UNDERSTANDING THE BER-PERFORMANCE OF SPACE-TIME BLOCK CODES Gerhard Gritsch, Hans Weinrichter, Markus Rupp Vienna University of Technology Institute of Communications and Radio Frequency Engineering Gusshausstrasse 25/389, A-1040 Vienna, Austria (ggritsch,jweinri,mrupp)@nt.tuwien.ac.at ABSTRACT In this paper we show that in the case of general Space-Time Block Codes (STBC) not only the Pairwise Error Probabil- ities (PEPs) between nearest neighboring code words have to be considered to evaluate the total Bit Error Ratio (BER) performance of the STBC. It turns out that code word pairs differing in a rather large large number of positions typi- cally lead to rank deficient Distance Matrices (DMs) caus- ing a severe diversity loss and thus deteriorate the total BER - performance in Rayleigh fading wireless Multiple Input- Multiple Output (MIMO) channels especially at high Signal to Noise Ratio (SNR) levels. In the special case of orthog- onal codes from the Generalized Complex Orthogonal De- sign (GCOD), nearest neighbor code word errors dominate the BER-curve and no diversity loss at high SNR occurs. 1. INTRODUCTION Recent research in the field of wireless communications heav- ily focuses on MIMO systems. Such systems promise a huge increase in capacity in case of wireless channels with rich scattering. To exploit this increase in capacity, cur- rently several transmission schemes are under investigation. Space-Time Coding is one method to take advantage of this potential. Several design criteria have already been pub- lished to optimize the design of Space-Time Codes (STCs), see for example [1][2]. One of the most famous design criteria are those de- rived by Tarokh et al [1]. In [1] an upper bound on the PEP of STCs is derived. This upper bound leads to power- ful code design criteria and thus to STBCs with good error performance. Evaluating the BER of a code, mostly nearest neighbor error events are considered. For example, if we have a STBC-word as shown in Eqn. (1): (1) which encodes four information symbols ( to ) to be transmitted over four transmit antennas within four time slots, then we usually consider only PEPs between code words differing only in a single information symbol. As- suming for example that in Eqn. (1) the code word is erroneously detected as : (2) If only one of the information symbols is erroneously de- coded, for example , but , and , then we speak of a single symbol error. We refer to such code word pairs and as nearest neighbor code words. If however more than one information symbol dif- fers, then we call this event a multiple symbol error. In this contribution we discuss the following question: Is the BER dominated by code word errors corresponding to single or to multiple information symbol errors? The rest of the paper is organized as follows. In Section 2 we introduce general STBCs and investigate their BER for single and multiple error events. Section 3 is devoted to or- thogonal STBCs. The paper closes with some conclusions. 2. NON ORTHOGONAL SPACE TIME BLOCK CODES 2.1. Introduction In this section we focus on general STBCs and we will an- swer the question posed in Section 1. In fact, we do not provide a general solution for this problem, but we show interesting results by an illustrative example, which is in- tended to be generalized in future work. As an example, we consider a cyclic STBC [3] for a MIMO system with four transmit ( ) and four receive antennas ( ) consisting of code words, which are shown in Eqn. (1). The information symbols to are assumed to be BPSK modulated symbols taken from the set . Then the number of valid code word pairs is 256 and thus not too large to visualize the essential aspects and results. We also want to emphasize that all considerations here are based on the ML-receiver principle. We assume slow, flat Rayleigh fading (block fading) MIMO channels in correspondence to Copyright 2003 IEEE. Published in the Proceeding of the SPAWC Workshop, 15-18.6.2003, Rome, Italy.