IEICE Communications Express, Vol.9, No.4, 100 The ﬁrst ...Keywords: quantum error-correcting codes, quantum deletion errors, dele-tion codes Classiﬁcation: Fundamental Theories

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,

1–33 Yayoi-cho, Inage-ku, Chiba, Japan

a) [email protected]

b) [email protected]

Abstract: A quantum error-correcting code for single deletion errors is

provided. To the authors’ best knowledge, this is the first code for deletion

errors.

Keywords: quantum error-correcting codes, quantum deletion errors, dele-

tion codes

Classification: Fundamental Theories for Communications

References

[1] V. I. Levenshtein, “Binary codes capable of correcting deletions, insertions, andreversals,” Sov. Phys. Dokl., pp. 707–710, 1966.

1 Introduction

In the classical coding theory, deletion error-correcting codes have been studied for

synchronization of communication since the pioneer work by Levenshtein [1].

However, no quantum code for deletion error has been constructed yet. This letter

provides the first quantum code for deletion errors that are defined as partial trace

operations. In particular, an encoding and a decoding are described.

2 Single quantum deletion error

For a square matrix A over a complex field C, TrðAÞ denotes the sum of the

diagonal elements of A. Set j0i; j1i 2 C2 as j0i :¼ ð1; 0ÞT ; j1i :¼ ð0; 1ÞT respec-

tively. We denote the set of all density matrices of order n by SðCnÞ. A density

matrix is used for representing a quantum state and is called a quantum message.

For an integer 1 � i � n and a square matrix A ¼ Px;y2f0;1gn ax;y �

jx1ihy1j � � � � � jxnihynj with ax;y 2 C, define the map Tri as TriðAÞ :¼Px;y2f0;1gn ax;y � Trðjxi ihyijÞ � jx1 ihy1 j � � � � � jxi�1ihyi�1j � jxiþ1ihyiþ1j � � � �

�jxnihynj. The map Tri is called the partial trace.© IEICE 2020DOI: 10.1587/comex.2019XBL0154Received December 3, 2019Accepted January 8, 2020Publicized January 22, 2020Copyedited April 1, 2020

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IEICE Communications Express, Vol.9, No.4, 100–104

Definition 2.1. For an integer 1 � i � n, we call Tri a single deletion error Di,

i.e.,

Dið�Þ :¼ Trið�Þ;where � 2 SðC2�nÞ is a quantum state.

3 Our quantum error-correcting code for single deletion errors

3.1 Encoding

For a quantum message � :¼ j ih j 2 SðC2Þ with unit vector j i :¼ �j0i þ �j1i 2C

2, we encode σ to � :¼ j�ih�j, where

j�i :¼ �ffiffiffi2

p ðj00001001i þ j01101111iÞ þ �ffiffiffi2

p ðj00001111i þ j01101001iÞ:

Remark that this encoding can be expressed by neither any CSS codes nor any

stabilizer codes.

3.2 Quantum states after the deletion errors

All the states Dið�Þ after single deletion errors for 1 � i � 8 are following:

D1ð�Þ¼ �

2j0001001ið�h0001001j þ �h0001111j þ �h1101001j þ �h1101111jÞ

þ �


þ �


þ �

2j1101111ið�h0001001j þ �h0001111j þ �h1101001j þ �h1101111jÞ;

D2ð�Þ ¼ D3ð�Þ¼ �

2j0001001ið�h0001001j þ �h0001111jÞ

þ �

2j0001111ið�h0001001j þ �h0001111jÞ

þ �

2j0101001ið�h0101001j þ �h0101111jÞ

þ �

2j0101111ið�h0101001j þ �h0101111jÞ;

D4ð�Þ¼ �


þ �


þ �


þ �


© IEICE 2020DOI: 10.1587/comex.2019XBL0154Received December 3, 2019Accepted January 8, 2020Publicized January 22, 2020Copyedited April 1, 2020

101


D5ð�Þ¼ �


þ �


þ �


þ �


D6ð�Þ ¼ D7ð�Þ¼ �

2j0000101ið�h0000101j þ �h0110101jÞ

þ �

2j0000111ið�h0000111j þ �h0110111jÞ

þ �

2j0110101ið�h0000101j þ �h0110101jÞ

þ �

2j0110111ið�h0000111j þ �h0110111jÞ;

and

D8ð�Þ¼ �


þ �


þ �


þ �

2j0110111ið�h0000100j þ �h0000111j þ �h0110100j þ �h0110111jÞ:

3.3 Decoding

Let P ¼ fP1; P2; . . . ; Pmg be a set of complex square matrices of order 2n. The set P

is called a projection measurement if and only if Pi is a projection matrix for any

1 � i � m andP

1�i�m Pi ¼ In holds, where In is the identity matrix of order 2n. For

a quantum state � 2 SðC2�nÞ, the probability that we obtain an outcome 1 � k � m

by the measurement P is given by TrðPk�Þ. The state ρ changes to ~� :¼Pk�Pk=TrðPk�Þ when the outcome k is obtained.

