國立交通大學 電信工程學系 碩士論文 利用適應性濾波器為設計用於多輸入多輸出之時空柵狀編碼系統之渦輪等化器 Adaptive Filter-Based Turbo Equalization for MIMO Space-Time Trellis Coded Systems 研究生 : 曾俊偉 指導教

Transcription

1 國立交通大學 電信工程學系 碩士論文 利用適應性濾波器為設計用於多輸入多輸出之時空柵狀編碼系統之渦輪等化器 Adaptve Flter-Based urbo Equalzaton for MIMO Space-me rells Coded Systems 研究生 : 曾俊偉 指導教授 : 紀翔峰 中華民國九十五年九月

2 利用適應性濾波器為設計用於多輸入多輸出之時空柵狀編碼系統之渦輪等化器 Adaptve Flter-Based urbo Equalzaton for MIMO Space-me rells Coded Systems 研究生 : 曾俊偉 指導教授 : 紀翔峰博士 Student:Jun-We zeng Advsor:Dr. Hsang-Feng Ch 國立交通大學 電信工程學系碩士班 碩士論文 A hess Submtted to Department of Communcaton Engneerng College of Electrcal and Computer Engneerng Natonal Chao ung Unversty n Partal Fulfllment of the Requrements for the Degree of Master n Communcaton Engneerng September 006 Hsnchu, awan, Republc of Chna 中華民國九十五年九月

3 利用適應性濾波器為設計用於多輸入多輸 出之時空柵狀編碼系統之渦輪等化器 研究生 : 曾俊偉 指導教授 : 紀翔峰博士 國立交通大學 電信工程學系碩士班 摘要 時空柵狀編碼是一個使用多根傳送天線的頻寬節省技術 當使用在寬頻的傳送系統時, 通道的頻率選擇特性會造成不可忽略的影響 信號之間的互相干擾 (ISI) 以及同通道中的外來干擾 (CCI) 都會降低時空柵狀編碼所帶來的的效能增長 在這個論文裡, 我們的目的是設計與建立一種接收的技巧, 讓時空柵狀編碼系統即使經歷了具有頻率選擇效應的多輸入多輸出通道, 仍能正確的接受資訊 我們設計了一個渦輪等化器 [3], 使用一根或兩根接受天線來接受多根傳送天線的訊號 等化的動作是用一個簡單的適應性濾波器為主的等化器 濾波器的係數是用一個名稱為 最少平均平方 [] 的適應性演算法來獲得 在每次遞迴時, 等化器和解碼器產生出外質的訊息, 然後這些外質的訊息會在下次遞迴的時候被拿來使

4 用, 就像渦輪碼一樣 我們用兩根 三根 四根的傳送天線來跑模擬, 藉此來觀察傳送天線數目對系統效能的增進 關於不同大小的交錯器對我們的渦輪等化器的影響, 我們也用模擬來觀察 這些模擬結果顯示出, 我們提出的渦輪等化器, 配上適當大小的交錯器, 能夠有效的對抗多路徑造成的不好影響以及同通道中的干擾

5 Adaptve Flter-Based urbo Equalzaton for MIMO Space-me rells Coded Systems Student:Jun-We zeng Advsor:Dr. Hsang-Feng Ch Department of Communcaton Engneerng Natonal Chao ung Unversty Hsnchu, awan Abstract Space-tme trells code (SC) [] s a bandwdth-effcent technque utlzng multple transmt antennas. When t s appled to wdeband systems, the channel s frequency-selectve characterstcs are not neglgble. he nter-symbol nterference (ISI) and the co-channel nterference (CCI) wll deterorate the system performance mprovement provded by SC. In ths thess, our goal s to develop a recevng scheme for space-tme trells coded system over frequency-selectve multple-nput-multple-output (MIMO) channels. We desgn the turbo-equalzaton

6 [3] wth one or two receve antennas to recover all symbols from multple transmt antennas. Equalzaton s performed usng a smple adaptve flter-based equalzer. he flter coeffcents of the equalzer are obtaned usng an adaptve algorthm called least-mean-squared (LMS) []. At each teraton, extrnsc nformaton s extracted from the detecton and decodng unts and s then used at the next teraton as n turbo-decodng. Smulatons of two, three and four transmt antennas are conducted to see the performance mprovement provded by the dfferent number of transmt antennas. Smulatons wth dfferent nterleaver szes are also conducted to observe the effect of the nterleavers on a turbo system. he smulaton results show that the proposed turbo-equalzer wth an approprate nterleaver sze can successfully combat the multpath effects and the co-channel nterference.

7 謝辭 在探究知識與理論的這條路上, 重要的不僅是我得到了什麼, 更重要的是我重新 回想了我一生所學 所愛與被愛, 在掙扎 矛盾與歡笑的交織中, 重新發現 了 解自己 這本論文是我學生生涯的一個段落, 也象徵著人生新里程碑的開始, 不論未來是 光明或黑暗, 我將帶著這個小小的結晶再出發 謝謝我的指導教授紀翔峰老師, 總是耐心的引領我一步一步完成我的研究, 還要 感謝其他在學路上給予我教導的師長, 讓我有足夠的力量迎向未來的挑戰 謝謝 93 實驗室所有學長 同學陪我走過的歡笑和淚水, 在我遇挫折時給我的 鼓勵 幫忙, 我永遠不會忘記那些一起奮戰的夜晚! 這篇謝辭更重要的是要特別感謝在我生命中扮演非常重要角色的人, 謝謝我的寶 貝品客, 因為妳, 我才有動力完成這篇論文, 妳讓我期待每一天的到來, 因為妳 會讓我的每個下一刻都是幸福開心的, 也希望我們會有更多美好的未來 最後, 感謝一直以來都很辛苦的媽媽和爸爸, 謝謝你們給我很大的空間和疼愛, 在我心目中你們是最棒最好的爸媽了! 擁有你們真的讓我感到很幸福! 我想我一定還沒有謝完, 真的很開心我可以成為這麼幸福的人, 我也會在你們的 關愛環繞下, 繼續我的下一個旅程 俊偉 006 年 9 月 7 日於交通大學

8 Content Chapter Introducton to MIMO Systems.. MIMO Channel..... MIMO Channel Model..... MIMO Channel Capacty Antenna Numbers versus Capacty MIMO Capacty over Frequency-Selectve Channels...7. Introducton to Recevers for MIMO Channel Motvaton Organzaton...0 Chapter Space-me rells Codes.... Dversty echnques..... me Dversty..... Frequency Dversty Space Dversty ransmt and Receve Dversty Combne Dfferent Dversty...4. Space-me Codng Space-me Bloc Codes Space-me rells Codes Space-me rells Codes rells descrpton...0 I

