正交分頻多工系統的盲道估測
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(2) 正交分頻多工系統的盲道估測 Blind Channel Estimation for OFDM Systems 研 究 生：冉瑞華. Student：Jui-Hua Jan. 指導教授：林清安. Advisor：Ching-An Lin. 國 立 交 通 大 學 電 機 與 控 制 工 程 學 系 碩 士 論 文. A Thesis Submitted to Department of Electrical and Control Engineering College of Electrical Engineering and Computer Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in. Electrical and Control Engineering June 2004 Hsinchu, Taiwan, Republic of China. 中華民國九十三年六月.
(3) 正交分頻多工系統的盲道估測. 學生：冉瑞華. 指導教授： 林清安. 國立交通大學電機與控制工程學系﹙研究所﹚碩士班. 摘. 要. 我們提出一個以有限字元(finite alphabet)特性為基礎的 OFDM 系統盲道估測方法. 這種我們稱為比率方法的 OFDM 系統通道估測 是利用相鄰聲調(adjacent tone)之相位變化緩慢的特性來消除相位的 不確定性. 比較起其他現存的方法, 這個方法減少許多運算量的需求. 模擬顯示, 這個方法的準確度, 不管是通道估測誤差或位元錯誤率 (bit error rate), 均與現存的方法接近.. i.
(4) Blind Channel Estimation for OFDM Systems student：Jui-Hua Jan. Advisors：Dr. Ching-An Lin. Department﹙Institute﹚of Electrical and Control Engineering National Chiao Tung University. ABSTRACT. We propose a blind channel estimation method for OFDM systems based on finite alphabet property. The method called ratio is to make use of the property of slow phase change between two adjacent tones to cancel phase ambiguity. The proposed method needs less computation load compared with other existing methods. According to simulations, the proposed method has performances close to other existing ones on both channel estimation error and bit error rate.. ii.
(5) 誌. 謝. 除了我的父母外, 我要特別感謝林清安老師. 我從他 身上學到許多做研究的態度與方法, 我在交大的日子裡也 學習到不少東西, 我會永遠懷念這段在交大的日子. 我還 要感謝實驗室的學長同學及學弟, 他們幫了我許多忙, 讓 我能如期完成學業. 在求學的過程有太多的人幫助過我, 在此也一並致上萬分謝意.. iii.
(6) 目 中文提要 英文提要 誌謝 目錄 表目錄 圖目錄 一、 二、 2.1 2.2 2.3 2.4 三、 3.1 3.2 3.3 四、 4.1 4.2 4.3. 錄. 4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 五、 5.1 5.2 六、. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------簡介---------------------------------------------OFDM 通訊傳輸技術---------------------------------歷史----------------------------------------------OFDM 傳輸原理 : Cyclic Prefix 技術-----------------OFDM 傳輸原理 : Zero Padding 技術------------------應用與特性----------------------------------------Finite Alphabet 特性------------------------------定理與假設----------------------------------------J 推導H (ρk) ---------------------------------------PSK 及 QAM 調變的準確度---------------------------通道估測-----------------------------------------問題陳述------------------------------------------一些相關的方法------------------------------------Modified Minimum Distance - Phase Directed Algorithm(MMD-PD)---------------------------------Clustered SubCarrier Algorithm(CSC)---------------以比率(Ratio)的方法做通道估測---------------------FIR 系統的頻率響應--------------------------------比率(Ratio)方法分析-------------------------------估測演算法----------------------------------------兩群相鄰連續的聲調(Tone)或是多群的問題------------藉由選擇一群非相鄰連續的聲調(Tone)來化減計算量---平均方法------------------------------------------模擬與比較---------------------------------------估測與系統效能模擬--------------------------------計算量比較----------------------------------------結論----------------------------------------------. 參考文獻. -----------------------------------------------------------. iv. i ii iii iv v vi 1 2 2 4 11 13 15 15 16 17 21 21 21 21 25 29 29 30 32 35 38 40 42 42 52 54 55.
(7) 表 目 錄. 表 1.. 16 QAM 的參數分析-------------------------------- 20. 表 2.. 三種不同的方法在 FIR 1 通道上計算量的比較--------- 53. 表 3.. 三種不同的方法在 FIR 2 通道上計算量的比較--------- 53. 表 4.. 三種不同的方法在 FIR 3 通道上計算量的比較--------- 53. v.
(8) 圖 目 錄. 圖 1.. 基頻複數等效的 OFDM 傳輸模型---------------------- 4. 圖 2.. 一個 FIR 系統的極零點與頻率響應圖. M=64.---------- 29. 圖 3.. 一個 FIR 系統的極零點與頻率響應圖. 圖中增益響應圖. 的快速變化是由於位於單位圓附近的零點過多所造成.M=64.---- 35 圖 4.. FIR 通道極零點與頻率響應圖, 此圖為一個說明用的例. 子. L=4 and M=64.--------------------------------------- 36 圖 5.. 使用平均方法將通道估測誤差及相對應的 BER 降低.z=9.. 調變方法為 BPSK.---------------------------------------- 41 圖 6.. 使用平均方法將通道估測誤差及相對應的 BER 降低.z=9.. 調變方法為 QPSK.---------------------------------------- 41 圖 7.. FIR 1 通道的極零點與頻率響應圖. L=3, M=64.------- 43. 圖 8.. FIR 2 通道的極零點與頻率響應圖. L=4, M=64.------- 43. 圖 9.. FIR 3 通道的極零點與頻率響應圖. L=4, M=64.------- 44. 圖 10. FIR 1 的 NRMSE 與相對應的 BER 效能模擬圖. L=3,. M=64, BPSK.------------------------------------------- 44 圖 11. FIR 1 的 NRMSE 與相對應的 BER 效能模擬圖. L=3,. M=64, QPSK.------------------------------------------- 45 圖 12. FIR 2 的 NRMSE 與相對應的 BER 效能模擬圖. L=4,. M=64, BPSK.------------------------------------------- 45 圖 13. FIR 2 的 NRMSE 與相對應的 BER 效能模擬圖. L=4,. M=64, QPSK.------------------------------------------- 46 vi.
(9) 圖 14. FIR 3 的 NRMSE 與相對應的 BER 效能模擬圖. L=4,. M=64, BPSK.------------------------------------------- 46 圖 15. FIR 3 的 NRMSE 與相對應的 BER 效能模擬圖. L=4,. M=64, QPSK.------------------------------------------- 47 圖 16. FIR 1 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=3, M=64, BPSK.--------------------------------- 49 圖 17. FIR 1 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=3, M=64, QPSK.--------------------------------- 49 圖 18. FIR 2 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=4, M=64, BPSK.--------------------------------- 50 圖 19. FIR 2 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=4, M=64, QPSK.--------------------------------- 50 圖 20. FIR 3 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=4, M=64, BPSK.--------------------------------- 51 圖 21. FIR 3 的 NRMSE 與相對應的 BER 效能模擬圖, 對應不同 的 I. L=4, M=64, QPSK.--------------------------------- 51. vii.
(10) 1. Aé޺ʡV¿§xí à, «wuʦm£óÉíä jÞOéOíZ‰, ¦mß“ ¥V.iíÊ0§AÅ, U)VÖíòx¦mß¹øøí|ÛÊA éí0ä, Wà×ð UàíWÚu (Mobile Phones), ðê2àV©(,æ˜í Ýú˚bPàEc˜ (ADSL,Asymmetric Digital Subscriber Line) bWœ, CuÊìõCqàVÌ(,æíÌ(– 昵 (Wireless Local Area Network Card), J£wFß¹. 7/ÔuWÚu, ¥áß¹o˛AÑAA nÞº í”.Û¹”, 7xXEÍÊ.iíyhª¥ç2. ¥<òx¦mß¹Fàí¦mxÍ ó çÖ/õÆ, ç2|ÏÆ5øíò§Ì(¦mfxÍ˚Ñ£>}äÖ (OFDM,Orthogonal Frequency Division Multiplexing). ¥¦mjÞFàíxÍxóçßíÍ$^ ?, Ĥ¿§× íû˝D«n. /âk……™Fxeíò^?Dò§f?‰, ¥ áxX(V\SÑdÑ×– 4C Åðítì™Ä, Wàr¹íbP;ßÈ (DAB,Digital Audio Broadcasting) DbP dÈ (DVB,Digital Video Broadcasting ) ¹uà OFDM dÑf™Ä, C 1Åí IEEE 802.11 Í dìíÌ(– æ˜6uà¤f™Ä.. OFDM ¥fxÍÊä$,OóçòíU^0, Ĥ?¾àä Tò^?. Í7Ö Í… OrÖíiõ£@à, …Ê…™íÔ42OrÖ&j²í½æ, Wà-šä0R (carrier frequency offset), ×í¼MúÌMŠ0ª (large peak to average power ratio), °¥“ J£ ¦−,¿ ½æ ¥<½æo˛\˜ínDû˝, 6ĤrÖíj²5−\×ðT|. ʤ ¹d2, †uø½æÕ2ʦ−,¿jÞVd«n, 1/â¦−,¿ÏÏJ£Í$íPj˜Ï0 (BER,Bit Error Rate) í_ÒÇVn¦−,¿íß;Dóɽæ. ¦−,¿¥jÞí½æ˛ rÖíû˝Ê«n, …¹†u‡úÌåj (finite alphabet) ¤Ô4Vd¿píû˝}&.. 1.
