Diagonal Space-time Research Articles

A note on the equivalence of energy–momentum distribution for diagonal and non-diagonal models in general relativity and teleparallel gravity

In this paper, it is shown that the super-potential defined by Einstein, Landau–Lifshitz, and Bergmann–Thompson in general relativity theory and in teleparallel gravity theory are equivalent for a general diagonal space–time metric. Therefore, the energy–momentum tensor defined by Einstein, Landau–Lifshitz, and Bergmann–Thompson is equivalent in general relativity theory and in teleparallel gravity theory for the general diagonal space–times. Two different examples are also given to ensure the validity of the established statements. However, this fact is not true in general for the non-diagonal space–times. Non-diagonal Phantom black hole metric and non-diagonal stationary axi-symmetric space–time are introduced as a counter-examples to prove the non-equivalence of the super-potential and energy–momentum tensor defined in general relativity theory and in teleparallel gravity theory.

Gravity as the Sole Fundamental Force

We had explained electromagnetism by gravity before a recent publication in this Journal, in which we further incorporated the nuclear strong force in the framework of gravity. This paper, summarizing our cumulative results, continues to integrate the nuclear weak force with gravity, where we go by the following line of logic: Planck’s formula shows energy E = frequency = probability = wave; hence quantum waves have energies and the Universe is a diagonal spacetime manifold containing {(particle pi, electromagnetic wave λ (pi))}. By Feynman’s analysis on electromagnetic mass, we assume that the distribution of E over (p, λ (p)) is (3/4 , 1/4)E. Then Newton’s gravitational acceleration formula yields E = 1.6 × the observed energy o f p, so that p exists only for a duration of 5/8 λ/c over the cycle [0, λ/c], such as evidenced in quantum tunneling, opening the possibility for λ (p) to be combined with other waves forming new particle(s) for t > 58/ λ/c. By the time ratios of two frames in General Relativity we deduce neutron’s lifetime, and by the Higgs mechanism we show neutron’s decay products.

Quantum Mechanics by General Relativity

In the framework of General Relativity we explain the creation of all particles, ordinary and anti, in two chiral directions, with multiple generations, as well as electromagnetism and the strong nuclear force. Quantum mechanics is well-known to have its foundational problems revolving around the wave-particle duality, which actually has an exact solution, viz., a diagonal spacetime manifold that admits any particle of energy coupled with its wave of energy co-existing at the same spacetime (t + it, x + iy, y + iz, z + ix). I.e., a photon can travel along x = ct with its associated electromagnetic wave spinning from y to z in circular motion as (y = cos t, z = sin t) ≡ eit. The construct of diagonal manifold, seemingly artificial, is fundamental in differential topology as it leads to the Euler characteristic. That Nature is inherently of duality cannot have a more evident example than that of the complex number x + iy, where 1 implies a linear motion in R and i = eπ2 i implies a circular motion along S 1. That the quantum wave itself possesses energy can be argued simply as: wave = probability = frequency = energy by Planck’s formula. By assigning energy entirely to particle, quantum mechanics has missed an entire copy of the Universe (the wave universe treated as the quantum vacuum).

Diagonal Space Time Block Coded Spatial Modulation

In this paper, a new Spatial Modulation (SM) scheme, called Diagonal Space Time Coded Spatial Modulation (DS-SM), is designed by embedding the Diagonal Space Time Code in SM. The DS-SM scheme still inherits advantages of SM while enjoying further benefits from spatial constellation (SC) designs. Based on rank and determinant criteria, a new set of four SC codewords is proposed for the DS-SM system with 4 transmit antennas to achieve the fourth-order diversity. Then a general design procedure for an even number of transmit antennas, larger than 4, is developed by cyclically shifting two rows of the SC codewords. Simulation results show that DS-SM surpasses several existing SM schemes at the same spectral efficiency and antenna configuration. DS-SM also exhibits better performance over the benchmark systems under spatially correlated channels. Complexity of DS-SM is also analyzed and compared with other SM schemes.

Spatially Modulated Diagonal Space Time Codes

A spatially modulated Diagonal Space Time Coding (SM-DC) scheme for four transmit antennas is presented in this letter. In the proposed SM-DC scheme, the diagonal code is employed as the core space-time block code (STBC), and the spatial constellation (SC) matrices designed by Le et al. are used to activate the transmit antennas. The SM-DC scheme can achieve the transmit diversity order of two by designing a proper precoding matrix. Moreover, it supports any number of activated transmit antennas from 1 to 4. Simulation results show that, at the cost of increased decoding complexity, the proposed SM-DC scheme outperforms several existing STBC-SM schemes with four transmit antennas.