9 .3. Generator Descrpton Desgn Crtera Decodng Algorthm Maxmum A posteror Probablty (MAP) Decoder BCJR Algorthm...5 Chapter 3 urbo Equalzaton Frequency-Selectve Channel Equalzer Overvew rells-based Flter-Based Lnear Equalzer (LE) Decson-Feedbac Equalzer (DFE) Flter Desgn Algorthm Zero-Forcng (ZF) Mnmum Mean Square Error (MMSE) Adaptve Equalzer Least Mean Square Algorthm Adaptve Decson Feedbac Equalzaton Equalzaton and Decodng Optmal Jont Equalzaton and Decodng Separate Equalzaton and Decodng Iteratvely Equalzaton and Decodng...4 II

10 Chapter 4 Adaptve Flter-Based urbo Equalzer wth Space-me Decoder ransmtter Space-me rells Encoder Interleavers Pulse Shapng Flter ranng Symbols Channel and Nose urbo Recever Equalzaton Sngle Receve Antenna wo Receve Antennas Mapper and DeMapper Decodng...6 Chapter 5 Smulaton Results and Comparsons Perfect Feedbac Improvement by Iteratons Improvement by wo Receve Antennas Effect of Interleavers...7 III

11 Chapter 6 Conclusons and Perspectves Conclusons Perspectves...75 Bblography...76 IV

12 Fgure Content Fgure. MIMO channel wth n transmt antennas and n receve R antennas Fgure. Equvalent MIMO channel after SVD...5 Fgure.3 Subchannel of OFDM systems...8 Fgure. Fgure. Delay dversty transmtter...5 Delay dversty transmtter...5 Fgure.3 Space-tme coded transmtter...6 Fgure.4 Space-tme bloc coded transmtter...6 Fgure.5 Fgure.6 Alamout space-tme bloc encoder...8 rells descrpton for a 4-state space-tme trells codes wth transmt antennas and 4-PSK sgnal constellaton...0 Fgure.7 4-PSK sgnal constellaton... Fgure.8 Fgure.9 Fgure 3. Fgure 3. Encoder structure of space-tme trells codes... A trells example wth 4 states...5 Equvalent dscrete baseband system...30 apped delay lne model of the channel...30 Fgure 3.3 Lnear equalzer...3 Fgure 3.4 Decson-feedbac equalzer...33 Fgure 3.5 Sgnal-flow graph representaton of the LMS algorthm...38 Fgure 3.6 ransmtter wth encoder and nterleaver...40 Fgure 3.7 Separate equalzaton and decoder...40 Fgure 3.8 Bloc dagram of turbo equalzer...4 V

13 Fgure 3.9 Fgure 4. Equalzer bloc n turbo equalzer...4 ransmtter of a space-tme trells coded system...46 Fgure 4. Squared rooted rased cosne flter wth rolloff factor 0.5 and truncated to be length Fgure 4.3 Pacet format....5 Fgure 4.4 Fgure 4.5 ransmtter of a space-tme trells coded system...53 Equalzer bloc dagram of a space-tme trells coded system Fgure 4.6 Fgure 4.7 Equalzer structure for one receve antenna...55 Equalzer structure for two receve antennas...58 Fgure 5. SER for transmt antennas and receve antenna and SC wth 3 states...64 Fgure 5. BER for transmt antennas and receve antenna and SC wth 3 states...64 Fgure 5.3 SER for 3 transmt antennas and receve antenna and SC wth 3 states...65 Fgure 5.4 BER for 3 transmt antennas and receve antenna and SC wth 3 states...66 Fgure 5.5 SER for 4 transmt antennas and receve antenna and SC wth 3 states...67 Fgure 5.6 BER for 4 transmt antennas and receve antenna and SC wth 3 states...67 Fgure 5.7 SER for transmt antennas and receve antenna and SC wth 3 states...68 Fgure 5.8 BER for transmt antennas and receve antenna and SC wth 3 states...69 VI

14 Fgure 5.9 SER for 3 transmt antennas and receve antenna and SC wth 3 states...70 Fgure 5.0 BER for 3 transmt antennas and receve antenna and SC wth 3 states...70 Fgure 5. SER for 4 transmt antennas and receve antenna and SC wth 3 states...7 Fgure 5. BER for 4 transmt antennas and receve antenna and SC wth 3 states...7 Fgure 5.3 BER for dfferent nterleaver szes at 5th teraton wth transmt antennas and receve antenna and SC wth 3 states...73 Fgure 5.4 SER for dfferent nterleaver szes at 4th teraton wth transmt antennas and receve antenna and SC wth 3 states...73 VII

15 able 4. able Content Generator sequences of 3 states and, 3 and 4 transmt antennas...47 able 4. Generator sequences of dfferent state numbers for transmt antennas...47 VIII

16 Chapter Introducton to MIMO Systems CHAPER Introducton to MIMO Systems he demands for hgh data rate transmsson are ncreasng rapdly recently. But, the tradtonal approaches to ncrease data rate, such as ncreasng bandwdth, or ncreasng transmsson rate, are becomng mpractcal due to the lmted resource. hs leads to consderable effort n fndng and developng new approaches n addton to the two aforementoned approaches. A popular approach s to develop a multple-nput-multple-output (MIMO) system. In ths chapter, we wll show how the advantages are obtaned from the MIMO system by dervng the fundamental capacty of MIMO channels and comparng t to that of the tradtonal sngle-nput-sngle-output (SISO) channels. At the end of ths chapter, we wll ntroduce the motvaton of ths wor and the organzaton of ths thess.. MIMO Channel.. MIMO Channel Model Assumng a MIMO channel wth n transmt antennas and n R receve antennas, and the path between each antenna s frequency-flat, then we can express ths system as: - -

17 Chapter Introducton to MIMO Systems r = Hx+ n (.) where r = r r n R s the n R receved vector, x = x x n s the n transmtted vector, n = n n n R s the n R addtve nose vector. H s the n R n fadng matrx of the form: h h n H = hn R h nrn (.) Let the total transmtted power be constraned to P, regardless of the number of transmt antennas n. he power can be represented as P= tr( R ) (.3) xx where tr() denotes the trace obtaned as the sum of dagonal elements, and denotes the covarance matrx obtaned by: xx H { } R xx R = E xx (.4) where E{ } denotes expectaton operaton. It s common to consder the transmtted sgnals to be zero mean ndependent dentcally dstrbuted (..d.) Gaussan varables. Let us consder the case when transmtted power s splt nto n parts evenly and dstrbuted to n transmt antennas, then equaton (.4) can be rewrtten as P R = I (.5) xx n n where I s the n n dentty matrx. n For normalzaton purposes, we assume that the receved power for each of n R receve antennas s equal to the total transmtted power P. hus we obtan a normalzaton constrant for the elements of H as - -

