__Trigonometric nomenclature for periodic waves__

- The basics. But note that in a discretized domain, L is the horizontal spatial scale of the domain, and x=n delta x.
- Spreadsheet example of discretized waves in periodic domain.

__Digital filters__

- Recursive and nonrecursive filters
- An example of a nonrecursive low-pass filter used by meteorologists - the Shapiro filter
- Spreadsheet example of 1-2-1 Shapiro filter
- Spreadsheet example of 1-8-28-56-186 Shapiro filter. Why is the result almost the same as the original? What has been removed?
- The classic Shapiro papers. Paper one. Paper two.
- Other nonrecursive common filters. Written by Ken Turkowki of Apple Computer. Notice that in electronic nomenclature, "decimation" represents the weight values, the frequency response is written such that no damping effect has a gain of 0 dB, and damping is represented by negative gain.
- Gibbs phenomenon.
- The classic Lanczos paper. Written by Claude Duchon. The Lanczos filter can perform lowpass or highpass fitlering, and is designed to reduce Gibbs phenomenon.
- Optional material - More on recursive exponential smoothing
- Optional material - Univariate time series methods

__Running mean (moving average)__

- Nonrecursive moving average, also known as a running mean. Smooths data, but does not remove any frequencies since all weights are equal. See Kalnay lecture for more details.
- Example application - the Oceanic Nino Index (ONI)
- Example application - Running mean of global temperatures since 1880

__Comparison of moving average, the one-two-one filter, and the Lanczos filter__

__Fourier (Harmonic) analysis__

- The basics (from "ask a mathematician")
- The mathematics (from Lamar University)
- The fundamental equations - a Fourier series; computing harmonic coefficient; computing signal power; computing explained variance
- Fourier series with a nonzero background average
- Trig identities used in Fourier series derivation. Youtube example for integrating sin(3x)sin(6x)dx
- Orthogonal relationships in Fourier series
- Example: decomposing two superimposed wave solutions, analytical solution
- Spreadsheet example: the same two superimposed waves decomposed with a background average. Power and explained variance also computed.
- Example application - ocean tide prediction

2) Example - Bridgeport, CT

3) Example - San Diego, CA

4) More realistic example - Seattle, WA

- Supplemental material - Signal power from Fourier analysis
- Supplemental material - Explained variance by Fourier analysis
- Supplemental material - another good summary from thefouriertransform.com
- Optional material - 1) The classic paper on spherical harmonics by Baer. 2) Advanced graduate school material from CSU professor David Randall.
- Optional material - more on time series analysis and some issues with Fourier analysis

__Computer language examples__

- ncl filters
- Matlab Lanczos filter
- Matlab digital filter
- Signal smoothing with Matlab
- Designing a digital filter with Matlab
- Shapiro filter for ROMS model output in FORTRAN
- Online calculator, running mean
- Matlab moving average
- Matlab Fourier analysis in one and two dimensions
- Fourier analysis with ncl
- Fourier analysis with IDL
- Software for harmonic analysis sea level data

__Recorded lectures__