(11) 2. OFDM ¦mfxX 2.1 vÍ OFDM ¥fxÍí–1oʇÿ\AT|, OuƒÛDívHn§ƒ×ðí·<£ «n, ¥uÄÑçvíZ;̶õÛAö£íÍ$, òƒÛÊ®áxXíA, àmUTÜ £ VLSI , U) OFDM ?\ö£íõÛ7“VUà. J-íH[|7×Ií OFDM ƪí¬˙ : • 1957 :Kineplex ql|Ö-šòäbWœ (multicarrier high frequency modem). • 1966 : Chang Ê Bell Lab. T|7 OFDM íd£¦)óÉù‚. • 1971 : Weinstein £ Ebert T|7Uà0§Z s² (FFT,Fast Fourier Transform) í xͣʈ–È (guard interval) VTÜ OFDM Í$íõÛDúG¯UÈß× (ISI,Intersymbol Interference) í½æ. • 1985 : Leonard J. Cimini T|7Êø_äbPW¦mí¦−2, à OFDM VT ÜÖ ˜fí^@Du°¦− (cochannel) ß×í à [4]. • 1987 : Alard D Lasalle T|7ÊbPÈ,Uà OFDM fxÍ. • 1995 : ETSI(European Telecommunication Standard Institute) SÑ7 OFDM dÑ bP;ßÈíf™Ä, 1/¥6uø_à OFDM fxÍÑ3í™Ä. • 1997 : bP dÈ\SÑAÑ™Ä. • 1998 : Magic Wand project ý7 OFDM àÊÌ(– æ˜íbWœ. • 1999 : OFDM \1ÅSÑAÑ@àÊÌ(– æ˜í™Ä, 7r¹6SÑdÑ™Ä. • 2000 : ²¾ OFDM(VOFDM,VectorOFDM) \T|dÑ ìíÌ(æ¦xÍ. • 2003 : IEEE 802.11g £ IEEE 802.16a sá™Äí„ì. 2.
(12) • 200X : IEEE 802.15.3 Öä OFDM(MultiBand OFDM), IEEE 802.11n £3G í Ì( ,Í $øÊ.˝íøV\T|.... 3.
(13) x (k). S/P. x (i). IF F T. s (i). CP A ddition. u (i). P/S. u (k). h (k) FIR C hannel order L y (i). y (k). P/S. z (i). FFT. CP Removal. v (i). v (k). S/P. A G N m (k). Ç 1: !äµb^íOFDM f_. 2.2 OFDM fŸÜ : =åí (Cyclic Prefix) xX. Ê«n OFDM Í$¦−,¿í½æ5‡, Bb.âblú OFDM fÍ$dø<Àí+3, Ç 1 uø_^í!ä OFDM fÍ$_, h(k) uÇ2 FIR ¦−í0§à@, 7¤ 0§à@í Z ²à-bç : H(z) =. L X. h(l)z −l. (1). l=0. w2Lu FIR ¦−í¼b. íl, BbzpmU x(k) %âå1 (serial-to-parallel) ‰ A-Þí :. . x(i) , . x(iM ) x(iM + 1) .. . x(iM + M − 1). . (2). M ×1. w2M ˚Ñ;| (tone) íbñ. QOBbú¤²¾mUd¥²×àZ s² (IDFT,Inverse Discrete Fourier Transform) í«7)ƒs(i) = F H x(i), w2F ˚Ñ×àZ s² (DFT,Dis-. 4.
(14) crete Fourier Transform) ä³, /ì2à F , . √1 e M. −j 2π(0)(0) M. .. .. ··· ... 2π(M −1)(0) √1 e−j M M. √1 e M. .. .. .. ···. −1) −j 2π(0)(M M. 2π(M −1)(M −1) √1 e−j M M. . (3). M ×M. 7H†uH[u 0 (Hermitian).. yVBbâ‹p CP(cyclic prefix) 7)ƒmUu(i); w2u(i)£s(i)íÉ[ªJâø_´åí ‹pä³ (CP addition matrix) V·H : u(i) , . u(iP ). u(iP + 1) .. . u(iP + P − 1). . O(P −M )×(2M −P ) = P ×1. . . |. IM. IP −M ·s(i) . {z CP addition matrix. (4). }. 6ÿuz, ‹p CP wõuøs(i) ²¾mU|(íP −M _¾‹pƒw²¾í|,j, 7/P −M > 0. Ê%¬1å (parallel-to-serial) í«(, øu(k)fƒ FIR ¦−, 7ʦ−í|« OÌMÑÉíµb‹A4íògÆm (AGN,Additive Gaussian Noise) m(k) í à. J,F· Hí FIR ¦−p£|íÉ[BbJ-bç [ý v(k) = h(k) ∗ u(k) + m(k) =. L X l=0. h(l)u(k − l) + m(k). (5). ¥³Bbcq0 < L ≤ P − M < M , wñíuÑ7Ê¢¯UÈß×. Ê%âå15(, Bb ªJ YƒmUv(i). ;W ä (5) F·Hí¦−_, BbªJ|-bç . v(iP ) = h(0)u(iP ) + h(1)u(iP − 1) + . . . + h(L)u(iP − L) + m(iP ) v(iP + 1) = h(0)u(iP + 1) + h(1)u(iP ) + . . . + h(L)u(iP + 1 − L) + m(iP + 1) v(iP + 2) = h(0)u(iP ) + h(1)u(iP − 1) + . . . + h(L)u(iP + 2 − L) + m(iP + 2) 5.
(15) .. . v(iP + P − 1) = h(0)u(iP + P − 1) + h(1)u(iP + P − 2) + . . . + h(L)u(iP + P − 1 − L) + m(iP + P − 1). µó. . v(i) , . 0 h(0) h(0) h(1) .. .. . . h(L − 1) h(L − 2) h(L) h(L − 1) 0 h(L) .. . 0 ... |. ... 0 ... .... v(iP + 1) v(iP + 2) .. . v(iP + P − 1). h(0) .... =. P ×1. .... 0. .... .... 0. 0 h(0). . 0. ... .... 0. h(0) . . .. 0. .... ... . h(L) {z. . .... ... ... ... v(iP ). .. .... . .... . h(0). · }. H0. 6. u(iP ) u(iP + 1). .. .. u(iP + P − 1). . P ×1. +.
(16) . 0. .... h(L). .... h(2). 0. .... 0. h(L). .... .. .. ... ... .. 0. .... 0. .... .. .. ... 0. .... |. 0. .. {z. H1. .. . h(1) u(iP − P ) h(2) u(iP − P + 1) .. . · h(L) .. 0 . .. . 0 u(iP − 1) }. . . P ×1. + . . m(iP ) m(iP + 1). .. .. m(iP + P − 1). . P ×1. , H0 u(i) + H1 u(i − 1) + m(i). QO, BbâÎ CP V)ƒz(i), wÉ[ªâø_=åíÎä³ (CP removal matrix) V·Hà- :. z(i) =. . OM ×(P −M ) IM {z | CP removal matrix. h(L) . . . h(0) .. ·v(i) = . } . . . h(L) . . . h(0) .. = . . . . h(L) . . . h(0). . . h(L) . . . h(0). M ×P. 7. · u(i) + q(i) M ×P. . . O(P −M )×(2M −P ) · . . IM. IP −M . P ×M. · s(i) + q(i).
(17) . . 0 h(0) h(1) h(0) . .. .. = . h(L) h(L) 0 . . . 0 0 {z |. h(1) .. . h(L) 0 ·s(i) + q(i) , Gc · s(i) + q(i) .. . 0 h(0) }. Gc. w2. q(i) ,. . OM ×(P −M ). . IM. M ×P. · m(i). 7Gc Bb˚Ñcì(circulant) ä³. , ç2íw2øá H1 u(i − 1) }¾¥, wŸÄuâkH1 ä³J£=åíÎä³íÔy!ZF¨A : . OM ×(P −M ) . . OM ×(P −M ). IM. . M ×P. · . IM. . M ×P. · H1 =. 0. .... h(L). .... h(2). 0. .... 0. h(L). .... .. .. ... ... .. 0. .... 0. .... .. .. ... 0. .... .. 0. .. . h(1) h(2) .. . h(L) 0 .. . 0. =0. (6). P ×P. |(, Bb‚à×àZ s²ä³«5(, ªJ)ƒmUy(i) , [y(iM ), y(iM + 1), ..., y(iM + M − 1)]T Ñ : y(i) = F · z(i) = F Gc s(i) + n(i) = Hx(i) + n(i) 8. (7).
