Accurate simple closed-form approximations to Rayleigh sum distributions and densities

Sums of Rayleigh random variables occur extensively in wireless communications. A closed-form expression does not exist for the sum distribution and consequently, it is often evaluated numerically or approximated. A widely used small argument approximation for the density is shown to be inaccurate for medium and large values of the argument. Highly accurate, simple closed-form expressions for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values and number of summands.     

Accurate closed-form approximations to Ricean sum distributions and densities

The statistical distribution of a sum of Ricean random variables occurs extensively in wireless communications. A closed-form expression does not exist for the sum distribution and, furthermore, the Ricean random variable does not have a closed-form characteristic function. For these reasons, it is somewhat difficult to numerically calculate the sum distribution. Highly accurate, closed-form approximations for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values, Rice factors and number of summands.     

An optimal lognormal approximation to lognormal sum distributions

Sums of lognormal random variables occur in many problems in wireless communications because signal shadowing is well modeled by the lognormal distribution. The lognormal sum distribution is not known in the closed form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a lognormal sum is well approximated by a lognormal random variable. Here, a new paradigm for approximating lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal lognormal approximation to a lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the optimal lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the lognormal characteristic function is also considered.     

Simple accurate lognormal approximation to lognormal sums

A simple accurate lognormal approximation to the sum of independent non-identical lognormal variates is derived by matching the first two moments of the inverse exact sum with those of the inverse lognormal approximation. Sample examples are given to illustrate the excellent agreement between exact and approximate sum statistics.     

Predictive densities for the lognormal distribution and their applications

Maximum likelihood predictive densities (MLPDs) for a future lognormal observation are obtained and their applications to reliability and life testing are considered. When applied to reliability and failure rate estimations, they give estimators that can be much less biased and less variable than the usual maximum likelihood estimations (MLEs) obtained by replacing the unknown parameters in the density function by their MLEs. When applied to lifetime predictions, they give prediction intervals that are shorter than the usual frequentist intervals. Using the MLPDs, it is also rather convenient to construct the shortest prediction intervals. Extensive simulations are performed for comparisons. A numerical example is given for illustration.     

Distinguishing between lognormal and Weibull distributions[time-to-failure data]

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When "standardized", the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation     

Wideband analyser for measurement of probability densities and distributions

A fast window comparator transforms an input random voltage to a random series of pulses with equal amplitude. For stationary processes, the d.c. component of the pulse voltage is proportional to the probability density of the input voltage. Inaccuracies due to finite switching times are considered, together with errors dependent on the nonzero window width.     

A simple, accurate method to calculate spread-spectrummultiple-access error probabilities

The exact calculation of error probabilities for direct-sequence spread-spectrum multiple-access (DS/SSMA) systems has been addressed in the literature. The exact calculation is computationally difficult, so emphasis has been on approximations and bounds. One particularly attractive approximation is to just use a signal-to-noise ratio in a Gaussian approximation, the `standard approximation'. Unfortunately, that approximation is not generally accurate enough. An improved Gaussian approximation with good accuracy has recently been presented. The author derives an accurate Gaussian approximation which is also computationally very simple     

New equivalent networks with simple closed-form expressions foropen and slit-coupled E-plane tee junctions

Analytical expressions are presented for the elements of new equivalent networks for open and slit-coupled E-plane tee junctions. The expressions are in simple closed form, and their accuracy is verified by comparisons with measurements. Physically based stored power considerations are also developed in context to indicate how appropriate changes in the geometry can be taken into account     

A simple and accurate high-pass filter and spectrometer using Huygen's principle

We have developed a new type of high pass filter and a new type of spectrometer. These devices are currently under testing and have yielded good agreement with theory. These devices seem to be easier to use and appear to be more accurate than devices now used.     

