4. Space-Time Block Coding
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- Mitchell Gilmore
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1 . Space-Time Block Coding. Intoduction. Delay Diveity Scheme. Alamouti Space-Time Code. Space-Time Block Code.5 Decoding of STBC.6 Simulation Reult.7 Impefect Channel Etimation: A Pefomance Analyi.8 Effect of Antenna Coelation on Pefomance.9 Dominant Eigenmode Tanmiion.0 Capacity of OSTBC Channel. Simulation Execie Refeence State Key Lab. of ISN, Xidian Univeity
2 . Intoduction Space-Time block coding i a imple yet ingeniou tanmit diveity technique in MIMO technology. We hall now dicu pace-time block code (STBC) and evaluate thei pefomance in MIMO fading channel. We hall fit examine the Alamouti code, which tated it all. Baically Alamouti popoed a imple cheme fo a x ytem that achieve a full diveity gain with a imple maximum likelihood decoding algoithm. We hall then examine highe-ode diveity ytem involving a lage numbe of antenna, but whoe baic appoach i deived fom the method popoed by Alamouti. The pemie in all of thee appoache ha been that we have pefect knowledge of the channel at the eceive and that the data team ae independent. In eality, howeve, thi i not tue. Theefoe, we hall then go on to examine the behavio of thee pace-time code in the peence of impefect channel etimate and coelated low Rayleigh fading channel. State Key Lab. of ISN, Xidian Univeity
3 . Delay Diveity Scheme Suppoe we aume M T = and M R =. Initially let u examine what will happen if we tanmit the ame ignal imultaneouly fom both antenna. If we aume a flat fading envionment, the eceived ignal may be expeed a E h h n (.) E i the aveage enegy available at the tanmitte ove a ymbol peiod and i evenly divided between the tanmit antenna. n i the ZMCSCG noie, epeenting additive white Gauian noie ample at the eceive. State Key Lab. of ISN, Xidian Univeity
4 We know fom pobability theoy that the um of two complex Gauian andom vaiable i alo complex Gauian. Hence, we detemine that i alo ZMCSCG with unit vaiance. Hence, whee h i ZMCSCG with E{ h } =. Theefoe, we can eadily infe that thi technique doe not impat diveity. h h E h n (.) State Key Lab. of ISN, Xidian Univeity
5 Ealy attempt to obtain tanmit diveity wee baed on the o-called delay diveity cheme. In the delay diveity cheme, thi appoach i implemented, but with a mao diffeence. We do not tanmit the ame ymbol imultaneouly fom both antenna, but with a delay between the tanmiion (i.e., we tanmit the data ignal fom the fit antenna and a delayed eplica of the ame ignal fom the econd antenna afte an inteval). State Key Lab. of ISN, Xidian Univeity
6 If we aume that thi delay i one ymbol inteval, the effective channel a een by the eceive now become two channel a given by hi hi h i i 0,,,... (.) whee h and h ae the channel gain between the two tanmit antenna and the ingle eceive antenna, epectively. We aume that h and h ae i.i.d. ZMCSCG andom vaiable with unit vaiance. Fom the point of view of the eceive, uch a channel look exactly like a twopath channel with independent path fading and equal aveage path enegy. If we now employ a maximum likelihood (ML) detecto at the eceive, we can captue full econd-ode diveity at the eceive. State Key Lab. of ISN, Xidian Univeity
7 The negative ide to thi appoach i that the method intoduce intefeence between ymbol and the complexity of the ML detecto ie exponentially with the numbe of tanmit antenna. State Key Lab. of ISN, Xidian Univeity
8 . Alamouti Space-Time Code.. Maximum Likelihood Decoding.. Maximum Ratio Combining.. Tanmit Diveity.. Summay of Alamouti Scheme State Key Lab. of ISN, Xidian Univeity
9 Figue. A block diagam of the Alamouti pace-time encode State Key Lab. of ISN, Xidian Univeity
10 The Infomation bit ae fit modulated uing an M-ay modulation cheme. The encode then take a block of two modulated ymbol and in each encoding opeation and give it to the tanmit antenna accoding to the code matix, S (.) In (.), the fit column epeent the fit tanmiion peiod and the econd column the econd tanmiion peiod. The fit ow coepond to the ymbol tanmitted fom the fit antenna and the econd ow coepond to the ymbol tanmitted fom the econd antenna. Elaboating futhe, duing the fit ymbol peiod, the fit antenna tanmit and the econd antenna tanmit. Duing the econd ymbol peiod, the fit antenna tanmit and the econd antenna tanmit being the complex conugate of. Thi implie that we ae tanmitting both in pace (aco two antenna) and time (two tanmiion inteval). Thi i pace-time coding. State Key Lab. of ISN, Xidian Univeity
