# RIS-Aided Interference Cancellation for Joint Device-to-Device and Cellular
Communications

Ly V. Nguyen and A. Lee Swindlehurst Email<EMAIL_ADDRESS><EMAIL_ADDRESS>This work was supported by the National Science Foundation under grants
CNS-2107182 and ECCS-2030029. Center for Pervasive Communications and
Computing, University of California, Irvine, CA, USA

###### Abstract

Joint device-to-device (D2D) and cellular communication is a promising
technology for enhancing the spectral efficiency of future wireless networks.
However, the interference management problem is challenging since the
operating devices and the cellular users share the same spectrum. The emerging
reconfigurable intelligent surfaces (RIS) technology is a potentially ideal
solution for this interference problem since RISs can shape the wireless
channel in desired ways. This paper considers an RIS-aided joint D2D and
cellular communication system where the RIS is exploited to cancel
interference to the D2D links and maximize the minimum signal-to-interference
plus noise (SINR) of the device pairs and cellular users. First, we adopt a
popular alternating optimization (AO) approach to solve the minimum SINR
maximization problem. Then, we propose an interference cancellation (IC)-based
approach whose complexity is much lower than that of the AO algorithm. We
derive a representation for the RIS phase shift vector which cancels the
interference to the D2D links. Based on this representation, the RIS phase
shift optimization problem is transformed into an effective D2D channel
optimization. We show that the AO approach can converge faster and can even
give better performance when it is initialized by the proposed IC solution. We
also show that for the case of a single D2D pair, the proposed IC approach can
be implemented with limited feedback from the single receive device.

##  I Introduction

With the proliferation of mobile users and devices, joint device-to-device
(D2D) and cellular communication has been envisioned as a promising technology
for enhancing spectral efficiency in future wireless networks [1]. In this
technology, devices and cellular users operate on the same spectrum, and
therefore significantly improve the network spectral efficiency. However, the
interference management problem is very challenging due to spectrum sharing
between the operating devices and cellular users [2, 3, 4]. Fortunately, the
D2D interference problem can be effectively addressed by reconfigurable
intelligent surfaces (RISs)–a recent innovative technology capable of shaping
the wireless channel in desired ways [5, 6]. For example, the works in [7, 8,
9] show that an RIS can be used to effectively null out or mitigate
interference in D2D systems. While the work in [7] considers a conventional
passive RIS structure which can only change the phase of the incoming signal,
the authors in [8, 9] show that a hybrid RIS structure with the capability of
simultaneously changing the phase and amplitude of the incoming signal can
provide better interference cancellation performance. However, these prior
works all consider systems with only D2D links, i.e., there were no active co-
channel cellular users present.

RIS-assisted joint D2D and cellular communication has been studied in several
recent papers. For example, the works in [10] and [11] perform analyses for
RIS-assisted D2D communications under Rician and Nakagami-$m$ fading channels,
respectively. The study in [10] investigates the effective capacity and
performs mode selection to determine whether the D2D communication is
established through an RIS or a base station (BS). The authors of [11] derived
closed-form expressions for the outage probability of the D2D links and the
resulting diversity order. Unlike [10, 11] which perform system analysis, some
other recent papers focus on system design and optimization [12, 13, 14, 15,
16, 17]. In particular, the sum-rate maximization problem was considered in
[12, 13, 14, 15] and deep reinforcement learning was used in [12, 14].
However, the works in [12, 14] assume that the BS is equipped with only one
antenna and only one D2D pair is allowed to share the spectrum with a single
cellular user. Multiple D2D pairs sharing the same spectrum with a cellular
user were considered in [13, 15], but it was still assumed that the BS has
only one antenna. The sum-rate maximization problem for systems with a multi-
antenna BS was studied in [16, 17], but again they assume only one D2D pair
sharing the spectrum with one cellular user.

Unlike the aforementioned works which consider the sum-rate maximization
problem with relaxed system assumptions, we study a different problem that
takes into account the fairness by maximizing the minimum signal-to-
interference-plus-noise ratio (SINR) over the cellular users and devices. We
also consider a more challenging RIS-assisted joint D2D and cellular
communication system where multiple cellular users share the same spectrum
with multiple D2D pairs and the BS is also equipped with multiple antennas.
The contributions of our paper is summarized as follows:

  * •

First, we adopt a popular alternating optimization (AO) approach and the semi-
definite relaxation (SDR) technique to optimize the combining matrix at the BS
and the RIS phase shift vector that maximize the minimum SINR over the
cellular users and devices.

  * •

Next, we propose an interference cancellation (IC)-based approach whose
complexity is much lower than that of the AO algorithm. More specifically, we
first derive a representation for the RIS phase shift vector that cancels the
interference to the D2D links. Based on this representation, we transform the
RIS phase shift optimization problem into an effective D2D channel
optimization problem. We show that the AO approach can converge faster and can
even give better performance when it uses the proposed IC solution as its
initial point.

  * •

We also show that for the case of a single D2D pair, the proposed IC approach
can be implemented with limited feedback from the receive device. Finally, we
numerically show the gains and benefits of the proposed approaches.

##  II System Model and Problem Formulation

TABLE I: Channel description and notation. Channel | Notation | Size
---|---|---
RIS $\to$ BS | $\mathbf{H}^{\mathtt{RB}}$ | $M\times N$
CU $\to$ RxD | $\mathbf{H}^{\mathtt{UD}}$ | $L\times K$
TxD $\to$ RxD | $\mathbf{H}^{\mathtt{DD}}$ | $L\times L$
CU $\to$ BS | $\mathbf{H}^{\mathtt{UB}}\overset{\Delta}{=}[\mathbf{h}^{\mathtt{UB}}_{1},\ldots,\mathbf{h}^{\mathtt{UB}}_{K}]$ | $M\times K$
TxD $\to$ BS | $\mathbf{H}^{\mathtt{DB}}\overset{\Delta}{=}[\mathbf{h}^{\mathtt{DB}}_{1},\ldots,\mathbf{h}^{\mathtt{DB}}_{L}]$ | $M\times L$
RIS $\to$ RxD | $\mathbf{H}^{\mathtt{RD}}\overset{\Delta}{=}[\mathbf{h}^{\mathtt{RD}}_{1},\ldots,\mathbf{h}^{\mathtt{RD}}_{L}]^{H}$ | $L\times N$
CU $\to$ RIS | $\mathbf{H}^{\mathtt{UR}}\overset{\Delta}{=}[\mathbf{h}^{\mathtt{UR}}_{1},\ldots,\mathbf{h}^{\mathtt{UR}}_{K}]$ | $N\times K$
TxD $\to$ RIS | $\mathbf{H}^{\mathtt{DR}}\overset{\Delta}{=}[\mathbf{h}^{\mathtt{DR}}_{1},\ldots,\mathbf{h}^{\mathtt{DR}}_{L}]$ | $N\times L$
cascaded TxD$\to$RxD | $(\mathbf{g}_{\ell,\ell^{\prime}}^{\mathtt{DD}})^{H}=(\mathbf{h}_{\ell}^{\mathtt{RD}})^{H}\operatorname{diag}{\left(\mathbf{h}_{\ell^{\prime}}^{\mathtt{DR}}\right)}$ | $1\times N$
cascaded CU$\to$RxD | $(\mathbf{g}_{\ell,k}^{\mathtt{UD}})^{H}=(\mathbf{h}_{\ell}^{\mathtt{RD}})^{H}\operatorname{diag}{\left(\mathbf{h}_{k}^{\mathtt{UR}}\right)}$ | $1\times N$
cascaded CU$\to$BS | $\mathbf{G}_{k}^{\mathtt{UB}}=\mathbf{H}^{\mathtt{RB}}\operatorname{diag}{\left(\mathbf{h}_{k}^{\mathtt{UR}}\right)}$ | $M\times N$
cascaded TxD$\to$BS | $\mathbf{G}_{\ell}^{\mathtt{DB}}=\mathbf{H}^{\mathtt{RB}}\operatorname{diag}{\left(\mathbf{h}_{\ell}^{\mathtt{DR}}\right)}$ | $M\times N$

###  II-A System Model

We consider an uplink RIS-assisted MIMO system where an $M$-antenna base
station serves $K$ users with the assistance of an $N$-element RIS. There are
also $L$ device pairs sharing the same spectrum with the cellular system. We
assume that the users and devices are all equipped with a single antenna and
the transmit power of user-$k$ and device-$\ell$ are denoted as
$p^{\mathtt{U}}_{k}$ and $p^{\mathtt{D}}_{\ell}$, respectively. Since the BS
has multiple antennas serving multiple users, a combining matrix
$\mathbf{W}=[\mathbf{w}_{1},\ldots,\mathbf{w}_{K}]^{H}$ is applied to the BS’s
received signal to recover the cellular users’ data. Here, $\mathbf{w}_{k}$ is
the combiner for user-$k$. The receive devices have only one antenna, so they
directly detect data from their received signal. The RIS is defined by a
vector $\boldsymbol{\phi}=[\phi_{1},\ldots,\phi_{N}]^{T}$ where
$|\phi_{i}|\leq 1$. Table I provides a description of all channels and their
corresponding mathematical definition. In this paper, we assume perfect
channel state estimation information (CSI) at the BS and the receive devices.
However, we do not assume knowledge of the individual channels to and from the
RIS but their cascaded versions, the estimation of which has been intensively
studied in the RIS literature. The receive devices feedback their CSI to the
BS for optimizing the combining matrix $\mathbf{W}$ and the RIS phase shift
vector $\boldsymbol{\phi}$. It is also assumed that the noise at the BS and at
the receive devices is distributed as
$\mathcal{CN}(\mathbf{0}_{M},\sigma^{2}_{\mathtt{B}}\mathbf{I}_{M})$ and
$\mathcal{CN}(0,\sigma^{2}_{\mathtt{D}})$, respectively.

