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 À×´ïѧ±¨ Journal of Radars ISSN 2095 -283X,CN 10-1030/TN      ¡¶À×´ïѧ±¨¡·ÍøÂçÊ×·¢ÂÛÎÄ  ÌâÄ¿£º  MIMO ϵͳ̽ͨһÌ廯ÐźžØÕóÉè¼Æ·½·¨ ×÷Õߣº  Ñî溣¬ÓàÏÔÏ飬ɳÃ÷»Ô£¬´Þ¹úÁú£¬¿×Áî½² ÊÕ¸åÈÕÆÚ£º  2022 -05 -07 ÍøÂçÊ×·¢ÈÕÆÚ£º  2022 -08 -30 ÒýÓøñʽ£º  Ñî溣¬ÓàÏÔÏ飬ɳÃ÷»Ô£¬´Þ¹úÁú£¬¿×Áî½²£®MIMO ϵͳ̽ͨһÌ廯ÐźžØÕóÉè¼Æ·½·¨[J/OL] £®À×´ïѧ±¨.  https://kns.cnki.net/kcms/detail/10.1030.tn.20220829.0911.002.html       ÍøÂçÊ×·¢£ºÔڱ༭²¿¹¤×÷Á÷³ÌÖУ¬¸å¼þ´Ó¼Óõ½³ö°æÒª¾­Àú¼Óö¨¸å¡¢ÅŰ涨¸å¡¢ÕûÆÚ»ã±à¶¨¸åµÈ½×¶Î¡£Â¼Óö¨¸åÖ¸ÄÚÈÝÒѾ­È·¶¨£¬ÇÒͨ¹ýͬÐÐÆÀÒé¡¢Ö÷±àÖÕÉóͬÒ⿯Óõĸå¼þ¡£ÅŰ涨¸åָ¼Óö¨¸å°´ÕÕÆÚ¿¯Ìض¨°æʽ£¨°üÀ¨ÍøÂç³ÊÏÖ°æʽ£©ÅÅ°æºóµÄ¸å¼þ£¬¿ÉÔݲ»È·¶¨³ö°æÄê¡¢¾í¡¢ÆÚºÍÒ³Âë¡£ÕûÆÚ»ã±à¶¨¸åÖ¸³ö°æÄê¡¢¾í¡¢ÆÚ¡¢Ò³Âë¾ùÒÑÈ·¶¨µÄÓ¡Ë¢»òÊý×Ö³ö°æµÄÕûÆÚ»ã±à¸å¼þ¡£Â¼Óö¨¸åÍøÂçÊ×·¢¸å¼þÄÚÈݱØÐë·ûºÏ¡¶³ö°æ¹ÜÀíÌõÀý¡·ºÍ¡¶ÆÚ¿¯³ö°æ¹ÜÀí¹æ¶¨¡·µÄÓйع涨£»Ñ§ÊõÑо¿³É¹û¾ßÓд´ÐÂÐÔ¡¢¿ÆѧÐÔºÍÏȽøÐÔ£¬·ûºÏ±à¼­²¿¶Ô¿¯ÎĵļÓÃÒªÇ󣬲»´æÔÚѧÊõ²»¶ËÐÐΪ¼°ÆäËûÇÖȨÐÐΪ£»¸å¼þÄÚÈÝÓ¦»ù±¾·ûºÏ¹ú¼ÒÓйØÊ鿯±à¼­¡¢³ö°æµÄ¼¼Êõ±ê×¼£¬ÕýȷʹÓúÍͳһ¹æ·¶ÓïÑÔÎÄ×Ö¡¢·ûºÅ¡¢Êý×Ö¡¢ÍâÎÄ×Öĸ¡¢·¨¶¨¼ÆÁ¿µ¥Î»¼°µØͼ±ê×¢µÈ¡£ÎªÈ·±£Â¼Óö¨¸åÍøÂçÊ×·¢µÄÑÏËàÐÔ£¬Â¼Óö¨¸åÒ»¾­·¢²¼£¬²»µÃÐÞ¸ÄÂÛÎÄÌâÄ¿¡¢×÷Õß¡¢»ú¹¹Ãû³ÆºÍѧÊõÄÚÈÝ£¬Ö»¿É»ùÓڱ༭¹æ·¶½øÐÐÉÙÁ¿ÎÄ×ÖµÄÐ޸ġ£ ³ö°æÈ·ÈÏ£ºÖ½ÖÊÆÚ¿¯±à¼­²¿Í¨¹ýÓ롶ÖйúѧÊõÆÚ¿¯£¨¹âÅ̰棩¡·µç×ÓÔÓÖ¾ÉçÓÐÏÞ¹«Ë¾Ç©Ô¼£¬ÔÚ¡¶ÖйúѧÊõÆÚ¿¯£¨ÍøÂç°æ£©¡·³ö°æ´«²¥Æ½Ì¨ÉÏ´´°ìÓëÖ½ÖÊÆÚ¿¯ÄÚÈÝÒ»ÖµÄÍøÂç°æ£¬ÒÔµ¥Æª»òÕûÆÚ³ö°æÐÎʽ£¬ÔÚÓ¡Ë¢³ö°æ֮ǰ¿¯·¢ÂÛÎĵļÓö¨¸å¡¢ÅŰ涨¸å¡¢ÕûÆÚ»ã±à¶¨¸å¡£ÒòΪ¡¶ÖйúѧÊõÆÚ¿¯£¨ÍøÂç°æ£©¡·Êǹú¼ÒÐÂÎųö°æ¹ãµç×ܾÖÅú×¼µÄÍøÂçÁ¬ÐøÐͳö°æÎISSN  2096-4188£¬CN 11-6037/Z £©£¬ËùÒÔÇ©Ô¼ÆÚ¿¯µÄÍøÂç°æÉÏÍøÂçÊ×·¢ÂÛÎÄÊÓΪÕýʽ³ö°æ¡£   1  MIMOϵͳ̽ͨһÌ廯ÐźžØÕóÉè¼Æ·½·¨ Ñî溢٠      ÓàÏÔÏé¢Ù       É³Ã÷»Ô¢Ú    ´Þ¹úÁú*¢Ù       ¿×Áî½²¢Ù ¢Ù(µç×ӿƼ¼´óѧÐÅÏ¢ÓëͨÐŹ¤³ÌѧԺ ³É¶¼ 611731) ¢Ú(±±¾©ÎÞÏßµç²âÁ¿Ñо¿Ëù ±±¾© 100854) Õª Òª£ºÓÉÓÚ¶àÊäÈë¶àÊä³ö£¨MIMO£©ÏµÍ³¾ßÓв¨ÐΡ¢¿Õ¼ä·Ö¼¯ºÍ¶à·¸´ÓõÈÓÅÊÆ£¬MIMO̽ͨһÌ廯£¨DFRC£©ÏµÍ³Í¨¹ý¹²ÏíÈíÓ²¼þ×ÊÔ´ÒÔͬʱʵÏÖÄ¿±ê̽²âºÍ±£ÃÜͨÐŹ¦ÄÜÊܵ½Á˼«´ó¹Ø×¢¡£±¾ÎÄÕë¶Ô»ùÓÚÔ¤±àÂë¾ØÕóµ÷ÖƵÄMIMO̽ͨһÌ廯ϵͳ£¬Ìá³öÁË»ùÓÚ½»Ìæ·½Ïò³Ë×Ó£¨ADMM£©µÄÒ»Ì廯ÐźžØÕóÉè¼Æ·½·¨¡£Í¨¹ýÓû§ºÍÇÔÌýÓû§²Î¿¼ÃÜÂë±¾Ô¼ÊøÏÂ×î´ó»¯·½Ïòͼ·åÖµÖ÷°êÅÔ°êµçƽ±È£¨PMSR£©£¬±£Ö¤ÁË̽²â·½ÏòͼÐÔÄܵÄͬʱ·ÀֹͨÐÅÐÅÏ¢±»ÇÔÌý¡£Õë¶ÔÔ¤±àÂë¾ØÕóͨÐŽâµ÷ÎÊÌ⣬Ìá³öÁË»ùÓÚ½»Ìæ·½Ïò³Í·££¨ADPM£©µÄÅÅÐòѧϰÓÅ»¯½âµ÷·½·¨£¬ÌáÉýÁËÒ»Ì廯²¨ÐÎÐÅÏ¢½âµ÷ЧÂÊ¡£ÊýÖµ·ÂÕæÑéÖ¤ÁËËùÌáÉè¼Æ·½·¨ÊµÏÖ̽ͨһÌ廯µÄÓÐЧÐÔ£¬ÓëÒÑÓÐËã·¨Ïà±È¿ÉʵÏÖ¶àÓû§Í¨Ðź͸ü¸ßµÄPMSR¡£ ¹Ø¼ü´Ê£ºÌ½Í¨Ò»Ì廯£»ÐźžØÕóÉè¼Æ£»½»Ìæ·½Ïò³Ë×Ó·¨£»ÅÅÐòѧϰ½âµ÷£»½»Ìæ·½Ïò³Í·£·¨ ÖÐͼ·ÖÀàºÅ£ºTN958                   ÎÄÏ×±êʶÂ룺A                        ÒýÓøñʽ£ºÑîæº, ÓàÏÔÏé, ɳÃ÷»Ô,µÈ. MIMOϵͳ̽ͨһÌ廯ÐźžØÕóÉè¼Æ·½·¨[J]. À×´ïѧ±¨,  Reference format: YANG Jing, YU Xianxiang, SHA Minghui, et al. DFRC Signal Matrix Design Method for MIMO System [J]. Journal of Radars  Dual Function Radar and Communication Signal Matrix Design Method for MIMO System YANG Jing¢Ù       YU Xianxiang¢Ù    SHA Minghui¢Ú     CUI Guolong*¢Ù       KONG Lingjiang¢Ù ¢Ù(School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China) ¢Ú(Beijing Institute of Radio Measurement, Beijing 100854, China) Abstract: Due to several advantages of the multi-input multi-output (MIMO) system in terms of