ being
in QAM constellation by its convex polytope $\mathcal{C}$ and hence, we can
get the symbol estimates by solving (4).

$\hat{\mathbf{x}}=\underset{\mathbf{x}\in\mathcal{C}}{\arg\min}\frac{1}{2}\left\lVert\mathbf{y}-\mathbf{Hx}\right\rVert^{2}_{2}.$
(4)

Solving (4) using conventional interior-point method would incur huge
complexity. Since the constraint set $\mathcal{C}$ is convex and non-smooth,
the authors in [36] apply the ADMM framework and rewrite the symbol detection
problem as (5).

$\hat{\mathbf{x}}=\underset{\mathbf{x},\mathbf{z}\in\mathcal{C}}{\arg\min}\frac{1}{2}\left\lVert\mathbf{y}-\mathbf{Hx}\right\rVert^{2}_{2}+g\left(\mathbf{z}\right)\hskip
5.69046pt\text{s.t.}\hskip 5.69046pt\mathbf{z}=\mathbf{x},$ (5)

where $g\left(\mathbf{z}\right)$ is the indicator function on set
$\mathcal{C}$ i.e. $\mathbbm{1}_{\\{\mathbf{z}\in\mathcal{C}\\}}$. The
augmented Lagrangian, in scaled form, for (5) is

$\mathcal{L}_{\rho}\left(\mathbf{x},\mathbf{z},\mathbf{\lambda}\right)=\frac{1}{2}\left\lVert\mathbf{y}-\mathbf{Hx}\right\rVert^{2}_{2}+\mathbbm{1}_{\\{\mathbf{z}\in\mathcal{C}\\}}+\frac{\rho}{2}\left\lVert\mathbf{z}-\mathbf{x}-\boldsymbol{\lambda}\right\rVert^{2}_{2},$
(6)

with $\boldsymbol{\lambda}$ being the scaled dual variable associated with the
equality constraint $\mathbf{z}=\mathbf{x}$. Simulation results in [36]
demonstrate that the ADMM based solution (ADMIN) outperforms linear detectors
such as MMSE-based solutions in terms of the achieved BER.

###  III-B Joint Channel Estimation and Symbol Detection approaches

The goal of JED is to simultaneously obtain the channel and symbol estimates
$\hat{\mathbf{H}}$, $\hat{\mathbf{X}}_{d}$ from the known received signal
$\mathbf{Y}$ and the short pilot sequence $\mathbf{X}_{t}$. The authors in
[18] formulate the simultaneous estimation of $\hat{\mathbf{H}}$ and
$\hat{\mathbf{X}}_{d}$ as

$\begin{split}\\{\hat{\mathbf{H}},\hat{\mathbf{X}}_{d}\\}=\underset{\mathbf{H}\in\mathbb{C}^{N\times
K},\mathbf{X}_{d}\in\mathbb{S}_{\beta}^{K\times
T_{d}}}{\arg\min}&\left\lVert\mathbf{Y}_{t}-\mathbf{H}\mathbf{X}_{t}\right\rVert^{2}_{F}\\\
+\left\lVert\mathbf{Y}_{d}-\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}_{F}&+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\mathbf{H}\right\rVert^{2}_{F}.\end{split}$
(7)

Solving (7) requires a combinatorial approach, due to the discrete set
$\mathbb{S}_{\beta}^{K\times T_{d}}$. It is common to instead solve (8) by
using the convex constraint set $\mathcal{C}$ in (3). Accordingly, the JED
optimization problem is given by:

$\begin{split}\\{\hat{\mathbf{H}},\hat{\mathbf{X}}_{d}\\}=&\underset{\mathbf{H},\mathbf{X}_{d}}{\arg\min}\left\lVert\mathbf{Y}_{t}-\mathbf{H}\mathbf{X}_{t}\right\rVert^{2}_{F}+\left\lVert\mathbf{Y}_{d}-\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}_{F}\\\
&+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\mathbf{H}\right\rVert^{2}_{F}\hskip
8.5359pts.t.\hskip 8.5359pt\mathbf{X}_{d}\in\mathcal{C}.\end{split}$ (8)

The authors of [18] use AM to solve (8) in an iterative manner with ZF for
symbol estimates $\hat{\mathbf{X}}_{d}$ and MMSE for channel
$\hat{\mathbf{H}}$, as summarized in Steps $1$ and $2$ of Algorithm 1.

input: $\mathbf{Y}$, $\mathbf{X}_{t}$ output:
$\hat{\mathbf{X}}_{d},\hat{\mathbf{H}}$ Initialize
$\tilde{\mathbf{H}}^{(0)}\leftarrow\mathbf{Y}_{t}\mathbf{X}^{*}_{t}\left(\mathbf{X}_{t}\mathbf{X}^{*}_{t}+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\mathbf{I}_{K}\right)^{-1}$.
for $l=1,2,\ldots,L$ do Step $\mathbf{1}$:
$\tilde{\mathbf{X}}^{(l)}_{d}\leftarrow\mathcal{P}_{\infty,\sqrt{\beta}}\left(\tilde{\mathbf{H}}^{(l-1)^{\dagger}}\mathbf{Y}_{d}\right)$.
Step $\mathbf{2}$:
$\tilde{\mathbf{H}}^{(l)}\leftarrow\left(\mathbf{Y}_{t}\mathbf{X}^{*}_{t}+\mathbf{Y}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}\right)\left(\mathbf{X}_{t}\mathbf{X}^{*}_{t}+\tilde{\mathbf{X}}^{(l)}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\mathbf{I}_{K}\right)^{-1}$.
end for Final symbol and channel estimates :
$\hat{\mathbf{X}}_{d}=\mathbb{S}_{\beta}\left(\tilde{\mathbf{X}}^{(L)}_{d}\right),\hat{\mathbf{H}}=\tilde{\mathbf{H}}^{(L)}$.

Algorithm 1 JED-AM [18]

In the next section, we propose our algorithms JED-ADMM and JED-U-ADMM.

##  IV ADMM-based techniques for Joint Channel Estimation and Symbol Detection

###  IV-A JED-ADMM

We observe from (5) that the optimization problem to obtain the symbol
estimates consists of a smooth term
$\left(\frac{\textstyle{1}}{\textstyle{2}}\left\lVert\mathbf{y}-\mathbf{H}\mathbf{x}\right\rVert^{2}_{2}\right)$
and a non-smooth term
$\left(\mathbbm{1}_{\\{\mathbf{z}\in\mathcal{C}\\}}\right)$. ADMM serves as a
powerful tool in such cases [35] by separating out the non-smooth part using
an auxiliary variable. It has been observed that auxiliary variables provide
better performance than existing methods without an auxiliary variable, in
image processing applications [37, 20] as well as for symbol detection in
wireless communication application [36, 32]. In the image processing problem,
the improvement in performance was due to updating the solution vector along
three variables (auxiliary variable, blurring kernel and latent sharp image),
as a result of the additional auxiliary variable, instead of the original two
variables. Thus, inspired by the technique in [37, 20] and the observed
superior performance of ADMM over ZF/MMSE-based detectors (with perfect CSI)
in [36], we propose to use ADMM for JED that updates over three variables
$\mathbf{H},\mathbf{X}_{d}$ and an additional auxiliary variable,
$\mathbf{Z}_{d}\in\mathbb{C}^{K\times T_{d}}$.

