OFDM Based Distributed Space Time Coding for Asynchronous Relay Networks

Recently Li and Xia have proposed a transmission scheme for wireless relay networks based on the Alamouti space time code and orthogonal frequency division multiplexing to combat the effect of timing errors at the relay nodes. This transmission scheme is amazingly simple and achieves a diversity order of two for any number of relays. Motivated by its simplicity, this scheme is extended to a more general transmission scheme that can achieve full cooperative diversity for any number of relays. The conditions on the distributed space time block code (DSTBC) structure that admit its application in the proposed transmission scheme are identified and it is pointed out that the recently proposed full diversity four group decodable DST-BCs from precoded co-ordinate interleaved orthogonal designs and extended Clifford algebras satisfy these conditions. It is then shown how differential encoding at the source can be combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full cooperative diversity in asynchronous wireless relay networks with no channel information and also no timing error knowledge at the destination node. Finally, four group decodable distributed differential space time block codes applicable in this new transmission scheme for power of two number of relays are also provided.

Low-Complexity Near-ML Decoding of Large Non-Orthogonal STBCs using Reactive Tabu Search

Non-orthogonal space-time block codes (STBC) with large dimensions are attractive because they can simultaneously achieve both high spectral efficiencies (same spectral efficiency as in V-BLAST for a given number of transmit antennas) as well as full transmit diversity. Decoding of non-orthogonal STBCs with large dimensions has been a challenge. In this paper, we present a reactive tabu search (RTS) based algorithm for decoding non-orthogonal STBCs from cyclic division algebras (CDA) having largedimensions. Under i.i.d fading and perfect channel state information at the receiver (CSIR), our simulation results show that RTS based decoding of 12 X 12 STBC from CDA and 4-QAM with 288 real dimensions achieves i) 10(-3) uncoded BER at an SNR of just 0.5 dB away from SISO AWGN performance, and ii) a coded BER performance close to within about 5 dB of the theoretical MIMO capacity, using rate-3/4 turbo code at a spectral efficiency of 18 bps/Hz. RTS is shown to achieve near SISO AWGN performance with less number of dimensions than with LAS algorithm (which we reported recently) at some extra complexity than LAS. We also report good BER performance of RTS when i.i.d fading and perfect CSIR assumptions are relaxed by considering a spatially correlated MIMO channel model, and by using a training based iterative RTS decoding/channel estimation scheme.

Low-Complexity Near-MAP Decoding of Large Non-Orthogonal STBCs Using PDA

Non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) are attractive because they can simultaneously achieve both high spectral efficiencies (same spectral efficiency as in V-BLAST for a given number of transmit antennas) as well as full transmit diversity. Decoding of non-orthogonal STBCs with hundreds of dimensions has been a challenge. In this paper, we present a probabilistic data association (PDA) based algorithm for decoding non-orthogonal STBCs with large dimensions. Our simulation results show that the proposed PDA-based algorithm achieves near SISO AWGN uncoded BER as well as near-capacity coded BER (within 5 dB of the theoretical capacity) for large non-orthogonal STBCs from CDA. We study the effect of spatial correlation on the BER, and show that the performance loss due to spatial correlation can be alleviated by providing more receive spatial dimensions. We report good BER performance when a training-based iterative decoding/channel estimation is used (instead of assuming perfect channel knowledge) in channels with large coherence times. A comparison of the performances of the PDA algorithm and the likelihood ascent search (LAS) algorithm (reported in our recent work) is also presented.

Belief Propagation Based Decoding of Large Non-Orthogonal STBCs

In this paper,we present a belief propagation (BP) based algorithm for decoding non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) having large dimensions. The proposed approachinvolves message passing on Markov random field (MRF) representation of the STBC MIMO system. Adoption of BP approach to decode non-orthogonal STBCs of large dimensions has not been reported so far. Our simulation results show that the proposed BP-based decoding achieves increasingly closer to SISO AWGN performance for increased number of dimensions. In addition, it also achieves near-capacity turbo coded BER performance; for e.g., with BP decoding of 24 x 24 STBC from CDA using BPSK (i.e.,n576 real dimensions) and rate-1/2 turbo code (i.e., 12 bps/Hz spectral efficiency), coded BER performance close to within just about 2.5 dB from the theoretical MIMO capacity is achieved.

A Training-Based Iterative Detection/Channel Estimation Scheme for Large Non-Orthogonal STBC MIMO Systems

In this paper, we propose a training-based channel estimation scheme for large non-orthogonal space-time block coded (STBC) MIMO systems.The proposed scheme employs a block transmission strategy where an N-t x N-t pilot matrix is sent (for training purposes) followed by several N-t x N-t square data STBC matrices, where Nt is the number of transmit antennas. At the receiver, we iterate between channel estimation (using an MMSE estimator) and detection (using a low-complexity likelihood ascent search (LAS) detector) till convergence or for a fixed number of iterations. Our simulation results show that excellent bit error rate and nearness-to-capacity performance are achieved by the proposed scheme at low complexities. The fact that we could show such good results for large STBCs (e.g., 16 x 16 STBC from cyclic division algebras) operating at spectral efficiencies in excess of 20 bps/Hz (even after accounting for the overheads meant for pilot-based channel estimation and turbo coding) establishes the effectiveness of the proposed scheme.

