Optimum Space-Time Block Codes over Time-Selective Channels Jun He and Pooi Yuen Kam Department of Electrical and Computer Engineering National University of Singapore, Republic of Singapore, S117576 Email: {hejun,elekampy}@nus.edu.sg Abstract—We analyze the average bit error probability (BEP) of orthogonal space-time block codes (STBC) over time-selective channels. Exact BEP results are obtained in closed form. Through the analysis, we reveal the relationship between the inter-symbol interference (ISI) and the row positions in a STBC matrix. We then introduce one proposition and two design criteria for code search/design. Optimum/near-optimum codes are found/designed with reduced ISI. I. I NTRODUCTION Orthogonal space-time block codes (STBC) [1] are com- monly used in MIMO systems, due to the simple maximum likelihood (ML) decoding structure. However, this decoding structure is based on the assumption that the channels are block-wise constant, that is not always true in practice. If the channels vary from symbol to symbol, the orthogonality will be corrupted by inter-symbol interference (ISI), so the conventional linear ML decoder [1] is no longer optimum. Considering the time-selective channels, [2]–[10] have pro- posed different decoders for orthogonal STBC. Reference [2] first proposed a suboptimum detection scheme with the conventional linear decoder. It retains the linear decoding structure, but has an irreducible error floor in the high signal- to-noise ratio (SNR) region. Later, an elegant zero-forcing (ZF) decoder for two transmit antennas is presented in [3], where the ISI is completely removed. The ZF decoder is extended to three and four transmit antennas cases in [4]– [6]. Besides the linear decoders above, there are also non- linear decoders, including parallel interference cancellation (PIC) decoder [6]–[8], ML decoder [8], successive interference cancellation (SIC) decoder [9] and decision-feedback decoder [10]. Instead of designing a new decoding scheme, a modified orthogonal STBC is developed in [11]. While keeping the full diversity order and the orthogonality, the modified orthogonal STBC reduces the ISI to a much lower level, compared with the original orthogonal STBC [1]. Among these works, however, surprisingly few have ana- lyzed the performance of STBC over time-selective channels. Due to the lack of theoretical analysis, little insight can be gained, and it remains unclear how and to what extent the code structure affects the performance of STBC over a time- selective channel. Most of the works simply apply the existing STBC to the time-selective channels, while these codes only provide sub-optimum performances. In this paper, through the performance analysis of orthog- onal STBC over the time-selective channel with the con- ventional linear decoder, we reveal the relationship between the ISI and the structure of the code matrix. Based on this relationship, we introduce one proposition and two design criteria, following which it is easy to design or search for better STBC that generate less ISI, compared with the original code matrix. With reduced ISI, the new codes not only improve the performance of the conventional linear decoder, but also benefit the PIC, SIC, decision-feedback and other decoders in existing works. II. SYSTEM MODEL For the purpose of illustration, we consider a system with four transmit and one receive antenna, transmitting with a modified G 4 encoder [1]. The method we use here to analyze the G 4 system can be easily applied to other orthogonal STBC systems with arbitrary numbers of antennas. Transmitting four information symbols s =[s 1 ,s 2 ,s 3 ,s 4 ] T in one STBC block, the original G 4 encoder generates an 8 × 4 code matrix, which is given as [1] G 4 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ s 1 s 2 s 3 s 4 −s 2 s 1 −s 4 s 3 −s 3 s 4 s 1 −s 2 −s 4 −s 3 s 2 s 1 s ∗ 1 s ∗ 2 s ∗ 3 s ∗ 4 −s ∗ 2 s ∗ 1 −s ∗ 4 s ∗ 3 −s ∗ 3 s ∗ 4 s ∗ 1 −s ∗ 2 −s ∗ 4 −s ∗ 3 s ∗ 2 s ∗ 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (1) The above encoder, however, generates high ISI over a time- selective channel (which we will explain later), and therefore, we use a modified G 4 encoder in this paper. The modified encoder simply interchanges the rows of the original code matrix, and is given by G op 4 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ s 1 s 2 s 3 s 4 −s ∗ 4 −s ∗ 3 s ∗ 2 s ∗ 1 −s ∗ 2 s ∗ 1 −s ∗ 4 s ∗ 3 −s 3 s 4 s 1 −s 2 −s ∗ 3 s ∗ 4 s ∗ 1 −s ∗ 2 −s 2 s 1 −s 4 s 3 −s 4 −s 3 s 2 s 1 s ∗ 1 s ∗ 2 s ∗ 3 s ∗ 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (2) This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. 978-1-4244-2324-8/08/$25.00 © 2008 IEEE.