Let us define a projection measurement P :¼ fP1; P2; . . . ; P9g as

P1 ¼ j0001001ih0001001j þ j0001111ih0001111j;P2 ¼ j1101001ih1101001j þ j1101111ih1101111j;P3 ¼ j0101001ih0101001j þ j0101111ih0101111j;P4 ¼ j0111001ih0111001j þ j0111111ih0111111j;P5 ¼ j0000111ih0000111j þ j0110111ih0110111j;P6 ¼ j0000001ih0000001j þ j0110001ih0110001j;P7 ¼ j0000101ih0000101j þ j0110101ih0110101j;P8 ¼ j0000100ih0000100j þ j0110100ih0110100j;© IEICE 2020

DOI: 10.1587/comex.2019XBL0154Received December 3, 2019Accepted January 8, 2020Publicized January 22, 2020Copyedited April 1, 2020

102


P9 ¼ I7 �X1�j�8

Pj:

Table I shows the probabilities corresponding to the single deletion Dið�Þ andthe outcome k.

Let us explain how to correct deletion error for each outcome k. Case k ¼ 1: In

this case, the obtained state ~� is

~� ¼ �j0001001ið�h0001001j þ �h0001111jÞþ �j0001111ið�h0001001j þ �h0001111jÞ:

Let U1 be a unitary matrix of order 27 that satisfies

U1j0001001i ¼ j0000000i; U1j0001111i ¼ j0000001i:Case k ¼ 2: In this case, the obtained state ~� is


Let U2 be a unitary matrix U2 that satisfies






Let U4 be a unitary matrix that satisfies

Table I. The probabilities corresponding to the single deletion Dið�Þand the outcome k

D1ð�Þ D2ð�Þ D3ð�Þ D4ð�Þ D5ð�Þ D6ð�Þ D7ð�Þ D8ð�Þk ¼ 1 0.5 0.5 0.5 0.5 0 0 0 0

k ¼ 2 0.5 0 0 0 0 0 0 0

k ¼ 3 0 0.5 0.5 0 0 0 0 0

k ¼ 4 0 0 0 0.5 0 0 0 0

k ¼ 5 0 0 0 0 0.5 0.5 0.5 0.5

k ¼ 6 0 0 0 0 0.5 0 0 0

k ¼ 7 0 0 0 0 0 0.5 0.5 0

k ¼ 8 0 0 0 0 0 0 0 0.5

k ¼ 9 0 0 0 0 0 0 0 0


103




Let U5 be a unitary matrix that satisfies










U8j0000100i ¼ j0000000i; U8j0110100i ¼ j0000001i:For each outcome k, we can use Uk as a recovery operator:

Tr1 � � � � � Tr1|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}6 times

ðUk ~�Uyk Þ ¼ �:

Thus we can obtain the original quantum message σ.

4 Conclusion

This letter gave the quantum code for single deletions. Our code does not belong

to previously known classes of quantum error-correcting codes. For future work,

the authors like to construct a class of quantum codes for deletion errors.

Acknowledgments

This paper is partially supported by KAKENHI 18H01435.


104


Performance comparison ofprobabilistic amplitudeshaping andmultidimensional modulation

Akira Nakaa)

Department of Electrical and Electronic Systems Engineering, Ibaraki University,

4–12–1 Naka-Narusawa, Hitachi, Ibaraki 316–8511, Japan


Abstract: We numerically evaluate Achievable Information Rate (AIR)

and Bit Error Rate (BER) performances of Probabilistic Amplitude Shaping

(PAS), eight-dimensional modulation with Bit-Interleaved Coded Modula-

tion Iterative Detection (BICM-ID), and conventional two-dimensional

16QAM (Quadrature Amplitude Modulation) for future high-speed optical

communication systems. We confirm that end-to-end BER performances of

three modulation formats are almost identical at a same transmission rate

when the error correction is used once, and the iterative detection makes

the performance of eight-dimensional modulation format better. Further, we

verify the BER error-free conditions can be estimated by Normalized

General Mutual Information (NGMI) for each modulation format.

