Gaussian Symbols Research Articles

A Note on the Approximate Formula for the Associated Legendre Function

The approximate formula for the associated Legendre function has been used for the determination of the wavelength and phase velocity of free oscillations of the Earth and long-period surface waves. The approximate formula is defined for several mathematical parameters. One of them is n >>1/ε, where n is the angular degree and εis a parameter related to the colatitude angle θ. In the present study the relationship between n and ε is defined as n0 = [ 1/ε ] +1, where the notation [ ] denotes the Gaussian symbol. If a condition n ≧ n0 (n>1/ε ) is assumed, the approximate formula may be reasonably applied for the angular degrees n ≧2 for π/6≦θ≦5π/6. This condition is in harmony with the practical usage implemented conventionally.

Metaplectic representation and ordering (in)dependence in Vasiliev’s higher spin gravity

We investigate the formulation of Vasiliev’s four-dimensional higher-spin gravity in operator form, without making reference to one specific ordering. More precisely, we make use of the one-to-one mapping between operators and symbols thereof for a family of ordering prescriptions that interpolate between and go beyond Weyl and normal orderings. This correspondence allows us to perturbatively integrate the Vasiliev system in operator form and in a variety of gauges. Expanding the master fields in inhomogenous symplectic group elements, and letting products be controlled only by the group, we specify a family of factorized gauges in which we are able to integrate the system to all orders, producing exact solutions, including but not restricted to ones presented previously in the literature; and then connect, at first order, to a family of rotated Vasiliev gauges in which the solutions can be represented in terms of Fronsdal fields. The gauge function responsible for the latter transformation is explicitly constructed at first order. The analysis of the system in various orderings is facilitated by an analytic continuation of Gaussian symbols, by means of which one can distinguish and connect the two branches of the metaplectic double cover and give a rationale to the properties of the inner Klein operators as Gaussian delta sequences defining analytic delta densities. As an application of some of the techniques here developed, we evaluate twistor space Wilson line observables on our exact solutions and show their independence from auxiliary constructs up to the few first subleading orders in perturbation theory.

High-Speed Post-Processing in Continuous-Variable Quantum Key Distribution Based on FPGA Implementation

In a continuous-variable quantum key distribution (CV-QKD) system, the computation speed of the post-processing procedure, including information reconciliation (IR) and privacy amplification (PA), inevitably affects the practical secret key rate. IR and PA can be implemented in parallel using low-density parity-check (LDPC) codes and hash functions, respectively. We achieve high-speed hardware-accelerated post-processing procedure for Gaussian symbols on a field-programmable gate array (FPGA) by taking advantage of its superior parallel processing ability. To this end, the sum-product algorithm decoders and a modified LDPC codes construction algorithm adapted to FPGA's characteristics are developed and employed. Two different structures including multiplexing and non-multiplexing are designed to achieve the trade-off between the speed and area of FPGAs, so that an optimal scheme can be adopted according to the requirement of a practical system. Simulation results show that the maximum throughput can reach 100 M symbols/s. We verified the correctness of the post-processing procedures when implemented on the Xilinx VC709 evaluation board, which is populated with the Virtex-7 XC7VX690T FPGA and provided some possible solutions to obtain better performance when more advanced FPGAs are available. The scheme can be applied readily for real-time key extraction and effectively reduce power consumption of the CV-QKD system.

Nonlinearity Mitigation in WDM Systems: Models, Strategies, and Achievable Rates

After reviewing models and mitigation strategies for interchannel nonlinear interference (NLI), we focus on the frequency-resolved logarithmic perturbation model to study the coherence properties of NLI. Based on this study, we devise an NLI mitigation strategy which exploits the synergic effect of phase and polarization noise compensation (PPN) and subcarrier multiplexing with symbol-rate optimization. This synergy persists even for high-order modulation alphabets and Gaussian symbols. A particle method for the computation of the resulting achievable information rate and spectral efficiency (SE) is presented and employed to lower-bound the channel capacity. The dependence of the SE on the link length, amplifier spacing, and presence or absence of inline dispersion compensation is studied. Single-polarization and dual-polarization scenarios with either independent or joint processing of the two polarizations are considered. Numerical results show that, in links with ideal distributed amplification, an SE gain of about 1 bit/s/Hz/polarization can be obtained (or, in alternative, the system reach can be doubled at a given SE) with respect to single-carrier systems without PPN mitigation. The gain is lower with lumped amplification, increases with the number of spans, decreases with the span length, and is further reduced by in-line dispersion compensation. For instance, considering a dispersion-unmanaged link with lumped amplification and an amplifier spacing of 60 km, the SE after 80 spans can be be increased from 4.5 to 4.8 bit/s/Hz/polarization, or the reach raised up to 100 spans (+25%) for a fixed SE.

