Geology 6600/7600 Signal Analysis 05 Oct 2015 © A.R. Lowry 2015 Last time: Assignment for Oct 23: GPS time series correlation Given a discrete function h defined on  –  sampled at intervals , the discrete time Fourier transform is defined on [– ,  ] (and otherwise periodic). Frequencies ≥ Nyquist frequency f N = 1/2  are aliased to lower f ! => amplitudes at frequencies    are mapped onto    – . Generally, continuous  aperiodic; discrete  periodic. In practice, we usually apply the FFT to discrete, finite data…

We can substitute for  =  (where  is the sample interval): Analogous to continuous random variables, correlation & power spectra are:

A discrete white noise stationary process: where  [l] is the discrete Kronecker delta function, And power spectrum: More generally, for stationary, zero-mean processes: Leopold Kronecker “God made the integers; all else is the work of man.”

Spectral Analysis: Spectral Analysis is generally performed by one of two approaches: Nonparametric (or “classical”) methods use the Fourier transform and (perhaps) one or more tapers applied to the data (examples include periodogram, tapered periodogram, multitaper power spectra) Parametric methods assume some model to describe the behavior or statistics of the data, and parameterize that model (e.g. maximum entropy method, autoregressive power spectra). In this course we will focus primarily on nonparametric spectral estimation. For more on the topic see Modern Spectral Estimation (Kay) or Kay & Marple (1981) (posted link)

Statistical Properties of the FT: Recall: The Fourier Transform of a random process is: The mean of the FT is: which, assuming f is wide-sense stationary, is (i.e. this has meaning only for  = 0 ). ~

The autocorrelation of the Fourier transform is: (can’t assume F is WSS just because f is!) Given f wide-sense stationary, So, F(  ) is generally nonstationary because it depends on frequency (u) ~ ~~

We’ve already noted that the autopower spectrum is the Fourier Transform of the autocorrelation function, & the autocorrelation function can be estimated (given ergodicity) from the signal via: Given an infinite record length, (provided R(  )  0 as   ). However given a finite record length, have limited overlap of x(t +  ) with x(t). This does not bias the estimate of R, but fewer realizations means increasing variance as   increases…

For example: 33-sample record R T (l) for l =  has N = 32 realizations here… R T (l) for l = 30 has just N = 3 realizations.