Lecture Digital signal processing: Lecture 8 - Zheng-Hua Tan

Lecture 8 - FIR Filter Design include all of the following content: FIR filter design, commonly used windows, generalized linear-phase FIR filter, the Kaiser window filter design method.

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Digital Signal Processing, VIII, Zheng-Hua Tan, 2006

1

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

zt@kom.aau.dk

Lecture 8: FIR Filter Design

Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/FFT

System analysis

Filter design

z-transform

MM1

MM2

MM9, MM10

MM3

MM6

MM4

MM7 , MM8

Sampling and reconstruction MM5

System structure System

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3

FIR filter design

Design problem: the FIR system function

Start from impulse response directly

Find

to approximate a desired frequency response

⎩

⎨

=

= ∑

=

−

otherwise

, 0

0 , ]

[

)

(

0

M n b

n

h

z b z

H

n

M

k

k k

M z M h z

h h z

H ( ) = [ 0 ] + [ 1 ] −1+ + [ ] −

Part I: FIR filter design

FIR filter design

Commonly used windows

Generalized linear-phase FIR filter

The Kaiser window filter design method

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5

Lowpass filter – as an example

Ideal lowpass filter

IIR filter: based on transformations of

continuous-time IIR system into discrete-continuous-time ones.

FIR filter: how? is non-causal, infinite! h [n ]

∞

<

∞

−

=

⎩

⎨

⎧

≤

<

=

n n

h

e H

c c

c j

lp

,

sin ] [

, 0

|

| , 1 ) (

π ω

π ω ω

ω ω ω

N c

c j

) / ( 1

1

| ) (

|

Ω Ω +

= Ω

poles

Design by windowing

Desired frequency responses are often

piecewise-constant with discontinuities at the boundaries

between bands, resulting in non-causal and infinte

impulse response extending from to , but

So, the most straightforward method is to truncate

the ideal response by windowing and do

time-shifting:

0 ] [

±∞

∞

−

] [

otherwise

, 0

|

| ], [ ]

[

M n g n

h

M n n h n

−

=

⎩

⎨

=

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7

Design by windowing

After Champagne & Labeau, DSP Class Notes 2002.

Design by rectangular window

In general,

For simple truncation, the window is the rectangular

window

∫−

−

π

θ ω θ

π H e H e d

e

2

1 )

⎩

⎨

=

otherwise

, 0

0 ,

1

]

[

] [ ] [

]

[

M n n

w

n w n h

n

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9

Convolution process by truncation

Fig 7.19

? t then wha ,

,

1

]

[ n = − ∞ < n < ∞

r

jω ∑∞ πδ ω π

−∞

=

+

= band

narrow be

should

)

( ejω

W

Requirements on the window

Requirements

reproduce the desired frequency response

filter) to minimize computation in the implementation of

the filter

Take the rectangular window as an example.

,

1

]

[ n = − ∞ < n < ∞

r

jω ∑∞ πδ ω π

−∞

=

+

= band narrow be

should

)

( ejω

W

) ( ejω

W

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11

Rectangular window

M+1

M=7

π

4

constant

Part II: Commonly used windows

FIR filter design

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13

Standard windows – time domain

Rectangular

Triangular

Hanning

Hamming

Blackman

⎩

⎨

=

otherwise

, 0 0 ), / 2 cos( 5 0 5 0 ] [ n n M n M w π ⎪ ⎩ ⎪ ⎨ ⎧ ≤ ≤ − ≤ ≤ = otherwise , 0 2 / , / 2 2 2 / 0 , / 2 ] [ n M M n M M n M n n w ⎩ ⎨ ⎧ ≤ ≤ = otherwise , 0 0 , 1 ] [ n n M w ⎩ ⎨ ⎧ − ≤ ≤ = otherwise

, 0 0 ), / 2 cos( 46 0 54 0 ] [ n n M n M w π ⎩ ⎨ ⎧ − + ≤ ≤ = otherwise

,

0

0 ), / 4 cos(

08 0 ) / 2 cos(

5 0 42

.

0

]

Standard windows – figure

Fig 7.21

Plotted for convenience In fact, the window is defined only at integer values of n.

