12015-9-171zhongguo liu_biomedical engineering_shandong univ. biomedical signal processing chapter 7...

123/4/19 1Zhongguo Liu_Biomedical Engineering_Shandong U

Biomedical Signal processingChapter 7 Filter Design

Techniques

Zhongguo Liu

Biomedical Engineering

School of Control Science and Engineering, Shandong University

Chapter 7 Filter Design Techniques

7.0 Introduction7.1 Design of Discrete-Time IIR Filters

From Continuous-Time Filters7.2 Design of FIR Filters by Windowing7.3 Examples of FIR Filters Design by

the Kaiser Window Method7.4 Optimum Approximations of FIR

Filters7.5 Examples of FIR Equiripple

Approximation7.6 Comments on IIR and FIR Discrete-

Time Filters

Filter Design Techniques

7.0 Introduction

Frequency-selective filters pass only

certain frequencies

Any discrete-time system that

modifies certain frequencies is called

a filter.

We concetrate on design of causal

Frequency-selective filters

Stages of Filter Design

The specification of the desired properties of the system.

The approximation of the specifications using a causal discrete-time system.

The realization of the system.Our focus is on second stepSpecifications are typically given in

the frequency domain.

Frequency-Selective Filters

Ideal lowpass filter

cjwlp ,0

nwnh c

lp ,sin

Ideal highpass filter

w wH e

w nh n n n

Ideal bandpass filter

others

wwweH ccjw

Ideal bandstop filter

others

wwweH ccjw

If input is bandlimited and sampling frequency is high enough to avoid aliasing, then overall system behave as a continuous-time system:

Linear time-invariant discrete-time system

,effjw w

H H j wT

continuous-time specifications are converted to discrete time specifications by:

Example 7.1 Determining Specifications for a Discrete-Time

FilterSpecifications of the continuous-time filter:1. passband2. stopband

20002001.0101.01 forjH eff

30002001.0 forjH eff

sT 410

22 5000

FilterSpecifications of the continuous-time filter:1. passband2. stopband

20002001.0101.01 forjH eff

30002001.0 forjH eff

sT 410

22 5000

1 0.01

2 0.001

2 (2000)p

2 (3000)s

sT 410T

FilterSpecifications of the discrete-time filter in

1 0.01 2 0.001

2 (2000)p 2 (3000)s

0.4p 0.6s

Filter Design Constraints

Designing IIR filters is to find the approximation by a rational function of z.

The poles of the system function must lie inside the unit circle(stability, causality).

Designing FIR filters is to find the polynomial approximation.

FIR filters are often required to be linear-phase.

Filter Design Techniques

7.1 Design of Discrete-Time IIR

Filters From Continuous-Time

Filters

7.1 Design of Discrete-Time IIR

Filters From Continuous-Time Filters

The traditional approach to the

design of discrete-time IIR filters

involves the transformation of a

continuous-time filter into a

discrete filter meeting prescribed

specification.

Three Reasons

1. The art of continuous-time IIR filter design is highly advanced, and since useful results can be achieved, it is advantageous to use the design procedures already developed for continuous-time filters.

Three Reasons

2. Many useful continuous-time IIR design method have relatively simple closed form design formulas. Therefore, discrete-time IIR filter design methods based on such standard continuous-time design formulas are rather simple to carry out.

Three Reasons

3. The standard approximation

methods that work well for

continuous-time IIR filters do not

lead to simple closed-form

design formulas when these

methods are applied directly to

the discrete-time IIR case.

Steps of DT filter design by transforming a prototype

continuous-time filter

The specifications for the continuous-time filter are obtained by a transformation of the specifications for the desired discrete-time filter.

Find the system function of the continuous-time filter.

Transform the continuous-time filter to derive the system function of the discrete-time filter.