多用户MIMO下行系统中的仿射投影解码方法

For the downlink of a multiuser (MU) multiple-input multiple-output (MIMO) system with M transmitting antennas at the base station and single receiving antenna at each user, new decoding methods, called affine projection decoders, are proposed in this paper and illustrated based on a diagonal space-time block code, under the assumption that the transmitter does not have channel state information (CSI) and transmits signals simultaneously to each user. In these decoders, the techniques for affine projection are applied, and user interference can be eliminated completely. In addition, the performances of these new decoders, such as the achievable rate regions and regions of degree of freedom (DoF), are analyzed. Simulations confirm our analysis.

Robust differential modulations for asynchronous cooperative systems

Accurate channel state information and time/frequency synchronization are challenging to acquire in mobile ad hoc cooperative set-ups, where resource-constrained relaying terminals rapidly (dis)join cooperation. This paper derives low-complexity differential modulations and cooperative transmission schemes so that detection at the receiver is feasible without channel knowledge or synchronization information. A simple distributed doubly-differential (DD) time-division multiplexing (TDM) scheme using diagonal space–time (ST) unitary codes is derived to bypass multiple carrier frequency offsets (CFOs) and collect space diversity. Precoded differential orthogonal frequency division multiplexing (OFDM) transmissions are derived to additionally bypass timing offsets and collect multipath diversity using either DD diagonal or single-differential (SD) orthogonal ST block codes (OSTBCs). DD diagonal mappings suffer from considerable coding gain and signal-to-noise ratio (SNR) loss due to unitary constellation constraints and noise enhancement at the receiver. SD-OSTBCs achieve higher performance with reduced decoding complexity, at the cost of reduced CFO mitigation range. Simple and robust non-coherent selective and adaptive transmissions are also included, showing that full space and multipath diversity can still be achieved even with decoding errors at relays. Simulations corroborate theoretical claims.

Transparency of strong gravitational waves

This paper studies diagonal spacetime metrics. It is shown that the overdetermined Einstein vacuum equations are compatible if one Killing vector exists. The stability of plane gravitational waves of the Robinson type is studied. This stability problem bares a fantastic mathematical resemblance to the stability of the Schwarzschild black hole studied by Regge and Wheeler. Just like for the Schwarzschild black hole, the Robinson gravitational waves are proven to be stable with respect to small perturbations. We conjecture that a bigger class of vacuum solutions are stable, among which are all gravitational solitons. Moreover, the stability analysis reveals a surprising fact: a wave barrier will be transparent to the Robinson waves, which therefore passes through the barrier freely. This is a hint of integrability of the 1+2 vacuum Einstein equations for diagonal metrics.

An Improved Complex Decoding Scheme for Diagonal Algebraic STBC over Fading Channels

We present a new decoding scheme of complex lattice for diagonal algebraic Space-time (ST) block codes in multiple-antennas systems. With multidimensional complex constellation symbols rotation and Hadamard transform, the constructed code is capable of achieving full diversity over a fast fading channels. At the receiver the received signals have structure similar to diagonal matrix, which is easy to decode. The Maximum-likelihood (ML) decoding of the proposed codes can be implemented by the improved complex sphere decoder, without doubling the channel matrix. It is shown that the complex lattice decoding scheme for proposed codes in a multiple-antennas system yields excellent performance and low decoding complexity.

On the Performance of Diagonal Lattice Space-Time Codes

There has been tremendous work done on designing space-time codes for the quasi-static multiple-input multiple output (MIMO) channel. All the coding design up-to-date focuses on either high-performance, high rates, low complexity encoding and decoding, or targeting a combination of these criteria . In this paper, we analyze in details the performance limits of diagonal lattice space-time codes under lattice decoding. We present both lower and upper bounds on the average decoding error probability. We first derive a new closed-form expression for the lower bound using the so-called sphere lower bound. This bound presents the ultimate performance limit a diagonal lattice space-time code can achieve at any signal-to-noise ratio (SNR). The upper bound is then derived using the union-bound which demonstrates how the average error probability can be minimized by maximizing the minimum product distance of the code. Combining both the lower and the upper bounds on the average error probability yields a simple upper bound on the the minimum product distance that any (complex) lattice code can achieve. At high-SNR regime, we discuss the outage performance of such codes and provide the achievable diversity-multiplexing tradeoff under lattice decoding.

Higher dimensional cylindrical or Kasner type electrovacuum solutions

We consider a D dimensional Kasner type diagonal spacetime where metric functions depend only on a single coordinate and electromagnetic field shares the symmetries of spacetime. These solutions can describe static cylindrical or cosmological Einstein-Maxwell vacuum spacetimes. We mainly focus on electrovacuum solutions and four different types of solutions are obtained in which one of them has no four dimensional counterpart. We also consider the properties of the general solution corresponding to the exterior field of a charged line mass and discuss its several properties. Although it resembles the same form with four dimensional one, there is a difference on the range of the solutions for fixed signs of the parameters. General magnetic field vacuum solution are also briefly discussed, which reduces to Bonnor-Melvin magnetic universe for a special choice of the parameters. The Kasner forms of the general solution are also presented for the cylindrical or cosmological cases.