18 Chapter Introducton to MIMO Systems h x r h n h nr x n h n n R r nr Fgure. MIMO channel wth n transmt antennas and n R receve antennas hs system s shown n fgure.. n hj = n, j =,, nr (.6) =.. MIMO Channel Capacty Before we start to derve the channel capacty of MIMO channels, we frst revew the Shannon s thrd theorem [7], the nformaton capacty theorem, as a remnder: he nformaton capacty of a contnuous channel of bandwdth B hertz, perturbed by addtve whte Gaussan nose of power spectral densty N / and 0 lmted n bandwdth to B, s gven by P log r C = B + bts per second NB 0 (.7) where P r represents the receved sgnal power. Let NB= σ be the total power of the nose, we rewrte the capacty formula 0 n as: P r C = Blog + bts per second σ n (.8) Now we are ready to derve the channel capacty of MIMO channel. In lght of - 3 -

19 Chapter Introducton to MIMO Systems sngular value decomposton (SVD) theorem [4], any n R n matrx H can be decomposed as: H H= UDV (.9) where D s an n R n non-negatve and dagonal matrx, U and V are n R n R and n n untary matrces, respectvely. he dagonal elements of D are the non-negatve square roots of the egenvalues of H HH. Furthermore, the column vectors of U are the egenvectors of H HH, and the column vectors of V are the egenvectors of H H H. he non-negatve square roots of the egenvalues of H HH are also referred to as the sngular values of H. We denote λ as the th element on the dagonal of D, and then we have the followng equatons: H HH y y (.0) = λ where y s an egenvector correspondng to egenvalue λ By untary matrces, we mply the followng equatons exst: UU VV H H = I = I Substtutng (.9) nto (.), and multplyng both sdes by (.) U H, we can rewrte equaton (.) as: Introducng the transformatons blow H H H H U r = U UDV x + U n H = DV x + H U n (.) H r' = U r H x' = V x H n' = U n (.3) we obtan an equvalent channel model: r' = Dx' + n ' (.4) We say that equaton (.4) s equvalent to equaton (.) s because V and U are untary matrces, therefore the transformaton that H represents s dentcal to - 4 -

20 Chapter Introducton to MIMO Systems D wth respect to dfferent bass. Due to the dagonal property of D, each element n r ' can be expressed as: r ' = λ x ' + n', =,, n (.5) R H If ran( H ) = γ, whch also means ran( HH ) = γ, then λ 0 for =,, γ and λ = 0 for = γ +, nr. Rewrte equaton (.3) and apply the property above leads to: r ' = λ x ' + n ; =, γ r ' = n ; = γ +, n R (.6) hs new equvalent model s shown n fgure. x ' λ r ' X ' x γ x γ + ' λ γ 0 ' r γ r γ + ' RX x n ' 0 r nr ' Fgure. Equvalent MIMO channel after SVD We have changed the system n fgure. nto fgure.. From fgure., we see that the orgnal MIMO channel can be expressed as γ parallel equvalent SISO channels, and each SISO channel capacty can be drectly computed usng the Shannon capacty formula revewed n equaton (.8). From ths vewpont, the capacty of MIMO channel s now easly obtaned by summng all the equvalent SISO channel capactes snce they are parallel: - 5 -

21 Chapter Introducton to MIMO Systems γ P r C = Blog + bts per second = σ (.7) where P r denotes the receved sgnal power of r ', and σ denotes the nose power of n '. P r can be obtaned usng followng equatons under our normalzaton constrant (.6) λ P P = tr( R ) = tr( U R U) = tr( R ) = tr( HR H ) = (.8) H H r r ' r ' r ' r ' r ' r ' x ' x ' n hus, the channel capacty can be wrtten as C = + λ P γ Blog = σ n γ λ P = B log + bts per second = σ n (.9) H Snce λ, =,, γ are the egenvalues of HH, to obtan all λ we have to fnd the roots of such equaton: H det( λ ) = 0 I HH (.0) We can also rewrte equaton (.0) by ntroducng the roots nto the equaton: herefore, we can equal these two equatons as blow: γ ( λ λ ) = 0 (.) = H det( λ ) = ( λ λ ) γ I HH (.) = Substtutng n σ for λ n (.), and multplyng both sdes a constant, we get P P λ P det( I+ HH ) = ( + ) (.3) n σ γ H = n σ he purpose of dong ths s to wrte the channel capacty formula n(.9) as C = Blog det + P H I HH (.4) nσ..3 Antenna Numbers versus Capacty From the capacty formula (.7), t s obvous that γ s an mportant factor n - 6 -

22 Chapter Introducton to MIMO Systems determnng the capacty. It represents the equvalent parallel SISO channel number. Snce H s a n R n matrx, the ran of H,γ, wll always be less or equal to mn( nr, n ). If n R n, the equvalent parallel SISO channel number of the MIMO channel s less or equal to the transmt antenna number. If n nr, the equvalent parallel SISO channel number of the MIMO channel s less or equal to the receve antenna number. It s often a fallacy to thn that more antennas there are, more equvalent SISO channels we must have. he fact s: t depends on the ran of the channel. A 3-to- MIMO system may gve less equvalent SISO channels than a -to- MIMO system. But for full-ran channels defned as the cases γ = mn( n, n ), of course, more antennas gve more capacty. herefore, a full-ran R channel s always consdered as a good channel for a MIMO system...4 MIMO Capacty over Frequency-Selectve Channels In ths secton, we consder the capacty of an OFDM-based MIMO channel, and then extend t to general frequency-selectve channel capacty. For a K-pont OFDM-based system, channel can be consdered as K parallel subchannels. If K s large enough, then each subchannel can be approxmated as frequency-nonselectve. herefore, a frequency-selectve MIMO channel n an OFDM-based system can be approxmated as K frequency-nonselectve MIMO subchannels f K s large enough as llustrated n fgure.3. he nstantaneous channel capacty of such system s gven n [5] as B C + K H log det( SNR ( ) K = I H H (.5) where H represents the th subchannel whch s a n R n matrx, and SNR represents the Sgnal-to-Nose Rato of the th subchannel at the receve antenna, and B s the bandwdth of the overall channel

23 Chapter Introducton to MIMO Systems Frequency-selectve Approxmately Frequency-nonselectve Fgure.3 Subchannel of OFDM systems By comparng equatons (.5) and (.4), t s not dffcult to see that ths formula s smply to sum the capactes over the K frequency-nonselectve subchannels. he dervaton seems plausble; however, t s an approxmaton because we approxmate each subchannel to be frequency-nonselectve. o derve the exact formula nstead of the approxmaton, Remann ntegral theorem can be appled. hat s, when K s approachng to nfnte, all subchannels (wth the bandwdth B ) become frequency-nonselectve. he summaton n (.5) becomes K ntegraton, and we obtan the exact MIMO frequency-selectve channel formula: H ( xx nn ) B log 0 det I H( ) S ( ) H ( ) S ( ) bts per second (.6) C = B + f f f f df where S ( f ) represents power spectrum matrx of x at frequency f obtan by xx S( f) = F{ R }, and H ( f ) s the frequency response of the MIMO channel at frequency f

