Peter D. Welch

The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms

The use of the fast Fourier transform in power spectrum analysis is described. Principal advantages of this method are a reduction in the number of computations and in required core storage, and convenient application in nonstationarity tests. The method involves sectioning the record and averaging modified periodograms of the sections.

Simulation Run Length Control in the Presence of an Initial Transient

This paper studies the estimation of the steady state mean of an output sequence from a discrete event simulation. It considers the problem of the automatic generation of a confidence interval of prespecified width when there is an initial transient present. It explores a procedure based on Schrubens Brownian bridge model for the detection of nonstationarity and a spectral method for estimating the variance of the sample mean. The procedure is evaluated empirically for a variety of output sequences. The performance measures considered are bias, confidence interval coverage, mean confidence interval width, mean run length, and mean amount of deleted data. If the output sequence contains a strong transient, then inclusion of a test for stationarity in the run length control procedure results in point estimates with lower bias, narrower confidence intervals, and shorter run lengths than when no check for stationarity is performed. If the output sequence contains no initial transient, then the performance measures of the procedure with a stationarity test are only slightly degraded from those of the procedure without such a test. If the run length is short relative to the extent of the initial transient, the stationarity tests may not be powerful enough to detect the transient, resulting in a procedure with unreliable point and interval estimates.

A spectral method for confidence interval generation and run length control in simulations

This paper discusses a method for placing confidence limits on the steady state mean of an output sequence generated by a discrete event simulation. An estimate of the variance is obtained by estimating the spectral density at zero frequency. This estimation is accomplished through a regression analysis of the logarithm of the averaged periodogram. By batching the output sequence the storage and computational requirements of the method remain low. A run length control procedure is developed that uses the relative width of the generated confidence interval as a stopping criterion. Experimental results for several queueing models of an interactive computer system are reported.

The Fast Fourier Transform and Its Applications

The advent of the fast Fourier transform method has greatly extended our ability to implement Fourier methods on digital computers. A description of the alogorithm and its programming is given here and followed by a theorem relating its operands, the finite sample sequences, to the continuous functions they often are intended to approximate. An analysis of the error due to discrete sampling over finite ranges is given in terms of aliasing. Procedures for computing Fourier integrals, convolutions and lagged products are outlined.

The fast Fourier transform algorithm: Programming considerations in the calculation of sine, cosine and Laplace transforms☆

Abstract In the organization of programming packages for computing Fourier and Laplace transforms, it is useful, both for conceptual understanding and for operational efficiency to consider the discrete complex Fourier transform as a kind of nucleus around which programming for special applications is performed. An advantage of these procedures is that the basic complex Fourier transform algorithm is systematic and can relatively easily be implemented in efficient subroutines, micro-programs and special hardware devices. Once this is done, programming for special properties of the data can efficiently be left to the user to implement on a general purpose computer. The problem of establishing the correspondence between the discrete transforms and the continuous functions with which one is usually dealing is described. The application of these results and the above-mentioned subroutines to the calculation and inversion of Laplace transforms is given with formulas and empirical results displaying the effect of optimal parameters on computational efficiency and accuracy.

Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals

The fast Fourier transform is a computational procedure for calculating the finite Fourier transform of a time series. In this paper, the properties of the finite Fourier transform are related to commonly used integral transforms including the Fourier transform and convolution integrals. The relationship between the finite Fourier transform and Fourier series is also discussed.

A fixed-point fast Fourier transform error analysis

This paper contains an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simulated fixed-point machine and their comparison with the error upper bound.

The finite Fourier transform

The finite Fourier transform of a finite sequence is defined and its elementary properties are developed. The convolution and term-by-term product operations are defined and their equivalent operations in transform space are given. A discussion of the transforms of stretched and sampled functions leads to a sampling theorem for finite sequences. Finally, these results are used to give a simple derivation of the fast Fourier transform algorithm.

Historical notes on the fast Fourier transform

The fast Fourier transform algorithm has a long and interesting history that has only recently been appreciated. In this paper, the contributions of many investigators are described and placed in historical perspective.

Statistical Results on Control Variables with Application to Queueing Network Simulation

The development and application of control variables for variance reduction in the simulation of a wide class of closed queueing networks is discussed. These networks allow multiple types of customers, priorities and blocking. Alternative methods of generating confidence intervals from independent replications of a simulation are investigated. A result is given which quantifies the loss in variance reduction caused by the estimation of the optimum control coefficients. This loss is an increasing function of the number of control variables. Good variance reduction is obtained providing that the number of control variables remains small.