Uniform Input Distribution Research Articles

Capacity analysis for pulse amplitude modulated visible light communications with dimming control

This paper focuses on the capacity-approaching, nonuniform signaling for the pulse amplitude modulated (PAM) visible light communications under the non-negativity, peak power, and dimmable average power constraints. The input distribution is characterized by three parameters, i.e., the intensities, the probabilities, and the number of mass points in the PAM constellation. In the open literature, no analytical expression can be used to obtain the capacity-achieving input distribution. In this paper, a computationally simple but capacity-approaching input distribution is alternatively derived by determining the three aforementioned parameters. The resulting input distribution can serve as a useful tool not to approach the channel capacity but to guide the practical system design. Numerical results substantiate that the derived input distribution is a capacity-approaching distribution and can offer a better performance gain in comparison with the commonly employed uniform input distribution.

Periodic flow boiling in a non-uniformly heated microchannel heat sink

Hydrodynamic and thermal characteristics of flow boiling in a non-uniformly heated microchannel were studied. Experiments were performed with a single microchannel and a series of microheaters to study the microscale boiling of water under axially non-uniform heat input conditions. A simultaneous real time visualization of the flow pattern was performed with the measurement of experimental parameters. Tests were performed over a mass flux of 309.8kg/m2s, and heat flux of 200–600kW/m2. Test results showed different fluctuations of heated wall temperature, pressure drop, and mass flux with variations of the heat input along the flow direction. The unique periodic flow boiling in a single microchannel was observed at all heat flux conditions except for the increasing heat input distribution case which is the nearly uniform effective heat input distribution condition. The instability is correlated with flow pattern transition. For the nearly uniform effective heating condition, no fluctuation of the wall temperature, pressure drop, or mass flux was observed. We can relieve the instability by increasing total heat input along the flow direction and predict the instability using the transition criteria and flow pattern map.

Coding for High-Density Recording on a 1-D Granular Magnetic Medium

In terabit-density magnetic recording, several bits of data can be replaced by the values of their neighbors in the storage medium. As a result, errors in the medium are dependent on each other and also on the data written. We consider a simple one-dimensional combinatorial model of this medium. In our model, we assume a setting where binary data is sequentially written on the medium and a bit can erroneously change to the immediately preceding value. We derive several properties of codes that correct this type of errors, focusing on bounds on their cardinality. We also define a probabilistic finite-state channel model of the storage medium, and derive lower and upper estimates of its capacity. A lower bound is derived by evaluating the symmetric capacity of the channel, i.e., the maximum transmission rate under the assumption of the uniform input distribution of the channel. An upper bound is found by showing that the original channel is a stochastic degradation of another, related channel model whose capacity we can compute explicitly.

Optimal Alphabets and Binary Labelings for BICM at Low SNR

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit $-1.59 \tr{dB}$. It is shown that the $\Eb/N_0$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.

Polar Codes for the Two-User Multiple-Access Channel

Arikan's polar coding method is extended to two-user multiple-access channels. It is shown that if the two users of the channel use the Arikan construction, the resulting channels will polarize to one of five possible extremals, on each of which uncoded transmission is optimal. The sum rate achieved by this coding technique is the one that correponds to uniform input distributions. The encoding and decoding complexities and the error performance of these codes are as in the single-user case: $O(n\log n)$ for encoding and decoding, and $o(\exp(-n^{1/2-\epsilon}))$ for block error probability, where $n$ is the block length.

On the Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels Using OOK Signaling

A new upper bound on the capacity of power- and bandwidth-constrained optical wireless links over gamma-gamma atmospheric turbulence channels with intensity modulation and direct detection is derived when on-off keying (OOK) formats are used. In this free-space optical (FSO) scenario, unlike previous capacity bounds derived from the classic capacity of the well-known additive white Gaussian noise (AWGN) channel with uniform input distribution, a new closed-form upper bound on the capacity is found by bounding the mutual information subject to an average optical power constraint and not only to an average electrical power constraint, showing the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the signal-to-noise ratio (SNR). Additionally, it is shown that an increase of the peak-to-average optical power ratio (PAOPR) provides higher capacity values. Simulation results for the mutual information are further demonstrated to confirm the analytical results under several turbulence conditions.

