Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 doi:10.4236/ijcns.2009.24029 Published Online July 2009 (http://www.SciRP.org/journal/ijcns/). Investigations into the Effect of Spatial Correlation on Channel Estimation and Capacity of Multiple Input Multiple Output System Xia LIU1, Marek E. BIALKOWSKI2, Feng WANG1 1Student Member IEEE, School of ITEE, The University of Queensland, Brisbane, Australia 2Fellow IEEE, School of ITEE, The University of Queensland, Brisbane, Australia Email: {xialiu, meb, fwang }@itee.uq.edu.au Received December 17, 2008; revised March 28, 2009; accepted May 25, 2009 ABSTRACT The paper reports on investigations into the effect of spatial correlation on channel estimation and capacity of a multiple input multiple output (MIMO) wireless communication system. Least square (LS), scaled least square (SLS) and minimum mean square error (MMSE) methods are considered for estimating channel properties of a MIMO system using training sequences. The undertaken mathematical analysis reveals that the accuracy of the scaled least square (SLS) and minimum mean square error (MMSE) channel estimation methods are determined by the sum of eigenvalues of the channel correlation matrix. It is shown that for a fixed transmitted power to noise ratio (TPNR) assumed in the training mode, a higher spatial correlation has a positive effect on the performance of SLS and MMSE estimation methods. The effect of accuracy of the estimated Channel State Information (CSI) on MIMO system capacity is illustrated by computer simulations for an uplink case in which only the mobile station (MS) transmitter is surrounded by scattering objects. Keywords: MIMO, Channel Estimation, Channel Capacity, Spatial Correlation, Channel Modelling 1. Introduction been investigated. It has been shown that the accuracy of the investigated training-based estimation methods is In recent years, there has been a growing interest in mul- influenced by the transmitted power to noise ratio (TPNR) in the training mode, and a number of antenna tiple input multiple output (MIMO) techniques in rela- elements at the transmitter and receiver. In particular, it tion to wireless communication systems as they can sig- has been pointed out that when TPNR and a number of nificantly increase data throughput (capacity) without the antenna elements are fixed, the SLS and MMSE methods need for extra operational frequency bandwidth. In order offer better performance than the LS method. This is due to make use of the advantages of MIMO, precise channel to the fact that SLS and MMSE methods utilize the chan- state information (CSI) is required at the receiver. The nel correlation in the estimator cost function while the reason is that without CSI decoding of the received sig- LS estimator does not take the channel properties into nal is impossible [1–5]. In turn, an inaccurate CSI leads account. to an increased bit error rate (BER) that translates into a It is worthwhile to note that the channel properties are degraded capacity of the system [6–8]. governed by a signal propagation environment and spa- Obtaining accurate CSI can be accomplished using tial correlation (SC) that is dependent on an antenna con- suitable channel estimation methods. The methods based figuration and a distribution of scattering objects that are on the use of training sequences, known as the training- present in the path between the transmitter and receiver. based channel estimation methods, are the most popular. The works in [9,10] have demonstrated superiority of In [9,10], several training-based methods including least SLS and MMSE estimation methods, which make use of square (LS) method, scaled least square (SLS) method channel correlation, over the LS method neglecting and minimum mean square error (MMSE) method have channel properties. However, no specific relationship Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 268 X. LIU ET AL. between spatial correlation and channel estimation accu- where HLOS denotes the LOS part as and HNLOS denotes racy has been shown. The works presented in [11,12] NLOS part. K is the Rician factor defined as the ratio of have reported on the relationship between spatial corre- power in LOS and the mean power in NLOS signal lation and estimation accuracy of MMSE method. How- component [17]. The elements of HLOS matrix can be ever, only simulation results, giving trends without any written as [18] further mathematical insight have been presented. rt 2 In this paper, we try to fill the existing void by pre- H LOS exp( Dj rt ) (3) senting the mathematical analysis explaining the effects of channel properties on SLS and MMSE channel esti- where Drt is the