Channel Esimaion for Wired MIMO Communicaion Sysems Final Repor Mulidimensional DSP Projec, Spring 2005 Daifeng Wang Absrac This repor addresses raining-based channel modeling and esimaion for a wired muliuser mulicarrier communicaions sysem. The special case of a muliple-inpu muliple-oupu (MIMO) channel is considered where he differen users ransmi a he same ime and over he same bandwidh. In he repor, I will inroduce he daa ransmission wih he mulicarrier modulaion and raining sequence designs based on he wired MIMO communicaion sysems. The repor also will presen a MIMO channel model. Then, an opimal channel esimaion mehod by raining sequences is presened and analyzed. I. Inroducion Communicaion sysems ha use muliple ransmiers and receivers are ofen called muliple-inpu muliple-oupu (MIMO) sysems. The bonded Asymmeric Digial Subscriber Line (ADSL) is a wired MIMO communicaion sysem [11]. The curren challenges for MIMO sysems are sill he ransmission power, bandwidh, and compuaional complexiy and connecion speeds [1]. To esimae an unknown channel is a very imporan and necessary work before ransmiing he real signals since he channel is commonly ime-varying. The channel esimaion can be performed by sending a known raining sequence, by ransmiing pilo signals, and by using cyclic saisics of he received signal [2] [3]. The wired communicaion channel is slowly ime-varying, so a preamble raining sequence is ofen used o esimae he channel. 1
In his repor, Secion II inroduces some key echniques and describes he mulicarrier daa ransmission and recepion for MIMO sysems. Secion III describes Training-Based MIMO channel models. Secion IV presens and analyzes raining sequence design mehods. Secion V presens and discusses an opimal channel esimaion mehod by raining sequences. Secion VI concludes and summarizes his repor. II. Background All ransmission channels are fundamenally analog and hus may exhibi a wide variey of ransmission effecs. Modulaion convers a sream of inpu bis ino equivalen analog signals ha are suiable for he ransmission line. A primary impairmen in communicaions is iner-symbol inerference (ISI) which is caused by he memory in he channel. To comba ISI, a receiver usually uses an equalizer o compensae for he spreading in ime and disorion in frequency caused by he channel [12]. Since he equalizer is designed in erms of he channel, i is very imporan o esimae he channel a he receiver for he equalizer. One echnique o avoid ISI, wihou sacrificing he ransmission rae, is Mulicarrier Modulaion (MCM). MCM divides a broadband channel ino narrowband subchannels ha have heir own cener carrier frequencies. There is no ISI in subchannels if each subchannel is ideally sampled and has consan gain. Because of MCM s robusness o mulipah, and he ease of implemening MCM using he fas Fourier ransform (FFT), he MCM concep is growing rapidly in pracical imporance. I has been implemened in several wireline and wireless high-speed daa communicaions sandards. The discree mulione modulaion (DMT) is a MCM applicaion in he wired communicaion sysem (e.g. ADSL and VDSL sysem) [4] [11]. A Mulicarrier Modulaion ransmier is shown below [5]: 2
Figure 1 Mulicarrier Modulaion Transmission The receiver is he dual of he ransmier assuming ha he receiver has adjused o he channel impairmens and ransmier imperfecions. The inpu of he S/P converer is a sequence of symbols of B bis each; he oupu for each symbol is N car groups of bn ( ) bi each. Tha is B= bn ( ). The groups of bn ( ) are hen consellaion-encoded, perhaps n N car filered, and hen modulaed ono N car subcarriers. III. MIMO Channel Modeling [4] TX 1 h11 RX 1 TX 2 h21 h12 RX 2 h22 Figure 2 MIMO wih wo ransmiers and wo receivers Consider a sysem ha employs wo-ransmi and wo-receive anennas simply. Two raining signals and are ransmied over four wired channels 3
T hij ( L) = hij (0) hij ( L 1), i or j = 1,2 (1) where () i T denoes he ranspose operaion. h 11 and h22 are main direc channel impulse responses beween he same ransmier and receiver. h 12 and h 21 are cross channel impulse responses beween he differen ransmier and receiver. Each channel is modeled as a finie-impulse response (FIR) filer wih L aps. The inpu raining sequences s 1 and s 2 belong o a finie-signal consellaion and are ransmied in daa blocks where each block consiss of N i informaion symbols and N raining symbols. For wo ransmiers, he receiver uses he 2N known raining symbols o esimae he unknown 2L channel coefficiens. The observed raining sequence oupu ha does no have inerference from informaion or preamble symbols can be expressed as [6] y1 h11( L) h12( L) y= Sh z [ S1( L, N) S2( L, N)] z y = + = + 2 h21( L) h22( L) (2) where y and z are of dimension 2( N L+ 1) 1, z is assumed o be addiive whie Gaussian noise(awgn). S 1 and S 2 are Toepliz marices of dimension ( N L+ 1) L, and si( L 1) si(0) si( L) si(1) Si( L, N) =, i = 1,2 si( N 1) si( N L) IV. Training Sequences Design A known raining sequence is ransmied o esimae he channel impulse response before daa ransmission in digial communicaion sysems. Training sequences are 4