Least Square Error Detection for Noncoherent Cooperative Relay Systems

In this paper, noncoherent cooperative amplify-and-forward (AF) half-duplex relay systems are considered. For such systems, an asymptotic formula of pairwise error probability (PEP) for the least square error (LSE) detector is derived by using the perturbation theory on the eigenvalues. The result shows that the full-diversity-gain function mimics coherent cooperative AF half-duplex relay systems, whereas the coding gain function mimics noncoherent multiple-input-multiple-output (MIMO) systems. In addition, it is proved that, for any given nonzero received signal, the unique blind identification of both the equivalent channel and the transmitted signals in a noise-free case is equivalent to full diversity with the LSE detector in a Gaussian noisy environment. In particular, for the noncoherent AF half-duplex protocol with three nodes, a full-diversity unitary diagonal distributed space-time block code (STBC) is designed by utilizing the full-diversity criterion established in this paper and recently developed uniquely factorable constellation. Furthermore, the generalized likelihood ratio test receiver is derived for this specific coded protocol. Simulation results show that the code presented in this paper substantially outperforms the differential code and the optimally precoded training scheme in the current literature.

Concatenated Alamouti codes using multi-level modulation and symbol mapping diversity technique

A family of space-time block codes (STBCs) for systems with even transmit antennas and any number of receive antennas is proposed. The new codeword matrix is constructed by concatenating Alamouti space-time codes to form a block diagonal matrix, and its dimension is equal to the number of transmit antennas. All Alamouti codes in the same codeword matrix have the same information; thus, full transmit diversity can be achieved over fading channels. To improve the spectral efficiency, multi-level modulations such as multi-quadrature amplitude modulation (M-QAM) are employed. The symbol mapping diversity is then exploited between transmissions of the same information from different antennas to improve the bit error rate (BER) performance. The proposed codes outperform the diagonal algebraic space-time (DAST) codes presented by Damen [Damen et al. IEEE Transactions on Information Theory, 2002, 48(3): 628–636] when they have the same spectral efficiency. Also, they outperform the 1/2-rate codes from complex orthogonal design. Moreover, compared to DAST codes, the proposed codes have a low decoding complexity because we only need to perform linear processing to achieve single-symbol maximum-likelihood (ML) decoding.

Some Designs and Normalized Diversity Product Upper Bounds for Lattice-Based Diagonal and Full-Rate Space–Time Block Codes

In this paper, we first present two tight upper bounds for the normalized diversity products (or product distances) of 2ntn2 diagonal space-time block codes from quadratic extensions on \BBQ (i) and \BBQ ( \mmbzeta6), where i =radic{-1} and \mmbzeta6=exp( i 2pi/6). Two such codes are shown to reach the tight upper bounds and therefore have the maximal normalized diversity products. We present two new diagonal space-time block codes from higher order algebraic extensions on \BBQ (i) and \BBQ ( \mmbzeta6) for three and four transmit antennas. We also present a nontight upper bound for normalized diversity products of 2ntn2 diagonal space-time block codes with QAM information symbols, i.e., in \BBZ [i ], from general 2ntn2 complex-valued generating matrices. We then present an nntnn-diagonal space-time code design method directly from 2n real integers based on extended complex lattices (of generating matrix size nntn2n) that are shown to have better normalized diversity products than the optimal diagonal cyclotomic codes do. We finally use the optimal 2ntn2 diagonal space-time codes from the optimal quadratic extensions to construct two 2ntn2 full-rate space-time block codes and find that both of them have better normalized diversity products than the Golden code does.

Optimal Normalized Diversity Product of $2\,\times\,2$ Lattice-Based Diagonal Space–Time Codes From QAM Signal Constellations

In this correspondence, we prove that the optimal normalized diversity product of lattice-based diagonal space-time block codes with Gaussian integer (or QAM) signal constellations, i.e., , and any generating matrices of complex entries (not necessarily algebraic extensions of as commonly used) is . This result implies that lattice-based diagonal space-time block codes with Gaussian integer signal constellations and generating matrices of entries from quadratic algebraic extensions of have already reached the optimal normalized diversity product.