Constraints of Transformation

to preserve the essential properties of the frequency response, the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane. jwezjs

planes planez Im Im

Constraints of Transformation

In order to preserve the property of stability, If the continuous system has poles only in the let half of the s-plane, then the discrete-time filter must have poles only inside the unit circle.

planes Im

planez Im

7.1.1 Filter Design by Impulse Invariance

The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system.

dcd nThTnh

dc TjHif ,0

wjHeHthen

wforTw d

2Relationship of frequencies

relation between frequencies

S plane Z plane

3 / dT

dc TjHif ,0

wjHeHthen

Aliasing in the Impulse Invariance

dc TjHif ,0

wthen H e H j

periodic sampling

[ ] ( ) | ( )c t nT cx n x t x nT

T ： sample period; fs=1/T:sample rateΩs=2π/T:sample rate

s c c cn n

x t x t s t x t t nT x nT t nT

Review

Time domain ： )(tx

Complex frequency domain ：

dtetxsX st)()(

Laplace transformLaplace transform

s pl ane

Relation between Laplace Relation between Laplace Transform and Z-transformTransform and Z-transform

Review

dtetxjX tj)()(

Fourier Transform

frequency domain ：

s-pl ane

Fourier Transform is the Laplace transform Laplace transform

when s have the value only in imaginary axis, s=jΩ

js Since

So 0 s j

dtetxsX st)()(

( ) ( ) ( ) ( ) ( )n n

x n x t t nT x nT t nT

For discrete-time signal ，

( ) ( ) st

x nT t nT e dt

( ) ( )snT sT

x nT e X e

[ ( )] ( ) stx n x n e dt

sTz e令：( ) ( )n

x n z X z

z-transform

of discrete-

time signal

the Laplace transformLaplace transform

( )sT j T T j T jz e e e e re

nznxzX )()(

so ：Tr e

relation betweens zand

Laplace transformLaplace transform continuous time signal

z-transformz-transform discrete-time signalrelation

[ ( )] ( ) ( )snT sT

x n x nT e X e

LsTz elet ：

2 sT f f

DTFT : Discrete Time Fourier Transform

( )sT j T T j T jz e e e e re

( ) ( )j j n

X e x n e

S plane Z plane

3 / sT

1|j jrz re e

2 2s sT f f f

z plane

Re[ ]z

Im[ ]z

0 2 / 2

:sf f f

s pl ane

If input is bandlimited and fs>2fmax , :

discrete-time filter design by impulse invariance

wforTw d

dc TjHif ,0

wthen H e H j w

dcd nThTnh

relation between frequencies

S plane Z plane

3 / dT

dc TjHif ,0

wjHeHthen

periodic sampling

[ ] ( ) | ( )c t nT cx n x t x nT

T ： sample period; fs=1/T:sample rateΩs=2π/T:sample rate

s c c cn n

S j kT

1 1 2*

2 2s c c sk

X j X j S j X j k dT

X j kT

Review

proof of

T ： sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate

S j kT

Review

S j kT

sjk tk

s(t) 为冲击串序列，周期为 T ，可展开傅立叶级数

1sjk t

2 ( )sjk t Fse k

periodic sampling

s c c cn n

x nT e

( ) j ts c

X j x nT t nT e dt

[ ] ( ) | ( )c t nT cx n x t x nT ( )j j n

X e x nT e

( ) ( ) ( )j j Ts TX j X e X e

1( )j T

X e X j kT

1 2( ) c

j kX X j

T T Te

( ) 0,j Tif X eT

1( ) c

jthen X X jT T

1s c s

X j X j kT

discrete-time filter design by impulse invariance

dc TjHif ,0

wthen H e H j w

dcd nThTnh

1 2( ) c

j kX X j

T T Te

( ) cjX X j

1 2( ) c

j kH H j

T T Te

1( ) c

jH H jT T

[ ] ( ) | ( )c t nT cx n x t x nT [ ] ( )ch n h nT

Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.

