contents- 9 장 추가 introduction preliminaries analog-to-digital filter transformations ...

Contents- 9Contents- 9 장 추가장 추가

Introduction

Preliminaries

Analog-to-Digital Filter Transformations

LowPass Filter Design Using Matlab

Frequency-Band Transformations

Comparison of FIR vs. IIR Filters

Contents

Introduction( 서론 )

Preliminaries( 예비 사항 )

- Absolute Specifications( 절대 사양 )

- Relative Specifications( 상대 사양 )

Properties of Linear-Phase FIR Filters( 선형 위상 필터의 특성 )

- Impulse Response( 임펄스 응답 )

- Frequency Response( 주파수 응답 )

- Zero Locations( 영점 위치 )

Introduction

디지털 신호 처리의 두 가지 시스템 형태

- Digital Filter( 디지털 필터 ): 시간 영역에서 신호 여과를 하는 시스템

- Spectrum Analyzer( 스펙트럼 분석기 ): 주파수 영역에서 신호를 표현

하는 시스템

Frequency Selective Type( 주파수 선택적 형태 )

- FIR 필터와 IIR 필터는 대부분 주파수 선택적 형태를 갖음 .

- 주로 다대역 (multi band), 저역통과 (low pass), 고역통과 (high pass)

대역통과 (band pass) 필터들을 설계

Introduction( 서론 ) - 1

Preliminaries - Absolute Specifications( 절대 사양 )

- Relative Specifications( 상대 사양 )

Preliminaries( 예비 사항 - 1)

디지털 필터의 설계 단계

- Specification( 사양 ): 필터를 설계하기 전에 몇 가지 사양을 가져야 함 .

ex) 정지대역의 끝 : 50dB, 통과대역의 끝 : 1dB

- Approximation( 근사 ): 사양 결정 후 , 사양을 근사시키는 작업 , 즉 , 주어

진 사양을 필터 표현으로 구현 .( 차분방정식 , 시스템 함수 , 또는 임펄스

응답의 형태를 갖는 필터 표현 )

- Implementation( 구현 ): 위 단계의 결과를 하드웨어나 컴퓨터 소프트웨 어로 구현


Absolute Specifications( 절대 사양 ) 과 Relative Specifications( 상대 사양 )

- 은 이상적인 통과대역 응답에서 감수할 수 있는 허용오차 ( 또는 리플 )- 은 이상적인 정지대역 응답에서 감수할 수 있는 허용오차 ( 또는 리플 )

1

2

- 는 dB 로 나타낸

통과대역 리플

- 는 dB 로 나타낸 정지대역 감쇠

pR

sA

(a) 절대 사양

(b) 상대 사양


예제 7.1

- 어떤 필터의 사양에서 통과대역 리플은 0.25dB 이고 , 정지대역 감쇠는 50dB 일때

, 을 구하라 .

Sol) 즉 , 절대 사양을 알 때 , 상대 사양에서의 허용오차를 구하라 . 이므로 ,

통과대역 리플 , 정지대역 리플 pR sA

)0(01

1log20

1

110

pR )1(01

log201

210

sA

1 2

25.01

1log20

1

110

pR

0144.0

9716.0101

1

0125.020

25.0

1

1log

1

0125.0

1

1

1

110

예제 7.2

- 통과대역 허용오차 =0.01, 정지대역 허용오차는 =0.001 이 주어질 때 , 통과

대역 리플 와 저지대역 감쇠 를 구하라 .

Sol) 즉 , 절대 사양을 알 때 , 상대 사양에서의 허용오차를 구하라 .

1

또한 , 이므로

0032.0

5.20144.01

log

500144.01

log20

501

log20

2

210

210

1

210

sA

2

pR sA

dBRp 1737.001.01

01.01log20 10

dBAs 09.4001.01

001.0log20 10


Analog-to-Digital Filter Transformations

OverviewOverview

Complex-Valued Mapping

Impulse Invariance Transformation

Finite Difference Approximation Transformation

Step Invariance Transformation

Bilinear Transformation

We sample at some sampling interval to obtain

Impulse Invariance TransformationImpulse Invariance Transformation

Sampling Operation : the analog and digital frequencies are related

by

Since on the unit circle and (analog frequency) on the

imaginary axis, we have the following transformation from the s-plane to

the z-plane:

