2019c9� $ Ê Æ Æ � 123ò 13Ï

Sept., 2019 Operations Research Transactions Vol.23 No.3

DOI: 10.15960/j.cnki.issn.1007-6093.2019.03.004

Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{∗

4æ¹1,†

Á� Ã�Ï&XÚ�O¥�Nõ¯K�ï��`z¯K. ��¡, ù`z¯K~~ä

kpÝ���55, ���¹eJu¦); ,��¡, §�qkg��AÏ(�, ~XÛ¹

�à5!�©5�. |^`z��{(ܯK�AÏ(�¦)Ú?nÃ�Ï&XÚ�O¯K

´Cc5Æâ.ïÄ�9:. �©­:?ØÃ�Ï&XÚ�O¥�ü�`z¯KÚ�'`z

�{, �)õ^rZ6&�����OKe�éÜDÑ/�ÂÅå¤/�OÚõÑ\õÑÑ

(Multi-Input Multi-Output, MIMO) uÿ¯K, Ì�0�y�`zEâ(ܯK�AÏ(

�3¦)Ú?nþãü�¯K��#?Ð.

'�c ��½tµ, Cþ�O�`z, õ^rZ6&�, éÜDÑ/�ÂÅå¤/�O,

O�E,5, ;tµ, MIMO uÿ, Ã�Ï&XÚ

¥ã©aÒ O221, TN92

2010 êÆ©aÒ 65K05, 90B18, 90C22, 90C30, 90C90

Two optimization problems in wireless communicationsystem design and related optimization methods∗

LIU Yafeng1,†

Abstract Many problems arising from wireless communication system design canbe formulated as optimization problems. On the one hand, these optimization problemsare often non-convex and highly nonlinear and thus are difficult to solve; on the otherhand, these problems have their own special structures such as (hidden) convexity andseparability. Recently applying mathematical optimization methods to solve/deal withthese problems while judiciously taking care of their special structures is a hot researchtopic. This (survey) paper aims to introduce two optimization problems in wirelesscommunication system design, max-min fairness linear transceiver design problem andMIMO detection problem, and related optimization methods. This paper will focus on theabove two problems and overview recent advances of applying mathematical optimizationtechniques to solve/deal with them by exploiting their special structures.

Keywords semi-definite relaxation, alternating optimization, multiuser interferencechannel, transceiver beamforming design, computational complexity, tight relaxation,MIMO detection, wireless communication system

ÂvFÏ: 2019-03-01

*Ä7�8:I[g,�ÆÄ7 (Nos. 11688101, 11671419, 11631013, 11571221),�®½g,Ä7­:;� (No. L172020), I[uU��Æ�êâú�ÑÖ²��M#A^«��8 (No. 2016-999999-65-01-

000696-01)

1. ¥I�Æ�êÆ�XÚ�ÆïÄ�, O�êÆ��Æó§O�ïĤ, �Æ�ó§O�I[­:¢�¿,�® 100190; State Key Laboratory of Scientific and Engineering Computing, Institute of Computa-

tional Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science,

Chinese Academy of Sciences, Beijing 100190, China

† Ï&�ö E-mail: yafliu@lsec.cc.ac.cn

48 4æ¹ 23ò

Chinese Library Classification O221, TN92

2010 Mathematics Subject Classification 65K05, 90B18, 90C22, 90C30, 90C90

0 Ú ó

�`z[1-3]?Øûü¯K��ZÀJ, �EϦ�Z)�O��{, ïÄùO��

{�nØ5�9¢SO�Ly. �`z3I�!²L!7K!ó§!�Ï!+n!êâ©

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^Å5DÑ&E.y�¿Âe�Ã�Ï&©u 19­V 90c�¿�|<ê�Z (Guglielmo

Marconi)��X�>^Å¢�,¦¤õ/|^>^Å¢yÃ�êâDÑ. 1948c, �à

(Claude Elwood Shannon) uLK�“A Mathematical Theory of Communication”�Ø

©, ïáÏ&�êÆnØ[7]. A�c5, <�Øä²{X#�Ã�Ï&�{��)ÚC

�, ¿�ÉXõ«õ��Ã�Ï&ÑÖ. Ã�Ï&Eâ�uÐ4�/Uõ<��Ñ1!

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[ÔÑuÐOy�­�|¤Ü©. �©­:?ØÃ�Ï&XÚ�O¥�`z¯KÚ`z

�{, Ì�0�y�`zEâ(ܯK�AÏ(�3Ã�Ï&XÚ5U`z�ü�Y~

9Ù�#?Ð.