waveform, space diversity, and multiplexing, the MIMO dual function radar and communication (DFRC) system, which is responsible for target detection and securing the communication by sharing the software and hardware resources, has attracted great attention. This paper addresses the MIMO DFRC system based on permutation matrix modulation and proposes a DFRC signal matrix design method based on the alternation direction method of multipliers. By maximizing the peak mainlobe to sidelobe level ratio (PMSR) of the beampattern with the constraints of the reference codebook for both users and eavesdroppers, the system guarantees excellent detection performance along with protecting the communication information from interception. Aiming at the  communication  demodulation  of  the  permutation  matrix,  a  permutation  learning demodulation method based on the alternating direction penalty method is proposed to improve                                                         ÊÕ¸åÈÕÆÚ£º2022-05-07  *ͨÐÅ×÷Õߣº´Þ¹úÁú cuiguolong@uestc.edu.cn      *Corresponding Author: CUI Guolong, cuiguolong@uestc.edu.cn »ù½ðÏîÄ¿£º¹ú¼Ò×ÔÈ»¿Æѧ»ù½ð(U19B2017, 62101097)£¬³¤½­Ñ§Õß½±Àø¼Æ»®£¬Öйú²©Ê¿ºó¿Æѧ»ù½ð(2020M680147, 2021T140096) Foundation Items: The National Natural Science Foundation of China (U19B2017, 62101097), The Chang Jiang Scholars Program, Postdoctoral Science Foundation under Grants (2020M680147, 2021T140096) ÔðÈÎÖ÷±à£ºÁº¾üÀû        Corresponding Editor: LIANG Junli ÍøÂçÊ×·¢Ê±¼ä£º2022-08-30 10:45:36ÍøÂçÊ×·¢µØÖ·£ºhttps://kns.cnki.net/kcms/detail/10.1030.tn.20220829.0911.002.html 2  the  demodulation  efficiency  of  the  co-use  waveform.  Numerical  simulations  verify  the effectiveness of the proposed methods to achieve dual function, capable of realizing multiuser communication and deriving higher PMSR compared with the existing counterparts. Key  words:  Dual Function  Radar  and  Communication  (DFRC);  signal  matrix  design; Alternation Direction Method of Multipliers (ADMM); demodulation via permutation learning; Alternating Direction Penalty Method (ADPM) 1  ÒýÑÔ Ëæ×ŵç×ÓϵͳÏòÐÅÏ¢»¯ºÍÖÇÄÜ»¯·¢Õ¹£¬ÐÂÒ»´ú¶à¹¦ÄÜÒ»Ì廯µç×Óϵͳͨ¹ý¹²ÓÃÐźš¢ÐŵÀ¡¢ÌìÏß¡¢Êý¾Ý´¦ÀíµÈÓ²¼þºÍÈí¼þ×ÊÔ´ÒÔʵÏֶ๦Äܹ²Ó㬳ÉΪ¹úÄÚÍâѧÊõ½çºÍ¹¤Òµ½çµÄÑо¿Èȵã[1 -4]¡£ÆäÖУ¬¶àÊäÈë¶àÊä³ö£¨Multi-Input Multi-Output, MIMO£©¼¼ÊõÓÉÓÚ¾ßÓпռä·Ö¼¯Ó븴ÓÃÔöÒæµÈÓÅÊÆ£¬ÔÚ̽ͨһÌ廯ϵͳÖпÉÔöǿϵͳÐÔÄÜ£¬´Ó¶øÒý·¢Á˹㷺¹Ø×¢[5 -8]¡£ MIMO̽ͨһÌ廯ϵͳ¸ù¾ÝÔØƵÊýÁ¿¿É·ÖΪÁ½ÖÖ£º£¨1£©¶àÔØƵMIMOÒ»Ì廯ϵͳ£ºÒÔÌøƵ£¨Frequency  Hopping, FH£©ºÍÕý½»Æµ·Ö¸´Óã¨Orthogonal  Frequency  Division Multiplexing, OFDM£©ÐźÅΪ´ú±í[9]£¬Í¨ÐÅÐÅÏ¢¿Éµ÷ÖÆÓÚƵÂÊ¡¢ÆµÂÊÔöÁ¿¡¢Ïàλ¡¢·½ÏòͼµÈ²ÎÊýÖУ»£¨2£©µ¥ÔØƵMIMOÒ»Ì廯ϵͳ£ºÍ¨¹ýÉè¼Æ·¢É䲨ÐξØÕó»òÕý½»²¨ÐμÓȨ¾ØÕó[10]£¬Óû§¶ËͨÐÅÐÅÏ¢¿É²ÉÓ÷ùÒƼü¿Ø£¨Amplitude  Shift  Keying, ASK£©¡¢ÏàÒƼü¿Ø£¨Phase  Shift Keying, PSK£©¡¢Õý½»µ÷·ù£¨Quadrature Amplitude Modulation, QAM£©µÈµ÷ÖÆ¡£ ¶ÔÓÚ¶àÔØƵMIMOÒ»Ì廯ϵͳ£¬Hassanien