(18) w2 s(i) = F H x(i) / . H , F Gc F H. H(e = . j 2π(0) M. . ) ... . H(ej. 2π(M −1) M. ). . (8). uø_úiä³ , 7 n(i) , F q(i) , ç2í H(ej. 2πk M. ) uâ ä (1) D z , ej. 2πk M. íÉ[ F)ƒí. 5(%â1åí«(,. ¹ª)ƒ|mUy(k).. bç (7) íÉ[uúF i ·A , ĤÑ7À–c, BbªJ ¥ i¥_Øù1/)ƒø _ÊLS i ·A í^ ä : y = Hx + n. (9). à‹Bbøx, y£n¥ú_²¾‹pøOíØùà x = [x(0), .., x(M − 1)]T , y = [y(0), ..., y(M − 1)]T £n = [n(0), ..., n(M − 1)]T , µó âkH ä³uø_úiä³, BbªJø ä (9) ZÑÞíbç :. . y(0) .. . y(M − 1). . . H(e = . 2π(0) j M. x(0) .. = . . . ) ... . H(ej. 2π(M −1) M. . ). 2π(0) j M. x(0) .. . x(M − 1). ) H(e .. . 2π(M −1) x(M − 1) H(ej M ). 9. . . + . n(0) .. . n(M − 1). . (10) . . + n , Xh + n . (11).
(19) w2. . . x(0) .. X, . 7. . 2π(0) j M. x(M − 1). ) H(e .. h, . 2π(M −1) H(ej M ). . (12). . √ = MV h . (13). w2 V , F ( : , 1 : L + 1) u*×àZ s²ä³F F}’|Víüä³, ú@ƒíuF ä³íF íJ£|‡ÞL + 1_W , 7. . . h(0) . h, .. h(L). (14). u FIR ¦−í0§à@²¾.. ʤBbú, (13) íÉ[ dø_Ì í}&. âk (1) /z = ej. H(e. j 2πk M. )=. L X. h(l)e−j. 2πk M. BbªJ)ƒ. 2πkl M. (15). l=0. Í(, BbªJUà ä (15) Vø hÇ7)ƒ-ÞíÉ[ : H(ej. 2π(0) M. ) = h(0) + h(1)e−j. 2π(0)(1) M. + h(2)e−j. 2π(0)(2) M. + · · · + h(JL)e−j. 2π(0)(L) M. H(ej. 2π(1) M. ) = h(0) + h(1)e−j. 2π(1)(1) M. + h(2)e−j. 2π(1)(2) M. + · · · + h(JL)e−j. 2π(1)(L) M. H(ej. 2π(2) M. ) = h(0) + h(1)e−j. 2π(2)(1) M. + h(2)e−j. 2π(2)(2) M. + · · · + h(JL)e−j. 2π(2)(L) M. .. . H(ej. 2π(M −1) M. ) = h(0) + h(1)e−j. 2π(M −1)(1) M. + h(2)e−j. 10. 2π(M −1)(2) M. + · · · + h(JL)e−j. 2π(M −1)(L) M.
(20) *-Þí äVõ, Bb¹ªJøFJU˝i FIR ¦−íä0à@YÕAø_²¾, Í(cÜAÞí²¾ä³W : . H(e. j 2π(0) M. H(ej H(ej. 2π(1) M. 2π(2) M. . ) ) ). .. . H(ej. 2π(M −1) M. ). . √ = M . √1 M. √1 M. √1 M. √1 M. 2π(1)(1) √1 e−j M M. 2π(1)(2) √1 e−j M M. √1 M. 2π(2)(1) √1 e−j M M. 2π(2)(2) √1 e−j M M. √1 M. ···. .. .. .. . 2π(M −1)(1) √1 e−j M M. =. √. √1 M. 2π(M −1)(2) √1 e−j M M. ··· ···. ···. 2π(1)(L) √1 e−j M M 2π(2)(L) √1 e−j M M. .. . 2π(M −1)(L) √1 e−j M M. M F ( : , 1 : L + 1)h. . . h(0) h(1) h(2) . .. h(L). (16). w2F u×àZ s²ä³ì2k (3).. âkÊ ä (10) C (11) ç2íf_!ZVõ, BbkuªJøc_ OFDM fÍ$í px(k)D|y(k)ŸA-bç : y(k) = H(ej. 2πk M. )x(k) + n(k). k ∈ [0, M − 1]. (17). w2n(k)u^í‹A4ògÆm.. 2.3 OFDM fxX : ^É (Zero Padding) xX. Î72.2 F Üí=åí OFDM fxÍ5Õ, OFDM ´øáéNíxÍ, ˚Ñ^É. í lBbø=åí‹pä³Z²A^Éä³ , 7 ä (4) ¹‰A : uz (i) =. . IM. . O(P −M )×M | {z } zero padding matrix. ·s(i). (18). w2í-™zu[ýUàíu^ÉíxÍ. Í(, FQYƒímUvz (i)ªJ\“Ñ : vz (i) , H0 uz (i) + H1 uz (i − 1) + m(i) = H0 uz (i) + m(i) 11. (19).
(21) w2H1 ¥øá}ÄÑwD^ÉíÔy!Z7¾¥ : H1 uz (i − 1) = . 0. .... h(L). .... h(2). 0. .... 0. h(L). .... .. .. ... ... .. 0. .... 0. .... .. .. ... 0. .... . h(1) h(2) .. . h(L) 0 .. . 0. .. 0. .. . · . . IM O(P −M )×M. . · s(i − 1) = 0. (20). P ×M. P ×P. Í7ÊQY«Bb6ÇÕUà7ø_QYä³, ˚½L‹pä³ (overlap-add matrix) V¦H ,Hí=åíÎä³ [1] : . zz (i) = IM |. . IP −M. ·vz (i) . (21). O(2M −P )×(P −M ) {z } overlap-add matrix. QO%â×àZ s²ä³í«(, BbªJ)ƒ|mUyz (i) : yz (i) = F · zz (i) = F · IM . =F · IM . IP −M. O(2M −P )×(P −M ). . IP −M. O(2M −P )×(P −M ) . · H0 · M ×P. 12. . · ( H0 uz (i) + m(i) ) M ×P. IM O(P −M )×M. . · F H x(i) + nz (i) P ×M. (22).
(22) = Hx(i) + nz (i). (23). 7ªJêÛ|mU ä (7) êrøš, w2H = F Gc F H 7 Gc u*-Þí äl| : . Gc = IM . . IP −M. O(2M −P )×(P −M ). . . · H0 · . IM O(P −M )×M. . (24) P ×M. M ×P. 7Æmnz (i)u*-Þí ä°) . nz (i) , F · IM . IP −M. O(2M −P )×(P −M ). . · m(i). (25). M ×P. ä (17) ˛éýOŸ…Êø_Ö˜¦−=1ç2í OFDM Í$, ˛\øÍím UTÜ xÍ, ²AM _Ö 1/Oî«¢Óï (flat fading gain) íÀ¦−_. wÓï¹Ñ H(ej. 2πk M. ). ¤Õ, øOVzBb·cqpmU x(k) DÆm m(k) uÖ ÌÉí, ¥¯¯øOí. UàÔ4Dòg.. 2.4 @àDÔ4. à°‡køÇáí Ü, OFDM Oóç˜í@à, …°vË(DÌ(í@à, 1À Üà- : • ú(í@àVz, àÛDððEEFUàíÝú˚bPàEc˜, ¹uUà OFDM íxX, OÄÖ‹7¦−|‹“mU 0í}º (bit loading), U)¦− Uà^0DøO OFDM ¢ .ó°, ¤xÍ˚5Ñ×àÖ;| (DMT,Discrete MultiTone). ´wFí@àW àÝ ò§bPàEc˜ (VDSL,Very High Speed Digital Subscriber Line) J£Š0(¦m (Power Line Communication,HomePNA 3.0). 13.