Keywords: probabilistic amplitude shaping, multi-dimensional modula-

tion, BICM-ID

Classification: Fiber-Optic Transmission for Communications

References

[1] G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matchedlow-density parity-check coded modulation,” IEEE Trans. Commun., vol. 63,no. 12, pp. 4651–4665, 2015. DOI:10.1109/TCOMM.2015.2494016

[2] F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rateadaptation and reach increase by probabilistically shaped 64-QAM: Anexperimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609, 2016. DOI:10.1109/JLT.2015.2510034

[3] J. Cho, “Probabilistic constellation shaping for optical fiber communications,”J. Lightw. Technol., vol. 37, no. 6, pp. 1590–1607, 2019. DOI:10.1109/JLT.2019.2898855

[4] E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherenttransmission systems,” J. Lightw. Technol., vol. 27, no. 22, pp. 5115–5126,2009. DOI:10.1109/JLT.2009.2029064

[5] J. Renaudier, A. Voicila, O. Bertran-Pardo, O. Rival, M. Karlsson, G. Charlet,and S. Bigo, “Comparison of set-partitioned two-polarization 16QAM formatswith PDM-QPSK and PDM-8QAM for optical transmission systems with error-correction coding,” ECOC’12, We1.5, 2012. DOI:10.1364/ECEOC.2012.We.1.C.5


105


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http://dx.doi.org/10.1109/JLT.2015.2510034












http://dx.doi.org/10.1364/ECEOC.2012.We.1.C.5







[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

[7] M. Nakamura, F. Hamaoka, A. Matsushita, K. Horikoshi, H. Yamazaki, M.Nagatani, A. Sano, A. Hirano, and Y. Miyamoto, “Coded eight-dimensionalQAM technique using iterative soft-output decoding and its demonstration inhigh baud-rate transmission,” J. Lightw. Technol., vol. 35, no. 8, pp. 1369–1375, 2017. DOI:10.1109/JLT.2017.2669919

[8] M. El-Hajjar and L. Hanzo, “EXIT charts for system design and analysis,” IEEECommun. Surveys Tuts., vol. 16, no. 1, pp. 127–153, 2014. DOI:10.1109/SURV.2013.050813.00137

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


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high-throughput optical fiber communication systems, though Constant Composi-

tion Distribution Matching (CCDM) based on arithmetic coding [1] can achieve

more ideal distribution with a long coding block. At a receiver, a soft-demapper

firstly calculates log-likelihood rations (LLRs) using bit-metric decoding (BMD)

technique for demodulation, where the supposed probability density of each symbol

is weighted with the probability of occurrence of the modulation symbol deter-

mined by the DM. The demodulated LLRs estimated by BMD are finally input to

DM−1, which has an inverse function of DM. Detailed configuration including

encoder/decoder is described in [1, 2, 3].

The transmitter also modulates a binary sequence encoded by Low-Density

Parity-check Code (LDPC) into 8D-SP4096-16QAM formats, where 216 symbols

consisted of 16 binary digits on four two-dimensional (2D) Gray-mapped 16QAM

planes are set-partitioned four times to form 212 (¼ 4;096) symbols [7]. Noted that

212 symbols per 8D correspond to 23 (¼ 8) symbols per 2D. At a receiver BICM-ID

is applied to the 8D-SP4096-16QAM symbols, where bit-metric LLRs are ex-

changed up to 10 round trips between a demodulator (soft-demapper) and a decoder

via interleaver and de-interleaver, called external iterations. Every bit-metric LLR

is repeatedly recalculated in the demapper, where the supposed probability density

of each bit consisting one symbol is weighted with LLRs of other bits in the same

symbol that are updated in the decoder every round.

In addition, the transmitter also modulates the LDPC encoded binary sequence

into a conventional Gray-mapped 2D-16QAM formats for comparison purpose,

which is detected at the receiver using bit-metric decoding technique again, but

without any weighing before LDPC decoding.

2.2 LDPC encoder/decoder and transmission rate

Encoder and Decoder use Low-Density Parity-check Code (LDPC) code defined by

Digital Video Broadcasting–Satellite–Second Generation (DVB-S2) with codeword

length of 64,800 for every modulation format. Each LDPC code is assumed to have

20 Inner iterations.

Transmission rate per 2D of PAS on 64QAM is given by

R ¼ 2 � k

nþ 1 � 3ð1 � RcÞ

� �ð1Þ

where ðk; nÞ is defined by a type of DM, Rc is code rate of LDPC. On the other

hands, transmission rates of 8D-SP4096-16QAM and 2D-16QAM are respectively

as follows;

