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The first quantum error-correcting code for singledeletion errors
Ayumu Nakayama1a) and Manabu Hagiwara2b)1 Graduate School of Science and Engineering, Chiba University,
1–33 Yayoi-cho, Inage-ku, Chiba, Japan2 Graduate School of Science, Chiba University,
Classification: Fiber-Optic Transmission for Communications
References
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[6] A. Naka, “BER performance analysis of multi-dimensional modulation withBICM-ID,” IEICE Commun. Express, vol. 6, no. 12, pp. 645–650, 2017. DOI:10.1587/comex.2017XBL0123
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1 Introduction
PAS modulation and multi-dimensional modulation with BICM-ID, which are ones
of coded modulation where encoders are combined with modulator, are promising
to construct high-speed optical systems with flexible transmission capacity. PAS
has been massively investigated in recent years, which uses the non-uniformly
distributed symbols on a conventional m-QAM constellation with a distribution
matcher (DM) to overcome a shaping gap of Shannon limit [1, 2, 3]. DM enables
PAS systems to have flexible transmission rate with high signal-to-noise ratio
(SNR) sensitivity characteristics.
Multi-dimensional modulation has been also actively studied as a power-
efficient modulation formats [4] and been demonstrated to provide variable capacity
with set-partitioning technique [5]. While multi-dimensional modulation formats
suffer from performance degradation due to non-Gray code mapping resulting from
multiple adjacent symbols, BICM-ID recovers the degradation [6].
In this paper, AIRs of the above two coded modulations, specifically, PAS on
64-QAMs having two types of DM with each Look-Up Table (LUT), and eight-
dimensional (8D) modulation based on 16 QAM are evaluated by numerical
calculation together with AIR evaluation of a conventional 16 QAM for compar-
ison. Further, BER performances of end-to-end section over DM and inverse DM
(DM−1) are quantitatively evaluated at several coding rates of FEC as well as the
BER performances over forward error correction (FEC) encoder/decoder section.
This allows us to compare the performance of each format at the same transmission
rates or net bitrates excluding FEC overhead and bitrate increase due to DM. And
finally, the obtained BER performances are analyzed with NGMI [3] derived from
the calculated AIR and FEC decoder characteristics.
2 Calculation model
2.1 Transmitter and receiver
Transmitter generates two types of PAS constellations on two-dimensional 64-
QAM with two types of DM, namely ðk; nÞ ¼ ð12; 10Þ and ð10; 10Þ, which
respectively transform uniformly distributed binary data blocks of length k into
Maxwell-Boltzmann distributed amplitude data blocks of length n with each
respective single LUT. The DM with a single LUT is a practical solution for
We numerically evaluated AIR and BER performances of PAS with a single LUT,
8D-SP4096-16QAM with BICM-ID, and 2D-16QAM for future high-speed optical
communication systems. We confirmed that BER performances of three modulation
formats are almost identical at a same transmission rate, when the error correction
is used once. Further, the error-free SNR conditions agree very well with the values
determined by the NGMI statically estimated from received LLR and MI input/
output characteristics of the LDPC.
Acknowledgments
This work was supported by JSPS KAKENHI Grant Number 19K004386.
Fig. 2. BER performances and their analysis(a) BER performances as a function of Signal-to-Noise Ratioper Symbol. (b) Normalized GMI (c) MI Input/Outputcharacteristics of LDPC FEC
Abstract: This letter presents an effective Q factor formula for self-reso-
nant spherical surface antennas. The self-resonant lossless Q factor and
radiation efficiency calculated using spherical wave expansion provide an
approximated expression for the effective Q factor. The resultant effective Q
factor is larger than that of the infinitesimal loop antenna and smaller than
that of the infinitesimal dipole antenna. Comparison of the result with the
Q factor of spherical helix antennas has shown good agreement. A simple
estimation formula can help design a small spherical helix antenna.
