digital signal processing, mm4.ppt - department of...

1

Digital Signal Processing, Fall 2009

L t 4 Filt D i n

Zheng-Hua Tan

D t t f El t i S t

Lecture 4: Filter Design

Digital Signal Processing, IV, Zheng-Hua Tan, 20091

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

2

Part I: Filter design

Filter design

IIR filter design

FIR filter design


Filter design process

Filter, in broader sense, covers any system.

Three design steps

SpecificationsProblem

Solution

RealizationApproximations

M it d

System functionPerformance constraints


Magnitude responsePhase response(frequency domain)Complexity

Structure IIR or FIRSubtype

3

Specifications – an example

Specifications for a discrete-time lowpass filter

pjeH 0 ,01.01|)(|01.01

sjeH ,001.0|)(|

001.0

01.0

2

1


Specifications of frequency response

Typical lowpass filter specifications in terms of tolerable Passband distortion as smallest as possible Passband distortion, as smallest as possible

Stopband attenuation, as greatest as possible

Width of transition band: as narrowest as possible

Improving one often worsens others a tradeoff

Increasing filter order improves all


Increasing filter order improves all

4

DT filter for CT signals

DT filter for the processing of CT signals Bandlimited input signal High enough sampling frequencyg g p g q y

Then, specifications (often given in frequency domain) conversion is straightforward

TjHeH

T

TeHjH

effj

Tj

eff

|| ),()(

/|| ,0

/|| ),()( Signal is band-limited;

T is small enough.


Fig. 7.1

Tjeff ||),()(

Specifications – an example

Specifications for a CT lowpass filter

)2000(20 ,01.01|)(|01.01 jHeff

Fig 7.2(a)(b)

)3000(2 ,001.0|)(| jHeff

001.0

01.0

2

1

T


)3000(2

)2000(22

s

p

5

Design a filter

Design goal: find system function to make frequency response meet the specifications (tolerances)

Infinite impulse response filter Infinite impulse response filter Poles insider unit circle due to causality and stability

Rational function approximation

Finite impulse response filter Linear phase is often required

Polynomial approximation


Polynomial approximation

E.g. IIR filter design

For rational system function

Nk

M

k

kk zb

zH 0)(

find the system coefficients such that the corresponding frequency response

id d i ti t d i d

k

kk za

1

1

jezj zHeH

|)()(


provides a good approximation to a desired response

H(z)•Rational system function•Stable•causal

)()( jdesired

j eHeH

6

FIR and IIR

IIR Rational system

f ti

FIR Polynomial system

functionfunction

Poles + zeros

Stable/unstable

Hard to control phase

Low order (4-20)

function

Zeros

Stable

Easy to get linear phase

High order (20-


Low order (4 20)

Designed on the basis of analog filter

2000)

Unrelated to analog filter

FIR or IIR

Whether FIR or IIR often depends on the phase requirements

Design principle Design principle If GLP is essential FIR

If not IIR preferable (can meet specifications with lower complexity)


7

Part II: IIR Filter design

Filter design

IIR filter design Analog filter design

IIR filter design by impulse invariance

FIR filter design


Design IIR filter based on analog filter

The mapping is direct

/||0

/|| ),()(

T

TeHjH

Tj

eff Signal is band-limited;

Advanced analog filter design techniques

Designing DT filter by transforming prototype CT filter:

|| ),()(

/|| ,0

TjHeH

T

effj

T is small enough.


filter: Transform (map) DT specifications to analog

Design analog filter

Inverse-transform analog filter to DT

8

Transformation method

Transform (map) DT specifications to analog


T


Inverse-transform to DT)(or )( thsH cc

][or )( nhzH

jj

j

tj

st

rez

dtethjH

dtethsH

js

)()(

)()(

• The imaginary axis of the s-plane


n

n

n

njj

znxzX

enxeX

][)(

][)( The imaginary axis of the s plane the unit circle of the z-plane• Poles in the left half of the s-plane poles inside the unit circle in the z-plane (stable)

Part II-A: Analog Filter design

Filter design



FIR filter design


9

Analog filter design

Butterworth

Chebyshev I

Cheb she II Chebyshev II

Ellipical


Butterworth lowpass filters

The magnitude response Maximally flat in the passband

Monotonic in both passband and stopband Monotonic in both passband and stopband

The squared magnitude response

Nc

c jH2

2

)/(1

1|)(|


10

Poles in s-plane


Part II-B: Design by impulse invariance

Filter design



FIR filter design


11

Filter design by impulse invariance

Impulse invariance: a method for obtaining a DT system whose is determined by the of a CT system.

)( jHc)( jeH

of a CT system.

In DT filter design, the specifications are provided in the DT, so Td has no role. Td is included for discussion

interval sampling design'' -

)(][

d

dcd

T

nThTnh


, d d

though. Td also has nothing to do with C/D and D/C conversion in Fig. 7.1, i.e. Td need not be the same as the sampling period T of the C/D and D/C conversion.

