High-Rate, Multisymbol-Decodable STBCs From Clifford Algebras

It is well known that space-time block codes (STBCs) obtained from orthogonal designs (ODs) are single-symbol decodable (SSD) and from quasi-orthogonal designs (QODs) are double-symbol decodable (DSD). However, there are SSD codes that are not obtainable from ODs and DSD codes that are not obtainable from QODs. In this paper, a method of constructing g-symbol decodable (g-SD) STBCs using representations of Clifford algebras are presented which when specialized to g = 1, 2 gives SSD and DSD codes, respectively. For the number of transmit antennas 2(a) the rate (in complex symbols per channel use) of the g-SD codes presented in this paper is a+1-g/2(a-9). The maximum rate of the DSD STBCs from QODs reported in the literature is a/2(a-1) which is smaller than the rate a-1/2(a-2) of the DSD codes of this paper, for 2(a) transmit antennas. In particular, the reported DSD codes for 8 and 16 transmit antennas offer rates 1 and 3/4, respectively, whereas the known STBCs from QODs offer only 3/4 and 1/2, respectively. The construction of this paper is applicable for any number of transmit antennas. The diversity sum and diversity product of the new DSD codes are studied. It is shown that the diversity sum is larger than that of all known QODs and hence the new codes perform better than the comparable QODs at low signal-to-noise ratios (SNRs) for identical spectral efficiency. Simulation results for DSD codes at variousspectral efficiencies are provided.

Error correction in nonbinary words

A simple error detecting and correcting procedure is described for nonbinary symbol words; here, the error position is located using the Hamming method and the correct symbol is substituted using a modulo-check procedure.

Design and characterization of in-plane MEMS yaw rate sensor

In this paper, we present the design and characterization of a vibratory yaw rate MEMS sensor that uses in-plane motion for both actuation and sensing. The design criterion for the rate sensor is based on a high sensitivity and low bandwidth. The required sensitivity of the yawrate sensor is attained by using the inplane motion in which the dominant damping mechanism is the fluid loss due to slide film damping i.e. two-three orders of magnitude less than the squeeze-film damping in other rate sensors with out-of-plane motion. The low bandwidth is achieved by matching the drive and the sense mode frequencies. Based on these factors, the yaw rate sensor is designed and finally realized using surface micromachining. The inplane motion of the sensor is experimentally characterized to determine the sense and the drive mode frequencies, and corresponding damping ratios. It is found that the experimental results match well with the numerical and the analytical models with less than 5% error in frequencies measurements. The measured quality factor of the sensor is approximately 467, which is two orders of magnitude higher than that for a similar rate sensor with out-of-plane sense direction.

Single-Symbol ML Decodable Precoded DSTBCs for Cooperative Networks

Single-symbol maximum likelihood (ML) decodable distributed orthogonal space-time block codes (DOST- BCs) have been introduced recently for cooperative networks and an upper-bound on the maximal rate of such codes along with code constructions has been presented. In this paper, we introduce a new class of distributed space-time block codes (DSTBCs) called semi-orthogonal precoded distributed single-symbol decodable space-time block codes (Semi-SSD-PDSTBCs) wherein, the source performs preceding on the information symbols before transmitting it to all the relays. A set of necessary and sufficient conditions on the relay matrices for the existence of semi-SSD- PDSTBCs is proved. It is shown that the DOSTBCs are a special case of semi-SSD-PDSTBCs. A subset of semi-SSD-PDSTBCs having diagonal covariance matrix at the destination is studied and an upper bound on the maximal rate of such codes is derived. The bounds obtained are approximately twice larger than that of the DOSTBCs. A systematic construction of Semi- SSD-PDSTBCs is presented when the number of relays K ges 4 and the constructed codes are shown to have higher rates than that of DOSTBCs.

A systematic design of high-rate full-diversity space-frequency codes for MIMO-OFDM systems

This paper presents a systematic construction of high-rate and full-diversity space-frequency block codes for MIMO-OFDM systems. While all prior constructions offer only a maximum rate of one complex symbol per channel use, our construction yields rate equal to the number of transmit antennas and simultaneously achieves full-diversity. The proposed construction works for arbitrary number of transmit antennas and arbitrary channel power delay profile. A key step in this construction is the generalization of the stacked matrix code design criteria given by Bolcskei et.al., (IEEE WCNC 2000). Explicit equivalence of our generalized code design criteria with the Hadamard-product based criteria of W. Su et.al., (lEEE Trans. Sig. Proc. Nov 2003) is established and new high-rate codes are constructed using our criteria.