The system function and are related through the

frequency-domain aliasing formula

)(tha T )(nh

)()( nThnh a

T or Tjj ee

sTez

jez js

)(zH )(sH a

k

a kTjsH

TzH )

2(

1)(

1. Given digital frequency, choose and determine the analog

frequencies

Design ProcedureDesign Procedure

2. Design an analog filter using the specifications

and

3. Using partial fraction expansion, expand into

T

Tand

Ts

sp

p

N

kTpk

ze

RzH

k1

11)(

)(sH a,p

sA

,s pR

)(sH a

4. Transform analog poles into digital poles to obtain the

digital filter

}{ kp }{ Tpke

N

k k

ka ps

RsH

1

)(

Example 1Example 1

Transform

Into a digital filter using the impulse invariance technique

in which =0.1

65

1)(

2

ss

ssH a

)(zH

T

We first expand using partial fraction expansion :

(Solution)

)(sH a

2

1

3

2

65

1)(

2

ssss

ssH a

31 p 22 p

T

21

1

1213 6065.05595.11

8966.01

1

1

1

2)(

zz

z

zezezH

TT

The poles are at and . Then from (8.25) and using

=0.1, we obtain

Example 2Example 2

Demonstrate the use of the imp_invr function on the system

function from Example 1

(Solution)

b = 1.0000 -0.8966

a = 1.0000 -1.5595 0.6065

Example 3Example 3

Design a lowpass digital filter using a Butterworth prototype to

satisfy,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

The desired filter is a 6th-order butterworth filter whose system

function is given in the parallel form

21

1

21

1

3699.00691.11

1454.11428.2

257.09973.01

6304.08587.1)(

zz

z

zz

zzH

)(zH

21

1

6949.02972.11

4463.02871.0

zz

z

Example 4Example 4

Design a lowpass digital filter using a Chebyshev-I prototype to

satisfy,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

The desired filter is a 4th-order Chebyshev-I filter whose system

function is

21

1

21

1

6549.05658.11

0239.00833.0

8392.04934.11

0246.00833.0)(

zz

z

zz

zzH

)(zH

Example 5Example 5

Design a lowpass digital filter using a Chebyshev-II prototype to

satisfy,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

Example 6Example 6

Design a lowpass digital filter using an elliptic prototype to

satisfy,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

Bilinear TransformationBilinear Transformation

This mapping is the best transformation method; it involves a

well-known function given by

2/1

2/1

1

121

1

sT

sTz

z

z

Ts

Another name for this transformation is the linear fractional

transformation because when cleared of fractions, we obtain

0122

zsT

szT

Which is linear in each variable if the other is fixed, or bilinear in s

and z.

Substituting in (8.27), we obtain 0

jeT

j

Tj

z

21

21

since the magnitude is 1. Solving for digital frequency as a function of analog frequency , we obtain

2tan

2

2tan2 1

Tor

T

This shows that is nonlinearly related to (or warped into) but that there is no aliasing. Hence in (8.28) we will say that is prewarped into

Example 6Example 6

Transform into a digital filter using the bilinear

transformation. Choose =165

1)(

2

ss

ssH a

T

Using (8.26), we obtain

(Solution)

1

1

1

1

1

1

12

1

12)(

z

zH

z

z

THzH a

T

a

61125

112

1112

1

12

1

1

1

1

zz

zz

zz

Simplifying,

2

21

1

21

2.01

05.01.015.0

420

23)(

z

zz

z

zzzH

Example 7Example 7

Transform the system function in Example 1 using the

bilinear function

)(sH a

(Solution)

2

21

2.01

05.01.015.0)(

z

zzzH

1. Choose a value for . This is arbitrary, and we may set


2. Prewarp the cutoff frequencies and ; that is, calculate

and using (8.28) :

T

1

1

1

12)(

z

z

THzH a

p p

s

4. Finally, set

and simplify to obtain as a rational function in

1T

s

2tan

2,

2tan

2 ss

pp TT

3. Design an analog filter to meet the specification and . We

have already described how to do this in the previous section.