Reformulation of the JED optimization problem using ADMM, as presented in
[38], for the original variables $\mathbf{H},\mathbf{X}_{d}$ and the auxiliary
variable $\mathbf{Z}_{d}$ becomes:

$\begin{split}\\{\hat{\mathbf{H}},\hat{\mathbf{X}}_{d},\hat{\mathbf{Z}}_{d}\\}=\underset{\mathbf{H},\mathbf{X}_{d},\mathbf{Z}_{d}}{\arg\min}&\left\lVert\mathbf{Y}_{t}-\mathbf{H}\mathbf{X}_{t}\right\rVert^{2}_{F}+\left\lVert\mathbf{Y}_{d}-\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}_{F}\\\
+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\mathbf{H}\right\rVert^{2}_{F}+&\mathbbm{1}_{\\{\mathbf{Z}_{d}\in\mathcal{C}\\}}\hskip
8.5359pts.t.\hskip 5.69046pt\mathbf{X}_{d}=\mathbf{Z}_{d}.\end{split}$ (9)

The term $\mathbbm{1}_{\\{\mathbf{Z}_{d}\in\mathcal{C}\\}}$ implies that
$\mathbf{Z}_{d}$ lies in the constraint set $\mathcal{C}$ described in (3).
The auxiliary variable $\mathbf{Z}_{d}$ is introduced to separate the non-
smooth part of the optimization problem i.e.,
$\mathbbm{1}_{\\{\mathbf{Z}_{d}\in\mathcal{C}\\}}$.

The augmented Lagrangian for the model in (9) is

$\begin{split}\tilde{\mathcal{L}}_{\rho}\left(\mathbf{X}_{d},\mathbf{Z}_{d},\mathbf{\Lambda^{{}^{\prime}}},\mathbf{H}\right)=&\left\lVert\mathbf{Y}_{t}-\mathbf{H}\mathbf{X}_{t}\right\rVert^{2}_{F}+\left\lVert\mathbf{Y}_{d}-\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}_{F}\\\
+\frac{\rho}{2}\left\lVert\mathbf{X}_{d}-\mathbf{Z}_{d}\right\rVert^{2}_{F}+&\text{tr}\left(\mathbf{\Lambda}^{{}^{\prime}*}\left(\mathbf{X}_{d}-\mathbf{Z}_{d}\right)\right)+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\mathbf{H}\right\rVert^{2}_{F},\end{split}$
(10)

where $\mathbf{\Lambda^{{}^{\prime}}}$ is the Lagrangian multiplier and
$\rho>0$ is the ADMM penalty parameter. For simplicity, we consider the scaled
ADMM [35] with
$\mathbf{\Lambda}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{\textstyle{\mathbf{\Lambda}^{{}^{\prime}}}}{\textstyle{\rho}}$
that translates the augmented Lagrangian in (10) to

$\begin{split}\mathcal{L}_{\rho}\left(\mathbf{X}_{d},\mathbf{Z}_{d},\mathbf{\Lambda},\mathbf{H}\right)=&\left\lVert\mathbf{Y}_{t}-\mathbf{H}\mathbf{X}_{t}\right\rVert^{2}_{F}+\left\lVert\mathbf{Y}_{d}-\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}_{F}\\\
-\frac{\rho}{2}\left\lVert\mathbf{\Lambda}\right\rVert^{2}_{F}+&\frac{\rho}{2}\left\lVert\mathbf{X}_{d}-\mathbf{Z}_{d}+\mathbf{\Lambda}\right\rVert^{2}_{F}+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\mathbf{H}\right\rVert^{2}_{F}.\end{split}$
(11)

We next minimize
$\mathcal{L}_{\rho}\left(\mathbf{X}_{d},\mathbf{Z}_{d},\mathbf{\Lambda},\mathbf{H}\right)$
in (11) over the variables successively in an iterative manner as in [38]. In
the $l^{th}$ iteration, we denote
$\tilde{\mathbf{X}}^{(l)}_{d},\tilde{\mathbf{Z}}^{(l)}_{d},\tilde{\mathbf{\Lambda}}^{(l)}_{d}\tilde{\mathbf{H}}^{(l)}_{d},$
as estimates of
${\mathbf{X}}_{d},{\mathbf{Z}}_{d},{\mathbf{\Lambda}}_{d},{\mathbf{H}}_{d}$,
respectively. The first step in the $l^{th}$ iteration is to estimate
$\tilde{\mathbf{X}}^{(l)}_{d}$ by solving the optimization function using the
estimates of other parameters from the $(l-1)^{th}$ iteration,

$\tilde{\mathbf{X}}^{(l)}_{d}=\underset{\mathbf{X}_{d}\in\mathbb{C}^{K\times
T_{d}}}{\arg\min}\hskip
5.69046pt\mathcal{L}_{\rho}\left(\mathbf{X}_{d},\tilde{\mathbf{Z}}^{(l-1)}_{d},\tilde{\mathbf{\Lambda}}^{(l-1)},\tilde{\mathbf{H}}^{(l-1)}\right),$
(12)

whose solution, for a given $\rho$, is given by solving for
$\nabla\mathcal{L}_{\rho}\left(\mathbf{X}_{d},\tilde{\mathbf{Z}}^{(l-1)}_{d},\tilde{\mathbf{\Lambda}}^{(l-1)},\tilde{\mathbf{H}}^{(l-1)}\right)=\mathbf{0}$
that yields

$\begin{split}\tilde{\mathbf{X}}^{(l)}_{d}\,=\,&\left(\tilde{\mathbf{H}}^{(l-1)^{*}}\tilde{\mathbf{H}}^{(l-1)}+\frac{\textstyle{\rho}}{\textstyle{2}}\mathbf{I}\right)^{-1}\\\
&\times\left(\tilde{\mathbf{H}}^{(l-1)^{*}}\mathbf{Y}_{d}+\frac{\textstyle{\rho}}{\textstyle{2}}\left(\tilde{\mathbf{Z}}^{(l-1)}_{d}-\tilde{\mathbf{\Lambda}}^{(l-1)}\right)\right).\end{split}$
(13)

Next, we estimate $\tilde{\mathbf{Z}}^{(l)}_{d}$ by solving the following
optimization function using
$\tilde{\mathbf{X}}^{(l)}_{d},\tilde{\mathbf{\Lambda}}^{(l-1)},\tilde{\mathbf{H}}^{(l-1)}$,

$\tilde{\mathbf{Z}}^{(l)}_{d}=\underset{\mathbf{Z}_{d}\in\mathcal{C}}{\arg\min}\hskip
5.69046pt\mathcal{L}_{\rho}\left(\tilde{\mathbf{X}}^{(l)}_{d},\mathbf{Z}_{d},\tilde{\mathbf{\Lambda}}^{(l-1)},\tilde{\mathbf{H}}^{(l-1)}\right).$
(14)

The solution is given by

$\tilde{\mathbf{Z}}^{(l)}_{d}=\mathcal{P}_{\infty,\sqrt{\beta}}\left(\tilde{\mathbf{X}}^{(l)}_{d}+\tilde{\mathbf{\Lambda}}^{(l-1)}\right).$
(15)

