# Performance Analysis and Optimization of Uplink Cellular Networks with
Flexible Frame Structure ††thanks: This research was supported by the U.S.
National Science Foundation under Grants CNS-1941348 and CNS-2008646.

Fatemeh Lotfi and Omid Semiari
Department of Electrical and Computer Engineering, University of Colorado,
Colorado Springs, CO, USA
Email<EMAIL_ADDRESS>

###### Abstract

Future wireless cellular networks must support both enhanced mobile broadband
(eMBB) and ultra-reliable low-latency communications (URLLC) to manage
heterogeneous data traffic for emerging wireless services. To achieve this
goal, a promising technique is to enable flexible frame structure by
dynamically changing the data frame’s numerology according to the channel
information as well as traffic quality-of-service requirements. However, due
to non-orthogonal subcarriers, this technique can result in an interference,
known as inter numerology interference (INI), thus, degrading the network
performance. In this work, a novel framework is proposed to analyze the INI in
the uplink cellular communications. In particular, a closed-form expression is
derived for the INI power in the uplink with a flexible frame structure and a
new resource allocation problem is formulated to maximize the network spectral
efficiency (SE) by jointly optimizing the power allocation and numerology
selection in a multi-user uplink scenario. The simulation results validate the
derived theoretical INI analyses and provide guidelines for the power
allocation and numerology selection.

##  I Introduction

Next-generation wireless networks are required to support new applications,
such as connected and autonomous vehicles or wireless extended reality, that
rely on both enhanced mobile broadband (eMBB) and ultra-reliable low-latency
communications (URLLC) [1]. To meet such strict service requirements, the 3rd
generation partnership project (3GPP) has proposed the notion of flexible
frame structure that allows dynamic selection of the numerology (i.e., sub-
carrier spacing and cyclic prefix (CP) length) across successive data frame
transmissions. Despite their advantages [2], multi-numerology systems do not
guarantee the subcarriers’ orthogonality, resulting in the so-called inter
numerology interference (INI). Without proper interference mitigation, the INI
can substantially impact the performance of URLLC services such as vehicular
communications.
The performance of multi-numerology systems and the impact of INI have been
recently studied for downlink scenarios [3, 4, 5, 6, 7, 8, 9, 10, 11]. In
particular, the authors in [3] investigate the INI in the context of filtered
orthogonal frequency division multiplexing (F-OFDM). In [4] the authors derive
a closed-form expression for the INI in the downlink. The work in [5] presents
INI analysis for the windowed-OFDM (W-OFDM) scheme. Both works in [3, 5] use
filtering and windowing techniques to control the out-of-band emission. The
authors in [11] propose a new scheme that reduces the scheduling latency by
leveraging standard numerology in conjunction with mini slots. An adaptive
quality of service (QoS) aware numerology selection scheme is presented in [6]
that aims to reduce the impact of INI in the downlink. In [7], the authors
formulate the cost function based on the long-term average SINR of a multi-
numerology system and propose a new resource allocation scheme that accounts
for the Doppler and delay spreads. The authors in [8] proposed an analytical
expression of peak-to-average power ratio (PAPR) distribution for a mixed-
numerology system and [12] presents two PAPR mitigation schemes for a multi-
numerology cyclic prefix OFDM (CP-OFDM) system. In [9], the authors present a
precoding scheme to mitigate INI in the downlink and, [10] considers multi-
numerology in non-orthogonal multiple access (NOMA) using successive
interference cancellation. The authors in [13] present an INI cancellation
method for the uplink based on the minimum mean square error (MMSE) detection
in OFDM systems. In [14], a radio access network slicing framework is
proposed, that by considering both radio frequency and baseband imparities,
defines numerology relationship among slices. While interesting, the works in
[3, 4, 5, 6, 7, 8, 9, 10], only consider the INI analysis in the downlink,
while as we show here, the INI can have a major impact on the uplink
performance. Also, the body of work in [3, 4, 5, 6, 7, 8, 9, 10, 14, 11, 13,
12] does not study the uplink performance for Discrete Fourier Transform-
spread OFDM (DFT-s-OFDM) which is the common modulation scheme for uplink
transmissions [15]. Compared to OFDM, the DFT-s-OFDM transmission results in a
lower PAPR for uplink signals and less power consumption at the user equipment
(UE).
The main contribution of this paper is a novel scheme for analysis and
optimization of spectral efficiency (SE) in uplink cellular networks with
flexible frame structure. First, we derive a closed-form expression for the
INI in the multi-numerology system with the uplink DFT-s-OFDM transmission
scheme. To the best of our knowledge, this is the first work that derives the
INI for uplink DFT-s-OFDM networks. Building on this analysis, we formulate a
new optimization problem that aims to maximize the minimum SE across the
network while accounting for the impact of INI. Then, to solve this problem,
we propose a new solution that yields the subcarrier powers allocation jointly
with the numerology for each UE to optimize the network’s performance.
Comprehensive simulation results are presented to validate the derived INI
analysis and to provide guidelines for joint power allocation and numerology
selection in uplink cellular networks. The paper is organized as follows.
Section II describes the system model. Section III presents the INI analysis
in the multi-user uplink scenario with multi-numerology support. Section IV
presents the problem formulation and the proposed solution to optimize the SE.
Simulation results are presented in Section V. Section VI concludes the paper.