11 Looking at the equation, whee i the infomation equence fom the fit antenna and i the infomation equence fom the econd antenna. (.5) A cloe examination of (.5) eveal that the equence ae othogonal (i.e., the inne poduct of and i zeo). Thi inne poduct i given by 0, (.6) State Key Lab. of ISN, Xidian Univeity
12 Figue. Alamouti two-antenna tanmit diveity cheme. State Key Lab. of ISN, Xidian Univeity
13 The fading coefficient fom antenna and ae defined by h (t) and h (t), epectively, at time t. If we aume that thee coefficient ae contant aco two conecutive ymbol tanmiion peiod, we obtain h h t h t T t h t T whee h i and θ i, i =, ae the amplitude gain and phae hift fo the path fom tanmit antenna i to the eceive antenna and T i the ymbol duation. h h h h e e (.7) State Key Lab. of ISN, Xidian Univeity
14 At the eceive the ignal afte paing though the channel can be expeed a h h n h h n (.8) whee n and n independent complex vaiable with zeo mean and unit vaiance, epeenting additive white Gauian noie ample at time t and t + T, epectively. State Key Lab. of ISN, Xidian Univeity
15 .. Maximum Likelihood Decoding Aume that the channel coefficient h and h can be ecoveed pefectly at the eceive. We ue thee coefficient a the CSI. State Key Lab. of ISN, Xidian Univeity
16 State Key Lab. of ISN, Xidian Univeity The combine combine the eceived ignal a (.9) and end them to the maximum likelihood detecto, which minimize the following deciion metic (.0) ove all poible value of and. Expanding thi and deleting tem that ae independent of the code wod, the above minimization educe to epaately minimizing fo detecting and fo decoding. ~ ~ n h n h h h h h n h n h h h h h h h h h h h h h h h h h
17 Equivalently, if we ue the notation d (x,y) = x-y = (x-y)(x-y), the deciion ule fo each combined ignal š, =, become: Pick ŝ i, iff h h d ~, h h d ~, i k i i (.) Fo PSK ignal (equal enegy contellation), thi implifie to d ~, i d ~, k i k (.) k k State Key Lab. of ISN, Xidian Univeity
18 .. Maximum Ratio Combining The eceived ignal ae h n 0 h n 0 and the combined ignal i ~ h h h h h n h n 0 0 (.) The maximum likelihood detecto decide ignal i uing exactly the ame ule in (.) o (.) fo PSK ignal. State Key Lab. of ISN, Xidian Univeity
19 Figue. Maximum atio combining with Tx and Rx State Key Lab. of ISN, Xidian Univeity
20 Note that the MRC ignal ŝ 0 in (.) i equivalent to the eulting combined ignal of the tanmit diveity cheme in (.), except fo a phae diffeence in the noie component that do not affect the effective SNR. Thi how that the diveity ode fom Alamouti two-antenna tanmit diveity (with one eceive antenna) i the ame a that of the two-banch MRC. State Key Lab. of ISN, Xidian Univeity
21 .. Tanmit Diveity The tanmiion in the Alamouti cheme ae othogonal. Thi implie that the eceive antenna ee two completely othogonal team. Hence, we obtain a tanmit diveity of two. State Key Lab. of ISN, Xidian Univeity
22 State Key Lab. of ISN, Xidian Univeity Conide two ditinct code equence S and Ŝ geneated by the input (, ) and (ŝ,ŝ ), epectively, whee (, ) (ŝ,ŝ ). The code wod diffeence matix i given by (.) Since the ow of the code matix ae othogonal, the ow of the code wod diffeence matix ae othogonal a well. The code wod ditance matix i given by (.5) ˆ ˆ ˆ ˆ ˆ, S S B ˆ ˆ 0 0 ˆ ˆ ˆ, ˆ, ˆ, S S B S S B S A S H
23 Since (, ) (ŝ,ŝ ), vey obviouly the ditance matixe of any two ditinct code wod have a full ank of two. In othe wod, the Alamouti cheme give u a tanmit diveity of M T =. The deteminant of matix A(S,Ŝ) i given by detas, Sˆ ˆ ˆ (.6) State Key Lab. of ISN, Xidian Univeity
24 The code wod ditance at (.5) ha two identical eigenvalue. The minimum eigenvalue i equal to the minimum quaed Euclidian ditance in the ignal contellation. Hence, the minimum ditance between any two tanmitted code equence emain the ame a in the uncoded ytem. Thi implie that the coding gain i. Thi i the diadvantage of the Alamouti cheme in that, unlike pace-time telli code, the cheme achieve full tanmit diveity gain without CSI at the tanmitte but ha no coding gain unle it i given the CSI. State Key Lab. of ISN, Xidian Univeity