AµÈÈË[11]Ìá³ö½«Í¨ÐÅÐÅÏ¢PSKµ÷ÖÆÔÚMIMOÀ×´ïÕý½»FHÐźÅÖС£ÎÄÏ×[12]·ÅËÉÁ˲»Í¬FHÐźŵÄÕý½»ÐÔÒÔÔö¼ÓͨÐÅÊý¾ÝÁ¿ºÍÓÅ»¯Ì½Í¨Ò»Ì廯ÐÔÄÜ£¬²¢ÔÚÎÄÏ×[13]ÖÐͨ¹ý¿ÕÓòµ÷ÖƽøÒ»²½ÌáÉýÊý¾Ý´«µÝËÙÂÊ¡£ÎÄÏ×[14]Ìá³öÁËMIMOϵͳ¶àÔز¨PM-FMCW¼Ü¹¹¡£Í¨¹ýÑ¡ÔñÕóÁÐÌìÏß¡¢µ÷ÖÆÏàλ¡¢Ôز¨ºÍÌìÏß·¢ÉäÐźÅÖû»Ñ¡Ôñµ÷ÖÆͨÐÅÐÅÏ¢£¬Ôö¼ÓÁËͨÐÅÐÅÏ¢Á¿£¬ÇÒÀ×´ï·Ö±æÂʽӽüÓڵȿ׾¶¿í´øÀ×´ï¡£ÎÄÏ×[15][16]²ÉÓöà×ÓÂö³åOFDMÐźÅʵÏÖÀ×´ïͨÐÅÒ»Ì廯£¬²¢Ìá³öÁË»ùÓÚͨÐÅÐÅÏ¢²¹³¥ºÍ½âÏà¸É´¦ÀíµÄÄ¿±ê¾àÀëºÍËٶȳ¬·Ö±æ¹À¼ÆËã·¨¡£¶ÔÓÚµ¥ÔØƵMIMOÒ»Ì廯ϵͳ£¬²¿·ÖMIMO̽ͨһÌ廯ϵͳÀûÓÃÕóÁпռ伸ºÎ½á¹¹£¬Í¨¹ý¶àÕý½»ÐźŵÄÏßÐÔ¼ÓȨ½«Í¨ÐÅÐÅϢǶÈëÆäÅÔ°ê·½Ïòͼ·ù¶È»ò£¨ºÍ£©ÏàλÖС£ÆäÖУ¬ÎÄÏ×[17]¿ÉͬʱʵÏÖÖ÷¡¢ÅÔ°êPSKÏÂÐÐÁ´Â·Í¨ÐÅ£¬ÇÒÓиü¸ßµÄ¾«¶ÈºÍÊý¾ÝÂÊ£¬ÎÄÏ×[18]½øÒ»²½½éÉÜÁËÈçºÎÀûÓýÓÊÕ²¨ÊøÐγÉÆ÷½«ÉÏÐÐͨÐÅÐźŴӽÓÊÕ»ìºÏÐźÅÖзÖÀë¡£ÎÄÏ×[19]Ìá³ö²ÉÓÃÖû»¾ØÕó»ìÅÅÌìÏ߼䷢ÉäÐźÅ˳ÐòÒÔµ÷ÖÆͨÐÅÐÅÏ¢£¬Éè¼ÆÕý½»ÐźżÓȨ¾ØÕóÒÔʵÏÖÀíÏëÀ״ﲨÊøͼ£¬¸Ãµ÷ÖÆ·½·¨¶ÔÀ×´ï·½ÏòͼÐÔÄÜÎÞÓ°Ïì¡£ÎÄÏ×[20]Õë¶Ô·ÖÀëºÍ¹²ÏíʽÌìÏßÕóÁУ¬Ìá³öÁË»ùÓÚ·½Ïòͼģ°åÆ¥ÅäµÄÒ»Ì廯²¨Êø³ÉÐÍ·½·¨£¬±£Ö¤ÁËÏÂÐÐÓû§µÄПÉÔë±È£¨Signal to Interference plus Noise Ratio, 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aΪPSKµ÷Öƶ˵ã·Ö²¼ÏòÁ¿£©£¬ÇÒÎcb±íʾÆä·ù¶È¡£µ±Í¨ÐÅÓû§½ÓÊÕÐÅÔë±È£¨Signal-to-Noise Ratio, SNR£©Ô½Ð¡£¬ÆäÎó·ûÂÊ£¨Symbol Error Rate, SER£©Ô½´ó¡£È»¶ø£¬ÈôͨÐÅ·½Ïò²¨ÊøÄÜÁ¿¹ý¸ß£¬Óû§É豸Ôò»á³öÏÖ½ÓÊÕÐźŷù¶È¹ý±¥ºÍ¡£Òò´Ë£¬ÎªÂú×㲻ͬ¾àÀëͨÐÅÓû§½ÓÊÕÐźŹ¦ÂÊÐèÇó£¬ÏÞÖÆcb·ù¶È·¶Î§Îª, 1, 2, ,c c c l b c C u = £ £¡£´ËÍ⣬ΪÁË×èֹλÓÚ, 1, 2, ,eavh hE=¼µÄÇÔÌýÓû§½ØȡͨÐÅÐÅÏ¢µÄͬʱ²»Ó°Ïì·¢Éä·½Ïòͼ£¬Áî( )eavhsÖи÷ÔªËØÏàͬ£¬¼´( ) , 1, 2, ,eavh h Kr hE= = s 1¡£ÆäÖУ¬ÎhrΪ( )eavhsµÄÔªËØÖµ¡£Ê½(13)-(14)¿ÉÖØдΪ  ( ) 1, , 1, 2, ,H comc c c c c b l b u cC = £ = £ Awa   (17)  ( ) , 1, 2, ,H eav h h K r h E == A w 1   (18) ×îºó£¬¸ÃMIMO̽ͨһÌ廯ϵͳµÄ·½ÏòͼÉè¼ÆÎÊÌâ¿É±»±íʾΪ  ( )( ),,021max log( ( )), 1, 2, ,, 1, 2, ,, 1, 2, ,H com H eav s.t.hc rbccc c ch h KJb c Cl b c Cr h EuÎìïï== ï£= í£ï== ïïï =DîwwAwA1waw   (19) ÆäÖУ¬logº¯ÊýÊÇΪÁ˱ãÓÚ¶Ô·ÖʽĿ±êº¯Êý´¦Àí[24]£¬DΪ·¢ÉäÄÜÁ¿¡£ÎÊÌâ(19)Ö¼ÔÚÒÖÖÆ·½Ïòͼ·åÖµÅ԰꣬ͬʱ¿ØÖÆÃÜÂë±¾ÏàλºÍ·¢ÉäÄÜÁ¿¡£½ÓÏÂÀ´½«ÒýÈë»ùÓÚADMMµÄµü´úËã·¨Çó½â¸ÃÎÊÌâ¡£ 3.2 ÎÊÌâÇó½â Ê×ÏÈ£¬Í¨¹ýÒýÈ븨Öú±äÁ¿, ,iy,sz,cx,hvºÍn£¬ÎÊÌâ(19)¿É±»µÈ¼Ûת»¯Îª[24]  6   ( )( )( )( ),,,2,,,2112min log,,,,, , 1, 2, ,, 1, 2, ,, , 1, 2, ,,Hmain Hside H com H eav  s.t. icisrbi i i is s s sc c c cc c ch h h h Kb c Cl b c CEurhì-ïïï= ³ Î ïï= £ Îïí= = =ïï£= ïï= = =ïï= = Dïî£zwnyy A w yz A w zx A w xv A w v 1n w na   (20) Òò´Ë£¬Ôö¹ãÀ­¸ñÀÊÈÕº¯Êý±»¶¨ÒåΪ ( ) ( )( ) ( )( ) ( )22112 222 1211242 2 231, , , , , , , , , , , log222 22,HH H comH eavIi s c h i s c h i i i iiSCs s s s c c c cscEh h h hhL=== =æö = - + - + -ç÷ èø æö æö + - + - + - + -ç÷ç÷ èøèø æö + - + - + - + -ç÷ èøååå åρw y z x v μ κ ξ λ nζ y A w μ μz A w ι ι x A w ξ ξv A w λ λ nwζ ζ   (21) ÆäÖУ¬1 2 3 4[]T= ρΪ³Í·£Òò×ÓÏòÁ¿£», , ,, μ ι ξ ζ λ  i s c hΪ¶Ôż±äÁ¿¡£ ²ÉÓÃADMMËã·¨µü´ú×îС»¯( ) , , , , , , , , , , , ,ρ μ ι ξ λ ζi s c h i s c h L wy z x v nÒÔµü´ú¸üÐÂ, , , , , , , , , , ι μ ν ξ λ    i s c h i s c h wy z x¡£Áî( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), , , , , , , , , , , , μ ι ξ λ ζ    tt i s c i s ctt t t t t t t t t th h w y z n xv±íʾ, , , , , , , , , , , , μ ι ξ λ ζ    i s c h i s c h wy z x n vµÄµÚt´Îµü´úÖµ£¬¾ßÌåµü´úÁ÷³ÌÈçËã·¨1Ëùʾ¡£×¢Òâµ½Èç¹ûÎÞÇÔÌýÓû§£¬¼´ÎÞÔ¼Êø(14)£¬ÈÔ¿Éͨ¹ýÉÏÊö˼·£¬Áî30 =²¢ºöÂÔʽ(26)ºÍʽ(31)Çó½â¹ý³Ì£¬ÒÔ½â¾ö¼ò»¯µÄ0ÎÊÌ⡣ͨ¹ýËã·¨1µÃµ½¼ÓȨÏòÁ¿wºó£¬Ëæ¼´¿ÉµÃMIMOÒ»Ì廯ϵͳµ÷ÖÆͨÐÅÐÅÏ¢iPʱµÄ·¢ÉäÐźžØÕóiS¡£Ëã·¨1µÄÊÕÁ²ÐԲμûÎÄÏ×[24]¶¨Àí1¡£ Ëã·¨1   »ùÓÚADMMµÄÒ»Ì廯ÐźžØÕóÉè¼Æ·½·¨  Alg. 1    DFRC waveform matrix design method based on ADMM ÊäÈ룺{ } { } { }{ }{ }{ }{ }{ }( ) ( ) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0)100, , , , , , , , , , , , , , s c s c iihhD y z x v μ ι ξ λ n ζ£» Êä³ö£ºMIMOÒ»Ì廯ϵͳ¼ÓȨÏòÁ¿w£» 1. 