24 Chapter Introducton to MIMO Systems. Introducton to Recevers for MIMO Channel here are many dfferent transmttng schemes for MIMO systems. he Space-tme trells code s a well-nown transmttng scheme whch ntroduces dversty and codng gan. However, as the transmsson bandwdth ncreases beyond the coherent bandwdth of the channel, ISI becomes a major performance-lmtng mparment. Equalzaton becomes ndspensable. he theory of equalzaton for sngle-nput sngle-output channels has been well-developed. Dfferent crtera and algorthms such as MMSE, BCJR, LMS and RLS are appled to obtan a varety of equalzers. In the recently years, the technques of turbo equalzaton are proposed and shown to provde much better performance than the tradton recevng schemes n whch the equalzaton and the decodng are conducted separately. he turbo equalzaton scheme performs equalzaton and decodng n an teratve manner and obtans the performance near the Shannon lmt. When MIMO channels are consdered, these desgn crtera and algorthms for the SISO systems reman vald and effectve. However, due to the nature of MIMO channel, some modfcatons or extra effort have to be made. rells-based equalzers suffer from the heavy computatonal complexty. hs problem can be mtgated by the use of several technques such as preflterng [5] and n-phase/quadrature-phase detecton [6]. Flter-based equalzers are low-complexty alternatves. Wth nown channel state nformaton (CSI), MMSE crteron can be derectly appled to the flter-based equalzer as t does n [3] for SISO channels. Wthout the CSI, MMSE solutons can be approached by the adaptve algorthms such as LMS and RLS as t does n [] for SISO channels. In ths thess, the channel state nformaton s assumed to be unnown at the recever, and we apply adaptve algorthm drectly to the equalzer. hs structure - 9 -

25 Chapter Introducton to MIMO Systems can be deemed as the MIMO verson of the one n []..3 Motvaton Achevng hgh bt rates over bandlmted wreless channels maes many applcatons possble. he use of multple-antennas brngs a bandwdth effcent soluton whle achevng hgh bt rate at the same tme. he fundamental phenomenon whch maes relable wreless transmsson dffcult s the multpath fadng. herefore, to overcome ths phenomenon s especally mportant. For some applcatons, the recevers are also requred to be small and power effcent. hus, a smple recever structure s attractve. However, the equalzaton for MIMO channels s generally consdered mpractcal due to the heavy computatonal complexty. In ths thess, our goal s to develop a low-complexty turbo equalzer for broadband MIMO systems..4 Organzaton hs thess s organzed as follows. Chapter ntroduces the space-tme trells codes, and ts advantages over sngle-nput-sngle-output trells codes. he encodng structure and the decodng algorthm are addressed n detals. Chapter 3 s the equalzer overvew. he flter-based equalzer s the canddate of our desgn and thus s the man topc of ths chapter, too. he deal of turbo equalzaton s ntroduced n chapter 3. In chapter 4, we propose the recever n combnaton of the topcs n chapter and 3. In ths chapter, the detals of the recever are gven, ncludng equalzaton, decodng, nterleavng, mappng, and demappng. In chapter 5, we show the smulaton results of our proposed system, and some comparson of the systems. Chapter 6 s the concluson we mae on the smulaton results and comparsons

26 Chapter Space-me rells Codes CHAPER Space-me rells Codes In chapter, we have shown the capacty a MIMO channel, but no system mplementaton has been mentoned. In ths chapter, we wll ntroduce the Space-me Codes whch s one approach to tae advantages of MIMO channel and provde hgh data rate or hgh ln qualty. Space-tme codes have a varety of dfferent structures, each of them has ther one advantages and dsadvantages. In ths chapter, we wll frst ntroduce brefly the three dverstes commonly used n communcaton systems. hen we move on to space-tme codes whch may combne several of the three dverstes. he famous space-tme bloc codes wll be brefly ntroduced, and then we focus on the space-tme trells codes whch are the channel codes we used n our proposed system n chapter 4.. Dversty echnques Many channels, especally wreless channels, suffer from attenuaton due to destructve addton of multpaths n the propagaton meda and due to the nterference from other users. Severe attenuaton maes t mpossble for the recever to determne the transmtted sgnal unless some less-attenuated replca of the transmtted sgnal s provded to the recever. hs resource s called dversty and t s an

27 Chapter Space-me rells Codes mportant contrbutor to relable wreless communcatons. Accordng to the doman where dversty s ntroduced, there are three categores of dversty: tme dversty, frequency dversty, space dversty... me Dversty he replcas of transmtted sgnals are provded to the recever n the form of redundancy n tme doman. Identcal messages are transmtted n dfferent tme slots, and the recever would receve several uncorrelated fadng sgnals. o be uncorrelated, the tme separaton between dentcal messages must be at least the coherent tme of the channel. he defnton of the coherent tme s the perod over whch the channel fadng process s correlated. In many systems, redundancy n tme doman s ntroduced by the error control codng (ECC), and an nterleaver s placed after error control codng to provde tme separaton greater than the coherent tme. However, n the recever, denterleavng process ntroduces message delay. For slow fadng channels, a larger nterleaver s requred to exceed the coherent tme, and therefore, a larger message delay s ntroduced. hs drawbac may be vtal to some delay-senstve applcatons, especally voce applcatons. Another drawbac of ths scheme s that there wll be a certan bandwdth effcency loss due to the redundancy n tme doman... Frequency Dversty he replcas of transmtted sgnals are provded to the recever n the form of redundancy n frequency doman. he frequency separaton s requred to be at least the coherent bandwdth to obtan uncorrelated fadng replcas n the recever. he defnton of the coherent bandwdth s smlar to the coherent tme: the frequency span over whch the channel fadng property s uncorrelated. Several mature

28 Chapter Space-me rells Codes communcaton systems ntroduce the frequency dversty to ncrease the data rate or mprove the ln qualty. Spread spectrum s one example, ths technque ncludes drect sequence spread spectrum (DSSS), frequency hoppng, multcarrer modulaton, and CDMA systems. A combnaton of error-control codng and OFDM can also be consdered as frequency dversty, because the tme dversty provded by ECC has been transferred nto frequency doman by OFDM modulator. hese technques use bandwdths that are far more than enough just to provde frequency dversty, thus, le tme dversty, t nduces a loss n bandwdth effcency due to the redundancy ntroduced n frequency doman...3 Space Dversty he replcas of transmtted sgnals are provded to the recever n the form of redundancy n spatal doman. It s typcally mplement usng multple antennas or antenna arrays arranged n space n a certan manner. herefore, space dversty s also called antenna dversty. he space separaton between antennas s requred to be at least the coherent dstance. Usually, a few wavelengths are enough for the antennas to experence dfferent fadngs. One advantage of ths technque s that unle tme and frequency dversty, t doesn t suffer from the loss n bandwdth effcency. hs advantage maes t very attractve to hgh data rate wreless communcatons...4 ransmt and Receve Dversty We can further classfy space dversty nto receve dversty and transmt dversty dependng on where the multple antennas are appled. he receve dversty s adapted n a varety of moble communcaton systems wth the am to both suppress co-channel nterference and mnmze the fadng effects. It s reasonable to apply 3

