430 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006 Transactions Papers Exact Error Performance of Square Orthogonal Space–Time Block Coding With Channel Estimation Parul Garg, Member, IEEE, Ranjan K. Mallik, Senior Member, IEEE, and Hari M. Gupta Abstract—Consider a wireless communication system in flat fading with transmit and receive antennas using space–time block coding, where code vectors are transmitted over symbol intervals, resulting in an code matrix. A least-squares estimate (LSE) as well as a minimum mean-square estimate (MMSE) of the channel matrix is obtained from a sequence of pilot code vectors. For the case of linear square (i.e., with ) orthogonal codes over constant envelope con- stellations, we obtain an expression for the exact decoding error probability (DEP) for coherent maximum-likelihood decoding. We also find the coding gain for high average signal-to-noise ratio (SNR) per diversity branch in the case of Rayleigh fading. A comparison between both channel-estimation techniques is done in terms of the average pilot-power-to-signal-power ratio (APPSPR). It is found that MMSE requires lower pilot power than LSE for the same DEP and the same average SNR per diversity branch. In addition, the error performance with LSE approaches that with MMSE, with an increase of average SNR per branch or an increase of APPSPR. Index Terms—Decoding error probability (DEP), flat fading, least-squares channel estimation, minimum mean-square channel estimation, space–time block coding. I. INTRODUCTION C ONSIDER a space–time (ST) coded system [1] with transmit and receive antennas. The codes are trans- mitted over symbol intervals. Coherent detection in an ST coded system requires accurate channel estimation at the re- ceiver. While analyzing the error performance of an ST coded system, perfect channel state information (CSI) is assumed, as in [2]. However, channel-estimation methods used in practice give rise to imperfections, due to imperfect channel-estimation algorithms [3], [4] or channel variations. In this paper, channel estimation is done by using a sequence of pilot symbols which are inserted after regular intervals of time, to obtain the random channel matrix . Both Paper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received July 19, 2004; revised February 24, 2005. This paper was presented in part at the IEEE International Conference on Communications, Seoul, Korea, May 2005. P. Garg is with the Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, New Delhi 110075, India (e-mail: parul_saini@yahoo.co.in). R. K. Mallik and H. M. Gupta are with the Department of Electrical En- gineering, Indian Institute of Technology—Delhi, New Delhi 110016, India (e-mail: rkmallik@ee.iitd.ernet.in; hmgupta@ee.iitd.ernet.in). Digital Object Identifier 10.1109/TCOMM.2006.869854 least-squares estimate (LSE) and minimum mean-square esti- mate (MMSE) are considered. For the case of linear square (i.e., with ) orthogonal ST block codes (STBCs) over con- stant envelope constellations and coherent maximum-likelihood (ML) decoding, we find an expression for the exact decoding error probability (DEP), which is defined as the probability of any code matrix being wrongly decoded as some other code ma- trix from a set of code matrices used for data transmission. The approach is as follows. We first find the decision variable of the receiver as a quadratic form, from which the probability of error of a single symbol, conditioned on the channel matrix , is obtained. We then obtain the error probability over a block of symbols, conditioned on . Finally, we average this condi- tional probability over to obtain the DEP. We also find the coding gain for high average signal-to-noise ratio (SNR) per di- versity branch in the case of Rayleigh fading (when the entries of are complex Gaussian with mean zero). A comparative study of the LSE-based and the MMSE-based channel-estima- tion techniques is done in terms of the average pilot-power-to- signal-power ratio (APPSPR). The paper is organized as follows. Section II describes the model of the ST coded system along with channel estimation using pilot symbols. The DEP for the cases of both LSE and MMSE is presented in Section III. Section IV gives the error analysis, along with asymptotic results and comparison between LSE and MMSE. Section V presents some numerical results. Concluding remarks are given in Section VI. II. MODEL Consider a wireless communication system in flat fading with transmit and receive antennas using ST block coding [5], [6]. An code matrix is used to transmit code vectors over symbol intervals. The complex baseband signal vector received at time index by the antennas after matched filtering is given by [1] (1) where denotes the code vector transmitted from the antennas, the random channel matrix, and the additive white Gaussian noise vector. The noise vectors for different time indexes are independent and identically dis- tributed (i.i.d.) complex circular Gaussian random vectors, each 0090-6778/$20.00 © 2006 IEEE