###  II-B Problem Formulation

The problem of interest is to maximize the minimum SINR:

$\displaystyle\operatorname*{maximize}_{\\{\boldsymbol{\phi},\,\mathbf{w}_{k}\\}}$
$\displaystyle\min_{k,\ell}\;\\{\mathrm{SINR}_{\ell}^{\mathtt{D}},\,\mathrm{SINR}_{k}^{\mathtt{U}}\\}$
(1) $\displaystyle\operatorname{subject\ to}$ $\displaystyle|\phi_{i}|\leq
1,\;\forall i=1,\ldots,N.$

where $\mathrm{SINR}_{\ell}^{\mathtt{D}}$ and $\mathrm{SINR}_{k}^{\mathtt{U}}$
are given in (2) and (3) and denote the SINR of the $\ell$-th device pair and
the SINR of user-$k$, respectively.

$\mathrm{SINR}_{\ell}^{\mathtt{D}}=\frac{p_{\ell}^{\mathtt{D}}|h_{\ell,\ell}^{\mathtt{DD}}+(\mathbf{g}_{\ell,\ell}^{\mathtt{DD}})^{H}\boldsymbol{\phi}|^{2}}{\sum_{\ell^{\prime}\neq\ell}p_{\ell^{\prime}}^{\mathtt{D}}|h_{\ell,\ell^{\prime}}^{\mathtt{DD}}+(\mathbf{g}_{\ell,\ell^{\prime}}^{\mathtt{DD}})^{H}\boldsymbol{\phi}|^{2}+\sum_{k=1}^{K}p_{k}^{\mathtt{U}}|h_{\ell,k}^{\mathtt{UD}}+(\mathbf{g}_{\ell,k}^{\mathtt{UD}})^{H}\boldsymbol{\phi}|^{2}+\sigma^{2}_{\mathtt{D}}}$
(2)

$\displaystyle\mathrm{SINR}_{k}^{\mathtt{U}}=\frac{p_{k}^{\mathtt{U}}|\mathbf{w}_{k}^{H}(\mathbf{h}_{k}^{\mathtt{UB}}+\mathbf{G}_{k}^{\mathtt{UB}}\boldsymbol{\phi})|^{2}}{\sum_{k^{\prime}\neq
k}p_{k^{\prime}}^{\mathtt{U}}|\mathbf{w}_{k}^{H}(\mathbf{h}_{k^{\prime}}^{\mathtt{UB}}+\mathbf{G}_{k^{\prime}}^{\mathtt{UB}}\boldsymbol{\phi})|^{2}+\sum_{\ell=1}^{L}p_{\ell}^{\mathtt{D}}|\mathbf{w}_{k}^{H}(\mathbf{h}_{\ell}^{\mathtt{DB}}+\mathbf{G}_{\ell}^{\mathtt{DB}}\boldsymbol{\phi})|^{2}+\|\mathbf{w}_{k}\|^{2}\sigma^{2}_{\mathtt{B}}}$
(3)

##  III Alternating Optimization Approach

This section develops an AO approach to solve the joint D2D and cellular
communication problem in (1). Here, we alternately optimize the BS combiners
$\\{\mathbf{w}_{k}\\}$ and the RIS phase shift $\boldsymbol{\phi}$.

###  III-A Optimizing $\\{\mathbf{w}_{k}\\}$ given $\boldsymbol{\phi}$

Let
$\mathbf{f}_{k}^{\mathtt{UB}}\overset{\Delta}{=}\mathbf{h}_{k}^{\mathtt{UB}}+\mathbf{G}_{k}^{\mathtt{UB}}\boldsymbol{\phi}$
and
$\mathbf{f}_{\ell}^{\mathtt{DB}}\overset{\Delta}{=}\mathbf{h}_{\ell}^{\mathtt{DB}}+\mathbf{G}_{\ell}^{\mathtt{DB}}\boldsymbol{\phi}$
define the effective channel from user-$k$ to the BS and from the transmit
device-$\ell$ to the BS, respectively. The optimal linear minimum mean squared
error (LMMSE) combiner at the BS for user-$k$ is given as

$\mathbf{w}_{k}=\bigg{(}\sum_{k^{\prime}=1}^{K}p^{\mathtt{U}}_{k^{\prime}}\mathbf{f}_{k^{\prime}}^{\mathtt{UB}}(\mathbf{f}_{k^{\prime}}^{\mathtt{UB}})^{H}+\sum_{\ell=1}^{L}p^{\mathtt{D}}_{\ell}\mathbf{f}_{\ell}^{\mathtt{DB}}(\mathbf{f}_{\ell}^{\mathtt{DB}})^{H}+\sigma^{2}_{\mathtt{B}}\mathbf{I}_{M}\bigg{)}^{-1}\mathbf{f}^{\mathtt{UB}}_{k}.$

###  III-B Optimizing $\boldsymbol{\phi}$ given $\\{\mathbf{w}_{k}\\}$

Let

$(\boldsymbol{\psi}^{\mathtt{UB}}_{k,k^{\prime}})^{H}\overset{\Delta}{=}\mathbf{w}_{k}^{H}\mathbf{G}_{k^{\prime}}^{\mathtt{UB}}\;\;\text{and}\;\;\alpha^{\mathtt{UB}}_{k,k^{\prime}}\overset{\Delta}{=}\mathbf{w}_{k}^{H}\mathbf{h}_{k^{\prime}}^{\mathtt{UB}},$

then we can express the term
$|\mathbf{w}_{k}^{H}(\mathbf{h}_{k^{\prime}}^{\mathtt{UB}}+\mathbf{G}_{k^{\prime}}^{\mathtt{UB}}\boldsymbol{\phi})|^{2}$
in (3) as follows:

$\displaystyle|\mathbf{w}_{k}^{H}(\mathbf{h}_{k^{\prime}}^{\mathtt{UB}}+\mathbf{G}_{k^{\prime}}^{\mathtt{UB}}\boldsymbol{\phi})|^{2}=\boldsymbol{\tilde{\phi}}^{H}\boldsymbol{\Psi}^{\mathtt{UB}}_{k,k^{\prime}}\boldsymbol{\tilde{\phi}}=\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{UB}}_{k,k^{\prime}})$
(4)

where $\boldsymbol{\tilde{\phi}}=[\boldsymbol{\phi}^{T},1]^{T}$,
$\boldsymbol{\tilde{\Phi}}=\boldsymbol{\tilde{\phi}}\boldsymbol{\tilde{\phi}}^{H}$,
and

$\boldsymbol{\Psi}^{\mathtt{UB}}_{k,k^{\prime}}=\begin{bmatrix}[1.2]\boldsymbol{\psi}^{\mathtt{UB}}_{k,k^{\prime}}(\boldsymbol{\psi}^{\mathtt{UB}}_{k,k^{\prime}})^{H}&\alpha^{\mathtt{UB}}_{k,k^{\prime}}\boldsymbol{\psi}^{\mathtt{UB}}_{k,k^{\prime}}\\\
(\boldsymbol{\psi}^{\mathtt{UB}}_{k,k^{\prime}})^{H}(\alpha^{\mathtt{UB}}_{k,k^{\prime}})^{*}&|\alpha^{\mathtt{UB}}_{k,k^{\prime}}|^{2}\end{bmatrix}.$

Similarly, by defining

$(\boldsymbol{\psi}^{\mathtt{DB}}_{k,\ell})^{H}\overset{\Delta}{=}\mathbf{w}_{k}^{H}\mathbf{G}_{\ell}^{\mathtt{DB}}\;\;\text{and}\;\;\alpha^{\mathtt{DB}}_{k,\ell}\overset{\Delta}{=}\mathbf{w}_{k}^{H}\mathbf{h}_{\ell}^{\mathtt{DB}},$

we can write the term
$|\mathbf{w}_{k}^{H}(\mathbf{h}_{\ell}^{\mathtt{DB}}+\mathbf{G}_{\ell}^{\mathtt{DB}}\boldsymbol{\phi})|^{2}$
in (3) as

