Copyright © 2005. Shi Ping CUC Chapter 8 FIR Filter Design Content Introduction Properties of Linear-Phase FIR Filters Window Design Techniques

Copyright © 2005. Shi Ping CUC

Chapter 8FIR Filter Design

Content

Introduction

Properties of Linear-Phase FIR Filters

Window Design Techniques


Introduction

The advantages of the FIR digital filter

The phase response can be exactly linear;

They are relatively easy to design since there are no stability problems;

They are efficient to implement;

The DFT can be used in their implementation.


Introduction

The advantages of a linear-phase response

Design problem contains only real arithmetic and not complex arithmetic;

Linear-phase filter provide no delay distortion and only a fixed amount of delay;

For the filter of length N (or order N-1) the number of operations are of the order of N/2.


The basic technique of FIR filter design

Window design techniques;

Frequency sampling design techniques;

Optimal equiripple design techniques.

Introduction

return



The system function and frequency response

The system function of FIR filters

Let h(n), n=0,1,…,N-1 be the impulse response of length N. Then the system function is

1

0

)()(N

n

nznhzH

It has (N-1) poles at the origin and N-1 zeros located anywhere in the z-plane.



The frequency response of FIR filters

)()(

1

0

)()(

)()(

jjj

M

n

njj

eHeeH

enheH

)( jeH Magnitude response function

Amplitude response function)(H



)( jeH )(H The difference between and

)( jeH is always positive and the associated phase response is a discontinuous function.

may be both positive and negative and the associated phase response is a continuous function.

)(H

Consider the following example:

1)2( ,1)1( ,1)0( hhh


jjj

n

njj

eee

enheH

)cos21(1

)()(

2

2

0

3/2

3/20 )(

0 ,cos21)( jeH

0 )(

cos21)(H



Linear-phase conditions

)(

)(

)()( )(jj eHeH

d

d )( A constant group delay

1

0

)()(N

n

nznhzHFor



)( For

The phase response is through the origin.

The phase response is not through the origin.

)1()( nNhnh

)1()( nNhnh

)( For



Frequency response of linear-phase FIR filters

1

0

)1(1

0

)1(

1

0

1

0

)()]([

)]1([)()(

N

m

mNN

m

mN

N

n

nN

n

n

zmhzzmh

znNhznhzH

)()( 1)1( zHzzH N



)()( 1)1( zHzzH N

1

0

)2

1()

2

1(

)2

1(

1

0

)1(

1)1(

]2

)[(

])[(2

1

)]()([2

1)(

N

n

nN

nN

N

N

n

nNn

N

zznhz

zzznh

zHzzHzH


Symmetric impulse response )1()( nNhnh

])2

1cos[()(

]2

[)(

)()(

1

0

)2

1(

)12

1()1

2

1(

1

0

)2

1(

nN

nhe

eenhe

zHeH

N

n

Nj

Nj

Nj

N

n

Nj

ez

jj

])2

1cos[()()(

1

0

nN

nhHN

n

)2

1()(

N

2

1

N


Antisymmetric impulse response )1()( nNhnh

])2

1sin[()(

]2

[)(

)()(

1

0

2)

2

1(

)12

1()1

2

1(

1

0

)2

1(

nN

nhe

eenhe

zHeH

N

n

jN

j

Nj

Nj

N

n

Nj

ez

jj

])2

1sin[()()(

1

0

nN

nhHN

n

2)

2

1()(

N

2

1

N



Type 1: symmetric impulse response, N is odd

The properties of amplitude function )(H

])2

1cos[()()(

1

0

nN

nhHN

n

)cos()(

)cos()2

1(2)

2

1(

])2

1cos[()(2)