##  II System Model

We consider an uplink multi-numerology cellular network consists of $K$ UEs in
a set $\mathcal{K}$ and a base station (BS). The uplink signals are modulated
based on the DFT-s-OFDM [2] scheme, as shown in Fig. 1. The uplink network
supports a flexible frame structure which allows UEs to dynamically select
different numerologies (i.e., subcarrier spacing and CP length) [2]. At the
transmitter of an arbitrary UE $i\in\mathcal{K}$, the modulated data stream at
the output of the fast Fourier transform (FFT) block can be written as

$\displaystyle S_{i}(k)=\sum_{n=0}^{N_{i}-1}x_{i}(n)e^{-j\frac{2\pi
nk}{N_{i}}},$ (1)

where the FFT size, $N_{i}$, is equal to the number of active subcarriers
allocated to the $i$-th UE and $S_{i}(k)$ is the FFT symbol of the $k$-th
subcarrier. Moreover, $x_{i}(n)$ is the modulated symbol of UE $i$. Note that
$S_{i}(k)$, are independent and identically distributed (i.i.d) complex
modulation symbols with zero mean and unit average power. The numerologies of
the UEs are determined as follow:

$\displaystyle M_{i}=N_{i}+N_{z_{i}},$ (2)
$\displaystyle\frac{N_{1}}{M_{1}}+\cdots+\frac{N_{K}}{M_{K}}=1,\vspace{-0.2cm}$
(3)

where $N_{z_{i}}$, $M_{i}$, and $\frac{N_{i}}{M_{i}}$ are, respectively, the
number of zero-padded subcarriers, the total number of subcarriers assigned to
UE $i$, and the fraction of bandwidth allocated to the $i$-th UE. By zero-
padding both the left and right sides of the signal subcarriers, the signal
can be mapped into the determined position within the system’s bandwidth. To
this end, a localized subcarrier mapping technique can be implemented whose
output can be written as

$\displaystyle X_{i}(k)=\begin{cases}S_{i}(k),&\text{$N_{i-1}+1\leqslant
k\leqslant N_{i-1}+N_{i}-1$},\\\ 0,&\text{otherwise}.\end{cases}$ (4)

