Performance of Space-Frequency Codes in MIMO Channels with Frequency Offset Dung Ngoc Dao and Chintha Tellambura Electrical and Computer Engineering, University of Alberta Edmonton, Alberta T6G 2V4, Canada Email: {dndung, chintha}@ece.ualberta.ca Telephone: (1-780) 492 7228, Fax: (1-780) 492 1811 Abstract—We investigate the pair-wise error probability (PEP) performance of space-frequency (SF) codes over quasi-static, frequency-selective Rayleigh fading channels with frequency offset (FO). The PEP expression shows that the conventional SF code design criteria are remained valid. If the normalized FO (NFO) is less than 1%, the performance loss is negligible. The loss increases with FO and with signal to noise ratio (SNR). We show that diversity can be used to mitigate the effects of FO. However, with large FO, the PEP is no longer exponentially decays with SNR. Therefore, we propose a SF coding scheme to combat effectively with even very high NFO (20%). Simulation results confirm our theoretical evaluations. I. I NTRODUCTION The space time (ST) codes proposed by Tarokh et al. [1] for coherent systems over quasi-static flat fading channels achieves the maximum diversity order (DO) d = MN , where M and N are the number of Tx (Tx) and Rx (Rx) antennas. The ST codes can achieve at least the designed DO in the FSF channels [1], but are not guaranteed to exploit all the available frequency diversity advantage of FSF channels. To overcome this, the authors in [2], [3], [4] and references therein propose new ST coding methods so that the maximum achievable DO is obtainable. It is found that in FSF channels, the maximum DO is d = LMN where L is the number of paths of channels. There are two approaches to design codes that can achieve this maximum DO [5]. In the first method, ST codes are constructed in the time domain, whereas in the second method called space-frequency (SF) coding, the source symbols are encoded across the subcarriers (or sub-carriers) of orthogonal frequency division multiplexing (OFDM) symbols in the frequency domain. Su et al. [6] derive SF code criteria which show an explicit relation between the SF code matrix and the characteristic parameters of FSF fading channels such as path delays and delay power distribution. Like other error control coding methods, the performance of SF codes depends on the propagation environment [7]. Moreover, the performance of SF codes and OFDM may be affected by underlying factors such as frequency offset (FO), timing error, to name a few. A residual FO exists due to carrier synchronization mismatch and Doppler shift [8]. Residual FO breaks down the orthogonality among sub- carriers, hence inter-carrier interference (ICI) is produced and bit error rate (BER) is increased consequently. Several papers have analyzed the effects of FO on the BER performance. For example in [9], BER is calculated for single input single output (SISO) OFDM systems with various modulation schemes. The authors in [10], [11] provide BER expressions of MIMO- OFDM employing Alamouti’s scheme [12]. However, to the best of our knowledge, the impact of FO on the SF code design criteria and the performance have not been investigated. This interesting question will be addressed in our paper. We analytically show that the SF code design criteria still hold in the case of FO. The performance loss is negligible if the NFO is small. This loss is increased rapidly with the increase of NFO and with signal-to-noise ratio (SNR). When NFO is large, PEP performance of SF codes no longer decays exponentially. To mitigate the effect of FO, we generalize the work of [13] for MIMO-OFDM with SF codes. The resulting SF codes not only achieve the same diversity and coding gains as the SF codes derived in [6] but also are capable of ICI self-cancellation. Simulations have showed that the ICI self- cancellation SF codes still yields good performance when the FO is extremely high, about 20%. The paper has seven sections. The model of MIMO-OFDM systems with FO is presented in Section II. Section III reviews the design criteria of SFC codes. In Section IV, we derive the PEP performance of SF codes in the presence of FO. In Section V, we propose a class of SF codes that reduce effectively FO effects to PEP. Section VI provides simulation results to support the theoretical analysis. We summarize the paper in Section VII. II. FREQUENCY OFFSET IN OFDM SYSTEMS This section briefly introduces the OFDM system with FO model to be analyzed for remaining parts of the paper. A. MIMO-OFDM System Model We consider the MIMO-OFDM system with M Tx and N Rx antennas. The number of subcarriers in the OFDM modulator is K. The L−path quasi-static Rayleigh fading channel model is assumed identically for any link between Tx antenna m (m =1, ..., M ) and Rx antenna n (n =1, ..., N ). The channel impulse response in the time domain is [14] h m,n (t, τ )= L−1  l=0 α m,n (t, l)δ(τ − τ l ) (1) 702 0-7803-8622-1/04/$20.00 ©2004 IEEE Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on December 22, 2009 at 16:14 from IEEE Xplore. Restrictions apply.