Power estimation in adiabatic circuits: a simple and accurate model

A simple procedure to evaluate the energy consumption of adiabatic gate circuits is proposed and validated. The proposed strategy is based on a linearization of the circuit and simplifying the analytical result obtained on the equivalent network. The approach leads to simple relationships which can be used for a pencil-and-paper evaluation or implemented on software. The accuracy of the results is validated by means of Spice simulations on an adiabatic full adder designed with a 0.8 μm technology     

A simple and accurate method for extracting substrate resistance ofRF MOSFETs

In this paper, a simple and accurate method was proposed for extracting substrate resistance of an RF MOSFET, the substrate of which is represented by a single resistor. The extraction results from the measured network parameters are presented for various bias conditions. Excellent agreement between the results of measurements and the model for the extracted substrate resistance was obtained up to 18 GHz. Also, the resistance extracted using the proposed method was shown to give scalable results     

Fitted Elmore delay: a simple and accurate interconnect delay model

In this brief, we present a new interconnect delay model called fitted Elmore delay (FED). FED is generated by approximating HSPICE delay data using a curve fitting technique. The functional form used in curve fitting is derived based on the Elmore delay (ED) model. Thus, our model has all the advantages of the ED model. It has a closed-form expression as simple as the ED model and is extremely efficient to compute. Interconnect optimization with respect to design parameters can also be done as easily as in the ED model. In fact, most previous algorithms and programs based on ED model can use our model without much change. Most importantly, FED is significantly more accurate than the ED model. The maximum error in delay estimation is at most 2% for our model, compared to 8.5% for the scaled ED model. The average error is less than 0.8%. We also show that FED can be more than 10 times more accurate than the ED model when applied to wire sizing.     

Analysis of tunable single-mode fiber directional couplers using simple and accurate relations

We present simple and accurate relations for the study of parallel and curved tunable single-mode fiber directional couplers employing step-index or graded-index fibers. Analytical results for the various coupler parameters such as coupling coefficient, effective interaction length, tunability, and channel wavelength separation in directional coupler wavelength filters are presented. These results must prove very useful for an easy, quick, and accurate prediction or design of tunable single-mode fiber directional couplers.     

A simple and accurate model for microstrip structures with slotted ground plane

A simple and accurate circuit model for microstrip filtering structures with slotted ground plane is presented. The unit cell model consists of a series inductance and shunt capacitance for the microstrip line, an ideal transmission line characterized by its impedance and electrical length for the slot, and an ideal transformer to model coupling to the slot. The coupling mechanism to the slots resonances and subsequent emergence of transmission gaps is explained for different positions of the microstrip line. Excellent agreement with measurement is demonstrated over the broad bandwidth of dc to 20 GHz.     

A simple and accurate model for microstrip structures with slotted ground plane

A simple and accurate circuit model for microstrip filtering structures with slotted ground plane is presented. The unit cell model consists of a series inductance and shunt capacitance for the microstrip line, an ideal transmission line characterized by its impedance and electrical length for the slot, and an ideal transformer to model coupling to the slot. The coupling mechanism to the slots resonances and subsequent emergence of transmission gaps is explained for different positions of the microstrip line. Excellent agreement with measurement is demonstrated over the broad bandwidth of dc to 20 GHz.     

SiO2 gate insulator defects,spatial distributions,densities, types,and sizes

Standard IC processes, as well as those involving the use of ionizing radiation, such as x-ray lithography etc., result in the generation of bulk defects, and interface states in the gate insulator, or underlying substrate, respectively, of insulated gate field effect transistors. Bulk defects are believed to be present as positively and negatively charged electron and hole traps, respectively, as well as neutral hole and “large” and “small” neutral electron traps. This paper provides a perspective of the current state of knowledge about the spatial distributions of large bulk defects, their areal densities, sizes, possible interrelationships among them, and the special cases of defects created by ion implanted silicon and oxygen, where knock-on effects have been simulated. It appears that bulk defects may all have their origin in neutral hole traps, (so-called E′ centers) and that when the insulator thickness is decreased to about 6-7 nm, defects are either no longer present, or, more likely, are incapable of trapping charge at room temperature because trapped carriers can either tunnel to one of the interfaces, or be annihilated by a reverse process. It appears possible also that the precursor of the several types of defects only forms at a “grown” silicon-silicon oxide interface. In theory, this would make it possible to grow defect free insulators by a combination of deposition and oxidation processes.     

Smooth, efficient real amplitude distributions with no edgebrightening for linear and circular near-Taylor sum patterns

The authors report the application of the simulated annealing optimisation technique to the synthesis of radiation patterns approximating linear and circular Taylor sum patterns by means of real aperture distributions with amplitudes that vary smoothly and show no edge brightening