(23) • úÌ(í@àVz, àÛD|ÏÆíÌ(– æ˜, àkqCìõÌ(,æí@à, CuJ ,Ñ!í¦m઼ ,æ˜ (ACIS,Advanced Cellular Internet Service), 0¦ OFDM(FlashOFDM), ä OFDM(WOFDM,WidebandOFDM), J£ 4G ,Ì(¦ mx. OFDM OrÖíÔ4, ªàkÌ(ò§í¦mf. ….OOò^0íä$‚àxÍ, £ò §í’ ef, …´ËÄÑÍ$Ôy!Z7?ú}ý¯UÈß×D-šÈß× (ICI,InterCarrier Interference) í?‰. 1/Ê“ÂjÞ, ½æ*ø_” µÆí“Â, ‰ÑM _q퓽 æ. ´âk VLSI íxX, Uí×àZ s²í«ªJà0§Z s²V qíõÛ. Ö Í…Öáiõ, OÄÑ……™Í$íÔy!Z, …6OrÖí½æ&j². WàÊ° ¥“jÞx òÜ>, J£-šä0R¸vÈR (timing offset) íòÜ>½æ. …´ O×í¼Mú ÌMŠ0ªí½æ, J£¦−,¿í½æ. 7Ê…¹dç2, Bbɇú¦−,¿Vd«n. Bbu‡úÌåj (finite alphabet) íÔ4Vd¦−,¿í}&. 1/Êdıí|(, Bb}TX FT|íj¶, ʦ−,¿í4?íiÿõ, J£óú@íÍ$ Pj˜Ï0 (BER,Bit Error Rate) í^?}&.. 14.
(24) 3. Ìåj (Finite Alphabet) Ô4 Ê…¹dç2, BbFb«níu OFDM ¦−,¿J£Í$íPj˜Ï0^?, OFDM ¦− ,¿íj¶VÖ, BbʤJÌåjÑ3íj¶d}&. Í7bà¤Ô4d¦−,¿5‡, BbÛ blVj„ø-ÌåjíÔ4.. 3.1 ìÜDcq J-íRûîu W [1]D [2]7V, Í7Ñ7Ì £êc4íŸÄ, Bbʤdø<n£zp. ílBblds_cq.. cq 1: pmUx(k)u*ø_x”Ìõb”írÙÇ (signal constellation) 2 ²¦|Ví, cqrÙÇíõbÑQ, †x(k) ∈ {sq }Q q=1 . cq 2: pmUx(k)u” œ}Ì” Ë*rÙÇ2íõ²¦|V.. Êcq 1 5- âkrÙÇí×ü (size=Q) uÌí, ĤBbªJâÇ. QQ. q=1 [x(k). − sq ]. V)ƒ-Þíbç : xQ (k) + α1 xQ−1 (k) + · · · + αQ = 0 w2 α1 , ..., αQ uâ{sq }Q q=1. (26). Vl|. w2α1 , ..., αQ ¥<[b.ªJ°vÑÉ, ¥_Ô4ªJ. 'p éí* ä(26) õ|V. ĤBbªJ*¥<.rÑÉí[bç2vƒø_ÝÉí[b. QO, BbVì2ø_àíb, J :. ì2 : Bbì2Juα1 , ..., αQ ¥<[bç2ø_.ÑÉí[bíØù. *¥_ì2BbZªJ Rû|ø<à/í ä. càBbì2ø PSK írÙÇà{x(k) = ej. (2q+1)π Q. 15. |Q−1 q=0 }. (27).
(25) µó;W ä (26) D (27) íÉ[ BbZªJ)ƒ-Þíbç : xQ (k) + 1 = 0. (28). kuBbªJêÛ, Ébu¯¯ ä (27) í PSK rÙÇ, wJ.kQ, /αJ = αQ = 1 .Wà, π. 3π. ú BPSK Vz, âkrÙÇÑ{ej 2 , ej 2 }, Bb;W,Þín ¹ªJ)ƒJ = Q = 2 /α2 = 1. π. 3π. 5π. 7π. ú QPSK 7k, âkrÙÇÑ{ej 4 , ej 4 , ej 4 , ej 4 }, Ĥ BbZªJ)ƒJ = Q = 4/α4 = 1. QOV5?ø- QAM írÙÇ, BbêÛ, úkLSõbíj££å QAM rÙÇVz,Jî Ñ 4 /α4 6= 0. Wàú 16QAM rÙÇVz,J = 4/ α4 = 272. úk 64QAM rÙÇV z,J = 4/αJ = 17472. úk 32 õíå QAM rÙÇVz,J = 4/α4 = 608.. ÊÅ—cq 1 £ 2 ¸ ,ÞFì2íbåJ5-, BbªJ)ƒ-Þíbç [2] : E{xJ (k)} = −. J · αJ Q. (29). ¤bç uúLSrÙÇ·A í, w2 E{·} H[íu‚Mí«ä. Wàú ä (27) Fì2 í PSK Vz, BbªJ)ƒE{xJ (k)} = −1. à‹BbyÔ ·<à-, wõ¥_Ô4úk,Þ í PSK Vz, ¹U.5?cq2 6A , £* ä (27) BbªJ)ƒ xJ (k) = −1.Ĥcq 2 ú PSK Vz1Ì à. 7ƒÛÊÑ¢ínç2, BbªJ·<ƒ, båJ DQíÉ[îÑJ ≤ Q, Ý BçUàœ×írÙÇv, Bb†J QíÔ4, Wà 16QAM C 64QAM. ¥JükkC± ükQíÔ4ø}ÊQ-Ví¦−,¿í½æç2rÆí½bíËP.. 3.2 Rû H J (ρk ) BblV5? ä (17) í OFDM ¦−fÍ$_, bç à- : y(k) = H(ρk )x(k) + n(k) , k ∈ [0, M − 1] 16. (30).
(26) 2πk. w2Ñ7Z–cBbì2ej M , ρk . Bbcq¦−uÝÓœí (deterministic), /n(k)uÌ MÑÉíµbògc (circular) Æm. cíÔ4¹â-bç V[ý [1]: E{nm (k)} = 0 ∀ integer m > 0. (31). QOBbzy(k)© AÐJŸ1¦‚M, BbªJ)ƒ-Þíbç : E{y J (k)} = H J (ρk )E{xJ (k)}. (32). w2âkcíÔ4,E{n1(k)}, E{n2 (k)}, · · ·òŸáíÆmîÑÉ. à‹Bbz ä (29) Hp ä (32), BbZªJlH J (ρk ), bç à-Fý: H J (ρk ) = −. Q E{y J (k)} J · αJ. k ∈ [0, M − 1]. (33). ¤ uúÊÅ—cq1£2J£Jíì2-íLSrÙÇ·A . Bbø}Ê-Þíqñç2nF T|í¦−,¿j¶.. 3.3 PSK £ QAM |‰Äü. Ê,Þínç2, Bb˚H J (ρk )u* (33) F)ƒí. õÒ,, BbuàJ-š…Ìíj Vl : I−1. X ˆ J (ρk ) = − Q { 1 H y J (i, k)} J · αJ I i=0. k ∈ [0, M − 1]. (34). w2ˆuH[,¿Mí<2, 7y(i, k)†uH[ y(i) = [y(iM ), ..., y(iM + M − 1)]T ¥_²¾mUí k + 1_jÖ. ¥³íIuN²¾mUFf£íbñ. ypüíVz, çBbf£I_²¾mUx(i)v,. 17.
(27) Bb}ÊÍ$|)ƒI_²¾mUy(i). BbøYƒímU[ýà-: y(0). y(M ) y(M + 1) y(1) y(M + 2) y(2) .. .. . . y(M − 1) y(2M − 1) | {z }| {z y(0) y(1) . y(2M ) y(2M + 1) y(2M + 2) .. . y(3M − 1) }| {z y(2). ... }. |. y((I − 1)M ). y((I − 1)M + 1) y((I − 1)M + 2) .. .. . (35). y((I − 1)M + M − 1) {z } y(I−1). ˆ J (ρk ) íj¶¹à-Þí äFý, k ∈ [0, M − 1] : Ĥl H J J J J ˆ J (ρk ) = − Q { y (k) + y (M + k) + y (2M + k) + · · · + y ((I − 1)M + k) } H J · αJ I. (36). ˆ J (ρk )}Ä. * ä (36) *¥_ äBbªJ'ògínj, Jf£í²¾mUbñÖ, ,¿MH ˆ J (ρk ), 7¤bM}\Q-V í¦−,¿F‚à, V BbªJ·<ƒ, BbuàI_²¾mUVlH l|&,¡b. 6ÿuz, ¤ OFDM í FIR ¦−ÊI_²¾mUf£ç2u ì.‰í, 7ÊI_ ²¾mUˇ5ÈneÑuÓœí. Ĥƒ-ø I_²¾mUYƒ5(, ZyŸlñ‡Fú@í ¦−¡b.. BbÛÊVdø<n, /cq ä (30) íÆmn(k)ÑÉ. càBbUà BPSK |‰, Bb ˆ 2 (ρ0 ). UàI = 2V,¿ø_ ;|í,¿M, £H. Q µó;WH J (ρk ) = − J·α E{y J (k)} = J. Q − J·α H J (ρk )E{xJ (k)} Ébpm7x(k)uêrÌG}Ó (exactly equally likely, ideal conJ. ˆ 2(ρ0 ) = − 2 H 2 (ρ0 )E{x ˆ 2 (ρ0 )} = H 2 (ρ0 ), 6ÿuö£í dition) í8”5 -, H 2 (ρ0 )í,¿H 2·1 ˆ 2(ρ0 ) = H 2 (ρ0 ). M. Jx(k).uêrÌG}Óíu, âk ä (27) í!ZBb´uªJ)ƒH Ĥœ0Ô4ú PSK 7k.§LS à. °šËBbV}& QAM, càBbUà 16QAM, w ˆ 4 (ρ0 ). µó;W,H°ší 2dmin = 2[9]. à‹BbUàI = 16V,¿ø_;|í,¿M, ¹H. 18.