29 Chapter Space-me rells Codes receve dversty at the base staton for upln (from mobles to base statons) communcaton because the power requrement and the dmenson requrement are easer to meet compared wth moble devces. For example, n GSM systems, multple antennas are used at the base staton to create upln receve dversty, compensatng for the relatvely low transmsson power from the moble. For downln (from base staton to mobles), t s much more dffcult to apply receve dversty at the mobles. Frstly, placng multple antennas n a portable moble devce s aganst the publc favor n a smaller devce. Secondly, multple antennas mean more power consumpton whch s also aganst the publc favor n a power-savng devce. herefore, transmt dversty s more adequate for the downln communcaton. However, n contrast to receve dversty whch s wdely appled n moble systems, transmt dversty has ganed lttle attenton and s less understood. he reason s that t s more dffcult to explot transmt dversty, and the dffculty s because the transmtted sgnals are mxed up before they arrve the recever. herefore, the recever requres extra sgnal processng to separate the transmtted sgnals before the transmt dversty can be exploted. hs sgnal processng s not always perfect, and may suffer from performance loss. In the next sesson, we wll ntroduce a transmt dversty technque called Space-me Codes, and n Chapter 4, we wll propose a sgnal processng to separate the transmtted sgnals...5 Combne Dfferent Dversty Not all forms of dversty can be avalable at all tmes. o obtan dversty, the resources provdng redundances must be uncorrelated. Otherwse, you smply obtan the same nformaton twce. For example, n slow fadng channels, tme dversty s not an opton due to large coherent tme. When delay spread s small, 4

30 Chapter Space-me rells Codes frequency dversty s not an opton due to large coherent bandwdth. When the platform s a small moble devce, space dversty s not sutable due to lmted dmensons. Nevertheless, combnng dfferent dverstes wll mprove the data rate or ln qualty f they are avalable. ransmt dversty and receve dversty can also be combned to provde more advantages.. Space-me Codng In 993, Wttneben proposed a delay dversty scheme [6]. hs scheme transmts the same nformaton from both antennas but wth a delay of one symbol nterval as shown n fgure.. he effect of ths process s to ntroduce an artfcal multpath channel to the recever whch changes a narrowband purely frequency-nonselectve fadng nto a frequency-selectve fadng channel. Wth ths multpath channel, the recever can utlze a trells-based equalzer and gan performance mprovement from t. In [7], t was shown that the use of a maxmum-lelhood sequence estmator at the recever s capable of provdng dual branch dversty. Informaton source Mapper Delay s Fgure. Delay dversty transmtter 5

31 Chapter Space-me rells Codes Fgure. Delay dversty transmtter hs framewor s dentcal to fgure. where the channel code s a repetton code wth code rate R =. In ths structure, t s natural to as f t s possble to desgn a channel code that s better than repetton code n order to mprove performance further. he answer s yes, and we use the term Space-me Codes to refer to these channel codes. In space-tme codng, the delay element before antenna unt s removed, and the structure s shown n fgure.3. Codng s performed n both spatal and temporal domans to ntroduce correlaton between sgnals transmtted from varous antennas at varous tme perods. Space-tme codng can acheve transmt dversty over spatally uncoded systems wthout sacrfcng the bandwdth. here are varous approaches n codng structures, ncludng space-tme bloc codes (SBC), and space-tme trells codes (SC). Antenna Antenna n Fgure.3 Space-tme coded transmtter 6

32 Chapter Space-me rells Codes Antenna a x [ x x ] X n p Antenna n Fgure.4 Space-tme bloc coded transmtter.. Space-me Bloc Codes Space-tme bloc codes operate on a bloc of nput symbols producng a matrx output whose columns represent tme and rows represent antennas. her ey feature s the provson of full dversty wth extremely low encoder/decoder complexty under frequency-nonselectve channels. he space-tme bloc code system s shown n fgure.4. Assumng n antennas are used, and p symbols per antenna are used to convey uncoded symbols. he code rate s gven by R p = (.) o descrbe the space-tme bloc codes, we use the transmsson matrx X whch s a n p matrx. he element of X n the th row and j th column, x, j, =,, n, j =,, p represents the sgnals transmtted from the antenna at tme j. he ey property of ths system s the orthogonalty between the sequences generated by the dfferent transmt antennas. hs feature was generalzed n [8] to an arbtrary number of transmt antennas by applyng the theory of orthogonal desgns. It s also shown n [8] that to acheve full transmt dversty, the code rate of a space-tme bloc code must be less than or equal to one, R whch requres an bandwdth expanson of R. Each element of the transmsson matrx X s a lnear combnatons of the 7

33 Chapter Space-me rells Codes modulated symbols x, x and ther conjugates x *,, * x,. o acheve the orthogonalty, X must satsfy the followng equaton: X X ( ) H = c x + x I n (.) where c s a constant, X H s the Hermtan of X and n I s an n n dentty matrx. hrough the orthogonal desgnng, the sgnal sequence from any two transmt antennas are orthogonal. hs property enables the recever to decouple the sgnals transmtted from dfferent antennas and consequently, a smple maxmum lelhood decodng, based only on lnear processng of the receved sgnals. Informaton Source a Modulator [ x x ] Space-tme Bloc Encoder [ x x] * x x * x x Antenna Antenna Fgure.5 Alamout space-tme bloc encoder he Alamout code [9], was the frst and the most famous space-tme bloc code. It s acheve a full dversty gan usng two transmt antennas and a smple maxmum-lelhood decodng algorthm. he transmsson matrx s gven by * x x = * x x X (.3) Fgure.5 shows an encoder structure for Alamout code. It s easy to see that the transmt sequence from antennas one and two are orthogonal, x x (.4) * * = xx x x = 0 he decodng of space-tme bloc codes was based on maxmum lelhood algorthm. Assumng one receve antenna and perfect channel state nformaton (CSI) s now at the recever. Usng Alamout code as an example, the maxmum 8