Ã�Ï&XÚ�O¥�Nõ¯K8�(.Ñ´`z¯K, �$^`z�{\±¦),

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Eâ3y¢)¹¥�%ÇuÐ,Ï&XÚ`z�êÆnØ��{w��é¢�,3,�

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55,���¹eJu¦);,��¡,§�qkg��AÏ(�,~XÛ¹�à5!�©

5!DÕ5�. |^`z��{(ܯK�AÏ(�¦)Ú?nÃ�Ï&XÚ�O¯K,

C��c²þzcÑkù�¡�ó�¼ IEEE &Ò?nƬ½ö IEEE Ï&Ƭ�ZØ

©ø. ­.Ͷ`z;[!\<��[�Æ��¬!&Ò?nÚÃ�Ï&º?Ïr IEEE

Transactions on Signal Processing c?Ì?Û���Ç�Ì�ïÄ��´`z�{��

O!©Û9Ù3Ã�Ï&Ú&Ò?n¥�A^.Û���Ç3þã����´a�ïÄ

¤J, l �¼� 2004!2009!2011 c IEEE &Ò?nƬ�ZØ©ø, 2015 c IEEE

Signal Processing Magazine �ZØ©ø, 2011 cî³&Ò?nƬ�ZØ©ø, Ú 2011

cISÏ&�¬�ZØ©ø. `z�{3Ã�Ï&XÚ�O¥�­�5dd����.

�©X­0�y�`zEâ(ܯK�AÏ(�3Ã�Ï&XÚ5U`z�ü�Y

~9Ù�#?Ð. ·�ò31 1 !0�õÑ\õÑÑ (Multi-Input Multi-Output, MIMO)

Z6&�¥����OKe�éÜDÑ/�ÂÅå¤/�O¯K9Ù�#ïÄ?Ð; 31

2!0� MIMOuÿ (MIMO Detection)¯K9Ù�#ïÄ?Ð.�©ý­ul`z��

Ý"ÀÃ�Ï&XÚ�O,kXeü�ý­:.�©�1��ý­:´'uÃ�Ï&XÚ

�O¥`z¯KO�E,5��x[8]. O�E,5�±�x¤�įK���J´§Ý,

´)ûÚ©Û¢S¯K�'�Ú½. �©�1��ý­:´�â¯K�AÏ(��Ok

���{?nÚ¦)�A�Ã�Ï&XÚ`z�O¯K, ¿é�'�{?1nØ©Û�.

'uÃ�Ï&XÚ`z�O�nã©Ù, �ë�©z [9-16].

ÎÒ�½: �©^��çNi1L«��þ, ��çNi1L«Ý. �½Ý A,

3Ï Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{ 49

AT L«§�=�, A† L«§��Ý=�, A−1 L«§�_, Ai,j L«§�1 (i, j) ��

�. �q�ÎÒ�·^u�þ. �½�þ x = [x1, x2, · · · , xK ]T, ‖x‖p ,

∑Kk=1 |xk|p L«

§� p �ê, Ù¥ p ∈ {2,∞}; Diag(x) L«d�þ x )¤�é�Ý. �½ (·��ê

�) HermitianÝ AÚ B, A � 0L« A´����½Ý; A •BL« AB �,,=∑i

∑j Ai,jBj,i; A⊗B L«§�� Kronecker ¦È. ��, �©^ e L«·��ê�� 1

�þ, ^ I L«·��ê�ü Ý, ^ 0 L«·��ê�� 0 ݽ�þ.

1 ����OKeéÜDÑ/�ÂÅå¤/�O

1.1 `z�.

õ^rZ6&�Xã 1 ¤«. 3õ^rZ6&�¥, kõ�DÑàÚõ��Âà. z

�DÑàÚ�ÂàCkØÓêþ�U�.z�DÑàF"�éA��Âàux&Ò.AO

/,�DÑà 1 (TX1)��Âà 1 (RX1)ux&Ò�Ó�, TX1 ��Ù¦��Âà�5

Z6;�L5,� RX1 � TX1 DÑ&Ò�Ó�,��Â�Ù¦DÑàDÑ�&Ò,ùé

u RX1 5`�´Z6. ù=´õ^rZ6&�.

ã1 õ^rZ6&�«¿ã

b�XÚ¥k K �^r, ·�^ K = {1, 2, · · · ,K} L«XÚ¥¤k^r�8Ü. �

Äü1ÅÏ&XÚ,=1 k �DÑà3�����ux��&Ò sk ∈ C�éA��Âàk. ·�^ Hkj ∈ CMk×Nj L«1 j �DÑà�1 k ��Âà�&�Ý, Ù¥ Mk Ú

Nk ©OL«1 k ��ÂàÚ1 k �DÑàCk�U��ê, @o�Âà k �Â�&Ò

�

yk = Hkkvksk +∑j 6=k

Hkjvjsj + zk,

Ù¥ vk ∈ CNk×1 L«DÑà k ¦^�DÑÅå¤/�þ, zk ∈ CMk×1 ´\5pdxD

(, Ñl��©Ù CN (0, σ2kI). - uk ∈ CMk×1 L«�Âà k ¦^��ÂÅå¤/�þ,

50 4æ¹ 23ò

K�Âà k ²L�5?n±����&Ò�

sk = u†kyk = u†kHkkvksk +∑j 6=k

u†kHkjvjsj + u†kzk.