(28) bç , Ébpm7x(k)uêrÌG}Óí8”5 -, BbZªJ)ƒ-Þí!‹ : ˆ 4 (ρ0 ) = − 16 H 4 (ρ0 )E{x ˆ 4 (ρ0 )} H 4 · 272. X ˆ 4 (ρ0 )} = 1 { E{x 16. X. m={1,3} n={1,3}. (37). [(m+ni)4 +(m−ni)4 +(−m+ni)4 +(−m−ni)4 ]} = −68 (38). ˆ 4 (ρ0 )}uE{x4 (ρ0 )}í,¿M. ~·< ä (37) D (38), BbZªJêÛH ˆ 4(ρ0 )íMk w2 E{x ö£íM, ¹H 4 (ρ0 ).. ,Þí!‹ªJêÛçx(k)í} ÓuÌGv, †Ê;|k = 0v©ø_Ê 16QAM íõ·É} |ÛøŸ, à¤nª?)ƒ, í!‹. Oucàx(k)1.uÌG}Ó, †Ê QAM 8”-ÿ.}d PSK øš)ƒêr£üíM. WàcqI = 16/ à‹³ÌG}Óíu, Dö£M|üíÏÏZ} êÞÊø_õ.|Û7wF 15 õíLSøõ|Û 2 Ÿí8”. Wà,i+1 ¥õ³|ÛOu 3-i ¥õ| ÛsŸ. Ñ7}&jZ, BblTXø_¡5í[dÑbW¡5 :. 19.
(29) [ 1. 16QAM í¡b}& group I. group II. group III. group IV. sum = −16. sum = 112. sum = 112. sum = −1296. (1 + i)4 = −4. (1 + 3i)4 = 28 − 96i. (3 + i)4 = 28 + 96i. (3 + 3i)4 = −324. (1 − i)4 = −4. (1 − 3i)4 = 28 + 96i. (3 − i)4 = 28 − 96i. (3 − 3i)4 = −324. (−1 + i)4 = −4. (−1 + 3i)4 = 28 + 96i. (−3 + i)4 = 28 − 96i. (−3 + 3i)4 = −324. (−1 − i)4 = −4. (−1 − 3i)4 = 28 − 96i (−3 − i)4 = 28 + 96i (−3 − 3i)4 = −324. µóúLS;|Vzç1 + i¥õ³|Û73 − i¥õ|ÛsŸv, BbªJ)ƒ-Þíbç : ˆ 4 } = 1 {−12 + 112 + 112 + (28 − 96i) − 1296} = −66 − 6i E{x 16. (39). ĤF)ƒíMZ}D ä (38) íMF.°. 7ÊI ì5-, õb.ÌG¤ÏÏ}×. Bb zçI = 16vFÉ|Ûø_õbÏÏí¡bM·l|Và- : {−66 ± 6i, −88, −70 ± 6i, −68 ± 12i, −90 ± 6i, −48, −46 ± 6i}. (40). BbªJêÛÉbø_rÙÇíõ.ÌG, ÿª?ï×í˜Ï, /çI ìv, pmU.ÌG ÏÏ}×. âkÊ3.1ç2Bb˛cqpmU x(k)uÌG}Óí, ĤJ,í½æZ.}êÞ. J BbyŸ·<,Hí}&ªJêÛ, JpmUx(k) uÌG}Óíu, †Ê,¿í¥¶}ÏϹÑÉ. 7'péí, çÆmn(k).ucòg/.Ñ Év, ñøíÏÏÄZÉÆmí^@7.. 20.
(30) 4. ¦−,¿ 4.1 ½æÏH. ílÊn¦−,¿í½æ‡, BblVõõ-Þí½æ : [H(ρk ) · {ej w2 {ej. 2πn J. 2πn J. }]J = H J (ρk ). n ∈ [0, J − 1] for each k ∈ [0, M − 1]. J−1 u…™FíóP.üìÄä ĤBbªJ*, êÛ{ej }|n=0. 2πn J. (41). }|J−1 n=0 ¥J _ó. P.üìÄä ·ªJ H(ρk)«7)ƒó°íH J (ρk ), Ĥ¥_½æZu¤dıí½-, ú k OFDM í¦−,¿óç×í½b4. 7BbʤT|Bb3bín3æ: ú©_;|k7k, à SÊ¥J_.üìÄäç2âH J (ρk )V°)H(ρk ) ?. 4.2 ø<óÉíj¶. Ñ7bj²¤óP.üìí½æ, ñ‡˛ø<j¶\T|. Ê [1]ídıç2, Uà7rÖí¦− ,¿j¶, wÍ$^?î´.˜, Wà Modified Minimum Distance(MMD) ¸ MMD-PD(Phase Directed Algorithm). Ê [2]ídıç2, T6†uT|7ø_óçh ˛íj¶Vd¦−[bí, ¿, Zj²7óP.üìí½æ. ÊQ-Vídıç2, Bb }l‡ú¤s¹dıíj¶Vdn, 7(ynBbFT|íj¶. J-·u;W3 ıí cqd}&, 1/·Õ2Ê BPSK D QPSK íj¶, QAM í!†D PSK ó°. /Ê|(í_Ò ı, Bb´}TXú.°íj¶íÍ$ _Òªœ, J£ø<Ç.. 4.3 Modified Minimum Distance - Phase Directed Algorithm(MMD-PD). 21.
(31) Ê [1]¹dı2, T6Uà7óçÖíj¶, /F_Ò|Ví¦−ÏÏ£Pj˜Ï0^?· óç í.˜. w2ø_j¶˚Ñ MMD-PD. ¤j¶lul MMD í¦−,¿, Q6ZठMMDí ,¿VTLH«, ||(í,¿. BbÌ íV}&¥_j¶. Êd MMD ,¿í¼ ¨, ;W ä (41) : [H(ρk ) · λk ]J = H J (ρk ). k ∈ [0, M − 1]. (42). Bb˛øƒú©_;|k7k , Bbîø_óPí.üìÄä, /¥_Ääú@ƒJ _ª?4. Ä ¤, cq FIR í¦−¼bL˛ø, à‹Bb*M _;|ç2L<²ÏL + 1_õ VTl (õÒ, ÊM _õç2”ÌG”²ÏL + 1_õVd«}ªœÄü, ŸÄuÄÑ×àZ s² ä³íÔ4), µó;W ä (13) íbçÉ[ BbZªJ)ƒ-Þíbç : . J 1/J λα [H (ρα )] √ .. = M Rh . λβ [H J (ρβ )]1/J | {z } J L+1 possible choices. (43). w2˝iíò²¾J L+1¥óÖª?, ŸÄu©_;|·J_ª?, /ò²¾ÅÑL + 1. R† u*×àZ s²ä³F F}’|Víüä³, Fú@ƒíuF ä³Dò²¾ó°í;|P0, J £F 䳇ÞíL + 1_W. Bb5FJ²ÏL + 1_;|íŸÄ, uÄѦ−í¼b ÑL(6ÿu L + 1_[b), ĤBýÛbL + 1_j˙ Vl, Í7BbõÒ,ªJ UàyÖí;| (j˙ ) Vl, 7)ƒyÄüíM, ÉuàÖí;|VlB b}ÛbyÖí«¾, s65ÈÑø¦ Ÿ (trade-off). Ê¥ší8”5-, BbªJ êÛ, JJ L·ïüív` (ex:J = 2 and L = 2), BíÉÛbdø<ý¾íl, Í7JJ L·‰×ív` (ex:J = 4 and L = 6), Bbÿöíu ÛbI‘óç×í«¾.. QO, Bb;W ä (43), l|FJ L+1 ª?íh. Í7b²ìµø_nu BbFbí£ü íh, Bbâ-Þíj¶Vzp. íl*H(ρk ) =. 22. PL. l=0. h(l)e−j. 2πkl M. É[ |ê. càBbì2ø.
