# Space-Time Wireless Communications

Space-time wireless communications refers to the physical layer-architecture that utilizes multiple transmit/receive antennas. Our work on this topic has been on the following aspects.

## Diversity embedded codes

Space-time coding refers to the use of multiple transmit and receive antennas and has been an extremely active research area over the past decade. Diversity order, which captures reliability in terms of error performance, and rate impose a fundamental trade-off in space-time coding. High-rate space-time codes come at a cost of lower diversity order, and high reliability (diversity order) results in a lower rate. Over the past few years, this trade-off has been quite well understood. However, if we design the overall system for a fixed rate-diversity operating point, we might be over-provisioning a resource which could be flexibly allocated to different applications. This alternate viewpoint motivated the work that we have pursued on opportunistic coding over the past few years. We proposed a new paradigm for the design of wireless links that makes it possible to design a high rate code with a high reliability code embedded within. This allows a form of communication where the high-rate code opportunistically takes advantage of good channel realizations whereas the embedded high-diversity code ensures that at least part of the information is received reliably. We have studied this problem both from the point-of-view of information-theoretic bounds on performance as well as explicit algebraic code designs.

**Multilevel diversity embedded codes:** We have constructed a class of space-time codes which can flexibly allocate rate and reliability to an application. We first introduced linear constructions and developed the basic ideas of diversity embedded codes. These linear constructions allow us to use sphere decoders for low-complexity decoding. We also explored other low-complexity decoding schemes along with successive interference cancelling decoders. We developed a class of constructions that can be viewed as a generalization of classical multi-level codes to the fading multiple antenna channel. We develop a correspondence between rank properties of binary matrix sets and lift these properties to the complex domain to give systematic constructions of a class of diversity embedded codes. These codes allow us to flexibly allocate rate and reliability for different streams. We have developed diversity embedded codes for broadband (ISI) channels by constructing maximal sets of Toeplitz binary matrices with rank guarantees. These constructions might be of independent interest.

**Successive refinement of diversity:** From an information-theoretic point of view we proved that when we have one degree of freedom (i.e., one transmit and many receive antennas or many transmit and one receive antennas), the rate-diversity trade-off is successively refinable. This shows that one can design ideal opportunistic coding strategies for wireless channels. We have further characterized the achievable performance for such schemes for some special cases where we have more than one degree of freedom. We have also extended these ideas to broadband (ISI) channels, and have shown that ideal opportunistic codes can also be designed for such channels. This was not obvious, since we had shown that parallel channels are not successively refinable, the ISI channel seems to have a significantly different characterization This observation could have an impact on the operation of broadband wireless links. We have related the general characterization of achievable tuples for diversity embedded codes to the (open) problem of broadcasting with degraded message sets. We have made some recent progress on this problem.

**Networking applications of diversity embedded codes:** We have also investigated networking aspects when we have diversity embedded code capabilities in the physical link. For example, we have applied the network utility maximization framework to address the distributed allocation of coding resources when physical layer links have diversity embedded capabilities. This could be important when applications with very different QoS requirements share the network resources. We have also investigated the delivery of images using a layered source coder matched with appropriate choices of diversity embedded codes. We have also studied aspects related to cell coverage extension and other applications.

## Space-time coding

There are significant challenges in bringing the theoretical advantages of space-time architectures to practice. Most of these challenges relate to code-design and signal processing problems for multiple antenna architectures.

**Inter-Symbol Interference Multiple-Access Channels:** We studied the multiple-access channel where users employ time-reversed space-time block codes (TR-STBC) which is suitable for ISI channels. One of the contributions is to identify key algebraic properties of the TR-STBC codes which allows us to design and analyze multiuser detectors. Using these, we showed that a diversity order of $2M_r(\nu+1)$ is achievable at full transmission rate for each user, when we have $M_r$ receive antennas, $M_t=2$ transmit antennas per user, channel memory of $\nu$ and an optimal multiuser maximum-likelihood (ML) decoder is used. Due to the decoding complexity of the ML detector we study the algebraic structure of linear multiuser detectors which utilize the properties of the STBC. We also do this when we have finite block length constraints using properties of quaternionic block circulant matrices.