periodic or aperiodic [7]. In eiher case, he power specrum of he raining sequence is approximaely a over he ransmission bandwidh. The suggesed raining sequence for channel esimaion in a DMT sysem is a pseudo-random binary sequence wih N samples [4]. The raining sequence is made periodic by repeaing N samples or adding a cyclic prefix. Obviously, he raining sequence design can be based on a ime domain or frequency domain. A ime-domain opimizaion mehod is inroduced in [8]. A disadvanage of he ime-domain mehod is ha an exhausive search for he opimal raining sequence of lengh N requires 2 N possible sequences. A frequency-domain mehod is proposed o reduce he compuaional cos in [9]. owever, he frequencydomain mehod canno always find he opimal periodic raining sequence in erms of he mean-squared channel esimaion error [8]. A summary and comparison for hese design mehods is shown below. A raining sequence is said o be perfec or opimal if i have impulse-like auocorrelaion and zero crosscorrelaion [6] and L is he channel aps: Domain Mehod Minimum Searching Opimal Sequence MSE Complexiy Time Periodic (LS)[8] Yes igh(2n) Yes Aperiodic [7] No Medium(N2) Yes L-Perfec (MIMO) Almos Low(Nlog 2 N) Someimes [6] Frequency Periodic [9] No Low(Nlog 2 N) Someimes Table1 Comparison of Training Sequence Design mehods So, he design is a kind of radeoff beween he searching complexiy and sequence performances. V. MIMO Channel Esimaion The objecive was o deermine a raining sequence which opimized he mean-squared esimaion error for a leas-squares ype channel esimaor. According o he model shown as above, a simple and inuiive mehod o esimae he MIMO channel impulse 5
responses, i.e. h11, h22, h12 and h 21, is o send he raining sequence s a only one ransmier by urning off anoher ransmier during one raining ime slo. This mehod is very low complexiy and even doesn need o design a good raining sequence. owever, i is very ime consuming. I needs wo ime slos o obain he esimaed channels. During he firs y 1,1 y 1,2 ime slos, h = 11, h12 s = s. During he second ime slos, y 2,1 y 2,2 h = 21, h22 s = s. Now, I use his idea combined wih radiional numerical mehods o esimae he MIMO channels. Tradiionally, he linear leas square channel esimaes ( î ) can be calculaed as [10] hˆ hˆ hˆ hˆ 11 12-1 ĥ= =(S S) S y 21 22 (3) where ( i) and () 1 i denoe he complex-conjugae (ermiian) ranspose and he inverse, respecively. The mean-squared error (MSE) for he ime-domain case is defined by 2-1 () i MSE ( h hˆ) ( h hˆ = E ) = 2 σ Tr((S S) ) where we assume whie noise wih auo-correlaion marix (4) 2 Rz = E zz = 2σ IN -L+ 1, I n denoes he ideniy marix of dimension n n, and Tr ( i ) denoes he race of a marix. The minimum mean-squared error (MMSE) is equal o 2 2 σ L S1 S1 S2 S 1 = N L + L+ 1 S1 S2 S2 S2 MMSE =, iff S S= ( 1)I N 2L (5) The raining sequences s 1 and s 2 ha saisfy (5) are considered as opimal sequences. So, (5) ells us ha he opimal sequences have an impulse-like auo-correlaion sequence and zero cross correlaion. 6
I can propose a mehod o encode he raining symbols wih zero correlaion. I ransmi * 1 1 he wo same consecuive raining sequences s 1 a each ransmier. I send [ S S ] a * 1 1 one ransmier and [ ] S S a anoher ransmier. The operaion denoed by ( ) i refers o ime-reversing a sequence. The ( ) * i denoes he complex conjugae of he sequences. According o (2), he received signals during wo ime slos can be expressed as * * y 1 S 1 S 1 h11( L) h12( L) y = + z 2 S1 S h 1 21( L) h22( L) S (6) If he sequence s 1 is symmeric abou is cener wih impulse-like auo correlaion, I can ge S S = ( N L+ 1)I2 which is he condiion o achieve he MMSE and opimal sequence in (5). L VI. Conclusion This repor inroduces sudied various mehods o design he raining sequences and esimae he channels for wired MIMO communicaion sysems. I am able o simplify he channel esimaion problem by designing he opimal raining sequences an impulse-like auo-correlaion sequence and zero cross correlaion. This mehod is easy o implemen and can ge he MMSE. References [1] A. Goldsmih, S. A. Jafar, N. Jindal and S. Vishwanah, Capaciy limis of MIMO channels, IEEE Transacions on seleced areas in Communicaions, vol.21, No.5, pp.684-702, Jun. 2003 [2] Y. Li, Simplified channel esimaion for OFDM sysems wih muliple ransmi anennas, IEEE Transacions on Wireless Communicaions, vol.1, No.1, pp.67-75, Jan. 2002 [3] A. Peropulu, R. Zhang and R. Lin, Blind OFDM Channel Esimaion Through Simple Linear Precoding, IEEE Transacions on Wireless Communicaions, vol.3, No.2, pp.647-655, Mar. 2004 [4] ITU-T G.992.3/4, ADSL2/2+ sandards [5] John A. C. Bingham, ADSL, VDSL and Mulicarrier Modulaion, Wiley, New York, 2000 [6] C. Fragouli, N. Al-Dhahir, W. Turin, Training-Based Channel Esimaion for Muliple-Anenna Broadband Transmissions," IEEE Trans. on Wireless Comm., vol.2, No.2, pp 384-391, March 2003 7
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