A non-orthogonal distributed space-time coded protocol - Part 1:Signal model and design criteria

In this two-part series of papers, a generalized non-orthogonal amplify and forward (GNAF) protocol which generalizes several known cooperative diversity protocols is proposed. Transmission in the GNAF protocol comprises of two phases - the broadcast phase and the cooperation phase. In the broadcast phase, the source broadcasts its information to the relays as well as the destination. In the cooperation phase, the source and the relays together transmit a space-time code in a distributed fashion. The GNAF protocol relaxes the constraints imposed by the protocol of Jing and Hassibi on the code structure. In Part-I of this paper, a code design criteria is obtained and it is shown that the GNAF protocol is delay efficient and coding gain efficient as well. Moreover GNAF protocol enables the use of sphere decoders at the destination with a non-exponential Maximum likelihood (ML) decoding complexity. In Part-II, several low decoding complexity code constructions are studied and a lower bound on the Diversity-Multiplexing Gain tradeoff of the GNAF protocol is obtained.

On the maximal rate of (n+1) x n and (n+2) x n complex orthogonal designs

For p x n complex orthogonal designs in k variables, where p is the number of channels uses and n is the number of transmit antennas, the maximal rate L of the design is asymptotically half as n increases. But, for such maximal rate codes, the decoding delay p increases exponentially. To control the delay, if we put the restriction that p = n, i.e., consider only the square designs, then, the rate decreases exponentially as n increases. This necessitates the study of the maximal rate of the designs with restrictions of the form p = n+1, p = n+2, p = n+3 etc. In this paper, we study the maximal rate of complex orthogonal designs with the restrictions p = n+1 and p = n+2. We derive upper and lower bounds for the maximal rate for p = n+1 and p = n+2. Also for the case of p = n+1, we show that if the orthogonal design admit only the variables, their negatives and multiples of these by root-1 and zeros as the entries of the matrix (other complex linear combinations are not allowed), then the maximal rate always equals the lower bound.

A Novel Construction of Complex Orthogonal Designs with Maximal Rate and Low-PAPR

Space-time block codes based on orthogonal designs are used for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have low decoding complexity. However, the rate of the square real/complex orthogonal designs tends to zero with increase in number of antennas, while it is possible to have a rate-1 real orthogonal design (ROD) for any number of antennas.In case of complex orthogonal designs (CODs), rate-1 codes exist only for 1 and 2 antennas. In general, For a transmit antennas, the maximal rate of a COD is 1/2 + l/n or 1/2 + 1/n+1 for n even or odd respectively. In this paper, we present a simple construction for maximal-rate CODs for any number of antennas from square CODs which resembles the construction of rate-1 RODs from square RODs. These designs are shown to be amenable for construction of a class of generalized CODs (called Coordinate-Interleaved Scaled CODs) with low peak-to-average power ratio (PAPR) having the same parameters as the maximal-rate codes. Simulation results indicate that these codes perform better than the existing maximal rate codes under peak power constraint while performing the same under average power constraint.

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.

Estimation of variables at ungaged locations by empirical orthogonal functions

A technique based on empirical orthogonal functions is used to estimate hydrologic time-series variables at ungaged locations. The technique is applied to estimate daily and monthly rainfall, temperature and runoff values. The accuracy of the method is tested by application to locations where data are available. The second-order characteristics of the estimated data are compared with those of the observed data. The results indicate that the method is quick and accurate.

Training-Symbol Embedded, High-Rate Complex Orthogonal Designs for Relay Networks

Distributed Space-Time Block Codes (DSTBCs) from Complex Orthogonal Designs (CODs) (both square and non-square CODs other than the Alamouti design) are known to lose their single-symbol ML decodable (SSD) property when used in two-hop wireless relay networks using the amplify and forward protocol. For such a network, a new class of high rate, training-symbol embedded (TSE) SSD DSTBCs are proposed from TSE-CODs. The constructed codes include the training symbols within the structure of the code which is shown to be the key point to obtain high rate along with the SSD property. TSE-CODs are shown to offer full-diversity for arbitrary complex constellations. Non-square TSE-CODs are shown to provide better rates (in symbols per channel use) compared to the known SSD DSTBCs for relay networks when the number of relays is less than 10. Importantly, the proposed DSTBCs do not contain zeros in their codewords and as a result, antennas of the relay nodes do not undergo a sequence of switch on and off transitions within every codeword use. Hence, the proposed DSTBCs eliminate the antenna switching problem.