25 .. Summay of Alamouti Scheme Alamouti futhe extended thi cheme to the cae of two tanmit antenna and M R eceive antenna and howed that the cheme povided a diveity ode of M R. State Key Lab. of ISN, Xidian Univeity
26 Chaacteitic of Alamouti cheme: No feedback fom eceive to tanmitte i equied fo CSI to obtain full tanmit diveity. No bandwidth expanion (a edundancy i applied in pace aco multiple antenna, not in time o fequency). Low complexity decode. Identical pefomance a MRC if the total adiated powe i doubled fom that ued in MRC. Thi i becaue, if the tanmit powe i kept contant, thi cheme uffe a -db penalty in pefomance ince the tanmit powe i divided in half aco two tanmit antenna. No need fo complete edeign of exiting ytem to incopoate thi diveity cheme. Hence, it i vey popula a a candidate fo impoving link quality baed on dual tanmit antenna technique, without any datic ytem modification. State Key Lab. of ISN, Xidian Univeity
27 . Space-Time Block Code.. STBC fo Real Signal Contellation.. STBC fo Complex Signal Contellation State Key Lab. of ISN, Xidian Univeity
28 The Alamouti cheme bought in a evolution of ot in multiantenna ytem by poviding full diveity of two without CSI at the tanmitte and a vey imple maximum likelihood decoding ytem at the eceive. Maximum likelihood decode povide full diveity gain of M R at the eceive. Hence, uch a ytem povide a guaanteed oveall diveity gain of M R, without CSI at the tanmitte. Thi i achieved by the key featue of othogonality between the equence geneated by the two tanmit antenna. Due to thee eaon, the cheme wa genealized to an abitay numbe of tanmit antenna by applying the theoy of oghogonal deign. The genealized cheme ae efeed to a pace-time block code (STBC). Thee code can achieve the full tanmit diveity of M T M R, while allowing a vey imple maximum likelihood decoding algoithm, baed only on linea poceing of eceived ignal. State Key Lab. of ISN, Xidian Univeity
29 Figue. Encode fo STBC State Key Lab. of ISN, Xidian Univeity
30 Let M T epeent the numbe of tanmit antenna and p epeent the numbe of time peiod fo tanmiion of one block of coded ymbol. Let u alo aume that the ignal contelltion conit of m point. Then each encoding opeation map a block of km infomation bit into the ignal contellation to elect k modulated ignal,,, k, whee each goup of m bit elect a contellation ignal. Thee k modulated ignal ae then encoded in a pace-time block encode to geneate M T paallel ignal equence of length p. Thi give ie to a tanmiion matix S of ize M T xp. Thee equence ae tanmitted though M T tanmit antenna imultaneouly in p time peiod. State Key Lab. of ISN, Xidian Univeity
31 The numbe of ymbol the encode take a it input in each encoding opeation i k. The numbe of tanmiion peiod equied to tanmit the entie S matix i p. The ate of the pace-time block code i defined a the atio between the numbe of ymbol the encode take a it input and the numbe of pace-time coded ymbol tanmitted fom each antenna. It i given by R k p (.7) The pectal efficiency of the pace-time block code i given by mr km b bit/ Hz B p / (.8) whee b and ae the bit and ymbol ate, epectively, and B i the bandwidth. State Key Lab. of ISN, Xidian Univeity
32 The entie of the tanmiion matix S ae o choen that they ae linea combination of the k modulated ymbol,,, k and thei conugate,,, k. The matix itelf i o contucted baed on othogonal deign uch that S S H c (.9) whee c i a contant, M T i the numbe of tanmit antenna, S H i the Hemitian of S, and I MT i an M T xm T identity matix. Thi appoach yield a diveity of M T. k I MT State Key Lab. of ISN, Xidian Univeity
33 Thee code tanmiion matixe ae clevely contucted that the ow and column of each matix ae othogonal to each othe. The code ate will vay depending on how the matix i contucted. Baed on (.7), we can have R =, which i a full ate. Thi implie that thee i no bandwidth expanion involved, wheea a code with ate R < implie a bandwidth expanion facto of /R. Code ate of unity (i.e., full ate) ae elatively eaily achievable if the matix i eal, but the choice fo full-ate code i moe eticted if the matix i complex. Uing (.9), the othogonality achieved in all cae enable u to achieve full tanmit diveity, iepective of the code ate and additionally allow the eceive to decouple the ignal tanmitted fom diffeent antenna. State Key Lab. of ISN, Xidian Univeity