34 Chapter Space-me rells Codes lelhood decoder choose a par of sgnals ( xˆ, x ˆ ) from the sgnal modulaton constellaton to mnmze the dstance metrc (, ˆ ˆ ) (, ˆ ˆ * ) d r hx h x d r hx * h x (.5) Wth some smple transformaton, the decson rule can be derved as ( ˆ ) xˆ = arg mn d x, x xˆ S ( ˆ ) xˆ = arg mn d x, x xˆ S (.6) where S s the set of all possble modulated symbol pars ( xˆ ˆ, x ) and x = hr+ hr * * x = h r + hr * * (.7) he decson rules dervaton can be extended to other cases wth other number of receve antennas, and they can be found n [8]. Space-tme bloc codes can acheve a maxmum possble dversty advantage wth a smple decodng algorthm. It s very attractve because of ts smplcty. However, no codng gan can be provded by space-tme bloc codes, and also, non-full rate space-tme bloc codes can ntroduce bandwdth expanson... Space-me rells Codes Space-tme trells codes operate on one nput symbol at a tme producng a sequence of vector symbols whose length represents antenna number. Le tradtonal trells coded modulaton (CM) for the sngle-antenna channel, space-tme trells codes provde codng gan. Snce they also provde full dversty gan, ther ey advantage over space-tme bloc codes s the provson of codng gan. Space-tme trells codes are nowadays wdely dscussed as t can smultaneously offer a substantal codng gan, spectral effcency, and dversty mprovement on flat fadng channels. he case for frequency-selectve channels are studed n [3][8], and the concluson of [3] s that the frequency-selectve channel doesn t affect the dversty 9

35 Chapter Space-me rells Codes provded by space-tme trells codes. However, compared wth space-tme bloc codes, ths code s far more dffcult to desgn, and also requres a computatonally ntensve encoder and decoder. he ey development was done by aroh, Seshadr and Calderban n 998 [], and some other mproved development was done n [0][][]. In next sesson, we wll tal about t n detal..3 Space-me rells Codes.3. rells descrpton he space-tme trells codes are descrbed by trells structures. Consderng a space-tme trells coded M-PSK modulaton wth n transmt antennas. he encoder taes a group of m= log M nformaton bts at tme t gven by ( m a,, a ) a = (.8) t t t and produces a group of mn coded bts (( ) ( )) n,,,,,,,,, n, c c c c c = (.9) t t t m t t m Each (,, ) ct,, ct n, =,, n are mapped nto symbol x t, and thus c t s mapped nto a group of n symbols gven by ( x,, x n ) x = (.0) t t t where each x, =,, n s an M-PSK modulated sgnal. At each tme t, t dependng on the state of the encoder and the nput bts, a transton branch s chosen. On the trells, each branch transton s labeled of ( x,, n t xt ) transmt antenna s used to send symbol smultaneous. whch means the x t, and all these transmssons are 0

36 Chapter Space-me rells Codes Fgure.6 rells descrpton for a 4-state space-tme trells codes wth transmt antennas and 4-PSK sgnal constellaton (0,) (-,0) 0 (,0) 3 (0,-) Fgure.7 4-PSK sgnal constellaton Fgure.6 shows an example of trells descrpton for a 4 states space-tme trells code wth transmt antennas and 4-PSK sgnal constellaton. 4-PSK sgnal constellaton s gven n fgure.7. he number pars a a / x x n front of each t t t t state are the labels of each branch transton startng from that state. he left-most number par s correspondng to the top-most branch transton of the state. As an llustraton, f the encoder s n the second state at tme t, the nput at ths tme s, then the encoder chooses the branch transton from the second state to the fourth state. hs branch transton s labeled /3 whch means antennas (,) wll transmt the symbol (,3) respectvely. he trells s a representaton whch can fully descrbe the space-tme trells codes. However, t s common to descrbe the space-tme trells codes as generator descrptons n the encoders.

37 Chapter Space-me rells Codes ( g 0,,, g0, ) n ( g,,, g, ) n c m c ( g,,, g, ) v v n m (,, m gv, gv, n ) m m n x,, x m (,, m g, g, n ) m (,, m g0, g0, n ) Fgure.8 Encoder structure of space-tme trells codes.3. Generator Descrpton he encoder structure of space-tme trells code s shown n fgure.8. he -th nput sequence ( 0 t ) a = a,, a,, =,, m s fed nto the -th shft regster and multpled by a generator sequence g, =,, m. he multpler outputs from all shft regsters are added up n modulo-m to gve the encoder output x= ( x x ) he generator sequence g, =,, m s of the form: ( 0, 0, n ) ( v, v, n ) = g,, g,, g,, g, =, m,,, 0. g (.) he total memory order of the encoder, denoted by v s gven by t m v = v (.) = where v, =,, m s the memory order for the -th lane of shft regster, and s gven by v v+ = log M (.3) he encoder output at tme t for transmt antenna s now gven as

38 Chapter Space-me rells Codes m v t = j, t j mod, =,, = j= 0 x g a M n (.4) As an example, let us consder a scheme of 4-state space-tme trells coded QPSK system wth transmt antennas and the generator sequences are ( 0 ),( 0) ( 0 ),( 0) g = g = (.5) he resultng trells structure s shown n fgure.6, whch s the same as the one we use as an example n the prevous sesson..3.3 Desgn Crtera For a gven encoder structure, a set of encoder coeffcents s determned by mnmzng the error probablty. It s shown n [] that the error rate for slow fadng channels, the upper bound s depend on the value of r whch s the ran of codeword dstance matrx (, ˆ ) matrx B( X, X ˆ ) A X X. It can be obtaned by utlzng the codeword dfference (, ˆ ) = (, ˆ ) H (, ˆ ) A X X B X X B X X (.6) where ˆX s the erroneous decson mad by decoder when the transmtted sequence was n fact X. herefore, n order to mnmze the error probablty, we have two dfferent crtera: Ran & determnant crtera: If rn R < 4, the mnmum ran r over all pars of dstnct codewords should be maxmzed. Also, the determnant of A( X, X ˆ ) along the pars of dstnct codewords wth the mnmum ran should maxmzed, too. race crtera: If 4 R rn, the mnmum trace of (, ˆ ) dstnct codewords should be maxmzed. 3 A X X among all pars of

39 Chapter Space-me rells Codes hese crtera were derved and summarzed n [] along wth the crtera for fast fadng channels. he above crtera was referred to as the aroh/seshadr/calderban (SC) codes. An mproved crteron was proposed and referred to as the Baro/Bauch/Hansmann (BBH) codes [4]..4 Decodng Algorthm.4. Maxmum A posteror Probablty (MAP) Decoder o recover the nformaton bts at the recever, t s natural to choose a recever that acheves the mnmum probablty of error Pa ( aˆ ) where ˆ a s the decson of -th nformaton bt. It s well nown that ths s acheved by settng a ˆ to the value whch maxmzes the a posteror probablty (APP) Pa ( aˆ ) = y gven the receved sequence y,.e. aˆ arg max P( a aˆ ) = = y (.7) cˆ he algorthms that acheve ths tas are commonly referred to as maxmum a posteror probablty (MAP) algorthms. When probabltes are concerned, t s often convenent to wor wth log-lelhood ratos (LLRs) rather than actual probabltes. he LLR for a varable a s defned as Pa ( = ) La ( ) = ln Pa ( = 0) (.8) he LLRs contans the same nformaton as the probabltes ( ) Pa = or Pa ( = 0). In fact, the sgn of La ( ) determnes whether Pa ( = ) s larger than Pa ( = 0) or t s on the contrary. herefore, by usng ths property of LLRs, the decson rules (.7) can be further wrtten as aˆ, La ( y) 0 = 0, La ( y) < 0 (.9) 4