The signal after the inverse FFT (IFFT) block will be

$\displaystyle\bar{x}_{i}(m)$
$\displaystyle=\frac{1}{M_{i}}\sum_{l=N_{i-1}+1}^{N_{i}-1}X_{i}(l)e^{j2\pi\frac{ml}{M_{i}}},$
(5)

where $1\leq m\leq M_{i}$. Finally, a cyclic prefix ($\tau_{\text{cp,i}}$),
equal to $M_{i}/16$, is appended to the UE $i$ symbols before concatenating
symbols. Let $\alpha_{i,j}=\frac{M_{i}}{M_{j}}$ be the numerology scaling
factor, where $i\neq j$, for $i,j\in\mathcal{K}$. For an arbitrary UE $i$, we
can partition the user set $\mathcal{K}$ into three disjoint subsets, based on
the values of numerology scaling factor. We define
$\mathcal{K}_{h,i}=\\{j\in\mathcal{K}\mid\alpha_{i,j}>1\\}$ as the subset of
UEs with higher numerology compared to UE $i$,
$\mathcal{K}_{l,i}=\\{j\in\mathcal{K}\mid\alpha_{i,j}<1\\}$ as a subset of UEs
with lower numerology and
$\mathcal{K}_{s,i}=\\{j\in\mathcal{K}\mid\alpha_{i,j}=1\\}$ as a subset of UEs
with the same numerology as UE $i$. Next, we will use the defined subsets to
analyze the INI.

Figure 1: A block diagram of a DFT-s-OFDMA.

Considering different numerology of UEs $i$ and $j$ and $\alpha_{i,j}$, the
symbol duration of UE $i$ is equal to $\alpha_{i,j}$ symbol duration of UE
$j$. So, in one symbol duration of UE $i$ the transmitted signals of UE $i$
and UE $j$ can be written as

$z_{i}(m)=\bar{x}_{i}(m),\,\,\,\,z_{j}(m)=\sum_{q=0}^{\alpha_{i,j}-1}\bar{x}_{j}(m-\nu_{q}),\\\
$ (6)

where $\nu_{q}=q(M_{j}+\tau_{\text{cp,j}})$. Next, we represent the received
signals at the BS under the flexible numerology setting. Let $h_{i}$
represents the channel impulse response for UE $i$’s link. As shown in Fig. 1,
the received signal of UE $i\in\mathcal{K}$ for one symbol duration can be
written as

$\displaystyle r_{i}(m)$ $\displaystyle=z_{i}(m)\ast
h_{i}(m)+\bigg{(}\sum_{j\in\mathcal{K}_{h,i}}z_{j}(m)\ast h_{j}(m)$ (7)
$\displaystyle+\sum_{j\in\mathcal{K}_{l,i}}z_{j}(m)\ast
h_{j}(m)\bigg{)}+w(m),$

where $z_{i}(m)$ is the desired transmitted symbol of UE $i$, and $w(m)$ is
the additive white Gaussian noise (AWGN) with zero mean and variance
$\sigma^{2}$. In (7), $z_{j}(m)$ is the $\alpha_{i,j}$ symbols of UE $j$, that
have been aligned with one symbol of the arbitrary UE $i$. Hence, the second
term in (7) will results in INI, as shown in the next section.

##  III Uplink Inter Numerology Interference Analysis

At the BS, the received signal for an arbitrary UE $i$ after CP removal and
passing through the FFT block can be written as

$\displaystyle
R_{i}(k)=\sum_{m=0}^{M_{i}-1}r_{i}(m)e^{-j2\pi\frac{mk}{M_{i}}}=\sum_{m=\tau_{\text{cp,i}}}^{M_{i}-1}\Big{(}\bar{x}_{i}(m)\ast
h_{i}(m)+$ (8)
$\displaystyle\sum_{j\in\mathcal{K}_{h,i}}\sum_{q=0}^{\alpha_{i,j}}\bar{x}_{j}(m-\nu_{q})\ast
h_{j}(m,q)+\sum_{j\in\mathcal{K}_{l,i}}\bar{x}_{i}(m)\ast h_{i}(m,q)$
$\displaystyle+w(m)\Big{)}e^{-j2\pi\frac{mk}{M_{i}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}{R_{d_{i}}(k)+R_{I_{h,i}}(k)+R_{I_{l,i}}(k)+W(k)}.$