Szekeres-type mappings of Kasner and Petrov type I(M+) purely magnetic spacetimes

We show that Szekeres-type mappings of Kasner spacetime generate a 2-variable Haddow magnetic spacetime of Petrov type I(M∞) and of Segré type . It is then shown that this spacetime is unique for the generating functions of 3- or 4-variable generalizations. Moreover, sufficient conditions on the variable dependence of the metric coefficients distinguishing types I(M∞) and I(M+) are established for diagonal purely magnetic spacetimes. These results are then combined by embedding the 2-variable Haddow magnetic spacetime within a diagonal spacetime such that the resulting spacetime is of Petrov type I(M+), in general. In turn, this leads to a family of non-vacuum purely magnetic spacetimes of Petrov type I(M+). It is then shown that either the anisotropic pressure tensor or heat-flux vector can be set to zero. Notably, the former class contains a sub-class of Haddow magnetic spacetimes also of Petrov type I(M+). Finally, the possibility of conformally relating the general family of non-vacuum purely magnetic spacetimes to vacuum is shown not to be possible.

On optimal multilayer cyclotomic space-time code designs

High rate and large diversity product (or coding advantage, or coding gain, or determinant distance, or minimum product distance) are two of the most important criteria often used for good space-time code designs. In recent (linear) lattice-based space-time code designs, more attention is paid to the high rate criterion but less to the large diversity product criterion. In this paper, we consider these two criteria together for multilayer cyclotomic space-time code designs. In a previous paper, we recently proposed a systematic cyclotomic diagonal space-time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas, where diagonal codes correspond to single-layer codes in this paper. In this paper, we first propose a general multilayer cyclotomic space-time codes. We present a general optimality theorem for these infinitely many cyclotomic diagonal (or single-layer) space-time codes over general cyclotomic number rings for a general number of transmit antennas. We then present optimal multilayer (full-rate) cyclotomic space-time code designs for two and three transmit antennas. We also present an optimal two-layer cyclotomic space-time code design for three and four transmit antennas. The optimality here is in the sense that, for a fixed mean transmission signal power, its diversity product is maximized, or equivalently, for a fixed diversity product, its mean transmission signal power is minimized. It should be emphasized that all the optimal multilayer cyclotomic space-time codes presented in this paper have the nonvanishing determinant property.

Systematic and optimal cyclotomic lattices and diagonal space-time block code designs

In this correspondence, a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory. This design provides infinitely many full diversity cyclotomic lattices for a given lattice size. Based on the packing theory and the concrete form of the design, optimal cyclotomic lattices are presented by minimizing the mean transmission signal power for a given minimum (diversity) product (or equivalently maximizing the minimum product for a given mean transmission signal power). The newly proposed cyclotomic lattices can be applied to both space-time code designs for multi-antenna systems and linear precode design for signal space diversity in single antenna systems over fast Rayleigh fading channels. Although there are some cyclotomic lattices/space-time codes existing in the literature, most of them are not optimal.

Diagonal Block Space-Time Code Design for Diversity and Coding Advantage Over Flat Fading Channels

The potential promised by multiple transmit antennas has raised considerable interest in space-time coding for wireless communications. In this paper, we propose a systematic approach for designing space-time trellis codes over flat fading channels with full antenna diversity and good coding advantage. It is suitable for an arbitrary number of transmit antennas with arbitrary signal constellations. The key to this approach is to separate the traditional space-time trellis code design into two parts. It first encodes the information symbols using a one-dimensional (M,1) nonbinary block code, with M being the number of transmit antennas, and then transmits the coded symbols diagonally across the space-time grid. We show that regardless of channel time-selectivity, this new class of space-time codes always achieves a transmit diversity of order M with a minimum number of trellis states and a coding advantage equal to the minimum product distance of the employed block code. Traditional delay diversity codes can be viewed as a special case of this coding scheme in which the repetition block code is employed. To maximize the coding advantage, we introduce an optimal construction of the nonbinary block code for a given modulation scheme. In particular, an efficient suboptimal solution for multilevel phase-shift-keying (PSK) modulation is proposed. Some code examples with 2-6 bits/s/Hz and two to six transmit antennas are provided, and they demonstrate excellent performance via computer simulations. Although it is proposed for flat fading channels, this coding scheme can be easily extended to frequency-selective fading channels.

On diagonal algebraic space-time block codes

Theoretical and practical aspects of diagonal algebraic space-time block codes over n transmit and m receive antennae are examined. These codes are obtained by sending a rotated version of the information symbols over the principal diagonal of the n /spl times/ n space-time matrix over n transmit antennae and n symbol periods. The output signal-to-noise ratios of two predecoding filters and two decoding algorithms are derived. Analysis of the information loss incurred by using the codes considered is used to clarify their structures, and the expected performances. Different algebraic real and complex rotations presented in the literature are analyzed and compared as regards the achieved coding gains, the complexities, performances, and peak-to-mean envelope power ratios.