R ¼ 3Rc ð2ÞR ¼ 4Rc ð3Þ

A set of two LDPC code rates are deliberately assumed for each modulation

format, which differs from one another, as described in Table I. For example,

Rc ¼ 4=5 and 2/3 are respectively assumed for ðk; nÞ ¼ ð12; 10Þ of PAS to form

transmission rates of 3.2 and 2.4 bit/symbol.© IEICE 2020DOI: 10.1587/comex.2019XBL0156Received December 9, 2019Accepted January 16, 2020Publicized January 27, 2020Copyedited April 1, 2020

107


2.3 Achievable information rate

AIR with BMD is derived for each modulation formats using a sufficiently large

number of samples by

RBMD ¼X

x2X HðxÞ �X

x2X HðxjyÞ

� �X

x2X Px log2 Px � 1

ns

Xm

k¼1Xns

n¼1 log2ð1 þ eð�1Þb�ðnÞ

k;bÞ ð4Þ

where Px is a probability of each symbol x, m is number of bits composing one

symbol, ns is number of samples, b is specific value of Bi with a value of 0 or 1, �ðnÞk;b

is LLR of kth bit in a symbol for binary value of b. To be noted, the value of the first

term in Eq. (5) depends on a type of DM but not on FEC code rate of Rc for PAS,

while the values of the first term respectively equal to a fixed number of 3 and 4 for

8D-SP4096-16QAM and 2D-16QAM.

3 Calculation result and discussion

3.1 Achievable information rate

Fig. 1 shows the obtained results of AIRs per two-dimension obtained by Eq. (4)

with numerical calculations for each modulation format as well as the AIRs

calculated by theoretical analysis with numerical integrations for conventional

BPSK, QPSK, 2D-16-QAM and PASs. Each dotted line of 2D-16QAM or PASs

by the numerical integration respectively corresponds well to the solid blue, red or

orange line obtained by numerical calculation, which proves the accuracy of the

numerical calculations. A green solid line of 8D-SP4096-16QAM asymptotically

approaches a value of 3 at large SNR as designed. Orange and red lines respectively

Table I. Transmission rate for each modulation format for numericalcalculations

Modulation format Rc Transmission Rate

PAS ðk; nÞ ¼ ð12; 10Þ 4/5, 2/3 3.2, 2.4

ðk; nÞ ¼ ð10; 10Þ 5/6, 2/3 3.0, 2.0

8D-SP4096-16QAM 4/5, 2/3 2.4, 2.0

2D-16QAM 4/5, 3/4 3.2, 3.0

Fig. 1. Achievable Information Rate per 2D-Symbol as a function ofSignal-to-Noise Ratio.


108


get closer to a solid black line representing Shannon limit in the range of about

12 dB or less, compared to the blue and green lines. This shows that both types of

PAS, which asymptotically approach 4.65 and 4.26 bit/symbol respectively, can

reduce the gap between Shannon limit and so-called constellation constrain

capacity. The reason why the two lines are separated from the lines obtained by

numerical integration in the range of 4 dB or less must be the performance loss of

SD FEC due to the asymmetric distribution of LLRs [3].

3.2 BER performance

Fig. 2(a) shows BER performances obtained by numerical calculations for each

modulation format at each LDPC code rate. Since the results described by circles

and squares in orange and red for PAS have almost the same characteristics as each

other at the respective four transmission rates, the DMs and DM−1s are shown not

to cause major BER deterioration. Further, the end-to-end BERs of PAS shown in

orange and red closed circles and 2D-QAM shown in blue triangles have almost the

same characteristics at transmission rate of 3.2 and 3.0 bit/symbol. Similarly, the

BERs of PAS shown in orange and red open circles are almost equal to the ones

of 8D-DP4096-16QAM for initial output of the decoder without external iteration

shown in green closed and open circles at transmission rate of 2.4 and 2.0

bit/symbol. These results show that end-to-end BER performances of three

modulation formats are almost identical at a same transmission rate when the error

correction is used once. According to our results, the BER performances of PAS do

not necessarily exceed those of other methods due to rate back-off that PAS

essentially has [1], despite of the better AIRs of PAS than the others. In addition,

the obtained results show that the BER performances improve as the number of

external iterations increases for 8D-DP4096-16QAM.

We analyze the obtained BER performances using the NGMI [3] derived from

the obtained AIR shown in Fig. 1 as follows;

RMI ¼ 1 � fHðxÞ � RBMDgm

� 1 � 1

m � nsXm

k¼1Xns

n¼1 log2ð1 þ eð�1Þb�ðnÞ

k;bÞ: ð5Þ

This parameter corresponds to mutual information (MI) for an LLR sequence [8].