In (8), the first term is the desired signal. However, the second and third
terms in (8), $R_{I_{h,i}}$ and $R_{I_{l,i}}$, represent, respectively, the
INI caused by UEs in $\mathcal{K}_{h,i}$ and $\mathcal{K}_{l,i}$. We note that
there will be no INI from the UEs in $\mathcal{K}_{s,i}$ since they have the
same numerology as UE $i$ and their transmissions will be orthogonal to UE
$i$’s link. From (8), we can derive the following results for the INI in the
uplink.

######  Proposition 1.

For a UE $i\in\mathcal{K}$, the received INI at subcarrier $n$ from other UEs
with higher and lower numerology (compared to UE $i$) in the uplink can be
written, respectively, as

$\displaystyle
I_{h,i}(n)=\frac{1}{N_{i}}\sum_{k=0}^{N_{i}-1}\sum_{j\in\mathcal{K}_{h,i}}\sum_{q=1}^{\alpha_{i,j}}\sum_{l=0}^{N_{j}-1}\frac{1}{M_{j}}X_{j}(\nu_{l})H_{j}(\nu_{l})\,\times$
$\displaystyle\hskip 99.58464ptQ_{i,j,h}^{I}(l,k,q){\rm
e}^{-j2\pi(\frac{\nu_{q-1}}{M_{i}}-\frac{n}{N_{i}})},$ (9) $\displaystyle
I_{l,i}(n)=\frac{1}{N_{i}}\sum_{k=M_{i}-N_{i}}^{M_{i}-1}\sum_{j\in\mathcal{K}_{l,i}}\sum_{q=1}^{\frac{1}{\alpha_{i,j}}}\sum_{l=0}^{N_{j}-1}\frac{1}{M_{j}}X_{j}(l)H_{j}(l)\,\times$
$\displaystyle\hskip 102.43008ptQ_{i,j,l}^{I}(l,k,q){\rm
e}^{j2\pi\frac{nk}{N_{i}}},$ (10)

where $\tau_{\text{cp},j}$ is the CP length of UE $j$ and

$\displaystyle\nu_{l}=l+M_{j}-N_{j},\hskip
14.22636pt\nu_{q}=q(M_{j}+\tau_{\text{cp},j}),$ (11)
$\displaystyle\kappa_{l,k}=\frac{l}{M_{j}}+\frac{N_{i}-k}{M_{i}},$ (12)
$\displaystyle
V_{q}=\begin{cases}M_{j}+(2-\alpha_{i,j})\tau_{\text{cp},j},&q=1\\\
M_{j}+\tau_{\text{cp},j},&q>1\end{cases}.$ (13)