The next step is to obtain $\tilde{\mathbf{\Lambda}}^{(l)}$ by solving

$\underset{\mathbf{\Lambda}\in\mathbb{C}^{K\times T_{d}}}{\arg\min}\hskip
5.69046pt\mathcal{L}_{\rho}\left(\tilde{\mathbf{X}}^{(l)}_{d},\tilde{\mathbf{Z}}^{(l)}_{d},\mathbf{\Lambda},\tilde{\mathbf{H}}^{(l-1)}\right),$
(16)

whose solution is given by

$\tilde{\mathbf{\Lambda}}^{(l)}=\tilde{\mathbf{\Lambda}}^{(l-1)}+\left(\tilde{\mathbf{X}}^{(l)}_{d}-\tilde{\mathbf{Z}}^{(l)}_{d}\right).$
(17)

The final step is to obtain a solution of $\tilde{\mathbf{H}}^{(l)}$ from

$\underset{\mathbf{H}\in\mathbb{C}^{N\times K}}{\arg\min}\hskip
5.69046pt\mathcal{L}_{\rho}\left(\tilde{\mathbf{X}}^{(l)}_{d},\tilde{\mathbf{Z}}^{(l)}_{d},\tilde{\mathbf{\Lambda}}^{(l)},\mathbf{H}\right),$
(18)

which is the MMSE solution

$\begin{split}\tilde{\mathbf{H}}^{(l)}=&\left(\mathbf{Y}_{t}\mathbf{X}^{*}_{t}+\mathbf{Y}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}\right)\\\
&\times\left(\mathbf{X}_{t}\mathbf{X}^{*}_{t}+\tilde{\mathbf{X}}^{(l)}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}+\frac{\textstyle{\sigma_{v}^{2}}}{\textstyle{\sigma_{h}^{2}}}\mathbf{I}_{K}\right)^{-1}.\end{split}$
(19)

The algorithm is summarized below with the appropriate choice of the initial
value of the parameters. The final estimate is obtained after $L$ iterations.

input: $\mathbf{Y}$, $\mathbf{X}_{t}$ output:
$\hat{\mathbf{X}}_{d},\hat{\mathbf{H}}$ Initialize
$\tilde{\mathbf{Z}}^{(0)}=\boldsymbol{0},\tilde{\mathbf{\Lambda}}^{(0)}=\boldsymbol{0}$,
$\tilde{\mathbf{H}}^{(0)}\longleftarrow\mathbf{Y}_{t}\mathbf{X}^{*}_{t}\left(\mathbf{X}_{t}\mathbf{X}^{*}_{t}+\frac{\textstyle{\sigma_{v}^{2}}}{\textstyle{\sigma_{h}^{2}}}\mathbf{I}_{K}\right)^{-1}$.
for $l=1,2,\dots,L$ do Step $\mathbf{1}$: update of $\mathbf{X}_{d}$ :
$\tilde{\mathbf{X}}^{(l)}_{d}\longleftarrow\left(\tilde{\mathbf{H}}^{(l-1)^{*}}\tilde{\mathbf{H}}^{(l-1)}+\frac{\textstyle{\rho}}{\textstyle{2}}\mathbf{I}\right)^{-1}\left(\tilde{\mathbf{H}}^{(l-1)^{*}}\mathbf{Y}_{d}+\frac{\textstyle{\rho}}{\textstyle{2}}\left(\tilde{\mathbf{Z}}^{(l-1)}_{d}-\tilde{\mathbf{\Lambda}}^{(l-1)}\right)\right)$.
Step $\mathbf{2}$: update of $\mathbf{Z}_{d}$ :
$\tilde{\mathbf{Z}}^{(l)}_{d}\longleftarrow\mathcal{P}_{\infty,\sqrt{\beta}}\left(\tilde{\mathbf{X}}^{(l)}_{d}+\tilde{\mathbf{\Lambda}}^{(l-1)}\right)$.
Step $\mathbf{3}$: update of $\mathbf{\Lambda}$ :
$\tilde{\mathbf{\Lambda}}^{(l)}\longleftarrow\tilde{\mathbf{\Lambda}}^{(l-1)}+\left(\tilde{\mathbf{X}}^{(l)}_{d}-\tilde{\mathbf{Z}}^{(l)}_{d}\right)$.
Step $\mathbf{4}$: update of $\mathbf{H}$ :
$\tilde{\mathbf{H}}^{(l)}\longleftarrow\left(\mathbf{Y}_{t}\mathbf{X}^{*}_{t}+\mathbf{Y}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}\right)\left(\mathbf{X}_{t}\mathbf{X}^{*}_{t}+\tilde{\mathbf{X}}^{(l)}_{d}\tilde{\mathbf{X}}^{(l)^{*}}_{d}+\frac{\textstyle{\sigma_{v}^{2}}}{\textstyle{\sigma_{h}^{2}}}\mathbf{I}_{K}\right)^{-1}$.
end for Final symbol and channel estimates :
$\hat{\mathbf{X}}_{d}=\mathbb{S}_{\beta}\left(\tilde{\mathbf{X}}^{(L)}_{d}\right),\hat{\mathbf{H}}=\tilde{\mathbf{H}}^{(L)}$.

Algorithm 2 Algorithm for Joint Channel Estimation and Symbol Detection using
ADMM (JED-ADMM)

###  IV-B Computational Complexity

The computational complexity in terms of Floating Point Operations (FLOPS) for
the proposed JED-ADMM and existing JED-AM is presented next. We note that the
multiplication of two matrices of size $m\times n$ and $n\times p$ requires
$mnp$ FLOPS and the inversion of a square matrix of size $n$ needs $n^{3}$
FLOPS. Both JED-ADMM and JED-AM use MMSE for the channel update. However, JED-
ADMM needs an extra $NKT_{d}$ FLOPS per iteration for the symbol detection
step. The summary of the complexity of the algorithms is given in Table III.
As an example, considering $N=16,K=16,T_{d}=512,T_{t}=K$, we observe that JED-
ADMM requires $557056$ FLOPS per iteration and JED-AM needs $425984$ FLOPS,
the former being $\approx 1.3$ times more expensive than the latter.

Computational complexity in terms of FLOPS for each step
---
proposed JED-ADMM in Algorithm 2 | JED-AM in Algorithm 1 [18]
initial estimate $\tilde{\mathbf{H}}^{(0)}$ | $NKT_{t}+K^{2}T_{t}+K^{2}N+K^{3}$ | initial estimate $\tilde{\mathbf{H}}^{(0)}$ | $NKT_{t}+K^{2}T_{t}+K^{2}N+K^{3}$
Steps in $l^{th}$ iteration | FLOPS | Steps in $l^{th}$ iteration | FLOPS
Step 1 : $\tilde{\mathbf{X}}^{(l)}_{d}$ | $K^{2}N+K^{3}+KNT_{d}+K^{2}T_{d}$ | Step 1 : $\tilde{\mathbf{X}}^{(l)}_{d}$ | $K^{3}+K^{2}N+KNT_{d}+2KT_{d}$
Step 2 : $\tilde{\mathbf{Z}}^{(l)}_{d}$ | $2KT_{d}$ | Step 2 : $\tilde{\mathbf{H}}^{(l)}_{d}$ | $NK(T_{t}+T_{d})+K^{2}(T_{t}+T_{d})+K^{2}N$
Step 3 : $\tilde{\mathbf{\Lambda}}^{(l)}_{d}$ | - |  | -
Step 4 : $\tilde{\mathbf{H}}^{(l)}_{d}$ | $NK(T_{t}+T_{d})+K^{2}(T_{t}+T_{d})+K^{2}N$ |  | -
Total no. of FLOPS over $L$ iterations
JED-ADMM | $(3K^{2}N+K^{2}(2T_{d}+2T_{t})+NK(2T_{d}+2T_{t})+2K^{3})L$ | JED-AM | $(3K^{2}N+K^{2}(2T_{d}+2T_{t})+NK(T_{d}+2T_{t})+2K^{3})L$

TABLE III: Comparison of FLOPS (multiplication) required for the baseline
algorithms with $L$ denoting the number of iterations in JED-AM and JED-ADMM.