Transformation from discrete to continuous

In the impulse invariance design procedure, the transformation is

Assuming the aliasing involved in the transformation is neglected, the relationship of transformation is

Continuous-time IIR filters

Butterworth filtersChebyshev Type I filtersChebyshev Type II filtersElliptic filters

Transformation from continuous to discrete

: k ds Tkpole s s z e

two requirements for transformation

nTskdcd nueATnueATthTnh dkdk

Example 7.2 Impulse Invariance with a Butterworth Filter

Specifications for the discrete-time filter:

3.0,17783.0

2.00,189125.0

dd TwTlet 1

Assume the effect of aliasing is negligible

3.0,17783.0

2.00,189125.0

0.89125 1, 0 0.2

0.17783, 0.3

0.2 0.89125

0.3 0.17783

0.30.2

2 20.2 1

10.89125

0.3 11

0.17783

j jH j

2 20.2 1

1 1.258930.89125

7032.0,6 cN

2 20.3 1

1 31.622040.17783

0.25893

30.62204

118.263782

70470.0,8858.5 cN

0.2 0.89125

0.3 0.17783

Example 7.2 Impulse Invariance with a Butterworth

Filter

1Nc c c

cH s H s H s

0,1, , 2 1k N 2 2 11 21 ,k

j N k NNc cs j e

6, 0.7032cN

0.182 0.679,j

0.497 0.497,j

0.679 0.182j

Plole pairs: cH s

4945.03585.14945.09945.04945.03640.0

12093.0222

ssssss

cc c c

cH s H s H s

0.182 0.679,j 0.497 0.497,j 0.679 0.182j Plole pairs: cH s

0.12093

0.182 0.679 0.182 0.679 0.497 0.497cH ss j s j s j

0.497 0.497 0.679 0.182 0.679 0.182s j s j s j

60.7032Nc

1 2 1 2

0.2871 0.4466 2.1428 1.1455

1 1.2971 0.6949 1 1.0691 0.3699

1.8557 0.6303

1 0.9972 0.2570

z z z z

0.12093

0.182 0.679 0.182 0.679 0.497 0.497cH ss j s j s j

0.497 0.497 0.679 0.182 0.679 0.182s j s j s j

1 11 1

N Nd k k

k kk d ks T s

T A AH z

e z e z

Basic for Impulse Invariance

To chose an impulse response for the discrete-time filter that is similar in some sense to the impulse response of the continuous-time filter.

If the continuous-time filter is bandlimited, then the discrete-time filter frequency response will closely approximate the continuous-time frequency response.

The relationship between continuous-time and discrete-time frequency is linear; consequently, except for aliasing, the shape of the frequency response is preserved.

7.1.2 Bilinear Transformation

Bilinear transformation can avoid the problem of aliasing.

Bilinear transformation maps onto

zH z H

Bilinear transformation:

js 221

anyforz 10

1 12 (1 ) 1dT s z z

11 2 ] 1 2d dT s z T s

jsaxisj

1z 1 2

jw j T

planes Im

planez Im

anyforz 10

2tan 2

2 2 sin 2

2cos 2d

2tan2 1dTw

/2 /2 /2

/2 /2 /22 ( )

jw jw jw

jw jw jwT

e e ee e e

prewarp

tan2)()(

cj jHeH

relation between frequency response of Hc(s), H(z)

Comments on the Bilinear Transformation

It avoids the problem of aliasing encountered with the use of impulse invariance.

It is nonlinear compression of frequency axis.

S plane Z plane

3 / dT

2tan2 1dTw

Comments on the Bilinear Transformation

The design of discrete-time filters using bilinear transformation is useful only when this compression can be tolerated or compensated for, as the case of filters that approximate ideal piecewise-constant magnitude-response characteristics.

Bilinear Transformation of

2tan 2

Comparisons of Impulse Invariance and Bilinear

Transformation

The use of bilinear transformation is restricted to the design of approximations to filters with piecewise-constant frequency magnitude characteristics, such as highpass, lowpass and bandpass filters.

Impulse invariance can also design lowpass filters. However, it cannot be used to design highpass filters because they are not bandlimited.

Comparisons of Impulse Invariance and Bilinear

Transformation

Bilinear transformation cannot design filter whose magnitude response isn’t piecewise constant, such as differentiator. However, Impulse invariance can design an bandlimited differentiator.