)(sH a,p

sA

,s pR

1z)(zH

Example 8Example 8

Design the digital Butterworth filter of Example 3. The

specifications are,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

)7149.03143.11)(3753.00541.11)(2342.09459.01(

)1(00057969.0)(

212121

61

zzzzzz

zzH

Example 9Example 9

Design the digital Chebyshev-I filter of Example 4. The

specifications are ,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

)6493.05548.11)(8482.04996.11(

)1(0018.0)(

2121

41

zzzz

zzH

Example 10Example 10

Design the digital Chebyshev-II filter of Example 5. The

specifications are ,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

)7183.01325.11)(1503.04183.01(

)0671.11)(5574.01(1797.0)(

2121

2121

zzzz

zzzzzH


Design the digital elliptic filter of Example 6. The specifications are

,2.0 p

(Solution)

dBRp 1

,3.0 s dBAs 15

)6183.01)(8612.04928.11(

)1)(4211.11(1214.0)(

121

121

zzz

zzzzH

Lowpass Filter Design Using Matlab

1. [b, a] = butter(N, wn)

2tan

2 1 Tcn

2. [b, a] = cheby1(N, Rp, wn)

/pn

3. [b, a] = cheby2(N, As, wn)

/sn

4. [b, a] = ellip(N, Rp, As, wn)

/pn

,2.0 p dBRp 1

,3.0 s dBAs 15


Digital Butterworth lowpass filter design :

(Solution)

)7149.03143.11)(3753.00541.11)(2342.09459.01(

)1(00057969.0)(

212121

61

zzzzzz

zzH


Digital Chebyshev-I lowpass filter design :

(Solution)

)6493.05548.11)(8482.04996.11(

)1(0018.0)(

2121

41

zzzz

zzH

Example14Example14

Digital Chebyshev-II lowpass filter design :

(Solution)

)7183.01325.11)(1503.04183.01(

)0671.11)(5574.01(1797.0)(

2121

2121

zzzz

zzzzzH


Digital elliptic lowpass filter design :

(Solution)

)6183.01)(8612.04928.11(

)1)(4211.11(1214.0)(

121

121

zzz

zzzzH

Comparison of Three FiltersComparison of Three Filters

,2.0 p dBRp 1

,3.0 s dBAs 15

This comparison in terms of order N and the minimum stopband

attenuations is shown in Table 8.1.

Prototype Order N Stopband Att.

Butterworth 6 15

Chebyshev-I 4 25

Elliptic 3 27

Frequency-Band Transformations

1

21

1

A

1

p s

)( jeH

0

1

21

1

A

1

ps

)( jeH

0

1

21

1

A

1

1s

1p

)( jeH

02p

2s

1

21

1

A

1

1p

1s

)( jeH

02s 2p

Lowpass Highpass

Bandpass Bandstop

Figure 8.20 Specifications of frequency-selective filters

Typical specifications for most commonly used types of frequency-selective digital

filters are shown below

Let be the given prototype lowpass digital filter

Let be the desired frequency-selective digital filter

using two different frequency variables, and , with

and , respectively.

Define a mapping of the form

Such that

)(ZH LP

)(zH

Z z LPH H

)( 11 zGZ

)( 11)()(

zGzLP ZHzH

1. G( ) must be a rational function in so that is implementable 1z )(zH

2. The unit circle of the Z-plane must map onto the unit circle of the z-plane

3. For stable filters, the inside of the unit circle of the Z-plane must also map onto the inside of the unit circle of the z-

plane

Let and be the frequency variables of Z and z

and on their respective unit circles

Then requirement 2 above implies that

''jeZ

1)()( 11 jeGzGZ

jez

and)(' )(

jeGjjj eeGe

or

)(' jeG The general form of the function G( ) that satisfies the above

requirements is a rational function of the all-pass type given by

n

k k

k

z

zzGZ

11

111

1)(

Where for stability and to satisfy requirement 31k

1

11

1

z

zZ

1

11

1

z

zZ

111

22

21

12

1

zz

zzZ

111

22

21

12

1

zz

zzZ

c'

]2/)'sin[(]2/)'sin[(

cc

cc

c'