òZ6w¤D(, @o1 k �^r�&ZD' (Signal-to-Interference-plus-Noise-Ratio,

SINR) �L«�

SINRk =|u†kHkkvk|2

σ2k‖uk‖2 +

∑j 6=k

|u†kHkjvj |2, k ∈ K. (1.1)

�âXÚ¥DÑàÚ�ÂàCk�U��ê,�òõ^rZ6&��Xe©a:�X

Ú¥z�DÑàÚ�ÂàÑCkõ�U��, = Mk > 2, Nk > 2 (k ∈ K) �, d&�¡

� MIMO Z6&�, Ù SINR L�ªXª (1.1) ¤«; �XÚ¥z�DÑàCkõ�U

�, z��ÂàCk 1 �U�, = Mk = 1, Nk > 2 (k ∈ K) �, d&�¡�õÑ\üÑÑ

(Multi-Input Single-Output, MISO) Z6&�.

3MIMOÚMISOZ6&�¥,z�DÑà�õÇ�å�L«� ‖vk‖2 6 pk, k ∈ K,Ù¥ pk L«1 k �DÑà�õÇþ�. �XÚ¥z��ÂàCkõ�U�, z�DÑà

Ck 1 �U��, = Mk > 2, Nk = 1 (k ∈ K) �, d&��üÑ\õÑÑ (Single-Input

Multi-Output, SIMO) Z6&�.

��y^r�m�ú²5, ·��Ä MIMO Z6&�¥��� SINR ��z (�

du���Ç��z) ¯K:

max{u,v}

G(u,v) , mink∈K{SINRk(uk,v)}

s.t. ‖uk‖2 = 1, ‖vk‖2 6 pk, k ∈ K,(1.2)

Ù¥ SINRk(uk,v) dª (1.1) �Ñ. 5¿�1 k �^r� SINR L�ª (1.1) Ø�6u

{uj}j 6=k , ¤±·�^ SINRk(uk,v) L«1 k �^r� SINR. Ú\9ÏCþ SINR =

mink∈K {SINRk} , þã�� SINR ��z¯K (1.2) �±�d/��

max{u,v}

SINR

s.t. SINR 6 SINRk, ‖uk‖2 = 1, ‖vk‖2 6 pk, k ∈ K.(1.3)

1.2 �#?Ð

�z��ÂàCk 1 �U� (Mk = 1, k ∈ K) �, = MISO Z6&�: ©z [17] J

ÑÏL��z_&Z'�\�Ú5%C¯K (1.2), ¿y²�±ÏLÀ�·��\�Xê

¼�¯K (1.2) ��`), dó�¼� 2007 c IEEE &Ò?nƬ�ZØ©ø; ©z

[18,19] JÑõ�ª�m�{�¦�¯K (1.2) ��Û�`).

�z�DÑàCk 1 �U� (Nk = 1, k ∈ K) �, = SIMO Z6&�, ·�Äg��

Xeõ�ª�m�)�(J, ù�(JÌ�´du¯K¥Û¹�à5[20]. duÙy²´

�E5�, ¤±Ó���ѯK (1.2) �DÑàk 1 �U���õ�ª�m�{; �

�©z [21].

½n 1.1 �z�DÑàCk 1 �U� (Nk = 1, k ∈ K) �, �� SINR ��z¯K

(1.2) �3õ�ª�mS¦).

3Ï Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{ 51

Ó�, ·��Äg��DÑàÚ�ÂàÑCkõ�U��¯K (1.2) �E,5(

J[23]: =�z�DÑàCk�U�ê�u½�u 2(3), z��ÂàCk�U�ê�u½

�u 3(2) �, �ä�½ SINR 8I��15¯K´r NP-J�. 'uDÑàÚ�ÂàC

k�U�êÑ�u 2 �¯K (1.2) �E,5, ù�¯K��] �û. �C, ·�£�

ù�¯K; ��©z[22].

½n 1.2 �½¯K (1.3) ¥ SINR 8I ζ. �z�DÑà/�ÂàCk�U�ê�

u½�u 2 �, �ä�½ ζ ��15¯K´r NP-J�. Ïd, �`z¯K (1.2) �´r

NP-J�.

L 1 ¥o( MIMO Z6&��� SINR ��z¯K (1.2) 3DÑàÚ�ÂàCk

ØÓU�ê��O�E,5. lL 1 ¥·���, ¯K (1.2) �E,5�XDÑàÚ�Â

àCk�U�ê8�Cz Cz. ��5ù, O\U�ê8�±JpÏ&XÚ�5U, �

���A`z¯K�¦)�5(J.