Gaussian distributions and phase space Weyl–Heisenberg frames

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a "phase space frame" associated with a Weyl-Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.

Decentralized Wireless Networks With Asynchronous Users and Burst Transmissions

This paper studies a decentralized wireless network of asynchronous transmitter–receiver pairs with burst transmissions. Each receiver learns about the number of active users, channel coefficients, and mutual delays based on locally available measurements. The estimates for the mutual delays are not perfect, however, they are reliable enough to guarantee successful decoding. Two signalling schemes are addressed, namely, randomized masking (RM) and reduced cycle transmission (RCT). Under RM, the $n$ symbols of a codeword are generated according to a Bernoulli–Gaussian distribution with activity factor $0 . This is in contrast to RCT where each codeword consists of $\lfloor \theta n\rfloor $ Gaussian symbols followed by $n-\lfloor \theta n\rfloor $ zeros. Assuming the transmitters are unaware of the number of users, channel coefficients, and mutual delays, the probability of outage under RM is considerably lower compared with RCT if the signal-to-noise ratio (SNR) is sufficiently large. A generalized RCT scheme is also examined where the $n-\lfloor \theta n\rfloor $ zero symbols are not necessarily located at the end of a codeword. In the asymptote of large SNR, the outage probability becomes vanishingly small under RM, however, it is bounded away from zero for generalized RCT regardless of the value of SNR.

On the Achievable Rate of Stationary Rayleigh Flat-Fading Channels With Gaussian Inputs

In this work, a discrete-time stationary Rayleigh flat-fading channel with unknown channel state information at transmitter and receiver side is studied. The law of the channel is presumed to be known to the receiver. For independent identically distributed (i.i.d.) zero-mean proper Gaussian input distributions, the achievable rate is investigated. The main contribution of this paper is the derivation of two new upper bounds on the achievable rate with Gaussian input symbols. One of these bounds is based on the one-step channel prediction error variance but is not restricted to peak power constrained input symbols like known bounds. Moreover, it is shown that Gaussian inputs yield the same pre-log as the peak power constrained capacity. The derived bounds are compared with a known lower bound on the capacity given by Deng and Haimovich and with bounds on the peak power constrained capacity given by Sethuraman et al.. Finally, the achievable rate with i.i.d. Gaussian input symbols is compared to the achievable rate using a coherent detection in combination with a solely pilot-based channel estimation.

Semi-Blind Key-Agreement over MIMO Fading Channels

In this paper, we study the fundamental limits of secret-key agreement over MIMO quasi-static fading channels. We provide closed-form expressions for the secret-key capacity in both the asymptotic high-power and low-power regimes. The optimal signaling strategy for the low-power regime is shown to be independent of the eavesdropper's channel and secret-key capacity is achieved by transmitting random Gaussian symbols along the direction corresponding to the maximal eigenvalue of the legitimate channel matrix. Hence, by beamforming and waterfilling over the main channel alone, one obtains a semi-blind key-agreement strategy in which the knowledge of the eavesdropper's channel is only required for privacy amplification. We also derive the probability that a target secret-key rate is not achieved by the optimal low-power signaling when assuming only statistical CSI about the eavesdropper's channel.

Improper Gaussian Signaling on the Two-User SISO Interference Channel

On a single-input-single-out (SISO) interference channel (IC), conventional non-cooperative strategies encourage players selfishly maximizing their transmit data rates, neglecting the deficit of performance caused by and to other players. In the case of proper complex Gaussian noise, the maximum entropy theorem shows that the best-response strategy is to transmit with proper signals (symmetric complex Gaussian symbols). However, such equilibrium leads to degrees-of-freedom zero due to the saturation of interference. With improper signals (asymmetric complex Gaussian symbols), an extra freedom of optimization is available. In this paper, we study the impact of improper signaling on the 2-user SISO IC. We explore the achievable rate region with non-cooperative strategies by computing a Nash equilibrium of a non-cooperative game with improper signaling. Then, assuming cooperation between players, we study the achievable rate region of improper signals. We propose the usage of improper rank one signals for their simplicity and ease of implementation. Despite their simplicity, rank one signals achieve close to optimal sum rate compared to full rank improper signals. We characterize the Pareto boundary, the outer-boundary of the achievable rate region, of improper rank one signals with a single real-valued parameter; we compute the closed-form solution of the Pareto boundary with the non-zero-forcing strategies, the maximum sum rate point and the max-min fairness solution with zero-forcing strategies. Analysis on the extreme SNR regimes shows that proper signals maximize the wide-band slope of spectral efficiency whereas improper signals optimize the high-SNR power offset.

