DIGITAL FILTERS h = time invariant weights (IMPULSE RESPONSE FUNCTION) 2M + 1 = # of weights N = # of data points

Impulse Response : Box Car filter Running Mean Moving Average

M = 48 M = 49 M = 50

Normalized SINC function windowed by the Lanczos window Impulse Response: Normalized SINC function windowed by the Lanczos window M is the filter length (# of weights or filter coefficients) N is the sampling frequency = 2π/Δt c is the cut-off frequency = 2π/Tc

repeat wrap

High-pass filtered : Original – Low-Pass

Frequency Response or Transfer Function or Admittance Function Fourier Transform of yn Convolution in time domain corresponds to multiplication in frequency domain Frequency Response or Transfer Function or Admittance Function

1 c N  Pass Band Stop H Low-pass:

High-pass: 1 c N  Pass Band Stop H

1 c1 N  Pass Band Stop H Band-pass: Stop Band c2

Band-pass filtered 1) High-pass to cut-off the upper bound period (e.g. 18 hrs) 2) Low-pass to cut-off the lower bound period (e.g. 4 hrs)

Frequency Response or Transfer Function Gibbs’ Phenomenon Frequency Response or Transfer Function (for Running Mean) H( ) M > M > M  / N

 (  ) Lynch (1997, Month. Wea. Rev., 125, 655)

Butterworth Filter http://cnx.org/content/m10127/latest/ q = 4 q = 10

Exercises http://www.falstad.com/dfilter/