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 bounds for the Marcum Q-function

New bounds are proposed for the Marcum Q-function, which is defined by an integral expression where the 0th-order modified Bessel function appears. The proposed bounds are derived by suitable approximations of the 0th-order modified Bessel function in the integration region of the Marcum Q-function. They prove to be very tight and outperform bounds previously proposed in the literature. In particular, the proposed bounds are noticeably good for large values of the parameters of the Marcum Q-function, where previously introduced bounds fail and where exact computation of the function becomes critical due to numerical problems     

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     

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     

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     

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     

Well-structured FHMA codebooks: a geometric approach

A geometric approach in the design of codebooks for OR frequency-hopping multiple-access (FHMA) channels is developed by treating a signal matrix as a finite set of distinct points. The relationship among the parameters of an interference-free j-distinguishable-point codebook, j⩾1, is established by using coordinate-free arguments. Geometry induced by such a codebook is characterized, and the design of a well-structured j -distinguishable-point codebook is related to a block design problem. It is shown that a well-structured 1-distinguishable-point codebook implies the axiom system of a finite affine plane     

Cortex segmentation: a fast variational geometric approach

An automatic cortical gray matter segmentation from a three-dimensional (3-D) brain images [magnetic resonance (MR) or computed tomography] is a well known problem in medical image processing. In this paper, we first formulate it as a geometric variational problem for propagation of two coupled bounding surfaces. An efficient numerical scheme is then used to implement the geodesic active surface model. Experimental results of cortex segmentation on real 3-D MR data are provided.     

Modeling of multileg sine-wave inverters: a geometric approach

Three fundamental sine-wave inverter topologies are analyzed: two-leg (one-phase, two-wire); three-leg (three-phase, three-wire); and four-leg (three-phase, four-wire). The topologies are “full-bridge” voltage-source inverters with LC filters suitable for producing sinusoidal output voltages. The switching states and corresponding output voltage vectors produced by each inverter are identified and presented along with an analysis of the geometric arrangement of these voltage vectors. A pattern of characteristics is established whereby the “qd” modeling forms commonly used with three-leg inverters are extended to address the expanded capabilities of the four-leg inverter. A unique 4×4 decoupling transformation matrix is presented for the four-leg inverter that enables direct transformation between the four-degree-of-freedom (DOF) leg-modulation space of the inverter and its corresponding 3-DOF output-voltage space. This is shown to be directly analogous to the well-known “abc-qd” transformation developed for the three-leg inverter. Fully decoupled models for each inverter are presented     

Fast mimo transmit antenna selection algorithms: a geometric approach

Motivated by matrix determinant properties, this letter develops a fast transmit antenna selection algorithm for MIMO systems: the G-circles method. This novel scheme is shown to achieve many advantages over other existing algorithms     

Communication on the Grassmann manifold: a geometric approach tothe noncoherent multiple-antenna channel

We study the capacity of multiple-antenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald (see ibid. vol.45, p.139-57, 1999). The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length T symbol periods. We compute the asymptotic capacity of this channel at high signal-to-noise ratio (SNR) in terms of the coherence time T, the number of transmit antennas M, and the number of receive antennas N. While the capacity gain of the coherent multiple antenna channel is min{M, N} bits per second per Hertz for every 3-dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be M* (1 - M*/T) bits per second per Hertz, where M*=min{M, N, [T/2]}. The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold     

Determination of the minimum sampling frequency in bandpass sampling by geometric approach and geometric programming

This paper presents a simple and fast approach to find a minimum sampling frequency for multi-band signals. Instead of neighbor and boundary conditions, constraints on the sampling frequency were derived by using the geometric approach to the bandpass sampling theorem. Reformulation of the constraints on the minimum sampling frequency enabled to represent the problem as an optimization problem which was structured by the geometric programming and mixed-integer nonlinear programming methods. The convex optimization problem was then solved by the proposed algorithm applying interior point approach in the line search framework. It was demonstrated that this unified structure directly linked the geometric approach of the bandpass sampling theorem to the optimization problem. The proposed method was verified through numerical simulations in terms of the minimum sampling frequency and the computational efficiency. Results illustrated the feasibility of the geometric approach and the proposed algorithm in the determination of the minimum sampling frequency by providing the savings in the number of iterations and the decrease in the valid minimum sampling frequency.     