(32) _²¾c, c , h ∗J h = [c(0), · · · , c(JL)]T (i.e. c =. h ∗ · · · h} ), BbªJ)ƒ-Þíb | ∗ h{z number of h is J. ç : H J (ρk ) =. JL X. c(l)e−j. 2πkl M. (44). l=0. ,Þíbç uâk×àZ s²íÔ4FUÍ. …í<2¹uzpÊv dì÷ (convolution) í«, ªJ^Ñä0 dÀí ¶«. µóâ¥_É[ , BbªJø ä (44) Ç7 )ƒ-Þí : H J (ρ0 ) = c(0) + c(1)e−j. 2π(0)(1) M. + c(2)e−j. 2π(0)(2) M. + · · · + c(JL)e−j. 2π(0)(J L) M. H J (ρ1 ) = c(0) + c(1)e−j. 2π(1)(1) M. + c(2)e−j. 2π(1)(2) M. + · · · + c(JL)e−j. 2π(1)(J L) M. H J (ρ2 ) = c(0) + c(1)e−j. 2π(2)(1) M. + c(2)e−j. 2π(2)(2) M. + · · · + c(JL)e−j. 2π(2)(J L) M. .. . H J (ρM −1 ) = c(0) + c(1)e−j. 2π(M −1)(1) M. + c(2)e−j. 2π(M −1)(2) M. + · · · + c(JL)e−j. 2π(M −1)(J L) M. Í(Bbø˝iíMYÕAø_ò²¾, 1â“7ªJ)ƒ-Þí : . J. H (ρ0 ) H J (ρ1 ) H J (ρ2 ) .. . H J (ρM −1 ). . . √ = M . √1 M. √1 M. √1 M. √1 M. 2π(1)(1) √1 e−j M M. 2π(1)(2) √1 e−j M M. √1 M. 2π(2)(1) √1 e−j M M. 2π(2)(2) √1 e−j M M. .. . √1 M. ···. .. . 2π(M −1)(1) √1 e−j M M. ,. 2π(M −1)(2) √1 e−j M M. √. ··· ···. ···. MUc. √1 M 2π(1)(JL) √1 e−j M M 2π(2)(JL) √1 e−j M M. .. . 2π(M −1)(JL) √1 e−j M M. . . c(0) c(1) c(2) .. . c(JL). (45). w2U , F ( : , 1 : JL + 1)/cqJL < M , 7H J (ρk )ªJâ (33) äVl. ÊRû| ä (45) 5(, BbZªJà…Vl²¾cà- : . 1 H −1 H c = √ (U U ) U M 23. H J (ρ0 ) .. . H J (ρM −1 ). . (46).
(33) QOƒBbŸVí½æ, b²ìÊJ L+1 ¥óÖª?4ç2µø_nuBb ;bíh, Bbª Jâ-ÞퟆVl[1] : h = arg min kc − h ∗J hk h. (47). w2éñåhu[ý%â ä(47) F°)íM. ʤøõ.âbT|íu, ?DÅ — ä (47) í j.ñø, }J_M°vÅ—. ĤâkóP.üìíÔ4, ¥J_M·ªJ“VdÑ£ üí,¿ M. Bb5FJ}J_M°vÅ— ä (47), uĤóP.üìÄäíÔ4FUÍ. ĤBbªJÊ ¥J_Mç2L<²ø_M, Í(;W äh =. √. M V h Zª°) MMD í¦−,¿¡bh.. QO-V, ‚àF|íMMD ,¿VdLH«. BbठMMD í,¿Vçd- äí á,¿M1âLH«Tò,¿Äü [1] : λˆk = arg min |Hi (ρk ) − λk [H J (ρk )]1/J |2 λk. k ∈ [0, M − 1]. (48). w2iuLHíØù, Hi (ρk )un°)í MMD ,¿, ʤdÑáLHM. H J (ρk )uâ ä (33) Fl|í. Ì LH¥ à- :. Step 1 : UàFl|Ví MMD ,¿Vçd ä (48) íá,¿MHi(ρk ). Step 2 : QO|Fíλk , 1)ƒ-íbç : λˆ0 [H J (ρ0 )]1/J .. h= . λMˆ−1 [H J (ρM −1 )]1/J w2[H J (ρk )]1/J = |H J (ρk )|1/J · ej. ]H J (ρk ) J. .. 24. .
(34) Step 3 : yhv í¦−¡bp- : 1 h(i) , √ (V H V )−1 V H h M yyhä í¦−¡bà- : h(i + 1) ,. √. M V h(i) = V (V H V )−1 V H h(i) = V V H h(i). , ¤M ZAÑ-øŸíLH¡b. Step 4 : ½µJ,íí¥ òƒY¹, ¹ú/_'üí£õb, Å— kh(i + 1) − h(i)k ≤ . 7| (b·<íu, ¥šF|Ví,¿´uø_|(&+íóP.üìÄä. WàçJ = 2v, .ü ìÄä¹Ñ{1, −1}. JuçJ = 4v, .üìÄä¹Ñ{1, −1, i, −i}. 7¥|(íóP. üìÄä, ÿÉ?àtð;| (pilot tones) V^k [1].. *J,í}&Võ, BbªJêÛ, çL'×ív`, Êd MMD ílu.õÒí, ÄÑçL' ×ív`,J L+1M6ÿóúí0§Ó×, Ĥblíª?¡bíbñ‰íÝ×. ¥_«¾} Ó O¦−¼bLÓ×7Ó×½æBbªJâJ-øb Üíj¶Vj², ¤j¶˚Ñ CSC(Clustered SubCarrier)[3].. 4.4 Clustered SubCarrier Algorithm(CSC). Ê [3]¹dıç2, T6Uà7ø_óçh˛í–1D;¶, V0§í°),¿, UFÛbíl ¾××íòü, 1/O.˜í¦−,¿ÏÏ^?J£Pj˜Ï0^?. BbJ-V}&FFT |íj¶. íl, Bbl5?øˇó¹©/í;|, k, ..., l , 1/¥<;|Oœ×íÓïà@ (magnitude response). FFT|íj¶, w| ½bí–1uz, cqçH J (ρk )íM'Äüív`, Î7óP.üìÄä λk = {ej. 2πn J. J−1 }n=0 ´„øí8”-, H(ρk )íMwõªJ'Äüí\l|V :. H(ρk ) = λk |H J (ρk )|1/J · ej 25. ]H J (ρk ) J. (49).
(35) 7¥<þ„øíóP.üìÄä, úk©_;|Vz·Jª?. Bb6˛ø−, úJ = 2ív`, óP.üìÄä¹Ñ{1, −1}. 7J = 4ívv`, .üìÄäÑ{1, −1, i, −i}. µó , Bbÿ²Ïw 2í/ø_.üìÄä (Wàús8”Vz, Bb·²Ï 1), Í(Zª JdÀ/0§íLH«.. püøõVz, ílBbÊ;|k,²Ï7”1”¥_óPÄä, ¹H(ρk ) = 1·|H J (ρk )|1/J ·ej 7 λk = 1.µóBbªJUà-íLH ä, w2[H J (ρk )]1/J = |H J (ρk )|1/J · ej λk+1 = arg min |H(ρk ) − λ[H J (ρk+1 )]1/J | , λ. λk+2 = arg min |H(ρk+1 ) − λ[H J (ρk+2 )]1/J | , λ. ]H J (ρk ) J. ]H J (ρk ) J. :. H(ρk+1 ) = λk+1 [H J (ρk+1 )]1/J H(ρk+2 ) = λk+2 [H J (ρk+2 )]1/J. .. . λl = arg min |H(ρl−1 ) − λ[H J (ρl )]1/J | , λ. H(ρl ) = λl [H J (ρl )]1/J. âkBbÊ;|k,F²íóP.üìÄäu”1”, 7J,íLH¢uóúkH(ρk)FT íLH, Ĥ Bb}Ê¥<FLH|VíMç2, O ;|køšíóPÔ4. BbÊJ, íLH äç2, U à7|ü (min,minimum) ퟆ, wŸÄuÄÑúk²¾hLSó¹ ís_ä [bVz, …bí bMøì}'Q¡. ¥uâk×àZ s²ä³«(F¨AíÔ4.. QOBbZªJYÕnFLH|Ví,¿MH(ρk ), ..., H(ρl ), ø…b§Aøò²¾(, ÇÕ yøFí.üìÄä#‹, , ĤªJ)ƒ-Þí ä : 1 · H(ρk ) 1 · H(ρ ) k+1 .. . 1 · H(ρl ). −1 · H(ρk ) −1 · H(ρ ) k+1 and .. . −1 · H(ρl ). 26. for J = 2 and . (50). ,.