Code Generation Paradigms On Optical Orthogonal And Other Codes In a Fiber Optic CDMA Lan

Code Division Multiple Access (CDMA) techniques, by far, had been applied to LAN problems by many investigators, An analytical study of well known algorithms for generation of Orthogonal codes used in FO-CDMA systems like those for prime, quasi-Prime, Optical Orthogonal and Matrix codes has been presented, Algorithms for OOCs like Greedy/Modified Greedy/Accelerated Greedy algorithms are implemented. Many speed-up enhancements. for these algorithms are suggested. A novel Synthetic Algorithm based on Difference Sets (SADS) is also proposed. Investigations are made to vectorise/parallelise SADS to implement the source code on parallel machines. A new matrix for code families of OOCs with different seed code-words but having the same (n,w,lambda) set is formulated.

The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

Analytical solutions to problems in finite elasticity are most often derived using the semi-inverse approach along with the spatial form of the equations of motion involving the Cauchy stress tensor. This procedure is somewhat indirect since the spatial equations involve derivatives with respect to spatial coordinates while the unknown functions are in terms of material coordinates, thus necessitating the use of the chain rule. In this classroom note, we derive compact expressions for the components of the divergence, with respect to orthogonal material coordinates, of the first Piola-Kirchhoff stress tensor. The spatial coordinate system is also assumed to be an orthogonal curvilinear one, although, not necessarily of the same type as the material coordinate system. We show by means of some example applications how analytical solutions can be derived more directly using the derived results.

Wavelength/Time orthogonal codes with FEC offer dual advantage

To meet the growing demands of the high data rate applications, suitable asynchronous schemes such as Fiber-Optic Code Division Multiple Access (FO-CDMA) are required in the last mile. FO-CDMA scheme offers potential benefits and at the same time it faces many challenges. Wavelength/Time (W/T) 2-D codes for use in FO-CDMA, can be classified mainly into two types: 1) hybrid codes and 2) matrix codes, to reduce the 'time' like property, have been proposed. W/T single-pulse-per-row (SPR) are energy efficient codes as this family of codes have autocorrelation sidelobes of '0', which is unique to this family and the important feature of the W/T multiple-pulses-per-row (MPR) codes is that the aspect ratio can be varied by trade off between wavelength and temporal lengths. These W/T codes have improved cardinality and spectral efficiency over other W/T codes and at the same time have lowest crosscorrelation values. In this paper, we analyze the performances of the FO-CDMA networks using W/T SPR codes and W/T MPR codes, with and without forward error correction (FEC) coding and show that with FEC there is dual advantage of error correction and reduced spread sequence length.

Quasi-Orthogonal Design and Performance Analysis of Amplify-And-Forward Relay Networks with Multiple-Antennas

This paper is on the design and performance analysis of practical distributed space-time codes for wireless relay networks with multiple antennas terminals. The amplify-andforward scheme is used in a way that each relay transmits a scaled version of the linear combination of the received symbols. We propose distributed generalized quasi-orthogonal space-time codes which are distributed among the source antennas and relays, and valid for any number of relays. Assuming M-PSK and M-QAM signals, we derive a formula for the symbol error probability of the investigated scheme over Rayleigh fading channels. For sufficiently large SNR, this paper derives closed-form average SER expression. The simplicity of the asymptotic results provides valuable insights into the performance of cooperative networks and suggests means of optimizing them. Our analytical results have been confirmed by simulation results, using full-rate full-diversity distributed codes.