**Non-intersecting subspaces over finite alphabet:** In the high SNR regime of non-coherent space-time codes, we communicate using subspaces in which the codewords lie. The reliability (diversity order) is determined by the maximal intersection between the subspaces of any two codewords in the codebook. Moreover, in many applications we are constrained to use a specific transmit alphabet ({\em e.g.,} QPSK). Therefore we formulated a question on what is the maximal rate we can get with these alphabet constraints when we desire to have the maximal diversity order. This translates to a mathematical question on the number of pairwise nonintersecting $M_t$-dimensional subspaces of an $m$-dimensional vector space $V$ over a field $\mathbb{F}$ if the generator matrices for the subspaces may contain only symbols from a given finite alphabet. Note that, two subspaces of a vector space are here called “nonintersecting” if they meet only in the zero vector. We have a complete answer if the alphabet is GF(q); we show that the number of nonintersecting subspaces is at most $(q^m-1)/(q^{M_t}-1)$, and that this bound can be attained if and only if $m$ is divisible by $M_t$. When we have PSK constellations, we have examined the case when we have $M_t=2$ tramsmit antennas, and have shown that the number of nonintersecting planes is at least $2^{r(m-2)}$ and at most $2^{r(m-1)-1}$ (the lower bound may in fact be the best that can be achieved).

**Differential Space-Time Transmission for Frequency-Selective Channels:** In fast time-varying channels, it is difficult to get accurate channel estimates. In some cases, it might be desirable to design non-coherent transmission/detection schemes. There have been several recent proposals for differential transmission over multiple antenna channels. Most of these have focussed on the “flat-fading” channel i.e. the channel is represented by a single tap (no frequency-selectivity). We have proposed space-time transmission schemes which allow full-rate and full-diversity non-coherent communications using two transmit antennas over fading frequency-selective channels. The first scheme operates in the frequency domain where it combines differential Alamouti space-time block-coding with OFDM. The second scheme operates in the time domain and employs differential time-reversal space-time block-coding to guarantee blind channel identifiability without the need for temporal oversampling or multiple receive antennas. We also describe their extensions to an arbitrary number of transmit antennas.

**Quaternionic space-time codes:** We designed full-rate, full-diversity orthogonal space-time block codes for 4 transmit antennas based on quaternionic algebra. This code is non-linear but can be designed to have no constellation expansion for QPSK modulation. The quaternionic structure can be further used to reduce the complexity of the optimal decoder. We also used this code to construct non-coherent differential codes.

## Space-time signal processing

**Interference suppression:** The frequency re-use in geographically separated cells causes co-channel interference (CCI) between users in different cells sharing the same frequencies.To combat time-variation adaptive equalization schemes have been proposed extensively in literature. Most of these are decision directed adaptation schemes and have severe error-propagation in dynamic environments. We proposed a detection scheme which uses a colored Gaussian metric to suppress CCI and adapts parameters conditioned on survivor sequences. This joint channel estimation with interference cancellation scheme is a robust structure which handles both time-variation and CCI. By exploiting the structure in the channel and tracking the interference covariance matrix, we obtain considerable gains in performance. The adaptive nature of this algorithm makes it tolerant to abrupt changes in interference characteristics, a problem frequently encountered in practice due to asynchronous interferers. Though the ultimate utility of an algorithm is its performance in practice, it is also important to glean insight into its characteristics through analysis. Therefore we studied the probability of error in detection over time-varying channels and showed that the phenomenon of error flooring is fundamentally caused by channel estimation errors. We also examined the interference suppression capabilities of the algorithm described above through its pairwise probability of error. This showed that its performance is similar to multi-channel MLSE detection in a reduced dimensional space. We also examined the error probability under channel mismatch and the impact of channel dynamics on performance.

