On computing the minimum distance for faster than Nyquist signaling

We consider the capacity of a band-limited additive white Gaussian noise channel, with bipolar inputs, a framework motivated by power-efficient faster than Nyquist signaling. We improve the available Ozarow-Wyner-Ziv (OWZ) lower bound on capacity by introducing a new communication scheme, exhibiting two advantages over the OWZ approach based on peak-power constrained pulse-amplitude modulation. We demonstrate analytically, a moderately improved information rate of the suggested technique, exhibiting also considerably less sign transitions of the bipolar signal. The gap between the known upper-bound based on spectral constrains of bipolar signals and the new achievable lower bound is reduced to 1.1 bits per Nyquist interval at the high signal-to-noise ratio regime.

Pulse transmission through a band-limited channel without intersymbol interference is governed by the Nyquist criterion. A particular pulse that has been widely used in practice corresponds to the raised cosine spectrum. This paper addresses the question of how to generate other Nyquist pulses. A worst case design criterion from the literature for analyzing pulse sensitivity to timing errors is reviewed and extended. Closed form expressions are derived for the Nyquist pulse generating a pulse with a triangular rolloff spectrum. The error performance, versus excess bandwidth, of the triangular pulse is compared with that of the standard raised cosine pulse. The triangular pulse is found to outperform the raised cosine pulse.

In 1975, J. E. Mazo showed the potential faster‐than‐Nyquist (FTN) gain of the single‐carrier binary signal. If the inter‐symbol interference is eliminated by an optimal detector, FTN single‐carrier binary signal can transmit 24.7% more bits than the Nyquist signal without loss of bit error rate performance, which is known as the Mazo limit. In this study, the authors apply sphere decoder (SD) with box optimisation (BO) to reduce inter‐carrier interference (ICI) in the FTN non‐orthogonal frequency division multiplexing (FTN–NOFDM) system. Compared with the commonly used SD algorithm, SD with BO can achieve the same ICI cancellation performance, and the computational complexity is significantly reduced. When the bandwidth compression factor is set to 0.802, the transmission rate of FTN–NOFDM is 24.7% faster than the Nyquist rate, and quadrature phase shift keying (QPSK)‐modulated FTN–NOFDM has almost the same performance as QPSK‐modulated orthogonal frequency division multiplexing (OFDM), which agrees well with Mazo limit. In the simulation, the QPSK‐modulated FTN–NOFDM with outperforms 16‐quadrature amplitude modulation (16‐QAM)‐modulated OFDM by about 1.5 dB, and the 16‐QAM‐modulated FTN–NOFDM with and outperforms 64‐ and 256‐QAM‐modulated OFDM by about 1.5 and 2 dB, respectively.

ABSTRACTIn this article, for the first time the generation, multiplexing, and detection of 75GBaud Nyquist QPSK signal transmission over two orthogonal Laguerre–Gaussian modes carrying orbital angular momentum (OAM) were experimentally demonstrated. To further increase spectrum efficiency, Nyquist signal was generated by all‐optical method to produce sinc‐shaped Nyquist pulses and received by full‐band signal coherent detection with digital signal processing. Two OAM modes with same absolute value of charge but opposite sign were generated, multiplexed, and demultiplexed. The bit error ratio performance of the received signal was under forward error correction threshold with optical signal‐to‐noise ratio of 20 and 22 dB for +8 mode and −8 mode, respectively. To our best knowledge, this is the highest baud rate ever achieved in OAM based wireless optical transmission systems, which clearly validate the feasibility and benefit of Nyquist signal transmission with OAM mode multiplexing to increase the spectrum efficiency and system capacity. © 2016 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:1866–1869, 2016

Let $k,n,m \in {\mathbb Z} ^{+}$ be integers such that $k\leq n \leq m$ , let $\mathrm {G}_{n,k}\in {\mathbb F} _{q^{m}}^{n}$ be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius $\tau $ , providing $\tau $ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in ${\mathbb F}_{q^{m}}^{n}$ . Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$ . As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when $n\mid m$ , we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +1$ , where $d$ is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +2$ , where $d$ is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of ${\mathbb F}_{q^{m}}^{n}$ of order $q^{kn}$ with $n$ dividing $m$ , such that $4n-3$ is a square in $\mathbb {Z}$ and containing $\mathrm {G}_{n,2}$ , is not efficiently list decodable at some values of the radius $\tau $ .

Based on Fresnel diffraction formula, the spatial frequency distribution of the holograms recorded with plane and spherical reference wave are analyzed. Using the sampling theorem and the condition of spectrum separation, the general recording conditions of the digital hologram are deduced. The minimum recording distances and the offsets of the reference wave for some practical recording configurations are given. The deduced recording conditions about off-axis Fresnel hologram are different from others, which are different each other. The separating condition of the frequency spectrum for Fourier transform holography is applied directly in their derivation. Moreover, they have applied different other approximations in their derivation. No approximation is applied to our deduction. Therefore, the results derived from our proposed method are accurate. The recording conditions of digital holography with general spherical reference wave are presented firstly on our knowledge. The recording conditions of in-line and off-axis lensless Fourier can be gotten directly as special case. The simulations and the experiments demonstrate that the method and the results are valid. The results also show that only the sampling and separation conditions are met simultaneously, the reconstructed image with high quality can be obtained.

