# Self-Calibration for Massive MIMO with Channel Reciprocity and Channel
Estimation Errors

De Mi1, Lei Zhang2, Mehrdad Dianati 31, Sami Muhaidat41, Pei Xiao1 and Rahim
Tafazolli1<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Institute
for Communication Systems, University of Surrey, United Kingdom  2School of
Engineering, University of Glasgow, United Kingdom  3Warwick Manufacturing
Group, University of Warwick, United Kingdom  4Department of Electrical and
Computer Engineering, Khalifa University, United Arab Emirates

###### Abstract

In time-division-duplexing (TDD) massive multiple-input multiple-output (MIMO)
systems, channel reciprocity is exploited to overcome the overwhelming pilot
training and the feedback overhead. However, in practical scenarios, the
imperfections in channel reciprocity, mainly caused by radio-frequency
mismatches among the antennas at the base station side, can significantly
degrade the system performance and might become a performance limiting factor.
In order to compensate for these imperfections, we present and investigate two
new calibration schemes for TDD-based massive multi-user MIMO systems, namely,
relative calibration and inverse calibration. In particular, the design of the
proposed inverse calibration takes into account a compound effect of channel
reciprocity error and channel estimation error. We further derive closed-form
expressions for the ergodic sum rate, assuming maximum ratio transmissions
with the compound effect of both errors. We demonstrate that the inverse
calibration scheme outperforms the traditional relative calibration scheme.
The proposed analytical results are also verified by simulated illustrations.

††We would like to acknowledge the support of the University of Surrey 5GIC
(www.surrey.ac.uk/5gic) members for this work. This work was also supported in
part by the European Commission under the 5GPPP project 5G-Xcast
(H2020-ICT-2016-2 call, grant number 761498). The views expressed in this
contribution are those of the authors and do not necessarily represent the
project.

##  I Introduction

Massive MIMO (multiple-input-multiple-output) is identified as a promising
technological paradigm which has been proposed in order to meet some of 5G
requirements, including data rates of 10-20 Gbps and a latency of less than 1
msec [1, 2, 3, 4]. Exploiting channel reciprocity, time-division-duplexing
(TDD) operation enables the channel state information (CSI) acquisition in
massive MIMO with an affordable overhead that is independent of the number of
base station (BS) antennas [5]. Most prior studies assume the perfect channel
reciprocity by constraining the time delay from the uplink (UL) to the
downlink (DL) is within the coherence time of the channel [6, 7]. However, the
assumption of the perfect reciprocity is unrealistic in practical systems even
within the coherence time, due to the fact that radio-frequency (RF)
transceivers introduce amplitude and phase mismatches between the UL and the
DL [8]. The imperfect channel reciprocity contaminates the estimate of the
effective channel response. This causes a significant degradation in the
performance of linear precoding schemes, due to their sensitivity to the CSI
accuracy. Our prior work [9] has thoroughly investigated such performance
degradation for two typical linear precoders, i.e., maximum radio transmission
(MRT) and zero-forcing (ZF), with considerations of the imperfect channel
estimation. The results in [9] show that both MRT and ZF are severely affected
by the compound effect of the reciprocity and estimation errors. Therefore, it
is of great interest to study suitable reciprocity calibration schemes in the
presence of both errors.

In principle, calibration schemes for a precoded TDD MIMO system contain the
estimation of transmit (Tx) and receive (Rx) RF frontends’ responses or
equivalently calibration coefficients and the design of the calibration matrix
to calibrate the precoders. For the calibration coefficients estimation, we
explicitly focus on the so-called self-calibration in massive MIMO systems
since it can be implemented at the BS side only and without exchanging
calibration pilots between the BS and user terminals (UTs) [10]. Prior studies
proposed different methods to realise self-calibration in massive MIMO systems
[11, 12]. In [11], the use of antenna coupling at the BS has been proposed to
measure the calibration coefficients. This method is very sensitive to the
scatterings near the BS antennas. A practical study in [12] presented a method
where an additional RF transceiver is used as a reference to exchange
calibration pilots with other BS antennas’ transceivers. However, the method
in [12] is sensitive to the placement of the reference transceiver. In order
to obtain reliable estimates of the calibration coefficients, additional
calibration circuits can be applied at the BS, as one example shown in [13].
The self-calibration scheme in [13, 12] is known as relative calibration,
which has been widely considered in the context of the massive MIMO system
[14, 15].

Although a number of recent progress has been developed from the perspective
of the reciprocity calibration, various issues still remain open, such that,
to the best knowledge of the authors, there have been no calibration schemes
taking into account the compound effect of the reciprocity and channel
estimation errors. In order to fill these research gaps among the existing
literature, we consider a low-cost calibration circuit that presented in [13],
and expand it into the TDD massive MU-MIMO system. Such design enables the BS
to estimate the RF responses reliably, also to perform the relative
calibration or what we call “inverse calibration” (in the sense that it is
based on the inverse of the calibration coefficients). More importantly, with
considerations of the compound effect of the reciprocity and estimation
errors, we provide an in-depth analysis of the performance of a precoded
massive MIMO system with the aforementioned two calibration algorithms. The
proposed analytical results are verified via Monte-Carlo simulations. The rest
of the paper is organised as follows. In Section II, we describe the TDD
massive MIMO system model with imperfect channel estimation and the
reciprocity error model due to the RF mismatch. In Section III, the
description of the relative and inverse calibration algorithms comes after a
discussion of the considered calibration circuit. The performance evaluation
of these two algorithms is given in Section IV. Simulation results and
conclusions are provided in Section V and Section VI, respectively. Some of
the detailed derivations are given in the appendices.

##  II System Model

A TDD massive MU-MIMO system is considered in this paper as illustrated in
Fig. 1. It consists of $M$ antennas at the BS, each antenna is connected with
an individual RF chain. In addition, $K$ single-antenna UTs ($M\gg K$) are
served in the same time and frequency resources. We assume that the time delay
from the UL channel estimation to the DL transmission is less than the
coherence time of the channel, ensuring that the propagation channels on the
UL and DL are equal.