Using Superposition Codebooks and Partial Decode-and-Forward in Low-SNR Parallel Relay Networks

A new communication scheme for Gaussian parallel relay networks based on superposition coding and partial decoding at the relays is presented. Some specific examples are proposed in which two codebook layers are superimposed. The first level codebook is constructed with symbols from a binary or ternary alphabet while the second level codebook is composed of codewords chosen with Gaussian symbols. The new communication scheme is a generalization of decode-and-forward, amplify-and-forward, and bursty-amplify-and-forward. The asymptotic low SNR regime is studied using achievable rates and minimum energy-per-bit as performance metrics. It is shown that the new scheme outperforms all previously known schemes for some channels and parameter ranges.

Achievable Rates of IID Gaussian Symbols on the Non-Coherent Block-Fading Channel Without Channel Distribution Knowledge at the Receiver

Consider receivers that do not exploit any knowledge of the channel distribution in non-coherent block-fading multiple-input multiple-output (MIMO) channels. The main contribution of this letter is to derive a semi-analytical semi-numerical lower bound to the achieveble rate of such receivers. The bound is of operational meaning since practical receivers can achieve the lower bound. Several numerical examples with Gaussian block fading channels, which illuminate that the lack of channel distribution at the receiver imposes a significant penalty if the coherence time (counted in symbol times) is not much larger than the number of transmit antennas, are provided.

Robust Lattice Alignment for $K$-User MIMO Interference Channels With Imperfect Channel Knowledge

In this paper, we consider a robust lattice alignment design for K-user quasi-static MIMO interference channels with imperfect channel knowledge. With random Gaussian inputs, the conventional interference alignment (IA) method has the feasibility problem when the channel is quasi-static. On the other hand, structured lattices can create structured interference as opposed to the random interference caused by random Gaussian symbols. The structured interference space can be exploited to transmit the desired signals over the gaps. However, the existing alignment methods on the lattice codes for quasi-static channels either require infinite SNR or symmetric interference channel coefficients. Furthermore, perfect channel state information (CSI) is required for these alignment methods, which is difficult to achieve in practice. In this paper, we propose a robust lattice alignment method for quasi-static MIMO interference channels with imperfect CSI at all SNR regimes, and a two-stage decoding algorithm to decode the desired signal from the structured interference space. We derive the achievable data rate based on the proposed robust lattice alignment method, where the design of the precoders, decorrelators, scaling coefficients and interference quantization coefficients is jointly formulated as a mixed integer and continuous optimization problem. The effect of imperfect CSI is also accommodated in the optimization formulation, and hence the derived solution is robust to imperfect CSI. We also design a low complex iterative optimization algorithm for our robust lattice alignment method by using the existing iterative IA algorithm that was designed for the conventional IA method. Numerical results verify the advantages of the proposed robust lattice alignment method.

Detection of Gaussian constellations in MIMO systems under imperfect CSI

This paper considers the problem of Gaussian symbols detection in MIMO systems in the presence of channel estimation errors. Under this framework we develop a computationally efficient approximations of the MAP detector. The new detectors are based on a relaxation of the discrete nature of the digital constellation and on the channel estimation error statistics. This leads to a non-convex program that is solved efficiently via a hidden convexity minimization approach. Additionally, we show that using a Bayesian EM approach, comparable BER performance to that of the MAP detector can be achieved. Next we extend the detection scheme to the case where the noise variance is unknown. We present a modified Bayesian EM approach with annealed Gibbs sampling to perform joint noise variance estimation and symbols detection. Simulation results in a random MIMO system show that the proposed algorithm outperforms the linear MMSE receiver in terms of BER.