A unified construction of space-time codes with optimal rate-diversity tradeoff

The problem of constructing space-time (ST) block codes over a fixed, desired signal constellation is considered. In this situation, there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code. The transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal-to-noise ratio (SNR). In the setting of a quasi-static channel model, let n(t) denote the number of transmit antennas and T the block interval. For any n(t) <= T, a unified construction of (n(t) x T) ST codes is provided here, for a class of signal constellations that includes the familiar pulse-amplitude (PAM), quadrature-amplitude (QAM), and 2(K)-ary phase-shift-keying (PSK) modulations as special cases. The construction is optimal as measured by the rate-diversity tradeoff and can achieve any given integer point on the rate-diversity tradeoff curve. An estimate of the coding gain realized is given. Other results presented here include i) an extension of the optimal unified construction to the multiple fading block case, ii) a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes, iii) the providing of a linear dispersion form for the underlying binary block codes, iv) a Gray-mapped version of the unified construction, and v) a generalization of construction of the S-ary case corresponding to constellations of size S-K. Items ii) and iii) are aimed at simplifying the decoding of this class of ST codes.

Minimum-Decoding-Complexity, maximum-rate Space-Time Block Codes from Clifford algebras

It is well known that Alamouti code and, in general, Space-Time Block Codes (STBCs) from complex orthogonal designs (CODs) are single-symbol decodable/symbolby-symbol decodable (SSD) and are obtainable from unitary matrix representations of Clifford algebras. However, SSD codes are obtainable from designs that are not CODs. Recently, two such classes of SSD codes have been studied: (i) Coordinate Interleaved Orthogonal Designs (CIODs) and (ii) Minimum-Decoding-Complexity (MDC) STBCs from Quasi-ODs (QODs). In this paper, we obtain SSD codes with unitary weight matrices (but not CON) from matrix representations of Clifford algebras. Moreover, we derive an upper bound on the rate of SSD codes with unitary weight matrices and show that our codes meet this bound. Also, we present conditions on the signal sets which ensure full-diversity and give expressions for the coding gain.

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.

On Computation of Minimum Distance of Linear Block Codes Above 1/2 Rate Coding

The minimum distance of linear block codes is one of the important parameter that indicates the error performance of the code. When the code rate is less than 1/2, efficient algorithms are available for finding minimum distance using the concept of information sets. When the code rate is greater than 1/2, only one information set is available and efficiency suffers. In this paper, we investigate and propose a novel algorithm to find the minimum distance of linear block codes with the code rate greater than 1/2. We propose to reverse the roles of information set and parity set to get virtually another information set to improve the efficiency. This method is 67.7 times faster than the minimum distance algorithm implemented in MAGMA Computational Algebra System for a (80, 45) linear block code.

Reduced ML-Decoding Complexity, Full-Rate STBCs for 4 Transmit Antenna Systems

For an n(t) transmit, n(r) receive antenna system (n(t) x nr system), a full-rate space time block code (STBC) transmits min(n(t), n(r)) complex symbols per channel use. In this paper, a scheme to obtain a full-rate STBC for 4 transmit antennas and any n(r), with reduced ML-decoding complexity is presented. The weight matrices of the proposed STBC are obtained from the unitary matrix representations of a Clifford Algebra. By puncturing the symbols of the STBC, full rate designs can be obtained for n(r) < 4. For any value of n(r), the proposed design offers the least ML-decoding complexity among known codes. The proposed design is comparable in error performance to the well known Perfect code for 4 transmit antennas while offering lower ML-decoding complexity. Further, when n(r) < 4, the proposed design has higher ergodic capacity than the punctured Perfect code. Simulation results which corroborate these claims are presented.

On Network-Error Correcting Convolutional Codes under the BSC Edge Error Model

Convolutional network-error correcting codes (CNECCs) are known to provide error correcting capability in acyclic instantaneous networks within the network coding paradigm under small field size conditions. In this work, we investigate the performance of CNECCs under the error model of the network where the edges are assumed to be statistically independent binary symmetric channels, each with the same probability of error pe(0 <= p(e) < 0.5). We obtain bounds on the performance of such CNECCs based on a modified generating function (the transfer function) of the CNECCs. For a given network, we derive a mathematical condition on how small p(e) should be so that only single edge network-errors need to be accounted for, thus reducing the complexity of evaluating the probability of error of any CNECC. Simulations indicate that convolutional codes are required to possess different properties to achieve good performance in low p(e) and high p(e) regimes. For the low p(e) regime, convolutional codes with good distance properties show good performance. For the high p(e) regime, convolutional codes that have a good slope ( the minimum normalized cycle weight) are seen to be good. We derive a lower bound on the slope of any rate b/c convolutional code with a certain degree.

