New exponential bounds and approximations for the computation of error probability in fading channels

We present new exponential bounds for the Gaussian Q function (one- and two-dimensional) and its inverse, and for M-ary phase-shift-keying (MPSK), M-ary differential phase-shift-keying (MDPSK) error probabilities over additive white Gaussian noise channels. More precisely, the new bounds are in the form of the sum of exponential functions that, in the limit, approach the exact value. Then, a quite accurate and simple approximate expression given by the sum of two exponential functions is reported. The results are applied to the general problem of evaluating the average error probability in fading channels. Some examples of applications are also presented for the computation of the pairwise error probability of space-time codes and the average error probability of MPSK and MDPSK in fading channels.     

Impurity-density distribution in the base region of drift transistors

The nature of the impurity grading in the base of drift transistors is studied by measuring the dependence of the base transit time on collector voltage. Provided that modulation of collector-depletion layer width caused by change of collector voltage occurs only in the base material, as in the case of an alloyed collector, it is possible to deduce the base-region field parameterm=Delta V/(kT/q)from such transit-time measurements. By this means, the validity of assumed distributions of impurity density may be verified; in particular, it may be established whether the impurity grading approximates an exponential or a complementary error function (erfc) form. Results are given for a number of drift-transistor samples, most of which are believed to have undergone impurity diffusion into the base material from a constant surface concentration during fabrication. In all cases, however, interpretation of measured data indicates a base impurity-density distribution approximating exponential rather than erfc form to be present.     

Construction of secure and fast hash functions using nonbinary error-correcting codes

This paper considers iterated hash functions. It proposes new constructions of fast and secure compression functions with nl-bit outputs for integers n>1 based on error-correcting codes and secure compression functions with l-bit outputs. This leads to simple and practical hash function constructions based on block ciphers such as the Data Encryption Standard (DES), where the key size is slightly smaller than the block size; IDEA, where the key size is twice the block size; Advanced Encryption Standard (AES), with a variable key size; and to MD4-like hash functions. Under reasonable assumptions about the underlying compression function and/or block cipher, it is proved that the new hash functions are collision resistant. More precisely, a lower bound is shown on the number of operations to find a collision as a function of the strength of the underlying compression function. Moreover, some new attacks are presented that essentially match the presented lower bounds. The constructions allow for a large degree of internal parallelism. The limits of this approach are studied in relation to bounds derived in coding theory.     

Approximations to complementary error function by method of least squares

A few simple approximations are derived for erfc (x) by method of least squares (MLS). The detailed error profiles are presented. It is shown how these approximations are useful in extracting the inverse of erfc(x). Finally a simple approximation with overall relative root mean square error (RRMS) of less than one percent is presented.     

A Useful Technique for Interference Analysis in Nakagami Fading

This paper presents a unified analytical method for computing the average E[g(x/(Sigma_k=1 ^K y_k + b))] of some arbitrary function g(.) when x is a gamma random variable and independent of the arbitrary random variables y₁, y₂, ..., y_k. This leads to new explicit expressions for the averages of some specific functions including, erfc(radicz), exp(-z), zⁿ, and In(1 + z), in terms of the moment generating functions of y₁, y₂, ..., y_K.     

Closed form and infinite series solutions for the MGF of a dual-diversity selection combiner output in bivariate Nakagami fading

Using a circular contour integral representation for the generalized Marcum-Q function, Q/sub m/(a,b), we derive a new closed-form formula for the moment generating function (MGF) of the output signal power of a dual-diversity selection combiner (SC) in bivariate (correlated) Nakagami-m fading with positive integer fading severity index. This result involves only elementary functions and holds for any value of the ratio a/b in Q/sub m/(a,b). As an aside, we show that previous integral representations for Q/sub m/(a,b) can be obtained from a contour integral and also derive a new, single finite-range integral representation for Q/sub m/(a,b). A new infinite series expression for the MGF with arbitrary m is also derived. These MGFs can be readily used to unify the evaluation of average error performance of the dual-branch SC for coherent, differentially coherent, and noncoherent communications systems.     

Efficient computation of erfc(x) for large arguments

A new, infinite series representation for the error function is developed. It is especially suitable for computing erfc(x) for large x. For instance, for any x⩾4, the error function can be evaluated with a relative error less than 10^-10 by using only eight terms. Similarly, the error function can be evaluated with a relative error less than 8×10^-7 for any x⩾2 using just six terms. An analytical bound is derived to show that the total error due to series truncation and undersampling rapidly decreases as x increases. Comparisons with two other series are provided     

Cauchy–Schwarz bound on the generalized Marcum Q‐function with applications

The Cauchy–Schwarz bounding technique is used to derive useful bounds on the generalized Marcum Q‐function and its complement. Three new exponential‐type bounds on Q_M (α, β) are derived, and these are found to be tight and useful for a number of applications of interest. One such example is the derivation of an upper bound on the average symbol error rate probability for noncoherent and differentially coherent communication systems over generalized fading channels. It is shown that these exponential‐type bounds are considerably tighter than the Chernoff bound (Rappaport SS. IEEE Trans. on Information Theory 1971; 17 : 497–498) counterpart. Numerical results also reveal that the tightness of one of the exponential‐type bounds is comparable to the bound obtained in Simon and Alouini (IEEE Trans. on Communications 2000: 359–366), while another is found to be superior than that obtained in Simon and Alouini over a wide range of arguments. Copyright © 2001 John Wiley & Sons, Ltd.     