0 t=£» 2.  Í¨¹ýÇó½âÒÔÏÂÎÊÌâ¸üÐÂ( ) ( ) { }( ) ( ) { }( ) ( ) { }( ) { }11 1 1 1 1 1, , , , , , ,++ + + + + + tt t t t t tsc i hw y z x v ( ) 1 + tn£º  { } { } ( { }{ }{ }{ }{ }( ) ( ))( ) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ): arg min , , , , , , , , , , , ,t t t t t t t t t t t ts c stc iihh L+ = ρww w y z x v μ ι ξ λ n ζ  (22)  { } { } { } ({ }{ }{ }{ }{ }( ) ( ))( 1) ( 1) ( 1) ( ) ( ),( ) ( ) ( )2, : arg min , , , ,, , , , , ,, 1, 2, , ;,,  s.t. it t t t tis it t tc s c i hhttiLiI+ ++=³=ρyy w y zxvμ ι ξ λ n ζy   (23)  7   { } { } { } ({ }{ }{ }{ }{ }( ) ( ))( 1) ( 1) ( 1) ( 1) ( 1),( ) ( ) ( )2, : a,rg min , , , ,, , , , , , ,, 1, 2, , ; s.t. st t t t tssit t tc s c i hhsttLsS+ + + + +=£=ρzz w y zxvμ ι ξ λ n ζz   (24)  { } { } { } ({ }{ }{ }{ }{ }( ) ( ))( 1) ( 1) ( 1) ( 1) ( 1) ( 1),( ) ( ) ( ) ( 1) ( ) ( )1, : arg min , , , ,, , , , , , ,, , 1, 2, , ;,  s.t. cct t t t t tc c s ibt t t t t tc s c i hhc c c cttcbLb l c u bC+ + + + + ++== £ = £ρxx w y zxv μ ι ξ λ n ζxa   (25)  { } { } { } ({ }{ }{ }{ }{ }( ) ( ))( 1) ( 1) ( 1) ( 1) ( 1) ( 1),( ) ( ) ( 1) ( 1) ( ) ( ), : arg min , , , ,, , , , , , ,, 1,,2, , s.t. hht t t t t ts i hh rtt t t t tc h s c i hhthKtrLr h E+ + + + + +++===ρvv w y zxvμ ι ξ λ n ζv1   (26)  ( ){ } { } ({ }{ }{ }{ }{ }( ))( 1) ( 1) ( 1) ( 1)( 1) ( ) ( )1( 1) ( 1) ( ) ( )2: arg min , , , ,, , , , , , , ,+s.t. t t t ts it t t t t t tc s c i hhttL+ + + ++ ++==Dρnn w y zxvμ ι ξ λ nζn   (27) 3.  Í¨¹ýÏÂÁй«Ê½¸üÐÂ{ }{ }{ }{ }( ) ( 1) ( 1) ( 1) ( 1 1 ), , , ,+ t ttsc i h++ μ ι ξ λ ζ£º  ( )( 1) ( ) ( 1) ( 1):H t t t ti i i i++ += + - μ μ y A w   (28)  ( )( 1) ( ) ( 1) ( 1):H t t t ts s s s+ + += + - ι ι z A w   (29)  ( )( 1) ( ) ( 1) ( 1):H com  t t t tc c c c+ + += + - ξ ξ x A w   (30)  ( )( 1) ( ) ( 1) ( 1):H eav  t t t th h h h++ += + - λ λ v A w   (31)  ( ) ( ) ( 1 ) 1 ) ( 1 + t tt t ++- =+ ζ ζ nw   (32) 4.  Èç¹ûÔ­ ʼ ¿É ÐÐ ÐÔ ÈÝ ²î( )2( 1) ( 1) ( 1)11HItt ti iiV++ +== - + åy A w ( )2( 1) ( 1)1HStt ss s++=-+ åz A w( )( 1)2( 1)1H com Ctt cc c++ =- + åx A w( )2( 1) ( 1)1H eav Et th hh+ +=- åv A w2( 1) ( 1)1tt++ + - £ nw£¬ÔòÊä³ö( ) 1 t+= ww£»·ñÔò:1 tt=+£¬»Øµ½²½Öè2¡£ £¨1£©¸üÐÂ( 1) + tw ÎÊÌâ(22)¿ÉµÈ¼Û±ä»»Îª  { } minHH-Âww Rw d w   (33) ÆäÖÐ  441()()1( ) ( ) ( ) ( )2( ) ( ) ( )22HH=+MMKM KMMMKMM´´Îδæö ç÷Îç÷ èøæö ç÷ =ç÷ èø=+ååB00BBaAaAIIR   (34)  ( ) ( )( ) 4( ) ( ) ( )ttÎ=++ å d A u n   (35)  8   123,, ( ) , ,,main  side com eav    Î ìï=ÎíïÎî   (36)  main side com eav,   ,,   ,(),   ,, +Î ìï+Î ï=í+Î ïï +Î îy μz ιux ξv λiisscchhfffff   (37) ÆäÖУ¬main side