Figure 1: A TDD massive MU-MIMO System with calibration circuits.

As shown in Fig. 1, we denote the UL and DL propagation channels by
$\mathbf{H}\in\mathbb{C}^{M\times K}$ and $\mathbf{H}^{T}$ respectively, whose
entries follow independent identically distributed (i.i.d.)
$C\mathcal{N}(0,1)$. We consider the same model of the BS RF frontends
response as that set up in [9]. Briefly speaking, $M\times M$ diagonal
matrices $\mathbf{H}_{br}$ and $\mathbf{H}_{bt}$ represent the effective
response matrices of the Rx and Tx RF frontends at the BS, whose $i$-th
diagonal entries are $h_{br,i}=A_{br,i}\textnormal{exp}(j\varphi_{br,i})$ and
$h_{bt,i}=A_{bt,i}\textnormal{exp}(j\varphi_{bt,i})$, respectively, where $A$
denotes amplitude, $\varphi$ denotes phase. We model both amplitude and phase
reciprocity errors as independent truncated Gaussian random variables [9].
Without loss of generality, the statistical magnitudes of these truncated
Gaussian distributed variables are assumed to be static within the considered
coherence time of the channel or even longer period, e.g., minutes [12].

We then consider the UL training protocol based on the minimum mean-square
error (MMSE) channel estimation as in [6], by taking into account the effect
of $\mathbf{H}_{br}$ and $\mathbf{H}_{bt}$. More specifically, in TDD massive
MIMO systems, UTs first transmit the orthogonal UL pilots of length $\tau_{u}$
to BS, where $\tau_{u}\geq K$. Therefore, the MMSE estimate of the actual UL
channel response $\mathbf{H}_{u}$ can be given by [6]

$\hat{\mathbf{H}}_{u}=a\mathbf{H}_{br}\mathbf{H}+b\mathbf{N}_{u},$ (1)

where the estimation-error-related parameters are given by

$a=\dfrac{\tau_{u}\rho_{u}}{\tau_{u}\rho_{u}+1},\>b=\dfrac{\sqrt{\tau_{u}\rho_{u}}}{\tau_{u}\rho_{u}+1}.$
(2)

In addition, the $M\times K$ noise matrix $\mathbf{N}_{u}$ is the channel
estimation noise matrix with i.i.d. $C\mathcal{N}(0,1)$ elements and is
independent of $\hat{\mathbf{H}}_{d}$, and $\rho_{u}$ denotes the expected UL
transmit SNR. Then the BS uses the transpose of $\hat{\mathbf{H}}_{u}$ as the
estimate of the DL channel $\hat{\mathbf{H}}_{d}$, i.e.,
$\hat{\mathbf{H}}_{d}=\hat{\mathbf{H}}_{u}^{T}$, whereas the actual effective
DL channel is $\mathbf{H}_{d}=\mathbf{H}^{T}\mathbf{H}_{bt}$. By comparing
$\hat{\mathbf{H}}_{d}$ and $\mathbf{H}_{d}$, we can rewrite the DL channel
estimate $\hat{\mathbf{H}}_{d}$ as

$\hat{\mathbf{H}}_{d}=a\mathbf{H}_{d}\mathbf{E}+b\mathbf{N}_{u}^{T},$ (3)

where $\mathbf{E}=\mathbf{H}_{bt}^{-1}\mathbf{H}_{br}$ denotes the channel
reciprocity error. We can see from (3) a compound effect of an additive
distortion, $\mathbf{N}_{u}$, caused by the imperfect channel estimation, and
a multiplicative distortion, $\mathbf{E}$, caused by the imperfect channel
reciprocity.

Let a $K\times 1$ vector $\mathbf{s}=[s_{1},\cdots s_{k},\cdots,s_{K}]^{T}$
denote the symbol to be transmitted to $K$ UTs, where the normalised symbol
power per user is assumed, i.e., $\mathbb{E}\left\\{\lvert
s_{k}\rvert^{2}\right\\}=1$, for $k=1,2,\cdots,K$. We also assume that the
symbols of different users are independent. The BS applies an $M\times K$
linear precoding matrix $\mathbf{W}$ to map the symbol vector $\mathbf{s}$
into an $M\times 1$ transmit signal vector to the BS antennas. We use
$\mathbf{x}$ to denote this transmit signal vector, which is given by

$\mathbf{x}=\sqrt{\rho_{d}}\lambda\mathbf{W}\mathbf{s},$ (4)

where $\rho_{d}$ denotes the average transmit power at the BS (note that the
expression (4) implies that the power is equally allocated to each UT in this
work), and $\lambda$ is a normalisation parameter to satisfy the transmission
power constraint at the BS such that

$\mathbb{E}\left\\{\lVert\mathbf{x}\rVert^{2}\right\\}=\mathbb{E}\left\\{\lVert\sqrt{\rho_{d}}\lambda\mathbf{W}\mathbf{s}\rVert^{2}\right\\}=\rho_{d}.$
(5)

Hence, $\lambda$ can be calculated as follows:

$\lambda=\sqrt{\dfrac{1}{\mathbb{E}\left\\{\textnormal{tr}\left(\mathbf{W}\mathbf{W}^{H}\right)\right\\}}}.$
(6)

Based on (4), the received signals of all $K$ UTs can be expressed in a vector
form as

$\mathbf{y}=\mathbf{H}_{d}\mathbf{x}+\mathbf{n}=\sqrt{\rho_{d}}\lambda\mathbf{H}^{T}\mathbf{H}_{bt}\mathbf{W}\mathbf{s}+\mathbf{n},$
(7)

where the $K\times 1$ vector $\mathbf{n}$ denotes the DL received noise for
all $K$ UTs, whose $k^{\text{th}}$ element $n_{k}\sim
C\mathcal{N}(0,\sigma_{k}^{2})$. We assume that $\sigma_{k}^{2}=1,\forall k$.
Therefore, $\rho_{d}$ can also be treated as the average input SNR for the DL
transmission. For the $k^{\text{th}}$ UT, we have