Keywords: Q factor, small antenna, radiation efficiency
Classification: Antennas and Propagation
References
[1] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,”IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005.DOI:10.1109/TAP.2005.844443
[2] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys.,vol. 19, no. 12, pp. 1163–1175, Dec. 1948. DOI:10.1063/1.1715038
[3] R. C. Hansen and R. E. Collin, “A new Chu formula for Q,” IEEE AntennasPropag. Mag., vol. 51, no. 5, pp. 38–41, Oct. 2009. DOI:10.1109/MAP.2009.5432037
[4] T. V. Hansen, O. S. Kim, and O. Breinbjerg, “Stored energy and quality factorof spherical wave functions – in relation to spherical antennas with materialcores,” IEEE Trans. Antennas Propag., vol. 60, no. 3, pp. 1281–1290, Mar.2012. DOI:10.1109/TAP.2011.2180330
[5] M. Gustafsson and S. Nordebo, “Optimal antenna currents for Q, super-directivity, and radiation patterns using convex optimization,” IEEE Trans.Antennas Propag., vol. 61, no. 3, pp. 1109–1118, Mar. 2013. DOI:10.1109/TAP.2012.2227656
[6] R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,”J. Res. Nat. Bur. Stand. Section D: Radio Propagation, vol. 64D, no. 1,pp. 1–12, Jan. 1960.
[7] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically smalldipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3114–3121, Oct. 2010. DOI:10.1109/TAP.2010.2055790
[8] K. Fujita and H. Shirai, “Theoretical limit of the radiation efficiency forelectrically small self-resonant spherical surface antennas,” IEICE Trans.Electron., vol. E100-C, no. 1, pp. 20–26, Jan. 2017. DOI:10.1587/transele.
[10] H. L. Thal, “Radiation efficiency limits for elementary antenna shapes,” IEEETrans. Antennas Propag., vol. 66, no. 5, pp. 2179–2187, May 2018. DOI:10.1109/TAP.2018.2809507
[11] K. Fujita, “Effective Q factor for spherical surface antennas,” Proc. iWAT,Nanjing, China, pp. 1–3, Mar. 2018. DOI:10.1109/IWAT.2018.8379124
[12] J. D. Jackson, Classical Electrodynamics, 3rd ed., John Wiley & Sons, NewJersey, 1999.
[13] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed., John Wiley &Sons, New Jersey, 2005.
[14] M. Capek and L. Jelinek, “Optimal composition of modal currents for minimalquality factor Q,” IEEE Trans. Antennas Propag., vol. 64, no. 12, pp. 5230–5242, Dec. 2016. DOI:10.1109/TAP.2016.2617779
1 Introduction
The bandwidth and radiation efficiency of electrically small antennas are both
strongly affected by a small antenna size. The channel capacity of the communi-
cation is limited by its bandwidth, thereby necessitating a wide operating band-
width. As the bandwidth is proportional to the reciprocal of the Q factor [1], the
bandwidth of electrically small antennas is evaluated using the Q factor.
The theoretical limit for the Q factor can be obtained both analytically and
numerically. Chu [2] demonstrated that a small antenna fabricated using lossless
materials cannot exceed the theoretical lower bound using the spherical wave
expansion. This result is limited to the case wherein the stored electric and magnetic
energy inside the circumscribing sphere is zero. The stored energy inside the sphere
can be increased by expanding the electromagnetic field inside the sphere [3, 4].
The Q factor of arbitrarily shaped antennas have also been numerically calculated
by discretizing the antenna surface and applying convex optimization [5].
The bandwidth of a small antenna should be evaluated by the effective Q factor
rather than the lossless Q factor because the low radiation efficiency of a small
antenna increases the effective bandwidth. The radiation efficiency and the effective
Q factor of a gain-optimized spherical antenna have been derived by Harrington
[6], but this publication does not mention the case for the maximum radiation
efficiency. The effective Q factor of the non-resonant small antenna is analytically
calculated using the radiation efficiency of the infinitesimal dipole and loop antenna
[7]. The radiation efficiency of these antennas is underestimated for the small self-
resonant antenna and is unsuitable for estimating the self-resonant effective Q
factor.
