110

N,,,n,xhxhyM

Mkkkn

M

Mkknkn

DIGITAL FILTERS

Box Car filter

Running Mean

Moving Average12

1

Mhk

12

0

M

k kh

1100

N,,,n,xxhyM

kknknkn

M = 48

M = 49

M = 50

Impulse Response

Nc

Nc

N

ck /k

)/ksin(

M/k

)M/ksin(h

M is the filter length (# of filter coefficients)

N is the sampling

frequency = 2π/Δt

c is the cut-off

frequency = 2π/Tc

Normalized SINC function windowed by the Lanczos window

repeat

wrap

High-pass filtered : Original – Low-Pass

Impulse Response

M

Mkknkn xhy

Frequency Response (Fourier Transform of Impulse Response)

M

Mn

tin

neyY

M

Mn

tikn

M

Mk

tik

knk exeh

XH

M

Mk

tik

kehH

Frequency Response or Admittance Function

c

c

@

@H

0

1Low-pass:

0

1

c N

PassBand

StopBand

H

c

c

@

@H

1

0High-pass:

0

1

c N

PassBand

StopBand

H

otherwise

@H cc

0

1 21 Band-pass:

0

1

c1 N

PassBand

StopBand

H

c2

StopBand

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)

H(

)

/ N

M

Mk

tik

kehH

Frequency Response or Admittance Function(Running Mean)

Gibbs’ Phenomenon

Hamming)M/kcos(..

LanczosM/k

)M/ksin(

46540

( )

Nc

Nc

N

ck /k

)/ksin(

M/k

)M/ksin(h

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

Butterworth Filter

q

c

H2

2

1

1

http://cnx.org/content/m10127/latest/

q = 1

q = 4 q = 10

Exercises

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