Butterworth Filter, Chebyshev Approximation, Elliptic Approximation

7.1.3 Example of Bilinear Transformation

6.0,001.0

4.0,01.199.0

0.0160.01

Example 7.3 Bilinear Transformation of a Butterworth Filter

0.89125 1, 0 0.2

0.17783, 0.3

2 0.20.89125 1, 0 tan

2 0.30.7783, tan

, 1dFor convenience we choose T

,17783.015.0tan2

,89125.01.0tan2

,17783.015.0tan2

,89125.01.0tan2

2 tan 0.1 11

0.89125

3tan 0.15 11

0.17783

305.5N

0.0160.01

Locations of Poles

0,1, , 2 1k N 2 2 11 21 ,k

j N k NNc cs j e

0.1998 0.7401,j

0.5418 0.5418,j

0.7401 0.1998j

Plole pairs: cH s

1Nc c c

cH s H s H s

6, 0.766cN

5871.04802.15871.00836.15871.03996.0

20238.0222

ssssss

2155.09904.01

3583.00106.117051.02686.11

10007378.0

cc c c

cH s H s H s

0.1998 0.7401,j 0.5418 0.5418,j 0.7401 0.1998j Plole pairs: cH s

Ex. 7.3 frequency response of discrete-time filter

Example 7.4 Butterworth Approximation (Hw)

6.0,001.0

4.0,01.199.014Norder

Example 7.4 frequency response

Chebyshev filters

C Chebyshev filter (type I)

1|)(| 22

)coscos()( 1 xNxVN

Chebyshev polynomial

Chebyshev filter (type II)

)]/([1

Example 7.5 Chebyshev Type I , II Approximation

6.0,001.0

4.0,01.199.08Norder

Type I Type II

Example 7.5 frequency response of Chebyshev

Type I Type II

elliptic filters

Elliptic filter

2Jacobian elliptic function

Example 7.6 Elliptic Approximation

6.0,001.0

4.0,01.199.06Norder

Example 7.6 frequency response of Elliptic

*Comparison of Butterworth, Chebyshev, elliptic filters: Example

-Given specification

0.4| | .011 |)(| 99.0 jeH ||0.6 0.001 |)(| jeH

6.0 ,4.0 001.0 ,01.0 s21 p

-Order

Butterworth Filter : N=14. ( max flat)

Chebyshev Filter : N=8. ( Cheby 1, Cheby 2)

Elliptic Filter : N=6 ( equiripple)

-Pole-zero plot (analog)

-Pole-zero plot (digital)

B C1 C2 E

(14) (8)

-Magnitude -Group delay

4.0 6.0 4.0 6.0

7.2 Design of FIR Filters by Windowing

FIR filters are designed based on directly approximating the desired frequency response of the discrete-time system.

Most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint.

Window MethodAn ideal desired frequency response

jwd enheH

dweeHnh jwnjw

Many idealized systems are defined by piecewise-constant frequency response with discontinuities at the boundaries. As a result, these systems have impulse responses that are noncausal and infinitely long.

cjwlp ,0

sin clp

w nh n

Window Method

otherwise

Mnnhnh d

nwnhnh d

otherwise

deWeHeH wjjwd

The most straightforward approach to obtaining a causal FIR approximation is to truncate the ideal impulse response.

Windowing in Frequency Domain

Windowed frequency response

deWeH21

eH jjd

The windowed version is smeared version of desired response

Window Method

2j wjw j jw

d dH e H e W e d H e

If nnw 1

jwnjw kwenweW 22

0 5 10510 1515

24 4 6

Choice of Window is as short as possible in duration.

This minimizes computation in the implementation of the filter.

approximates an impulse. jweW

Mjw jwn jwn

W e w n e e

2sin 1 21

1 sin 2

jw MjwM

otherwise

Window Method

jw eHdeWeHeH

then would look like , except where changes very abruptly.

jweH jwd eH

jwd eH

nw jweW0w

If is chosen so that is concentrated in a narrow band of frequencies around

jwdH e

Rectangular Window for the rectangular window has a

generalized linear phase. jweW

As M increases, the width of the “main lobe” decreases. 4 1

While the width of each lobe decreases with M, the peak amplitudes of the main lobe and the side lobes grow such that the area under each lobe is a constant.