A recent investigation has revealed that the upper bound of the radiation
efficiency for the small self-resonant antenna can be obtained using the spherical
wave expansion [8] and the equivalent circuit method [9, 10]. The Q factor of the
lossless antenna increases monotonically as the antenna size decreases. In contrast,
the effective Q factor of small antennas with lossy material approaches zero owing
where Z0 is the free-space impedance. Qeff is plotted in Fig. 2 with the η of
Eq. (10), the Qsr of Eq. (8), and the approximated Qsr of Eq. (9) against the antenna
size. For this calculation, antennas were assumed to be made of copper (� ¼5:8 � 107 S/m) and have a radius of R ¼ 0:04m. In the region of kR > 0:05, the
Qeff is similar to the Qsr owing to the high radiation efficiency, whereas a significant
difference is observed in the region of kR < 0:05.
Eq. (10) can be expanded in the Laurent series and approximated by
Pl
Pr�
ffiffiffiffiffiffiffiffi!"02�
r3
ðkRÞ4 þ3
10
1
ðkRÞ2� �
ð11Þ
where "0 denotes the free-space permittivity. With the aid of Eqs. (9) and (11), the
approximated effective Q factor can be expressed as
Qeff � 10kR þ 11ðkRÞ3
10ðkRÞ4 þffiffiffiffiffiffiffiffi!"0
2�
sð30 þ 3ðkRÞ2Þ
: ð12Þ
This newly derived formula has an error of less than 2% for the exact value in the
region of kR < 0:5 because the approximated radiation efficiency and effective Q
factor is accurate in the same region.
4 Numerical validation
Fig. 3(a) shows the numerical validation of Qeff . The effective Q factors of the
infinitesimal dipole and loop antenna [7] are indicated by Qlbe and Qlbm, respec-
tively. Qlbe is close to Qeff , whereas Qlbm is smaller than Qeff . This is due to the
relatively large radiation efficiency of the infinitesimal dipole antenna and the
extremely small efficiency of the infinitesimal loop antenna. QCarl in Fig. 3(a) is the
effective Q factor calculated by the equivalent circuit method [9]. Two current
sheets radiating inside and outside of the sphere are assumed for QCarl, whereas
Qeff is calculated with one current sheet. This area of the current sheet causes a
Fig. 2. Effective Q factor Qeff and radiation efficiency η.
Theoretical system capacityof multi-user MIMO-OFDMTHP in the presence ofterminal mobility
Ryota Mizutani1, Yukiko Shimbo1, Hirofumi Suganuma1,Hiromichi Tomeba2, Takashi Onodera2, and Fumiaki Maehara1a)1 Graduate School of Fundamental Science and Engineering, Waseda University,
3–4–1 Ohkubo, Shinjuku-ku, Tokyo 169–8555, Japan2 Telecommunication and Image Technology Laboratories, Corporate Research and
carrier interference (ICI), system capacity, mod-Λ channel, terminal mobility
Classification: Wireless Communication Technologies
References
[1] Q. H. Spencer, C. B. Peel, A. L. Swindlehurst, and M. Haardt, “An introductionto the multi-user MIMO downlink,” IEEE Commun. Mag., vol. 42, no. 10,pp. 60–67, Oct. 2004. DOI:10.1109/MCOM.2004.1341262
[2] Wireless LAN medium access control (MAC) and physical layer (PHY)specifications: Enhancements for very high throughput for operation in bandsbelow 6GHz, IEEE Std. 802.11ac, Dec. 2013.
[3] 3GPP TS 36.211 v10.5.0, “Evolved universal terrestrial radio access (E-UTRA);Physical channels and modulation,” June 2012.
