ANTIPODAL PARAUNITARY PRECODING FOR OFDM APPLICATION See-May Phoong, Kai-Yen Chang Dept. of EE & Grad. Inst. of Comm. Engr. National Taiwan Univ. Taipei, Taiwan, ROC Yuan-Pei Lin Dept. Electrical and Control Engr. National Chiao Tung Univ. Hsinchu, Taiwan, ROC ABSTRACT Paraunitary (PU) matrices have found many applications. In this paper, a special class of PU matrices, namely the antipodal PU (APU) matrices, is used as precoding matrices for OFDM systems. Both the zero-forcing and MMSE receivers will be derived for pre- coded OFDM systems with APU precoding matrices. The perfor- mance of such precoded OFDM systems will be analyzed. We will show that using a APU precoding matrix, we are able to average the noise variance in both the time and frequency domains, and this obtains time and frequency diversity. Experiments show that pre- coded OFDM systems with MMSE receivers have a much better bit error rate performance than the conventional OFDM system. 1. INTRODUCTION Multirate systems and filter banks have played an important role in various areas of signal processing [1]. Of particular interest is the class of paraunitary (PU) matrices. One attractive feature of these matrices is their energy conservation property which can avoid the noise or error amplification problem. In the past, the design and complete parameterization of PU matrices have been successfully derived. In this paper, we are going to study a special class of PU matrices, namely the antipodal paraunitary (APU) matrices. An polynomial matrix is APU if all the entries of are and it satisfies 1 The tilde notation denotes , where is transpose- conjugation and is the complex conjugation. For the special case of constant (memoryless) matrices, APU matrices reduce to scaled Hadamard matrices. Various methods have been proposed for the construction of APU matrices [2] [3]. The application of APU ma- trices in synchronous spread spectrum communications has been explored [2] and promising results have been demonstrated. In this paper, we will apply APU matrices to linearly precoded OFDM systems. Linearly precoded OFDM systems have been studied by a number of researchers [4] [5] [6]. When the OFDM system has a DFT precoding matrix, it was shown to be the same This work was supported in parts by National Science Council, Tai- wan, ROC, under NSC 92-2219-E-002-015 and 92-2213-E-009-022, Min- istry of Education, Taiwan, ROC, under Grant # 89E-FA06-2-4, and the Lee and MTI Center for Networking Research. 1 One can generalize the definition of APU matrices to include complex matrices. In this case, all the entries of the coefficient matrices will have equal magnitude. as the so-called the single carrier with frequency domain equal- izer (SC-FDE) system, which was first introduced in [7]. In [4], it was shown that the SC-DFE system has the maximum diversity gain among all linearly precoded OFDM systems. In [5] [6], BER minimized precoder for OFDM system was considered. For high SNR transmission, the SC-FDE system is optimal in the sense that it minimizes the bit error rate among OFDM systems with any or- thogonal precoding matrix. In these studies, the precoders are con- stant matrices and the resulting precoded OFDM systems belong to the class of block transmission systems. OFDM systems with APU precoding matrices are overlapped block transmission systems; a block of data symbols is transmitted over several blocks of transmitted signals. By doing so, we are able to average the noise variance in both the time and frequency domains and this achieves time and frequency diversity. Both the zero-forcing and MMSE receivers will be derived. Performances of the proposed systems will be analyzed and compared with the conventional OFDM system. 2. PRECODED OFDM SYSTEMS Fig. 1 shows the block diagram of a precoded OFDM system. In a precoded OFDM transmitter, the th input block consisting of modulation symbols, e.g. QAM symbols, are first passed through an by precoding matrix . The output of is given by In this paper, we consider only APU precoding matrices. APU precoding matrices enjoy two main advantages. Firstly, they have very low complexity. Their implementation involves only addi- tions and there exists an efficient butterfly structure for a broad class of APU matrices [3]. Secondly, as we will show later, APU matrices have the ability to average the noise variance over both the time and frequency domains. We assume that the APU pre- coding matrix is normalized so that In other words, all the entries of are . After taking the -point IDFT of , we get: where is the DFT matrix with its th entry given by . Before is transmitted, a cyclic prefix (CP) of length is added. Note that unlike the con- ventional block transmission system, the transmitted block now contains information of blocks of input vectors