$\displaystyle|\mathbf{w}_{k}^{H}(\mathbf{h}_{\ell}^{\mathtt{DB}}+\mathbf{G}_{\ell}^{\mathtt{DB}}\boldsymbol{\phi})|^{2}$
$\displaystyle=\boldsymbol{\tilde{\phi}}^{H}\boldsymbol{\Psi}^{\mathtt{DB}}_{k,\ell}\boldsymbol{\tilde{\phi}}=\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{DB}}_{k,\ell})$
(5)

where

$\boldsymbol{\Psi}^{\mathtt{DB}}_{k,\ell}=\begin{bmatrix}[1.2]\boldsymbol{\psi}^{\mathtt{DB}}_{k,\ell}(\boldsymbol{\psi}^{\mathtt{DB}}_{k,\ell})^{H}&\alpha^{\mathtt{DB}}_{k,\ell}\boldsymbol{\psi}^{\mathtt{DB}}_{k,\ell}\\\
(\boldsymbol{\psi}^{\mathtt{DB}}_{k,\ell})^{H}(\alpha^{\mathtt{DB}}_{k,\ell})^{*}&|\alpha^{\mathtt{DB}}_{k,\ell}|^{2}\end{bmatrix}.$

Using (4) and (5), the SINR of user-$k$ in (3) is written as

$\mathrm{SINR}_{k}^{\mathtt{U}}=\frac{p_{k}^{\mathtt{U}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{UB}}_{k,k})}{\sum_{k^{\prime}\neq
k}p_{k^{\prime}}^{\mathtt{U}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{UB}}_{k,k^{\prime}})+\sum_{\ell}p_{\ell}^{\mathtt{D}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{DB}}_{k,\ell})+\zeta_{k}}$
(6)

where $\zeta_{k}=\|\mathbf{w}_{k}\|^{2}\sigma^{2}_{\mathtt{B}}$. We also have

$\mathrm{SINR}_{\ell}^{\mathtt{D}}=\frac{p_{\ell}^{\mathtt{D}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{DD}}_{\ell,\ell})}{\sum_{\ell^{\prime}\neq\ell}p_{\ell^{\prime}}^{\mathtt{D}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{DD}}_{\ell,\ell^{\prime}})+\sum_{k}p_{k}^{\mathtt{U}}\operatorname{tr}(\boldsymbol{\tilde{\Phi}}\boldsymbol{\Psi}^{\mathtt{UD}}_{\ell,k})+\sigma^{2}_{\mathtt{D}}}$
(7)

where

$\displaystyle\boldsymbol{\Psi}^{\mathtt{DD}}_{\ell,\ell^{\prime}}$
$\displaystyle=\begin{bmatrix}[1.2]\mathbf{g}^{\mathtt{DD}}_{\ell,\ell^{\prime}}(\mathbf{g}^{\mathtt{DD}}_{\ell,\ell^{\prime}})^{H}&h^{\mathtt{DD}}_{\ell,\ell^{\prime}}\mathbf{g}^{\mathtt{DD}}_{\ell,\ell^{\prime}}\\\
(\mathbf{g}^{\mathtt{DD}}_{\ell,\ell^{\prime}})^{H}(h^{\mathtt{DD}}_{\ell,\ell^{\prime}})^{*}&|h^{\mathtt{DD}}_{\ell,\ell^{\prime}}|^{2}\end{bmatrix},$
$\displaystyle\boldsymbol{\Psi}^{\mathtt{UD}}_{\ell,k}$
$\displaystyle=\begin{bmatrix}[1.2]\mathbf{g}^{\mathtt{UD}}_{\ell,k}(\mathbf{g}^{\mathtt{UD}}_{\ell,k})^{H}&h^{\mathtt{UD}}_{\ell,k}\mathbf{g}^{\mathtt{UD}}_{\ell,k}\\\
(\mathbf{g}^{\mathtt{UD}}_{\ell,k})^{H}(h^{\mathtt{UD}}_{\ell,k})^{*}&|h^{\mathtt{UD}}_{\ell,k}|^{2}\end{bmatrix}.$

Therefore, problem (1) can be written as

$\displaystyle\operatorname*{maximize}_{\\{\xi,\;\boldsymbol{\tilde{\Phi}}\succeq\mathbf{0}\\}}$
$\displaystyle\xi$ (8) $\displaystyle\operatorname{subject\ to}$
$\displaystyle\text{\eqref{eq:SINR_Uk}}\geq\xi\;\forall k$
$\displaystyle\text{\eqref{eq:SINR_Dl}}\geq\xi\;\forall\ell$
$\displaystyle|\boldsymbol{\tilde{\Phi}}_{i,i}|\leq 1,\;\forall i=1,\ldots,N$
$\displaystyle\boldsymbol{\tilde{\Phi}}_{N+1,N+1}=1.$

This is a fractional programming problem and thus can be solved by the
Dinkelbach algorithm [18]. Given a solution $\boldsymbol{\hat{\tilde{\Phi}}}$
of (8), we need to convert $\boldsymbol{\hat{\tilde{\Phi}}}$ into a feasible
solution $\boldsymbol{\hat{\phi}}$, which can be done by the Gaussian
randomization approach [19]. Specifically, we first generate a set of Gaussian
random vectors as
$\boldsymbol{{\tilde{\phi}}}_{i}\sim\mathcal{CN}(\mathbf{0},\boldsymbol{\hat{\tilde{\Phi}}})$,
$i=1,\ldots,I$. Since the elements of $\boldsymbol{{\tilde{\phi}}}_{i}$ may
not satisfy the constraints $|\tilde{\phi}_{i,n}|\leq 1$ for $n=1,\ldots,N$
and $\tilde{\phi}_{i,N+1}=1$, we need to first normalize
$\boldsymbol{{\tilde{\phi}}}_{i}$ to obtain
$\boldsymbol{{\bar{\phi}}}_{i}=\boldsymbol{{\tilde{\phi}}}_{i}/\tilde{\phi}_{i,N+1}$,
then we can obtain a feasible candidate
$\boldsymbol{\hat{\phi}}_{i}=[\hat{\phi}_{i,1},\ldots,\hat{\phi}_{i,N}]^{T}$
where $\hat{\phi}_{i,n}=e^{j\measuredangle(\bar{\phi}_{i,n})}$ if
$|\bar{\phi}_{i,n}|>1$, otherwise $\hat{\phi}_{i,n}=\tilde{\phi}_{i,n}$.
Finally, a solution $\boldsymbol{\hat{\phi}}$ of problem (8) can be obtained
as

$\boldsymbol{\hat{\phi}}=\operatorname*{arg\;max}_{\\{\boldsymbol{\hat{\phi}}_{1},\ldots,\boldsymbol{\hat{\phi}}_{I}\\}}\;\;\min\,\\{\mathrm{SINR}_{\ell}^{\mathtt{D}},\,\mathrm{SINR}_{k}^{\mathtt{U}}\\}.$
(9)

##  IV Interference Cancellation Approach

The AO approach presented above is efficient but its computational complexity
is high because the BS combiners and the RIS phase shift vector are
alternately optimized. In addition, each AO iteration requires use of the
Dinkelbach algorithm which is an iterative algorithm itself, and each of its
iterations also requires the solution of an SDR optimization problem. Here, we
propose an IC approach whose computational complexity is significantly lower
than that of the AO algorithm since the proposed IC approach only needs to
solve one SDR problem. Details of the proposed IC approach are presented as
follows.

Let $\mathbf{F}^{\mathtt{DD}}\in\mathbb{C}^{L\times L}$ be the effective
channel from the transmit devices to the receive devices, and define

$\displaystyle\mathbf{f}^{\mathtt{DD}}=\operatorname{vec}{\left((\mathbf{F}^{\mathtt{DD}})^{T}\right)}$
$\displaystyle=\operatorname{vec}{\left((\mathbf{H}^{\mathtt{DD}})^{T}\right)}+\begin{bmatrix}[0.8]\mathbf{G}^{\mathtt{DD}}_{1}\\\
\vdots\\\ \mathbf{G}^{\mathtt{DD}}_{L}\end{bmatrix}\boldsymbol{\phi}$
$\displaystyle=\mathbf{h}^{\mathtt{DD}}+\mathbf{G}^{\mathtt{DD}}\boldsymbol{\phi},$
(10)

where
$\mathbf{G}^{\mathtt{DD}}_{\ell}=[\mathbf{g}^{\mathtt{DD}}_{\ell,1},\ldots,\mathbf{g}^{\mathtt{DD}}_{\ell,L}]^{H}$.
We use a permutation matrix $\mathbf{P}$ to separate the vector
$\mathbf{f}^{\mathtt{DD}}$ into two sub-vectors
$\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}$ and
$\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}$ which contain the diagonal and off-
diagonal elements of $\mathbf{F}^{\mathtt{DD}}$, respectively. The diagonal
elements of $\mathbf{F}^{\mathtt{DD}}$ represent the useful channels for the
D2D pairs, while the off-diagonal elements represent the interference channels
from other D2D pairs. Hence we use the subscripts ‘$\mathrm{sig}$’ and
‘$\mathrm{itf}$’ to indicate the D2D signal and interference, respectively.
This means we can write