In addition,

$\displaystyle Q_{i,j,h}^{I}(l,k,q)={\rm
e}^{j\pi\kappa_{l,k}(V_{q}-1)}\frac{\sin\big{(}\pi\kappa_{l,k}V_{q}\big{)}}{\sin\big{(}\pi\kappa_{l,k}\big{)}},$
(14) $\displaystyle Q_{i,j,l}^{I}(l,k,q)=\begin{cases}Q_{l,1}^{I}{\rm
e}^{j2\pi(M_{j}-\tau_{\alpha})\zeta_{l,k}}+Q_{l,2}^{I},&\text{$q=1$}\\\
Q_{l,3}^{I}{\rm
e}^{j2\pi\zeta_{l,k}(\nu_{q}-M_{i})},&\text{$q>1$}\end{cases},$ (15)
$\displaystyle Q_{l,1}^{I}={\rm
e}^{j\pi\zeta_{l,k}(\tau_{\alpha}-1)}\frac{\sin\big{(}\pi\zeta_{l,k}\tau_{\alpha}\big{)}}{\sin\big{(}\pi\zeta_{l,k}\big{)}},$
(16) $\displaystyle Q_{l,2}^{I}={\rm
e}^{j\pi\zeta_{l,k}(M_{i}-\tau_{\alpha}-1)}\frac{\sin\big{(}\pi\zeta_{l,k}(M_{i}-\tau_{\alpha})\big{)}}{\sin\big{(}\pi\zeta_{l,k}\big{)}},$
(17) $\displaystyle Q_{l,3}^{I}={\rm
e}^{j\pi\zeta_{l,k}(M_{i}-1)}\frac{\sin\big{(}\pi\zeta_{l,k}M_{i}\big{)}}{\sin\big{(}\pi\zeta_{l,k}\big{)}},$
(18) $\displaystyle\zeta_{l,k}=\frac{l}{M_{j}}-\frac{k}{M_{i}},\hskip
14.22636pt\tau_{\alpha}=(\frac{1}{\alpha_{i,j}}-1)\tau_{\text{cp,i}}.$ (19)

###### Proof:

The proof is presented in Appendix A. ∎

Using the results in Proposition 1, the total INI at the UE $i$’s link will be

$I_{i}(n)=I_{h,i}(n)+I_{l,i}(n),$ (20)

where $I_{h,i}(n)$ and $I_{l,i}(n)$ given in (9) and (10), respectively. Next,
using this analysis, we will formulate a new optimization problem to maximize
the minimum SE across the uplink network while accounting for the INI.

##  IV Joint Numerology Selection and Power Allocation in Uplink Cellular
Communications

Given the derived INI, the SE for UE $i$’s link will be

$\displaystyle
S_{e_{i}}(n)=(1-P_{\text{out}})\log_{2}\Big{(}1+\frac{p_{d,i}(n)}{p_{I,i}(n)+\sigma^{2}}\Big{)},$
(21)

where $p_{d,i}(n)$ and $p_{I,i}(n)=|I_{i}(n)|^{2}$ are defined, respectively,
as the received desired power and the INI power. Also, $P_{\text{out}}$
represents the outage probability. To optimize the SE across the uplink
network we define $\Lambda(\boldsymbol{p},\boldsymbol{M})$ as

$\displaystyle\Lambda(\boldsymbol{p},\boldsymbol{M})=\min(\boldsymbol{S_{e}}(n)),$
(22)

where $\boldsymbol{S_{e}}$ is a vector of $S_{e_{i}}$, $i\in\mathcal{K}$.
Also, $\boldsymbol{p}$ and $\boldsymbol{M}$ represent the vector of
subcarriers’ powers and the vector with elements $M_{i}$ with
$i\in\mathcal{K}$, respectively. Our goal is to optimize the system
performance by maximizing $\Lambda$, while taking into account the INI. To
this end, we formulate the following problem that aims to maximize the minimum
SE across the uplink network:

$\displaystyle\operatorname*{argmax}_{\boldsymbol{p},\boldsymbol{M}}$
$\displaystyle\hskip
19.91684pt\Lambda\left({\boldsymbol{p}},{\boldsymbol{M}}\right),$ (23) s.t.,
$\displaystyle\hskip
19.91684pt{S}_{e_{i}}(n)\geq{\lambda}_{i},\,\,\,i\in\mathcal{K},$ (24)
$\displaystyle\hskip 19.91684pt0\leq{p}_{x_{i}}\leq{p}_{\max},$ (25)
$\displaystyle\hskip 19.91684pt{M}_{i}=2^{\mu_{i}},\hskip
5.69046pt\mu_{i}\in\\{6,7,8,9,10\\},$ (26)