Solid lines in Fig. 2(b) show the calculated NGMI rates at soft-demapper output or

LDPC decoder input as a function of SNR.

On the other hand, Fig. 2(c) shows MI input/output characteristics of the

LDPC, which is estimated by the method shown in [8]. The dotted lines respec-

tively show minimum values of MI input leading to error-free transmission under

each LDPC code rate, whose MI output values approach 1. By comparing the solid

line in Fig. 2(b) and the dotted line in Fig. 2(c) at each condition, SNR that makes

error-free can be estimated, as shown in dotted lines in Fig. 2(b).

The estimated SNRs in Fig. 2(b) agree very well with the SNRs in Fig. 2(a).

For example, both BERs of PAS for ðk; nÞ ¼ ð10; 10Þ at LDPC code rate of 5/6 and

2D-16QAM at LDPC code rate of 3/4 turn into error-free around SNR of 11.1 dB.

These results indicate correctness of numerical calculations for AIR and BER in

this paper.© IEICE 2020DOI: 10.1587/comex.2019XBL0156Received December 9, 2019Accepted January 16, 2020Publicized January 27, 2020Copyedited April 1, 2020

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4 Conclusion

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


110


Effective Q factor formula forsmall spherical surfaceantennas

Keisuke Fujitaa)

Maebashi Institute of Technology,

460–1 Kamisadori, Maebashi, Gunma 371–0816, Japan


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.


111


http://dx.doi.org/10.1109/TAP.2005.844443




http://dx.doi.org/10.1063/1.1715038

http://dx.doi.org/10.1063/1.1715038

http://dx.doi.org/10.1063/1.1715038

http://dx.doi.org/10.1109/MAP.2009.5432037

















http://dx.doi.org/10.1587/transele.E100.C.20


E100.C.20[9] C. Pfeiffer, “Fundamental efficiency limits for small metallic antennas,” IEEE

Trans. Antennas Propag., vol. 65, no. 4, pp. 1642–1650, Feb. 2017. DOI:10.1109/TAP.2017.2670532

[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


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to the low radiation efficiency [11]. This finding poses the question of how large

the effective Q factor of the transition region should be to ensure a value between

an extremely small antenna size and an intermediate one. To answer this question,

we herein investigate the effective Q factor of the self-resonant spherical surface

antenna for all small antenna regions using the lossless self-resonant Q factor and

the radiation efficiency. These results may help the antenna designer by providing

not only a design guideline for spherical helix antennas but also a rough estimate

for general small antennas.

2 Lossless Q factor for self-resonant spherical surface antennas

We assumed a spherical current sheet with a radius R fabricated from a good

conductor having conductivity σ as the spherical surface antenna. The center of the

spherical surface antenna coincides with the origin of the spherical coordinate

system. The current distribution on the sphere and the radiated electromagnetic

fields can be expressed using the vector spherical wave expansion [12] and

classified into TMnm or TEnm modes, where n and m denote the indices in the

radial and azimuthal directions, respectively. As a small antenna was considered,

it was assumed that the lowest-mode (n ¼ 1) current and electromagnetic fields are

excited and higher-order modes (n � 2) are suppressed. The index m was fixed at

zero because it did not affect the stored energy and the Q factor.

The Q factor of the lossless and self-resonant antenna Qsr is defined as [13]

Qsr ¼ !We þWm

Pr; ð1Þ

where ω is the angular frequency, Pr is the total radiated power, and We and Wm

denote the stored electric and magnetic energy, respectively. The radiated power

and stored energy can be separated into the TM mode part and the TE mode part as

follows:

Pr ¼ PTMr þ PTE

r ð2ÞWe ¼ WTM

e þWTEe ð3Þ

Wm ¼ WTMm þWTE

m : ð4ÞSubstituting Eqs. (2)–(4) into Eq. (1), the Q factor can be rewritten as

Qsr ¼ ðQTMe þ QTM

m Þ � PTMr

PTMr þ PTE

r

þ ðQTEe þ QTE

m Þ � PTEr

PTMr þ PTE

r

ð5Þ

where

QTMe ¼ !

WTMe

PTMr

; QTMm ¼ !

WTMm

PTMr

; QTEe ¼ !

WTEe

PTEr

; QTEm ¼ !

WTEm

PTEr

: ð6Þ

The Q factor is the sum of contributions from both the TM and TE modes. Each

QTM,TEe;m is calculated as mentioned in Ref. [4] in terms of the spherical Bessel

functions. The self-resonant Q factor Qsr has been described in Ref. [11] by

Eq. (5); however, Qsr in the elementary function was not described in this work.