Considering a larger MIMO system $N=64,K=80,T_{d}=512,T_{t}=K$, we find that
JED-ADMM needs $14315520$ FLOPS, which is $\approx 1.22$ times more than that
of JED-AM. Hence, we note from the above examples that as $N$ and $K$
increase, this difference becomes negligible because the matrix inversion
steps dominate the complexity.
A concise version of our work on JED-ADMM for overloaded MIMO $\left(N\leq
K\right)$ systems was presented in [38]. We demonstrate, in Experiment $1$ of
Section VI-B1, that the proposed JED-ADMM algorithm outperforms JED-AM for
properly chosen values of the ADMM penalty $\rho$. This variation in
performance of JED-ADMM, depending on the value of $\rho$, motivates the
development of JED-U-ADMM as detailed in Section V.

##  V Unfolding Framework for Deep Learning based Joint Estimation and
Detection

We observed from Figs. 4, 5 and 6 that the choice of $\rho$ and number of
iterations are critical for JED-ADMM. To mitigate the decline in performances
that could be caused by poor choice of these parameters, we propose to use
deep unfolding on JED-ADMM framework by introducing trainable parameters in
the update equations.

###  V-A Unfolding of JED-ADMM

We follow the approach of [13, 6] and introduce only scalar learnable
quantities to propose am unfolded network JED-U-ADMM, that is derived from the
unfolding of the ADMM iterations of JED-ADMM. In Steps $1$-$4$ of JED-ADMM
(Algorithm 2) the right hand side of the equation is essentially an affine
combination of two or more terms with the weighting coefficient set to $1$. To
allow for flexibility in the framework, we introduce additional parameters
$\theta_{l},\alpha_{l}$ that do not necessarily restrict the weighting
coefficients to $1$. Furthermore, we replace the fixed quantity
$\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$ with
$\rho_{l}$ and $\gamma_{l}$ which can be learnt from the data. We then learn
$\rho_{l},\theta_{l},\alpha_{l},\gamma_{l}$ from data.

We note that the system model as described in (1) has complex-valued
variables. We recast the model using real-valued variables for ease of
implementation using PyTorch [39], as:

$\overline{\mathbf{Y}}=\overline{\mathbf{H}}\,\overline{\mathbf{X}}+\overline{\mathbf{V}},$
(20)

where

$\displaystyle\overline{\mathbf{Y}}=$
$\displaystyle\left[Re\left(\mathbf{Y}\right)^{T}Im\left(\mathbf{Y}\right)^{T}\right]^{T},\hskip
5.69046pt\overline{\mathbf{X}}=\left[Re\left(\mathbf{X}\right)^{T}Im\left(\mathbf{X}\right)^{T}\right]^{T}.$
$\displaystyle\overline{\mathbf{H}}=\begin{bmatrix}Re\left(\mathbf{H}\right)&-Im\left(\mathbf{H}\right)\\\
Im\left(\mathbf{H}\right)&Re\left(\mathbf{H}\right)\end{bmatrix}.$

Similar to the pilot and data portions of $\mathbf{X}$ in (1), we denote the
respective real counterparts of pilot and data using the overline
$\overline{(.)}$, as $\overline{\mathbf{X}}_{t}$, $\overline{\mathbf{X}}_{d}$,
$\overline{\mathbf{Y}}_{t}$, $\overline{\mathbf{Y}}_{d}$. The corresponding
real auxiliary variable is now denoted by $\overline{\mathbf{Z}}_{d}$ and
$\overline{\mathbf{\Lambda}}$ denotes the real dual variable for the ADMM
updates.

Solving for the Lagrangian in a similar manner as in (11) with the real-valued
variables of (20), we obtain

$\begin{split}\mathcal{L}_{\rho}\left(\overline{\mathbf{X}}_{d},\overline{\mathbf{Z}}_{d},\overline{\mathbf{\Lambda}},\overline{\mathbf{H}}\right)=&\left\lVert\overline{\mathbf{Y}}_{t}-\overline{\mathbf{H}}\,\overline{\mathbf{X}}_{t}\right\rVert^{2}_{F}+\left\lVert\overline{\mathbf{Y}}_{d}-\overline{\mathbf{H}}\,\overline{\mathbf{X}}_{d}\right\rVert^{2}_{F}\\\
-\frac{\rho}{2}\left\lVert\mathbf{\overline{\Lambda}}\right\rVert^{2}_{F}+&\frac{\rho}{2}\left\lVert\overline{\mathbf{X}}_{d}-\overline{\mathbf{Z}}_{d}+\overline{\mathbf{\Lambda}}\right\rVert^{2}_{F}+\frac{\sigma_{v}^{2}}{\sigma_{h}^{2}}\left\lVert\overline{\mathbf{H}}\right\rVert^{2}_{F}.\end{split}$
(21)

We next minimize
$\mathcal{L}_{\rho}\left(\overline{\mathbf{X}}_{d},\overline{\mathbf{Z}}_{d},\overline{\mathbf{\Lambda}},\overline{\mathbf{H}}\right)$
in (21) over the variables successively using a multi-layered neural network.
In the $l^{th}$ layer, we denote
$\tilde{\overline{\mathbf{X}}}^{(l)}_{d},\tilde{\overline{\mathbf{Z}}}^{(l)}_{d},\tilde{\overline{\mathbf{\Lambda}}}^{(l)}_{d},\tilde{\overline{\mathbf{H}}}^{(l)}_{d},$
as estimates obtained after that layer.

Successive minimization over the variables for
$\mathcal{L}_{\rho}\left(\overline{\mathbf{X}}_{d},\overline{\mathbf{Z}}_{d},\overline{\mathbf{\Lambda}},\overline{\mathbf{H}}\right)$
yield similar update equations as Steps $1$-$4$ in Algorithm 2 with
$\tilde{\mathbf{X}}^{(l)}_{d}$, $\tilde{\mathbf{Z}}^{(l)}_{d}$,
$\tilde{\mathbf{\Lambda}}^{(l)}$, $\tilde{\mathbf{H}}^{(l)}$ being replaced by
their real counterparts $\tilde{\overline{\mathbf{X}}}^{(l)}_{d}$,
$\tilde{\overline{\mathbf{Z}}}^{(l)}_{d}$,
$\tilde{\overline{\mathbf{\Lambda}}}^{(l)}$,
$\tilde{\overline{\mathbf{H}}}^{(l)}$, respectively.

