A Painless (but Quantitative) Introduction to Wavelets, with Planetary Examples

Joe Harrington

Just as Fourier analysis did two centuries ago, wavelet analysis is opening up entirely new ways to approach data. Fourier analysis gives us the power spectrum of a signal. The signal is assumed to be static and its sampling is assumed to be fundamentally periodic, like a set of temperature sensors around a planet's equator. All variation in the signal is assumed to be due to the static spectral content. But real signals are rarely periodically sampled, every signal has a beginning and an end in time, and most interesting signals change in time. After all, we expect the power spectrum of a symphony to change as the orchestra plays different notes! Replacing hundreds of imperfect windowing schemes, wavelet analysis cleanly determines the time-dependent power spectrum. Not only can you see what notes the orchestra played, but you can determine which instruments played them, and when. Further, wavelet analysis gives a quantitative means for determining when a signal is aliased, and for analyzing errors. Using this information, one can intelligently average a 2D wavelet power spectrum in time and frequency, and produce a far better 1D power spectrum than can the Fourier transform. I present a straightforward introductory paper on wavelets by Torrence and Compo (1998), which includes software in FORTRAN, IDL, and Matlab, and sample datasets. This talk should make the paper easy to read, after which you should be able to start using wavelet analysis on your own data. The paper and software are available here: http://paos.colorado.edu/research/wavelets/ .