L 1 ����OKeéÜDÑ/�ÂÅå¤/�O¯K (1.2) �O�E,5hhhhhhhhhhhhhhhh�Âà

DÑàNk = 1 Nk > 2

Mk = 1 õ�ª�) õ�ª�)[18,19]

Mk > 2 õ�ª�)[21] r NP-J[22]

·��JÑü�Cþ�O�`z�{¦)�� SINR ��z¯K (1.2): 1���

{´°(Ì��Iþ,�{ (Exact Cyclic Coordinate Ascent Algorithm, ECCAA), z�Ú

S�°(¦)f¯K;1���{´�°(Ì��Iþ,�{ (Inexact Cyclic Coordinate

Ascent Algorithm, ICCAA), Ù¥kf¯K�°(¦). Cþ�O�`z�{�Ä�g

�´Äk·�/ò¯K�Cþ©¬, ,� (°(½ö�°() �O/3��¬�m?1`

z/S�, zg`z/S���#Ù¥�¬Cþ, Ù¦¬Cþ�½ØC.

AO/, éu¯K(1.2), òÙCþ©�ü¬un = (un1 , · · · ,unK)Úvn = (vn1 , · · · ,vnK),

Ù¥ n > 0 L«S��I. �½ vn, Ï� SINRk =� uk �', ¤±�±ÏL¦) K �

�¯K

max{uk}

|u†kHkkvnk |2

σ2k‖uk‖2 +

∑j 6=k

|u†kHkjvnj |2

s.t. ‖uk‖2 = 1

(1.4)

��¯K (1.2) � vn �½��). ½Â

Mk(v) =

∑j∈K

Hkjvj (Hkjvj)†

+ σ2kI

−1 , k ∈ K,K¯K (1.4) ��`) unk ��5��þ�Ø� (Linear Minimum Mean Square Error,

LMMSE) �ÂÅå¤/�þ

unk = unk/‖unk‖, uk = Mk(vn)Hkkvnk . (1.5)

52 4æ¹ 23ò

�½ un, ¯K (1.2) 'uCþ v C�Xe MISO Z6&�¥�`z¯K:

max{v}

mink∈K

{| (unk )

†Hkkvk|2

σ2k +

∑j 6=k | (unk )

†Hkjvj |2

}s.t. ‖vk‖2 6 pk, k ∈ K.

(1.6)

dL 1 ��, ¯K (1.6) �±3õ�ª�mS¦��Û�`).

�{ 1.1 ECCAA

Ú 1 Щz: �½Ð©: v0 ÚÊ�OK ε. - n = 0.

Ú 2 O� un: �½ vn, O��`� LMMSE �ÂÅå¤/�þ un (�ª (1.5)).

Ú 3 O� vn+1: �½ un, ¦)¯K (1.6) ���`�DÑÅå¤/ vn+1.

Ú 4 ª�u�: �G(un,vn+1)−G(un,vn−1)

max {G(un,vn−1), 1}6 ε (1.7)

÷v�, ª��{; ÄK, � n := n+ 1, =Ú 2.

���¹e, =¦z�f¯KÑ°(¦), E,ØU�yCþ�O�{�Âñ5[23].

8I¼ê�1wÜ©±9�å¥Cþ��©5éu�{�Âñ5�~­�. N´�E�

~y²,�8I¼ê�1wÜ©½ö�å¥�CþØ�©�,Cþ�O�{�)�:��

UÂñ���ÿÂ�:.¦+¯K (1.2)�8I¼ê´�1w��8I¼ê¥Cþ uÚ

v ÍÜ3�å, ·�|^Cþ uk �Ñy3 SINRk ¥�AÏ(�y²þã�O�`z

�{ ECCAA �)�:�Âñ�¯K (1.2) ��� KKT (Karush-Kuhn-Tucker) :[24].

½n 1.3 - ECCAA �{¥ ε = 0. ECCAA �)�:� {(un,vn)} ½ª�u¯K(1.2) ��� KKT :, ½§�?¿à: (u, v) ´¯K (1.2) � KKT :.

�?�ÚJp ECCAA �{�O�k�5, ·�3© [25] ¥JÑ���°(�

Cþ�O�`z�{¦)¯K (1.2), ÙzgS��°(/¦)'uCþ v �`z¯K.

äN/, © [25] JÑÏL¦)f¯K

max{v, θ}

θ

s.t.(unk )

†Hkkvk − θ√

σ2k +

∑j 6=k | (unk )

†Hkjvj |2

>√G2n, k ∈ K,

‖vk‖2 6 pk, k ∈ K,

(1.8)

�#Åå¤/�þ vn+1, Ù¥ G2n ´¯K (1.2) 3: (un,vn) ?�¼ê�. 5¿�¯K

(1.8) ´����I5y (Second Order Cone Program, SOCP) ¯K, ¤±�3õ�ª

�mS¦�Ù�Û�`). ��#DÑÅå¤/�þ vn+1, ECCAA I�¦)�X�

SOCP �15¯K, ICCAA �I�¦)�� SOCP ¯K, ¤±4�/ü$zg�#

Cþ vn+1 �E,5. � ECCAA �{aq, ·��y² ICCAA ��ÛÂñ5[25].