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15

Standard windows – magnitude

Standard windows – comparison

Magnitude of side lobes vs width of main lobe

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17

Part III: Linear-phase FIR filter

FIR filter design

Linear-phase FIR systems

Generalized linear-phase system

Causal FIR systems have generalized linear-phase

if h[n] satisfies the symmetry condition

M n

n h n

M

h

M n

n h n

M

h

, , 1 , 0 ], [ ] [

or

, , 1 , 0 ], [ ] [

=

−

=

−

=

−

constants real

are and

of function real

a is )

(

) ( ) (

β

α

ω ω

β ωα ω

ω

j

j j j j

e

A

e e A e

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19

Types of GLP FIR filters

1

or

, , 1 , 0 ], [ ]

[

−

=

−

ε

n

M

h

Type IV Type III

Type II Type I

M odd

M even

1

=

ε

1

−

=

ε

Generalized linear phase FIR filter

Often aim at designing causal systems with a

generalized linear phase (stability is not a problem)

symmetric about M/2,

is a real, even function of w

Æ the resulting frequency response will have a

2 /

) ( ) ( ej We ej e j M

W ω = ω −ω

⎩

⎨

= otherwise

, 0

0 ], [ ]

w

) ( jω

e e W

] [ ]

hd − = d

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21

Linear-phase lowpass filter – an example

Desired frequency response

so

apply a symmetric window Æ a linear-phase system

∞

<

∞

−

=

⎩

⎨

⎧

≤

<

n M

n

M n n

h

e e

H

c lp

c

c M

j j

lp

, ) 2 / (

)]

2 / ( sin[

]

[

, 0

|

| , )

(

2 /

π ω

π ω ω

ω ω

] [ ]

hlp − = lp

] [ ) 2 / (

)]

2 / ( sin[

] [ ] [ ]

M n

M n n

w n h n

−

=

π ω

Window method approximations

even and real is ) (

) ( ) (

even and real is ) (

) ( ) (

2 /

ω

ω ω ω

ω

ω ω ω

j e

M j j e j

j e

M j j e j

d

e W

e e W e

W

e H

e e H e

H

−

=

∫−

−

=

π π

θ ω θ

ω

ω ω ω

θ

e A

e e A e

H

j e j e j

e

M j j e j

) ( ) ( 2

1 ) (

) ( ) (

) ( 2 /

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Key parameters

To meet the requirement of FIR filter, choose

Trail and error is not a satisfactory method to design

filters Æ a simple formalization of the window

method by Kaiser

Part IV: Kaiser window filter design

FIR filter design

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The Kaiser window filter design method

An easy way to find the trade-off between the

main-lobe width and side-main-lobe area

The Kaiser window

the first kind is an adjustable design parameter.

adjusted to trade side-lobe amplitude for main-lobe

width (not possible for preceding windows!)

2 /

otherwise

, 0

0 , )

(

] ) ] / ) [(

1 ( [ ]

2 / 1 2 0

M

M n I

n I

n

w

=

⎪

⎩

⎪

⎨

⎧

≤

−

=

α

β

α α β

)

0 ⋅

I

0

≥ β

β

The Kaiser window

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27

Design FIR filter by the Kaiser window

Calculate M and to meet the filter specification

Define then

Stopband cutoff frequency by:

Transition width

M must satisfy

δ

ω) | ≥ 1 − (

| H ej

β δ

β

δ

ω) | ≤ (

| H ej

⎪

⎩

⎪

⎨

⎧

<

≤

− +

>

−

=

21

, 0 0 50 21

), 21 ( 07886 0 21) -0.5842(A 50

), 7 8 ( 1102 0 0.4 A A A A A β p s ω ω ω = − Δ p ω δ 10 log 20 − = A s ω ω Δ − = 285 2 8 A M (Peak error is fixed for other windows) (Rectangular) A lowpass filter Specifications Specifications for a discrete-time lowpass filter π ω ω π ω ω ω ω 6 0 , 001 0 | ) ( | 4 0 0 , 01 0 1

| ) (

| 01

.

0

1

=

≥

≤

=

≤

≤ +

≤

−

s j

p j

e

H

e H

001 0

01 0

2

1

=

= δ δ

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29

Design the lowpass filter by Kaiser window

Designing by window method indicating , we

must set

Transition width

The two parameters:

Cutoff frequency of the ideal lowpass filter

Impulse response

60 log

−

A

001 0

= δ

π ω

ω

2

1 δ

37

, 653 5 = = M β ⎪ ⎩ ⎪ ⎨ ⎧ ≤ ≤ − − ⋅ − − = otherwise

,

0

0 , )

(

] ) ] / ) [(

1 ( [ ) (

) ( sin

]

2 / 1 2 0

M n I

n I

n

n n

h

c

β

α α β

α π

α ω

π ω

ω

ωc = ( s+ p) / 2 = 0 5

Design the lowpass filter

What is the group delay?

M/2=18.5

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Summary

FIR filter design

Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/ FFT

System analysis

Filter design

z-transform

MM1

MM2

MM6

MM4

MM7, MM8

Sampling and reconstruction MM5

System structure System

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