A filter is a linear operation that converts one time series into another (Chatfield, 1995). Low-pass, band-pass and high-pass filters are used to separate different signals from a time series.

Application of a low pass filter to a climate time series removes high frequency fluctuations from the time series. Figure 1 shows a time series of daily maximum temperature 30 year (1971 - 2000) averages, containing high frequency signal, and a smooth fit to this using Equation 1. This smooth fit is an example of a low pass filter.

Time series plot of Tmax  for one year (Dec 1970 through December 1971) with 30 year (1971-2000) daily average and the low pass filter
Figure 1. Fit of the Low Pass filter to 30 year daily averages of the daily maximum temperature time series at Bismarck, North Dakota. This is shown together with observations for December 1970 through December 1971
Tday equal T annual mean + sum for four cycles k of product of a and sin of (product k, 2 pi and day divided by 365) + sum for 4 cycles k of productc b and cos of (product k, 2pi and day divided by 365). Equation 1

Tday is the ordinate of the Low pass filter
day is the day of the year
Tannual mean is the 30 year (1971 - 2000) average of the daily temperature
a and b are the coefficients



Chatfield, C, 1995, The analysis of Time Series, 4th edition. Chapman & Hall