'u MIMO Z6&�éÜDÑ/�ÂÅå¤/�OÚõÇ��?�Ú�ó�, �)

�\E,�|µ (~X MIMO 2Â&�!É��ä!�VóXÚ)Ú�O (~XÄÕg·

AÜ�!ÄÕ-¹!ÄÕ-^r'é) �, �ë� [26-39]; 'uCþ�O�`z�{¦)

Ã�Ï&XÚ�O¯K��õó�, �ë� [34,37,38,40-43];'u�O�`z�{�nã

©Ù9Ù3��+�¥�A^, �ë� [3,44,45].

3Ï Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{ 53

2 MIMO uÿ

2.1 `z�.

MIMO uÿ¯K´y�êiÏ&¥�Ä�¯K��[46,47]. êÆþ, MIMO &��Ñ

\ÑÑ'X�L«�

r = Hx∗ + v, (2.1)

Ù¥ r ∈ Cm ��Âà�Â���þ; H ∈ Cm×n �E�&�Ý; x∗ ∈ Cn L«DÑ�ý¢&Ò�þ; v ∈ Cm �\5D(. XJ¦^ M -Phase-Shift Keying (M -PSK) �N��

ª, @o x∗ �z��©þ x∗i Aáu��k�8Ü, =

x∗i ∈{

exp (iθ) | θ =2jπ

M, j = 0, 1, · · · ,M − 1

}, i = 1, 2, · · · , n,

Ù¥ i�JÜü ÷v i2 = −1.b�&�&E H®�, MIMOuÿ¯KF"ÄuÂ��

&Ò r ¡EDÑ�&Ò x. êÆþ, MIMO uÿ¯K�ï��

minx∈Cn

‖Hx− r‖22s.t. |xi|2 = 1, arg (xi) ∈ A, i = 1, 2, · · · , n,

(2.2)

Ù¥ arg (·) L«Eê�Ë�, A = {0, 2π/M, · · · , 2(M − 1)π/M} .�ª (2.1) ¥�D( v Ñlpd©Ù�, þã4�z�. (2.2) �du��q,�

O�.. N´�y, MIMO uÿ¯K (2.2) ��1:�ê� Mn. ¢Sþ, MIMO uÿ¯

K (2.2) ´r NP-J�[48], ¤±Ø�3õ�ª�m��{U¦)�Ù�Û�`) (Ø�

P=NP).

2.2 �#?Ð

Äu��½tµ (SemiDefinite Relaxation, SDR) ��{´¦)¯K (2.2) �~k�

��{��. Äu SDR ��{Ø�äkõ�ª�m���E,Ý, �3¢S¥�Ðy

ÑéÐ�uÿØèÇ. Äu SDR ��{�Ä�g�´ò¯K (2.2) ktµ��� (E½

ö¢�) SDR, ,�ÏL��{ (�Åz��{½öéuª��{) ���¯K���

�1).

�Ù¥·�X­0�'u MIMOuÿ¯K (2.2) SDR��#?Ð.�Qã�B,·

�Ú\Xe�PÒ: -

Q = H†H, c = −H†r;

- s = [s1, s2, · · · , sM ]T ∈ CM , Ù¥

sj = cos

(2(j − 1)π

M

)+ i sin

(2(j − 1)π

M

), j = 1, 2, · · · ,M ;

- sR = Re(s), sI = Im(s), Ù¥ÎÒ Re(·) Ú Im(·) ©OL«�AEÝ/E�þ/Eê

�¢ÜÚJÜ.

Ú\ n× n EÝ X = xx†, ¯K (2.2) ��dC/�

minx,X

Q •X + 2Re(c†x)

54 4æ¹ 23ò

s.t. Xi,i = 1, i = 1, · · · , n,arg (xi) ∈ A, i = 1, · · · , n,X = xx†,

Ù¥Cþ x ∈ Cn, X ∈ Cn×n. ¯K (2.2) �DÚ (��þ��) E SDR ´

minx,X

Q •X + 2Re(c†x)

s.t. Xi,i = 1, i = 1, · · · , n,

X � xx†,

(2.3)

Ù¥Cþ x ∈ Cn, X ∈ Cn×n. duÙé�AÏ(�, S:�{�3 O(n3.5

)�mS¦�

SDR (2.3)��`)[49].ê�¢�L²,�¥)è (Sphere Decoding)�{[50-52]�',Äu

SDR��{3ØèÇ5UÚ�mE,5þk�~Ð�²ï[49]. ¥)è�{´¦) MIMO

uÿ¯K (2.2) �Ͷ��{��. lêÆ�Ý, ¥)è�{´�«¿©|^¯K (2.2)

AÏ(��©|½.�{, ¤±Ù�±�y¦�¯K (2.2) ��Û�`), �´¥)è�

{���Ú²þE,ÝÑ´�ê�[48,53].