34 .. STBC fo Real Signal Contellation Squae tanmiion matixe (.9) i cucial to ou deign. The numbe of tanmit antenna M T =,, o 8. Thee code ae full ate and full tanmit diveity of M T. State Key Lab. of ISN, Xidian Univeity
35 State Key Lab. of ISN, Xidian Univeity The tanmiion matixe fo M T = tanmit antenna (.0) The tanmiion matixe fo M T = tanmit antenna (.) S S
36 State Key Lab. of ISN, Xidian Univeity The tanmiion matixe fo M T =8 tanmit antenna (.) S
37 If we deie to contuct full code ate R = tanmiion cheme fo any numbe of antenna, we can do o by following anothe et of ule applicable to both quae and nonquae matixe. Thi ule ay that fo M T tanmit antenna, the minimum value of tanmiion peiod p to achieve the full ate i given by p min whee the minimization i taken ove the et c, d 0 cd c, 0 d, and 8c d M T (.) (.5) State Key Lab. of ISN, Xidian Univeity
38 State Key Lab. of ISN, Xidian Univeity Baed on (.) and (.5), we contuct nonquae matixe of ize, 5, 6, and 7 fo eal numbe, yielding full diveity and full ate. (.6) (.7) S S
39 State Key Lab. of ISN, Xidian Univeity (.8) (.9) S S
40 .. STBC fo Complex Signal Contellation Complex othogonal deign matixe ae defined a matixe of ize M T xp with complex entie of,,, k and thei conugate and atifying (.9). Such matixe povide the full tanmit diveity of M T with a code ate of k/p. State Key Lab. of ISN, Xidian Univeity
41 State Key Lab. of ISN, Xidian Univeity Complex tanmiion matix of ize M T = with a full code ate of (.) The Alamouti cheme i unique in that fo complex entie, it i the only uch matix with a code ate of unity. Complex tanmiion matix of ize M T = and M T = incopoating a code ate of /. (.) (.) G G G
42 State Key Lab. of ISN, Xidian Univeity The deie fo highe code ate lead u to moe complex linea poceing. The following ae ize and code with ate /: (.) (.5) The each i till on code with ate geate than 0.5. Thi i till an open field fo futhe eeach. H H
43 State Key Lab. of ISN, Xidian Univeity.5 Decoding of STBC The decode fo G minimize the deciion metic fo decoding, the deciion metic fo decoding, the deciion metic fo decoding, the deciion metic fo decoding. The decoding of thee code i imila to the one popoed fo Alamouti cheme.,, 7, 6, 5,,, m i i m,, 8, 6, 5,,, m i i m,, 8, 7, 5,,, m i i m,, 8, 7, 6,,, m i i m
44 State Key Lab. of ISN, Xidian Univeity The decode fo G minimize the deciion metic fo decoding, the deciion metic fo decoding, the deciion metic fo decoding, the deciion metic fo decoding.,, 8, 7, 6, 5,,,, m i i m,, 8, 7, 6, 5,,,, m i i m,, 8, 7, 6, 5,,,, m i i m,, 8, 7, 6, 5,,,, m i i m
45 .6 Simulation Reult Simulation model: In thi ection, we how imulation eult petaining to the pefomance of STBC on Rayleigh fading channel. In the imulation, it i aumed that the eceive ha pefect CSI and that the fading between tanmit and eceive antenna i mutually independent. We futhe aume that the total tanmitted powe level fom two antenna fo the Alamouti cheme i the ame a the tanmit powe fom the ingle tanmit antenna fo the MRC eceive diveity cheme and that it i nomalized to one. The SISO channel i hown fo compaion. State Key Lab. of ISN, Xidian Univeity
46 Figue.5 The BER pefomance of 6QAM Alamouti cheme with diffeent antenna combination in low Rayleigh fading channel State Key Lab. of ISN, Xidian Univeity
47 Simulation eult: x Alamouti cheme v. x MRC cheme The x Alamouti cheme and the x MRC cheme have the ame diveity of ode, a the lope of both the cuve ae the ame. The x Alamouti cheme appea to have le gain than the x MRC. Thi i becaue the powe fom the Alamouti cheme i divided equally between the antenna (i.e., db le pe antenna than the powe fom the MRC cheme, which ha only one antenna). x Alamouti cheme v. x MRC cheme Both have the ame diveity of ode. The x Alamouti cheme will be db infeio to the x MRC cheme. State Key Lab. of ISN, Xidian Univeity
