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1

FilteringFiltering

Filtering is one of the most widely used

complex signal processing operations

The system implementing this operation is

called a filter

A filterpasses certain frequency

components without any distortion and

blocks other frequency components

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2

FilteringFiltering

The range of frequencies that is allowed to

pass through the filter is called the

passband, and the range of frequencies thatisblocked by the filter is called the

stopband

In most cases, the filtering operation foranalog signals is linear

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3

FilteringFiltering

The filtering operation of a linear analog

filter is described by the convolution

integral

where x(t) is the input signal, y(t) is the

output of the filter, and h(t) is the impulse

response of the filter

=

dxthty )()()(

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4

FilteringFiltering

A lowpass filterpasses all low-frequencycomponents below a certain specified

frequency , called the cutoff frequency,

and blocks all high-frequency componentsabove

A highpass filterpasses all high-frequency

components a certain cutoff frequencyand blocks all low-frequency components

below

cf

cf

cf

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5

FilteringFiltering

Abandpass filterpasses all frequencycomponents between 2cutoff frequencies,

and , where , and blocks all

frequency components below the frequencyand above the frequency

Abandstop filterblocks all frequency

components between 2cutoff frequencies,and , where , and passes allfrequency components below the frequency

and above the frequency

1cf 2cf 21 cc ff 0

In practice

2/2/,)(

})/2(1{][

0

2

0 MnMI

MnInw

=

+=

=1

2

0 ]!

)2/(

[1)( r

r

r

u

uI0)(0 >uI

+=

20

1

20 ]!

)2/([1)(

r

r

ru

uI

)(0 uI

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57Copyright 2005, S. K. Mitra

Adjustable Window FunctionsAdjustable Window Functions

controls the minimum stopbandattenuation of the windowed filter response

is estimated using

Filter order is estimated using

where is the normalized transitionbandwidth

+

=

,0

),21(07886.04.0)21(5842.0

),7.8(1102.0

sss

21for

5021for

50for

s

ss

)(285.2

8

= sM

FIR Fil D i E l

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58

FIR Filter Design ExampleFIR Filter Design Example

Specifications: , ,

dB

Thus

Choose M= 24

01.01020/ == s

s

40=s

= 3.0p

= 5.0s

=+= 4.02/)( spc

3953.31907886.0)19(5842.0 4.0 =+=

2886.22

)2.0(285.2

32==

M