Network architecture: The network architecture of JED-U-ADMM network is
obtained by unfolding each of the iterations of JED-ADMM. There are $L$
cascaded layers in the network, with the same architecture, but with different
trainable parameters $\Theta_{l}$ in each of the $l^{th}$ layer. We provide a
pictorial description of the JED-U-ADMM in Fig. 3.

For the purpose of unfolding, we introduce trainable parameters in each of the
update equations in the following manner:

  * •

In Step $1$ for the MMSE-like update of $\tilde{\overline{\mathbf{X}}}_{d}$,
we replace the fixed ADMM penalty parameter $\rho$ with $\rho_{l}$ that can
vary with the layers $l$.

  * •

To incorporate the non-linearity of neural networks, we use the $\tanh$
activation function for the update of the auxiliary variable
$\tilde{\overline{\mathbf{Z}}}_{d}$ in Step $2$, instead of
$\mathcal{P}_{\infty,\sqrt{\beta}}$.

  * •

In Step $3$, the error term
$\left(\tilde{\overline{\mathbf{X}}}^{(l)}_{d}-\tilde{\overline{\mathbf{Z}}}^{(l)}_{d}\right)$
has a coefficient of $1$. We can modify it to having $\alpha_{l}$ as the
coefficient and let the data decide the optimal value of $\alpha_{l}$.

  * •

For the MMSE-like update of $\tilde{\overline{\mathbf{H}}}$ in Step $4$, we
replace the fixed regularization parameter
$\frac{\textstyle{\sigma_{v}^{2}}}{\textstyle{\sigma_{h}^{2}}}$, that is
dependent on the noise variance and might not always be known, with a
trainable parameter $\gamma_{l}$.

We summarize the above steps for JED-U-ADMM in Algorithm 3. Note that for
$\rho_{l},\gamma_{l}=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$,
$\alpha_{l}=1$ and by replacing the $\tanh$ function with
$\mathcal{P}_{\infty,\sqrt{\beta}}$ (the standard projection function in
literature), we get back the JED-ADMM algorithm in Algorithm 2.

Figure 3: Block diagram of the JED-U-ADMM architecture. There are $L$ cascaded
layers, each with $4$ trainable parameters
$\rho_{l},\theta_{l},\alpha_{l},\gamma_{l}$. Steps $1,2,3,4$ refer to the
corresponding steps in Algorithm 3.

Training : The loss function for backpropagation is defined as
$\mathbf{L}_{\Theta}=\left\lVert\overline{\mathbf{X}}_{d}-\tilde{\overline{\mathbf{X}}}^{(L)}_{d}\right\rVert^{2}_{F}$.

In the training phase of JED-U-ADMM, we start with an initial value of
$\Theta_{l}$ which is updated in accordance with the loss function
$\mathbf{L}_{\Theta}$ in each training epoch. The update rule for $\Theta_{l}$
is determined by the Adam optimizer [40]. Once the network is trained for
enough number of epochs, the weights ( i.e., $\Theta_{l}$ values ) are frozen
and we use this trained network to detect/estimate the unknown data. Hence,
basically the equations governing JED-U-ADMM are obtained following the
framework of JED-ADMM where the parameters $\Theta_{l}$ are obtained in a
data-driven manner.

Computational complexity : In the training phase, since JED-U-ADMM is a neural
network, there is an accompanying training cost which depends on the number of
trainable parameters. For the JED-U-ADMM architecture summarized in Algorithm
3, it can be seen that we have distinct trainable parameters for each of the
$L$ layers leading to the training of $4L$ parameters.

input: $\overline{\mathbf{Y}}$, $\overline{\mathbf{X}}_{t}$ output:
$\hat{\overline{\mathbf{X}}}_{d},\hat{\overline{\mathbf{H}}}$ Trainable
parameters :
$\Theta_{l}=\\{\rho_{l},\theta_{l},\alpha_{l},\gamma_{l}\\},\hskip
5.69046ptl=1,2,\ldots,L$.  Initialize
$\tilde{\overline{\mathbf{Z}}}_{d}^{(0)}=\boldsymbol{0},\tilde{\overline{\mathbf{\Lambda}}}^{(0)}=\boldsymbol{0}$,
$\tilde{\overline{\mathbf{H}}}^{(0)}\longleftarrow\overline{\mathbf{Y}}_{t}\overline{\mathbf{X}}^{*}_{t}\left(\overline{\mathbf{X}}_{t}\overline{\mathbf{X}}^{*}_{t}+\gamma_{0}\mathbf{I}_{K}\right)^{-1}$.
for $l=1,2,\ldots,L$ do Step $\mathbf{1}$: update of
$\overline{\mathbf{X}}_{d}$ :
$\tilde{\overline{\mathbf{X}}}^{(l)}_{d}\longleftarrow\left(\tilde{\overline{\mathbf{H}}}^{(l-1)^{*}}\tilde{\overline{\mathbf{H}}}^{(l-1)}+\rho_{l}\mathbf{I}\right)^{-1}\left(\tilde{\overline{\mathbf{H}}}^{(l-1)^{*}}\overline{\mathbf{Y}}_{d}+\rho_{l}\left(\tilde{\overline{\mathbf{Z}}}^{(l-1)}_{d}-\tilde{\overline{\mathbf{\Lambda}}}^{(l-1)}\right)\right)$.
Step $\mathbf{2}$: update of $\overline{\mathbf{Z}}_{d}$ :
$\tilde{\overline{\mathbf{Z}}}^{(l)}_{d}\longleftarrow\tanh\left(\frac{\textstyle{\tilde{\overline{\mathbf{X}}}^{(l)}_{d}+\tilde{\overline{\mathbf{\Lambda}}}^{(l-1)}_{d}}}{\textstyle{|\theta_{l}|}}\right)$.
Step $\mathbf{3}$: update of $\overline{\mathbf{\Lambda}}$ :
$\tilde{\overline{\mathbf{\Lambda}}}^{(l)}\longleftarrow\tilde{\overline{\mathbf{\Lambda}}}^{(l-1)}+\alpha_{l}\left(\tilde{\overline{\mathbf{X}}}^{(l)}_{d}-\tilde{\overline{\mathbf{Z}}}^{(l)}_{d}\right)$.
Step $\mathbf{4}$: update of $\overline{\mathbf{H}}$ :
$\tilde{\overline{\mathbf{H}}}^{(l)}\longleftarrow\left(\overline{\mathbf{Y}}_{t}\overline{\mathbf{X}}^{*}_{t}+\overline{\mathbf{Y}}_{d}\overline{\mathbf{X}}^{(l)^{*}}_{d}\right)\left(\overline{\mathbf{X}}_{t}\overline{\mathbf{X}}^{*}_{t}+\tilde{\overline{\mathbf{X}}}^{(l)}_{d}\overline{\mathbf{X}}^{(l)^{*}}_{d}+\gamma_{l}\mathbf{I}_{K}\right)^{-1}$.
end for Final symbol and channel estimates :
$\hat{\overline{\mathbf{X}}}_{d}=\mathbb{S}_{\beta}\left(\tilde{\overline{\mathbf{X}}}^{(L)}_{d}\right),\hat{\overline{\mathbf{H}}}=\tilde{\overline{\mathbf{H}}}^{(L)}$.