com eav =¡£Òò´Ë£¬¸ÃÎÊÌâµÄ±Õʽ½âΪ  ( 1) 1/2+-=tw R d   (38) £¨2£©¸üÐÂ( 1) ( 1) ( 1) ( 1), ,, + + + + t t t tis yz ºöÂÔÎ޹س£ÊýÏÎÊÌâ(23)ºÍÎÊÌâ(24)¿É·Ö±ðµÈ¼Ûת»»Îª[24]  2() 1,2min log2, 1, 2, ,  s.t. itiiiiI- + -³ = ¼yyyy   (39) ºÍ  2() 1,2min log2, 1, 2, ,  s.t. stssssS+-£ = ¼zzzz   (40) ÆäÖУ¬( )( ) ( 1) ( ) H t t ti i i+- = y A w μ,( )( ) ( 1) ( ) H t t ts s s+- = z A w ι¡£ Ê×ÏÈÕë¶ÔÎÊÌâ(39)£¬Ò»µ©»ñµÃ( 1) + t£¬( 1){}+ tiy¼´¿Éͨ¹ý()tiy·ÅËõµÃµ½£¬¼´  2( ) ( ) ( 1)()1()(()1),,+++ì>ï=íïïîÆäËût t tiititiittyyyyy   (41) È»ºó½«Ê½(41)·´´ú»Øʽ(39)£¬¼´¿ÉµÃµ½¹ØÓÚµÄÓÅ»¯ÎÊÌâ  1min ( ) f   (42) ÆäÖУ¬  ( )2() 111ˆ ( ) log( )2== - + - åItii if y   (43)  2()0,ˆ1,ì> ï=íïî ÆäËûtiiy   (44) ÏàËƵģ¬¸ø¶¨( 1) + t£¬( 1){}+ tsz¿Éͨ¹ý()tsz·ÅËõµÃµ½£¬¼´  ()( 1)2( ) ( 1)( 1)(()),,++ +ì>ï=íïïî ÆäËûtt stt stttssszzzzz   (45) ½«Ê½(45)·´´ú»Øʽ(40)¿ÉµÃ  2min ( ) f   (46) ÆäÖУ¬  9   ( )2() 121( ) log( )2== + - åStss sf z   (47)  2()0,1,ì£ ï=íïî ÆäËûtssz   (48) ×¢Òâµ½ÎÊÌâ(42)ºÍÎÊÌâ(46)¾ßÓÐÏàËƽṹ£¬¾ùΪÎÞÔ¼ÊøÓÅ»¯ÎÊÌ⣬¿Éͨ¹ýÎÄÏ×[24]Öз½·¨ÇóµÃÆä±Õʽ½â¡£ £¨3£©¸üÐÂ( 1) ( 1) ( 1) ( 1),, + + + +,t t t tcc hh br xv ÎÊÌâ(25)¿ÉÒÔÖØдΪ  2()1min, 1, 2, , s.t.ctcc bc c cbu l b c C-£=£ax   (49) ÆäÖУ¬( )( ) H com  ( 1) ( ) t t tc c c+=-q x A w ξ¡£×¢Ò⵽Ŀ±êº¯ÊýÊǹØÓÚcbµÄ¶þ´Îº¯Êý£¬Áî  { }()1(21)1H tctcb+Â=axa   (50) Ôò¸ÃÎÊÌâ±Õʽ½âΪ  ( 1) ( 1)( 1) ( 1)( 1),,,+++++ì £ï=<íï>ïî£tt c c c ctt c c c ctc c cb l bb l b lbuuu   (51) ÔòÓÐ( 1) ( 1)1++=tt ccb a x¡£ ÏàËƵأ¬ÎÊÌâ(26)¿ÉÖØдΪ  2()min -hthK hrr1v   (52) ÆäÖУ¬( )( ) ( 1) ( ) H eav  t t th hh+=- v A w λ¡£¸ÃÎÊÌâµÄ±Õʽ½âΪ  ()( 1)T tt K hhrK+=1v   (53) ÔòÓÐ( 1) ( 1) ++=ttK hhr 1 v¡£ £¨4£©¸üÐÂ( ) 1 + tn ÎÊÌâ(27)¿ÉµÈ¼Ûת»¯Îª  ( )22min=D-s.t.tnnnn   (54) ÆäÖÐ( ) ( ) ( ) 1 +=-ζt t tnw¡£Æä±Õʽ½âΪ  ( )( )=Dttnnn   (55) 3.3 ¼ÆË㸴ÔӶȺÍÊÕÁ²ÐÔ Ëã·¨1µÄ¼ÆË㸴ÔӶȷÖÎöÈçÏ£º1 -RºÍd¿ÉÔÚËã·¨¿ªÊ¼Ç°¼ÆËã²¢±£´æ£¬Æä¼ÆË㸴ÔӶȷֱðΪ2 2.373( ( ) ) + OQM KM[24]ºÍ() OQMK£¬ÆäÖÐ= + + + Q I S C EΪԼÊø¸öÊý¡£Ã¿´Îµü´úÖУ¬Ê½(22)-ʽ(31)¿ÉÓüòµ¥µÄ±Õʽ½âÀ´½â¾ö¡£Òò´Ë£¬Ö»ÐèÒª»ù±¾µÄ¾ØÕóµ½ÏòÁ¿µÄ³Ë·¨¡£¾ßÌåµÄ£¬Ê½(22)¡¢Ê½(23)-ʽ(26)¡¢Ê½(28)-ʽ(31)µÄ¼ÆË㸴ÔӶȷֱðΪ22() O K M,() O KQºÍ() O KMQ¡£ 10  Òò´Ë£¬Ëã·¨1µÄ×ܼÆË㸴ÔÓ¶ÈΪ2 2.373 2 20( ( ) ( )) + + + OQM KM T K M KMQ£¬ÆäÖÐ0TΪµü´ú´ÎÊý¡£ 4  »ùÓÚADPMµÄÅÅÐòѧϰ½âµ÷·½·¨ Õë¶ÔͨÐŽÓÊÕÐźŽâµ÷ÎÊÌ⣬±¾½ÚÌá³öÁË»ùÓÚADPMµÄÅÅÐòѧϰ½âµ÷·½·¨ÒÔÓÅ»¯Çó½âÖû»¾ØÕó¡£¼ÙÉèÿ¸öͨÐŽÓÊÕ»ú¶¼ÍêÈ«ÒÑÖªÕý½»²¨ÐξØÕóΦºÍ1a£¬½âµ÷¹ý³Ì¿É½¨Ä£ÎªÔ¤±àÂë¾ØÕóPµÄÓÅ»¯ÎÊÌâ[19]  ( )2minT com  s.t. icc-ÎPx P sP   (56) ÆäÖУ¬{ | ( , ) {0,1}, , }T  K K K Kij = Î = = P P P 1 1 1 P 1¶¨ÒåÁËÖû»¾ØÕ󼯺ϡ£ ×¢Òâµ½ÎÊÌâ(56)ÊÇÒ»¸ö»ìºÏ-²¼¶ûÓÅ»¯ÎÊÌ⣬ÎÄÏ×[19]ÖÐͨ¹ýÇî¾Ù·¨½â¾ö¸ÃÎÊÌ⣬Ȼ¶øµ±KºÜ´óʱÇî¾Ù·¨ÎÞ·¨Ó¦¶Ôά¶È±¬Õ¨¡£ÎªÊµÏÖ¿ìËÙÓÐЧÐÅÏ¢½âµ÷£¬Ê×ÏȽ«Ä¿±êº¯ÊýÖØвÎÊý»¯Îª  ( )22T com T ic c cbb - = - + x P s g p   (57) ÆäÖУ¬( ) { }22 12*, , vec( ) = + =Â Ä Î = Îcom i i K Kc c c bPx g x p P a¡£½øÒ»²½µØ£¬ÒýÈ븨Öú±äÁ¿212, ÎKuu£¬ÎÊÌâ(56)¿É±»µÈ¼Ûת»¯Îª  21122min{0,1}, 1, 2, ,T s.t. iKKp i K-Î=====pgpp uuuuB1pC1   (58) ÆäÖУ¬,TT K K K K = Ä = Ä B 1 I C I 1¡£Òò´Ë£¬Ôö¹ãÀ­¸ñÀÊÈÕº¯Êý¿É±»¶¨ÒåΪ  ( )22 12 1 2 1 2 1 2, , , ,22 T= - + - + + - +κ κp κ κ g p p u p u u u   (59) ÆäÖУ¬ºÍΪ¶Ôż±äÁ¿¡£ Áî1 2 1 2, , , , , κ κl l l l l lp u u±íʾµÚl´Îµü´ú1 2 1 2, , , , , κ κ p u uµÄÖµ£¬»ùÓÚADPMµÄÅÅÐòѧϰÓÅ»¯½âµ÷·½·¨Á÷³ÌÈçËã·¨2Ëùʾ¡£ADPMºÍADMMËã·¨µÄÖ÷ÒªÇø±ðÔÚÓڳͷ£Òò×Ó ÊÇ·ñËæµü´ú¸Ä±ä¡£ÓÉÓÚÎÊÌâ(60)ÊÇ»ìºÏ²¼¶ûÓÅ»¯ÎÊÌ⣬lpÒ×ÏÝÈë¾Ö²¿×îÓŽ⣬Èô²ÉÓÃADMMËã·¨Ôò´Ëºó¸÷ÔªËØÔÚµü´ú¹ý³ÌÖÐÎÞ¸üС£Òò´Ë²ÉÓÃÁËADPMËã·¨¿ò¼ÜÇó½â½âµ÷ÓÅ»¯ÎÊÌ⣬Áî10+-=  ll ppʱ£¬Ôö´ó³Í·£Òò×Ól£¬´Ó¶øÔöÇ¿Ô­±äÁ¿Ó븨Öú±äÁ¿µÄÔ¼Êø²¢¼ÓËÙÊÕÁ²¡£ Ëã·¨2   »ùÓÚADPMµÄÅÅÐòѧϰÓÅ»¯½âµ÷·½·¨ Alg. 2  The permutation learning demodulation method based on ADPM ÊäÈ룺0 0 0 0 0 01 2 1 2 2 3, , , , , , , , , , p κ κ g B C uu£» Êä³ö£ºÎÊÌâ(56)µÄ×îÓŽâp£» 1. 0 l =£» 2.  Í¨¹ýÇó½âÒÔÏÂÎÊÌâ¸üÐÂ11112 ,, +++ lll p uu£º  11   ( )11 2 1 22: arg min , , , ,{0,1}, 1, 2, , s.t. ll l l l li p i K+=Î=pu p κ u p κ   (60)  ( )111 1 1 2 1 21: arg min , , , ,  s.t. ll l l l lK++==up κ κBuuu 1u   (61)  ( )21 1 12 1 2 1 22: arg min , , , ,  s.t. ll l l l lK+ + +==uuCuup1u   (62) 3.  Í¨¹ýÏÂÁй«Ê½¸üÐÂ1 1 112,, l l l + + +κ κ£º  121,0,   l l lll++ì-= ï=íïî ÆäËûpp   (63)  ( )1 1 1 11 1 1:l l l l l + + + += + - κ u κ p   (64)  ( )1 1 1 12 2 2:l l l l l + + + += + - κ u κ p   (65) 4.  Èç¹ûԭʼ¿ÉÐÐÐÔÈݲî22 1 1 1 1 12 1 2 3 Vl l l l l + + + + += - + - £ u ppu£¬ÔòÊä³ö1 l += pp£»·ñÔò:1 ll=+£¬»Øµ½²½Öè2¡£ £¨1£©¸üÐÂ1 + lp ºöÂÔ³£ÊýÏÎÊÌâ(60)¿ÉÖØдΪ  12min ( ) s.t.  {0,1}, 1, 2, ,ifp i K Î=pp   (66) ÆäÖУ¬( )221 2 2,1()TTKl l l li i iif p g p== - = - å p p p g p£¬( ) 2 1 2 1 2 = + + + + κ κl l l l l lgg uu¡£ ¹Û²ìÎÊÌâ(66)¿ÉÖª£¬Ä¿±êº¯ÊýºÍÔ¼Êø¶ÔÓÚpÖи÷ÔªËØÏ໥¶ÀÁ¢£¬Ôòʽ(66)µÄ×îÓŽâΪ  0, 0.51,£ ì=íî ÆäËû   iipp   (67) ÆäÖУ¬2/ (2 ) =ll pg¡£ £¨2£©¸üÐÂ1112,++ ll uu ÎÊÌâ(61)¿É±»ÖØдΪ  121 111min+-+=κ s.t. lllKuuupB1   (68) ½«µÈʽԼÊø²¢ÈëÔö¹ãÀ­¸ñÀÊÈÕº¯ÊýÖУ¬¿ÉµÃ  ( ) ( )21 11 1 1 1 1 1,TllKlL+= - + + -κp υ B uu υ 1 u   (69) ÔòÓÉKKTÌõ¼þ¿ÉµÃÓÅ»¯ÎÊÌâ(68)µÄ±Õʽ½â  1 1 1 11 1/Tll l l lKllK+ + +æö = - + + - ç÷ èø κ BκBp u p B 1   (70) ÁíÒ»·½Ã棬ÎÊÌâ(62)¿ÉÒÔÖØдΪ  12   221 222min+-+=κ s.t. lllKuuupC1   (71) ÀàËƿɵÃÎÊÌâ(62)µÄ×îÓŽâΪ  1 1 T 1 22 2/ll l l lKllK+ + +æö = - + + - ç÷ èø rr κ CκCp u p C 1   (72) £¨3£©¼ÆË㸴ÔÓ¶È Ëã·¨2µÄ¼ÆË㸴ÔÓ¶ÈÓëµü´ú´ÎÊýÒÔ¼°PµÄ´óСÓйء£Ã¿´Îµü´úÖУ¬pºÍ¸üÐÂ1112,ll++ κ κËùÐè¼ÆË㸴ÔÓ¶ÈΪ2() OK£¬¸üÐÂ11+ luºÍ12+ luËùÐè¼ÆË㸴ÔÓ¶ÈΪ3() OK¡£Òò´Ë£¬Ã¿´Îµü´úËùÐè×ܼÆË㸴ÔÓ¶ÈΪ3() OK¡£ 5  ·ÂÕæʵÑé ±¾½ÚÖ÷Òª´Ó·¢É䲨Êø·½Ïòͼ¡¢ÐÇ×ùͼ¡¢SER ÒÔ¼°ÊÕÁ²ÐԵȷ½Ãæ¶ÔËùÌá³öµÄËã·¨¿ò¼Ü½øÐÐÐÔÄÜÆÀ¹À¡£¿¼ÂǾßÓÐ10 = M¸ö·¢ÉäÌìÏßÇÒ¼ä¸ôΪ°ë²¨³¤µÄ¾ùÔÈÏßÕóMIMO̽ͨһÌ廯ϵͳ£¬¿Õ¼ä½Ç¼ä¸ô¾ùÔÈ»®·ÖÀëÉ¢¸ñµã£¬[ 10 ,10 ]main°°=-¡£ÁîËã·¨1Í˳öÌõ¼þΪ8110-=´K£»³Í·£Òò×ÓΪ121 ==£»µ±1 E³Ê±£¬31 r =£¬·ñÔò30 =£»410 =¡£wµÄÄÜÁ¿ÉèΪD=M¡£ 5.1 µ¥Í¨ÐÅÓû§Ì½Í¨Ò»Ì廯ÐÔÄÜ µ±µ¥Í¨ÐÅÓû§²ÉÓÃPSKÃÜÂë±¾ÇÒ²»¿¼ÂÇͨÐŽÓÊÕÐźŷù¶ÈÔ¼Êøʱ£¬¿ÉÓÉ·¢Éä·½Ïòͼ²»±äºÍÑ¡Ôñ£¨Transmit Radiation Pattern Invariance and Selection, TRPIS£©·½·¨[19]ʵÏÖ¡£ÆäÖУ¬TRPISµÄ²¨ÊøÐγÉȨÏòÁ¿ÓÉÎÄÏ×[26]Éú³É£¬ÆäĸȨÏòÁ¿²ÉÓÃ×îС×î´ó×¼ÔòÓÅ»¯£¬ÅÔ°ê±£³ÖÔÚ20 dBÒÔÏ¡£ Òò´Ë£¬Ê×ÏÈ¿¼ÂÇÎÞÇÔÌýÓû§£¨0 = E£©Ê±£¬Ò»¸öͨÐŽÓÊÕ»ú£¨¼´1 = C£©Î»ÓÚ160°=-com£¬68 , 52 18 , 11 11 ,18tran° ° ° ° ° °é ù é ù é ù = - - - -ë û ë û ë û£¬90 , 69 51 , 19side ° ° ° °é ù é ù = - - - -ë û ë û19 , 90°° éù