40 Chapter Space-me rells Codes he man problem of the MAP approach s that the APPs calculaton s computaton-ntensve. o show ths, we use Bayes rule and the theorem of total probablty [5] on Pa ( aˆ ) = y, and obtan Pa ( = aˆ y) = P( a y) = a: a = aˆ a: a = aˆ P( y a) P( a) P( y) (.0) he computaton loadng n ths equaton s too heavy. herefore we ntroduce algorthms whch requre less computaton complexty than (.0)..4. BCJR Algorthm hs algorthm was proposed n974 by Bahl, Coce, Jelne, and Ravv [6], and s now nown as the name BCJR algorthm composed of the ntals of four authors. hs s a algorthm to effcently compute the APPs of nterest Pa ( = aˆ y ). r 3 r 3 r r r r r 0 r 0 Fgure.9 A trells example wth 4 states We denote the par (, j ) as the branch transton from state r to state r j. Defne a set β of (, j, ) and each par n the set s a vald branch transton to the chosen trells. For example, the trells n fgure.9 has the set We also denote s β = {( 00 ),( 0 ),( ),( 3 ),( 0 ),( ),( 3 ),( 33) } (.) S as the state of the encoder when -th set of nformaton bts s the nput. We denote by {,, v } S = r0 r as the set of all possble states where v 5

41 Chapter Space-me rells Codes s the memory order of the encoder. We begn wth the computaton of the probablty that the transmtted sequence path n the trells contaned the branch transton from state r to state r j when the -th set of nformaton bts s the nput,.e., to compute Ps ( = r, s = rj y ). Applyng the chan rule for jont probabltes,.e., Pab (, ) = PaPb ( ) ( a), we can obtan Ps (, s+, ) = P( ) P(s,s + ) he left hand sde of ths equaton can be wrtten as y y y (.) y = ( ) (.3) Ps (, s, ) P s, s,( y, y ), y,( y, y ) N + Applyng the chan rules agan on (.3), we obtan the ey decomposton (, +, y ) = (,(, ) ) ( +,,( +, N),(, ) ) = P( s, y, y ) P( s+, y s) P( y+, yn s+ ) P s s P s y y P s y y y s y y α ( s ) γ ( s, s ) β ( s ) (.4) It s easy to see that the term α ( s ) can be computed from α ( s ) and γ ( s s ), and therefore t can be extended to a recursve computaton:, ( s ) = ( s ) ( s s ) (.5) α α γ, s S wth ntal states α ( r ) = and zero for other states. Lewse, the term β ( s ) 0 0 has also a recursve computaton formula: wth end states ( ) and ( s ) N ( s ) = ( s ) ( s, s ) (.6) β β γ s+ S β r N 0 = and zeros for other states. he assgned values of α ( ) β are because an encoder always starts at the zeros state and end at the zero state. For all, the probablty of α ( s ) and β ( s ) must be normalzed to whch means v = 0 v = 0 α β ( r ) ( r ) = = 0 s (.7) From (.5) and (.6), we now that f we have the nowledge of all ( ) γ s, s +, = 0,, N, then we can compute all P( s, s, y ), 0,, N. he + = 6

42 Chapter Space-me rells Codes term γ ( s, s + ) can be further decomposed nto ( s, s ) = P( s s ) P( y s, s ) (.8) γ Wth specfc state numbers, (.8) can be wrtten as ( s r, s r ) (, j) (, j), (, ) P a = a P y x = x j β = = = 0, (, j) β γ + j (.9) where a, j represent the set of nformaton bts correspondng to branch transton from state r to state r j. Usually, the nformaton bts are assumed to be ndependent dentcally dstrbuted (..d.), whch maes (, j) P( a ) = m. As for P y x = x term, how to calculate t s dependent on the system. For the smple system wth only one decoder and AWGN channel, ths term s smply the pdf. of the nose dstrbuton,.e. ( y x ) Py ( x) = exp πσ σ (.30) where σ s the power of the nose. In ths thess, we wll propose a turbo equalzer system, and ths term wll be provded by the equalzer. Our goal s to compute the APPs P( a = aˆ y ). Snce we have derved the formula for P(s,s + y ), we can sum the APPs + P(s,s ) y over all branches that correspond to a = aˆ. hs dea s actually the theorem of total probablty [5]. ( ˆ = y) = ( =, + = j y) P a a P s r s r = = (, j) β: a = a, j (, j) β: a = a, j (, j) β: a = a, j ( =, + = j, y) P s r s r P ( y) ( r ) ( r, r ) + ( r ) α γ β j j P ( y) (.3) Fnally, the LLRs of the APPs are 7

43 Chapter Space-me rells Codes ( aˆ y) L a = = = (, j) β: a =, j (, j) β: a = 0, j (, j) β: a =, j (, j) β: a = 0, j ( r ) ( r, r ) + ( r ) α γ β j j P ( y) ( r ) ( r, r ) + ( r ) α γ β j j P ( y ) (.3) ( r ) ( r, r ) + ( r ) α γ β j j ( r ) ( r, r ) + ( r ) α γ β j j he turbo equalzer system wll also requre the computaton of L( c = cˆ y ), and ths s done smlarly as follows, ( cˆ y) L c = = = (, j) β: c =, j (, j) β: c = 0, j (, j) β: c =, j (, j) β: c = 0, j ( r ) ( r, r ) + ( r ) α γ β j j P ( y) ( r ) ( r, r ) + ( r ) α γ β j j P ( y ) (.33) ( r ) ( r, r ) + ( r ) α γ β j j ( r ) ( r, r ) + ( r ) α γ β j j hese LLRs wll be used n the turbo equalzer system descrbed n Chapter 3. 8