where $\lambda_{i}$ is the minimum required SE for data transmission,
$p_{x_{i}}$ is the subcarriers’ power of UEs $i$ and ${p}_{\max}$ is the
maximum allowed power per subcarrier. Note that the objective function and the
first constraint are non-convex. To solve this problem, our proposed solution
is based on relaxing the non-convex constraint and solving the problem as a
convex optimization problem. The proposed solution can be described as follow.
For a given $\boldsymbol{M}$, we first optimize the transmit power based on
the successive convex optimization technique[16]. Then, given the obtained
transmit power, we find the optimal numerology factors in $\boldsymbol{M}$ by
simply searching over the limited possible values presented in (26).
To convert the problem into a convex optimization, note that in (21), the
outage probability can be shown to be convex [17]. Moreover, the log part can
be written by expanding it as a difference between two concave functions as
follow:

$\displaystyle S_{e_{i}}(n)=$
$\displaystyle(1-P_{\text{out}})\Big{(}\log_{2}(p_{I,i}(n)+\sigma^{2}+p_{d,i}(n))$
$\displaystyle\hskip 128.0374pt-S_{e_{i},sub}\Big{)},$ (27)

where,

$\displaystyle S_{e_{i},sub}=\log_{2}\Big{(}p_{I,i}(n)+\sigma^{2}\Big{)}.$
(28)

Since in (27) both the subtrahend ($\text{SE}_{i,sub}$) and minuend are
concave, there is no guarantee that their difference is also concave. Here, we
can define an upper bound for $S_{e_{i},sub}$ based on its first-order Taylor
approximation. By minimizing this upper bound, we can effectively maximize the
SE [16]. Thus, we derive the following convex upper bound at the given local
point $\hat{p}_{x_{i,r}}$:

$\displaystyle
S_{e_{i},sub}\leq\Big{(}D_{i}(p_{x_{i}}-\hat{p}_{x_{i,r}})+\Upsilon_{i}\Big{)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}S_{e_{i},sub}^{up},$
(29)

where

$\displaystyle\Upsilon_{i}=\\{S_{e_{i},sub}\mid
p_{x_{i}}=\hat{p}_{x_{i,r}}\\},$ (30) $\displaystyle D_{i,p}=\\{\frac{\partial
S_{e_{i},sub}}{\partial p_{x_{i}}}\mid p_{x_{i}}=\hat{p}_{x_{i,r}}\\},$ (31)

and $\hat{p}_{x_{i,r}}$ is the given transmit power in the $r$-th iteration of
the convex optimization. As a result, with the upper bound in (29), problem
(23) for any given $M_{i}$ parameters, is a convex optimization problem that
can be solved efficiently by an existing convex optimization approach [16].

##  V Numerical Results

We consider a multi-numerology uplink system with three UEs where UE 1 uses
subcarrier spacing of $\Delta f_{1}=60$ kHz with $M_{1}=256$ as its FFT order,
while UE 2 and UE 3 may switch between other numerologies with
$\alpha_{1,2}=\\{2,4,8\\}$ and $\alpha_{1,3}=\\{2,4,8\\}$. We let $\tau_{{\rm
cp},1}=M_{1}/16$ and use quadrature amplitude modulation (QAM) with $64$
symbols and unity average power. We further consider Rayleigh fading channel
and a maximum transmit power of $100$ mW for uplink transmissions.

Figure 2: INI power vs. subcarrier index with respect to the subcarrier
spacing granularity of UE 1.

Fig. 2 compares the INI power per subcarrier obtained from simulations with
the derived theoretical results for $\alpha_{1,2}=2$. From Fig. 2, we first
note that the simulations validate the derived analyses in Proposition 1. We
also observe that the edge subcarriers of both numerologies are more impacted
by the INI, as compared to the middle subcarriers. The main reason behind this
result is that the sinc-like signals corresponding to each subcarrier decays
with a rate of $1/f$ [3]. Moreover, the sinc functions of the two adjacent
BWPs are not orthogonal at the sampling points anymore, due to using different
numerologies in the uplink. Hence, the maximum level of INI is observed at the
border of the two BWPs or edge subcarriers.