'�ª (2.2) Ú (2.3) ��, ª(2.3) ò�å X = xx† tµ��å X � xx† ¿¿K

¤k�Ë��å arg (xi) ∈ A, i = 1, 2, · · · , n. N´�y

X � xx† ⇐⇒[

1 x†

x X

]� 0.

�JpþãE SDR ��þ, ·�3© [54] ¥JÑ3ÙÄ:þV\k�Ë��[55,56]. ä

N/,�â x�½Â, n�¥ xi A��u�þ s¥�,��©þ. ÏLtµþã�|Ü

�å, ·���Xe\r�E SDR:

minx,X,t

Q •X + 2Re(c†x)

s.t. Xi,i = 1, i = 1, · · · , n,X � xx†,

x = St,

At = en, t > 0,

(2.4)

Ù¥Cþ x ∈ Cn, X ∈ Cn×n, t ∈ RnM ,

S = In ⊗ sT, A = In ⊗ eTM . (2.5)

3ª (2.4) ¥, t = [tT1 , tT2 , · · · , tTn]T Ú ti = [ti,1, ti,2, · · · , ti,M ]T ∈ RM .

©z [54] y²þãE SDR ÚXe�¢ SDR �d

miny,Y

Q •Y + 2cTy

s.t. Yi,i + Yn+i,n+i = 1, i = 1, 2, · · · , n,

Y � yyT,

(2.6)

3Ï Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{ 55

Ù¥Cþ y ∈ R2n, Y ∈ R2n×2n,

Q =

[Re(Q) −Im(Q)

Im(Q) Re(Q)

], c =

[Re(c)

Im(c)

], y =

[Re(x)

Im(x)

]. (2.7)

Äuª (2.6), ©z [54] ?�ÚJÑ��\r�¢ SDR. ½Â 3× 3 �Ý

Yi =

1 yi yn+i

yi Yi,i Yi,n+i

yn+i Yn+i,i Yn+i,n+i

, i = 1, 2, · · · , n (2.8)

Ú

Pj =

1

Re(sj)

Im(sj)

[ 1 Re(sj) Im(sj) ] , j = 1, 2, · · · ,M. (2.9)

�⪠(2.7) ¥ y �½Â, n�¥ª (2.8) ¥½Â� Yi A��u Pj (j = 1, 2, · · · ,M) ¥

�,��, =

Yi ∈ {P1, P2, · · · ,PM} , i = 1, 2, · · · , n.

ÏLtµþã�|Ü�å±9¿K�P{��å,©z [54]JÑXe�'u MIMOu

ÿ¯K (2.2) � SDR:

miny,Y,t

Q •Y + 2cTy

s.t. Yi =

M∑j=1

ti,jPj , i = 1, 2, · · · , n,

At = en, t > 0,

Y � yyT,

(2.10)

Ù¥Cþ y ∈ R2n, Y ∈ R2n×2n, t ∈ RMn, Q Ú c 3ª (2.7) ¥�Ñ, Yi 3ª (2.8) ¥�

Ñ, Pj 3ª (2.9)¥�Ñ, A 3ª (2.5)¥�Ñ.du Yi �é¡5,�å Yi =∑Mj=1 ti,jPj

��d/L«�Xe 5 ��5�å:

yi = tTi sR, yn+i = tTi sI , Yi,i = sTRDiag(ti)sR,

Yn+i,n+i = sTI Diag(ti)sI , Yi,n+i = sTRDiag(ti)sI .

(2.11)

'u MIMOuÿ¯K (2.2)� SDR,��­��nدK´Ù3�o^�e´;�,

=3�o^�e¦)�A� SDR�±���¯K��`).'u M = 2� SDR;5�

(Jéõ, �'(J�ë�©z [49,57-60]. AO/, ©z [60] y²Xe(Ø.

½n 2.1 éu M = 2, XJª (2.1) ¥�ëê H Ú v ÷v

λmin

(Re(H†H)

)> ‖Re(H†v)‖∞, (2.12)

Ù¥ λmin (·) L«�'Ý���A��, KXe¢ SDR 'u¯K (2.2) ´;�:

minx,X

Re(Q) •X + 2Re(c)Tx

s.t. Xi,i = 1, i = 1, 2, · · · , n,X � xx†.