$y_{k}=\sqrt{\rho_{d}}\lambda\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{k}s_{k}+\\!\\!\sum_{i=1,i\neq
k}^{K}\\!\\!\sqrt{\rho_{d}}\lambda\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{i}s_{i}+n_{k},$
(8)

where the $M\times 1$ vectors $\mathbf{h}_{k}$ and $\mathbf{w}_{k}$ are the
$k^{\text{th}}$ column of $\mathbf{H}$ and $\mathbf{W}$ respectively. Note
that the received signal $y_{k}$ in (8) is decomposed into three terms. The
first two terms accounts for the desired signal for the $k^{\text{th}}$ UT and
the inter-user interference from other $K-1$ UTs, respectively, while the last
term represents the receiver noise. Since $\mathbf{W}$ is a function of the DL
channel estimate instead of the actual DL channel response, the performance of
linear precoding schemes is affected by the imperfect channel reciprocity. To
address this issue, we introduce the calibration scheme in the following
section.

##  III Self-Calibration

To compensate for the imperfection of channel reciprocity, an $M\times M$ pre-
precoding calibration matrix $\mathbf{B}$ can be introduced to compensate for
the non-reciprocity [15], such that $\hat{\mathbf{H}}_{d}$ in (3) becomes

$\hat{\mathbf{H}}_{d,CL}=a\mathbf{H}_{d}\mathbf{E}\mathbf{B}+b\mathbf{N}_{u}^{T}\mathbf{B},$
(9)

where $\hat{\mathbf{H}}_{d,CL}$ represents the estimate of the DL channel
response after applying calibration, which could be used to calculate the DL
precoding matrix $\mathbf{W}$. The majority of the reported results on the
reciprocity calibration has proposed the design concept of the calibration
matrix $\mathbf{B}$ without considering the effect of channel estimation error
[12, 14, 15], e.g., in the case with $a\approx 1$ and $b\approx 0$. In such
case, the minimum requirement to calibrate the BS antennas is that
$\mathbf{EB}=c\mathbf{I}_{M}$, where the scalar $c\in\mathbb{C}_{\neq 0}$ is
multiplied by all calibration factors. The scalar $c$ is arbitrary.
$\mathbb{C}_{\neq 0}$ denotes the set of non-zero complex numbers. Thus, this
does not change the direction of the precoding beamformer [16].

The acquisition of the calibration matrix $\mathbf{B}$ contains two steps: 1)
the estimation of $\mathbf{H}_{bt}$, $\mathbf{H}_{br}$, and 2) finding
$\mathbf{B}$ based on the estimates of $\mathbf{H}_{bt}$ and
$\mathbf{H}_{br}$. In the following, we first present a calibration circuit
used for the first step. We then discuss two calibration algorithms obtained
based on the considered circuit design, and their relationship with the
channel estimation error.

###  III-A RF Frontend Response Measurement

Motivated by the calibration method for conventional MIMO systems in [13], we
present a circuit design as shown in Fig. 1, to achieve the measurement of the
effective response matrix of the BS RF frontend for the massive MIMO system.
Particularly, switching units attached to each antenna have three modes:
“Tx/Rx” mode (the antenna connects to Tx or Rx RF frontend), “Link” mode (Tx
and Rx RF frontends are connected) and “Null” mode (no connection); A
reference signal source is split and equally injected at each Rx Rf frontend
by couplers. Then the measurement of $\mathbf{H}_{bt}$ and $\mathbf{H}_{br}$
can be carried out, which contains two steps which we call “self connection”
and “half connection”.

During the self connection, all switching units that attached to the BS
antennas are set to be in “Link” mode. The reference signal source is
disconnected. The individual baseband of each BS antenna, taking $i$-th
antenna element as an example, estimates the product of $h_{bt,i}$ and
$h_{br,i}$ by sending a known signal $p_{i}$ simultaneously, such that
$r_{i}=h_{br,i}h_{bt,i}p_{i}+u_{i}$ received at the baseband of the $i$-th
antenna, where the thermal noise $u_{i}$ has a negligible value due to the
fact that the calibration SNR is usually sufficiently high, e.g., 20 dB in
[13]. Thus the estimate of $\mathbf{H}_{bt}\mathbf{H}_{br}$, denoted by
$\mathbf{R}_{\text{self}}$, is given by

$\mathbf{R}_{\text{self}}=\text{diag}\left(r_{1}/p_{1},\cdots,r_{i}/p_{i},\cdots,r_{K}/p_{K}\right).$
(10)

During the half connection, all switching units that attached to the BS
antennas are set to be in “Null” mode. The reference signal source is equally
injected at all Rx RF frontends. Let an $M\times 1$ vector
$\mathbf{p}_{\text{ref}}$ be the reference signal vector with duplicate
entries. Then the collective received signal vector at all the BS antenna
baseband is given by
$\mathbf{r}_{h}=\mathbf{H}_{br}(\mathbf{p}_{\text{ref}}+\mathbf{u}_{h})$ ,
where $\mathbf{r}_{h}$ contains the received signals at each baseband that are
sampled at the same time [13]. Again, the effect of the measurement noise
$\mathbf{u}_{h}$ is assumed to be trivial in this work due to the assumption
on the high calibration SNR. Hence, the estimate of $\mathbf{H}_{br}$, denoted
by $\mathbf{R}_{\text{half}}$, is given by

$\mathbf{R}_{\text{half}}=\text{diag}(\mathbf{r}_{h})\left(\text{diag}(\mathbf{p}_{\text{ref}})\right)^{-1}.$
(11)

Note that, as discussed in Section II, the reciprocity-error-related
parameters, or equivalently $\mathbf{H}_{bt}$ and $\mathbf{H}_{br}$, are
relatively static, i.e., they change in a much slower rate compared to the
variations of the channel state. Hence, once the measurement of the BS RF
responses, i.e., $\mathbf{R}_{\text{self}}$ and $\mathbf{R}_{\text{half}}$, is
reliably obtained, it can be applied within the the coherence time of the
channel or even longer period [12].