High-Rate Space-Time Coded Large MIMO Systems: Low-Complexity Detection and Performance

Large MIMO systems with tens of antennas in each communication terminal using full-rate non-orthogonal space-time block codes (STBC) from Cyclic Division Algebras (CDA) can achieve the benefits of both transmit diversity as well as high spectral efficiencies. Maximum-likelihood (ML) or near-ML decoding of these large-sized STBCs at low complexities, however, has been a challenge. In this paper, we establish that near-ML decoding of these large STBCs is possible at practically affordable low complexities. We show that the likelihood ascent search (LAS) detector, reported earlier by us for V-BLAST, is able to achieve near-ML uncoded BER performance in decoding a 32x32 STBC from CDA, which employs 32 transmit antennas and sends 32(2) = 1024 complex data symbols in 32 time slots in one STBC matrix (i.e., 32 data symbols sent per channel use). In terms of coded BER, with a 16x16 STBC, rate-3/4 turbo code and 4-QAM (i.e., 24 bps/Hz), the LAS detector performs close to within just about 4 dB from the theoretical MIMO capacity. Our results further show that, with LAS detection, information lossless (ILL) STBCs perform almost as good as full-diversity ILL (FD-ILL) STBCs. Such low-complexity detectors can potentially enable implementation of high spectral efficiency large MIMO systems that could be considered in wireless standards.

Low-Complexity Detection and Performance in Multi-Gigabit High Spectral Efficiency Wireless Systems

Recently, we reported a low-complexity likelihood ascent search (LAS) detection algorithm for large MIMO systems with several tens of antennas that can achieve high spectral efficiencies of the order of tens to hundreds of bps/Hz. Through simulations, we showed that this algorithm achieves increasingly near SISO AWGN performance for increasing number of antennas in Lid. Rayleigh fading. However, no bit error performance analysis of the algorithm was reported. In this paper, we extend our work on this low-complexity large MIMO detector in two directions: i) We report an asymptotic bit error probability analysis of the LAS algorithm in the large system limit, where N-t, N-r -> infinity keeping N-t = N-r, where N-t and N-r are the number of transmit and receive antennas, respectively. Specifically, we prove that the error performance of the LAS detector for V-BLAST with 4-QAM in i.i.d. Rayleigh fading converges to that of the maximum-likelihood (ML) detector as N-t, N-r -> infinity keeping N-t = N-r ii) We present simulated BER and nearness to capacity results for V-BLAST as well as high-rate non-orthogonal STBC from Division Algebras (DA), in a more realistic spatially correlated MIMO channel model. Our simulation results show that a) at an uncoded BER of 10(-3), the performance of the LAS detector in decoding 16 x 16 STBC from DA with N-t = = 16 and 16-QAM degrades in spatially correlated fading by about 7 dB compared to that in i.i.d. fading, and 19) with a rate-3/4 outer turbo code and 48 bps/Hz spectral efficiency, the performance degrades by about 6 dB at a coded BER of 10(-4). Our results further show that providing asymmetry in number of antennas such that N-r > N-t keeping the total receiver array length same as that for N-r = N-t, the detector is able to pick up the extra receive diversity thereby significantly improving the BER performance.

Coherent states on Hilbert modules

We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over C*-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert C*-modules which have a natural left action from another C*-algebra, say A. The coherent states are well defined in this case and they behave well with respect to the left action by A. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive definite kernel between two C*-algebras, in complete analogy to the Hilbert space situation. Related to this, there is a dilation result for positive operator-valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory. Some possible physical applications are also mentioned.

On the Successive Refinability of MIMO channels: DMT and codes

Diversity embedded space time codes are high rate codes that are designed such that they have a high diversity code embedded within them. A recent work by Diggavi and Tse characterizes the performance limits that can be achieved by diversity embedded space-time codes in terms of the achievable Diversity Multiplexing Tradeoff (DMT). In particular, they have shown that the trade off is successively refinable for rayleigh fading channels with one degree of freedom using superposition coding and Successive Interference Cancellation (SIC). However, for Multiple-Input Multiple-Output (MIMO) channels, the questions of successive refinability remains open. We consider MIMO Channels under superposition coding and SIC. We derive an upper bound on the successive refinement characteristics of the DMT. We then construct explicit space time codes that achieve the derived upper bound. These codes, constructed from cyclic division algebras, have minimal delay. Our results establish that when the channel has more than one degree of freedom, the DMT is not successive refinable using superposition coding and SIC. The channels considered in this work can have arbitrary fading statistics.

Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design

We consider single-source single-sink (ss-ss) multi-hop relay networks, with slow-fading links and single-antenna half-duplex relay nodes. While two-hop cooperative relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this paper, we identify two families of networks that are multi-hop generalizations of the two-hop network: K-Parallel-Path (KPP)networks and layered networks.KPP networks, can be viewed as the union of K node-disjoint parallel relaying paths, each of length greater than one. KPP networks are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the DMT of these families of networks completely for K > 3. Layered networks are networks comprising of layers of relays with edges existing only between adjacent layers, with more than one relay in each layer. We prove that a linear DMT between the maximum diversity dmax and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4.For multiple-antenna KPP and layered networks, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks.For arbitrary multi-terminal wireless networks with multiple source-sink pairs, the maximum achievable diversity is shown to be equal to the min-cut between the corresponding source and the sink, irrespective of whether the network has half-duplex or full-duplex relays. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable.Along the way, we derive the optimal DMT of a generalized parallel channel and derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. We also give alternative and often simpler proofs of several existing results and show that codes achieving full diversity on a MIMO Rayleigh fading channel achieve full diversity on arbitrary fading channels. All protocols in this paper are explicit and use only amplify-and-forward (AF) relaying. We also construct codes with short block-lengths based on cyclic division algebras that achieve the optimal DMT for all the proposed schemes.Two key implications of the results in the paper are that the half-duplex constraint does not entail any rate loss for a large class of cooperative networks and that simple AF protocols are often sufficient to attain the optimal DMT

Asymptotically optimal cooperative wireless networks without constellation expansion

In this work, we construct a unified family of cooperative diversity coding schemes for implementing the orthogonal amplify-and-forward and the orthogonal selection-decode-and-forward strategies in cooperative wireless networks. We show that, as the number of users increases, these schemes meet the corresponding optimal high-SNR outage region, and do so with minimal order of signaling complexity. This is an improvement over all outage-optimal schemes which impose exponential increases in signaling complexity for every new network user. Our schemes, which are based on commutative algebras of normal matrices, satisfy the outage-related information theoretic criteria, the duplex-related coding criteria, and maintain reduced signaling, encoding and decoding complexities

Maximum Rate of Unitary-Weight, Single-Symbol Decodable STBCs

It is well known that the space-time block codes (STBCs) from complex orthogonal designs (CODs) are single-symbol decodable/symbol-by-symbol decodable (SSD). The weight matrices of the square CODs are all unitary and obtainable from the unitary matrix representations of Clifford Algebras when the number of transmit antennas n is a power of 2. The rate of the square CODs for n = 2(a) has been shown to be a+1/2(a) complex symbols per channel use. However, SSD codes having unitary-weight matrices need not be CODs, an example being the minimum-decoding-complexity STBCs from quasi-orthogonal designs. In this paper, an achievable upper bound on the rate of any unitary-weight SSD code is derived to be a/2(a)-1 complex symbols per channel use for 2(a) antennas, and this upper bound is larger than that of the CODs. By way of code construction, the interrelationship between the weight matrices of unitary-weight SSD codes is studied. Also, the coding gain of all unitary-weight SSD codes is proved to be the same for QAM constellations and conditions that are necessary for unitary-weight SSD codes to achieve full transmit diversity and optimum coding gain are presented.

Algebraic Approaches to Space-Time Code Construction for Multiple-Antenna Communication

A major challenge in wireless communications is overcoming the deleterious effects of fading, a phenomenon largely responsible for the seemingly inevitable dropped call. Multiple-antennas communication systems, commonly referred to as MIMO systems, employ multiple antennas at both transmitter and receiver, thereby creating a multitude of signalling pathways between transmitter and receiver. These multiple pathways give the signal a diversity advantage with which to combat fading. Apart from helping overcome the effects of fading, MIMO systems can also be shown to provide a manyfold increase in the amount of information that can be transmitted from transmitter to receiver. Not surprisingly,MIMO has played, and continues to play, a key role in the advancement of wireless communication.Space-time codes are a reference to a signalling format in which information about the message is dispersed across both the spatial (or antenna) and time dimension. Algebraic techniques drawing from algebraic structures such as rings, fields and algebras, have been extensively employed in the construction of optimal space-time codes that enable the potential of MIMO communication to be realized, some of which have found their way into the IEEE wireless communication standards. In this tutorial article, reflecting the authors’interests in this area, we survey some of these techniques.

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S (-1)(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szego kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Muller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in a'', (m) . Some consequences of this more general result are then explored in the case of several natural function algebras.

Hilbert W*-modules and coherent states

Hilbert C*-module valued coherent states was introduced earlier by Ali, Bhattacharyya and Shyam Roy. We consider the case when the underlying C*-algebra is a W*-algebra. The construction is similar with a substantial gain. The associated reproducing kernel is now algebra valued, rather than taking values in the space of bounded linear operators between two C*-algebras.