Modulated codes (MC) are error correction codes (ECC) defined on the complex field and therefore can be naturally combined with an intersymbol interference (ISI) channel. It has been previously proved that for any finite tap ISI channel there exist MC with coding gain comparing to the uncoded AWGN channel. In this chapter, we first consider space-time MC for memory channels, such as multiple transmit and receive antenna systems with ISI. Similar to MC for single antenna systems, the space-time MC can be also naturally combined with a multiple antenna system with ISI, which provides the convenience of the study. Some lower bounds on the capacities C and the information rates Ii.i.d of the MC coded systems are presented. We also introduce an MC coded zero-forcing decision feedback equalizer (ZF-DFE) where the channel is assumed known at both the transmitter and the receiver. The optimal MC design based on the ZF-DFE are presented.
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Space-time coding for multiple transmit and receive antenna communication systems has recently attracted considerable attentions, see for example (Eittneben (1993), Winters, Salz, Gitlin (1994), & Telatar (1995), Foschini & Gans (1998), Tarokh, Seshadri, & Calderbank (1998), Tarokh, Naguib, Seshadri, & Calderbank (1999)), which is mainly because of the significant capacity increase from the diversities. Such studies include, for example, the capacity studies (Telatar (1995), Foschini & Gans (1998), Winters, Salz, Gitlin (1994)), space-time trellis coded modulation (TCM) schemes (Tarokh, Seshadri, & Calderbank (1998), Tarokh, Naguib, Seshadri, & Calderbank (1999)), and the combination of the space time coding and signal processing (Tarokh, Seshadri, & Calderbank (1998), Tarokh, Naguib, Seshadri, & Calderbank (1999)). Most studies for such systems so far are for memoryless channels that may fit slow fading environment well, where all the paths from different transmit and receive antennas are assumed constants and treated as independent random variables. A recent study on multiple transmit and receive antenna systems with memory can be found in (Ariyavisitakul, Winters, & Lee (1999)), where no space-time coding was considered.
In applications, MIMO-OFDM has been considered to be one of the best choices in the next generation of wireless communications. IEEE 802.16e has received some proposals on MIMO precoding with limited feedbacks (Kambourov (2006), Zhang et al (2004)). In MIMO-OFDMA systems, multiuser precoding with limited number of users has been proposed to increase the system capacity of users (Liu & Zhang (2007)). In the literature, there has been a lot of works focusing on the precoding techniques. The main reason is that if the wireless channels vary relatively slow and can be predicated or estimated by some methods in a relatively short time slot, the estimation and/or the prediction of the channel characteristics can be kept a relatively high accuracy when employing in the subsequent time slots. In this case, the precoding method can be used to reduce the effect of the multiuser interference and the channel fading, so that the system capacity can be greatly improved.
In this chapter, we are interested in multiple transmit and receive antenna channels with memory, where there are intersymbol interferences (ISI) for each pair of transmit and receive antennas. Note that, unlike those in the discussions in wireless ad hoc networks, we would not employ the statistics model to characterize the intersymbol interference but the algebraic model. Here we assume that all the ISI channels for all the different pairs are known at both the transmitter and the receiver. This assumption might be too strong for wireless communications. We have two reasons for such interest. The first reason is that the channel model here may fit some communication systems, such as multi-head and multi-track recording systems, such as (Soljanin & Georghiades (1995)), where there are ISI. The second reason is that, although the following study is based on the knowledge of all the ISI channels, the generalization of the study to unknown ISI channels might be possible in the WiMAX or some with fixed MIMO-OFDMA system.