Dual diversity over correlated log-normal fading channels

A great deal of attention has been devoted in the literature to studying the bit error rate (BER) performance of diversity combining systems in the presence of Rayleigh, Rice, and Nagakami-m fading. By comparison, the literature is relatively sparse in comparable analyses over log-normal channels which typically characterize shadowing from indoor obstacles and moving human bodies. One reason for this disparity stems from the difficulty in evaluating the exact average BER when log-normal variates are involved, using, for example, the moment-generating function (MGF) approach, due to the inability of expressing the MGF itself in a simple closed form. Since it is possible to evaluate the marginal and joint statistical moments as well as the cumulative distribution function (CDF) associated with a log-normal distribution in closed form, we rather focus here on other performance measures, namely, average combined output signal-to-noise ratio, amount of fading, and outage probability. The first two performance measures depend only on the moments, whereas the outage probability depends solely on the cdf. Closed-form expressions (in terms of known functions), single-integral representations, or upper and lower bounds are obtained for these measures corresponding to maximal-ratio combining, selection combining, and switch-and-stay combining schemes, allowing for the possibility of correlation between the two branches. Numerical evaluations of these expressions illustrating the performances of each individual diversity type as well as comparisons among them are also presented.     

Tight bounds for the first order Marcum Q‐function

In this paper, we develop new bounds for the first order Marcum Q‐function, which are extremely tight and tighter than any of the existing bounds to the best of our knowledge. The key idea of our approach is to derive refined approximations for the 0th order modified Bessel function in the integration region of the Marcum Q‐function. The new bounds are very tight and can serve as an effective means in bit error rate (BER) performance analysis for non‐coherent demodulation in digital communication. Copyright © 2010 John Wiley & Sons, Ltd.     

On the generalization ability of on-line learning algorithms

In this paper, it is shown how to extract a hypothesis with small risk from the ensemble of hypotheses generated by an arbitrary on-line learning algorithm run on an independent and identically distributed (i.i.d.) sample of data. Using a simple large deviation argument, we prove tight data-dependent bounds for the risk of this hypothesis in terms of an easily computable statistic M/sub n/ associated with the on-line performance of the ensemble. Via sharp pointwise bounds on M/sub n/, we then obtain risk tail bounds for kernel perceptron algorithms in terms of the spectrum of the empirical kernel matrix. These bounds reveal that the linear hypotheses found via our approach achieve optimal tradeoffs between hinge loss and margin size over the class of all linear functions, an issue that was left open by previous results. A distinctive feature of our approach is that the key tools for our analysis come from the model of prediction of individual sequences; i.e., a model making no probabilistic assumptions on the source generating the data. In fact, these tools turn out to be so powerful that we only need very elementary statistical facts to obtain our final risk bounds.     

Time-frequency representation of electrocorticograms in temporallobe epilepsy

Three time-frequency distributions are evaluated in terms of their efficacy in representing nonstationary electrocorticograms (ECoG's) in human temporal lobe epilepsy. The results of a new method, the exponential distribution, are compared with those of the spectrogram and the Wigner distribution. It is shown that the exponential distribution represents a considerable improvement over the spectrogram in terms of resolution and markedly reduces cross-terms present in the Wigner distribution. Exponential distribution representations of ECoG's from different stages of an epileptic record are developed as contour maps. These high-resolution representations offer a lucid display of temporal-spectral features of the rapidly varying signals that constitute ECoG's recorded in temporal lobe epilepsy.     

Design of time-frequency representations using a multiform,tiltable exponential kernel

A novel Cohen's (1981) class time-frequency representation with a tiltable, generalized exponential kernel capable of attaining a wide diversity of shapes in the ambiguity function plane is proposed for improving the time-frequency analysis of multicomponent signals. The first advantage of the proposed kernel is its ability to generate a wider variety of passband shapes, e.g., rotated ellipses, generalized hyperbolas, diamonds, rectangles, parallel strips at arbitrary angles, crosses, snowflakes, etc., and narrower transition regions than conventional Cohen's class kernels; this versatility enables the new kernel to suppress undesirable cross terms in a broader variety of time-frequency scenarios. The second advantage of the new kernel is that closed form design equations can now be easily derived to select kernel parameters that meet or exceed a given set of user specified passband and stopband design criteria in the ambiguity function plane. Thirdly, it is shown that simple constraints on the parameters of the new kernel can be used to guarantee many desirable properties of time-frequency representations. The well known Choi-Williams (1989) exponential kernel, the generalized exponential kernel, and Nuttall's (1990) tilted Gaussian kernel are special cases of the proposed kernel     

