The Design of Linear Space-Time Codes for Quasi-static Flat-fading Channels

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Varadarajan, Badri
Barry, John R.
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The reliability and data rate of wireless communication have traditionally been limited by the presence of multipath fading in wireless channels. However, dramatic performance improvements can be obtained by the use of multiple transmit and receive antennas. Specifically, multiple antennas increase reliability by providing diversity gain, namely greater immunity to deep channel fades. They also increase data rates by providing multiplexing gain, i.e., the ability to multiplex multiple symbols in one signaling interval. Harvesting the potential benefits of multiple antennas requires the use of specially designed space-time codes at the transmitter front-end. Space-time codes introduce redundancy in the transmitted signal across two dimensions, namely multiple transmit antennas and multiple signaling intervals. In this work, we focus on linear space-time codes, which linearly combine the real and imaginary parts of their complex inputs to obtain transmit vectors for multiple signaling intervals. We aim to design optimum linear space-time codes. Optimality metrics and design principles for space-time codes are shown to depend strongly on the codes' function in the overall transmitter architecture. We consider two cases, depending on whether or not the space-time code is complemented by a powerful outer error-control code. In the absence of an outer code, the multiplexing gain of a space-time code is measured by its rate, while its diversity gain is measured by its raw diversity order. To maximize multiplexing and diversity gains, the space-time code must have maximum possible rate and raw diversity order. We show that there is an infinite set of maximum-rate codes, almost all of which also have maximum raw diversity order. However, different codes in this set have different error rate for a given input alphabet and SNR. Therefore, we develop analytical and numerical optimization techniques to find the code in this set which has the minimum union bound on error rate. Simulation results indicate that optimized codes yield significantly lower error rates than unoptimized codes, at the same data rate and SNR. In a concatenated architecture, a powerful outer code introduces redundancy in the space-time code inputs, obtaining additional diversity. Thus, the raw diversity order of the space-time inner code is only a lower limit to the total diversity order of the concatenated transmitter. On the other hand, we show that the rate of the space-time code places an upper limit on the multiplexing ability of the concatenated architecture. We conclude that space-time inner codes should have maximum possible rate but need not have high raw diversity order. In particular, the serial-to-parallel converter, which introduces no redundancy at all, is a near-optimum space-time inner code. This claim is supported by simulation results. On the receiver side, we generalize the well known sphere decoder to develop new detection algorithms for stand-alone space-time codes. These new algorithms are extended to obtain efficient soft-output decoding algorithms for space-time inner codes.
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