###  III-B Design of the Calibration Matrix

Based on the measurement in (10) and (11), the calibration matrix $\mathbf{B}$
can be calculated. The study in [13] considers the relative calibration
scheme, where the calibration matrix, denoted by $\mathbf{B}_{\text{RC}}$, is
given by
$\mathbf{B}_{\text{RC}}=\mathbf{R}_{\text{self}}\left(\mathbf{R}_{\text{half}}^{2}\right)^{-1}.$
As discussed at the beginning of this section, the widely-used relative
calibration ignores the effect of the imperfect channel estimation, which can
result in the estimation error amplification, and additionally cause the
enhancement of the inter-user interference. More specifically, it can be seen
from (9) that the use of the calibration matrix $\mathbf{B}_{\text{RC}}$ may
amplify the power of the estimation noise (equivalently, channel estimation
error), which can even outweigh the benefit of calibration in certain cases,
such as in the low region of $\rho_{u}$. This motivates us to design a
calibration matrix without amplifying the estimation error.

In this work, we present a calibration scheme to compensate for the effect of
the channel reciprocity error, as well as to reduce the noise power of the UL
channel estimation, or equivalently reduce the estimation noise variance. To
this end, we consider a calibration matrix, denoted by
$\mathbf{B}_{\text{IC}}$, which is given by

$\mathbf{B}_{\text{IC}}=\mathbf{R}_{\text{half}}^{\ast}\left(\mathbf{R}_{\text{half}}\mathbf{R}_{\text{self}}^{\ast}\right)^{-1}.$
(12)

In the ideal scenario, e.g., in the sufficiently high calibration SNR regime,
the calibration matrix $\mathbf{B}_{\text{IC}}$ is equivalent to the inverse
of the product of $\mathbf{H}_{bt}^{\ast}$ and $\mathbf{H}_{br}$. Thus, we
name this calibration scheme as “Inverse Calibration”. Such a scheme can
ensure that noise power of UL channel estimation after calibration is equal to
or even less than that before calibration, since the expected value of the
product of $\mathbf{H}_{bt}^{\ast}$ and $\mathbf{H}_{br}$ is greater than or
equal to one [9]. We shall evaluate the performance of the inverse calibration
and the traditional relative calibration in the following section.

##  IV Performance Evaluation

In this section, we analyse the ergodic sum rate to evaluate the performance
of the inverse calibration and the widely-used relative calibration. We
consider the simplest precoder, i.e., maximum ratio transmission. Note that
our theoretical analysis contends with the compound effects on the system
performance of the additive channel estimation error and multiplicative
channel reciprocity error.

Recall that in (8), we denote the desired signal power of the $k^{\text{th}}$
UT by $P_{s}$, and the its inter-user interference by $P_{I}$, where $P_{s}$
and $P_{I}$ are given by

$P_{s}=\lvert\sqrt{\rho_{d}}\lambda\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{k}s_{k}\rvert^{2},$
(13) $P_{I}=\left|\sum_{i=1,i\neq
k}^{K}\sqrt{\rho_{d}}\lambda\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{i}s_{i}\right|^{2},$
(14)

respectively. Thus, the ergodic rate for the $k^{\text{th}}$ UT, denoted by
$R_{k}$, can be given by

$R_{k}=\mathbb{E}\left\\{\text{log}_{2}\left(1+\dfrac{P_{s}}{P_{I}+\sigma_{k}^{2}}\right)\right\\}.$
(15)

Let $R_{K}$ denote the ergodic sum rate of all $K$ UTs. Considering the
approximation derived in [9, 17], $R_{K}$ is given by

$R_{K}=KR_{k}\approx
K\text{log}_{2}\left(1+{\mathbb{E}}\left\\{P_{s}\right\\}{\mathbb{E}}\left\\{\dfrac{1}{P_{I}+\sigma_{k}^{2}}\right\\}\right),$
(16)

where

$\mathbb{E}\left\\{\frac{1}{P_{I}+\sigma_{k}^{2}}\right\\}=\dfrac{1}{\mathbb{E}\\{P_{I}+\sigma_{k}^{2}\\}}+\mathcal{O}\left(\dfrac{\textnormal{var}(P_{I}+\sigma_{k}^{2})}{\mathbb{E}\\{P_{I}+\sigma_{k}^{2}\\}^{3}}\right).$
(17)

In this paper, we take the simplest precoding algorithm [5], i.e., MRT, as an
example. Recall (9), when the MRT is used at the BS, the precoding matrix
$\mathbf{W}$ can be given by

$\mathbf{W}_{\textnormal{mrt}}=\hat{\mathbf{H}}_{d,CL}^{H}\>.$ (18)

Let $\lambda_{\textnormal{mrt}}$ represent the normalisation parameter of the
MRT precoding scheme, to meet the power constraint at the BS. With no
calibration (NC), i.e., $\mathbf{B}=\mathbf{I}_{M}$, the $k^{\text{th}}$ UT’s
output SINR (signal-to-interference-plus-noise ratio) with uncalibrated MRT
precoder has been derived in [9]. The analytical result of the output SINR in
[9] is obtained based on (17), thus, it can be used to obtain the ergodic sum
rate in (16), as follows:

######  Lemma 1.

Consider a TDD massive MIMO system modelled in Section II, with uncalibrated
MRT precoder at the BS. The closed-form expression of the ergodic sum rate for
$K$ UTs, $R_{K,\textnormal{mrt}}^{\textnormal{NC}}$ , is given by

$\displaystyle R_{K,\textnormal{mrt}}^{\textnormal{NC}}\approx$ $\displaystyle
K\textnormal{log}_{2}\left(\\!1+\rho_{d}A_{t}\left(\dfrac{a^{2}A_{r}((M-1)A_{I}+2)\\!+b^{2}}{a^{2}A_{r}+b^{2}}\right)\right.$
$\displaystyle\left.\times\left(\dfrac{K^{2}+\rho_{d}K(K-1)(\rho_{d}A_{t}^{2}+2A_{t})}{(\rho_{d}(K-1)A_{t}+K)^{3}}\right)\right),$
(19)

where the estimation-error-related parameters $a$ and $b$ are given by (2),
and the reciprocity-error-related parameters $A_{t}$, $A_{r}$ and $A_{I}$ are
given in [9].