Algorithm 3 Proposed Algorithm for Unfolding based Joint Channel Estimation
and Symbol Detection using ADMM (JED-U-ADMM)

Shared parameter architecture : A variation could be to allow some of the
parameters to be shared across the layers. The reasoning for this alternate
unfolded architecture arises from the intuition that it is possible that the
values of some of the parameters do not vary significantly across the layers.
In such cases, parameter sharing can lead to reduction of training complexity
while maintaining similar performance. Hence, we propose the following shared
architecture, henceforth referred to as ‘JED-U-ADMM : shared Params’. The
parameters $\alpha_{l},\theta_{l},\gamma_{l}$ are shared across all the $L$
layers which effectively replaces these $3L$ parameters with just $3$
trainable parameters i.e. $\alpha,\theta,\gamma$. Thus, in the Steps $2,3,4$
of Algorithm 3, $\theta_{l}=\theta$, $\alpha_{l}=\alpha$ and
$\gamma_{l}=\gamma$ respectively. This reduces the total trainable parameters
to only $L+4$ trainable parameters. Note that for the initial
$\tilde{\overline{\mathbf{H}}}^{(0)}$ update, we use $\gamma_{0}$. In the
estimation phase, the computational complexity of JED-ADMM and JED-U-ADMM is
the same (Table III).

In the next section we study the performance of the proposed methods for
different scenarios.

##  VI Simulation Results

We study the BER performance of our proposed algorithms JED-U-ADMM, JED-ADMM
and JED-AM.

###  VI-A Simulation scenario:

In this paper, we consider an uplink scenario where the BS with $N$ antennas
receives signals from $K$ UEs each with a single transmit antenna. We assume
the UEs are geographically well separated and the signals propagate through
independent channels.

Transmitter parameters: The pilot matrix $\mathbf{X}_{t}$ consists of columns
from the Discrete Fourier Transform (DFT) matrix and the data matrix
$\mathbf{X}_{d}$ is sampled from a $4$-QAM $\left(\beta=4\right)$
constellation. Thus, the constellation set is given by
$\mathbb{S}_{\beta}=\\{1+j,1-j,-1+j,-1-j\\}$ and the average energy per
symbol, denoted by $E_{s}$ equals $2$. The pilot length $T_{t}$ for obtaining
an initial CSI was set to the optimal pilot length required i.e., $T_{t}=K$.

Channel: The channel is assumed to be constant over $T=300$ transmissions. We
consider two cases of the channel: i.i.d fading Rayleigh channel,
$\mathbf{H}\sim\mathcal{CN}\left(\mathbf{0},\frac{1}{K}\mathbf{I}\right)$ and
correlated channels
$\mathbf{H}=\mathbf{R}_{r}^{\frac{1}{2}}\mathbf{H}_{w}\mathbf{R}_{t}^{\frac{1}{2}}$,
with $\mathbf{H}_{w}$ being an i.i.d Rayleigh fading channel. The correlation
matrices are $\mathbf{R}_{r}$ in (22) and $\mathbf{R}_{t}=\mathbf{I}_{K}$ at
the receiver and transmitter respectively [41].

$\mathbf{R}_{r}=\begin{bmatrix}1&\rho_{c}&...&\rho^{K-2}_{c}&\rho^{K-1}_{c}\\\
\rho_{c}&1&\rho_{c}&...&\rho^{K-2}_{c}\\\ ...&...&...&...&...\\\
\rho^{K-1}_{c}&\rho^{K-2}_{c}&...&...&1\end{bmatrix}.$ (22)

In this paper, we consider an uplink channel model which justifies the
introduction of a correlation matrix at the BS antennas at the receiver. The
users are assumed to be geographically separated and hence uncorrelated.

Receiver algorithms: For JED-ADMM, the receiver implements the iterative
algorithms described in Algorithm 2 where $L$ refers to the number of
iterations. $\rho$ is chosen empirically in a manner as in [36]. For the
unfolded JED-U-ADMM, we use Algorithm 3 to train the network over $100/200$
epochs to get the trained values of $\Theta_{l}$. In the initial $T_{t}$
transmissions when the transmitter sends pilots to the receiver to obtain an
initial CSI, we choose a pilot structure such that only one antenna sends a
pilot at a particular time instant and the rest of the antennas are turned
off. During the detection phase we assume data transmitted over all antennas
and use the trained $\Theta_{l}$ to detect symbols. For each channel
realization, $T=(512+T_{t})$ bits are sent and $2\times 10^{4}$ channel
realizations are used. We use the PyTorch [39] environment with Adam
optimizer.

Performance Metric: The performance metric to compare the algorithms is chosen
to be the bit error rate (BER). We compute the BER based on errors observed in
$5\times 10^{6}$ bits from each of the transmitter antennas. In all the
experiments, we define the Signal-to-Noise Ratio (SNR) per receiver antenna as

$\text{$\text{SNR}$}=\frac{\mathbf{E}\left[\left\lVert\mathbf{H}\mathbf{X}_{d}\right\rVert^{2}\right]}{\mathbf{E}\left[\left\lVert\mathbf{V}_{d}\right\rVert^{2}\right]}=\frac{E_{s}}{\sigma^{2}_{v}}.$
(23)

###  VI-B Experiments

We consider experiments to study the performance of the proposed JED-U-ADMM,
JED-ADMM and JED-AM [18] methods.

####  VI-B1 Study the performance of JED-AM and JED-ADMM

We first study the performance of JED-ADMM and JED-AM under various MIMO
configurations and channel conditions.

###### Experiment 1

Study the effect of varying $\rho$
In this experiment, we study how the choice of the ADMM penalty parameter
$\rho$ affects the BER performance of JED-ADMM. Three different MIMO
configurations are considered - $N>K$, $N=K$ and $N<K$. For $N<=K$, $20$
iterations of JED-ADMM is considered while $100$ iterations were taken for
$N>K$. We consider two values of
$\rho=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$ and
$\rho=4\times\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$.

Figure 4:  $N=K$ : This plot shows the variation of BER vs SNR as $\rho$ is
varied for $20$ iterations of JED-ADMM and JED-AM in a $32\times 32$ MIMO for
independent Rayleigh channels. Figure 5:  $N>K$ : This plot shows the
variation of BER vs SNR as $\rho$ is varied for $100$ iterations of JED-ADMM
and JED-AM in a $32\times 16$ MIMO for independent Rayleigh channels. Figure
6:  $N<K$ : This plot shows the variation of BER vs SNR as $\rho$ is varied
for $20$ iterations of JED-ADMM and JED-AM in a $64\times 80$ MIMO for
independent Rayleigh channels.

Fig. 4 demonstrates that the choice of $\rho$ affects the BER performance, and
a lower value of $\rho$ leads to an improvement in BER. For example, at
$\text{SNR}=20$ dB, with
$\rho=4\times\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$,
we get BER $\approx 10^{-2}$ while for
$\rho=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$, the BER
value drops to $\approx 2\times 10^{-3}$. However, we still achieve an
improvement over JED-AM irrespective of the chosen value of $\rho$. For $N>K$
in Fig. 5, we observe that for
$\rho=4\times\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$,
the performance of JED-ADMM and JED-AM are very close while for
$\rho=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$, JED-ADMM
gives $\approx 0.5$ dB improvement in terms of SNR at high $\text{SNR}>10$ dB.
We also see from Fig. 6 that for $N<K$, JED-ADMM with
$\rho=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$ gives an
order of magnitude lesser BER at $\text{SNR}=24$ dB than JED-AM. On the other
hand, JED-ADMM with
$\rho=4\times\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$
performs similarly as JED-AM.