48 The Alamouti cheme doe not poe CSI of the channel. Hence, it cannot povide aay gain at the tanmitte but it doe povide diveity gain at the tanmitte becaue the data team ae othogonal. The MRC cheme doe not poe diveity gain at the tanmitte becaue thee i only one antenna, but it doe poe aay gain at the eceive becaue it poee CSI at the eceive. State Key Lab. of ISN, Xidian Univeity
49 Thee ae thee type of gain that we ae dicuing: Coding gain: Thi i the gain povided by tempoal coding, like convolutional coding and block coding. Aay gain: Thi efe to the aveage inceae in the SNR at the eceive that aie fom the coheent combining effect of multiple antenna at the eceive o tanmitte o both. Thi i the gain povided by CSI, which i ealized by udiciou weighting of the ignal fom each antenna baed on CSI. State Key Lab. of ISN, Xidian Univeity
50 Diveity gain: Thi i the gain povided by patial diveity aco channel, eithe at the tanmitte o eceive o both and i neceay fo combating fading. The tipulation in thi cae i that the gap aco the antenna bank unde quetion ( eithe at the tanmitte o eceive o both) hould be epaated by ditance exceeding the coheent ditance fo that channel. Thi enue that the channel between each tanmit-eceive antenna pai fade independently. Nomally, ytem equie CSI to achieve diveity gain, but Alamouti cheme doe not equie CSI to achieve tanmit diveity becaue it i baed on othogonal data team. The diveity ode i equal to the poduct of the numbe of tanmit and eceive antenna. State Key Lab. of ISN, Xidian Univeity
51 When CSI i available eithe at the tanmitte o eceive, we obtain both aay gain and diveity gain at the epective poition. The fome diectly inceae the SNR wheea the latte educe deep fade, impoving quality of eception and theeby alo inceaing SNR. Hence, the net inceae in SNR i a contibution of both aay gain a well a diveity gain. Aay gain: The aay gain i calculated baed on the antenna aay geomety and i given in any tandad textbook on antenna. The aay gain can alo be expeed in db uing the elationhip 0log 0 (x). Diveity gain State Key Lab. of ISN, Xidian Univeity
52 The natue of coding and diveity gain i diffeent. Diveity gain manifet itelf in inceaing the magnitude of the lope of the cuve, wheea coding gain hift the eo ate cuve to the left. We mut be caeful when looking at the amount of coding gain fom the cuve, becaue it alo include aay gain. The SNR advantage due to diveity gain inceae with diveity ode and lowe taget eo ate. Coding gain i typically contant at high SNR. Thi i becaue at high SNR the ignal ae aleady tong enough to be coectly eceived and theefoe the peence/abence of tempoal coding doe not affect the ituation and the coding become contant. State Key Lab. of ISN, Xidian Univeity
53 Simulation model: We examine the pefomance of STBC uing vaiable numbe of tanmit/eceive antenna on Rayleigh fading channel. Duing thee imulation, we hall vay the type of modulation and the code ize, with a view to maintaining contant pectal efficiency o a to enue a fai compaion. We aume that the eceive know the pefect CSI. State Key Lab. of ISN, Xidian Univeity
54 Figue.6 BER pefomance fo STBC of bit//hz on Rayleigh fading channel with one eceive antenna State Key Lab. of ISN, Xidian Univeity
55 The BER fo STBC with bit//hz and a vaiable numbe of tanmit antenna i hown in Fig..6. The pefomance of an uncoded 8PSK i plotted fo compaion. An 8PSK modulation with a full-ate code will yield a pectal efficiency of bit//hz. Thi i hown fo a x ytem. Thee and fou antenna ytem employ ate / code uing H and H code, epectively. Thi, with 6QAM modulation, yield a pectal efficiency of bit//hz. In all cae we employ the ame tanmiion ate (i.e., bit//hz). We obeve that at the BER of 0-5, the code G i bette by about 7 db and.5 db than the code G and G, epectively. State Key Lab. of ISN, Xidian Univeity
56 Figue.7 BER pefomance fo STBC of bit//hz on Rayleigh fading channel with one eceive antenna State Key Lab. of ISN, Xidian Univeity