2sin 1 2

sin 2jw jwM

w MW e e

Rectangular Window will oscillate at the

discontinuity.

deWeH wjjwd

The oscillations occur more rapidly, but do not decrease in magnitude as M increases.

The Gibbs phenomenon can be moderated through the use of a less abrupt truncation of the Fourier series.

Rectangular WindowBy tapering the window smoothly to zero

at each end, the height of the side lobes can be diminished.

The expense is a wider main lobe and thus a wider transition at the discontinuity.

7.2 Design of FIR Filters by Windowing Method

To design an ilowpass FIR Filters

cjwlp ,0

sin clp

w nh n

Review

dh n h n w n 1, 0

n Mw n

otherwise

deWeHeH wjjwd

2cw n M

7.2.1 Properties of Commonly Used Windows

Rectangular

otherwise

Bartlett (triangular)

Hanning

otherwise

MnMnnw

0,2cos5.05.0

otherwise

MnMnnw

0,2cos46.054.0

Hamming

Blackman

otherwise

,00,4cos08.0

2cos5.042.0

Frequency Spectrum of Windows

(a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman , (M=50)

(a)-(e) attenuation of sidelobe increases, width of mainlobe increases.

biggest ， high oscillations at discontinuity

smallest ， the sharpest transition

Table 7.1

7.2.2 Incorporation of Generalized Linear Phase

In designing FIR filters, it is desirable to obtain causal systems with a generalized linear phase response.

otherwise

MnnMwnw

The above five windows are all symmetric about the point ,i.e.,2M

Their Fourier transforms are of the form

2Mjwjwe

jw eeWeW woffunctionevenandrealaiseW jw

causalnwnhnh d :

d d dif h M n h n h n h n w n

2Mjwjwe

jw eeAeH

:h M n h n generalized linear phase

phaselineardgeneralizenhnMh

nwnhnhnhnMhif ddd

2Mjwjwo

jw eejAeH

Frequency Domain Representation

nhnMhif dd 2Mjwjwe

jwd eeHeH

j wjH e H W de e

j wjw jwewhere A H W de e e

j w j w Mj j MH W de e e e

2Mjwjwe

jw eeWeW w n w M n

2jw jwMeA e e

dh n h n w n

Example 7.7 Linear-Phase Lowpass Filter

The desired frequency response is

jwlp ,0

lph M n

Mnwnh c

jwM jwnh n dw

w n Mfor n

pjw 01

magnitude frequency response

1 0jwp p

H e w w

s pw w w

1020logp p

0.1 0.15jwH e w

1 0.05 0 0.25jwH e w

0.1s pw w w 1020log 0.05 26p dB

1020logs s

20s dB

smallest ， the sharpest transition

biggest ， high oscillations at discontinuity

7.2.3 The Kaiser Window Filter Design Method

I nn Mw n

otherwise

2,where M

0 :I u zero order modified Bessel function of the first kind

: 1,length M :parametershape:two parameters

Trade side-lobe amplitude for main-lobe width

Figure 7.24

As increases, attenuation of sidelobe increases, width of mainlobe increases.

As M increases, attenuation of sidelobe is preserved, width of mainlobe decreases.

(a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6

Table 7.1

Transition width is a little less than mainlobe width

Increasing M wile holding constant causes the main lobe to decrease in width, but does not affect the amplitude of the side lobe.

ComparisonIf the window is tapered more, the side lobe

of the Fourier transform become smaller, but the main lobe become wider.