$\begin{bmatrix}[1.2]\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}=\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}+\begin{bmatrix}[1.2]\mathbf{G}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{G}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}\boldsymbol{\phi}$ (11)

where

$\begin{bmatrix}[1.2]\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}=\mathbf{P}\mathbf{f}^{\mathtt{DD}},\;\;\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}=\mathbf{P}\mathbf{h}^{\mathtt{DD}},\;\;\begin{bmatrix}[1.2]\mathbf{G}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{G}^{\mathtt{DD}}_{\mathrm{itf}}\end{bmatrix}=\mathbf{P}\mathbf{G}^{\mathtt{DD}}.$

Similarly, let $\mathbf{F}^{\mathtt{UD}}\in\mathbb{C}^{L\times K}$ be the
effective channel from the cellular users to the receive devices, so we can
also write

$\displaystyle\mathbf{f}^{\mathtt{UD}}=\operatorname{vec}{\left((\mathbf{F}^{\mathtt{UD}})^{T}\right)}$
$\displaystyle=\operatorname{vec}{\left((\mathbf{H}^{\mathtt{UD}})^{T}\right)}+\begin{bmatrix}[0.8]\mathbf{G}_{1}^{\mathtt{UD}}\\\
\vdots\\\ \mathbf{G}_{L}^{\mathtt{UD}}\end{bmatrix}\boldsymbol{\phi}$
$\displaystyle=\mathbf{h}^{\mathtt{UD}}+\mathbf{G}^{\mathtt{UD}}\boldsymbol{\phi},$
(12)

where
$\mathbf{G}_{\ell}^{\mathtt{UD}}=[\mathbf{g}_{\ell,1}^{\mathtt{UD}},\ldots,\mathbf{g}_{\ell,K}^{\mathtt{UD}}]^{H}$.

Combining (11) and (12), we have

$\begin{bmatrix}[1.2]\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{f}^{\mathtt{UD}}\end{bmatrix}=\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{h}^{\mathtt{UD}}\end{bmatrix}+\underbrace{\begin{bmatrix}[1.2]\mathbf{G}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{G}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{G}^{\mathtt{UD}}\end{bmatrix}}_{\mathbf{A}}\boldsymbol{\phi},$ (13)

where $\mathbf{A}\in\mathbb{C}^{L(K+L)\times N}$ is a matrix representing the
RIS-aided channel. Assuming that $N\geq L(K+L)$, then using (13) we can write
the phase shift vector $\boldsymbol{\phi}$ in the following expression:

$\boldsymbol{\phi}=\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}\left(\begin{bmatrix}[1.2]\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{f}^{\mathtt{UD}}\end{bmatrix}-\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}\\\
\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{h}^{\mathtt{UD}}\end{bmatrix}\right).$ (14)

Let
$[\mathbf{B},\mathbf{C}]\overset{\Delta}{=}\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}$
where $\mathbf{B}\in\mathbb{C}^{N\times L}$ and
$\mathbf{C}\in\mathbb{C}^{N\times L(K+L)-L}$, then decompose the right-hand
side of (14) as follows:

$\displaystyle\boldsymbol{\phi}$
$\displaystyle=\mathbf{B}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}-\mathbf{B}\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{C}\left(\begin{bmatrix}[1.2]\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{f}^{\mathtt{UD}}\end{bmatrix}-\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{h}^{\mathtt{UD}}\end{bmatrix}\right).$ (15)

The representation above shows a relationship between the RIS phase shift
$\boldsymbol{\phi}$ and the effective device-related channels including the
useful D2D channels $\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}$ and the other
interference links $\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}$ and
$\mathbf{f}^{\mathtt{UD}}$. If we let
$\mathbf{f}^{\mathtt{DD}}_{\mathrm{itf}}=\mathbf{0}$ and
$\mathbf{f}^{\mathtt{UD}}=\mathbf{0}$, the RIS solution (15) reduces to an IC
$\boldsymbol{\phi}$ solution expressed in terms of
$\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}$ as

$\displaystyle\boldsymbol{\phi}_{\mathrm{IC}}$
$\displaystyle=\mathbf{B}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}-\mathbf{B}\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}-\mathbf{C}\begin{bmatrix}[1.2]\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}}\\\
\mathbf{h}^{\mathtt{UD}}\end{bmatrix}$
$\displaystyle=\mathbf{B}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{d}$
(16)

where
$\mathbf{d}=-\mathbf{B}\mathbf{h}^{\mathtt{DD}}_{\mathrm{sig}}-\mathbf{C}[(\mathbf{h}^{\mathtt{DD}}_{\mathrm{itf}})^{T},\,(\mathbf{h}^{\mathtt{UD}})^{T}]^{T}$.
This means that if we could design the RIS phase shifts with (16), the
interference to the receive devices will be completely cancelled. In this
case, the SINR at the receive device-$\ell$ would simplify to
$\mathrm{SINR}_{\ell}^{\mathtt{D}}=p^{\mathtt{D}}_{\ell}|f_{\mathrm{sig},\ell}^{\mathtt{DD}}|^{2}/\sigma_{\mathtt{D}}^{2}$.

With the $\boldsymbol{\phi}_{\mathrm{IC}}$ representation in (16), we can use
the Cauchy–Schwarz inequality to find the following upper bound for the SINR
of user-$k$:

$\displaystyle\mathrm{SINR}_{k}^{\mathtt{U}}$ $\displaystyle\leq
p^{\mathtt{U}}_{k}\|\mathbf{h}^{\mathtt{UB}}_{k}+\mathbf{G}^{\mathtt{UB}}_{k}\boldsymbol{\phi}_{\mathrm{IC}}\|^{2}/\sigma_{\mathtt{B}}^{2}$
(17)
$\displaystyle=p^{\mathtt{U}}_{k}\|\mathbf{h}^{\mathtt{UB}}_{k}+\mathbf{G}^{\mathtt{UB}}_{k}(\mathbf{B}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{d})\|^{2}/\sigma_{\mathtt{B}}^{2}$
(18)
$\displaystyle=p^{\mathtt{U}}_{k}\|\mathbf{Z}_{k}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{q}_{k}\|^{2}/\sigma_{\mathtt{B}}^{2},$
(19)

where $\mathbf{Z}_{k}=\mathbf{G}^{\mathtt{UB}}_{k}\mathbf{B}$ and
$\mathbf{q}_{k}=\mathbf{h}^{\mathtt{UB}}_{k}+\mathbf{G}^{\mathtt{UB}}_{k}\mathbf{d}$.
Therefore, we propose the following IC-based optimization problem

$\displaystyle\operatorname*{maximize}_{\\{\xi,\;\mathbf{f}_{\mathrm{sig}}^{\mathtt{DD}}\\}}$
$\displaystyle\xi$ (20) $\displaystyle\operatorname{subject\ to}$
$\displaystyle
p^{\mathtt{U}}_{k}\|\mathbf{Z}_{k}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{q}_{k}\|^{2}/\sigma_{\mathtt{B}}^{2}\geq\xi\;\forall
k$ $\displaystyle
p^{\mathtt{D}}_{\ell}|f_{\mathrm{sig},\ell}^{\mathtt{DD}}|^{2}/\sigma_{\mathtt{D}}^{2}\geq\xi\;\forall\ell$
$\displaystyle|\mathbf{b}_{i}^{H}\mathbf{f}_{\mathrm{sig}}^{\mathtt{DD}}+d_{i}|^{2}\leq
1\;\forall i=1,\ldots,N,$

where the constraint
$|\mathbf{b}_{i}^{H}\mathbf{f}_{\mathrm{sig}}^{\mathtt{DD}}+d_{i}|^{2}\leq 1$
is equivalent to the constraint $|\phi_{i}|^{2}\leq 1$. Hence, we have
transformed the RIS phase shift $\boldsymbol{\phi}$ optimization problem into
an effective D2D channel optimization (20) by exploiting the IC representation
(16).

