Integral representation and bounds for Marcum Q-function

A new expression for the Marcum Q-function involving an integral over a fixed interval is given. Tight upper and lower bounds are then derived and applied to the performance evaluation of noncoherent and differentially coherent detection of digital modulation over Nakagami fading channels     

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.     

Exponential-type bounds on the generalized Marcum Q-function withapplication to error probability analysis over fading channels

Strict upper and lower bounds of exponential-type are derived for the generalized (mth order) Marcum Q-function which enable simple evaluation of a tight upper bound on the average bit-error probability performance of a wide class of noncoherent and differentially coherent communication systems operating over generalized fading channels. For the case of frequency selective fading with arbitrary statistics per independent fading path, the resulting upper hound on performance is expressed in the form of a product of moment generating functions of the instantaneous power random variables that characterize these paths     

A new twist on the Marcum Q-function and its application

A new form of the Marcum (1950) Q-function is presented that has both computational and analytical advantages. The new form is particularly useful in simplifying and rendering more accurate the analysis of the error probability performance of uncoded and coded partially coherent, differentially coherent, and noncoherent communication systems in the presence of fading. It also enables simple upper and lower bounds to be found analogous to the Chernoff bound on the Gaussian Q-function     

Computing and bounding the first-order Marcum Q-function: a geometric approach

A geometric interpretation of the first-order Marcum Q-function, Q(a,b), is introduced as the probability that a complex, Gaussian random variable with real mean a, takes on values outside of a disk C_O,b of radius b centered at the origin O. This interpretation engenders a fruitful approach for deriving new representations and tight, upper and lower bounds on Q(a,b). The new representations obtained involve finite-range integrals with pure exponential integrands. They are shown to be simpler and more robust than their counterparts in the literature. The new bounds obtained include the generic exponential bounds which involve an arbitrarily large number of exponential functions, and the simple erfc bounds which involve just a few erfc functions, together with exponential functions in some cases. The new generic exponential bounds approach the exact value of Q(a,b) as the number of exponential terms involved increases. These generic exponential bounds evaluated with only two terms and the new simple erfc bounds are much tighter than the existing exponential bounds in most cases, especially when the arguments a and b are large. Thus, in many applications requiring further analytical manipulations of Q(a,b), these new bounds can lead to some closed-form results which are better than the results available so far.     

Relations between Rice Ie-function and Marcum Q-function withapplications to error rate calculations

Useful relations connecting the Rice Ie-function and the Marcum Q-function are presented. A function P(U,V), closely related to the Ie-function, is applied to bit and symbol error probability calculations and shown to lead to expressions that are simpler and more elegant than corresponding results written in terms of the Q-function     

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.     

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     

New performance bounds for turbo codes

We derive a new upper bound on the word- and bit-error probabilities of turbo codes with maximum-likelihood decoding by using the Gallager bound. Since the derivation of the bound for a given interleaver is intractable, we assume uniform interleaving as in the derivation of the standard union bound for turbo codes. The result is a generalization of the transfer function bound and remains useful for a wider range of signal-to-noise ratios, particularly for some range below the channel cutoff rate. The new bound is also applicable to other linear codes     

New lower bounds for covering codes

Some new lower bounds on |C| for a binary linear [n, k]R code C with n+1=t(R +1)-r(0⩽r<R+1, t>2 odd) or with n+1=t(R+1)-1(t>2 even) are obtained. These bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t[n, k], the smallest covering radius of any [n, k] code     

New convergence bounds for Bayes estimators

Using some recently derived spectral expressions for distance measures between vector Gaussian stationary processes, upper bounds are given for the mean-square-error performance of the Bayes estimate of a parameter on the basis of vector Gaussian observations. The result is directly applicable to evaluating the performance of certain adaptive estimation schemes and to finite-state Markov chain systems.     

New lower bounds for constant weight codes

Some new lower bounds are given for A(n,4,w ), the maximum number of codewords in a binary code of length n , minimum distance 4, and constant weight w. In a number of cases the results significantly improve on the best bounds previously known     

High-order exponential approximations for the Gaussian Q-function obtained by genetic algorithm

This article presents new and accurate approximations for the Gaussian Q-function, which is an important tool in the field of communication theory. A genetic algorithm was employed to find the optimum coefficients of the proposed approximations. The accuracy of the new approximations was examined and compared with those of previous works in the literature. It was shown that the presented high-order exponential approximations provide better accuracy compared to the previously introduced approximations, in general, at the cost of slightly increased mathematical complexity.     

A simpler form of the Craig representation for the two-dimensionaljoint Gaussian Q-function

We derive a simpler form for the Craig (1991) representation of the two-dimensional joint Gaussian Q-function which dispenses with the trigonometric factor that precedes the exponentials in the integrands and furthermore results in an exponential argument that is precisely in the same simple form as that in the Craig representation of the one-dimensional Gaussian Q-function. As such, the entire dependence on the correlation parameter now appears only in the limits of integration. The resulting single integral form is particularly useful in evaluating the outage probability for dual diversity selection combining over correlated identically and nonidentically distributed log normal channels     

New asymptotic bounds for self-dual codes and lattices

We give an independent proof of the Krasikov-Litsyn bound d/n/spl lsim/(1-5/sup -1/4/)/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(/spl radic/n).     

New lower bounds for asymmetric and unidirectional codes

New lower bounds on the sizes of asymmetric codes and unidirectional codes are presented. Various methods are used, three of them of special interest. The first is a partitioning method that is a modification of a method used to construct constant weight codes. The second is a combining codes method that is used to obtain a new code from a few others. The third method is shortening by weights that is applied on symmetric codes or on codes generated by the combining codes method. Tables for the sizes of codes of length n⩽23 are presented     

New minimum distance bounds for certain binary linear codes

Let an [n, k, d]-code denote a binary linear code of length n, dimension k, and minimum distance at least d. Define d(n, k) as the maximum value of d for which there exists a binary linear [n, k, d]-code. T. Verhoeff (1989) has provided an updated table of bounds on d(n, k) for 1⩽k⩽n⩽127. The authors improve on some of the upper bounds given in that table by proving the nonexistence of codes with certain parameters     

New tight bounds on the pairwise error probability for unitary space-time modulations

We present two upper bounds and one lower bound on the pairwise error probability (PEP) of unitary space-time modulation (USTM) over the Rayleigh fading channel. The two new upper bounds are the tightest so far, and the new lower bound is the tightest at low signal-to-noise ratio. Some implications for USTM constellation design are also pointed out.     

New upper bounds on generalized weights

We derive new asymptotic upper bounds on the generalized weights of a binary linear code of a given size. We also prove some asymptotic results on the distance distribution of binary codes