2

1()(

2/)1(

0

2/)1(

1

2/)3(

0

nna

mmN

hN

h

nN

nhN

hH

N

n

N

m

N

n

mnN

2

1



)cos()()(2/)1(

0

nnaHN

n

)2

1()0(

N

ha

2

1,,2,1 ),

2

1(2)(

Nnn

Nhna

The middle sample



Type 2: symmetric impulse response, N is even

])2

1cos[()()(

1

0

nN

nhHN

n

)]2

1(cos[)()(

2/

1

nnbHN

n

2,,2,1 ),

2(2)(

Nnn

Nhnb



Type 3: antisymmetric impulse response, N is odd

])2

1sin[()()(

1

0

nN

nhHN

n

)sin()()(2/)1(

1

nncHN

n

2

1,,2,1 ),

2

1(2)(

Nnn

Nhnc



Type 4: antisymmetric impulse response, N is even

])2

1sin[()()(

1

0

nN

nhHN

n

)]2

1(sin[)()(

2/

1

nndHN

n

2,,2,1 ),

2(2)(

Nnn

Nhnd



Zero locations of linear-phase FIR filters

)()( 1)1( zHzzH N

If has a zero at)(zH jrezz 1

Then for linear phase there must be a zero at

jerz

z 11

1

For a real-valued filter, there must be zeros at

jrezz 1je

rzz

11

1

return



Basic window design idea

Choose a proper ideal frequency-selective filter (which always has a noncausal, infinite-length impulse response);

Then truncate (window) its impulse response to obtain a linear-phase and causal FIR filter.

The emphasis is on

Selecting an appropriate ideal filter;

Selecting an appropriate windowing function.



Denote an ideal frequency-selective filter by )( jd eH

|| ,0

|| ,1)(

c

cj

jd

eeH

)(

)](sin[21

)()( 1

nn

deeeHFnhc

ccnjjjdd

c

c

)()( nRnw N


21

otherwise ,0

10 ),()()()(

N

Nnnhnwnhnh d

d

otherwise ,0

10 ,)

21

(

)2

1(sin

)( NnN

n

Nn

nhc

cc

Windowing


The effect of Window function

deWeHeH jjd

j )()(21

)( )(

)2

1()

21

(1

0

)()

2sin(

)2

sin()(

N

j

R

NjN

n

njjR eW

N

eeeW

1

0

)()(N

n

njj enweW

)2

sin(

)2

sin()(

N

WR


)

21

()()(

N

j

dj

d eHeH

|| ,0

|| ,1)(

c

cdH

)2

1()

21

(

))(2

1()

21

(

)()()(21

)()(21

)(

N

j

Rd

Nj

Nj

R

Nj

dj

eHdWHe

deWeHeH

dWHH Rd )()(

21

)(



The conclusion

The periodic convolution produces a smeared version of the ideal response )( j

d eH

Since the window has a finite length equal to N, its response has a peaky main lobe whose width is proportional to 1/N, and has side lobes of smaller heights.

)(nw

The main lobe produces a transition band in whose width is responsible for the transition width. This width is then proportional to 1/N. The wider the main lobe, the wider will be the transition width.

)( jeH

The side lobes produces ripples that have similar shapes in both the passband and stopband.



Windowing functions Rectangular window

)2

sin(

)2

sin()(

)()( ),()()

21

(

N

W

eWeWnRnw

R

Nj

Rj

RN

This is the simplest window function but provides the worst performance from the viewpoint of stopband attenuation. The width of main lobe is N/4



Gibbs phenomenon

The truncation of the infinite length will introduce ripples in frequency response . The oscillatory behavior near the band edge of the filter is called the Gibbs phenomenon.

)(H)(nhd

When the N is increased:The transition band of the filter will decreaseBut the relative amplitude of the peaky values will remain constant.


Bartlett window

1

2

1 ,

1

22

2

10 ,

1

2

)(Nn

N

N

n

Nn

N

n

nw

)1 ,1( ,

2sin

4sin

2)(

)2

1(

2

NNNe

N

NeW

Nj

j

Since the Gibbs phenomenon results from the fact that the rectangular window has a sudden transition from 0 to 1 (or 1 to 0), Bartlett suggested a more gradual transition in the form of a triangular window. The width of main lobe is N/8


Hanning window

This is a raised cosine window function given by:

)1(

)2

()2

(25.0)(5.0)(

N

NW

NWWW RRR

The width of main lobe is:N

8

)(1

2cos1

2

1)( nR

N

π nnw N


Hamming window

)(1

2cos46.054.0)( nR

N

π nnw N

This is a modified version of the raised cosine window.