The problem in (20) can also be easily solved using SDR. Specifically, let

$\boldsymbol{\Omega}_{k}=\begin{bmatrix}[1.2]\mathbf{Z}_{k}^{H}\mathbf{Z}_{k}&\mathbf{Z}_{k}^{H}\mathbf{q}_{k}\\\
\mathbf{q}_{k}^{H}\mathbf{Z}_{k}&\|\mathbf{q}_{k}\|^{2}\end{bmatrix}\;\text{and}\;\boldsymbol{\Upsilon}_{i}=\begin{bmatrix}[1.2]\mathbf{b}_{i}\mathbf{b}_{i}^{H}&\mathbf{b}_{i}d_{i}\\\
\mathbf{b}_{i}^{H}d_{i}^{*}&|d_{i}|^{2}\end{bmatrix},$

so that

$\displaystyle\|\mathbf{Z}_{k}\mathbf{f}^{\mathtt{DD}}_{\mathrm{sig}}+\mathbf{q}_{k}\|^{2}=\operatorname{tr}(\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}\boldsymbol{\Omega}_{k})$
$\displaystyle|\mathbf{b}_{i}^{H}\mathbf{h}_{\mathrm{sig}}^{\mathtt{DD}}+d_{i}|^{2}=\operatorname{tr}(\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}\boldsymbol{\Upsilon}_{i}),$

where
$\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}=\mathbf{\tilde{f}}^{\mathtt{DD}}_{\mathrm{sig}}(\mathbf{\tilde{f}}^{\mathtt{DD}}_{\mathrm{sig}})^{T}$
with
$\mathbf{\tilde{f}}^{\mathtt{DD}}_{\mathrm{sig}}=[(\mathbf{{f}}^{\mathtt{DD}}_{\mathrm{sig}})^{T},1]^{T}$.
Problem (20) can be now relaxed to a convex optimization problem:

$\displaystyle\operatorname*{maximize}_{\\{\xi,\;\mathbf{F}_{\mathrm{sig}}^{\mathtt{DD}}\succeq\mathbf{0}\\}}$
$\displaystyle\xi$ (21) $\displaystyle\operatorname{subject\ to}$
$\displaystyle
p^{\mathtt{U}}_{k}\operatorname{tr}(\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}\boldsymbol{\Omega}_{k})/\sigma_{\mathtt{B}}^{2}\geq\xi\;\forall
k$ $\displaystyle
p^{\mathtt{D}}_{\ell}\Re\\{\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig},\ell,\ell}\\}/\sigma_{\mathtt{D}}^{2}\geq\xi\;\forall\ell$
$\displaystyle\operatorname{tr}(\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}\boldsymbol{\Upsilon}_{i})\leq
1\;\forall i=1,\ldots,N$
$\displaystyle\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig},L+1,L+1}=1,$

which again can be solved using SDR. Finally, we need to recover an IC
solution $\boldsymbol{\hat{\phi}}_{\mathrm{IC}}$ from
$\mathbf{\tilde{F}}^{\mathtt{DD}}_{\mathrm{sig}}$, which can also be done
using the Gaussian randomization method. Given a
$\boldsymbol{\hat{\phi}}_{\mathrm{IC}}$, the corresponding LMMSE combiner at
the BS can be computed.

It can be seen that unlike the AO approach which needs to solve a series of
SDR problems, the IC approach involves only one SDR optimization and therefore
significantly reduces the computational complexity. We will show later that if
the AO approach is initialized with $\boldsymbol{\hat{\phi}}_{\mathrm{IC}}$,
it converges more rapidly with better performance compared to using a random
initialization.

Limited feedback for single D2D pair: To implement the IC method for the case
of multiple device pairs, it is required that each receive device-$\ell$
forwards its estimated CSI including two cascaded channel matrices
$\mathbf{G}_{\ell}^{\mathtt{DD}}\in\mathbb{C}^{L\times N}$,
$\mathbf{G}_{\ell}^{\mathtt{UD}}\in\mathbb{C}^{K\times N}$ and two direct
channel vectors $\mathbf{H}^{\mathtt{DD}}_{\ell,:}\in\mathbb{C}^{1\times L}$,
$\mathbf{H}^{\mathtt{UD}}_{\ell,:}\in\mathbb{C}^{1\times K}$ via a control
link to the BS, which results in a total number of $(L+K)(N+1)$ complex
coefficients to be fed-back.

However, for the case of a single device pair, the representation in (14) can
be obtained at the receive device as follows:

$\displaystyle\boldsymbol{{\phi}}$
$\displaystyle=\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}\left(\begin{bmatrix}[1.2]f^{\mathtt{DD}}\\\
\mathbf{f}^{\mathtt{UD}}\end{bmatrix}-\begin{bmatrix}[1.2]h^{\mathtt{DD}}\\\
\mathbf{h}^{\mathtt{UD}}\end{bmatrix}\right)$ (22)

where $\mathbf{A}=[(\mathbf{g}^{\mathtt{DD}})^{T},\,\mathbf{G}^{\mathtt{UD}}]$
and we have dropped the device index $\ell$ since there is only D2D pair here.
By setting $\mathbf{f}^{\mathtt{UD}}=\mathbf{0}$, the
$\boldsymbol{\phi}_{\mathrm{IC}}$ representation that cancels the interference
at the device is

$\displaystyle\boldsymbol{{\phi}}_{\mathrm{IC}}$
$\displaystyle=\mathbf{b}f^{\mathtt{DD}}+\mathbf{d},$ (23)

where $\mathbf{b}$ is the first column of
$\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}$ and
$\mathbf{d}=-\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}[h^{\mathtt{DD}},(\mathbf{h}^{\mathtt{UD}})^{T}]^{T}$.
This means the receive device can calculate the matrix
$\mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}$ using its local CSI, then
obtain the two corresponding vectors $\mathbf{b}$ and $\mathbf{d}$, which will
be forwarded to the BS to solve the RIS phase shift optimization problem:

$\displaystyle\operatorname*{maximize}_{\\{\xi,\;{f}^{\mathtt{DD}}\\}}$
$\displaystyle\xi$ (24) $\displaystyle\operatorname{subject\ to}$
$\displaystyle
p^{\mathtt{U}}_{k}\|\mathbf{z}_{k}{f}^{\mathtt{DD}}+\mathbf{q}_{k}\|^{2}/\sigma_{\mathtt{B}}^{2}\geq\xi\;\forall
k$ $\displaystyle
p^{\mathtt{D}}|f^{\mathtt{DD}}|^{2}/\sigma_{\mathtt{D}}^{2}\geq\xi$
$\displaystyle|b_{i}f^{\mathtt{DD}}+d_{i}|^{2}\leq 1\;\forall i=1,\ldots,N.$

Thus, instead of forwarding the $(K+1)(N+1)$ complex coefficients from
$\mathbf{g}^{\mathtt{DD}}\in\mathbb{C}^{N\times 1}$,
$\mathbf{G}^{\mathtt{UD}}\in\mathbb{C}^{K\times N}$, $h^{\mathtt{DD}}$, and
$\mathbf{h}^{\mathtt{UD}}\in\mathbb{C}^{K\times 1}$, the receive device finds
$\mathbf{b}$ and $\mathbf{d}$ and forwards these two vectors to the BS. This
only requires transmission of $2N$ complex coefficients, but still guarantees
that the BS can solve the RIS phase shift optimization problem.

(a) Minimum SINR performance

(b) Computational complexity comparison

Figure 1: Performance and computational complexity comparison versus transmit
power with $N=64$ RIS elements.

(a) Minimum SINR performance

(b) Computational complexity comparison

Figure 2: Performance and computational complexity comparison versus number of
RIS elements $N$ at $30$-dBm transmit power.

##  V Numerical Results

In this section, we present numerical results to show the superiority and
benefits of the proposed IC and IC-AO approaches where IC-AO indicates the AO
approach that uses the IC solution $\boldsymbol{\hat{\phi}}_{\mathrm{IC}}$ as
an initialization. For the regular AO method, we use a random initial solution
for $\boldsymbol{\phi}$. We consider $K=2$ users, $L=2$ device pairs, and
$M=8$ antennas at the BS. If not specifically stated, the number of RIS
elements is set to $64$. We position the BS and the RIS at the locations
$(x,y)=(0,0)$ and $(x,y)=(100,30)$, respectively. The cellular users and
transmit devices are randomly located within a circular area whose center is
$(x,y)=(200,0)$ with a radius of $25$ m, and we also locate the receive
devices randomly also within a circular area centered at $(x,y)=(50,0)$ with
the same radius. The transmit power of the cellular users and the transmit
devices are set to be the same. The noise power is set to $-169$ dBm/Hz and a
bandwidth of $1$ MHz is assumed. The large-scale fading is modeled as
$\beta=\beta_{0}(d/d_{0})^{-\eta}$, where $\eta$ is the path loss exponent and
$d$ is the distance. We set $\beta_{0}=-30$ dB as the path loss at the
reference distance $d_{0}=1$m. The path loss exponent of the user-RIS, TxD-
RIS, RIS-RxD, and RIS-BS channel links is set to $2.2$. However, the path loss
exponent of the user-BS and TxD-BS channels is set to $4$ and the path loss
exponent of the user-RxD and TxD-RxD channels are set to $5$, to model their
weak propagation.

