BNL # 49035 rev. 5/94 BASIC Program for Power Spectrum Estimation E. L. Church and P. Z. Takacs May 1993. Abstract A standard data set is presented to be used in testing routines that are developed for evaluating the one-dimensional periodogram spectral estimate of a series of uniformly spaced data points representing measurements of a surface profile. These data were generated using the GWBASIC program "FFTPSD" which is included in the Appendix of this report. The original purpose of this data set was to allow us to test and debug our own periodogram estimation routines, written in various programming languages, against a standard data set. At the request of numerous individuals, we are making it available to others who may wish to do the same. We have also included a summary explanation of periodogram estimators and the various considerations that go into generating computer algorithms for power spectrum estimation. Please address any comments or questions to: Nomenclature The Power Spectral Density, S(f x ), is the most important statistical property of the profiles of an optical surface. Statistical quantities, however, are defined in terms of averages taken over an infinite set of equivalent surfaces. In statistical terminology the set is an ensemble, the ensemble averaging is denoted by the symbol <...>, and a single member of the ensemble is called a realization. This presents us with a practical problem: in the real world we have only a finite number of profile data sets (realizations) -- perhaps only one --, the data refer to only a finite profile length, and consist of values taken at discrete points on the profile. How do we estimate the profile spectrum from one or more finite, sampled data sets? This is done by defining an algebraic function of the profile data which approximates the profile spectrum, and evaluating it for the particular data sets on hand. In statistical terms this procedure is called statistical estimation and the function is called an estimator. The spectral estimator and its values are denoted by ˆ S ( f x ) , where the hat distinguishes it from the true spectrum, S ( f x ) . E. L. Church email: echurch@ieee.org P.Z. Takacs Building 535B Brookhaven National Laboratory Upton, NY 11973-5000 631 282-2824 FAX 631 282-5773 email: takacs@bnl.gov