8

)1(

)2

()2

(23.0)(54.0)(

N

NW

NWWW RRR


Blackman window

)(1

4cos08.0

1

2cos5.042.0)( nR

N

π n

N

π nnw N

This is a 2-order raised cosine window.


12

)1(

)4

()4

(04.0

)2

()2

(25.0)(42.0)(

N

NW

NW

NW

NWWW

RR

RRR


Kaiser window

)()(

12

11

)(0

2

0

nRI

Nn

I

nw N

This is one of the most useful and optimum windows.

Where is the modified zero-order Bessel function, and is a parameter that can be chosen to yield various transition widths and stopband attenuation. This window can provide different transition widths for the same N.

)(0 I

window Blackman 5.8

window Hamming 44.5

windowr rectangula 0


The design equations for Kaiser window

Given spsp AR , , ,

ps The norm transition width:

286.2

95.7sANThe filter order N:

dB21 0

dB5021dB )21(07886.0)21(5842.0

dB50 )7.8(1102.04.0

s

sss

ss

A

AAA

AA


Summary of window function characteristics

Window name

Window function Filter

Peak value of side lobe

The width of main lobe

Transition width

Min. stopband attenuation

Rectangular -13 dB -21 dB

Bartlett -25 dB -25 dB

Hanning -31 dB -44 dB

Hamming -41 dB -53 dB

Blackman -57 dB -74 dB

N4

N8

N8

N8

N12

N8.1

N2.4

N2.6

N6.6

N11



Design procedure

Given the ideal frequency response )( jd eH

Compute the impulse response of ideal filter)(nhd

Determine the window shape and N from the minimum stopband attenuation and the transition width

ps sA

Compute the impulse response of the designed filter)()()( nwnhnh d

Compute the frequency response of the designed filter and verify the performance

)( jeH



Examples of FIR linear-phase filter design

Digital FIR lowpass filter

Design a digital FIR lowpass filter:

dB 50

sec)/rad(1032

sec)/rad(105.12

sec)/rad(105.12

3

3

4

s

s

p

sample

A

Example


Solution:

Compute the digital frequencies

2.0 ,3.02

4.02

,2.02

pssp

c

sample

ss

sample

pp

Derive the frequency response of ideal FIR lowpass filter

||3.0 ,0

3.0|| ,

|| ,0

|| ,)(

j

c

cj

jd

eeeH


Compute the impulse response of the ideal filter

Determine the window shape and N

)(

)](3.0sin[

)(

)](sin[

2

1)(

n

n

n

ndeenh

c

ccnjjd

c

c

dB 50sA

162

1 ,33 ,2.0

6.6

NN

N

Hamming


Compute the impulse response of the designed filter

)(16

cos46.054.0

)(1

2cos46.054.0)(

33 nRπ n

nRN

π nnw N

)16(

)]16(3.0sin[)(

n

nnhd

)(16

cos46.054.0)16(

)]16(3.0sin[

)()()(

33 nRπ n

n

n

nwnhnh d


Compute the frequency response of the designed filter

)]([DTFT)( nheH j

Verify the performance of the designed filter

dB 46

dB 073.0

s

p

A

R

It is not satisfied by this design

Let N = 34 and redesign

dB 52

dB 048.0

s

p

A

R



Digital FIR highpass filter

An ideal FIR highpass filter can be obtained from two ideal FIR lowpass filters, provided they have the same phase response.