Relationship btw frequency responses

Impulse response sampling:

Frequency response)(][ dcd nThTnh

k21

)(][ nTxnx c Frequency response

if the CT filter is bandlimited

then

/|| ,0)(

)2

()(

dc

dk dc

j

TjH

kT

jT

jHeH

k

cj

T

kj

TjX

TeX )

2(

1)(

||)()( jHeH ffj


This is also the way to get CT filter specifications from

|| ),()(d

cj

TjHeH

dj TeH /relation theapplyingby )(

|| ),()(T

jHeH eff

12

Aliasing in the impulse invariance design

)2

()( kT

jT

jHeHdk d

cj

)(][ dcd nThTnh

TT dk d


The continuous-time filter may be designed to exceed the specifications, particularly in the stopband.

Impulse invariance with a Butterworth filter

Specifications

||30177830|)(|

2.0||0 ,1|)(|89125.0j

j

H

eH

Since the sampling interval Td cancels in the impulse invariance procedure, we choose Td=1, so

Magnitude function for a CT Butterworth filter

||3.0 ,17783.0|)(| jeH

||30177830|)(|

2.0||0 ,1|)(|89125.0

jH

jHc

/|| ,0

/|| ),()(

T

TeHjH

j

Tj

eff


Due to the monotonic function of Butterworth filter, we have

||3.0 ,17783.0|)(| jHc

17783.0|)3.0(|

89125.0|)2.0(|

jH

jH

c

c

|| ),()(T

jHeH effj

13


Squared magnitude function of a Butterworth filter

177830|)30(|

89125.0|)2.0(|

jH

jHc

Nc jH2

2

)/(1

1|)(|

(1)(2)

22

22

)17783.0

1()

3.0(1

)89125.0

1()

2.0(1

N

c

N

c

17783.0|)3.0(| jHcc )/(1

7032.0c

6N(5)

(3)

(6)


70474.0

8858.5

c

N(4)

12

2

)7032.0/(1

1

)/(1

1)()(

js

jssHsH

Nc

cc


12 poles for the squared magnitude functionfunction

The system function has the three pole pairs in the left half of the s-plane


14


To construct Hc(s) from the magnitude-squared function Hc(s)Hc(-s) , we choose the poles on the left-half-plane part of the s-plane to obtain a stable and causal filter.

)4945.03585.1)(4945.09945.0)(4945.03640.0(

12093.0)(

222

sssssssHc

4466028710 1z

Express Hc(s) as a partial fraction expansion, perform the transformation of Eq. (7.12):


)2570.09972.01(

6303.08557.1

)3699.00691.11(

1455.11428.2

)6949.02971.11(

4466.02871.0)(

21

1

21

1

21

zz

z

zz

z

zz

zzH



15

Part III: FIR filter design

Filter design

IIR filter design

FIR filter design Commonly used windows

Generalized linear-phase FIR filter

The Kaiser window filter design method


FIR filter design

Design problem: the FIR system function

)(0

zbzHM

k

kk

Start from impulse response directly

otherwise ,0

0 ,][

0

Mnbnh n

k

MzMhzhhzH ][...]1[]0[)( 1


Find

the degree M and

the filter coefficients h[k]

to approximate a desired frequency response

16

Lowpass filter – as an example

Ideal lowpass filter

nn

nheH ccj sin][

|| ,1)(

IIR filter: based on transformations of CT IIR system into DT ones.

nn

nheHc

lp ,][ ,0

)(

Nc jH2

2 1|)(|


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

Nc

c j2)/(1

|)(|

poles

Design by windowing

Desired frequency responses are often piecewise-constant with discontinuities at the boundaries between bands, resulting in non-causal and infintebetween bands, resulting in non causal and infinteimpulse response extending from to , but

So, the most straightforward method is to truncate the ideal response by windowing and do time-shifting:

0][ , nhn d


][][

otherwise ,0

|| ],[][

Mngnh

Mnnhng d

17

Design by windowing

After Champagne & Labeau, DSP Class Notes 2002.


Design by rectangular window

In general,

][][][ nwnhnh d

For simple truncation, the window is the rectangular window

otherwise ,0

0 ,1][

][][][

Mnnw

d


deHeHeH jjd

j )()(2

1)( )(

18

Convolution process by truncation

Fig. 7.19


?t then wha, ,1][ nnw )2(2)( reWr

j

band narrow be should )( jeW

Requirements on the window

, ,1][ nnw )2(2)( reWr

j

bandnarrowbeshould)( jeW

Requirements approximates an impulse to faithfully

reproduce the desired frequency response

w[n] as short as possible in duration (the order of the filter) to minimize computation in the implementation of

bandnarrow be should)(eW

)( jeW


) p pthe filter

Conflicting

Take the rectangular window as an example.