###### Proof.

See the proof of Theorem 1 in [9]. ∎

The result in (1) quantifies the compound effect of the reciprocity and
estimation errors on the ergodic achievable sum rate of the MRT precoded
system without calibration.

When the inverse calibration is applied at the BS for the MRT precoder, the
precoding matrix $\mathbf{W}_{\text{mrt}}$ can be expressed as

$\displaystyle\mathbf{W}_{\textnormal{mrt}}^{\textnormal{IC}}$
$\displaystyle=\left(a\mathbf{H}_{d}\mathbf{E}\mathbf{B}_{\text{IC}}+b\mathbf{N}_{u}^{T}\mathbf{B}_{\text{IC}}\right)^{H}$
$\displaystyle=a\mathbf{H}_{bt}^{-1}\mathbf{H}^{\ast}+b\mathbf{H}_{bt}^{-1}\left(\mathbf{H}_{br}^{\ast}\right)^{-1}\mathbf{N}_{u}^{\ast},$
(20)

Then the corresponding normalisation parameter
$\lambda_{\textnormal{mrt}}^{\textnormal{IC}}$, the expected values of the
desired signal power and the interference power of the $k^{\text{th}}$ UT,
i.e., ${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{IC}}\right\\}$ and
${\mathbb{E}}\left\\{1/\left(P_{I,\text{mrt}}^{\text{IC}}+\sigma_{k}^{2}\right)\right\\}$,
can be derived. Details are provided in Appendix B. These values are used to
calculate the ergodic sum rate in the following proposition:

######  Proposition 1.

Assuming that the same conditions are held as in the Lemma 1, while the
inverse calibration is applied at the BS. The ergodic sum rate for $K$ UTs,
$R_{K,\textnormal{mrt}}^{\textnormal{IC}}$ , is given by

$\displaystyle R_{K,\textnormal{mrt}}^{\textnormal{IC}}\approx$ $\displaystyle
K\textnormal{log}_{2}\left(1+\rho_{d}\left(\dfrac{a^{2}(M-1)+b^{2}E_{\bar{2}}^{r}}{a^{2}+b^{2}E_{\bar{2}}^{r}}\right)\right.$
$\displaystyle\\!\\!\\!\left.\times\left(\dfrac{(KE_{\bar{2}}^{t})^{2}+\rho_{d}K(K-1)(\rho_{d}+2E_{\bar{2}}^{t})}{(\rho_{d}(K-1)+KE_{\bar{2}}^{t})^{3}}\right)\right),$
(21)

where the reciprocity-error-related parameters $E_{\bar{2}}^{r}$ and
$E_{\bar{2}}^{t}$ are given by (27) and (28) respectively, in Appendix A.

###### Proof.

See Appendix B. ∎

When the traditional relative calibration is applied at the BS, the precoding
matrix $\mathbf{W}_{\text{mrt}}$ becomes

$\displaystyle\mathbf{W}_{\textnormal{mrt}}^{\textnormal{RC}}$
$\displaystyle=\left(a\mathbf{H}_{d}\mathbf{E}\mathbf{B}_{\text{RC}}+b\mathbf{N}_{u}^{T}\mathbf{B}_{\text{RC}}\right)^{H}$
$\displaystyle=a\mathbf{H}_{bt}^{\ast}\mathbf{H}^{\ast}+b\mathbf{H}_{bt}^{\ast}\left(\mathbf{H}_{br}^{\ast}\right)^{-1}\mathbf{N}_{u}^{\ast}.$
(22)

Similar to the case of the inverse calibration,
$\lambda_{\textnormal{mrt}}^{\textnormal{IC}}$,
${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{IC}}\right\\}$ and
${\mathbb{E}}\left\\{1/(P_{I,\text{mrt}}^{\text{IC}}+\sigma_{k}^{2})\right\\}$
can be derived (see Appendix B). Then the corresponding closed-form expression
of the ergodic sum rate can be given as follows:

######  Proposition 2.

Assuming that the same conditions are held as in the Lemma 1, while the
relative calibration is applied at the BS. The ergodic sum rate for $K$ UTs,
$R_{K,\textnormal{mrt}}^{\textnormal{RC}}$ , is given by

$\displaystyle\\!\\!\\!R_{K,\textnormal{mrt}}^{\textnormal{RC}}\approx$
$\displaystyle
K\textnormal{log}_{2}\left(\\!1\\!+\\!\rho_{d}\left(\dfrac{a^{2}((M\\!-\\!1)A_{t}^{2}\\!+\\!2E_{4}^{t})\\!+b^{2}E_{4}^{t}E_{\bar{2}}^{r}}{a^{2}+b^{2}E_{\bar{2}}^{r}}\right)\right.$
$\displaystyle\\!\\!\\!\left.\times\left(\dfrac{K^{2}A_{t}^{2}\\!+\\!\rho_{d}K(K\\!-\\!1)E_{4}^{t}(\rho_{d}E_{4}^{t}\\!+\\!2A_{t})}{(\rho_{d}(K-1)E_{4}^{t}+KA_{t})^{3}}\right)\right),$
(23)

where the reciprocity-error-related parameters $E_{4}^{t}$ is given by (A) in
Appendix A.

###### Proof.

See Appendix B. ∎

Comparing (1) with (2), it can be seen that the inverse calibration completely
removes phase reciprocity error, and leaves the MRT precoder with the
negligible amplitude-error-related parameters, i.e., $E_{\bar{2}}^{t}$ and
$E_{\bar{2}}^{r}$. On the contrary, the relative calibration introduces the
residual amplitude error into the MRT system, i.e., $A_{t}^{2}$ and
$E_{4}^{t}$ in (2) (based on the results in Appendix A, we have
$E_{4}^{t}>A_{t}^{2}>A_{t}$). This in turn can cause the aforementioned
estimation error amplification. Consequently, the compound effect of the
residual amplitude reciprocity error and the “amplified” channel estimation
error may result in a significant performance loss in the system with the
relative calibration.