0

0 c

0 c


The frequency response of an ideal FIR highpass filters

c

cj

jd

eeH

|| ,0

|| ,)(

The impulse response of an ideal FIR highpass filters

)(

)](sin[

)(

)](sin[2

1

)(2

1)(

)( )(

n

n

n

n

dede

deeHnh

c

njnj

njjd

c

c


Example

Design a digital FIR highpass filter :dB 5.0 ,6.0

dB 60 ,4.0

pp

ss

R

A

Solution:


2.0 ,5.02

6.0 ,4.0

spsp

c

ps


Derive the impulse response of ideal FIR highpass filter

)27(

)]27(5.0sin[

)27(

)]27(sin[

)(

)](sin[

)(

)](sin[)(

n

n

n

n

n

n

n

nnh c

d


272

155 ,55 ,2.0

11

N

N

dB 60sA Blackman

Note: the N must be odd for FIR highpass filters



)()()( nwnhnh d

)(54

4cos08.0

54

2cos5.042.0

)(1

4cos08.0

1

2cos5.042.0)(

55 nRπ nπ n

nRN

π n

N

π nnw N



)]([DTFT)( nheH j


dB 71

dB 0039.0

s

p

A

R

It is satisfied by this design



Digital FIR bandpass filter

An ideal FIR bandpass filter can be obtained from two ideal FIR lowpass filters, provided they have the same phase response.

0 1c 2c

0 1c

0 2c


The frequency response of an ideal FIR bandpass filters

otherwise ,0

|| ,)( 21 cc

jj

d

eeH


)(

)](sin[

)(

)](sin[2

1

)(2

1)(

12

)( )(

2

1

1

2

n

n

n

n

dede

deeHnh

cc

njnj

njjd

c

c

c

c


Example

Design a digital FIR bandpass filter :

dB 1 ,35.0

dB 60 ,2.0

11

11

pp

ss

R

A

Solution:


15.0)](),min[(

725.02

,275.02

2211

222

111

pssp

spc

spc

dB 60 ,8.0

dB 1 ,65.0

22

22

ss

pp

A

R


Derive the impulse response of ideal FIR bandpass filter

)5.36(

)]5.36(275.0sin[

)5.36(

)]5.36(725.0sin[

)(

)](sin[

)(

)](sin[)( 12

n

n

n

n

n

n

n

nnh cc

d


5.362

174 ,74 ,15.0

11

N

N

dB 60sA Blackman



)()()( nwnhnh d

)(73

4cos08.0

73

2cos5.042.0

)(1

4cos08.0

1

2cos5.042.0)(

74 nRπ nπ n

nRN

π n

N

π nnw N



)]([DTFT)( nheH j


dB 73 dB, 73

dB 003.0 dB, 003.0

21

21

ss

pp

AA

RR



Digital FIR bandstop filter

An ideal FIR bandstop filter can be obtained from three ideal FIR lowpass filters, provided they have the same phase response.

0 1c 2c

0 1c

0 2c

0


The frequency response of an ideal FIR bandpass filters

otherwise ,0

|| ,|| 0 ,)( 21

cc

jj

d

eeH


)(

)](sin[

)(

)](sin[

)(

)](sin[2

1

)(2

1)(

12

)( )( )(

2

1

1

2

n

n

n

n

n

n

dedede

deeHnh

cc

njnjnj

njjd

c

c

c

c


Example

Design a digital FIR bandstop filter :

dB 40 ,4.0

dB 5.0 ,3.0

11

11

ss

pp

A

R

Solution:


1.0)](),min[(

65.02

,35.02

2211

222

111

spps

spc

spc

dB 5.0 ,7.0

dB 40 ,6.0

22

22

pp

ss

R

A


Derive the impulse response of ideal FIR bandstop filter


312

163 63 ,62 ,1.0

2.6

NN

N

dB 40sA Hanning

)(

)](35.0sin[

)(

)](65.0sin[

)(

)](sin[

)(

)](sin[

)(

)](sin[

)(

)](sin[)( 12

n

n

n

n

n

n

n

n

n

n

n

nnh cc

d

Note: the N must be odd for FIR bandstop filters



)()()( nwnhnh d

)(61

2cos1

2

1

)(1

2cos1

2

1)(

63 nRπ n

nRN

π nnw N



)]([DTFT)( nheH j


dB 44 dB, 44

dB 0884.0 dB, 0884.0

21

21

ss

pp

AA

RR


return


0 2/3 1

-1

0

3

Magnitude Response

Frequency in pi units

0 2/3 1

-2/3

0

1/3

Piecewise Phase Response


an

gle

in p

i un

its

0 2/3 1

-1

0

3

Amplitude Response


0 2/3 1

-2/3

0

1/3

Phase Response


an

gle

in p

i un

its

return


0 0.5 1 1.5 2-10

-8

-6

-4

-2

0phase Response

Frequency in pi units0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

h(n), N is odd

n

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

h(n), N is even

n

10N

return

2

1

)(

N

)1()( nNhnh


0 0.5 1 1.5 2

-8

-6

-4

-2

00.5

Phase Response

Frequency in pi units0 1 2 3 4 5 6 7 8 9 10

0

h(n), N is odd

n

0 1 2 3 4 5 6 7 8 9

0

h(n), N is even

n

10N

return

2 ,

2

1

)(

N

)1()( nNhnh


0 5 10

-5

0

5

10

n

h(n

)

Type-1 Impulse Response

0 0.5 1 1.5 2-20

-10

0

10

20

frequency in pi units

H(w

)

Type-1 Amplitude Response

-1 0 1

-1

-0.5

0

0.5

1

Real Part

Ima

gin

ary

Pa

rt

Type-1 Pole-Zero Plot

10

return

2 , ,0 偶对称


0 5 10

-5

0

5

10

n

h(n

)


0 0.5 1 1.5 2-40

-20

0

20

40


H(w

)


-1 0 1

-1

-0.5

0

0.5

1

Real Part

Ima

gin

ary

Pa

rt


11

return

奇对称

0|)( H

不能用来设计高通、带阻滤波器


0 5 10

-5

0

5

10

n

h(n

)


0 0.5 1 1.5 2-40

-20

0

20

40


H(w

)


-1 0 1

-1

-0.5

0

0.5

1

Real Part

Ima

gin

ary

Pa

rt


10

return

2 , ,0 奇对称0|)( 2,,0 H

不能用来设计低通、高通、带阻滤波器


0 5 10

-5

0

5

10

n

h(n

)


0 0.5 1 1.5 2-10

0

10

20

30


H(w

)


-1 0 1

-1

-0.5

0

0.5

1

Real Part

Ima

gin

ary

Pa

rt


11

return

2 , ,0 偶对称0|)( 0H

不能用来设计低通滤波器


jIm[z]

Re[z]0

return

Quadruplet


-10 -5 0 5 10 15 20 25 30-0.05

0

0.05

0.1

0.15

0.2

)(nhd

return

)()( nRnw N


-10 -5 0 5 10 15 20 25 30-0.05

0

0.05

0.1

0.15

0.2

)(nhd

)()( nRnw N

return


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20

)2

sin(

)2

sin()(

N

WR

20N

return

N2

N2


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000


)(RW

)(H

return

20N)(dH 5.0c


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000


)(RW

)(H

5.0 ,20 cN


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000

Amplitude response


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000

Magnitude response


)(H

)( jeH

return

ps

ps


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Rectangular window: N=25

n-1 -0.5 0 0.5 10

5

10

15

20

25

Magnitude response


-1 -0.5 0 0.5 1

0

10

20

Amplitude response

frequency in pi units-1 -0.5 0 0.5 1

0

500

1000

1500

2000

Amplitude response of filter: wc=0.5pi


N4

N4


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Rectangular window: N=25

n-1 -0.5 0 0.5 1

-40

-30

-20

-10

0

Magnitude response


dB

-1 -0.5 0 0.5 1

0

10

20

Amplitude response


-50

-40

-30

-20

-10

0

Magnitude response of filter: wc=0.5pi


dB

return

N

8.1


-1 -0.5 0 0.5 1

0

500

1000

1500

2000

N = 7


0

500

1000

1500

2000

N = 21


-1 -0.5 0 0.5 1

0

500

1000

1500

2000

N = 51


0

500

1000

1500

2000

N = 101


Gibbs


Copyright © 2005. Shi Ping CUCreturn

(a) (b)