ëû¡£¼ÙÉèÒ»Ì廯ϵͳ̽²âÖ÷°ê·½ÏòÄÜÁ¿½Ï´ó£¬Í¨ÐÅÓû§·½ÏòËùÐèÂú×ãÄÜÁ¿Ô¼Êø²ÎÊýΪ110.1 £¬ 05 = 0 = . l u¡£Í¼2(a)-ͼ2(c)ÃèÊöÁË4,8,16 = KʱPSKºÍQAMµ÷ÖÆÏÂËùÌáADMMËã·¨£¨¼ò¼ÇΪPSK-ADMM,QAM-ADMM£©£¬ÒÔ¼°PSKµ÷ÖÆÏÂTRPISËã·¨£¨¼ò¼ÇΪPSK-TRPIS£©ËùµÃ·¢Éä·½Ïòͼ¡£ÆäÖУ¬Í¨ÐÅ·½ÏòµÄÄÜÁ¿ÉÏÏÞºÍÏÂÏÞ·Ö±ð²ÉÓÓ+”ºÍ“×”±ê¼Ç£¬²¢½«²»Í¬·½·¨ËùµÃPMSRÖµºÍÔËÐÐʱ¼ä·Ö±ð¼Ç¼ÓÚ±í1ºÍ±í2ÖС£TRPISËã·¨´Ó121--M¸ö¸´Êý¸ùÖÐÑ¡ÔñK¸ö£¬ÓëĸȨÏòÁ¿Ïà³Ë£¬ÐγÉÏàͬµÄ·½ÏòͼºÍͨÐÅ·½Ïò²»Í¬µÄÏàλ½Ç¡£Òò´Ë£¬TRPISÔÚ²»Í¬KÏÂËùµÃ·½ÏòͼÏàͬ£¬ÇÒʱ¼äÏà½ü¡£·ÂÕæ½á¹û±íÃ÷£¬²»Í¬KÏÂPSK-ADMMºÍQAM-ADMM¾ù¿É¿ØÖÆͨÐÅ·½Ïò·¢ÉäÄÜÁ¿ÇÒËùµÃ·½ÏòͼÏàËÆ£¬²¢¿ÉÔÚ²»É趨·½Ïòͼģ°åϺķÑÓëTRPISËã·¨Ïà½üʱ¼ä»ñµÃ¸ü¸ßµÄPMSR¡£  13  PSK-ADMM QAM-ADMM PSK-TRPIS ÏÂÏÞ ÉÏÏÞ(a) K=4 (b) K=8 (c) K=16 ͼ2. ¼ÙÉè160com °=-ʱ·¢Éä·½Ïòͼ¡£ Fig. 2.   The transmit beampattern with 160°=-com.  ±í1 ²»Í¬Ëã·¨ËùµÃPMSR Tab. 1 PMSR derived by different methods PMSR (dB)  PSK-ADMM  QAM-ADMM  PSK-TRPIS 4 = K  18.7693  18.7715 17.8392 8 = K  18.7693  18.7735 16 = K  18.7535  18.8135 ±í2 ²»Í¬Ëã·¨ËùÐèʱ¼ä Tab. 2 Computing time required by different methods Time (s)  PSK-ADMM  QAM-ADMM  PSK-TRPIS 4 = K  8.354  8.434  8.276 8 = K  5.502  5.182  8.281 16 = K  9.221  7.398  8.282 4,8,16 = KʱPSK-ADMM,QAM-ADMMºÍPSK-TRPISËùµÃÐÇ×ùͼ£¨¼´¼«×ø±êÏÂÃÜÂë±¾·ù¶È( ) 1comsËæÏàλ( ) ( ) 1argcomsµÄ·Ö²¼£©·Ö±ðÈçͼ3(a)-ͼ3(c)Ëùʾ¡£ÆäÖУ¬( )21comsÓëͼ2ÖÐ1()comPµÄ´óСһÖ¡£ÓÉÓÚPSK-ADMMºÍQAM-ADMMµÄÉè¼ÆÖоù¿¼ÂÇÁËÔ¼Êø(17)£¬Òò´ËËùµÃµÄ²Î¿¼×ÖµäΪPSKºÍQAM¶Ëµã·Ö²¼µÄ·ÅËõ£¬ÇÒÂú×ãͨÐÅ·½ÏòÄÜÁ¿Ô¼Êø1 1 1£ £ lbu¡£¶øPSK-TRPISËã·¨ÓÉÓÚ²ÉÓÃÁ˸´¸ùÑ¡Ôñ·½°¸£¬ÓëPSK-ADMMÏà±È£¬( ) 1coms¶Ëµã·Ö²¼ÂÔÓÐÆ«²î£¬ÇÒÎÞ·¨Áî( ) 1coms¸÷ÔªËØʵÏÖ²»Í¬·ù¶È£¬Òò´Ë²»ÄܲÉÓÃ8QAMºÍ16QAMµ÷ÖÆ¡£  14  (a) K=4(b) K=8(c) K=16PSK-ADMM QAM-ADMM PSK-TRPIS ͼ3. ͨÐÅÐÇ×ùͼ¡£ Fig. 3.  Communication constellation diagram.  ½ÓÏÂÀ´£¬Ëæ»ú·¢Éä410¸ö·ûºÅ¶ÔÓ¦Ò»Ì廯ÐźŲ¢ÆÀ¹ÀËùÌáË㷨ͨÐÅÐÔÄÜ¡£4,8,16 = Kʱ×ܵķûºÅ¸öÊý·Ö±ðΪ4!=24,8!=40320ºÍ16!>2^{13}£¬ÎÄÏ×[19]ÖÐËùÌáÇî¾Ù·¨ÒÔÇó½â½âµ÷ÎÊÌâ(56)ÔÚK½Ï´óʱÎÞ·¨Ó¦¶Ôά¶ÈÔÖÄÑÎÊÌ⣬Òò´ËÎÞ·¨ÊµÏÖÓÐЧ¿ìËÙ½âµ÷¡£¶¨ÒåµÚc¸öͨÐÅÓû§½ÓÊÕ¶ËSNRΪ22/ = SNRcccn b£¬±¾ÎÄËùÌáËã·¨1£¨PSK-ADMMºÍQAM-ADMM£©ºÍTRPISËã·¨¾ù²ÉÓñ¾ÎÄËùÌáËã·¨2 ÒÔ»Ö¸´Öû»Ë³Ðò£¬Ïà¹Ø²ÎÊýÉèΪ00.01 =,21.01 =ºÍ4310-=¡£²»Í¬Ò»Ì廯ÐźÅͨÐÅÓû§¶Ë·ûºÅSERËæSNRµÄ±ä»¯ÇúÏßÈçͼ4Ëùʾ¡£ÆäÖУ¬“KPSK”£¬K=4,8,16±ê¼ÇµÄºÚɫʵÏß±íʾÔÚ½ÓÊն˲ÉÓÃPSKÖ±½Ó½âµ÷(10)²¢»Ö¸´1aÖû»Ë³ÐòËùµÃSER½á¹û¡£ÕýÈçÔ¤ÆÚ£¬SNRµÄÔö¼ÓʹµÃSERÐÔÄܸÄÉÆ¡£Ëã·¨2ÀûÓÃÁËÖû»¾ØÕóÌØÐÔ£¬Í¨¹ý×î´ó»¯×Öµä1aÓëicPxµÄ»¥Ïà¹ØʵÏÖÁËSNRÔöÒ棬Òò´ËÏà±ÈPSKÖ±½Ó½âµ÷£¬ÔÚÏàͬSNRÏ¿ɵøüСSER£¬ÑéÖ¤ÁËËùÌáËã·¨2ÔÚÅÅÐòѧϰÓÅ»¯ÎÊÌâÉϵÄÓÐЧÐÔ¡£ÓÉÓÚPSKÐźÅÐÇ×ùͼºÍQAMÐźžØÐÎÐÇ×ùͼµã¼ä×îС¾àÀëÌØÐÔ£¬µ±K=4,8ʱ£¬PSK-ADMM,QAM-ADMMºÍPSK-TRPISËã·¨ËùµÃSERËæSNR±ä»¯ÇúÏßÏà½ü£»µ±K=16ʱ£¬QAM-ADMMÐÔÄÜÓÅÓÚPSK-ADMM£¬ÓÉÓÚTRPISËùµÃ²Î¿¼ÃÜÂë±¾Ïàλ¾ßÓÐÒ»¶¨Æ«²î£¬Òò´ËSER ¸ßÓÚPSK-ADMM¡£  Í¼4.   ²»Í¬Ëã·¨SERËæSNR±ä»¯ÇúÏß Fig. 4.  SER versus SNR for different algorithms ÒÔ4QAM-ADMMΪÀý£¬Í¼5Ãè»æÁËSNR=4 dBʱ²»Í¬½Ç¶È½âµ÷SER¡£ÓÉÓÚͨÐÅÐÅϢǶÈëÔÚ60°-ÖУ¬Òò´ËÔڸ÷½Ïò¾ßÓÐ×îµÍµÄÎóÂëÂÊ¡£Èçͼ6Ëùʾ£¬( ) 65°sÓë( ) 60°- sÔªËØ·Ö²¼£¨¼´( ) sËæ( ) ( ) arg s±ä»¯£©ÏàËÆ£¬Òò´Ë¸Ã·½Ïò½âµ÷Ò²¿ÉµÃ½ÏСÎóÂëÂÊ¡£Òò´Ë£¬Èô¸Ã·½Ïò´æÔÚÇÔÌýÓû§£¬Ôò¿ÉÔڽϴóÐÅÔë±ÈϽâµ÷µ÷ÖÆÐÅÏ¢¡£  15   Í¼5.   SERËæ½Ç¶È±ä»¯ÇúÏß Fig. 5.  