On the maximal rate of non-square STBCs from complex orthogonal designs

A Linear Processing Complex Orthogonal Design (LPCOD) is a p x n matrix epsilon, (p >= n) in k complex indeterminates x(1), x(2),..., x(k) such that (i) the entries of epsilon are complex linear combinations of 0, +/- x(i), i = 1,..., k and their conjugates, (ii) epsilon(H)epsilon = D, where epsilon(H) is the Hermitian (conjugate transpose) of epsilon and D is a diagonal matrix with the (i, i)-th diagonal element of the form l(1)((i))vertical bar x(1)vertical bar(2) + l(2)((i))vertical bar x(2)vertical bar(2)+...+ l(k)((i))vertical bar x(k)vertical bar(2) where l(j)((i)), i = 1, 2,..., n, j = 1, 2,...,k are strictly positive real numbers and the condition l(1)((i)) = l(2)((i)) = ... = l(k)((i)), called the equal-weights condition, holds for all values of i. For square designs it is known. that whenever a LPCOD exists without the equal-weights condition satisfied then there exists another LPCOD with identical parameters with l(1)((i)) = l(2)((i)) = ... = l(k)((i)) = 1. This implies that the maximum possible rate for square LPCODs without the equal-weights condition is the same as that or square LPCODs with equal-weights condition. In this paper, this result is extended to a subclass of non-square LPCODs. It is shown that, a set of sufficient conditions is identified such that whenever a non-square (p > n) LPCOD satisfies these sufficient conditions and do not satisfy the equal-weights condition, then there exists another LPCOD with the same parameters n, k and p in the same complex indeterminates with l(1)((i)) = l(2)((i)) = ... = l(k)((i)) = 1.

BER analysis of space-time block codes from generalized complex orthogonal designs for M-PSK

In this paper, we present an analysis for the bit error rate (BER) performance of space-time block codes (STBC) from generalized complex orthogonal designs for M-PSK modulation. In STBCs from complex orthogonal designs (COD), the norms of the column vectors are the same (e.g., Alamouti code). However, in generalized COD (GCOD), the norms of the column vectors may not necessarily be the same (e.g., the rate-3/5 and rate-7/11 codes by Su and Xia in [1]). STBCs from GCOD are of interest because of the high rates that they can achieve (in [2], it has been shown that the maximum achievable rate for STBCs from GCOD is bounded by 4/5). While the BER performance of STBCs: from COD (e.g., Alamouti code) can be simply obtained from existing analytical expressions for receive diversity with the same diversity order by appropriately scaling the SNR, this can not be done for STBCs from GCOD (because of the unequal norms of the column vectors). Our contribution in this paper is that we derive analytical expressions for the BER performance of any STBC from GCOD. Our BER analysis for the GCOD captures the performance of STBCs from COD as special cases. We validate our results with two STBCs from GCOD reported by Su and Xia in [1], for 5 and 6 transmit antennas (G(5) and G(6) in [1]) with rates 7/11 and 3/5, respectively.

BER analysis of space-time block codes from generalized complex orthogonal designs for M-PSK

In this paper, we present an analysis for the bit error rate (BER) performance of space-time block codes (STBC) from generalized complex orthogonal designs for M-PSK modulation. In STBCs from complex orthogonal designs (COD), the norms of the column vectors are the same (e.g., Alamouti code). However, in generalized COD (GCOD), the norms of the column vectors may not necessarily be the same (e.g., the rate-3/5 and rate-7/11 codes by Su and Xia in [1]). STBCs from GCOD are of interest because of the high rates that they can achieve (in [2], it has been shown that the maximum achievable rate for STBCs from GCOD is bounded by 4/5). While the BER performance of STBCs: from COD (e.g., Alamouti code) can be simply obtained from existing analytical expressions for receive diversity with the same diversity order by appropriately scaling the SNR, this can not be done for STBCs from GCOD (because of the unequal norms of the column vectors). Our contribution in this paper is that we derive analytical expressions for the BER performance of any STBC from GCOD. Our BER analysis for the GCOD captures the performance of STBCs from COD as special cases. We validate our results with two STBCs from GCOD reported by Su and Xia in [1], for 5 and 6 transmit antennas (G(5) and G(6) in [1]) with rates 7/11 and 3/5, respectively.

BER analysis of space-time block codes from generalized complex orthogonal designs for M-PSK

In this paper, we present an analysis for the bit error rate (BER) performance of space-time block codes (STBC) from generalized complex orthogonal designs for M-PSK modulation. In STBCs from complex orthogonal designs (COD), the norms of the column vectors are the same (e.g., Alamouti code). However, in generalized COD (GCOD), the norms of the column vectors may not necessarily be the same (e.g., the rate-3/5 and rate-7/11 codes by Su and Xia in [1]). STBCs from GCOD are of interest because of the high rates that they can achieve (in [2], it has been shown that the maximum achievable rate for STBCs from GCOD is bounded by 4/5). While the BER performance of STBCs: from COD (e.g., Alamouti code) can be simply obtained from existing analytical expressions for receive diversity with the same diversity order by appropriately scaling the SNR, this can not be done for STBCs from GCOD (because of the unequal norms of the column vectors). Our contribution in this paper is that we derive analytical expressions for the BER performance of any STBC from GCOD. Our BER analysis for the GCOD captures the performance of STBCs from COD as special cases. We validate our results with two STBCs from GCOD reported by Su and Xia in [1], for 5 and 6 transmit antennas (G(5) and G(6) in [1]) with rates 7/11 and 3/5, respectively.

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.

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.

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.

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.

Degrees of freedom of relay aided MIMO X network with orthogonal components

We consider the one-way relay aided MIMO X fading Channel where there are two transmitters and two receivers along with a relay with M antennas at every node. Every transmitter wants to transmit messages to every other receiver. The relay broadcasts to the receivers along a noisy link which is independent of the transmitters channel. In literature, this is referred to as a relay with orthogonal components. We derive an upper bound on the degrees of freedom of such a network. Next we show that the upper bound is tight by proposing an achievability scheme based on signal space alignment for the same for M = 2 antennas at every node.