Computing the unmeasured: an algebraic approach to Internet mapping

Distance estimation is important to many Internet applications. It can aid a World Wide Web client when selecting among several potential candidate servers or among candidate peer-to-peer servers. It can also aid in building efficient overlay or peer-to-peer networks that react dynamically to changes in the underlying Internet. One of the approaches to distance (i.e., time delay) estimation in the Internet is based on placing tracer stations in key locations and conducting measurements between them. The tracers construct an approximated map of the Internet after processing the information obtained from these measurements. This work presents a novel algorithm, based on algebraic tools, that computes additional distances, which are not explicitly measured. As such, the algorithm extracts more information from the same amount of measurement data. Our algorithm has several practical impacts. First, it can reduce the number of tracers and measurements without sacrificing information. Second, our algorithm is able to compute distance estimates between locations where tracers cannot be placed. To evaluate the algorithm's performance, we tested it both on randomly generated topologies and on real Internet measurements. Our results show that the algorithm computes up to 50%-200% additional distances beyond the basic tracer-to-tracer measurements.     

A Simple Polynomial Approximation to the Gaussian Q-function and Its Application

A simple polynomial approximation to the Gaussian Q-function is proposed, based on the observation that a Gaussian random variable can be well approximated by a sum of uniform random variables. The approximation can be used to obtain accurate explicit approximations to problems that otherwise do not have explicit solutions or approximate explicit solutions. As an example, an explicit expression for the average symbol error rate of M-ary pulse amplitude modulation in lognormal channels is derived using the new approximation, and the approximate symbol error rate is shown to be very close to the exact value.     

Statistical inference under multiterminal rate restrictions: adifferential geometric approach

A statistical inference problem for a two-terminal information source emitting mutually correlated signals X and Y is treated. Let sequences Xⁿ and Yⁿ of n independent observations be encoded independently of each other into message sets M_X and M_Y at rates R₁ and R ₂ per letter, respectively. This compression causes a loss of the statistical information available for testing hypotheses concerning X and Y. The loss of statistical information is evaluated as a function of the amounts R₁ and R ₂ of the Shannon information. A complete solution is given in the case of asymptotically complete data compression, R₁, R₂→0 as n→∞. It is shown that the differential geometry of the manifold of all probability distributions plays a fundamental role in this type of multiterminal problem connecting Shannon information and statistical information. A brief introduction to the dually coupled e-affine and m-affine connections together with e -flatness and m-flatness is given     

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     

Scattering from composite materials: a first-order model

The nonspecular electromagnetic scattering from finite composite laminates is investigated. The composite material is modeled as planar lamina composed of undirectional collimated fibers with regular spacings between the elements. The fibers, perfectly or partially conductive, are assumed to be embedded in a resin matrix translucent to electromagnetic radiation. For the partially conductive case, the Schelkunoff Ansatz is used. The case of two-plied laminates with skewed fiber orientation is discussed. The mathematical formulation is based on the electric-field integral equation solved with an entire-domain Galerkin expansion. Results are obtained for laminates with both finite and infinite numbers of elements. For the latter, the Floquet-Galerkin solution for periodic structures is used. The effect of truncation of the panels is discussed for arbitrary angles of illumination. It is shown that in many cases implementation of the Floquet-Galerkin solution using diagonal system matrices yields accurate results for the nonspecular cross sections of the laminates. The theoretical results are confirmed by experimental data     

Note on `The calculation of the probability of detection and thegeneralized Marcum Q-function'

The author presents corrections to his original paper (see ibid., vol.35, no.2, p.389-400, 1989). The corrections concern computational cases using the steepest descent integration technique. It is pointed out that, for certain specific parameter ranges, the calculation error is too large to be accounted for by accumulated round-off error     

Relations between the Rice Ie-function and the Marcum Q-functionwith applications 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