Optimizing Training Lengths and Training Intervals in Time-Varying Fading Channels

In time-varying faded channels the transmissions are organized into frames where the channel estimation is mainly training-based. The optimal design of the training structure is formulated here by finding the training length (the optimal number of contiguous pilots) and the training interval (the interval among two successive training phases) to maximize system throughput. The optimal balance of training and payload depends on the combination of Doppler frequency and frame length. The level of the signal to noise ratio and the fading dynamics constrain the quality of the estimate from training. It is shown that the length of the training can be conveniently traded for lower training intervals to reduce the estimate out-dating. For fast-varying fading and for high enough signal to noise ratio, there is a definite advantage in fragmenting the frame with dispersed segments of training symbols of smaller length rather than having a highly reliable channel estimate by concentrating all the training symbols at the beginning of the frame. Extensive simulations corroborate the design criteria. System throughput is maximized either for noisy binary transmission and for Gaussian input symbol distribution (i.e., by using information theoretic analysis).

A Fuzzy Qualitative Framework for Connecting Robot Qualitative and Quantitative Representations

This paper proposes a novel framework for describing articulated robot kinematics motion with the goal of providing a unified representation by combining symbolic or qualitative functions and numerical sensing and control tasks in the context of intelligent robotics. First, fuzzy qualitative robot kinematics that provides theoretical preliminaries for the proposed robot motion representation is revisited. Second, a fuzzy qualitative framework based on clustering techniques is presented to connect numerical and symbolic robot representations. Built on the k-\bb AGOP operator (an extension of the ordered weighted aggregation operators), k-means and Gaussian functions are adapted to model a multimodal density of fuzzy qualitative kinematics parameters of a robot in both Cartesian and joint spaces; on the other hand, a mixture regressor and interpolation method are employed to convert Gaussian symbols into numerical values. Finally, simulation results in a PUMA 560 robot demonstrated that the proposed method effectively provides a two-way connection for robot representations used for both numerical and symbolic robotic tasks.

On the effect of receiver estimation error upon channel mutual information

Our goal in this paper is to study the effect of the receiver structure upon the achievable data rates. We consider transmission of linearly precoded data symbols over a frequency selective block fading multiple input multiple output (MIMO) wireless channel. To encompass a number of transmission schemes, we study this problem utilizing affine precoding, which is a unified model of linearly precoded data symbols with superimposed training. We focus on Bayesian receivers that estimate both the unknown fading coefficients and the data symbols. The receiver may adopt either of the following strategies to retrieve the data symbols: strategy (i) the receiver obtains joint Bayesian channel and symbol estimates, strategy, (ii) the receiver obtains a Bayesian channel estimate initially and the channel measurement is utilized to estimate the data symbols. For both strategies, we provide lower bounds on the mutual information between the data symbols and their corresponding estimates, and we relate these bounds to the symbol Cramer-Rao bound (CRB) matrices. In contrast to strategy (ii), for strategy (i) the lower bound does not depend on either the channel estimate or the covariance of the channel estimation error. For strategy (ii) we show that asymptotically (as the size of the transmission block grows) there is no loss of information after the maximum a posteriori (MAP) estimate of Gaussian symbols. We also provide guidelines to design affine precoders that maximize the derived lower bounds under the total average transmit power constraint.

Error rate in a multiple-frequency-shift system

Based on the assumption of a fading-free signal and an additive white Gaussian noise, the element and symbol error rates in a multiple-frequency-shift (m.f.s.) system and the output signal/noise ratio in a frequency-modulation (f.m.) and a pulse-code-modulation/frequency-shift (p.c.m.f.s.) system are evaluated for wide ranges of system parameters. It is shown that the required intrinsic signal/noise ratio for a given symbol error rate in an m.f.s. system can be reduced by increasing the number of frequencies in the keying. The possibility of improving the threshold of an f.m. system beyond that of a conventional one, by frequency-modulating the carrier with sampled values and demodulating the modulated wave with a band-dividing demodulator, is shown. A brief discussion on the threshold effects in the frequency-lock and phase-lock f.m. demodulators suggests that the threshold of these feedback demodulators cannot be improved beyond that of a band-dividing one. It is shown that the threshold in a p.c.m.f.s. system can be reduced by increasing the base in the coding. The comparison of a p.c.m.f.s. system with a band-dividing f.m. system shows the inferiority of the former to the latter, in so far as the minimum-power criterion of system comparison is concerned.