0 10 20 300

0.2

0.4

0.6

0.8

1

Triangular window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 1

0

5

10

15

20Amplitude response


-40

-30

-20

-10

0



dB

return

N

8 N

2.4


0 10 20 300

0.2

0.4

0.6

0.8

1

Hanning window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 10

5

10

15

Amplitude response


-60

-40

-20

0



dB

return

N

8 N

2.6


0 10 20 300

0.2

0.4

0.6

0.8

1

Hamming window: N=35

n-1 -0.5 0 0.5 1

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 10

5

10

15



-60

-40

-20

0



dB

return

N

8 N

6.6


0 10 20 300

0.2

0.4

0.6

0.8

1

Blackman window: N=35

n-1 -0.5 0 0.5 1

-80

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 1

0

5

10

15

Amplitude response


-80

-60

-40

-20

0



dB

return

N

12N

11


-10 0 100

0.2

0.4

0.6

0.8

1

Kaiser window: N=35,beta=7.865

n-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 1

0

5

10

15



-100

-80

-60

-40

-20

0



dB N

10


-10 0 100

0.2

0.4

0.6

0.8

1

Kaiser window: N=35,beta=9.5

n-1 -0.5 0 0.5 1

-100

-80

-60

-40

-20

0

Magnitude response


dB

-1 -0.5 0 0.5 1

0

5

10

15



-100

-80

-60

-40

-20

0



dB

return


0 5 10 15 20 25 30

0

0.1

0.2

0.3

Ideal Impulse Response

n 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Hamming Window

n

0 5 10 15 20 25 30

0

0.1

0.2

0.3

Actual Impulse Response

n 0 0.2 0.4 0.6 0.8 1-100

-80

-60

-40

-20

0Magnitude Response in dB

pi

dB

33N

return


0 5 10 15 20 25 30

0

0.1

0.2

0.3


n 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Hamming Window

n

0 5 10 15 20 25 30

0

0.1

0.2

0.3


n 0 0.2 0.4 0.6 0.8 1-100

-80

-60

-40

-20


pi

dB

34N

return


0 10 20 30 40 50-0.4

-0.2

0

0.2

0.4

0.6Ideal Impulse Response

n 0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Blackman Window

n

0 10 20 30 40 50-0.4

-0.2

0

0.2

0.4

0.6Actual Impulse Response

n 0 0.2 0.4 0.6 0.8 1-120

-100

-80

-60

-40

-20


pi

dB

55N

return


1s 1p 2p 2s

pR

sA

0 2.0)(

35.0 65.0 8.0

dB 60

dB 1

1


0 20 40 60

-0.2

-0.1

0

0.1

0.2

0.3


n 0 20 40 600

0.2

0.4

0.6

0.8

1

Blackman Window

n

0 10 20 30 40 50 60 70

-0.2

-0.1

0

0.1

0.2

0.3


n 0 0.2 0.4 0.6 0.8 1

-100

-80

-60

-40

-20


pi

dB

74N

return


1s1p 2p2s

pR

sA

0 3.0)(

4.0 6.0 7.0

dB 40

dB .50

1


0 10 20 30 40 50 60-0.2

0

0.2

0.4

0.6

0.8Ideal Impulse Response

n 0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Hanning Window

n

0 10 20 30 40 50 60-0.2

0

0.2

0.4

0.6

0.8Actual Impulse Response

n 0 0.2 0.4 0.6 0.8 1-70

-60

-50

-40

-30

-20

-10


pi

dB

63N

return

Documents

Copyright © 2005. Shi Ping CUC Chapter 8 FIR Filter Design Content Introduction Properties of Linear-Phase FIR Filters Window Design Techniques