Thus, we observe that the choice of $\rho$ affects the BER performance of JED-
ADMM which motivates us to consider letting the data decide the optimal value
of $\rho$ in a trainable setting. This is the motivation for the unfolding of
JED-ADMM iterations to give us the model-based deep learning JED-U-ADMM
network.

###### Experiment 2

Study the effect of varying number of iterations, for a given $\rho$, in the
presence of independent MIMO channels with $N=K$. We study the BER performance
of the proposed JED-ADMM algorithm as a function of the number of iterations
for the case of independent and identically distributed (i.i.d) Rayleigh
channels when the number of receive antennas equals the number of transmit
antennas, $N=K$ for $N=16,32$. The value of the ADMM penalty parameter
$\rho=4\times\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$ is
kept constant throughout all the iterations.

(a) (b)

Figure 7:  This plot shows the variation of BER vs SNR depending on the number
of iterations of JED-ADMM and JED-AM for (a) $16\times 16$ and (b) $32\times
32$ MIMO for independent Rayleigh channels.

In Fig. 7, we observe that with the increase in number of iterations, JED-ADMM
yields a decreasing BER across all values of SNR. For example, in Fig. 7 (a),
at $\text{SNR}=20$ dB, $50$ iterations of JED-AM gives us a BER of $\approx
10^{-2}$, whereas $50$ iterations of JED-ADMM produces BER of $\approx
10^{-3}$, which is an order of magnitude lower. We also note from Figs. 7 (a)
and 7 (b) that for a given number of iterations and SNR, the performance is
better for lower values of $N=K$.

It is to be noted, from Fig. 7, that even as low as $10$ iterations of JED-
ADMM surpass $50$ iterations of JED-AM. Hence, even though the number of FLOPS
per iteration is slightly higher for JED-ADMM than JED-AM (Table III), we see
an overall reduction in complexity owing to the lesser number of iterations
needed. Accordingly, for a $16\times 16$ MIMO as depicted in Fig. 7 (a), $50$
iterations of JED-AM requires $21299200$ FLOPS whereas $10$ iterations of JED-
ADMM requires only $5570560$. Thus JED-ADMM reduces the overall FLOPS count by
$\approx 75\%$ and yet achieves a lower BER.

###### Experiment 3

Study the effect of correlation coefficient $\rho_{c}$ at the receiver with
$N=K$
We investigate how JED-ADMM performs for correlated channels, that follow the
Kronecker correlation channel model [41] with the correlation coefficient
$\rho_{c}=0.5$ and $0.9$ as per (22), for the $N=K$ MIMO configuration.

(a) (b)

Figure 8:  This plot shows the effect of the receiver correlation coefficient
$\rho_{c}$ on the variation of BER vs SNR for (a) $16\times 16$ and (b)
$32\times 32$ MIMO for JED-ADMM and JED-AM for correlated channels [41].

We see from Fig. 8, as the correlation is decreased from $\rho_{c}=0.9$ to
$\rho_{c}=0.5$, we see almost an order of magnitude lower BER for both JED-
ADMM and JED-AM and that JED-ADMM performs better than JED-AM for both cases.
For all values of $\rho_{c}$ considered and both the MIMO configurations of
Figs. 8 (a) and (b), JED-ADMM consistently shows better BER performance than
JED-AM. Also, for a given SNR = $20$ dB and $N=K=16$, JED-ADMM achieves BER
$\approx 10^{-3}$ whereas JED-AM achieves BER $\approx 10^{-2}$ i.e., an order
of magnitude improvement in BER by JED-ADMM. It is to be noted from Figs. 8
(a) and (b), that as the correlation coefficient $\rho_{c}$ decreases, the gap
between the performance of JED-ADMM and JED-AM also increases. For
$\rho_{c}=0.5$ and at an $\text{SNR}=20$ dB, JED-ADMM provides a considerably
lesser BER, an order of magnitude lower, than JED-AM.

####  VI-B2 Study the performance of JED-ADMM and proposed deep unfolded JED-
U-ADMM

In the following experiments, we study our deep unfolded version JED-U-ADMM,
in Algorithm 3, and JED-ADMM algorithm.

###### Experiment 4

Study the effect of sharing parameters in the presence of independent MIMO
channels with $N=K$. We now study how using shared parameters
$\left(\alpha,\theta,\gamma\right)$ across all the $L$ layers affects the BER
performance of the unfolded network i.e. the ‘JED-U-ADMM : shared Params’
architecture. We consider $10$ layers in a $16\times 16$ MIMO system with
i.i.d Rayleigh fading channel. Such a technique leads to having only $10+4=14$
trainable parameters instead of $40$ trainable parameters (if there was no
sharing i.e. the ‘JED-U-ADMM : unshared Params’ architecture).

Figure 9:  This plot shows the effect of using shared parameters across the
layers of the unfolded network for a $16\times 16$ MIMO system for independent
Rayleigh channels.

It can be observed in Fig. 9 that the loss in BER performance caused by the
lesser number of trainable parameters in the ‘JED-U-ADMM : shared Params’
architecture is negligible, even at higher $\text{SNR}>16$ dB. The reduction
in number of trainable parameters is also accompanied by a reduction in the
computational complexity of the unfolded network during the training phase.
Hence, to simplify the architecture, in the subsequent experiments, we use
only the ‘JED-U-ADMM : shared Params’ architecture.

###### Experiment 5

Study the effect of the number of layers on the unfolded network in the
presence of independent MIMO channels with $N=K$.
In this experiment, we investigate how the BER performance of ‘JED-U-ADMM :
shared Params’ changes as the number of layers is varied from $5$ to $20$. We
consider i.i.d Rayleigh fading channel in a $16\times 16$ MIMO system.

Figure 10:  This plot shows the effect of varying the number of layers in JED-
U-ADMM in a $16\times 16$ MIMO system for independent Rayleigh channels.

In Fig 10, we observe that as the number of layers in increased from $5$ to
$15$ the BER performance steadily improves. However, the network with $20$
layers provides marginal improvement and therefore, in the future set of
experiments with ‘JED-U-ADMM : shared Params’, we only consider the cases for
$5$ and $10$ layers.

###### Experiment 6

Study the effect of unfolding in the presence of independent MIMO channels
with $N=K$.
We study the BER performance of the unfolded algorithm JED-U-ADMM as a
function of the number of layers of the neural network when the number of
transmit and receive antennas are same in the presence of i.i.d Rayleigh
fading channels.

(a) (b)

Figure 11:  This plot shows the effect of deep unfolding on the variation of
BER vs SNR for (a) $16\times 16$ and (b) $32\times 32$ MIMO for independent
Rayleigh channels.