Figure 3: The CDF of average SIR for UE 1.

Fig. 3 shows the cumulative distribution function (CDF) of the signal to INI
power ratio (SIR) for randomly selected $\alpha_{1,2}$ and $\alpha_{1,3}$ and
different average power per subcarrier. Fig. 3 shows that with random
numerology selection, the SIR can be degraded while increasing the average
power per subcarrier. Therefore, it is meaningful to control the INI by
jointly optimizing the power allocation and the numerology selection.

Figure 4: Average $\Lambda$ vs. the average power per subcarrier.

Fig. 4 shows the effect of multi-user interference on the average $\Lambda$ as
the minimum SE across the UEs, with $\alpha_{1,2}=\\{2,4,8\\}$,
$\alpha_{1,3}=2$. The results are also compared with uniform power allocation
with $\alpha_{1,2}=4$ and $\alpha_{1,3}=2$. The results show that for
$\alpha_{1,2}=4$, the performance is better than $\alpha_{1,2}=\\{2,8\\}$, and
thus, highest or lowest $\alpha$ does not necessarily yield the optimal
performance. Fig. 4 also shows that the proposed solution can yield up to 30%
performance gain compared to uniform power allocation.

##  VI Conclusions

In this paper, we have proposed a novel scheme to analyze and optimize the
performance of uplink cellular communications with flexible frame structure.
We have presented the theoretical derivations for the INI power in the uplink
and have formulated a new resource allocation problem that maximizes the
minimum SE across the network by jointly optimizing the power allocation and
the numerology selection. The simulation results have corroborated the derived
INI analyses and have provided guidelines for joint power allocation and
numerology selection in uplink communications.

##  Appendix A proof of proposition 1

Let $X_{i}$ and $H_{i}$ represent the FFT of the transmitted signal and the
channel frequency response of an arbitrary UE $i$, respectively. From (8), the
interference term resulting from INI can be written as

$\displaystyle
R_{I_{h,i}}(n)=\sum_{j\in\mathcal{K}_{h,i}}\sum_{m=\tau_{\text{cp,i}}}^{M_{j}+\tau_{\text{cp},j}-1}\frac{1}{M_{j}}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(\nu_{l})H_{j}(\nu_{l})e^{\frac{j2\pi}{M_{j}}\nu_{l}m}\Big{)}$
$\displaystyle
e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{j\in\mathcal{K}_{h,i}}\sum_{q=2}^{\alpha_{i,j}}\sum_{m=\nu_{q-1}}^{\nu_{q}-1}\frac{1}{M_{j}}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(\nu_{l})e^{-\frac{j2\pi}{M_{j}}\nu_{q-1}\nu_{l}}$
$\displaystyle
H_{j}(\nu_{l})e^{\frac{j2\pi}{M_{j}}\nu_{l}m}\Big{)}e^{-\frac{j2\pi}{M_{i}}mk}.$
(32)

Similarly, for a neighbor UEs $j$ with lower numerology, the $I_{l,i}$ can be
written as

$\displaystyle
R_{I_{l,i}}(k)=\sum_{j\in\mathcal{K}_{l,i}}\sum_{m=M_{j}-\tau_{\alpha}}^{M_{j}-1}\frac{1}{M_{j}}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}$
$\displaystyle
e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{j\in\mathcal{K}_{l,i}}\sum_{m=0}^{M_{i}-\tau_{\alpha}-1}\frac{1}{M_{j}}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}$
$\displaystyle
e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{j\in\mathcal{K}_{l,i}}\sum_{q=2}^{\frac{1}{\alpha_{i,j}}}\sum_{m=\nu_{q}-M_{i}}^{\nu_{q}-1}\frac{1}{M_{j}}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)$
$\displaystyle e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}e^{-\frac{j2\pi}{M_{i}}mk}.$
(33)