[5] X. Wang, X. Hou, H. Jiang, A. Benjebbour, Y. Saito, Y. Kishiyama, J. Qiu, H.Shen, C. Tang, T. Tian, and T. Kashima, “Large scale experimental trial of 5G
mobile communication systems—TDD massive MIMO with linear and non-linear precoding schemes,” Proc. IEEE 27th Annu. Int. Symp. Pers., Indoor,Mobile Radio Commun. (PIMRC 2016), pp. 1–5, Sept. 2016. DOI:10.1109/PIMRC.2016.7794572
[6] F. Hasegawa, H. Nishimoto, N. Song, M. Enescu, A. Taira, A. Okazaki, and A.Okamura, “Non-linear precoding for 5G NR,” Proc. 2018 IEEE Conf. StandardsCommun. Networking (CSCN 2018), pp. 1–7, Oct. 2018. DOI:10.1109/CSCN.2018.8581859
[7] C. Windpassinger, R. F. H. Fischer, T. Vencel, and J. B. Huber, “Precoding inmultiantenna and multiuser communications,” IEEE Trans. Wireless Commun.,vol. 3, no. 4, pp. 1305–1316, July 2004. DOI:10.1109/TWC.2004.830852
[8] K. Zu, R. de Lamare, and M. Haardt, “Multi-branch Tomlinson-Harashimaprecoding design for MU-MIMO systems: Theory and algorithms,” IEEE Trans.Commun., vol. 62, no. 3, pp. 939–951, Mar. 2014. DOI:10.1109/TCOMM.2014.012514.130241
[9] H. Suganuma, Y. Shimbo, N. Hiruma, H. Tomeba, T. Onodera, and F. Maehara,“Theoretical system capacity of multi-user MIMO THP in the presence ofterminal mobility,” Proc. IEEE 88th Veh. Technol. Conf. (VTC 2018-Fall),pp. 1–5, Aug. 2018. DOI:10.1109/VTCFall.2018.8690890
1 Introduction
In recent years, multi-user multiple-input multiple-output (MU-MIMO) has become
a promising technique for high-speed and high-capacity wireless communication
systems, as simultaneous transmission can be realized via a single antenna mounted
on a mobile station (MS) [1]. Moreover, to further enhance the system capacity,
MU-MIMO is normally applied to orthogonal frequency division multiplexing
(OFDM), which has been adopted for IEEE 802.11ac [2] and LTE-Advanced [3].
To realize MU-MIMO, precoding techniques are essential and are categorized
into two approaches: linear precoding (LP) and non-linear precoding (NLP). NLP
provides the better system capacity than LP because it reduces noise enhancement
and has thus emerged as a candidate technique to realize 5G systems [4, 5, 6]. Of
the various NLP schemes, Tomlinson-Harashima precoding (THP) is considered
a practical approach because the perturbation vector can be generated by a simple
modulo operation [5, 7, 8].
In this letter, we investigate the theoretical system capacity of MU-MIMO-
OFDM THP in the presence of terminal mobility. The primary objective of our
investigation is to grasp the exact theoretical capacity of MU-MIMO-OFDM THP
under time-selective fading channels caused by terminal mobility, which ought
to be considered in mobile wireless communications. In a previous study, we
successfully derived the exact system capacity of MU-MIMO THP under time-
selective fading channels [9], and therefore this study extends our previous work to
OFDM-based broadband wireless systems. More specifically, the effect of both the
multi-user interference (MUI) and inter-carrier interference (ICI) caused by time-
selective fading is analyzed considering the application of THP to OFDM-based
systems, and its effect is included in the mod-Λ channel [9]. This makes it possible
to provide an exact system capacity analysis based on the adoption of OFDM as
well as the modulo loss peculiar to THP in the presence of terminal mobility.
Moreover, our investigation enables a fair comparison in terms of the system
capacity between THP and LP without time-consuming computer simulations, and
it demonstrates the superiority of THP over LP even in the presence of terminal
mobility.