In Fig. 1, we show a comparison of the minimum SINR performance and
computational complexity as the transmit power changes. It can be seen from
Fig. 1(a) that the AO and IC-AO approaches give the same minimum SINRs, which
are about $2.5$ to $3$ dB higher than that of the IC approach. However, the
computational complexity of the IC approach is much lower than that of AO and
IC-AO since the IC approach requires only one iteration while AO and IC-AO
require many more iterations as seen in Fig. 1(b). It is also observed that
although the AO and IC-AO approaches give the same SINR performance, IC-AO has
a lower computational complexity since it requires a smaller number of
iterations.

Next, in Fig. 2, we also show a comparison for the minimum SINR performance
and computational complexity but with a fixed transmit power of $30$ dBm and
the number of RIS elements $N$ is varied. The minimum SINRs for AO and IC-AO
are still about $3$-dB higher than that of the IC approach. However, it is
interesting that as the number of RIS elements $N$ increases, the IC-AO
approach tends to perform better than AO even though IC-AO still requires a
smaller number of iterations. This indicates that IC-AO not only converges
faster than AO but also converges to better local solutions.

Finally, we show an SINR performance comparison between the AO and IC-AO
approaches in Fig. 3 as the maximum number of iterations changes. It is clear
that IC-AO converges much faster than AO and if we set a small maximum number
of iterations, the performance of IC-AO will be significantly higher than that
of AO. For example, if we set the maximum number of iterations to $5$, the IC-
AO approach has already nearly converged and its minimum SINRs are about
$4$-dB, $3.5$-dB, and $3$-dB higher than that of AO for $N=80$, $60$, $40$,
respectively.

Figure 3: Minimum SINR performance versus maximum number of iterations.

##  VI Conclusion

This paper has considered a minimum SINR maximization problem in an RIS-aided
joint D2D and cellular communication system where the RIS was exploited to
simultaneously support the cellular users and the devices. We first presented
an AO approach based on the SDR technique to solve the min-max SINR problem.
We then derived a representation for the RIS phase shift vector that cancels
the interference to the D2D links. Based on the IC representation, we proposed
an IC algorithm whose complexity is much lower than that of the AO approach.
We also showed that the IC solution can help the AO approach converge faster
and achieve better performance. It was also pointed out that for the case of a
single D2D pair, the proposed IC approach can be accomplished with limited
feedback from the device.

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Complex Field Inflation

Ruopeng Zhang and Sibo Zheng

Department of Physics, Chongqing University, Chongqing 401331, China

###### Abstract

We report first study of complex field inflation. Although understood as a
specific two-field inflation, a complex field inflation is able to make more
robust model predictions on primordial curvature perturbation. Explicitly we
discuss the model realizations of complex chaotic and exponential inflation in
various large-field contexts. Both complex field models contain a single
complex scalar together with only two free parameters. Using numerical handles
aimed to calculate primordial curvature perturbation from multifield
inflation, we show that both models are compatible with current Planck data,
and the individual surviving parameter space can be substantially or fully
probed by future CMB-S4 experiments.

##  1 Introduction

Observations from cosmic microwave background (CMB) experiments [1, 2, 3, 4]
have placed strong constraints on amplitude, spectrum index and non-Gaussianty
of primordial curvature perturbation originated from inflation [5, 6, 7] \- a
paradigm to resolve both horizon and flatness problem in early Universe. These
measurements are expected to be improved in higher precisions by future
experiments such as cosmic microwave background (CMB-S4) [8] as well as large
scale structure (LSS) [9] which can access to energy frontiers far beyond
reaches of foreseeable ground-based colliders.

So far, the measurements on the primordial curvature perturbation have
provided us remarkable insights on single-field inflation. In this context, a
critical fact - the primordial large-scale curvature perturbation is frozen
[10, 11] between horizon crossing and re-entering either in a radiation- or
matter-dominated period - allows us to directly compare a fundamental single-
field model to the CMB data. For example, the latest Planck 2018 data strongly
disfavors sing-field chaotic inflation with index larger than two, and a
combined Planck 2018 plus BK15 data has already excluded single-field natural
inflation [2], among others.

Nevertheless, the established correspondence in the single-field case is no
longer valid in multifield case whether the multifield inflation is slow-roll
or not. Because the presence of new isocurvature (entropy) [12, 13]
perturbation in multifield inflation leads to a non-conversation of the
curvature (adiabatic) perturbation during inflation, as long as classical
trajectory in field space is not a straight line. As a result, a varying
curvature perturbation complexes the direct comparison between a multifield
inflation and the CMB data. The situation is further worsen by a larger number
of model parameters in multifield than single-field case. Because it makes
theoretical predictions of the primordial curvature perturbations in the
former case less robust. In this study we consider the question of how
powerful predictions on the primordial curvature perturbation can be regained
in the multifield context.

As an attempt to this question, in this work we advocate complex field
inflation from a perspective different from earlier attempts [14, 15, 16, 17].
A complex field inflation naturally takes place in various contexts of new
physics such as supersymmetry, supergravity and superstring. Although
understood as a specific two-field version, a complex field inflation is able
to make more robust predictions on the primordial curvature perturbation, as
the number of free parameters therein can be obviously reduced in virtue of
rational symmetry. The rest of this work is organized as follows. In Sec.2, we
briefly review the primordial curvature perturbation in complex field
inflation. Then, in Sec.3 we consider model realizations of complex chaotic
and exponential inflation, in each of which we explicitly use a shift symmetry
as in supergravity- or superstring-based single-field inflation. Both of the
illustrative models contain a complex scalar together with only two free
parameters. Sec.4 is devoted to a numerical calculation of the primordial
curvature perturbation in these models, where we will report their current
status and prospects in light of future CMB-S4 experiments [8]. Finally, we
conclude in Sec.5.

##  2 Inflation In 2D Field Space

In this section we briefly review two-field inflation with the two-dimensional
(2D) field space composed of $\phi_{I}$, $I=1,2$. We will clarify the
circumstances where a conservation of the primordial curvature (adiabatic)
perturbation is violated by the presence of the isocurvature (entropy)
perturbation.

###  2.1 Backgrounds

We begin with the two-field Lagrangian

$\displaystyle{}S=\int
d^{4}x\sqrt{-g}\left[\frac{M^{2}_{P}}{2}R-\sum_{I}g^{\mu\nu}\partial_{\mu}\phi_{I}\partial_{\nu}\phi_{J}-V(\phi_{I})\right],$
(1)

where $g_{\mu\nu}$ is Friedmann-Robertson-Walker metric, $R$ is curvature,
$M_{P}$ is reduced Planck mass, $\phi_{I}$ denote the two independent scalar
fields, and $V$ is inflation potential. From Eq.(1) the equations of the
background fields are given as,

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{I}\dot{\phi}_{I}^{2}+V,$ (2)
$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{I}\dot{\phi}_{I}^{2},$ (3) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\ddot{\phi}_{I}+3\dot{\rho}\dot{\phi}_{I}+V_{I},$ (4)

where $H$ is Hubble parameter, “dot” means derivative over time, and
$V_{I}=\partial V/\partial\phi_{I}$.

To ensure an exponential expansion during inflation, we introduce following
slow-roll parameters as in the single-field case

$\displaystyle\epsilon_{I}\equiv\frac{1}{2}\left(\frac{V_{I}}{V}\right)^{2}\approx\frac{\dot{\phi}_{I}^{2}}{2H^{2}},$
(5)

where the equations of the background fields in Eqs.(2)-(4) have been used.
Furthermore, in order to satisfy the slow-roll requirements
$\mid\ddot{\phi}_{I}\mid<<3H\mid\dot{\phi}_{I}\mid$, we define the analogies
of $\eta$ parameter

$\displaystyle\eta_{I}\equiv\frac{V_{II}}{V}\approx\epsilon-\frac{\ddot{\phi}_{I}}{H\dot{\phi}_{I}}-\delta^{IJ}\frac{V_{IJ}}{3H^{2}}\frac{\dot{\phi}_{J}}{\dot{\phi}_{I}},$
(6)

where $V_{IJ}=\partial^{2}V/\partial\phi_{I}\partial\phi_{J}$ and
$\epsilon=\sum_{I}\epsilon_{I}$. The smallness of $\eta_{I}$ can be satisfied
by imposing $\mid\dot{\phi}_{I}\mid\sim\mid\dot{\phi}_{J}\mid$ and $\mid
V_{IJ}\mid<<H^{2}$ for $I\neq J$, which are “natural” for the two-field
inflation under consideration. Because both scalar masses should be light
compared to $H$, and for $\mid\dot{\phi}_{I}/\dot{\phi}_{J}\mid$ either far
larger or smaller than unity the effective number of the inflaton fields is
actually reduced such as in quasi-single-field inflation [18, 19].

