146 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 1, JANUARY 2004 Error Probability Minimizing Pilots for OFDM With M-PSK Modulation Over Rayleigh-Fading Channels Xiaodong Cai, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE Abstract—Orthogonal frequency division multiplexing (OFDM) with pilot symbol assisted channel estimation is a promising tech- nique for high rate transmissions over wireless frequency-selective fading channels. In this paper, we analyze the symbol error rate (SER) performance of OFDM with -ary phase-shift keying ( -PSK) modulation over Rayleigh-fading channels, in the pres- ence of channel estimation errors. Both least-squares error (LSE) and minimum mean-square error (MMSE) channel estimators are considered. For prescribed power, our analysis not only yields exact SER formulas, but also quantifies the performance loss due to channel estimation errors. We also optimize the number of pilot symbols, the placement of pilot symbols, and the power allocation between pilot and information symbols, to minimize this loss, and thereby minimize SER. Simulations corroborate our SER performance analysis, and numerical results are presented to illustrate our optimal claims. Index Terms—Channel estimation, error probability, orthogonal frequency division multiplexing (OFDM), pilots. I. INTRODUCTION O RTHOGONAL FREQUENCY DIVISION MULTI- PLEXING (OFDM) provides an effective and low- complexity means of eliminating intersymbol interference for transmissions over frequency-selective fading channels [1], [2]. Channel state information (CSI) is required for the OFDM receiver to perform coherent detection, or diversity combining, if multiple transmit and receive antennas are deployed. In practice, CSI can be reliably estimated at the receiver by inserting training (a.k.a. pilot) symbols at the transmitter. Pilot symbol assisted channel estimation is especially attractive for wireless links [3], where the channel is time-varying. In [4]–[7] the channel correlation in the time and frequency domains was exploited for pilot-based channel estimation in OFDM systems. Channel estimation using pilot symbols in only one OFDM block was advocated in [8] and [9]. Interpolating schemes were investigated in [10], [11], and joint multipath delay and tap estimation of OFDM channels was studied in [12]. While many channel estimators have been developed for OFDM, error probability analysis in the presence of channel estimation errors has received relatively less attention. Only Manuscript received May 20, 2003; revised September 23, 2003. This work was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The authors are with the Department of Electrical and Computer Engi- neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: caixd@ece.umn.edu; georgios@ece.umn.edu). Digital Object Identifier 10.1109/TVT.2003.819624 recently, BER approximations for -ary phase shift keying ( -PSK) and -ary quadrature amplitude modulation (QAM) were provided for OFDM with channel estimation errors [13], [14]. In this paper, we will derive exact symbol error rate (SER) expressions for pilot assisted OFDM transmissions with -PSK modulation over Rayleigh-fading channels. Our SER analysis also quantifies the performance loss due to channel estimation error and the transmit pilot power. Based on this SER analysis, we will optimize the design of pilots to minimize the performance loss caused by channel estimation errors. For prescribed power, this will lead us to pilots that minimize error probability. Optimizing pilots for wireless OFDM systems has been considered recently, based on: maximizing a lower bound on ergodic capacity [15], [16], or, minimizing the channel mean-square error (MSE) [17]–[19]. Pilot optimization for single-carrier transmissions has also been investigated in [15], [20]–[22] based on these two criteria, and in [23] by minimizing the Cramér–Rao bound on the channel MSE. As error probability directly determines the reliability of a communication link, our use of SER as a criterion is certainly of practical interest. The rest of this paper is organized as follows. Section II de- scribes the system model, and analyzes the average SER perfor- mance in the presence of channel estimation errors. Pilot sym- bols are optimized to minimize SER in Section III. Simulations and numerical results are presented in Section IV, and conclu- sions are drawn in Section V. Notation: Superscripts , , and stand for transpose, con- jugate, and Hermitian transpose, respectively; denotes ex- pectation. Column vectors (matrices) are denoted by boldface lower (upper) case letters; represents the identity matrix; stands for a diagonal matrix with on its diag- onal; and denotes the trace of matrix . We use to denote that is a complex Gaussian distributed vector with mean , and covariance . II. MODELING AND ERROR PROBABILITY ANALYSIS In this section, we will present the signal model, and analyze the SER performance of OFDM in the presence of channel es- timation errors. A. Signal Model The OFDM transmission system under consideration is de- picted in Fig. 1. Information and pilot symbols are modulated on a set of subcarriers, and transmitted over a frequency-selec- tive fading channel through a single transmitter antenna. After demodulation at the receiver end, where we allow for multiple 0018-9545/04$20.00 © 2004 IEEE