57 The pefomance of a ituation yielding a pectal efficiency of bit//hz with two, thee, and fou tanmit antenna and one eceive antenna on a Rayleigh fading channel i hown in Fig..7. If we deploy two antenna, we need to ue a ate code G, and with QPSK modulation to enue a pectal efficiency of bit//hz. With ate / code G and G, we need to employ 6QAM modulation to achieve the ame tanmiion ate. We note that at a BER of 0 -, the code with fou tanmit antenna gain about db elative to the code with two and thee antenna, epectively. At highe BER thi diffeence i even highe. State Key Lab. of ISN, Xidian Univeity
58 Figue.8 BER pefomance fo STBC of bit//hz on Rayleigh fading channel with one eceive antenna State Key Lab. of ISN, Xidian Univeity
59 The pefomance yielding a pectal efficiency of bit//hz i hown in Fig..8. x ytem: BPSK modulation with ate code G. x ytem: QPSK modulation with ate / code G. x ytem: QPSK modulation with ate / code G. At the BER of 0 -, the code with fou antenna i upeio by about.5 db and.5 db to the code with two and thee antenna, epectively. State Key Lab. of ISN, Xidian Univeity
60 An inceae in tanmit diveity impove the pefomance. Thi i a vey impotant infeence fom a commecial point of view, becaue handheld mobile alway poe a lot of poblem in achieving antenna diveity at the eceive. Hence, tanmit diveity at the bae tation hold a lot of pomie. Thi type of diveity i eaie to implement and conequently, a lot of impotance i being attached to attaining thi type of diveity. We can eaily inceae the ize of the code fom two to thee and to fou, with a vey little inceae in decoding complexity, due to the fact that only linea poceing i equied fo decoding. State Key Lab. of ISN, Xidian Univeity
61 .7 Impefect Channel Etimation: A Pefomance Analyi.7. Leat Squae Etimation.7. Minimum Mean Squae Etimation.7. Channel Etimation Algoithm Uing the FFT Method State Key Lab. of ISN, Xidian Univeity
62 It i aumed that the channel i contant ove the duation of a fame and independent between fame. Thee ae two pinciple technique, baed on the manne of tuctuing the taining equence, ued in channel etimation. Peamble tuctue: We append a packet of taining ymbol that ae tictly pilot ymbol. Obviouly, the moe the ymbol, the bette the etimate. Thi method i effective only in low fading channel, a the channel i equied to emain tatic fo the duation of the peamble and the data ymbol. Pilot tuctue: The packet conit of both pilot and infomation ymbol. Thee i a need to compomie between the numbe of pilot eential fo coect etimation and the numbe of infomation data neceay to maintain the deied data thoughput. Thi method allow fo tacking a fat moving channel but i le accuate. State Key Lab. of ISN, Xidian Univeity
63 Since we ae dealing with a multiantenna ytem, it become neceay to diciminate between the pilot (i.e., we need to know at the eceive a to which pilot ae coming fom which antenna). We achieve thi by making the taining equence fom the individual antenna othogonal to each othe. Othogonal in time: We tanmit fom each antenna by tun. Othogonal in fequency: Each antenna will ue one tone, diffeent fom the tone fom othe antenna. Othogonal in coding: We chooe taining ymbol fo each antenna, uch that they ae othogonal to each othe. State Key Lab. of ISN, Xidian Univeity
64 .7. Leat Squae Etimation The channel etimate can be obtained by minimizing the eo matix E k pe ymbol whee E k i given by H E R S R S (.6) k k k whee R k i the eceived vecto with the kth ymbol, S k i the kth tanmitted ample, and η k i the channel coefficient fo the kth ymbol. The leat quae olution fo the channel etimate i obtained a k, LS k H H S S S R k k k k (.7) In cae the peamble ae deigned uch that they ae othogonal, then the matix S i unitay. In uch a cae, (.7) implifie to H ~ S R S R W (.8) k k k k k k k whee W i the additive white Gauian noie matix Ŵ k =S - k W k. k k k State Key Lab. of ISN, Xidian Univeity
65 .7. Minimum Mean Squae Etimation The channel fading coefficient ae etimated by ineting pilot equence in the tanmitted ignal. In geneal, with M T tanmitting antenna we need M T diffeent pilot equence P,P,,P MT, which ae othogonal to each othe. Thee pilot equence ae tanmitted a a peamble of k ymbol. P P, P,..., P i i, i, i, k (.9) The eceived ignal at antenna at time t i given by MT h P t, i i, t t i (.0) whee h,i i the fading coefficient fo the path fom tanmit antenna i to eceive antenna and n t i the noie ample at eceive antenna at time t. n State Key Lab. of ISN, Xidian Univeity