###  2.2 Scalar Perturbations

Let us now consider the scalar perturbations $\delta\phi_{I}$. Along a
classical trajectory in the 2D field space, one can transform the scalar
perturbations as

$\displaystyle\left(\begin{array}[]{c}\delta\sigma\\\ \delta s\\\
\end{array}\right)=\left(\begin{array}[]{cccc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{c}\delta\phi_{1}\\\ \delta\phi_{2}\\\
\end{array}\right)$ (13)

where $\cos\theta=\dot{\phi}_{1}/\sqrt{\dot{\phi}_{1}^{2}+\dot{\phi}_{2}^{2}}$
and $\sin\theta=\dot{\phi}_{2}/\sqrt{\dot{\phi}_{1}^{2}+\dot{\phi}_{2}^{2}}$.
In terms of this transformation, the gauge-invariant curvature perturbation
[20] reads

$\displaystyle\zeta=-\frac{H}{\dot{\sigma}}\delta\sigma,$ (14)

where $\dot{\sigma}=\cos\theta\dot{\phi}_{1}+\sin\theta\dot{\phi}_{2}$. From
Eq.(14) one finds that the new variables $\delta\sigma$ and $\delta s$
correspond to the curvature (adiabatic) and isocurvature (entropy)
perturbation, respectively.

Following [21, 22, 23], the curvature perturbation in Eq.(14) on large scales
evolves as

$\displaystyle\dot{\zeta}=\frac{2H}{\dot{\sigma}}\dot{\theta}\delta s,$ (15)

Although the evolution of the curvature perturbation in Eq.(15) is a linear
treatment, it still reveals the condition for the non-conservation of the
curvature perturbation. Since the angle $\theta$ introduced in Eq.(13)
characterizes the classical trajectory of inflation in the 2D field space,
$\zeta$ varies whenever it is not a straight line. As a result, the curvature
perturbation is altered due to the entropy perturbation either in
inflationary111Under the slow roll conditions in Eq.(4), $\dot{\theta}\approx
0$ at the leading order of slow roll parameters, which seems to validate the
conservation of $\zeta$. However, it doesn’t hold at the next-leading order.
or post-inflationary222In reheating the role of the entropy perturbation is
not only played by the scalar degrees of freedom of multifield inflation but
also their decay products [24, 25, 26, 22]. stage.

##  3 Complex Field Inflation

In this section we illustrate how to construct two kinds of complex field
inflation in large-field contexts.

###  3.1 Chaotic Inflation

As mentioned above, complex chiral field is a building block of
supersymmetry-, supergravity- or superstring-based inflation, see e.g,
ref.[27] for a review. This topic is further divided into two different
classes - the D-term [28] and F-term [29] inflation, where in the later case a
realistic construction of inflation potential has to avoid the so-called
“$\eta$” problem.

We restrict to a realization of complex chaotic inflation [30], whose original
two-field version is given as

$\displaystyle V(\phi_{I})=\sum_{I}\lambda_{I}\phi^{p}_{I},$ (16)

where $\lambda_{I}$ are two real parameters and the index $0<p\leq 4$. Instead
of Eq.(16), the potential for the complex chaotic inflation becomes

$\displaystyle V(\phi,\phi^{*})=\lambda\left(\phi^{p}+\phi^{*p}\right),$ (17)

where there are only two model parameters $\lambda$ and $p$ as well.

To realize the inflation potential in Eq.(17), we make use of a shift symmetry
as in refs.[31, 32, 33] to avoid the well-known $\eta$ problem. We consider a
supergravity model with the following superpotential and Kahler potential

$\displaystyle W$ $\displaystyle=$ $\displaystyle W_{0},$ $\displaystyle K$
$\displaystyle=$ $\displaystyle\ln[-(z^{-n}+z^{*-n})]+\delta K(z,z^{*}),$ (18)

where $z$ is the original complex scalar field, $W_{0}$ is a constant term,
$n$ is a real parameter with range $0<n<1$, the shift symmetry is identified
as

$\displaystyle z^{-n}\rightarrow z^{-n}+iR,$ (19)

with $R$ a real parameter, and the small breaking term of the shift symmetry

$\displaystyle\delta K(z,z^{*})=-\frac{c}{zz^{*}},$ (20)

where $c$ a positive real parameter. From Eq.(3.1) $c$ has to satisfy $c\mid
z\mid^{-2}<n\ln(\mid z\mid)$.

In the parameter range $\mid z\mid>>1$ in unit of Planck mass, we have from
Eq.(3.1)

$\displaystyle K_{z}$ $\displaystyle=$ $\displaystyle\frac{\partial
K}{\partial z}=\frac{cz^{*}}{(zz^{*})^{2}}-\frac{nz^{-n-1}}{z^{-n}+z^{*-n}},$
$\displaystyle K_{z^{*}}$ $\displaystyle=$ $\displaystyle\frac{\partial
K}{\partial z^{*}}=\frac{cz}{(zz^{*})^{2}}-\frac{nz^{*-n-1}}{z^{-n}+z^{*-n}},$
$\displaystyle K_{zz^{*}}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}K}{\partial z\partial
z^{*}}=\frac{c}{(zz^{*})^{2}}-\frac{n^{2}(zz^{*})^{-n-1}}{(z^{-n}+z^{*-n})^{2}}.$
(21)

So the $c$-terms in the above derivatives are the dominant contributions if
one further imposes $c\mid z\mid^{-2}>n$. Substituting Eq.(3.1) into the
supergravity Lagrangian [27] yields

$\displaystyle\mathcal{L}\approx-\frac{c}{(zz^{*})^{2}}\sqrt{-g}g^{\mu\nu}\partial_{\mu}z\partial_{\nu}z^{*}-\left(\frac{c}{zz^{*}}-3\right)\mid
W_{0}\mid^{2}e^{\ln[-(z^{-n}+z^{*-n})]}.$ (22)

Using a canonical normalization with $\phi=\sqrt{c}/z$ we finally obtain the
complex chaotic potential

$\displaystyle V\approx\lambda(\phi^{n}+\phi^{*n}),$ (23)

with $\lambda=3c^{-n/2}\mid W_{0}\mid^{2}$ in the mildly fine-tuned parameter
range $n<c\mid z\mid^{-2}<3$. In this complex chaotic inflation, the super-
Planck field range $\phi\sim\mathcal{O}(1)$ and the amplitude of the curvature
perturbation can be adjusted by the range of $c$ and the magnitude of $W_{0}$
respectively.

###  3.2 Exponential Inflation

We now discuss a realization of complex exponential inflation. Exponential
inflation appears either in the context of supergravity [28] or superstring
[34, 35]. The two-field exponential potential takes a form

$\displaystyle V(\phi_{I})=\sum_{I}V_{I}\left(1-e^{-q_{I}\phi}\right),$ (24)

where $V_{I}$ and $q_{I}$ are the model parameters. In virtue of Eq.(24) the
complex field version reads

$\displaystyle
V(\phi_{I})=V_{0}[\left(1-e^{-q\phi}\right)+\left(1-e^{-q\phi^{*}}\right)],$
(25)

where there are only two model parameters $V_{0}$ and $q$.

Here, we stick to a supergravity model with following Kahler potential

$\displaystyle
K(\varphi,\varphi^{*})=\tanh^{-1}(\varphi)\tanh^{-1}(\varphi^{*}),$ (26)

and Lagrangian

$\displaystyle\mathcal{L}=K_{\varphi\varphi^{*}}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi^{*}-\lambda\left(\bar{\Psi}_{R}\varphi^{n}\Psi_{L}+\bar{\Psi}_{L}\varphi^{*n}\Psi_{R}\right),$
(27)

where $\lambda$ is Yukawa coupling constant, and $L$ and $R$ referring to the
left- and right-hand part of fermion $\Psi$ respectively. Such Yukawa
interaction can be either fundamental $(n=1)$ or arises from supergravity-
induced effect $(n>1)$. For the fermion $\Psi$, we further assume that it is
charged under a strong gauge dynamics with characteristic scale

$\displaystyle\left<\bar{\Psi}\Psi\right>=\Lambda^{3},$ (28)

where $\Lambda$ is assumed to be at least a few times of $M_{P}$.

To capture the low-energy effective theory we firstly determine the metric in
terms of Eq.(26), which gives

$\displaystyle
K_{\varphi\varphi^{*}}=\frac{1}{1-\varphi^{2}}\frac{1}{1-\varphi^{*2}}.$ (29)

Substituting Eq.(29) into Eq.(27) suggests a canonical normalization of the
field

$\displaystyle\varphi=\tanh(\phi/\sqrt{2}).$ (30)

In the field range $M_{P}<\phi<\Lambda$, one finds $\varphi\rightarrow 1$ in
Eq.(30) and the effective fermion mass is relatively light for
$\lambda<<\Lambda$. So, below $\Lambda$ scale the fermion condensate in
Eq.(28) occurs and gives rise to an effective potential

$\displaystyle V\approx
V_{0}[(1-2ne^{-\sqrt{2}n\phi})+(1-2ne^{-\sqrt{2}n\phi^{*}})]$ (31)

with $V_{0}=\lambda\Lambda^{3}$. Note, in the field range $\varphi\rightarrow
1$, $K^{-1}_{\varphi\varphi^{*}}\approx 0$ which suppresses the supergravity
correction to Eq.(31).