Multicode MIMO for High Data Rate Mobile Ad-hoc Networks

In a mobile ad-hoc network scenario, where communication nodes are mounted on moving platforms (like jeeps, trucks, tanks, etc.), use of V-BLAST requires that the number of receive antennas in a given node must be greater than or equal to the sum of the number of transmit antennas of all its neighbor nodes. This limits the achievable spatial multiplexing gain (data rate) for a given node. In such a scenario, we propose to achieve high data rates per node through multicode direct sequence spread spectrum techniques in conjunction with V-BLAST. In the considered multicode V-BLAST system, the receiver experiences code domain interference (CDI) in frequency selective fading, in addition to space domain interference (SDI) experienced in conventional V-BLAST systems. We propose two interference cancelling receivers that employ a linear parallel interference cancellation approach to handle the CDI, followed by conventional V-BLAST detector to handle the SDI, and then evaluate their bit error rates.

Training-Embedded, Single-Symbol ML-Decodable, Distributed STBCs for Relay Networks

Recently, a special class of complex designs called Training-Embedded Complex Orthogonal Designs (TE-CODs) has been introduced to construct single-symbol Maximum Likelihood decodable (SSD) distributed space-time block codes (DSTBCs) for two-hop wireless relay networks using the amplify and forward protocol. However, to implement DSTBCs from square TE-CODs, the overhead due to the transmission of training symbols becomes prohibitively large as the number of relays increase. In this paper, we propose TE-Coordinate Interleaved Orthogonal Designs (TE-CIODs) to construct SSD DSTBCs. Exploiting the block diagonal structure of TE-CIODs, we show that the overhead due to the transmission of training symbols to implement DSTBCs from TE-CIODs is smaller than that for TE-CODs. We also show that DSTBCs from TE-CIODs offer higher rate than those from TE-CODs for identical number of relays while maintaining the SSD and full-diversity properties.

X-Codes: A Low Complexity Full-Rate High-Diversity Achieving Precoder for TDD MIMO Systems

We consider a time division duplex multiple-input multiple-output (nt × nr MIMO). Using channel state information (CSI) at the transmitter, singular value decomposition (SVD) of the channel matrix is performed. This transforms the MIMO channel into parallel subchannels, but has a low overall diversity order. Hence, we propose X-Codes which achieve a higher diversity order by pairing the subchannels, prior to SVD preceding. In particular, each pair of information symbols is encoded by a fixed 2 × 2 real rotation matrix. X-Codes can be decoded using nr very low complexity two-dimensional real sphere decoders. Error probability analysis for X-Codes enables us to choose the optimal pairing and the optimal rotation angle for each pair. Finally, we show that our new scheme outperforms other low complexity precoding schemes.

A Low ML-Decoding Complexity, Full-Diversity, Full-Rate MIMO Precoder

Precoding for multiple-input multiple-output (MIMO) antenna systems is considered with perfect channel knowledge available at both the transmitter and the receiver. For two transmit antennas and QAM constellations, a real-valued precoder which is approximately optimal (with respect to the minimum Euclidean distance between points in the received signal space) among real-valued precoders based on the singular value decomposition (SVD) of the channel is proposed. The proposed precoder is obtainable easily for arbitrary QAM constellations, unlike the known complex-valued optimal precoder by Collin et al. for two transmit antennas which is in existence for 4-QAM alone and is extremely hard to obtain for larger QAM constellations. The proposed precoding scheme is extended to higher number of transmit antennas on the lines of the E - d(min) precoder for 4-QAM by Vrigneau et al. which is an extension of the complex-valued optimal precoder for 4-QAM. The proposed precoder's ML-decoding complexity as a function of the constellation size M is only O(root M)while that of the E - d(min) precoder is O(M root M)(M = 4). Compared to the recently proposed X- and Y-precoders, the error performance of the proposed precoder is significantly better while being only marginally worse than that of the E - d(min) precoder for 4-QAM. It is argued that the proposed precoder provides full-diversity for QAM constellations and this is supported by simulation plots of the word error probability for 2 x 2, 4 x 4 and 8 x 8 systems.