**Inter-carrier interference in MIMO-OFDM:** There are important signal processing problems in multicarrier transmission over time-varying channels. We developed a model for such a transmission scheme and focussed particularly on MIMO OFDM. Using this method, we analyzed the impact of time-variation within a transmission block (time variation could arise both from Doppler spread of the channel and from synchronization errors). To mitigate the effects of such time-variations, we proposed a time-domain approach. We designed ICI-mitigating block linear filters, and we examined how they are modified in the context of space-time block-coded transmissions. Our approach reduces to the familiar single-tap frequency-domain equalizer when the channel is block time-invariant. Channel estimation in rapidly time-varying scenarios becomes critical, and we proposed a scheme for estimating channel parameters varying within a transmission block. Along with the channel estimation scheme, we also examined the issue of pilot tone placement and showed that in time-varying channels it may be better to group pilot tones together into clumps equispaced onto the FFT grid; this placement technique is in contrast to the common wisdom for time-invariant channels.

## Information theory for space-time coding

There has been a recent explosion of interest in using multiple antennas both at the transmitter and the receiver. This has been fueled by information-theoretic results by Foschini (1996), Telatar (1995) and space-time coding schemes proposed by Tarokh, Seshadri and Calderbank (1998).

**Rate-growth of linear detectors:** One of the questions we asked was whether the rate advantages shown by Foschini et al can be obtained by using simpler receiver structures, such as linear receivers (akin to multiuser detection schemes). We showed that the linear growth in the rate with the number of antennas, asymptotically as the number of antennas being large, can also be achieved by such low-complexity schemes. However, the linear growth assumes that the channel gain becomes unbounded resulting in unbounded achievable rates. Consequently we examined the normalized channel (where the average gain is unity), and found that the mutual information grows linearly with signal-to-noise ratio (SNR) as the diversity elements become large.This is analogous to the infinite bandwidth result in Gaussian channels (Gallager, Information theory and reliable communications, 1968). Additionally we showed that the linear gain in the number of spatial diversity elements (on both the transmitter and the receiver) also occurs when the SNR becomes very large. We derived the cut-off rate of the diversity channel when the transmitter knows only the spatial correlation behavior of the fading channel. Using the cut-off rate we evaluated the gains in using coding over multiple sensors, i.e. space-time coding. These results demonstrate that the rate advantages of multiple antennas can be still obtained using lower complexity linear detection schemes and such schemes have a rich literature in the context of multiuser detection (Verdu, Multiuser detection, 1998).

**Fast time-varying ISI channels:** If the transmission blocklength is much smaller than the coherence time (time between fades of a wireless channel), then the channel is block time-invariant and one can derive the achievable rate for such a channel with multiple transmit and receive antennas.When there is fast time-variation there is an inherent tension between the block time-invariance model and letting the block size be large for coding arguments to hold. Therefore we studied the impact of fast time-variation (i.e. time-variation within a transmission block) for multicarrier schemes and in particular for OFDM. This could also occur in the presence of a frequency offset. In such channels, the Fourier basis is not in general an eigenbasis and this loss of orthogonality causes Inter-Carrier Interference (ICI). We derived the throughput rates both when optimal (joint) decoding is done and when the ICI is considered a part of the noise (as is usually done in practice). The degrading effect of ICI on the throughput was quantified and these tradeoffs are studied in the context of packet size and transceiver design.This also gives insights into the role of equalization in time-varying ISI channels.We have also explored signal processing techniques to mitigate the effects of ICI, due to channel time-variation.

## Channel characterization

In order to develop algorithms and architectures for wireless communications, one needs to first understand the nature of the transmission medium itself. We developed a channel model suitable for multiple antenna architectures based on the plane-wave propagation of the radio signals. This model also enables us to isolate the characteristics of the wireless channel which we can exploit using signal processing algorithms.

It might be unrealistic to assume that the channel state information is available at the transmitter in a Frequency Division Duplex (FDD) system with channel time-variation. However, the channel statistics such as the covariance (spatial and temporal) could be fairly accurately determined using the received signal in two-way communication schemes. We studied the transmit beamforming problem where we estimate the channel covariance using the received signal. We formulated the problem as an optimization problem to maximize signal-to-interference and noise ratio (SINR). Using this we devised interference avoidance schemes which result in enhanced performance. Results suggest significant gains in capacity which could be translated to lower frequency re-use factor. This work also resulted in a US patent (Patent Number US6101399) and a European patent (Patent Number EP832509A1).