##  V Simulation Results

In this section, we perform Monte-Carlo simulations to corroborate the
analysis presented in Section IV, and compare the performance of the
calibration schemes discussed in the paper under different scenarios, in order
to provide valuable insights into the practical system design. Unless
otherwise specified, the statistical magnitudes of both amplitude and phase
reciprocity errors are identical to that defined in [9, 18], with quadruple
notations, such that
$(\alpha_{bt,0},\sigma_{bt}^{2},[a_{t},b_{t}])=(\alpha_{br,0},\sigma_{br}^{2},[a_{r},b_{r}])=(0\
\textnormal{dB},1,[-4\ \textnormal{dB},4\ \textnormal{dB}])$, and
$(\theta_{bt,0},\sigma_{\varphi_{t}}^{2},[\theta_{t,1},\theta_{t,2}])=(\theta_{br,0},\sigma_{\varphi_{r}}^{2},[\theta_{r,1},\theta_{r,2}])=(0^{\circ},1,[-50^{\circ}\\!,\
50^{\circ}])$, as considered in [9]. We also consider a reference scenario,
namely, “Perfect Channel Reciprocity”, for the case that
$\sigma_{bt}^{2}=\sigma_{bt}^{2}=\sigma_{\varphi_{t}}^{2}=\sigma_{\varphi_{t}}^{2}=0$.
The orthogonal UL pilots are of length $\tau_{u}=K$, and the ergodic sum rate
is measured in bits/s/Hz.

First, we investigate the ergodic sum rate of MRT with IC and RC, for
different DL SNR regimes. As a benchmark, we consider perfect channel
reciprocity and no calibration, NC. We use $M=100$, $K=10$, and reciprocity-
error-related parameters as mentioned before. Fig. 2 illustrates that, for the
MRT precoded system, IC nearly eliminates the effect of reciprocity error, and
outperforms RC. Thus, we conclude that IC is more efficient than RC for MRT.
More specifically, increasing the DL transmit power 10 times, e.g., $\rho_{d}$
from 0 dB to 10 dB, the performance of IC increases by 36% (i.e., 9
bits/s/Hz), while only 12% improvement (i.e., 3 bits/s/Hz) for RC. This can
also be approved analytically based on the comparison between (1) and (2). As
mentioned earlier, both IC and RC can remove the phase reciprocity error.
However, RC suffers from the strong residual amplitude error, which can cause
estimation error amplification that could outweigh the benefit of using RC. We
further analyse this effect of estimation error amplification in RC in Fig. 3
and 4.

Figure 2: Ergodic Sum Rate versus DL SNR in the presence of the high level
reciprocity error and channel estimation error with $\rho_{u}=0\ \text{dB}$.
Figure 3: Ergodic Sum Rate of MRT in the presence of different combinations of
amplitude and phase reciprocity errors.

In Fig. 3, we assume $M=100$, $K=10$, $\rho_{d}=10\ \text{dB}$, and a low
level estimation error with $\rho_{u}=10\ \text{dB}$. Let
$\sigma_{A}^{2}=\sigma_{P}^{2}$ be the x-axis that captures the aggregated
effect of both amplitude and phase reciprocity errors, i.e.,
$(\alpha_{bt,0},\sigma_{bt}^{2},[a_{t},b_{t}])=(\alpha_{br,0},\sigma_{br}^{2},[a_{r},b_{r}])=(0\
\textnormal{dB},\sigma_{A}^{2},[-4\ \textnormal{dB},4\ \textnormal{dB}])$, and
the phase error has
$(\theta_{bt,0},\sigma_{\varphi_{t}}^{2},[\theta_{t,1},\theta_{t,2}])=(\theta_{br,0},\sigma_{\varphi_{r}}^{2},[\theta_{r,1},\theta_{r,2}])=(0^{\circ},\sigma_{P}^{2},[-50^{\circ}\\!,\
50^{\circ}])$. We consider the case with the phase error only by setting
$\sigma_{A}^{2}=0$. Fig. 3 shows that: a) both IC and RC eliminate the phase
error, with a 12.5 % increase in the sum rate compared with NC (i.e., 4
bits/s/Hz more than 32 bits/s/Hz of NC at $\sigma_{A}^{2}=0$ and
$\sigma_{P}^{2}=0.2$); b) the performance gain of RC over NC decreases by 50 %
in the case with the aggregated effect of the amplitude and phase error (i.e.,
only 2 bits/s/Hz improvement compared to NC at
$\sigma_{A}^{2}=\sigma_{P}^{2}=0.2$, which is 50 % of the previous 4 bits/s/Hz
improvement in (a)), whereas the gain of IC over NC increases by 50 % in this
case (i.e., around 6 bits/s/Hz improvement compared to NC at
$\sigma_{A}^{2}=\sigma_{P}^{2}=0.2$); c) RC may not work properly in the
presence of the compound effect of both reciprocity and estimation errors, see
$\sigma_{A}^{2}$, $\sigma_{P}^{2}<0.02$. We now take a closer look at the
third observation, c, as follows.

Similar parameters are considered as in Fig. 3. In addition, higher levels of
channel estimation error are introduced, e.g., $\rho_{u}=0\ \text{dB}$ in Fig.
4(a) and $\rho_{u}=-5\ \text{dB}$ in Fig. 4(b). It can be seen that the gain
of the relative calibration vanishes in the case of the severe estimation
error, whereas the inverse calibration, due to its greater robustness to the
compound effect of the reciprocity error and estimation error, still works
effectively with only a minor performance degradation.

Figure 4: Ergodic Sum Rate versus Reciprocity Error Variance with the
different level of channel estimation error.