This work addresses the problem of increasing the robustness to bit errors for two description scalar quantizers. To this aim a permutation is applied to the indices of each description. We show how to construct permutation pairs that increase the minimum Hamming distance of the set of valid index pairs to 3 when the redundancy is sufficiently high. Additionally, for the case when one description is known to be correct we propose a new performance criterion, denoted by d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">side,min</sub> . This represents the minimum Hamming distance of the set of valid indices of one description, when the index of the other description is fixed. We develop a technique for constructing permutation pairs achieving d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">side</sub> , <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> ≥ h based on a linear (R, [log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> m]) channel code of minimum Hamming distance h + 1, where R is the rate of each description and m is the number of diagonals occupied by the valid index pairs in the matrix of the initial index assignment.

The sampling theorem enables a band-limited signal to be expressed in terms of a set of sample point values, which occur at the Nyquist rate. The sampling theorem has been generalized to include nonuniform sampling and the use of derivatives of the signal. In the present paper, a sampling theorem has been developed which utilizes information related to the zeros of the signal. The concept of complex zeros is introduced to show that the zeros occur at the Nyquist rate. This sampling theorem can be of use for enabling the transmission of binary signals, such as facsimile and infinitely-clipped speech, over a continuous band-limited channel. The result indicates the desirability of developing a completely general theory of sampling applicable to the various situations which may arise in practice.

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We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distributed rather densely in the domain and the characteristic sizes of the cavities and the minimal distances between the cavities satisfy certain simple condition, we show that a solution to our problem tends to zero as the perforation becomes finer. Our main result are order sharp estimates for the \(L_2\)- and \(W_2^1\)-norms of the solution uniform in the \(L_2\)-norm of the right hand side in the equation.

The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x1,…, xk+1 ∈ Gk+1 for which the simple commutator [x1, …, xk+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.

Two delay-and-sum beamformers for 3-D synthetic aperture imaging with row-column addressed arrays are presented. Both beamformers are software implementations for graphics processing unit (GPU) execution with dynamic apodizations and third-order polynomial subsample interpolation. The first beamformer was written in the MATLAB programming language and the second was written in C/C++ with the compute unified device architecture (CUDA) extensions by NVIDIA. Performance was measured as volume rate and sample throughput on three different GPUs: a 1050 Ti, a 1080 Ti, and a TITAN V. The beamformers were evaluated across 112 combinations of output geometry, depth range, transducer array size, number of virtual sources, floating-point precision, and Nyquist rate or in-phase/quadrature beamforming using analytic signals. Real-time imaging defined as more than 30 volumes per second was attained by the CUDA beamformer on the three GPUs for 13, 27, and 43 setups, respectively. The MATLAB beamformer did not attain real-time imaging for any setup. The median, single-precision sample throughput of the CUDA beamformer was 4.9, 20.8, and 33.5 Gsamples/s on the three GPUs, respectively. The throughput of CUDA beamformer was an order of magnitude higher than that of the MATLAB beamformer.

Let X be a subgroup of the full automorphism group of the Hamming graph H ( m , q ) , and C a subset of the vertices of the Hamming graph. We say that C is an ( X , 2) -neighbour-transitive code if X is transitive on C , as well as C 1 and C 2 , the sets of vertices which are distance 1 and 2 from the code. It has been shown that, given an ( X , 2) -neighbour-transitive code C , there exists a subgroup of X with a 2 -transitive action on the alphabet; this action is thus almost-simple or affine. This paper completes the classification of ( X , 2) -neighbour-transitive codes, with minimum distance at least 5 , where the subgroup of X stabilising some entry has an almost-simple action on the alphabet in the stabilised entry. The main result of this paper states that the class of ( X , 2) neighbour-transitive codes with an almost-simple action on the alphabet and minimum distance at least 3 consists of one infinite family of well known codes.

In previous continuous-variable quantum key distribution (CV-QKD) systems, the transmission rate is limited by the Nyquist criterion. Since the data symbols transmit without intersymbol interference (ISI) under this limitation, how to improve the secret key rate is still a notoriously hard problem. Faster-than-Nyquist (FTN) technology has been viewed as a potential technique in future communication systems, because of its higher spectrum efficiency and faster symbol rate. In this paper, a CV-QKD system based on FTN is proposed. The transmission of an FTN system is simulated, and then we analyze the Mazo limit and the constrained capacities of the system, and finally compare the secret key rate of FTN signals and Nyquist signals. Our proposed scheme is especially beneficial for the high-speed scenario. Simulation results demonstrate that such a CV-QKD system can increase channel capacity and enhance key rate; it can also improve the safe transmission distance of CV-QKD.

For one-dimensional (1-D) recording channels, the detector performance can be assessed by the distance properties of the channel, which are defined in terms of the maximum likelihood decoding trellis. Error events with minimum and near minimum distances play an important role in the performance of the recording system particularly at moderate to high signal-to-noise ratio (SNR). In this paper, we analyze the distance properties of two-dimensional (2-D) inter-symbol interference channels, in particular the 2-D PR1 channel which is an extension of the 1-D PR1 channel. The minimum distance of this channel is proved to be 2 and a complete characterization of the distance-2 error events is provided. Also, the error events with squared-Euclidean distance 6 are partially characterized. Analogous to 1-D channels, error-state diagrams for 2-D channels can be constructed to characterize the error events. We propose an efficient error event search algorithm operating on the errorstate diagram that is applicable to any 2-D channel.