66 State Key Lab. of ISN, Xidian Univeity The eceived ignal and noie vecto at antenna can be epeented a (.) The minimum mean quae etimation (MMSE) of h,i i given by (.) whee e,i i the etimation eo due to the noie and i given by (.) Since n t i a zeo-mean complex Gauian andom vaiable with ingle-ided powe pectal denity N 0, the etimation eo e,i ha a zeo mean and ingle-ided powe pectal denity N 0 /k. The advantage of MMSE ove LSE i that it i a mean etimato (i.e., it even aveage out the noie). i i i i i i i i e h P P n h P P h,,,, ~, i i i P P n e k k n n n n,...,,,...,,
67 Figue.9 Pefomance of STBC with bit//hz uing impefect channel etimation and two eceive antenna State Key Lab. of ISN, Xidian Univeity
68 Simulation model: We ue a ate code G with QPSK modulation. The channel model i a low-fading Rayleigh channel with contant coefficient ove a fame of 0 ymbol. The pilot equence ineted in each fame ha a length of 0 ymbol in the peamble mode. Simulation eult: Due to impefect channel etimation, the code pefomance i degaded by about db compaed with the ideal. Thi degadation alo include the degadation caued due to lo of ignal enegy by appending the pilot equence. The pefomance degadation i alo linked to the numbe of tanmit antenna. If the numbe of tanmit antenna inceae, the enitivity of the ytem to channel etimation eo inceae. State Key Lab. of ISN, Xidian Univeity
69 .7. Channel Etimation Algoithm Uing the FFT Method Extending Eq. (.8) to a et of ymbol, we obtain fo N ymbol (giving ie to a block matix M T xm R ), auming that all S k ae noningula (I.e., they can be inveted). We call thee channel etimate coae channel etimate. The channel etimate ae futhe impoved by fit taking N-point invee fat Fouie tanfom (IFFT) of the M T xm R coae channel etimate vecto to convet them to the time domain. Thee M T xm R length-n vecto ae then paed though a ectangula window uch that h m g 0 0 m whee G i the maximum tap length of the channel. m m G G (.7) State Key Lab. of ISN, Xidian Univeity
70 The time domain fine channel etimate ĥ m ae then conveted uing fat Fouie tanfom (FFT) to fine channel etimate in fequency domain uch that ~ ~ W q M, l M h q, FFT l m q, l Becaue of the window opeation, the noie vaiance in the fine channel etimate i educed to σ G/N. q, l T R (.8) State Key Lab. of ISN, Xidian Univeity
71 Figue.0 Pefomance of STBC with bit//hz uing impefect channel etimation uing FFT method and two eceive antenna State Key Lab. of ISN, Xidian Univeity
72 Simulation model: Thi method wa applied to a fame of 6 ymbol with a peamble of 6 pilot in a low Rayleigh fading channel. One ue LSE and the othe with FFT. The equence wee othogonal in time. QPSK modulation fo a ate code G. Simulation eult: At a BER of 0 - thee i a pefomance gap of db fom the ideal if we ue the LSE method. Howeve, thi i impoved to 0.7 db fom the ideal if we ue the FFT method. The pefomance will be even bette if we ue MMSE intead of LSE in the fit tep. State Key Lab. of ISN, Xidian Univeity
73 .8 Effect of Antenna Coelation on Pefomance Aume that the tanmit antenna ae not coelated but the eceive antenna ae coelated. The eceive coelation matix i given by R whee α i the coelation facto between the eceive antenna. (.9) State Key Lab. of ISN, Xidian Univeity
74 Figue. Pefomance of STBC with bit//hz on coelated low Rayleigh fading channel with two tanmit and two eceive antenna State Key Lab. of ISN, Xidian Univeity
75 Simulation eult The coelation pogeively degade the pefomance by 0.5 db, 0.5 db,. db and db at a BER of 0 - fo a coelation facto of 0.5, 0.5, 0.75 and elative to the ideal cae, epectively. State Key Lab. of ISN, Xidian Univeity
76 .9 Dominant Eigenmode Tanmiion The poeion of complete channel knowledge at the tanmitte lead u to the vey inteeting concept of dominant eigenmode tanmiion. Conide a M T xm R ytem and channel etimation uing feedback. Thi feedback will involve ome delay ( o lag), δ lag. Since wiele channel ae time-vaying, we need whee T c i the coheence time. Theefoe, δ lag /T c detemine channel accuacy at the tanmitte. lag T c (.50) State Key Lab. of ISN, Xidian Univeity