(36) . 1 · H(ρk ) 1 · H(ρ ) k+1 .. . 1 · H(ρl ). . . −1 · H(ρk ) −1 · H(ρ ) k+1 and .. . −1 · H(ρl ). µó*h = [H(ρ0 ), · · · , H(ρM −1 )]T =. . . i · H(ρk ) i · H(ρ ) k+1 and .. . i · H(ρl ) √. . . −i · H(ρk ) −i · H(ρ ) k+1 and .. . −i · H(ρl ). . for J = 4 (51). M V h¥_ äíÉ[ BbZªJ°)FJóú@. ív ¦−0§à@²¾. 7¥<v í²¾, Bb¹ªJ*h =. √. M V híÉ[ V° )ä í. ¦−,¿¡b. ~R<, J,í}&ubÊç|H J (ρk )|1/J , ..., |H J (ρl )|1/J ¥<M·ï×ív`l |VíMnÄü [2]. 7Ê¥øˇó¹©/í;|í8”5-, ¥<Fl|VíJ_hîª JdÑ £üí¦−,¿.. cqBb.Éøˇó¹©/í;|µóBbEͪJ‚à°ší–1V,¿¦−. 5?cà B bsˇó¹©/í;|, /¥<;|îOï×íÓïà@ : k, ..., l ¸m, ..., n. cqBbì2Þí ä :. . H(ρk ) .. H1 , . H(ρl ). . . H(ρm ) .. and H2 , . H(ρn ). . (52). w2H1 DH2 uâ,HÉ5?øˇ;|íj¶Fl|Ví, 7-™íbåH[íu;| ˇíØù. µóBbªJ‚à ä (52), øFóP.üìÄäË‹,H1DH2 , 1øs6¯9A ø_œÅíò ²¾7)ƒ-í ä : . . . . 1 · H1 1 · H1 −1 · H1 −1 · H1 , , , forJ = 2 1 · H2 −1 · H2 1 · H2 −1 · H2. 27. (53).
(37) . . 1 · H1 1 · H1 , −1 · H2 1 · H2 . . . . 1 · H1 1 · H1 , , −i · H2 i · H2 . −1 · H1 −1 · H1 −1 · H1 −1 · H1 , , , −i · H2 i · H2 −1 · H2 1 · H2 . . . . . i · H1 i · H1 i · H1 i · H1 , , , −i · H2 i · H2 −1 · H2 1 · H2 . . . . . −i · H1 −i · H1 −i · H1 −i · H1 forJ = 4 , , , −i · H2 i · H2 −1 · H2 1 · H2 Í(BbZªJyøŸíUàh =. √. (54). M V híÉ[ V l|FJóú@íh. ¥³Bbø}J g. ª?í¦−,¿h, w2íguH[O;|ˇíbñ. Ñ7bÊ¥<ª?í,¿ç2vƒ£üí,¿M, Bb´ u‚à7 ä (47) íj¶Vl. BbêÛ, Å— ä (47) í¦−hEÍJª?M. Ä ¤BbZÓ<íÊ¥Jª?4-²¦ø_,¿, Í(y‚à ä (13) Vl|óú@íä í, ¿h. ʤ´ubR<øõ, ¥š|VíMEÍOø_c ñíóP.üìÄä, 7¥_Ääí +É?âtð;|V®A.. BbªJ*,Þí}&êÛ, CSC ¥_j¶TX7óçÀ/0§íƶV°),¿M, 1 / .}d MMD íj¶øš, OøO¦−¼bLÓ‹, l¾ÿ} O0§Ó‹íÔ4, Uí CSCí j¶0§/Z. l¾¥øjÞ, ÄÑ MMD ÛbJ L+1 íª?4í«¾, 7 CSC É ÛbJ g í «¾, â¤ZªJnj«¾íÏæ. Í7, c CSC O.˜íiõ, …´u<&Z¾5T. ÄÑBbêÛ, ÊÉøˇó¹©/í;|v, Ñ7ı?°)œ Äüí,¿M, Ê5?M = 64v [8], ¦Bb×Ûb 40 _;|n?®ƒ,¿Mh Äüíñí. 7ç;|ˇíbñÖv, ;|íb. 28.
(38) ñ6}.iÓ‹, ĤӋ7ä³í&( Z}Ó‹óú@í«¾.. Ê-Þíı³, BbZVnFT|íj¶, V«n¤j¶íŸÜ£Ô4. BbFT|íj¶Ê –1, wFsj¶éN, Ou?Dº¯ø<ƶí^ZVy‹Ë±Q CSCí«¾.. 4.5 Jª0 (Ratio) íj¶d¦−,¿ 4.5.1 FIR Í$íä0à@. 1.4. 1. 1.2 0.8. 1 k’. 0.6. 0.8 0.6. 0.4. Imaginary Part. d4. θ2. 0.2 4. 0. 4. −0.2. 0.4. d. 2. 1. 0.2 d. 0. θ1. 3. θo. θ. d. 0. 10. 20. 30. 40. 50. 60. 70. 10. 20. 30. 40. 50. 60. 70. θ3. 0. −0.4. -100. −0.6. -200. −0.8. -300. −1. -400 −1. −0.5. 0 Real Part. 0.5. 1. -500 0. Ç 2: ø_FIR Í$í”ÉõDä0à@Ç. M =64.. ÊÇá ÜBbíj¶5‡, BblV+3ø- FIR Í$í×ü£²Pà@íÔ4. íl¡5Ç 2, ÇÑø FIR Í$í z Þ”ÉõÇD×ü£óPà@Ç. ·<úF¼b ×kÉí FIR Í$, w”õ.ìÕ2ÊŸõ. µó;WÇ 2, BbªJ)ƒ-Þí!‹:. !‹ 1 : *Ç2, úk/_;|k 0 7k, H(ej Q4. i=1. di , 7wóPà@[ýÑ ]H(ρk0 ) =. 2πk0 M. P4. j=1 θj. 29. ) = H(ρk0 ) í×üíà@[ýÑ|H(ρk0 )| =. − 4 · θo . w2θo uH[”õíóPi. µóBb.
(39) ÿªJ*¥<}&)ƒ FIR Í$í×ü£óPà@à- : |H(ρk0 )| = ]H(ρk0 ) =. L Y. i=1 L X i=1. di. (55). θi − L · θ o. (56). w2θo u”õíóPi, 7Lu FIR ¦−í¼b. di u;|k 0 DÉõí ×, 7θi uw óú@íÉõ óPi.. !‹ 2 : *Ç2BbªJêÛ, çFIR Í$í×üà@×ív`, wFú@íóPà@íó P‰ “ªœîM, Í7Ju×üà@ªœüív`, wFú@íóPà@ÿª?}ø_éNó P.©/, CóP” &” íÔ4. ĤBbÑ7fǤóP.©/íÔ4, BbZz·<‰Õ 2ƒ×íÓï Pàí¶M. ;W¥<O×íÓïà@í;|, 1‚àwó¹íóPà@íóPî ‰“, Bbı ?‚à¥_ÔõV,wFíóPà@M, JZªJøFíM´Ÿ, )ƒ|(í, ¿. ĤBbZ 7ú_!‹.. !‹ 3 : Bb²Ïµ<OòÓïà@í;|Vdl, ¹×í|H J (ρk )|1/J , ¥uÄÑ…bO îMíóP‰“. ‚àó¹s_;|5ÈóPîM‰íÔ4VøFíóPøø´Ÿ.. 4.5.2 ª0 (Ratio) j¶}&. Q-VBbVnàS‚àóP5ÈíÉ[Vd¦−,¿. BbT|-Þíj¶. Bbì2ª0∆Hk ,. H(ρk+1 ) , H(ρk ). w2k ∈ [0, M − 2]. µó¹ªJ)ƒ-Þí ä : J. ∆HkJ = |∆HkJ | · ej(]∆Hk ) = |∆HkJ | · ej(J· ]∆Hk ) 30. (57).