(2.13)

56 4æ¹ 23ò

'u M > 2 � SDR ;5©Û�(J3©z¥�~��. AO/, ©z [60] ¥JÑ

��úm¯K, =éu M > 2 ��/, ´Ä�3'u¯K (2.2) ;� SDR. ·��C3

©z [54] ¥£�ù�úm¯K, y²ª (2.4) Ú (2.10) ¥� SDR Ñ´;�. ùü�

SDR �´'u�� MIMO uÿ¯K (2.2) ®��@�3nØþk;5�y� SDR. ä

N(JXe.

½n 2.2 éu?¿� M > 3, XJª (2.1) ¥�ëê H Ú v ÷v

λmin

(H†H

)sin( πM

)>∥∥H†v∥∥∞ , (2.14)

Kª (2.4) Ú (2.10) ¥� SDR 'u¯K (2.2) Ñ´;�.

'uª (2.4)Ú (2.10)¥� SDR±9½n 2.2��?ØXe.Äk, � M = 2�,

(2.9) ¥½Â�Ýòz�

P0 =

1 1 0

1 1 0

0 0 0

Ú P1 =

1 −1 0

−1 1 0

0 0 0

.¤± (2.11) ¥��5�åòz�

Yi,i = 1, yi = ti,1 − ti,2, Ú yn+i = Yn+i,i = Yn+i,n+i = 0.

�ÑÝ Y ¥�"¬, N´�yª (2.10)¥� SDRòz�ª (2.13)¥� SDR. ¤±�

M = 2 �, SDR (2.10) 3^�ª (2.12) ¤á�'u¯K (2.2) �´;�.

Ùg, ª(2.12) Ú (2.14) ¥� (¿©) ^�ÑkéÐ�ó§ÚÔn)º. ±ª (2.14)

�~: (2.14) ¥�� λmin

(H†H

)L�&��Ð�, �&�Ý H ¥���O'��

�, λmin

(H†H

)��, ^� (2.14) �J÷v; (2.14) ¥�� sin (π/M) �x M XÛK

�¯K�JÝ, M ��, ��ü�ÎÒ�ål 2 sin (π/M) ��, ^� (2.14) �J÷v;

(2.14) ¥��∥∥H†v∥∥∞ �xD(�K�, D( v é&�Ý H ���mK��� (=∥∥H†v∥∥∞ ��),^� (2.14)�J÷v.{ ó�,�N���ê��!&���!D( (é

&�Ý��mK�)��, (2.14)¥�^��J÷v;�� (2.14)¥�^��N´÷v.

2g, ·�?Ø8c���©¥�Ñ� SDR �m±9�©z [61] ¥ SDR �'X.

N´�y, SDR (2.10)' SDR (2.4)r, SDR (2.4)' SDR (2.3)Ú (2.6)r, SDR (2.3)

Ú (2.6) �d. ©z [61] ¥JÑ� SDR ´ÄuXe�*: éu x∗ ¥�?Û x∗i Ñ�3

ti ∈ RM ¦�Ù¥�k��©þ� 1 Ù{©þÑ� 0 �÷v x∗i = tTi s. Äud, ©z [61]

ò¯K (2.2) �d��

mint

tTQt + 2cTt

s.t. At = en, t > 0,

t ∈ {0, 1}Mn,

Ù¥

Q = STQS, c = STc, S =

(Re(S)

Im(S)

). (2.15)

3Ï Ã�Ï&XÚ�O¥�ü�`z¯KÚ�'`z�{ 57

Äuþã�dC/¿|^�þ t �AÏ(�, ©z [61] JÑXe�'u¯K (2.2) �

SDR∗:

mint,T

Q •T + 2cTt

s.t. At = en, t > 0,

T � ttT,

Ti,i = Diag(ti), i = 1, 2, · · · , n,

(2.16)

Ù¥Cþ t ∈ RMn, T ∈ RMn×Mn, Ti,i ∈ RM×M ´Ý T �1 i �é�¬. �����

å Ti,i = Diag(ti) �¦ Ti,i ´��é�Ý�Ùé���� ti. ÃØ´3/ª�´í�

�ªþ, SDR (2.10)Ú SDR (2.16)ÑØ��.k¿g�´,·�3©z [62]¥y²ùü

� SDR �m��d'X.

½n 2.3 SDR (2.10) Ú SDR (2.16) ��18�m÷vXe�'X:

STST = Y Ú St = y, (2.17)

Ù¥ S 3 (2.15) ¥�Ñ. ¤± SDR (2.10) Ú SDR (2.16) ´�d�.

þã^� (2.17)�y�A�ü� SDR�8I¼ê���.(ܽn 2.2Ú 2.3,·�

��: � (2.1) ¥�ëê H Ú v ÷v (2.14) �, SDR (2.16) 'u¯K (2.2) ´;�. ½

n 2.3 Ó��« SDR (2.16) ¥äk�½P{5. l (2.17) ¥�±��, p��m (T, t)

¥�k^&E3$��m (y,Y, t)¥Ñ���±.AO/, SDR (2.16)¥ÝCþ��

ê´ Mn×Mn SDR (2.10) ¥ÝCþ��ê´ 2n× 2n. ¤±¦) SDR (2.10) ò¬

'¦) SDR (2.16) 3ê�þk��õ (AO´� M ��u 2 �).