The ratio of the TE-to-TM radiated power of the resonance antenna [8, 14] for Qsr

is given by© IEICE 2020DOI: 10.1587/comex.2019XBL0158Received December 15, 2019Accepted January 17, 2020Publicized January 30, 2020Copyedited April 1, 2020

113


PTMr

PTEr

¼ �j1ðkRÞy1ðkRÞ þ kRy0ðkRÞj1ðkRÞj1ðkRÞy1ðkRÞ � kRy1ðkRÞj0ðkRÞ

ð7Þ

where jn and yn are the spherical Bessel functions of the first kind and second kind

of the n-th order, and k denotes the free space wavenumber. On substituting Eq. (7)

into (5) and using Rayleigh’s formulas [12], Qsr becomes the explicit form of

elementary functions; this can be rewritten as

Qsr ¼ 1

2½f4ðkRÞ4 � 12ðkRÞ2 þ 2g sin4ðkRÞ þ f10ðkRÞ3 � 8ðkRÞg cosðkRÞ sin3ðkRÞ

þ f4ðkRÞ6 � 6ðkRÞ4 þ 12ðkRÞ2 � 2g sin2ðkRÞþ f�2ðkRÞ7 þ 4ðkRÞ5 � 5ðkRÞ3 þ 4ðkRÞg cosðkRÞ sinðkRÞ� 2ðkRÞ6 þ ðkRÞ4 � 2ðkRÞ2�� ½f2ðkRÞ5 � ðkRÞ3g sin2ðkRÞ þ f2ðkRÞ4 � ðkRÞ6g cosðkRÞ sinðkRÞ � ðkRÞ5��1:

ð8Þ

The self-resonant Q factor Qsr in the exact form in Eq. (8) can be approximated by

Qsr � 1

ðkRÞ3 þ11

10

1

kRð9Þ

where kR � 1. This expression coincides with the formula using the equivalent

circuit method [9].

Fig. 1(a) shows the single and self-resonant Q factor of the lossless spherical

antenna. This figure indicates that the Qsr is approximately a third and two-thirds of

the TE and TM single-mode Q factors (QTE and QTM), respectively. The approxi-

mated formula in Eq. (9) is shown in Fig. 1(b) and is close to the Chu limit QChu

for the small antenna region (kR 0:5). Qsr includes the stored energy inside the

antenna, whereas the Chu limit does not consider the internal energy.

3 Effective Q factor for spherical surface antenna

The effective Q factor for self-resonant spherical surface antennas Qeff is defined as

Qeff ¼ �Qsr, where η denotes the radiation efficiency of the spherical surface

antenna. η can be rewritten as � ¼ ð1 þ Pl=PrÞ�1 where Pl=Pr is the ratio of the

dissipated power to the radiated power. Pl=Pr is calculated under the assumption of

uniform current distribution in the skin depth D [8] as

(a) Single-mode and self-resonant Q factor. (b) Self-resonant and Chu Q factor.

Fig. 1. Q factor of lossless self-resonant spherical surface antenna Qsr.


114


Pl

Pr¼ k

Z0�

1

ðkRÞðkR � kDÞðkDÞ þ ðkDÞ33

PTMr

PTEr þ PTM

r

� j0ðkRÞ �j1ðkRÞkR

� ��2þ PTE

r

PTMr

fj1ðkRÞg�2" #

ð10Þ

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.


115


difference in the radiation efficiency and the effective Q factor. Both Qeff and QCarl

have the order of ðkRÞ1.The effective Q factor of one-, two-, and four-arm spherical helix antennas Qsph

shown in Fig. 3(b) were also calculated numerically and are plotted in Fig. 3(a).

The method of moment (NEC2 engine) was used for this numerical simulation. As

the current distribution of these antennas was similar to the ideal spherical current

sheet, the result approached Qeff as the number of wires increased.

It is obvious that the formula can estimate the effective Q factor of a spherical

helix antenna. As the spherical expansion limits the shape of the antenna to a

sphere, the formula is limited to spherical-type antennas. The resultant formula,

however, may provide a rough estimate for general small antennas because a

spherical surface antenna is considered the simplest and most well-analyzed model

of small antennas.

5 Conclusion

In this paper, we described the exact self-resonant Q factor Qsr and approximated an

effective Q factor Qeff using the stored energy calculated by spherical wave

expansion. The approximated expression of Qeff was derived for the first time. It

was confirmed that Qsr is smaller than the single-mode Q factor and is close to the

Chu limit. The simulated results of the spherical helix antennas demonstrated that

the effective Q factor of these antennas approaches Qeff . Moreover, comparisons

between Qeff and the previous results obtained via the equivalent circuit method

validate the value obtained for Qeff . In a future study, the radiation efficiency and

the effective Q factor will be measured and compared with these results.