Then, the received signal at subcarrier $n$ after subcarrier demapping and
IFFT can be written as

$\displaystyle\tilde{R_{i}}(n)=\frac{1}{N_{i}}\sum_{k=0}^{N_{i}-1}\Big{(}R_{d_{i}}(k)+R_{I_{i}}(k)+W(k)\Big{)}e^{j\frac{2\pi}{N_{i}}kn},$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}{d_{i}(n)+I_{i}(n)+\tilde{w}(n)}.$
(34)

The INI term from (32) and (33) can be written as

$I_{h,i}(n)=\frac{1}{N_{i}}\sum_{k=0}^{N_{i}-1}\sum_{j\in\mathcal{K}_{h,i}}\frac{1}{M_{j}}\Bigg{(}\sum_{m=\tau_{\text{cp,i}}}^{M_{j}+\tau_{\text{cp,j}}-1}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(\nu_{l})H_{j}(\nu_{l})$
$\displaystyle
e^{\frac{j2\pi}{M_{j}}\nu_{l}m}\Big{)}e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{q=2}^{\alpha_{i,j}}\sum_{m=\nu_{q-1}}^{\nu_{q}-1}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(\nu_{l})e^{-\frac{j2\pi}{M_{j}}\nu_{q-1}\nu_{l}}$
$\displaystyle
H_{j}(\nu_{l})e^{\frac{j2\pi}{M_{j}}\nu_{l}m}\Big{)}e^{-\frac{j2\pi}{M_{i}}mk}\Bigg{)}e^{\frac{j2\pi}{N_{i}}nk}=\frac{1}{N_{i}}\sum_{k=0}^{N_{i}-1}\sum_{j\in\mathcal{K}_{h,i}}\sum_{q=1}^{\alpha_{i,j}}\sum_{l=0}^{N_{j}-1}$
$\displaystyle\frac{1}{M_{j}}X_{j}(\nu_{l})H_{j}(\nu_{l})e^{-j2\pi(\frac{\nu_{q-1}}{M_{i}}-\frac{n}{N_{i}})}Q_{i,j,h}^{I}(l,k,q).$
(35)

In (34), the lower and upper limits of the IFFT summation are defined
according to (4). In the following, the received INI signal from neighbor UE
$j$ with lower numerology can be written as

$\displaystyle
I_{l,i}(n)=\frac{1}{N_{i}}\sum_{k=0}^{N_{i}-1}\sum_{j\in\mathcal{K}_{l,i}}\frac{1}{M_{j}}\Bigg{(}\sum_{m=M_{j}-\tau_{\alpha}}^{M_{j}-1}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)$
$\displaystyle
e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{m=0}^{M_{i}-\tau_{\alpha}-1}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}$
$\displaystyle
e^{-\frac{j2\pi}{M_{i}}mk}+\sum_{q=2}^{\frac{1}{\alpha_{i,j}}}\sum_{m=\nu_{q}-M_{i}}^{\nu_{q}-1}\Big{(}\sum_{l=0}^{N_{j}-1}X_{j}(l)H_{j}(l)e^{-\frac{j2\pi}{M_{j}}ml}\Big{)}$
$\displaystyle
e^{-\frac{j2\pi}{M_{i}}mk}\Bigg{)}e^{\frac{j2\pi}{N_{i}}nk}=\frac{1}{N_{i}}\sum_{k=M_{i}-N_{i}}^{M_{i}-1}\sum_{j\in\mathcal{K}_{l,i}}\sum_{q=1}^{\frac{1}{\alpha_{i,j}}}\sum_{l=0}^{N_{j}-1}\frac{1}{M_{j}}X_{j}(l)$
(36) $\displaystyle H_{j}(l)Q_{i,j,l}^{I}(l,k,q)e^{\frac{j2\pi}{N_{i}}nk}.$

This concludes the proof. ∎

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