##  VI Conclusion

In this paper, we have presented and investigated two calibration schemes,
i.e., inverse calibration and relative calibration, in the TDD massive MU-MIMO
system. A low-cost calibration circuit has been considered, which enables the
BS to select these two calibration algorithms. The performance of both
calibration algorithms has been evaluated, by taking into account the compound
effect of the multiplicative reciprocity error and the additive estimation
error. Particularly, we have derived closed-formed expressions for the ergodic
sum rate of the considered system, assuming MRT with the compound effect of
both errors. We have demonstrated that the inverse calibration in general
outperforms the traditional relative calibration. Analytical results perfectly
match the simulated results for different scenarios, including large or
practical number of BS antennas, different number of UTs and different
combinations of the reciprocity error and estimation error. The comprehensive
performance analysis has been given to provide a useful guidance for the
selection of the calibration schemes in the massive MIMO system, which
inevitable in practice.

For the way forward, one can consider the application of the inverse
calibration in different scenarios, e.g., with different precoding schemes. It
can also be of interest to carry out a field trial of the presented
calibration circuits and algorithms, thanks to the simplicity of their design
principles.

## Appendix A

Recall the probability density function of a truncated Gaussian distributed
variable $x\sim\mathcal{N}_{\text{T}}(\mu,\sigma^{2}),x\in[a,b]$ (also
$\alpha$, $\beta$ and $Z$) in [9], the $l^{\text{th}}$ ($l\geq 0$) non-central
moment of $x$, denoted by $E_{l}^{x}$, is given by [19]

$E_{l}^{x}=\mathbb{E}\left\\{x^{l}\right\\}=\sum_{i=0}^{l}\binom{l}{i}\sigma^{i}\mu^{l-i}L_{i}.$
(24)

We can now calculate the $l^{\text{th}}$ ($l\geq 0$) non-central moment of a
truncated Gaussian distributed variable, such as

$\displaystyle\\!\\!\\!\\!E_{4}^{t}$
$\displaystyle=\mathbb{E}\left\\{A_{bt,i}^{4}\right\\}$
$\displaystyle=\mu^{4}\\!+\\!4\sigma\mu^{3}\\!\left(\\!\frac{\phi(\beta)\\!-\\!\phi(\alpha)}{Z}\\!\right)\\!+\\!6\sigma^{2}\mu^{2}\\!\left(\\!1\\!+\\!\frac{\alpha\phi(\alpha)\\!-\\!\beta\phi(\beta)}{Z}\\!\right)$
$\displaystyle\qquad+4\sigma^{3}\mu\left(\frac{(\alpha^{2}+2)\phi(\alpha)-(\beta^{2}+2)\phi(\beta)}{Z}\right)$
$\displaystyle\qquad+\sigma^{4}\left(3+\frac{(\alpha^{3}+3\alpha)\phi(\alpha)\\!-\\!(\beta^{3}+3\beta)\phi(\beta)}{Z}\right).$
(25)

To calculate the expected value of the inverse square of the truncated
Gaussian distributed variable $x$, denoted by $E_{\bar{2}}^{x}$ where the
subscript $(\cdot)_{\bar{2}}$ is used for the 2-nd inverse moment of $x$, we
have

$\displaystyle E_{\bar{2}}^{x}$
$\displaystyle=\mathbb{E}\left\\{\frac{1}{x^{2}}\right\\}=\int_{a}^{b}x^{-2}\dfrac{1}{\sigma
Z}\phi\left(\frac{x-\mu}{\sigma}\right)\,\mathrm{d}x$
$\displaystyle=\dfrac{1}{\sqrt{2\pi}\sigma
Z}\int_{a}^{b}x^{-2}\textnormal{exp}\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)\,\mathrm{d}x.$
(26)

The integral in (A) is a nonelementary antiderivative and not able to be
further simplified when $\mu\neq 0$. Consider the particular case in this
work, the value of $E_{\bar{2}}^{r}$ can be measured based on the measurement
of the response of the BS RF frontends, i.e., $\mathbf{R}_{\textnormal{self}}$
and $\mathbf{R}_{\textnormal{half}}$, such that

$E_{\bar{2}}^{r}=\frac{1}{M}\textnormal{tr}\left(\mathbf{R}_{\textnormal{half}}^{-1}(\mathbf{R}_{\textnormal{half}}^{\ast})^{-1}\right).$
(27)

Similarly, we have

$\displaystyle
E_{\bar{2}}^{t}=\frac{1}{M}\textnormal{tr}\left(\mathbf{R}_{\textnormal{self}}^{2}(\mathbf{R}_{\textnormal{half}}\mathbf{R}_{\textnormal{half}}^{\ast})^{-1}\right).$
(28)

## Appendix B

###  B-A Inverse Calibration

  1. (1)

$\lambda_{\textnormal{mrt}}^{\textnormal{IC}}$: Recall (6) and (IV), the
denominator inside of the square root in
$\lambda_{\textnormal{mrt}}^{\textnormal{RC}}$ can be given by

$\displaystyle\mathbb{E}\left\\{\textnormal{tr}\left(\mathbf{W}_{\textnormal{mrt}}^{\textnormal{IC}}(\mathbf{W}_{\textnormal{mrt}}^{\textnormal{IC}})^{H}\right)\right\\}$
$\displaystyle\overset{\textnormal{(a)}}{=}{\mathbb{E}}\left\\{\textnormal{tr}\left(a^{2}\mathbf{H}_{bt}^{-1}\mathbf{H}^{\ast}\mathbf{H}^{T}(\mathbf{H}_{bt}^{\ast})^{-1}\right)\right\\}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+{\mathbb{E}}\left\\{\textnormal{tr}\left(b^{2}(\mathbf{H}_{bt}\mathbf{H}_{br}^{\ast})^{-1}\mathbf{N}_{u}^{\ast}\mathbf{N}_{u}^{T}(\mathbf{H}_{bt}^{\ast}\mathbf{H}_{br})^{-1}\right)\right\\}$
(29)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=MKE_{\bar{2}}^{t}\left(a^{2}+b^{2}E_{\bar{2}}^{r}\right),$
(30)

where (a) is conditioned on the independence between $\mathbf{H}$ and
$\mathbf{N}_{u}$. Thus we have

$\lambda_{\textnormal{mrt}}^{\textnormal{IC}}=\sqrt{\dfrac{1}{MKE_{\bar{2}}^{t}\left(a^{2}+b^{2}E_{\bar{2}}^{r}\right)}}.$
(31)