新度量指标及安全准则新定义

引入了密码安全的一些新度量指标，讨论了这些指标之间的关系。以此为基础得到布尔函数安全准则的新定义，给出非线性度新的上界，说明了这些上界与一般上界之间的关系。     

Moment space upper and lower error bounds for digital systems with intersymbol interference

A method for the evaluation of upper and lower bounds to the error probability of a linear pulse-amplitude modulation (PAM) system with bounded intersymbol interference and additive Gaussian noise is obtained via an isomorphism theorem from the theory of moment spaces. These upper and lower bounds are seen to be equivalent to upper and lower envelopes of some compact convex body generated from a set of kernel functions. Depending on the selection of these kernels and their corresponding moments, different classes of bounds are obtained. In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches. We study in detail the exponential moment case and obtain a family of new upper and a family of new lower bounds. Within each family, expressions for these bounds are given explicitly as a function of an arbitrary real-valued parameter. For two channels of interest, upper and lower bounds are evaluated and compared. Results indicate these bounds to be tight and useful.     

Closed‐form solutions to SEP integrals over generalized fading distributions η−μ and κ−μ using approximate computing

The fading channels often involve complex expressions, when it comes to computing the integrals required for performance evaluation of various digital modulation schemes. In this paper, usefulness of exponential‐based approximations to the Gaussian‐Q function in computing these integrals is discussed. We present generic symbol error probability (SEP) expressions over η?μ and κ?μ fading distributions, which can be tailored for any digital modulation technique using different approximations. The resulting expressions thus obtained comprise only elementary mathematical functions thereby avoiding complex evaluations of hypergeometric functions. We explore all the exponential‐based approximations proposed till date and conclude that apart from being mathematically simple, they also lead to accurate expressions for performance analysis of various digital modulation schemes.     

Tight geometric bound for Marcum Q-function

A new geometric bound is proposed for the Marcum Q (MQ)-function. The proposed bounds are derived by the geometric interpretation of the first-order MQ-function, Q(a,b). It has been shown by previous researchers that Q(a,b) can be viewed as the probability that a complex, Gaussian random variable Z with real, nonzero mean lies outside of a circular region. Based on the interpretation, the new multi-sectors bound proves to be very tight and outperforms all previously proposed bounds.     

New sequences for asynchronous frequency-hopping multiplex

A new method for generating large sets of sequences for frequency-hopping multiplex, when arbitrary or bounded mutual asynchronity is present, is presented. The sequences are of length Q, with elements from GF(Q); Q is prime. The maximal number of sequences in the sets is very close to the theoretical bounds.     

Jensen-cotes upper and lower bounds on the gaussian Q-function and related functions

We present new families of lower and upper bounds on Q-functions. First, we consider the Craig form of the Gaussian Q-function Q(ξ) and shown that its integrand ϕ(ϕ; ξ) can be partitioned into a pair of complementary convex and concave segments. This property is then exploited in conjunction with the Jensen inequality and the Newton-Cotes' quadrature rule to produce a complete family of upper and lower bounds on Q(ξ), which can be made arbitrarily tight by finer segmentation. The basic idea is then utilized to derive families of upper and lower bounds also for the squared Gaussian Q-function Q/²(ξ), the 2D joint Gaussian Q-function Q(x, y, p), and the generalized Marcum Q-function Q_M(x, y). The bounds are shown to be tighter than alternatives found in the literature, and in some cases the lower bounds provided find no equivalent in current literature. The generality of the principle is the elegant point of the method and the resulting Jensen-Cotes bounds are easy to implement and evaluate since only elementary transcendental functions are involved. As an example of application to the analysis of communication systems, we consider the bit error rates (BER's) of decode-and-forward (DF) cooperative relaying schemes with coherent and differential phase-shift keying (PSK) modulations, which have been shown to have an intricate dependence on the Gaussian Q-function, complicated by crossproducts, irrational functional arguments and multiple numerical integrations. In that example the bounds substantially reduce the complexity required to evaluate the expressions, retaining tightness despite multiple numerical integrations with infinite limits.i     

Formulas for the distribution of sums of independent exponential random variables

This paper derives a new type of formula for the probability that, among a collection of items with s-independent exponential times to failure, a certain subset of them fails in a given order before a certain time, and all the remaining items survive beyond that time. This formula is in the form of a power series that satisfies a certain constant coefficient linear differential equation with specified initial conditions. This provides an alternative to existing closed-form formulas of the "exponomial" variety, viz., a nonlinear combination of exponential terms, where the coefficients of the exponential terms are polynomials in the mission time. Some results are given which quantify the computation effort required to achieve a specified accuracy using partial sums of the infinite series; a simple example illustrates these results. This approach can be very efficient for system reliability analysis where the product of the mission time and the sum of the failure rates down any path leading to system failure is small. Further work is needed to expand the practical applicability of this approach to cases where some rates are large and/or the mission time is long.