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Journal of Time Series Econometrics
Volume 1, Issue 1 2009 Article 1

Statistical Fourier Analysis: Clarifications and Interpretations
Stephen D.S.G. Pollock, University Leicester

Recommended Citation: Stephen D.S.G. Pollock (2009) "Statistical Fourier Analysis: Clarifications and Interpretations," Journal of Time Series Econometrics: Vol. 1 : Iss. 1, Article 1. Available at:http://www.bepress.com/jtse/vol1/iss1/art1 DOI: 10.2202/1941-1928.1004 ©2009 Berkeley Electronic Press. All rights reserved.

Statistical Fourier Analysis: Clarifications and Interpretations
Stephen D.S.G. Pollock

Abstract
This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose thestructure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domainand in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti-aliasing filtering and sub sampling. These techniques are alsodescribed. It is argued that band-limited processes are more prevalent in statistical and econometric time series than is commonly recognised. KEYWORDS: Fourier analysis, band-limited processes, ideal filters, circulant matrices

Pollock: Statistical Fourier Analysis

1. Introduction
Statistical Fourier analysis is an important part of modern time-series analysis, yet it frequently poses animpediment that prevents a full understanding of temporal stochastic processes and of the manipulations to which their data are amenable. This paper provides a survey of the theory that is not overburdened by inessential complications, and it addresses some enduring misapprehensions. Amongst these is a misunderstanding of the effects in the frequency domain of linear filtering operations. It is commonlymaintained, for example, that an ideal frequency-selective filter that preserves some of the elements of a time series whilst nullifying all others is incapable of being realised in finite samples. This paper shows that such finite-sample filters are readily available and that they possess time-domain and frequency-domain representations that are both tractable. Their representations are directly relatedto the classical Wiener–Kolmogorov theory of filtering, which presupposes that the data can be treated as if they form a doubly-infinite or a semi-infinite sequence. A related issue, which the paper aims to clarify, concerns the nature of continuous stochastic processes that are band-limited in frequency. The paper provides a model of such processes that challenges the common supposition that thenatural primum mobile of any continuous-time stochastic process is a Wiener process, which consists of a steam of infinitesimal impulses. The sampling of a Wiener process is inevitably accompanied by an aliasing effect, whereby elements of high frequencies are confounded with elements of frequencies that fall within the range that is observable in sampling. Here, it is shown, to the contrary, thatthere exist simple models of continuous bandlimited processes for which no aliasing need occur in sampling. The spectral theory of time series is a case of a non-canonical Fourier theory. It has some peculiarities that originally caused considerable analytic difficulties, which were first overcome in a satisfactory manner by Norbert Wiener (1930) via his theory of a generalised harmonic analysis. The...
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