# Fourier and Wavelet Processing Concepts

Fourier filtering and Fourier operations are fundamental to signal processing in geophysics.

You may have used Fourier filtering before, but do you really understand what is going on and what the limitations are?  Waveforms change after filtering and unless you understand what is happening, your interpretation may be incorrect.

Most courses, particularly University courses, use mathematics to explain Fourier processes.

No Mathematics in this course!!!

Well maybe just a little.  However, in this course, mathematics will be virtually non-existent.  The whole Fourier and wavelet processes will be explained by simple to understand diagrams and in an Excel spreadsheet which participants can take away with them.  Mathematics will only be displayed after the concepts have been explained so the formula can be explained.

Processes that are explained by simple examples in the spreadsheet include:

• Necessary Fourier terms such as amplitude, phase and decibels (dB) and aliasing.
• The Fourier Theorem which is explained through construction of various waveforms (triangle, square, etc) by summing sin and cos waves of various amplitude and phase (see above).
• Fourier Filtering using manipulation of various sin and cos waveforms.
• Phase and what happens when you introduce phase into filters.
• The difference between Full Fourier Spectra, Power Spectra and short term Fourier analysis.
• Convolution and deconvolution, their application and uses.

• The necessary Wavelet terms such as amplitude and scale. Discrete vs Continuous wavelet analysis.
• Wavelet filters and differences between wavelet and Fourier filtering including short term Fourier.
• Using Wavelets to find discontinuities.
• Using Wavelets to remove noise.

Participants will be shown how to calculate the Fourier spectrum in Excel and how to scale this appropriately as well as calculate the Power Spectrum.  Building on the earlier simple examples, this will give an understanding of real world profiles and their spectra as well as some of the limitations of FFTs.

Using the Excel FFT function, participants will learn about

• Padding – why we need it and how it works.
• Real world filters such as Butterworth etc and examples of High Pass, Low Pass, Band Pass filters.
• Why real-world filters need a ramp, e.g. Hamming (or similar), rather than a sharp cut off.
• Calculating the Power Spectral Density (PSD) and making noise estimates from the PSD
• How wavelet filters are applied in the real world.

Visual Examples will continue to also show the following in simple non-mathematical language:

• 2D Fourier processes.
• 2D Fourier filtering including directional filters and how they work.
• Different types of wavelets and their uses.

Participants will be encouraged to test what they have learned on their own real or synthetic data in the spreadsheet.  So please bring some small sets (1024 or 2048 points) of real 1D profiles.