Äuþã\r� SDR,·��uÐk��©|½.�{¦)MIMOuÿ¯K (2.2).

'uù�¡�?Ð,�ë�©z [63].���J�´,3�^���¹e,~XÑ\ÚÑÑ

�ê8��½XÚ�&D'�$�,·�uÐ�Äu\r SDR�©|½.�{'¥)è

�{3¦)�Ýþkêþ?�Jp. ·�uÐ�©|½.�{�±í2^5¦)Xe�

����àE�g5y¯K:

minx∈Cn

1

2x†Qx + Re

(c†x)

s.t. `i 6 |xi| 6 ui, i = 1, 2, · · · , n,arg (xi) ∈ Ai, i = 1, 2, · · · , n,

Ù¥ |xi| L«Eê xi �ÌÝ, ui Ú `i (i = 1, 2, · · · , n) ´ 2n �¢ê÷v ui > `i > 0,

Ai (i = 1, 2, · · · , n) ´ n �lÑ/ëY�8Ü.

'u¦) MIMO uÿ¯K (2.2) �Ù¦k��{!�#?б9nã, �ë�©z

[47,48,50-53,63-68]; 'u��½5y/��½tµ9Ù3&Ò?nÚÃ�Ï&¥A^�n

ãÚ�#?Ð, �ë�©z [11,69,70],

∗SDR (2.16) Ú [61] ¥� SDR (Model III) ���«O3u SDR (Model III) ¥|^z��þ ti ¦Ú�u 1 �5�����Cþ.

58 4æ¹ 23ò

3 ( å �

Ã�Ï&XÚ�O¥�Nõ¯K8�(.Ñ´`z¯K. |^`z��{(ܯK

�AÏ(�¦)Ú?nÃ�Ï&XÚ�O¯K´Cc5Æâ.ïÄ�9:. �©X­0

�üaÃ�Ï&XÚ�O¥�¯K, =õ^rZ6&�����OKe�éÜDÑ/�

ÂÅå¤/�O¯KÚ MIMO uÿ¯K, Ì�0�ùüa¯K�`z�.±9|^`

zEâ(ܯK�AÏ(�3¦)Ú?nþãüa¯K��#?Ð.

Cc5,15�(5G)£ÄÏ&XÚ®¤�­.��SÆâ.Úó�.�ïÄ9:[71,72].

�'u�c�£ÄÏ&XÚ, 5G ØI�?�ÚJp^rN��Ç, �I�|±pë

��ݱ¢y£Äêâ�ÖþZ�O�!DÑà��Âà�Ï&�òü$ 90%!�¢y

ªÌ�ÇÚõÇ�ÇJ,���8I. �¢yd8I, 5G £ÄÏ&3Ã�DÑEâÚÃ

��äe��¡?1�·5�EâM#.ùC���¡U¢yþã8I,,��¡

��ïÄö�JÑ#��ä]Ô5�Ä:nØÚ'��ƯK. ù¯K¥éõÑ´

`z¯K, ~X�Ã��\� (Cloud Radio Access Network) ¥���`z[73,74]!�5

�U�� (Massive MIMO) ¥� 1 'Aý?è (One-Bit Precoding) �O[75,76]!ÑÖ

���ä��ä�¡ (Network Slicing)[77,78]!Ôé�¥��Å�\uÿ (Random Access

Detection)[79,80] �, )ûù¯K�Ì�(JÚ]Ô3u�Op�Ø%ûü`z�{é

Ï&] ?1Ä�©�±·A¯�Cz�Ã��¸! r±96þG�. o ó�, `z

�{òUY3 5G £ÄÏ&XÚ�O¥�üØ�O��­��Ú.

�u�öY²k�, �©J�k¢¦Ú ¹�?, �Öö1µ��.

��

�©¤�9�ö�ïÄó�Ñ´�Ü�ö�å�¤�. AO/, �©'u MIMO Z

6&�����OKe�éÜDÑ/�ÂÅå¤/�O�ó�Ì�´Ú�ö����Í

ôïÄÚ�l¥©�Æ (�e) Û���ÇÜ��¤�; �©'u MIMO uÿ¯K�

�½tµ�ó�Ì�´Úu�>å�Æ´§Æ¬Ü��¤�. 3d�¤kÜ�ö�¿L

«a�. �ö�/dŬAOa�¥I�Æ�êÆ�XÚ�ÆïÄ��æ��¬Ú`z

�K|�P�ÓÆ�.

ë � © z

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