Fig. 11 demonstrates that unfolding leads to significant improvement in BER.
For $16\times 16$ MIMO in Fig. 11 (a), we observe that JED-U-ADMM with $10$
layers yields an SNR improvement of $2.9$ dB in achieving a BER of $10^{-2}$,
whereas for $32\times 32$ MIMO in Fig. 11 (b), we get an SNR improvement of
$4$ dB for a BER of $10^{-2}$. Comparing in terms of BER obtained at a given
SNR, we see that for the $16\times 16$ MIMO in Fig. 11 (a), JED-U-ADMM needs
$10$ layers to reach BER $\approx 10^{-3}$ at $\text{SNR}=16$ dB, which is an
order of improvement over JED-ADMM that requires $20$ iterations to give a BER
$\approx 10^{-2}$ at the same SNR. We also note that for $16\times 16$ MIMO in
Fig. 11 (a), JED-U-ADMM with $5$ layers has almost similar performance as JED-
ADMM with $20$ iterations and $\rho=\sigma^{2}_{v}/\sigma^{2}_{h}$. On the
other hand, Fig. 11 (b) shows that for $32\times 32$ MIMO system, even $5$
layers of JED-U-ADMM shows more than $2$ dB SNR improvement for SNR $>14$ dB.
Thus, as we increase the system size, deep unfolding seems to perform much
better than conventional iterative methods.

###### Experiment 7

Study the effect of varying the ratio $N/K$ in an unfolded network in the
presence of independent MIMO channels.
This experiment showcases the performance of JED-U-ADMM when we operate in the
standard MIMO regime of $N>K$ and in the overloaded MIMO regime of $N<K$.

(a) (b)

Figure 12:  This plot shows the variation of BER vs SNR for (a) $N>K$ :
$32\times 16$ MIMO and (b) $N<K$ : $64\times 80$ for independent Rayleigh
channels.

For $N>K$ (the standard MIMO regime), we had observed from Fig. 5 that JED-
ADMM needs a large number of iterations of about $100$ to outperform JED-AM.
However, with deep unfolding, we can improve the performance, especially at
high SNR values by about $1$ dB, as compared to JED-ADMM, with even $10$
layers as shown in Fig. 12 (a).
For the overloaded MIMO in Fig. 12 (b), we see that JED-U-ADMM needs only $10$
layers to consistently yield a lower BER than JED-ADMM. At SNR = $20$ dB, JED-
U-ADMM with $10$ layers gives a $2$ dB improvement in SNR compared to $20$
iterations of JED-ADMM. As an example, at $\text{SNR}=22$ dB, we need $20$
iterations of JED-ADMM yield a BER of $10^{-2}$ whereas even $10$ layers of
JED-U-ADMM gives us BER $<10^{-3}$.

###### Experiment 8

Study the effect of correlation $\rho_{c}$ at the receiver antennas for
unfolded network with $N\geq K$. In this experiment, we demonstrate the
performance of JED-ADMM and JED-U-ADMM, for the case of correlated channels.
We consider the Kronecker correlation channel model [41] with the correlation
coefficient $\rho_{c}$ at the receiver.

(a) (b)

Figure 13:  This plot shows the variation of BER vs SNR for (a) $16\times 16$
and (b) $32\times 16$ MIMO where the channels are correlated at the Rx with
$\rho_{c}$ (22) for correlated channels [41].

We observe from Fig. 13 that as the correlation at the receiver antennas
increases, both JED-ADMM and JED-U-ADMM degrade in performance. For the case
of $16\times 16$ MIMO, we compare $5$ layers of JED-U-ADMM with $20$
iterations of JED-ADMM in 13 (a). We see that for $\rho_{c}=0.1$, using
unfolding results in slightly lesser BER of JED-U-ADMM than JED-ADMM, with the
difference being more prominent at higher $\text{SNR}>16$ dB. However, for
$\rho_{c}=0.5$ and $0.9$, the BER curves of JED-U-ADMM and JED-ADMM are almost
the same.
In Fig. 13 (b), we present the BER curves for $32\times 32$ with $10$ layers
of JED-U-ADMM and $20$ iterations of JED-ADMM. For $\rho_{c}=0.1$ and $0.5$,
we see that unfolding improves the BER performance throughout the range of SNR
values presented, whereas for $\rho_{c}=0.9$, the improvement is visible only
for $\text{SNR}>10$ dB.

###### Experiment 9

Study the effect of unfolding for a fixed $K=16$ and varying $N$ in the
presence of independent MIMO channels.
We study the performance of the proposed algorithms by fixing the number of
UEs $K=16$ and vary the number of BS antennas $N=8,16,32,48,64$.

Figure 14:  This plot shows the variation of BER vs SNR for $K=16$,
$N=8,16,32,48,64$ MIMO for independent Rayleigh channels.

As $N$ increases, the system transitions from an overloaded scenario to the
standard MIMO yielding a lower BER. As $N$ is increased from to $16$ to $32$,
the BER falls drastically from $\approx 10^{-2}$ to $\approx 10^{-5}$ at
$\text{SNR}=14$ dB. Upon further increasing $N$ to $48$ and $64$, we note that
the margin of improvement in performance gradually reduces.

We now present a summary of the observations in the next section.

##  VII Discussion and Conclusion

In this paper, we propose two algorithms for Joint Channel Estimation and
Symbol Detection, JED-ADMM and its unfolded version, JED-U-ADMM. The ADMM
based algorithms for symbol detection exploit the non-smooth constraint
arising from the QAM structure of the data symbols which significantly improve
the BER performance compared to the state-of-the-art JED-AM. We provide a
summary our findings, as per the experiments conducted.

Considering the fully loaded MIMO $N=K$, our proposed algorithm JED-ADMM, in
Algorithm 2, results in an SNR improvement of about $3$ dB for BER = $10^{-2}$
for $\rho=\frac{\textstyle{\sigma^{2}_{v}}}{\textstyle{\sigma^{2}_{h}}}$
chosen, as shown in Fig. 4. The unfolded algorithm JED-U-ADMM, in Algorithm 3,
further provides $3$-$4$ dB SNR improvement over JED-ADMM as shown in Fig. 11
(a), thereby demonstrating the power of model-based neural network
architectures. For the standard MIMO regime $N>K$, JED-U-ADMM needs only $10$
iterations to outperform $100$ iterations of JED-ADMM, by giving an SNR
improvement of $1$ dB whereas JED-ADMM requires $(\sim 100)$ iterations to
yield an an SNR improvement of $0.5$ dB over JED-AM, as depicted in Fig. 5. In
the overloaded scenario $N<K$, we note from Fig. 6 that JED-ADMM continues to
exhibit waterfall decrease in BER whereas JED-AM saturates around BER of
$2\times 10^{-2}$. Fig. 12 (b) depicts an additional SNR improvement of $2$ dB
by JED-U-ADMM. Furthermore, under correlated channel model in Fig. 8, JED-ADMM
markedly lowers the BER, especially for lower values of $\rho_{c}$. The
unfolded network JED-U-ADMM further leads to slight improvement in BER
performance over JED-ADMM in Fig. 13.

Hence, we conclude that both of our proposed algorithms JED-ADMM and JED-U-
ADMM enhance the BER performance in MIMO communication systems over the
existing JED technique in [18]. The deep unfolded network JED-U-ADMM provides
us with an additional SNR gain over the iterative JED-ADMM in much fewer
iterations $\leq 10$, thus reducing the computational complexity. The unfolded
network contains few trainable parameters and has simple update equations. In
our future work, we plan to extend the proposed scheme to higher orders of
modulation.

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