19

Rectangular window

M+1

M=7

4constant


Part III-A: Commonly used windows

Filter design

IIR filter design





20

Standard windows – time domain

Rectangular

2/0/2 MM

otherwise ,0

0 ,1][

Mnnw

Triangular

Hanning

otherwise ,0

0 ),/2cos(5.05.0][

MnMnnw

otherwise ,0

2/ ,/22

2/0 ,/2

][ MnMMn

MnMn

nw


Hamming

Blackman

,

otherwise ,0

0 ),/2cos(46.054.0][

MnMnnw

otherwise ,0

0 ),/4cos(08.0)/2cos(5.042.0][

MnMnMnnw

Standard windows – figure

Fig. 7.21


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

21

Standard windows – magnitude


Standard windows – comparison

Magnitude of side lobes vs width of main lobe


Independent of M!

22

Part III-B: Linear-phase FIR filter

Filter design







Linear-phase FIR systems

Generalized linear-phase system

)()( jjjj eeAeH

Causal FIR systems have generalized linear-phase if h[n] satisfies the symmetry condition

constants real are and

, offunction real a is )(

jeA


MnnhnMh

MnnhnMh

,...,1,0 ],[][

or

,...,1,0 ],[][

23

Generalized linear phase FIR filter

Often aim at designing causal systems with a generalized linear phase (stability is not a problem) If the impulse response of the desired filter is If the impulse response of the desired filter is

symmetric about M/2,

Choose windows being symmetric about the point M/2

2/)()( Mjje

j eeWeW

otherwise ,0

0 ],[][

MnnMwnw

][][ nhnMh dd


is a real, even function of w

the resulting frequency response will have a generalized linear phase

)( je eW

2/)()( Mjje

j eeAeH even and real is )( je eA

Linear-phase lowpass filter – an example

Desired frequency response

e

eH cMj

jlp

|| ,)(

2/

so

apply a symmetric window a linear phase system

nMn

Mnnh c

lp

c

lp

,)2/(

)]2/(sin[][

,0)(

][][ nhnMh lplp


apply a symmetric window a linear-phase system

][)2/(

)]2/(sin[][][][ nw

Mn

Mnnwnhnh c

lp

24

Window method approximations

)()(

even and real is )(

)()(

2/

2/

Mjjj

je

Mjje

jd

eeWeW

eH

eeHeH

even and real is )(

)()(j

e

e

eW

eeWeW

eeAeH Mjje

j )()( 2/


deWeHeA j

ej

ej

e )()(2

1)( )(

Key parameters

To meet the requirement of FIR filter, choose Shape of the window

Duration of the window Duration of the window

Trail and error is not a satisfactory method to design filters a simple formalization of the window method by Kaiser


25

Part III-C: Kaiser window filter design

Filter design








An easy way to find the trade-off between the main-lobe width and side-lobe area

The Kaiser window The Kaiser window

is the zeroth order modified Bessel function of

2/

otherwise ,0

0 ,)(

])]/)[(1([

][ 0

2/120

M

MnI

nI

nw

)(I


is the zeroth-order modified Bessel function of the first kind. is an adjustable design parameter.

The length (M+1) and the shape parameter can be adjusted to trade side-lobe amplitude for main-lobe width (not possible for preceding windows!)

)(0 I0

26

The Kaiser window


Design FIR filter by the Kaiser window

Calculate M and to meet the filter specification The peak approximation error is determined by

Define then

log20A (Peak error is fixed for other windows)Define then

Passband cutoff frequency is determined by: 1|)(| jeH

21 ,0.0

5021 ),21(07886.021)-0.5842(A

50 ),7.8(1102.00.4

A

AA

AA

p

10log20A (Peak error is fixed for other windows)

(Rectangular)


Stopband cutoff frequency by:

Transition width

M must satisfy

|)(|

|)(| jeH

ps s

285.2

8AM

27

A lowpass filter Specifications

Specifications for a discrete-time lowpass filter

4.00 ,01.01|)(|01.01 pjeH

6.0 ,001.0|)(| sjeH

001.0

01.0

2

1


Design the lowpass filter by Kaiser window

Designing by window method indicating , we must set

Transition width

001.0 20

21

Transition width

The two parameters:

Cutoff frequency of the ideal lowpass filter

I l

60log20 10 A

2.0 ps

37 ,653.5 M

5.02/)( psc


Impulse response

otherwise ,0

0 ,)(

])]/)[(1([

)(

)(sin

][ 0

2/120 Mn

I

nI

n

n

nhc

28

Design the lowpass filter

What is the group delay?

M/2=18 5M/2 18.5


Summary Filter design







29

Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

digital signal processing, mm4.ppt - department of...

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