77 The eceived ignal vecto i given by y E M (.5) T whee y i the M R x eceived ignal vecto, H i the M R xm T channel tanfe function, w i the M T x complex weight vecto, and n i patially white ZMCSCG noie. The weighted um of all antenna output at the eceive i given by H z g y (.5) whee g i a M R x vecto of complex weight. The SNR at the eceive, η, i given by g Hw n H Hw M g (.5) T F whee. F i Fobeniou nom of a matix, which i the um of nom of all the matix element and ρ i the SNR at the eceive fo an SISO channel. Note that w =M T to maintain total aveage tanmitted enegy. State Key Lab. of ISN, Xidian Univeity
78 Hence, maximizing the SNR at the eceive i equivalent to maximizing g H Hw F / g F. The ingula value decompoition (SVD) of H i given by H H UV (.5) Fom (.5) we note that η i maximized when w M T and g ae the input and output ingula vecto, epectively, coeponding to the maximum ingula value σ max of H. Uing thee value, the effective input-out elation fo the channel educe to z E n max (.55) whee n i ZMCSCG noie with vaiance N 0. Singula value of H ae quae oot of eigenvalue of HH H. Theefoe, σ max=λ max, whee λ max i the maximum eigenvalue of HH H, the SNR at the eceive i given by (.56) max The aay gain of dominant eigenmode tanmiion i given by E{λ max }. State Key Lab. of ISN, Xidian Univeity
79 .0 Capacity of OSTBC Channel The capacity of an othogonal pace-time block code (OSTBC) channel i given by C H OSTBC F (.58) M T Fom (.7), the capacity of a MIMO channel when the channel i unknown to the tanmitte i given by E H C log det I HH M R M N T 0 log log C OSTBC k M M T T H F k M T (.59) State Key Lab. of ISN, Xidian Univeity
80 The capacity of OSTBC channel i infeio to the channel with optimal coding except in the cae of Alamouti cheme wheein the code ate = cauing C = C OSTBC. Howeve, the outage popetie of OSTBC will be upeio to the outage obtained with optimal coding fo a given tanmiion ate, becaue OSTBC fundamentally impove the link. State Key Lab. of ISN, Xidian Univeity
81 Table. Aay gain and diveity ode fo diffeent multiple antenna configuation Configuation Expected Aay Gain Diveity Ode SIMO(CU) M R M R SIMO(CK) M R M R MISO(CU) M R MISO(CK) M T M R MIMO(CU) M R M R M T MIMO(CK) E(λ max ) M R M T CU: Channel unknown to the tanmitte CK: Channel known to the tanmitte State Key Lab. of ISN, Xidian Univeity
82 . Simulation Execie Ty out the execie in thi chapte. Implement a x cheme uing dominant eigenmode tanmiion. How doe it compae with the Alamouti (G ) cheme? State Key Lab. of ISN, Xidian Univeity
83 Refeence Alamouti, S. M., A Simple Tanmit Diveity Technique fo Wiele Communication, IEEE Jounal Select. Aea Commun., Vol. 6, No. 8, Octobe 998, pp Sehadi, N., and J. Winte, Two Signaling Scheme fo Impoving the Eo Pefomance of Fequency- Diviion-Duplex (FDD) Tanmiion Sytem Uing Tanmitte Antenna Diveity, Int. J. Wiele Infomation Netwok, Vol., Januay 99, pp State Key Lab. of ISN, Xidian Univeity
84 Taokh, v., H. Jafakhani, and A. R. Caldebank, Space-Time Block Coding Fom Othogonal Deign, IEEE Tan. Infom. Theoy, Vol. 5, No. 5, July 999, pp Taokh, v., H. Jafakhani, and A. R. Caldebank, Space-Time Block Coding fo Wiele: Pefomance Reult, IEEE J. Select. Aea Commun., Vol. 7, No., Mach 999, pp Mody, A. N., and G. L. Stube, Paamete Etimation fo OFDM With Tanmit Receive Diveity, IEEE Vehicula Technology Confeence, Rhode, Geece, May 00. Mody, A. N., and G. L. Stube, Synchonization fo MIMO-OFDM Sytem, IEEE Global Communication Confeence, San Antonio, TX, Novembe 00. State Key Lab. of ISN, Xidian Univeity
85 Siew, J., et al., A Channel Etimation Method fo MIMO-OFDM Sytem, London Communication Sympoium, London, Septembe 00. Taokh, V., et al., Space-Time Code fo High Data Rate Wiele Communication: Pefomance Citeia in the Peence of Channel Etimation Eo, Mobility, and Multiple Path, IEEE Tan. Commun., Vol. 7, No., Febuay 999, pp Paula, A., R. Naba, and D. Goe, Intoduction to Space-Time Wiele Communication, Cambedge, UK: Cambidge Univeity Pe, 00. Golub G., and C. Van Loan, Matix Computation, Baltimoe, MD: John Hopkin Univeity Pe, nd edition, 989. State Key Lab. of ISN, Xidian Univeity
86 Haibi B., and B. Hochwald, High-Rate Code That Ae Linea in Space and Time, IEEE Tan. Inf. Theoy, Vol. 8, No. 7, July 00, pp State Key Lab. of ISN, Xidian Univeity
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