Acknowledgments

A portion of this study has been supported by JSPS KAKENHI Grant Number

JP18K13760.

(a) Comparison of effective Q factorscalculated via various methods.

(b) Example of spherical helixantenna. (four arms)

Fig. 3. Validation of effective Q factor using the analytical andnumerical method.


116


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

Development BU, Sharp Corporation,

1–9–2 Nakase, Mihama-ku, Chiba 261–8520, Japan


Abstract: This letter investigates the theoretical system capacity of multi-

user multiple-input multiple-output (MU-MIMO) orthogonal frequency di-

vision multiplexing (OFDM) Tomlinson-Harashima precoding (THP) in the

presence of terminal mobility. Considering that MU-MIMO THP has been

adopted for OFDM-based mobile broadband systems, analyzing the effects

of time-selective fading and the modulo loss peculiar to THP on system

capacity is essential. In this study, we theoretically derive both the multi-user

interference (MUI) and inter-carrier interference (ICI) resulting from terminal

mobility and incorporate these into the modulo loss analysis based on the

mod-Λ channel. The theoretical results obtained by the proposed analysis are

compared with those of linear precoding, which demonstrates the applicabil-

ity of THP to OFDM-based mobile broadband systems.

Keywords: MU-MIMO-OFDM THP, multi-user interference (MUI), inter-

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.

[4] ITU-R Report M.2320-0, “Future technology trends of terrestrial IMT systems,”Nov. 2014.

[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

© IEICE 2020DOI: 10.1587/comex.2019XBL0160Received December 20, 2019Accepted January 22, 2020Publicized February 7, 2020Copyedited April 1, 2020

117


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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


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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]

Fig. 1. System configuration of MU-MIMO-OFDM THP.


119


Hðt; fÞ ¼ K�tHðt � �t; fÞ þM�t; ð1Þwhere K�t ¼ diagðk1;�t; � � � ; kNr;�tÞ 2 R

Nr�Nr and M�t 2 CNr�Nt denote the time

correlation matrix and uncorrelated channel error matrix. Here, the diagonal

element of K�t is given by ki;�t ¼ J0ð2�fDi�tÞ, where J0ð�Þ and fDi

are Bessel

function of the first kind of order 0 and the maximum Doppler frequency of the i-th

MS, respectively. Moreover, each element of M�t follows the complex Gaussian

distribution with mean 0 and variance ð1 � k2i;�tÞ�2h .

Assuming that the time difference between the CSI for precoding and precoded

data transmission is �t as shown in Fig. 1, the received time domain signal vector

rðtÞ 2 CNr is expressed as

rðtÞ ¼ 1

N

XN�1

k¼0g�1k ej

2�kTstHðt; k=TsÞFkVk þ zðtÞ

¼ 1

N

XN�1

k¼0g�1k ej

2�kTstðK�tHðt � �t; k=TsÞ þM�tÞFkVk þ zðtÞ; ð2Þ

where zðtÞ 2 CNr is the noise vector. Moreover, it should be noted that

Hðt � �t; k=TsÞ denotes the CSI matrix which matches with the FF filter Fk.

After conducting FFT processing, we can represent the received frequency

domain signal vector of the p-th sub-carrier Yp ¼ ½Y1;p; � � � ; YNr;p�T 2 CNr by

Yp ¼ gpXN�1

n¼0e�j

2�pnN r n

TsN

� �

¼ 1

N

XN�1

n¼0K�tH n

TsN

� �t;p

Ts

� �þM�t

� �FpVp

þ gpN

XN�1

k¼0k≠p

XN�1

n¼0g�1k ej

2�ðk�pÞnN K�tH n

TsN

� �t;k

Ts

� �þM�t

� �FkVk þ gpZp

¼ 1

N

XN�1

n¼0K�tH n

TsN

� �t;p

Ts

� �� FpVp þM�tFpVp

þ gpN

XN�1

k¼0k≠p

XN�1

n¼0g�1k ej


TsN

� �t;k

Ts

� �þM�t

� �FkVk þ gpZp; ð3Þ

where Zp ¼ ½Z1;p; � � � ; ZNr;p�T 2 CNr denotes the noise vector of the p-th sub-

carrier. In Eq. (3), since the FF filter Fp is originally determined by the CSI matrix

which represents the preceding channel condition for �t from actual data trans-

mission HðnTs=N � �t; p=TsÞ, the first term of Eq. (3) corresponds to the desired

signal component. Consequently, the received signal Yp can be rewritten as [9]

Yp ¼ K�tXp þM�tFpVp

þ gpN

XN�1

k¼0k≠p

XN�1

n¼0g�1k ej


TsN

� �t;k

Ts

� �þM�t

� �FkVk þ gpZp; ð4Þ

where Xp ¼ ½X1;p; � � � ; XNr;p�T 2 CNr denotes the original modulated signal vector

of the p-th sub-carrier.