Just like the previous examples, the overall coefficient $V_{0}$ is fixed by
the amplitude of the primordial curvature perturbation, whereas the other
observables vary as the index $n$ changes.

$\ln(10^{10}\mathcal{P}_{\zeta})=(3.044\pm 0.014)$ (68 $\%$ CL)
---
$n_{s}=0.965\pm 0.004$ (68 $\%$ CL)
$r<0.10$ (95 $\%$ CL)
$f^{\rm{equil}}_{\rm{NL}}=-26\pm 47$ (68 $\%$ CL)
Table 1: Constraints on observables of the primordial curvature perturbation
from Planck 2018 data [2, 3, 4].

##  4 Numerical Analysis

Having shown the viability of the model realizations of above two models of
complex field inflation, we proceed to discuss whether these models are
compatible with current experimental data. To calculate the observables of the
primordial curvature perturbation in these models, we use a public numerical
code - transport approach [36, 37]. The transport approach, combined with
PyTransport [38], is able to calculate observables of the primordial curvature
perturbation such as power spectrum, tensor-to-scalar index and nonlinear
parameter related to non-Gaussianty of bispectrum in multifield inflation
whether the kinetic term for the complex scalar field is canonical or non-
canonical.

The aforementioned observables of the primordial curvature perturbation, which
are strongly constrained by the latest Planck 2018 data as shown in Table.1,
are defined as following. Firstly, the amplitude of the power spectrum is
given by

$\displaystyle\left<\zeta(\overrightarrow{k}_{1})\zeta(\overrightarrow{k}_{2})\right>=(2\pi)^{3}\delta^{(3)}(\overrightarrow{k}_{1}+\overrightarrow{k}_{2})P_{\zeta}(k)$
(32)

with $\mathcal{P}_{\zeta}(k)=k^{3}P_{\zeta}(k)/(2\pi^{2})$, with the pivot
scale $k_{*}=0.05$ Mpc-1. The explicit value of $\mathcal{P}_{\zeta}(k)$ fixes
an overall magnitude of inflation potential. Secondly, the spectrum index
which describes the dependence of $\mathcal{P}_{\zeta}(k)$ on $k$ is defined
as

$\displaystyle n_{s}-1=\frac{d\ln\mathcal{P}_{\zeta}(k)}{d\ln k}.$ (33)

Thirdly, the tensor-to-scalar ratio is simply given by

$\displaystyle r=\frac{P_{T}}{P_{\zeta}},$ (34)

where $P_{T}$ is the tensor power spectrum. Finally, the nonlinear parameter
$f_{\rm{nl}}$ which quantifies the non-Gaussianty of bispectrum with
equilateral configuration reads

$\displaystyle f_{\rm{nl}}=\frac{5}{18}\frac{B_{\zeta}(k)}{P^{2}_{\zeta}(k)},$
(35)

where the bispectrum $B_{\zeta}(k)$ is given by

$\displaystyle\left<\zeta(\overrightarrow{k}_{1})\zeta(\overrightarrow{k}_{2})\zeta(\overrightarrow{k}_{3})\right>=(2\pi)^{3}\delta^{(3)}(\overrightarrow{k}_{1}+\overrightarrow{k}_{2}+\overrightarrow{k}_{3})B_{\zeta}(k).$
(36)

###  4.1 Chaotic Inflation

The parameter space of the complex chaotic inflation in Eq.(17) is rather
sensitive to the parameter $p$. Because for certain value of $p$ the
requirement of positivity of $V$ is so severe that viable field ranges are
significantly suppressed, while for some $p$ this requirement is nearly
absent. To illustrate such feature in the complex chaotic inflation, in the
following we consider two specific choices $p=\\{0.25,0.5\\}$.

Fig.1 shows the predictions of the primordial curvature perturbation
consistent with current Planck 2018 data in Table.1 in the complex chaotic
inflation in field ranges $\rm{Re}\phi\subset[3,11]$ and
$\rm{Im}\phi\subset[2,11]$ for $p=0.5$. This figure shows that the samples are
also consistent with the constraint from BICEP2/Keck Array experiments plus
Planck [39] (in blue) at 95$\%$ CL, and can be fully excluded by the future
CMB-S4 sensitivity [8] (in red) at 95$\%$ CL as they are distributed in a
narrow region in the plane of $n_{s}-r$. Compared to $p=0.5$, the case with
$p=0.25$, which is not explicitly shown, has been confirmed to be excluded by
the Planck 2018 data in Table.1.

Combing the two cases illustrates that the predictions of the complex chaotic
inflation obviously differ from those of the single-field chaotic inflation
[2], as both of the index choices in the later circumstance are still
compatible with the Planck 2018 data in Table.1.

Figure 1: Observables of the primordial curvature perturbation consistent with
the Planck 2018 data in Table.1 in the complex chaotic inflation in the field
ranges $\rm{Re}\phi\subset[3,11]$ and $\rm{Im}\phi\subset[2,11]$ for $p=0.5$,
as compared with the constraint from BICEP2/Keck Array experiments plus Planck
[39] (in blue) and the future CMB-S4 sensitivity [8] (in red) at 95$\%$ CL.

###  4.2 Expotential Inflation

The parameter space of the complex exponential inflation in Eq.(31) is shaped
by two facts. Earlier we have seen a fact that the initial value of $\phi$ has
to be at least a few times of Planck mass. Another fact is that given an
explicit value of the index $n$ the initial value of $\phi$ has to be upper
bounded, as too large $\phi$ results in too small slow roll parameters for the
scale $k_{*}$, which in turn yields $n_{s}$ too close to unity. We expect that
when $n$ increases the critical upper bound value of $\phi$ decreases, leading
to a shrinking parameter space. To manifest this point, we consider two
specific cases with $n=\\{0.5,1\\}$.

Fig.2 and 3 show the predictions of the primordial curvature perturbation for
$n=0.5$ and $n=1$ respectively, where both of the field ranges are chosen to
cover a complete period in the complex field space. Explicitly, the number of
samples is of order $\sim 100$ and $\sim 10$ for the case with $n=0.5$ and
$n=1$ respectively. The trend in the number of the samples implies that the
upper bound value of $n$ is close to $1$. As for observation, Fig.2 tells that
the parameter space in the case with the smaller $n=0.5$ can be substantially
excluded by the future CMB-S4 limit, while Fig.3 shows that the parameter
space with respect to the larger $n=1$ is beyond the reach of the same CMB-S4
limit.

In the light of the phenomenological study on the two models we have verified
that $i)$ complex field inflation can indeed make robust predictions of the
primordial curvature perturbation, $ii)$ complex field inflation can be
consistent with current Planck data, and $iii)$ distinctive signal of complex
field inflation, which differs from the original single-field inflation, can
be tested by the future CMB measurements on $n_{s}$ and $r$, despite that the
magnitude of $f_{\rm{nl}}$ is probably too small to be detected.

Figure 2: Observables of the primordial curvature perturbation consistent with
the Planck 2018 data in Table.1 in the complex exponential inflation in the
field ranges $\rm{Re}\phi\subset[4,8]$ and $\rm{Im}\phi\subset[4,12.9]$ for
$n=0.5$. The experimental limits are the same as in Fig.1.

Figure 3: Same as Fig.2 in the field ranges $\rm{Re}\phi\subset[4,8]$ and
$\rm{Im}\phi\subset[4,8.5]$ for $n=1$ instead.

##  5 Conclusion

This work is devoted to advocate complex field inflation which makes more
robust predictions on the primordial curvature perturbation than its original
two-field version. In order to illustrate the viability of model realization
of complex field inflation, we have considered two representative examples,
i.e, the complex chaotic and exponential inflation. Similar to the
supergravity- or superstring-based single field model, a shift symmetry has
been used. We then numerically calculate the predictions of the primordial
curvature perturbation in these models based upon on the numerical code
transport approach. It turns out that both two models are compatible with the
latest Planck limits. The distinctive signal in individual surviving parameter
space, which is different from the single-field prediction, can be
substantially or fully probed by the future CMB experiments.

Along this study there are several directions which deserve further
investigations. In this work a model realization suffers from the fine-tuning
issue, which may be improved in circumstances due to an alternative shift
symmetry instead of the adopted one. Also, more examples of complex field
inflation can be constructed especially in the small-field context which we
did not consider here. Lastly, there is an important technique needed to be
developed. Here, we have implicitly assumed an instantaneous reheating which
follows the inflationary stage. Strictly speaking, the primordial curvature
perturbation is inevitably changed by a reheating process which is highly
likely to be nonlinear. To our knowledge there is no public numerical tool to
handle a general reheating process.

## Acknowledgments

The authors thank David Mulryne for correspondence about the numerical code
transport approach. This research is supported in part by the National Natural
Science Foundation of China under Grant No. 11775039.

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