Filter Design by Kaiser Window

pjw 01

ps www 10log20A

Filter Design by Kaiser Window

ps www 10log20A

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

2285.2

I nn Mw n

otherwise

Example 7.8 Kaiser Window Design of a Lowpass Filter

6.0,001.0

4.0,01.199.0

ps www 10log20A

I nw n

n Mn I

otherwise

5.182 Mwhere

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

6.0,001.0

4.0,01.199.0

001.0,min,001.0,01.0

,6.0,4.0

w wcutoff frequency w

2 :step

3:step100.2 20log 60

0.5653 37

s pw w w A

0.5653

5.182 Mwhere 2

I nw n

n Mn I

otherwise

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

0.2 20log 60s p

w w w A

I nw n

h n n Mn I

Ex. 7.8 Kaiser Window Design of a Lowpass Filter

7.3 Examples of FIR Filters Design by the Kaiser Window

MethodThe ideal highpass filter with

generalized linear phase

cjwhp ,

Mjwjwhp eHeeH 2

Mnnh c

hph n h n w n

Example 7.9 Kaiser Window Design of a Highpass Filter

Specifications:

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

Example 7.9 Kaiser Window Design of a Highpass Filter

Specifications:

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

7.3.2 Discrete-Time Differentiator

wejweH Mjwjwdiff ,2

Mnnhdiff ,

nwnhnh diff

phaselineardgeneralizeIVtypeorIIItypenMhnh :

Example 7.10 Kaiser Window Design of a Differentiator

Since kaiser’s formulas were developed for frequency responses with simple magnitude discontinuities, it is not straightforward to apply them to differentiators.

Suppose 4.210 M

Group Delay

Phase:

Group Delay:

samplesM

Group Delay

Phase:

Group Delay:

Noninteger delay

samplesM

7.4 Optimum Approximations of FIR Filters

Goal: Design a ‘best’ filter for a given MIn designing a causal type I linear phase

FIR filter, it is convenient first to consider the design of a zero phase filter.

Then insert a delay sufficient to make it causal.

nhnh ee

2, MLenheAL

functionperiodicevenrealwnnhheAL

jwe ,,:cos20

.2 samplesMLbyitdelaying

bynhfromobtainedbecansystemcausalA e

nMhMnhnh e 2

2Mjwjwe

jw eeAeH

Designing a filter to meet these specifications is to find the (L+1) impulse response values

Lnnhe 0,

In Packs-McClellan algorithm,

is fixed, and is variable. 21,,, andwwL sp

Packs-McClellan algorithm is the dominant method for optimum design of FIR filters.

wnwTwn n coscoscoscoscos 1 1coscos0coscos0cos 1

0 wwTw

wwwTw coscoscos1coscos1cos 11

1cos2cos2cos 22 wwTw

coscoscos2

coscos

cos3cos4cos1cos2cos2

cos2coscos23cos32

xaxPwhere

xPwawnnhheA

coscos20

functionweightingtheiswWwhere

eAeHwWwE

functionerrorionapproximatanDefinejw

pjwd ,0

Minimax criterion

Within the frequency interval of the passband and stopband, we seek a frequency response that minimizes the maximum weighted approximation error of

jwe eA

jwd eAeHwWwE

FwLnnhe maxmin

Other criterions

00:1 min dwwEH

0:2 min dwwEH

FwLnnhe maxmin0:

Let denote the closet subset consisting of the disjoint union of closed subsets of the real axis x.

Alternation Theorem

is an r th-order polynomial.

kk xaxP

denotes a given desired function of x that is continuous on

pF is a positive function, continuous on pF xWP

The weighted error is xPxDxWxE PPP

xEE PFx P

The maximum error is defined as

Alternation Theorem

A necessary and sufficient condition that be the unique rth-order polynomial that minimizes is that exhibit at least (r+2) alternations; i.e., there must exist at least (r+2) values in such that

xPE xEP

221 rxxx

ExExE iPiP 1

1,,2,1 ri

and such that

Example 7.11 Alternation Theorem and Polynomials

Each of these polynomials is of fifth order.

The closed subsets of the real axis x referred to in the theorem are the regions

11.01.01 xandx

7.4.1 Optimal Type I Lowpass Filters

For Type I lowpass filter

The desired lowpass frequency response

Weighting function

kk wawP

coscos

wwwwwD

ppp coscos1,0

01coscos,1cos

wwwwKwW

coscos1,1

01coscos,1

The weighted approximation error is

The closed subset is

wPwDwWwE PPP coscoscoscos

wwandww sp0

sp wwandww cos11coscos

The alternation theorem states that a set of coefficients will correspond to the filter representing the unique best approximation to the ideal lowpass filter with the ratio fixed at K and with passband and stopband edge and if and only if exhibits at least (L+2) alternations on , i.e., if and only if alternately equals plus and minus its maximum value at least (L+2) times.