]2/)'cos[(]2/)'cos[(

cc

cc

)1/(21 KKu

)1/()1(2 KK

]2/)cos[(

]2/)cos[(

u

u

2tan

2cot cuK

cutoff frequency of new filter

cutoff frequency of new filter

lower cutoff frequency

upper cutoff frequency

)1/(21 KKu

)1/()1(2 KK

]2/)cos[(

]2/)cos[(

u

u

2tan

2cot cuK

lower cutoff frequency

upper cutoff frequency

Type of Transformation Transformation Parameters

Lowpass

Highpass

Bandpass

Bandstop


In Example 13 we designed a Chebyshev-I lowpass filter with

specifications

and determined its system function

Design a highpass filter with the above tolerances but with

passband beginning at

,2.0' p dBRp 1

,3.0' s dBAs 15

)6493.05548.11)(8482.04996.11(

)1(0018.0)(

2121

41

zzzz

zzH LP

,6.0 p

(Solution)

From Table 8.2

38197.0]2/)'cos[(]2/)'cos[(

cc

cc

Hence

1

1

38197.01

38197.0)()(

z

zzLP ZHzH

)4019.00416.11)(7657.05661.01(

)1(02426.02121

41

zzzz

z

which is the desired filter


Use the zmapping function to perform the lowpass-to-highpass

transformation in Example 16

(Solution)

)4019.00416.11)(7647.05661.01(

)1(0243.0)(

2121

41

zzzz

zzH

The system function of the highpass filter is

Which is essentially identical to that in Example 8.25


Use the highpass filter of Example 17 as an example

The passband-edge frequencies were transformed using the parameter

Let and determine from using the formula from Table 8.2.

38197.0]2/)3.0cos[(

]2/)3.0cos[(

s

s

38197.0?s

4586.0s 2.0' p p

Now can be determined from and

Where

and , or

Continuing our highpass filter example, let and be the band-edge frequencies. Let us choose . Then

from (8.30), and from (8.31)

as expected.

11 z

e sj

s

sjeZ '

38197.0

6.0p

s'

1

1

1

z

zZ

sjeZ 4586.0s

2.0' p

3.038197.01

38197.038197.0

4586.0

j

j

s e

e


Design a highpass digital filter to satisfy

Use the Chebyshev-I prototype.

,6.0 p

(Solution)

dBRp 1,4586.0 s dBAs 15

)4019.00416.11)(7647.05661.01(

)1(0243.0)(

2121

41

zzzz

zzH

The system function is

Which is identical to that in Example 8.26

Matlab ImplementationMatlab Implementation

[b, a] = BUTTER(N, wn, ‘high’) designs an Nth-order highpass filter with

digital 3-dB cutoff frequency wn in units of

[b, a] = BUTTER(N, wn,) designs an order 2N bandpass filter if wn is a two-

element vector, wn=[w1 w2], with 3-dB passband w1 < w < w2 in units of

[b, a] = BUTTER(N, wn, ‘stop’) is an order 2N bandstop filter if wn=[w1, w2]

with 3-dB stopband w1 < w < w2 in units of

[N, wn]=buttord(wp, ws, Rp, As)

The parameters wp and ws have some restrictions, depending on the type of filter:

For lowpass filter wp < ws

For highpass filter wp < ws

For bandpass filter wp and ws are two-element vectors, wp=[wp1, wp2] and ws=[ws1,

ws2], such that ws1 < wp1 < wp2 < ws2

For bandstop filters wp1 < ws1 < ws2 < wp2

Now using the buttord function in conjunction with the butter function, we can

design any Butterworth IIR filter


In this example we will design a Chebyshev-I highpass filter

whose specifications were given in Example 18.

(Solution)

)4019.00416.11)(7647.05661.01(

)1(0243.0)(

2121

41

zzzz

zzH

The cascade form system function

Is identical to the filter designed in example 8.27


In this example we will design an elliptic bandpass filter whose

specifications are given in the following Matlab script :

(Solution)

The designed filter is an 8th-order filter

)9399.05963.01)(7929.02774.01)(7929.02774.01)(9399.05963.01(

)1(0197.0)(

21212121

81

zzzzzzzz

zzH


Finally we will design a Chebyshev-II bandstop filter whose

specifications are given in following Matlab script.

(Solution)

The cascade form system function

)7602.089361)(3916.047131)(2145.02132.01)(4614.08901.01)(8031.03041.11(

)1(1558.0)(

2121212121

101

zzzzzzzzzz

zzH

contents- 9 장 추가 introduction preliminaries analog-to-digital filter transformations ...

Documents