Space Code Design in Delay Diversity Transmission for PSK Modulation Meixia Tao and Roger S. Cheng Center for Wireless Information Technology Electrical & Electronic Engineering Department The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong Abstract— Delay diversity transmission is well-known as a simple and efficient transmission scheme to exploit transmit diversity in wireless com- munications. The interpretation of a spatial repetition code transmitted by multiple antennas in a delayed fashion opens up the possibility of better code design to enhance the performance. This paper considers the design criteria fordifferent channel models. It is shown that all flat Rayleigh fad- ing models, regardless of the time selectivity, share a common design cri- terion, which is to maximize the minimum effective product distance. This resultfurtherverifiestheflexibilityandrobustnessofdelaydiversitytrans- mission. Nonbinaryspatialblockcodeconstructionisthenproposedovera ringofintegersforPSKmodulation. Someoptimalsystematiclinearblock codesareprovidedthroughexhaustivesearchforupto64PSKmodulation and with up to 5 transmit antennas. Comparing with the repetition codes employedinoriginaldelaydiversitytransmission,thenewlydesignedcodes canachievesignificantcodingimprovementwithoutanyincreaseofdecod- ingcomplexity. I. I NTRODUCTION In the last decade, a considerable amount of work has been undertaken to combat fading by employing multiple transmit and receive antennas in wireless communications. Receive diversity essentially can be achieved using maximum-ratio- combing (MRC). In contrast, transmit diversity is less straight- forward to achieve. Delay diversity (DD) transmission [1] [2] is a simple transmit diversity scheme which transmits delayed copies of the informa- tion signal sequence on multiple antennas. In this scheme, the system with M transmit antennas is observed at the receiver as a virtual single-transmit-antenna system with frequency-selective fading of M taps. Thus transmit diversity of order M can be achieved by equalizing the virtual channel through maximum likelihood sequence estimator (MLSE)[3]. The original DD is proposed for flat fading channels. Recent result [6] has shown that it can be easily extended to a frequency- selective fading channel by increasing the delay step between antennas from one symbol interval to L symbol intervals, where L is the number of the actual channel taps. A combined antenna diversity and frequency diversity of order ML can be achieved. Since the invention of space-time coding (STC) [4], DD has been interpreted as a class of space-time trellis codes (STTC) and is referred to as delay diversity codes (DDC). It turns out that the nice structure of DDC assures full rank but may not maximize the determinant. The rank and the determinant quan- tify the transmit antenna diversity order and the coding advan- tage, respectively, in quasi-static flat Rayleigh fading channels [4]. Another interesting way to interpret DD is to consider it as This work is supported in part by the Hong Kong RGC. Emails of the authors are: mxtao@ee.ust.hk, eecheng@ee.ust.hk a spatial repetition code transmitted on different antennas in a delayed fashion. Hence, carefully designed spatial block codes, rather than repetition codes, should enhance the coding advan- tage. In this paper, we consider the design criteria of the spatial block codes at high signal-to-noise ratio (SNR) for different flat Rayleigh fading channels. We show that design criterion for rapid fading is the same as that for quasi-static fading [5], which is to maximize the minimum effective product distance of the block code. As the quasi-static fading and rapid fading are just the two extremes of a practical time-varying fading channel, this result implies that all flat Rayleigh fading channels, regardless of the time-selectivity, share a common design criterion. In other words, the best block code in DD designed for one channel is also the best for any other channels. Having the design crite- rion, we then consider the code construction, mainly for PSK modulation. Notice that, both BPSK and QPSK modulations have been studied in [5] based on permutation. However it is not easy to extend this method to higher modulations. In this paper, we propose a systematic linear block code construction method based on modulo-P integer rings, with P being a non- prime integer. Through exhaustive search, optimal systematic linear block codes are found for up to 64PSK modulation and with up to M =5 transmit antennas. The remainder of this paper is organized as follows. In Sec- tion II, we describe the system model of the generalized DD. In Section III, we derive the common design criterion of the spatial block codes used in DD over all flat Rayleigh fading channels. Code construction, as well as some search results, are provided in Section IV. In Section V, we provide some simulation results, and draw some conclusions in Section VI. II. SYSTEM MODEL A. Transmitter Fig. 1 depicts the simplified baseband system model for the delay diversity scheme with M transmit antennas. The mod- ulated information data sequence is denoted by {x[t]}. Each information signal x[t] is first encoded by an (M, 1) block code with output vector [x 1 [t],...,x M [t]] on the same constellation. For systematic block code, we have x 1 [t]= x[t]. For repetition code, we have x m [t]= x[t], for all m =1,...,M . To ensure no loss of diversity gain, the minimum Hamming distance of the (M, 1) block code should be M . This is also the sufficient con- dition to achieve full diversity gain [5]. We shall emphasis it by the following requirement. 0-7803-7467-3/02/$17.00 ©2002 IEEE. 444