distance between t-th transmit antenna and mation methods. It is shown that for a fixed TPNR, the r-th receive antenna. Assuming that the components of accuracy of SLS and MMSE methods is determined by NLOS are jointly Gaussian, HNLOS can be written as [19,20], the sum of eigenvalues of channel correlation matrix, 2/12/1 RHRH (4) which in turn characterizes the signal propagation condi- NLOS TgR tions. In addition, we report on the effect of spatial cor- where Hg is a matrix with i.i.d Gaussian entries. relation on both the channel estimation and capacity of Here, the Jakes fading model [21,22] is used to de- MIMO system. In the work presented in [13–16], it has scribe the spatial correlation matrices RR at the receiver been shown that the existence of spatial correlation leads and RT at the transmitter. An uplink case between a base to the reduced MIMO channel capacity. However, these station (BS) and a mobile station (MS) is assumed, as conclusions rely on the assumption of perfect CSI avail- shown in Figure 1. able to the receiver. In practical situations, obtaining The BS antennas are assumed to be located at a large perfect CSI can not be achieved. Therefore, in this paper height above the ground where the influence of scatterers we take imperfect knowledge of CSI into account while close to the receiver is negligible. In turn, MS is assumed evaluating MIMO capacity. to be surrounded by many scatterers distributed within a The rest of the paper is organized as follows. In Sec- “circle of influence”. For this case, the signal correlation BS tion 2, a MIMO system model is introduced. In Section 3, coefficients at the receiver BS and transmitter MS, ρR MS LS, SLS and MMSE channel estimation methods are and ρT , can be obtained from [22] and are given as: described and the channel estimation accuracy analysis is MS() T J [2/]T (5) given. Section 4 shows derivations for the lower bound Tmn0 mn of MIMO channel capacity when the channel estimation 22 BS( R )Jj [R cos( )]exp(R sin( )) errors are included. Section 5 describes computer simu- Rmn 0mmnax mn lation results. Section 6 concludes the paper. (6) T R 2. System Description & Channel Model where, δmn and δmn are the antenna spacing distances between m-th and n-th antennas at transmitter and re- ceiver, respectively; λ is the wavelength of the carrier; We consider a flat block-fading narrow-band MIMO sys- γmax is the maximum angular spread (AS); θ is the AoA tem with Mt antenna elements at the transmitter and Mr of LOS and J0 is the Bessel function of 0-th order. Using antenna elements at the receiver. The relationship between BS T MS R BS ρR (δmn ) and ρT (δmn ), the correlation matrices RR the received and transmitted signals is given by (1): MS and RT for BS and MS links can be generated as YHSV (1) s BS() BS BS ( BS ) RR11 1Mr where Ys is the Mr × N complex matrix representing the BS RR (7) received signals; S is the Mt × N complex matrix repre- BS BS BS BS RM()1 RMM ( ) senting transmitted signals; H is the Mr × Mt complex rr r channel matrix and V is the M × N complex zero-mean r white noise matrix. N is the length of transmitted signal. The channel matrix H describes the channel properties which depend on a signal propagation environment. Here, the signal propagation is modeled as a sum of the line of sight (LOS) and non-line of sight (NLOS) components. As a result, the channel matrix is represented by two terms and given as [17,18], 1 K HHNLOS HLOS (2) 11KK Figure 1. Jakes model for the considered MIMO channel. Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 INVESTIGATIONS INTO THE EFFECT OF SPATIAL CORRELATION ON CHANNEL ESTIMATION 269 AND CAPACITY OF MULTIPLE INPUT MULTIPLE OUTPUT SYSTEM MS MS MS MS 3.2. SLS Method TT()11 ()1M t MS The SLS method reduces the estimation error of the LS RT (8) MS() MS MS ( MS )method. The improvement is given by the scaling factor TMtt1 TMMtγ which can be written as tr{} RH 3. Training-Based Channel Estimation (13) MSELS tr{} R H For a training based channel estimation method, the rela- The estimated channel matrix is given as [9], [10] tionship between the received signals and the training ˆ tr{} RH † sequences is given by Equation (1) as H SLS 21H YP (14) nrMtrPP{( ) } trR { H } YHPV (9) 2 Here, σn is the noise power; RH is the channel correla- H Here the transmitted signal S in (1) is replaced by P, tion matrix defined as RH=E{H H} and tr{.} implies the which represents the Mt × L complex training matrix trace operation. The SLS estimation MSE is given as [9,10] (sequence). L is the length of the training sequence. The 2 goal is to estimate the complex channel matrix H from ˆ MSESLS E{} H H LS the knowledge of Y and P.