120


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

operation zmod, which is given by

pðzmodÞ ¼X1

l¼�1

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ðPMUI þ PICI þ PNÞ

p exp � ðzmod þ l�Þ22ðPMUI þ PICI þ PNÞ

� �: ð11Þ

3 Numerical results

We demonstrate the theoretical results of MU-MIMO-OFDM THP in terms of both

the signal-to-interference-plus-noise ratio (SINR) and system capacity to clarify

the impact of terminal mobility and then compared it to MU-MIMO-OFDM LP. In

our performance evaluation, spatially uncorrelated Rayleigh fading is assumed for

the MIMO channel, where each channel follows a 16-ray exponentially decaying

multipath channel. Here, the delay spread normalized by the sampling rate Tsam

(¼ Ts=N) is set to be �rms ¼ 1:0Tsam. Moreover, the perfect CSI feedback is

assumed and its feedback error and delay are negligible. To enhance the trans-

mission performance of THP, the ordering process [8, 9] is adopted.

Fig. 2 shows the cumulative distribution function (CDF) of the SINR with

parameters of the normalized maximum Doppler frequency fDTsam and number of


121


sub-carriers N, where the MIMO antenna configuration and average CNR are set to

be 8 � 8 and 25 dB, respectively. From Fig. 2, we can see that because THP

effectively suppresses the effect of the MUI and ICI as well as the noise enhance-

ment, THP achieves better SINR than LP even in the presence of terminal mobility.

Moreover, it is observed that the SINR is degraded with an increase in the number

of sub-carriers N regardless of the precoding scheme because the effect of ICI

becomes critical.

Fig. 3 shows a performance comparison between THP and LP in terms of the

sum-rate versus the normalized maximum Doppler frequency fDTsam with a

parameter of the MIMO antenna configuration, where the number of sub-carriers

N ¼ 256 and the average CNR is set to be 25 dB. From Fig. 3, it can be seen that

the performance gap between THP and LP decreases in the range of fDTsam > 10�4

regardless of the MIMO antenna configuration. This is because terminal mobility

escalates the effect of the modulo loss with the MUI and ICI. Moreover, the

superiority of THP over LP is enlarged with an increase in the MIMO antenna

configuration as a result of the space diversity effect.

4 Conclusion

In this letter, we theoretically analyzed the exact system capacity of MU-MIMO-

OFDM THP in the presence of terminal mobility. Considering the application of

THP to OFDM transmission, we derived the effect of both the MUI and ICI due to

terminal mobility and this effect was incorporated into the system capacity analysis

based on the mod-Λ channel. Numerical results showed that THP still achieves the

higher system capacity than LP even in the presence of terminal mobility. The study

showed that the proposed approach provides a comprehensive performance eval-

uation of MU-MIMO-OFDM THP when considering possible effects such as the

MUI, ICI, and modulo loss without any time-consuming computer simulations. In

general, the proposed analysis can be used to verify the applicability of THP to

OFDM-based mobile broadband systems.

Acknowledgments

The authors would like to thank Y. Hamaguchi of Sharp Corporation for his

continuing support.

Fig. 2. CDF of SINR. Fig. 3. Sum-rate versus fDTsam.


122


Errata

The following editorial correction has been found in Vol. 8, No. 12, and should be

corrected as follows.

Wrong

p. 534

handler forwards pn.

p. 534

Handler drops pn.

p. 534

pn must be retransmitted.

p. 535

Y-axis information of Fig. 3(b).

Correct

p. 534

handler forwards p.

p. 534

Handler drops p.

(b) Twelve sessions

Fig. 3. Total goodput of all sessions

© IEICE 2020DOI: 10.1587/comex.2020XBL8001

Published April 1, 2020

123


p. 534

p must be retransmitted.

p. 535

(b) Twelve sessions

Fig. 3. Total goodput of all sessions

© IEICE 2020DOI: 10.1587/comex.2020XBL8001

Published April 1, 2020

124


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IEICE Communications Express, Vol.9, No.4, 100 The ﬁrst ...Keywords: quantum error-correcting codes, quantum deletion errors, dele-tion codes Classiﬁcation: Fundamental Theories