SER versus angle  Í¼6. ( )65°sÓë( )60°- sÔªËØ·Ö²¼ Fig. 6.  The element distributions of ( )65°s and ( )60°- s ͼ7(a)չʾÁ˲»Í¬KÏÂËã·¨1·Ö±ð²ÉÓÃPSKºÍQAMµ÷ÖÆËùµÃԭʼ¿ÉÐÐÐÔÈݲî()1tVËæÔËÐÐʱ¼ä±ä»¯Çé¿ö¡£Í¼7(b)½øÒ»²½Ãè»æÁ˵±SNR=4 dBʱËã·¨2½âµ÷¸÷·¢ÉäÐźÅËùµÃ2lVËæÔËÐÐʱ¼ä±ä»¯ÇúÏß¡£·ÂÕæ½á¹û˵Ã÷±¾ÎÄËùÌáËã·¨¾ùÖð½¥ÊÕÁ²£¬Òò´ËÂú×ãÔ¼ÊøÌõ¼þ¡£ÆäÖУ¬Ëã·¨2ÔÚK=16ʱÔËÐÐʱ¼äСÓÚ0.25 s£¬µ±K=4,8ʱ£¬ÔËÐÐʱ¼äСÓÚ0.05  s£¬Òò´Ë¸ÃËã·¨ÊÊÓÃÓÚͨÐÅϵͳʵʱÐÔÐèÇó¡£     (a) ()1tVËæʱ¼ä±ä»¯                                                             (b) 2lVËæʱ¼ä±ä»¯ (a) ()1tV versus time                                                          (b) 2lV versus time ͼ7.   Ëã·¨1ÓëËã·¨2¼ÆË㸴ÔÓ¶È Fig. 7.  The computational complexities of Alg. 1 and Alg. 2  5.2 ÇÔÌý϶àͨÐÅÓû§Ì½Í¨Ò»Ì廯ÐÔÄÜ·ÖÎö  16  ¿¼ÂÇÇÔÌýÓû§Î»ÓÚ165°=eav£¬Í¨ÐŽÓÊÕ»ú·Ö±ðλÓÚ160°=-comºÍ240°=com£¬68 , 52 3 18 , 11 11 ,18 , 48 2 tran° ° ° ° ° ° ° ° é ù é ù é ù é ù = - - È - - È Èë û ë û ë û ë û£¬90 , 69side °° éù = - - Èëû 51 , 19°° éù-- ëû19 ,31 49 ,90° ° ° °é ù é ù ÈÈë û ë û¡£¼ÙÉèÁ½¸öͨÐÅÓû§·½ÏòÓëÒ»Ì廯ϵͳ¾àÀ벻ͬ£¬ËùÐèÂú×ãÄÜÁ¿Ô¼Êø²ÎÊý·Ö±ðΪ11 2 2 0.1, 2 , 0.05, = = = 1 = lluu¡£´Ëʱ£¬TRPISËã·¨ÎÞ·¨Í¬Ê±ÓÅ»¯·¢É䲨ÊøÐÔÄܺͿØÖÆͨÐźÍÇÔÌýÓû§²Î¿¼ÃÜÂë±¾¡£±¾½ÚÒÔK= 4ÏÂPSKµ÷ÖÆΪÀý£¬ÑéÖ¤ÁËËùÌáËã·¨µÄÓÐЧÐÔ¡£ ͼ8(a)-ͼ8(b)·Ö±ðÃè»æÁ˲ÉÓÃËã·¨1ËùµÃ·¢Éä·½ÏòͼºÍ( ) { } ,,Îcom eavce s·Ö²¼¡£ÓÉ·ÂÕæ½á¹û¿ÉµÃ£¬Í¼8(a)ÖÐPMSRΪ16.54 dB£¬Éè¼ÆµÄÒ»Ì廯¼ÓȨ¾ØÕó¿ÉʵÏÖ²»Í¬·½ÏòͨÐÅÓû§·¢É䲻ͬÄÜÁ¿Í¨ÐÅÐźŵÄͬʱÒÖÖÆ·½ÏòͼÅÔ°êµçƽ£¬´Ó¶øÂú×ãÁËÀ×´ï̽²âºÍ²»Í¬¾àÀëµÄͨÐÅÓû§SNRÐèÇó¡£ÓÉͼ8(b)¿ÉµÃ£¬Ëã·¨1¿É¾«È·¿ØÖÆÃÜÂë±¾( ) { } ,,Îcom eavce s·Ö²¼£¬ÒÔ×èÖ¹ÇÔÌý²¢ÔÚͨÐÅ·½ÏòʵÏÖ½âµ÷¡£     (a)·¢Éä·½Ïòͼ                                                          (b)( ) ( ) ¡¢com eavce ss·Ö²¼ (a) The transmit beampattern                        (b) The distribution of ( )comcs and ( )eaves. ͼ8. ̽ͨһÌ廯ÐÔÄÜ Fig. 8.  DFRC performance ͼ9չʾÁË1, 1, 2, =com eavcc·½Ïò²ÉÓÃËã·¨2µÄ½âµ÷SERËæSNR±ä»¯ÇúÏß¡£ÕýÈçÔ¤ÆÚ£¬ÔÚÏàͬµÄµ÷ÖÆ·Ö²¼1aºÍ½âµ÷·½·¨Ï£¬²»Í¬·½ÏòÓû§SERÐÔÄܼ¸ºõÏàͬ£¬¶øÓÉÓÚÇÔÌý·½ÏòÃÜÂë±¾¸÷ÔªËØÍêÈ«Ïàͬ£¬ÇÔÌý·½ÏòÎÞ·¨½âµ÷ͨÐÅÐÅÏ¢¡£  Í¼9. SERËæSNR±ä»¯ÇúÏß Fig. 9.  SER versus SNR 6  ½áÓï Õë¶Ô»ùÓÚÔ¤±àÂë¾ØÕóµ÷ÖƵÄMIMO̽ͨһÌ廯ϵͳ£¬±¾ÎÄÌá³öÁË»ùÓÚADMMµÄÒ»Ì廯ÐźžØÕóÓÅ»¯Éè¼Æ·½·¨ºÍ»ùÓÚADPMµÄÅÅÐòѧϰÓÅ»¯½âµ÷·½·¨¡£Ê×ÏÈ£¬½¨Á¢ÁËͨÐźÍÇÔ 17  ÌýÓû§·½ÏòÃÜÂë±¾Ô¼ÊøÏÂ×î´ó»¯·½ÏòͼPMSRÓÅ»¯ÎÊÌ⣻Ȼºó£¬ÒýÈëÁ˸¨Öú±äÁ¿½«¶þ´Î·ÖʽñîºÏÎÊÌâת»»Îª¶à¸öÓбÕʽ½âµÄ¶þ´ÎÓÅ»¯×ÓÎÊÌ⣬Ìá³öÁË»ùÓÚADMMµÄÒ»Ì廯¼ÓȨ¾ØÕóÓÅ»¯Éè¼Æ·½·¨²¢·ÖÎöÁ˸ÃËã·¨µÄ¼ÆË㸴ÔӶȺÍÊÕÁ²ÐÔ¡£×îºó£¬»ùÓÚͨÐÅ·¢Éäµ÷Öƽâµ÷»úÀí£¬½¨Á¢ÁËÅÅÐòѧϰ»ìºÏ²¼¶ûÓÅ»¯ÎÊÌ⣬Ìá³öÁË»ùÓÚADPMµÄÅÅÐòѧϰÓÅ»¯½âµ÷·½·¨ÌáÉýÁ˽âµ÷ЧÂÊ¡£·ÂÕæ½á¹ûÆÀ¹ÀÁËËùÌáÒ»Ì廯ÐźžØÕóÉè¼ÆºÍ½âµ÷·½·¨µÄÓÐЧÐÔ¡£ºóÐøÑо¿¹ý³Ì½«¿¼ÂÇ¿í´øMIMO[27 ,28]ºÍÐŵÀ¹À¼ÆÎó²îϵÄÎȽ¡MIMOÒ»Ì廯ϵͳ²¨ÐÎÉè¼Æ[29]£¬ÒÔ¼°·ÇÍêÈ«Õý½»²¨Ðμ¯´øÀ´µÄÐÔÄÜËðʧ[30]¡£ ²Î¿¼ÎÄÏ× [1]  Áõ·²,Ԭΰ½Ü,Ô­½øºê,µÈ. À×´ïͨÐÅƵÆ×¹²Ïí¼°Ò»Ì廯£º×ÛÊöÓëÕ¹Íû[J]. À×´ïѧ±¨, 2021, 10(3): 467-484. doi: 10.12000/JR20113. LIU  Fan,  YUAN  Weijie,  YUAN  Jinhong, et  al.  Radar-communication  spectrum  sharing andintegration:  Overview  and  prospect[J].  Journal  of  Radars,  2021,  10(3):  467–484.  doi: 10.12000/JR20113. 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