44 Chapter 3 urbo Equalzaton CHAPER 3 urbo Equalzaton For wdeband systems, because the bandwdth of the transmtted sgnals s large and often exceeds the coherent bandwdth of the channel, the channels usually have frequency-selectve characterstcs. he selectvty n frequency s aroused by the multpath fadng channel n whch transmsson paths have dfferent delays and fadngs. hs channel can be charactered by a tapped-delay lne flter. he objectve of an equalzer s to elmnate the dstorton nduced by the frequency-selectve channel. Structures of a varety of equalzers are surveyed n ths chapter. he algorthms to derve the optmal flter coeffcents are also brefly ntroduced here. At the end of ths chapter, a promsng equalzaton scheme ncorporated wth the turbo prncple s ntroduced. 3. Frequency-Selectve Channel Fgure 3. shows an equvalent dscrete baseband system ncludng transmtter and recever. Here we focus on channel and equalzer only. he system does not nclude any codng layer whch s commonly found n practcal communcaton systems. o add codng layers nto ths system s straghtforward, that s to replace 9

45 Chapter 3 urbo Equalzaton the nformaton source wth the coded bts. Informaton bts a n fgure 3. s mapped nto modulated symbols x accordng to a chosen modulaton. A transmt flter and a recever flter are used to meet the spectrum requrement and to mtgate the effect of frequency-selectve channel. he objectve of the equalzer at the recever s to recover the symbols x from the output of the receve flter y. he relaton between x and y s (( ) ) y = x g h g + n v (3.) where s the operaton of convoluton. We defne a new channel h to be h = g h g R (3.) and rewrte (3.) to be y = h x+ n (3.3) where n = n v. he h represents the total effect of the transmt flter, channel, and the receve flter. It s the channel that equalzers have to cope wth, and can be charactered by the tapped-delay lne model as shown n fgure 3.. hus, channels can be descrbed by the mpulse response ( h [ 0, ], h [ L] ) channel order. R where L s the a x d r y g h g R n ˆx â Fgure 3. Equvalent dscrete baseband system h[ 0] h [ ] h [ ] hl [ ] Fgure 3. apped delay lne model of the channel 30

46 Chapter 3 urbo Equalzaton 3. Equalzer Overvew 3.. rells-based Due to the tapped delay lne model of the channel, we can derve a trells descrpton for the channel. Wth ths trells descrpton, the BCJR algorthm we presented n.4. can also be appled readly to detect the symbols. o apply BCJR algorthm n the equalzer, the ntal values of α ( s ) and β ( s ) need to be modfed accordng to the property of channels. Usually, the channel starts at the zero state and ends at arbtrary state, whch leads to 0 0 N N, = 0 α0( r ) = 0, 0 β ( r) = for all N (3.4) Or, f the channel was preoccuped by the prevous transmsson, the value α ( s ) wll be ( r) α 0 = for all. hs approach of equalzaton requres the nowledge of channel state nformaton (CSI). hs requres another effort n channel estmatng, and the accuracy of the estmate wll affect the detecton performance n a certan level. However, the problem of an trells-based equalzer s the complexty. he state number of the equvalent trells s dependent on the channel order L, the sgnal constellaton M-PSK, and transmt antenna n n a MIMO system. It can be calculated by 0 0 n state number = ( M ) L (3.5) For example, f the transmtter uses transmt antennas and 4-PSK modulaton, and the channel order s 5, then we wll have a equvalent trells of state number ( 4 ) 5 = hs s a huge number for a decoder to buld, not to menton the one wth more antennas or wth hgher constellaton. herefore, ths approach s 3

47 Chapter 3 urbo Equalzaton only practcal when M, L, and n are very small. 3.. Flter-Based Due to the hgh complexty requred by trells-based equalzer, we turn to a much smpler approach, flter-based equalzaton. he trells-based approach does not recover the sgnals, but t calculates the APPs and then chooses the one wth max APPs to be the estmate sgnals. he flter-based does not calculate the APPs, but nstead, t tres to recover the sgnals of nterest and maes decsons on them whether n soft or hard decsons. he structures of flters can be categorzed nto lnear flters or decson-feedbac flters. f [ 0] f [ ] f [ ] f [ L] Fgure 3.3 Lnear euqalzer 3... Lnear Equalzer (LE) o compensate for the channel dstorton, we may employ a lnear flter wth adjustable coeffcents f. he flter coeffcents are adjusted on the bass of measurements of the channel characterstcs. here are dfferent crtera to derve the flter coeffcents and wll be addressed n A lnear equalzer s shown n fgure 3.3. Assumng the flter order s L f, the lnear operaton of the equalzaton can be expressed as H x ˆ = f y (3.6) 3

48 Chapter 3 urbo Equalzaton where y s the receved vector y y Lf he tme delay τ between adjacent taps may be selected as large as s, the symbol nterval, n whch case the FIR equalzer s called a symbol-space equalzer. In ths case the nput to the equalzer s the sampled receved sequence at a samplng rate equal to s s selected such that τ >. On the other hand, when the tme delay τ between adjacent taps s, the channel equalzer s sad to have fractonally spaced taps and t s called a fractonally spaced equalzer. he advantages of a fractonally spaced equalzer are that t provdes the functon of match flterng and t s less senstve to symbol samplng tmng. he dsadvantage s the computaton load ncrease. y f xˆ b x Fgure 3.4 Decson-feedbac equalzer 3... Decson-Feedbac Equalzer (DFE) he lnear flter equalzers descrbed above are very effectve on channels where the ISI s not severe. A decson-feedbac equalzer s a nonlnear equalzer that employs prevous decsons to elmnate the ISI caused by prevously detected symbols on the current symbol to be detected,.e., to elmnate the post-cursor part of ISI. he DFE conssts of two flters as shown n fgure 3.4. he frst flter s called a feedforward flter and s generally a fractonally spaced FIR flter. hs flter s dentcal n structure to the LE descrbed above. he second flter s a feedbac flter. 33

49 Chapter 3 urbo Equalzaton It s also mplemented as an FIR flter wth symbol-spaced taps b. Its nput s the set of prevously detected symbols. In a smple system where equalzer process one symbol only once, the nput of the feedbac flter s usually the hard decson of prevous equalzed symbols. In the turbo system we proposed n chapter 4, the nput to the feedbac flter are the estmate symbols produced from the decoder n the prevous teraton. Assumng the feedbac flter order s the output of a DFE can be expressed as L b, and the feedforward flter order s L f, H H x ˆ = f y b x (3.7) where y s the receved vector y y Lf and x s the feedbac flter nput of the form x x Lb. Snce DFE was ntroduced by Austn n 967, t has receved consderable attenton from many researchers due to ts mproved performance over the lnear equalzer and reduced mplementaton complexty as compared to the optmal trells-based equalzer we mentoned n 3... However, due to the feedbac of prevously detected symbols, a DFE suffers from error propagaton. Especally when SNR s low, the prevously detected symbols are erroneous, and thus the DFE under ths condton may not outperform the LE Flter Desgn Algorthm Flterng theorem s well-establshed and has a rch hstory. herefore, there are lots of algorthms and crtera to determne the flter coeffcents. he zero-forcng crteron and the mnmum mean square error crteron are the most famous two Zero-Forcng (ZF) 34