Such approximations are called equiripple approximations.

pwsw )(coswEP

)(coswEP

The alternation theorem states that the optimum filter must have a minimum of (L+2) alternations, but does not exclude the possibility of more than (L+2) alternations.

In fact, for a lowpass filter, the maximum possible number of alternations is (L+3).

Because all of the filters satisfy the alternation theorem for L=7 and for the same value of , it follows that and/or must be different for each ,since the alternation theorem states that the optimum filter under the conditions of the theorem is unique.

Property for type I lowpass filters from the alternation

theorem

The maximum possible number of alternations of the error is (L+3)

Alternations will always occur at andAll points with zero slop inside the

passband and all points with zero slop inside stopband will correspond to alternations; i.e., the filter will be equiripple, except possibly at and

7.4.2 Optimal Type II Lowpass Filters

For Type II causal FIR filter:The filter length (M+1) is even, ie, M is oddImpulse response is symmetricThe frequency response is

Mnnh 0

21,,2,1,212

MnnMhnbwhere

wnheeH

wnnbwnwnb

wPweeH Mjwjw cos2cos2

21cos0

MLandwawPwhereL

nMhnbnbnbafind k 212~

For Type II lowpass filter,

wwwwDeH

0,2cos

7.4.3 The Park-McClellan Algorithm

From the alternation theorem, the optimum filter will satisfy the set of equation

2,,2,11 1 LieAeHwW ijwe

jwe eA

wxwhere

Guessing a set of alternation frequencies

1 cos,1

bwhere

2,,2,1 Liforwi slpl wwww 1,and

cos,cos

kii ik

wxxxxd

CxxdwPeA

For equiripple lowpass approximation

Filter length: (M+1)

ps wwwwherew

13log10 2110

7.5 Examples of FIR Equiripple Approximation

7.5.1 Lowpass Filter

6.0,001.0

4.0,01.199.0

errorionapproximatunweighted

Comments

M=26, Type I filterThe minimum number of

alternations is (L+2)=(M/2+2)=157 alternations in passband and 8

alternations in stopbandThe maximum error in passband and

stopband are 0.0116 and 0.0016, which exceed the specifications.

7.5.1 Lowpass Filter

M=27, , Type II filter, zero at z=-1The maximum error in passband and

stopband are 0.0092 and 0.00092, which exceed the specifications.

The minimum number of alternations is (L+2)=(M-1)/2+2=15

7 alternations in passband and 8 alternations in stopband

Comparison

Kaiser window method require M=38 to meet or exceed the specifications.

Park-McClellan method require M=27Window method produce

approximately equal maximum error in passband and stopband.

Park-McClellan method can weight the error differently.

7.6 Comments on IIR and FIR Discrete-Time Filters

What type of system is best, IIR or FIR?

Why give so many different design methods?

Which method yields the best result?

7.6 Comments on IIR and FIR Discrete-Time Filters

Closed-Form

Formulas

Generalized Linear Phase

IIR Yes No Low

FIR No Yes High

Their Fourier transforms are concentrated around

They have a simple functional form that allows them to be computed easily.

The Fourier transform of the Bartlett window can be expressed as a product of Fourier transforms of rectangular windows.

The Fourier transforms of the other windows can be expressed as sums of frequency-shifted Fourier transforms of rectangular windows.(Problem7.34)

Homework

Simulate the frequency response (magnitude and phase) for Rectangular, Bartlett, Hanning, Hamming, and Blackman window with M=21 and M=51

23/4/19152Zhongguo Liu_Biomedical Engineering_Shandong U

Chapter 5 HW7.2, 7.4, 7.15,

12015-9-171zhongguo liu_biomedical engineering_shandong univ. biomedical signal processing chapter 7...

discrete filter

time iir filters

fir discretetime filters

design of discrete

continuoustime system

filter design techniques7

design of fir filters

causal discretetime

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