On the sum capacity of a T‐user N‐frequency multiple access channel with noise

AbstractIn this paper, we investigate the sum capacity of a noisy multiple access channel (MAC): the T‐user N‐frequency noiseless MAC in cascade with a noisy channel. The noisy channel includes frequency‐selective Rayleigh fading and Additive White Gaussian Noise (AWGN). The receiver may use multiuser detection (MUD) or single user detection (SUD). We show that when using SUD, the uniform distribution achieves sum capacity. Considering MUD and assuming that all users have the same input distribution, we show through numerical optimisation, that, for most relevant values of T and N, a product of uniform input distributions maximises a tight upper bound on sum capacity. Numerical results were obtained and it was observed that, for fixed N, there is a value of T which maximises sum capacity. As expected, the results also show that there is a significant gain in sum capacity when MUD is used. Copyright © 2009 John Wiley & Sons, Ltd.

Spectral properties of the Preisach hysteresis model with random input. I. General results

We derive exact results for the spectral density S(omega) of the output of the Preisach model, a standard model for complex, nonlocal hysteresis. We obtain general results for uncorrelated input signals with arbitrary input and Preisach densities. It is shown analytically that uncorrelated input signals are transformed into output exhibiting long-time correlations. For the simplest example of uniform input and Preisach distributions we prove that correlations decay asymptotically with a t(-3) power law corresponding to a logarithmic low frequency divergence of the second derivative of the spectrum S(omega) . A simpler expression for symmetric Preisach models is also obtained, which is discussed in detail in Part II, showing that long-time tails or even 1/f noise are general features of this class of models.

An induced additive-noise model for memoryless Rayleigh-fading channels

This correspondence investigates a noncoherent discrete-time memoryless Rayleigh-fading channel. A logarithmic transform converts it into an induced channel with additive noise that is independent of the channel input. From this perspective, it is natural and convenient for us to revisit several known results and gain new insights. In particular, we specify a class of simple, log-scale uniform channel input distributions that performs well for moderate to high signal-to-noise ratio (SNR). Furthermore, a continuous-amplitude log-scale uniform distribution is asymptotically capacity-achieving in the sense that the difference between the resulting mutual information and the channel capacity approaches zero as SNR becomes large. We also extend the induced additive-noise channel approach to memoryless multiple-antenna channels possibly with transmit, but without receive, spatial correlation.

Influence of Distributional Shape of Substance Parameters on Exposure Model Output

Uncertainty of environmental concentrations is calculated with the regional multimedia exposure model of EUSES 1.0 by considering probability input distributions for aqueous solubility, vapor pressure, and octanol-water partition coefficient, K(ow). Only reliable experimentally determined data are selected from available literature for eight reference chemicals representing a wide substance property spectrum. Monte Carlo simulations are performed with uniform, triangular, and log-normal input distributions to assess the influence of the choice of input distribution type on the predicted concentration distributions. The impact of input distribution shapes on output variance exceeds the effect on the output mean by one order of magnitude. Both are affected by influence and uncertainty (i.e., variance) of the input variable as well. Distributional shape has no influence when the sensitivity function of the respective parameter is perfectly linear. For nonlinear relationships, overlap of probability mass of input distribution with influential ranges of the parameter space is important. Differences in computed output distribution are greatest when input distributions differ in the most influential parameter range.