2 System capacity analysis of MU-MIMO-OFDM THP
Fig. 1 shows the system configuration of MU-MIMO-OFDM THP, where Nt, Nr,
and N denote the number of transmit antennas, MSs with one received antenna
element, and sub-carriers. In Fig. 1, the feedforward (FF) and feedback (FB) filters
of THP can be implemented in each sub-carrier by an LQ decomposition [8, 9] so
as to retain spatial orthogonality among multiple MSs. Especially in THP, the
modulo operation is performed to limit the transmit power increased by the addition
of an interference subtraction vector generated by the FB filter. Moreover, because
the transmit power is changed by the FF filter, a power normalization factor is
required. In the p-th sub-carrier, the power normalization factor is given by gp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrðFpCVp
FHp Þ=ðNr�2
x Þp
, where Fp 2 CNt�Nr is the FF filter, CVp
2 CNr�Nr is the
covariance matrix of the transmit signal after the modulo operation Vp 2 CNr , and
�2x denotes the modulated signal power. After the precoder output is converted into
the time domain signal with the length of Ts by means of IFFT processing, these
signals are transmitted from each transmit antenna.
In general, terminal mobility creates time-selective fading, which causes a
mismatch between the channel state information (CSI) for precoding and actual
channel condition in data transmission. This mismatch destroys the space-fre-
quency orthogonality in precoding, which leads to both the MUI and ICI. In this
letter, we derive both the MUI and ICI resulting from terminal mobility in MU-
MIMO-OFDM and then incorporate its impact into the system capacity analysis
based on the mod-Λ channel.
In time-selective Rayleigh fading channels, the channel matrix between the
j-th transmit antenna and i-th MS Hðt; fÞ 2 CNr�Nt is correlated with the preceding
channel condition Hðt � �t; fÞ 2 CNr�Nt , which is represented by [9]
With a focus on the i-th MS, the received signal Yi;p is represented by
Yi;p ¼ ki;�tXi;p þmi;�tFpVp
þ gpN
XN�1
k¼0k≠p
XN�1
n¼0g�1k ej
2�ðk�pÞnN ki;�thi n
TsN
� �t;k
Ts
� �þmi;�t
� �FkVk þ gpZi;p; ð5Þ
where mi;�t and hið�; �Þ are the i-th row vectors of M�t and Hð�; �Þ, respectively.From Eq. (5), the received signal Yi;p contains the desired signal, MUI, ICI, and
noise components, and in consequence, the powers of these terms are calculated as
PD ¼ E½jki;�tXi;pj2� ¼ k2i;�t�2x ; ð6Þ
PMUI ¼ E½jmi;�tFpVpj2� ¼ trðFpFHp Þð1 � k2i;�tÞ�2
h�2v ; ð7Þ
PICI ¼ EgpN
XN�1
k¼0k≠p
XN�1
n¼0g�1k ej
2�ðk�pÞnN ki;�thi n
TsN
� �t;k
Ts
� �þmi;�t
� �FkVk
��������
��������
22664
3775
¼ g2pk2i;�tNr�
2h�
2x
N2NðN � 1Þ � 2
XN�1
n¼1ðN � nÞJ0 2�fDi
TsN
n
� �" #; ð8Þ
PN ¼ E½jgpZi;pj2� ¼ g2p�2n; ð9Þ
where �2v and �2
n are the transmit signal power after the modulo operation and noise
power, respectively.
The system capacity as well as the effect of the modulo loss peculiar to THP
can be derived by the mod-Λ channel [9], which is represented by
Csum ¼ 1
N
XN�1
p¼0
XNr
i¼12 log2 � þ
Z �2
��2
pðzmodÞ log2 pðzmodÞdzmod !
½bps=Hz�; ð10Þ
where τ denotes the modulo width. Moreover, pðzmodÞ (��=2 < zmod < �=2) is the
probability distribution function of the white Gaussian noise after the modulo