(40) w2]∆Hk = ]H(ρk+1 ) − ]H(ρk ) .7∆Hk íóPOJª?4, ?¹Ñ-Þí ä : ]∆Hk +. 2mπ J. m ∈ [0, J − 1]. (58). ¥³Bbc5?óPí3M, ¹[−π, π]íM. µóBbÿªJâu´∆Hk íóPi?DÅ—-Þ íbç Vv|¥J_óPMç2¨_nu£üí : −. π 2mπ π < ]∆Hk + < J J J. (59). w2J =2 C 4 . ¥_Ÿ†u;Wó¹ís_;|íóPOîM퉓ícqFRû|Ví, 7 JóP‰¬×†.?Å—, . 7*,Þ)ní)ø, óP}Ê×Óïà@íË¡OîMí ‰ “. Ĥ²Ïµ<O×Óïà@í;|Vd,Þíln.}|˜. Ĥ, Êâ ä (59) ° ) H(ρk+1 )DH(ρk )5È£üíóPÏM5(, J]H(ρk )íM#ì , Bb¹ªâ-Þí äV° )H(ρk+1 ) : ]H(ρk+1 ) = ]∆Hk + ]H(ρk ). (60). w2]∆Hk uH[H(ρk+1)DH(ρk )5È, Ê Jª?4ç2£üíóPÏM. ~·<, ÖÍ]H(ρk )M ucq#ì˛ø, Ouõ Ò,BbÉ?ø−]H(ρk )MíJª?4ç2íw2ø_, ÄÑ* H J (ρk )M bR H(ρk )MŸ…ÿOø_xJª?4íóP.üìÄä [1]. 7/y½bíu, Ê¥<J ª?4ç2,]∆Hk Mîu ì .‰í. ĤBbZªJâ ä (60) V°)]H(ρk+1 ), 7¤°)í MD ]H(ρk )Oó°íóPÔ4. YÎ¥j¶, BbªJøøíø ]H(ρk+2 ) ]H(ρk+3 )... ¥ <M#l|V, 1/· ]H(ρk )íMó°íóPÔ4. âkú©_;|kVz, ¦−í× üà@ ªJ'üìÌOËâ- äl|V : |H(ρk )| = |H J (ρk )|1/J. (61). BbZªJø¥<ÓïD,ÞFlíóP¯9, AÑø_êcíà@¡b, 1ø¡b§Aøò². 31.
(41) ¾à- :. . H(ρk ) H(ρ ) k+1 .. . H(ρk+m ). . . J. 1/J. j]H(ρk ). |H (ρk )| · e |H J (ρ )|1/J · ej]H(ρk+1 ) k+1 = .. . |H J (ρk+m )|1/J · ej]H(ρk+m ). . (62). (m+1)×1. w2k, ..., k + muBbcìF²Ïí;|. 7¥šíbç , BbZªJ‚à ä (13) íÉ[ Vl|h : . |H J (ρk )|1/J · ej]H(ρk ) |H J (ρ )|1/J · ej]H(ρk+1 ) k+1 1 H −1 H h = √ (K K) K .. M . |H J (ρk+m )|1/J · ej]H(ρk+m ) w2. . |H J (ρk )|1/J · ej]H(ρk ) |H J (ρ )|1/J · ej]H(ρk+l ) k+1 .. . |H J (ρk+m )|1/J · ej]H(ρk+m ). . (63). (m+1)×1. . √ = M Kh . (64). /K , V (k : k + m , : ). ¥³Bbcq¦−í¼bL˛ø. |(Bbâ ä (13) ZªJ|| (FÛbí¦−ä ²¾,¿h. Í7yøŸ í·<ƒ, ¥_,¿´uø_„øíóP.üìÄä &tð;|Vj². 7Bbí,¿j¶ƒ¤ZªJ!!.. 4.5.3 ,¿Æ¶. Ê-Þíqñç2, BbbVÌ zpFT|j¶íÆ¥ , J£ø<óÉí½æ.. 32.
(42) Step 1. š…Ì. ílBbÛø ä (33) íH J (ρk )Ml|V. bl¥_¡b, BbSàš…Ìíj¶, ä à- : I−1. X ˆ J (ρk ) = − Q { 1 H y J (i, k)} J · αJ I i=0. k ∈ [0, M − 1]. (65). 'ògí* ä (65) Võ, ÉbBbàÖíf£²¾mUbñI, BbøìªJíƒyÄü í,¿ ˆ J (ρk ). MH. Step 2. ²Ï×ÓïM |H(ρk )|. -ø¥BbÛb©v_çí;|Vdl. Í7;|í©vj ˛Êl‡dı2í!‹ 3 Fn , ¹u‘²xœ×Óïà@í;|Vdl : ˆ k )| = |H ˆ J (ρk )|1/J |H(ρ. (66). Í(BbÉÛz·<‰[Ê¥<;|,ÞÿW7. yVZªW-Þíl.. Step 3. øˇó¹©;|íÌóP.üì4. cqBbøˇó¹©/í;|, k 0 , ..., k 0 +m, 1/¥<;|Oï×íÓïà@. íl, Bbl ˆ k0 ). ~·<, BbÉÛbÓZ²Ïw2íø _.üìÄ乪, Í(‚à ä (57) (59) l]H(ρ ˆ k0 l|V. QO]H(ρ ˆ k0 +1 )ZªJ‚à (60) Vdl. 7¥<l |VíM BbªJz]∆H ˆ k0 )øšO°šíóPÔ4. ¥ší¥ BbªJøòªW- , òƒF²í;|·l · ]H(ρ êH. püøõVz, JBb²Ï7øˇó¹ ©/í;|, k 0 , ..., k 0 + m, w2mu/_£cb /k 0 < k 0 + m < M . µóBbøÆ¥ Ì |à- :. 33.
(43) ˆ k0 ) = 1. ll]H(ρ. ˆ J (ρ 0 ) ]H k J. +. 2πn M J. , 1/ÉÓ<²¦w2íø_n¹ª,n ∈ [0, J − 1].. ˆ k0 M. 2. à ä (57) (59) Vl]∆H ˆ k0 +1 )M. 3. ‚à ä (60) Vl]H(ρ ˆ k0 +2 ), ..., ]H(ρ ˆ k0 +m ). 4. ½µ2 ƒ3 òƒFí;|íóP·lêH, )ƒ]H(ρ. ˆ kíM, w2 k ∈ [k 0 , k 0 + m − 1], ŸÄuÄÑBbʲ; *2 2BbªJ£üílí]∆H |ív`, F²Ïí;|x×íÓïà@, Ĥ?‚à ä (59) Vl, ´†ª?}|˜. 7 ˆ k0 ), ..., ]H(ρ ˆ k0 +m ) ¥<óúH(ρ ˆ k0 )Fl|VíM· …O°šíóPÔ4, ¹…bcñ ]H(ρ 7kÉ ø_óP.üìÄä.. ˆ k0 ), ..., H(ρ ˆ k0 +m ) íÓïà Ê£ülê¥<óPM5(, BbZ‚à ä (66) l|ú@H(ρ ˆ k0 )|, ..., |H(ρ ˆ k0 +m )|, Í(ø¥<MD˛lßíóP à@¯97)ƒFÛbíä0¡b, à @,|H(ρ -Fý : ˆ k) ˆ k ) = |H(ρ ˆ k )| · e]H(ρ H(ρ. k ∈ [k 0 , k 0 + m]. (67). Step 4. |üj (Least Square) l. * ä (67) )øBbªJUà ä (13) Vl|,¿h : ˆ k0 ) H(ρ √ . ˆ .. = MWh ˆ k0 +m ) H(ρ m+1. 34. (68).
(44) . 1 H −1 H ˆ = √ (W W ) W h M . . ˆ k0 ) H(ρ .. . ˆ k0 +m ) H(ρ. . (69). m+1. w2 W , F (k 0 : k 0 + m , 1 : L + 1) u*×àZ s²ä³~’|Víüä³, OF í k 0 , ..., k 0 + míJ£‡ÞíL + 1_W. ä2éñåu[ý%â|üj l°)íM. ~ ·<øõ, Ñ7b?U|üjƶªJÏW, BbÛ b(W H W )¥_ä³uÅ–ä³. ¹uz, W ¥_ä³íbñ.âb×kCk…íW bñ, C[ýÑm ≥ L. µóBbZªJâ-Þí äV°)h ˆ= h. √. ˆ MV h. (70). ˆ â, °)íh´ø_cñíóP.üìÄä& tð;|Vj². '½bíøõ, ÿu CSC íj¶øš, çÊÉøˇó¹©/í;|v, J M = 64†BbÛb× 40 _;|Vdl, àŸ ˆ n?)ƒœÄüí,¿M h.¥_½æv`ª?}ݶ,, à°Ç 3 í FIR ¦ −Fý. Ç2í FIR ¦−âkØÖíÉõÔ¡ÀPÆ, ĤJøˇó¹©/í;|íj¶V²† 40 _;|} ˚Ø.. 1.5. 5 4. 1. 3 2. 0.5 Imaginary Part. 1 0 0. 7. 0. 10. 20. 30. 40. 50. 60. 70. 10. 20. 30. 40. 50. 60. 70. 0. −0.5. −500 −1. −1000 −1500. −1.5 −2.5. −2. −1.5. −0.5 −1 Real Part. 0. 0.5. 1. −2000 0. Ç 3: ø_FIR Í$í”ÉõDä0à@Ç. Ç2Óïà@Çí0§‰“uâkPkÀPÆË¡ íÉ õ¬ÖF¨A. M =64.. 35.
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