How wrong can it be? Understanding uncertainty in petroleum systems modelling

Petroleum systems modelling is used in industry to understand and predict the interplay between the main elements of a petroleum system. This is done using numerical modelling because it cannot possibly be done experimentally given the geological time- and space scales. Such models describe the geological processes forward in time and, therefore, generate a single unique solution for a given input dataset. The evidence from data is, in most cases, insufficient to remove any ambiguities. Therefore, it is more interesting to know how much this uncertainty can be reduced, or how wrong it still might be, rather than knowing the right answer. Given the uncertainty of model outcomes, the appropriate solution should, therefore, be probabilistic rather than deterministic, where the variance expresses the remaining uncertainty in the outcome covering all possible valid geological scenarios. This paper describes how experimental design and response surface analysis can be used to perform sensitivity and uncertainty analysis in petroleum generation and migration modelling, focusing on trapped volumes derived from its drainage area. Map properties are parameterized as end-member scenarios, either as discrete or linearly varying maps. Regression coefficients indicate a good representation of trapped hydrocarbon volume by the response surface. In the example presented, hydrocarbon volume is most sensitive to fault characteristics, total organic carbon content and, to a lesser degree, to thickness of the source rock and heat flow. Well data allow a significant reduction of uncertainties, which can be shown by the probability distribution that becomes narrower but remains similar in form. The volume probability distribution of the prospect is multimodal for a discrete parameter scenario and log-normal for both discrete or uniform input distributions. Volume distributions result directly from the deterministic modelling without any prior statistical assumption on the form of the distribution.

An Efficient Sequential Learning Algorithm for Growing and Pruning RBF (GAP-RBF) Networks

This paper presents a simple sequential growing and pruning algorithm for radial basis function (RBF) networks. The algorithm referred to as growing and pruning (GAP)-RBF uses the concept of "Significance" of a neuron and links it to the learning accuracy. "Significance" of a neuron is defined as its contribution to the network output averaged over all the input data received so far. Using a piecewise-linear approximation for the Gaussian function, a simple and efficient way of computing this significance has been derived for uniformly distributed input data. In the GAP-RBF algorithm, the growing and pruning are based on the significance of the "nearest" neuron. In this paper, the performance of the GAP-RBF learning algorithm is compared with other well-known sequential learning algorithms like RAN, RANEKF, and MRAN on an artificial problem with uniform input distribution and three real-world nonuniform, higher dimensional benchmark problems. The results indicate that the GAP-RBF algorithm can provide comparable generalization performance with a considerably reduced network size and training time.

How different feature spaces may be represented in cortical maps

This paper explores how different high-dimensional feature spaces might be represented in cortical maps, subject to continuity and completeness constraints. Spaces explored included products of circular variables (such as orientation), products of linear dimensions encoding scalar features (such as spatial frequency) and products of binary features. Maps were generated using the Kohonen algorithm, with uniform or non-uniform stimulus distributions. A 2D retina was always assumed to be present. Simulations were run with and without annealing. For uniform input distributions, coverage uniformity (Swindale 1991 Biol. Cybern. 65 415-24) was used to measure how well the map was able to represent the feature space. For non-uniform distributions a weighted measure of coverage uniformity was calculated. Good coverage could be achieved for up to five or six cyclic variables but was substantially worse for a similar number of uniformly distributed scalar features. For annealed maps of multi-dimensional stimuli with Gaussian distributions, the distribution of receptive field centres and the distribution of total activity evoked on the cortex matched the stimulus distribution well. For annealed maps of non-uniformly distributed binary features there was an approximately linear relationship between the area of a map devoted to a specific feature and the probability of occurrence of the feature during development. Deviations from uniform retinotopy often led to improved coverage.

How different feature spaces may be represented in cortical maps

This paper explores how different high-dimensional feature spaces might be represented in cortical maps, subject to continuity and completeness constraints. Spaces explored included products of circular variables (such as orientation), products of linear dimensions encoding scalar features (such as spatial frequency) and products of binary features. Maps were generated using the Kohonen algorithm, with uniform or non-uniform stimulus distributions. A 2D retina was always assumed to be present. Simulations were run with and without annealing. For uniform input distributions, coverage uniformity (Swindale 1991 Biol. Cybern. 65 415–24) was used to measure how well the map was able to represent the feature space. For non-uniform distributions a weighted measure of coverage uniformity was calculated. Good coverage could be achieved for up to five or six cyclic variables but was substantially worse for a similar number of uniformly distributed scalar features. For annealed maps of multi-dimensional stimuli with Gaussian distributions, the distribution of receptive field centres and the distribution of total activity evoked on the cortex matched the stimulus distribution well. For annealed maps of non-uniformly distributed binary features there was an approximately linear relationship between the area of a map devoted to a specific feature and the probability of occurrence of the feature during development. Deviations from uniform retinotopy often led to improved coverage.