  2. (2)

${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{IC}}\right\\}$: Recall (13) and
$\mathbb{E}\left\\{\lvert s_{k}\rvert^{2}\right\\}=1$, we have

${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{IC}}\right\\}={\mathbb{E}}\left\\{\lvert\sqrt{\rho_{d}}\lambda_{\textnormal{mrt}}^{\textnormal{IC}}\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{k,\textnormal{mrt}}^{\textnormal{IC}}\rvert^{2}\right\\},$
(32)

where

$\displaystyle{\mathbb{E}}\left\\{\lvert\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{k,\textnormal{mrt}}^{\textnormal{IC}}\rvert^{2}\right\\}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!={\mathbb{E}}\left\\{\lvert\mathbf{h}_{k}^{T}\mathbf{H}_{bt}(a\mathbf{H}_{bt}^{-1}\mathbf{h}_{k}^{\ast}+b(\mathbf{H}_{bt}\mathbf{H}_{br}^{\ast})^{-1}\mathbf{n}_{u,k}^{\ast})\rvert^{2}\right\\}$
(33)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=a^{2}{\mathbb{E}}\left\\{\lvert\mathbf{h}_{k}^{T}\mathbf{h}_{k}^{\ast}\rvert^{2}\right\\}+b^{2}{\mathbb{E}}\left\\{\lvert\mathbf{h}_{k}^{T}(\mathbf{H}_{br}^{\ast})^{-1}\mathbf{n}_{u,k}^{\ast}\rvert^{2}\right\\}$
(34)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=a^{2}M(M+1)+b^{2}ME_{\bar{2}}^{r}.$
(35)

Substituting (31) and (35) into (32), we have

${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{IC}}\right\\}=\dfrac{\rho_{d}\left(a^{2}(M+1)+b^{2}E_{\bar{2}}^{r}\right)}{KE_{\bar{2}}^{t}\left(a^{2}+b^{2}E_{\bar{2}}^{r}\right)}.$
(36)

  3. (3)

${\mathbb{E}}\left\\{1/\left(P_{I,\text{mrt}}^{\text{IC}}+\sigma_{k}^{2}\right)\right\\}$:
Consider that the symbols of different users are independent, we first
calculate ${\mathbb{E}}\left\\{P_{I,\text{mrt}}^{\text{IC}}\right\\}$ as

$\displaystyle{\mathbb{E}}\left\\{P_{I,\text{mrt}}^{\text{IC}}\right\\}$
$\displaystyle={\mathbb{E}}\left\\{\left|\sum_{i=1,i\neq
k}^{K}\sqrt{\rho_{d}}\lambda_{\textnormal{mrt}}^{\textnormal{IC}}\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{i,\textnormal{mrt}}^{\textnormal{IC}}\right|^{2}\right\\}$
(37) $\displaystyle=\sum_{i=1,i\neq
k}^{K}{\mathbb{E}}\left\\{\left|\sqrt{\rho_{d}}\lambda_{\textnormal{mrt}}^{\textnormal{IC}}\mathbf{h}_{k}^{T}\mathbf{H}_{bt}\mathbf{w}_{i,\textnormal{mrt}}^{\textnormal{IC}}\right|^{2}\right\\}$
(38)
$\displaystyle\overset{\textnormal{(b)}}{=}\frac{\rho_{d}(K-1)}{KE_{\bar{2}}^{t}},$
(39)

where (b) can be carried out in the similar way as that from (32) to (36). We
then calculate $\textnormal{var}(P_{I,\text{mrt}}^{\text{IC}}+\sigma_{k}^{2})$
by following the technique as in [9, Theorem 1], such that

$\textnormal{var}\left(P_{I,\text{mrt}}^{\text{IC}}+\sigma_{k}^{2}\right)=\frac{\rho_{d}^{2}(K-1)}{\left(KE_{\bar{2}}^{t}\right)^{2}}.$
(40)

Substituting (39) and (40) into (17), we have

${\mathbb{E}}\left\\{\frac{1}{P_{I,\text{mrt}}^{\text{IC}}\\!+\\!\sigma_{k}^{2}}\right\\}\\!=\\!\dfrac{KE_{\bar{2}}^{t}\left((KE_{\bar{2}}^{t})^{2}\\!+\\!\rho_{d}K(K\\!-\\!1)(\rho_{d}\\!+\\!2E_{\bar{2}}^{t})\right)}{(\rho_{d}(K-1)+KE_{\bar{2}}^{t})^{3}}.$
(41)

Now we can arrive at (1) in Proposition 1 by substituting (36) and (41) in
(16).

###  B-B Relative Calibration

For the sake of simplicity, we list the main results for the relative
calibration scheme as below. The derivation of these results follows the
similar technique as in the previous subsection.

$\lambda_{\textnormal{mrt}}^{\textnormal{RC}}=\sqrt{\dfrac{1}{MKA_{t}\left(a^{2}+b^{2}E_{\bar{2}}^{r}\right)}},$
(42)
${\mathbb{E}}\left\\{P_{s,\text{mrt}}^{\text{RC}}\right\\}=\dfrac{\rho_{d}\left(a^{2}((M\\!-\\!1)A_{t}^{2}+2E_{4}^{t})\\!+\\!b^{2}E_{4}^{t}E_{\bar{2}}^{r}\right)}{KA_{t}\left(a^{2}+b^{2}E_{\bar{2}}^{r}\right)},$
(43)
$\displaystyle\\!\\!\\!\\!\\!\\!{\mathbb{E}}\left\\{\frac{1}{P_{I,\text{mrt}}^{\text{RC}}+\sigma_{k}^{2}}\right\\}$
$\displaystyle=\dfrac{KA_{t}\left(K^{2}A_{t}^{2}+\rho_{d}K(K-1)E_{4}^{t}(\rho_{d}E_{4}^{t}+2A_{t})\right)}{(\rho_{d}(K-1)E_{4}^{t}+KA_{t})^{3}}.$
(44)

Substituting (43) and (44) in (16), we have (2) in Proposition 2.

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