Finite Difference Output Feedback Control of a Class of Distributed Parameter Processes

Low-dimensional nonlinear feedback control schemes for distributed parameter processes are addressed. The main design procedure involves the use of a feasible numerical method to establish a finite difference-based differential equation system and to utilize the pure nonlinear feedback design for achieving uniform output regulation. When the almost uniform input distribution at steady-state conditions can be planed, the controller reduction can be involved in the proposed control design guidelines. Based on the nonlinear inverse design procedure, the low-dimensional and approximate feedback control law is developed such that the stable and bounded output tracking can be guaranteed. With the aid of internal model control strategy, the low-dimensional output feedback control scheme can ensure the asymptotic output regulation. Finally, those control methodologies are shown to be robust and effective in controlling a nonisothermal tubular reactor at possibly realistic situations.

On the decoding of convolutional codes on an unknown channel

An algorithm is proposed for universal decoding of convolutional/trellis codes employed over unknown channels. On discrete memoryless channels and at rates below the channel's computational cutoff rate (for a uniform input distribution), the algorithm achieves an asymptotic complexity-performance tradeoff similar to the tradeoff achieved by the Viterbi (1979) algorithm, but with the benefit that the algorithm's implementation does not require knowledge of the channel law. The algorithm is also applicable to channels with memory, and in particular to intersymbol interference (ISI) channels, to channels with nonlinear ISI, and even to general finite-state channels.

Approaching capacity of a continuous channel by discrete input distributions

In this paper memoryless channels with general alphabets and input constraint are considered. Sufficient conditions are given that channel capacity can be approached by discrete input distributions or by uniform input distributions with finite support. As an example, the additive white Gaussian noise channel is considered.

Expected complexity of graph partitioning problems

We study the expected time complexity of two graph partitioning problems: the graph coloring and the cut into equal parts. If k = o(√ n log n ) , we can test whether two vertices of a k-colorable graph can be k-colored by the same color in time O( k log n) per pair of vertices with O( k 4 log 3 n)-time preprocessing in such a way that for almost all k-colorable graphs the answer is correct for all pairs of vertices. From this we obtain a sublinear (with respect to the number of edges) expected time algorithm for k-coloring of k-colorable graphs (assuming the uniform input distribution). Similarly, if c ⩽ ( 1 8 − ε)n 2, ε > 0 is a constant, and G is a graph having a cut of the vertex set into two equal parts with at most c cross-edges, we can test whether two vertices belong to the same class of some c-cut in time O(log n) per vertex with O(log 3 n)-time preprocessing in such a way that for almost all graphs having a c-cut the answer is correct for all pairs of vertices. The methods presented in the paper can also be used for other graph partitioning problems, e.g. the largest clique or independent subset.

Computational complexity of sequential sequence estimation for intersymbol interference channels

The computational complexity of a sequential algorithm (SA) developed for intersymbol interference (ISI) channels is analyzed. To determine the computational complexity, the finite-state machine that models the channel and white matched filter system, of which the SA is a part, is interpreted as a special convolutional encoder followed by a binary symbol to Q-ary symbol mapping. It follows that the computational distribution is Pareto, and that there exists a computational cutoff rate R/sub comp/. For the uncoded data considered, the rate is fixed and the R/sub comp/ criterion translates into a signal-to-noise ratio (SNR) criterion. An upper bound on SNR/sub comp/ is found analytically by assuming a uniform input distribution. Iteration equations developed by S. Arimoto (1976) are adapted to find the true SNR/sub comp/ numerically. >