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Butterworth Filter

Spring 2009© Ammar Abu-Hudrouss -Islamic

University Gaza

Slide ٢Digital Signal Processing

What are the function of Filters ?

Filters can be classified according to range of signal frequencies in the passband

Lowpass filter

Highpass filter

Bandpass filter

Stopband (bandreject) filter

A filter is a system that allow certain frequency to pass to its output and reject all other signals

Filter types

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Slide ٣Digital Signal Processing

Filter types

Slide ٤Digital Signal Processing

Filter types according to its frequency response

Butterworth filterChebychev I filter

Chebychev II filter Elliptic filter

Filter types

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Slide ٥Digital Signal Processing

Butterworth filter

Ideal lowpass filter is shown in the figure

The passband is normalised to one.

Tolerance in passband and stopband are allowed to enable the construction of the filter.

Slide ٦Digital Signal Processing

Lowpass prototype filterLowpass prototype filter: it is a lowpass filter with cutoff frequency p=1.

Lowpass prototype filter

Frequency Transformation

Lowpass filter

Highpass filter

Bandpass filter

Bandreject filter

The frequency scale is normalized by p. We use = / p.

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Slide ٧Digital Signal Processing

Lowpass prototype filterNotationIn analogue filter design we will use

s to denote complex frequency

to denote analogue frequency

p to denote complex frequency at lowpass prototype frequencies.

to denote analogue frequency at the lowpass prototype frequencies.

Slide ٨Digital Signal Processing

Magnitude Approximation of Analog Filters

The transfer function of analogue filter is given as rational function of the form

The Fourier transform is given by

nmsdsdsddscscsccsH nn

mmo

2

210

221

n

nn

mm

mo

js djdjddcjcjccsHH

2

210

221)(

jejHH )(

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Slide ٩Digital Signal Processing

Magnitude Approximation of Analog Filters

Analogue filter is usually expressed in term of

Example Consider the transfer function of analogue filter, find

jHjHjH *2

22

12

ssssH

jsjs sss

ssssHsHjH

221

221

222

42

12 4

2

jH

2

jHjH

Slide ١٠Digital Signal Processing

In order to approximate the ideal filter

1) The magnitude at = 0 is normalized to one

2) The magnitude monotonically decreases from this value to zero as ∞.3) The maximum number of its derivatives evaluated at = 0 are zeros.

This can be satisfied if

Butterworth filter

nn

mmo

DDDCCCCH 22

44

22

22

44

222

1)(

Will have only even powers of , or

NND

H 22

2

11)(

2jH

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Slide ١١Digital Signal Processing

The following specification is usually given for a lowpassButterworth filter is

1) The magnitude of H0 at = 0

2) The bandwidth p.

3) The magnitude at the bandwidth p.

4) The stopband frequency s.

5) The magnitude at the stopband frequency s.

6) The transfer function is given by

Butterworth filter

NNDHH 22

02

1)(

Slide ١٢Digital Signal Processing

To achieve the equivalent lowpass prototype filter

1) We scale the cutoff frequency to one using transformation = / p.

2) We scale the magnitude to 1 to one by dividing the magnitude by H0 .

The transfer function become

We denotes D2N as 2 where is the ripple factor, then

Butterworth filter

NND

H 2'2

2

11)(

NH 22

2

11)(

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Slide ١٣Digital Signal Processing

If the magnitude at the bandwidth = p = 1 is given as (1 - p)2

or −Ap decibels,

the value of 2 is computed by

If we choose Ap = -3dB 2 = 1. this is the most common case and gives

Butterworth filter

pAHp

2)(log20 2

1

pA 211log10

110 1.02 pA

NH 22

11)(

Slide ١٤Digital Signal Processing

If we use the complex frequency representation

The poles of this function occurs at

Or in general

Poles occurs in complex conjugates

Poles which are located in the LHP are the poles of H(s)

Butterworth filter

Njpp

HpH2/

22

11)(

even,2,...,2,1odd,2,...,2,1

2/12

2/2

nNkenNke

pNk

Nk

k

Nkep NNkjk 2,...,2,12/)12(

Nkep NNkjk ,...,2,12/)12(

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Slide ١٥Digital Signal Processing

When we found the N poles we can construct the filter transfer function as

The denominator polynomial D (p) is calculated by

Butterworth filter

pDpH 1

n

kkpppD

1

Slide ١٦Digital Signal Processing

Butterworth filter

Another method to calculate D (p ) using

The coefficients dk is calculated recursively where d0 = 1

k

kpppD )(

nn pdpdpdpD 2

211)(

Nkd

Nk

kd kk ,,3,2,1

2sin

2/1cos1

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Slide ١٧Digital Signal Processing

Butterworth filter

The minimum attenuation as dB is usually given at certain frequency s.

The order of the filter can be calculated from the filter equation

s (rad/sec)

H()dB

Nsss AH

2

2

1log10

)(log10

s

AsN

log2

110log 10/

Slide ١٨Digital Signal Processing

Design Steps of Butterworth Filter

1. Convert the filter specifications to their equivalents in the lowpass prototype frequency.

2. From Ap determine the ripple factor .

3. From As determine the filter order, N.

4. Determine the left-hand poles, using the equations given.

5. Construct the lowpass prototype filter transfer function.

6. Use the frequency transformation to convert the LP prototype filter to the given specifications.

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Slide ١٩Digital Signal Processing

Butterworth filter

Example:Design a lowpass Butterworth filter with a maximum gain of 5 dB and a cutoff frequency of 1000 rad/s at which the gain is at least 2 dB and a stopband frequency of 5000 rad/s at which the magnitude is required to be less than −25dB.

Solution:p = 1000 rad/s , s = 5000 rad/s,

By normalization, p = p/ p = 1 rad/s,

s = s / p = 5 rad/s,And the stopband attenuation As = 25+ 5 =30 dB

The filter order is calculated by

3146.2)5log(2

)110log( 10/

sA

N

Slide ٢٠Digital Signal Processing

Butterworth filter

The pole positions are: 3,2,16/22 kep kj

k

866.05.0,1,866.05.0,, 3/43/2 jjeeep jjjk

866.05.01866.05.0)( jppjppD

11)( 2 ppppD

122)( 23 ppppD

Hence the transfer function of the normalized prototype filter of third order is

1221)( 23

ppp

pH

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Slide ٢١Digital Signal Processing

Butterworth filter

To restore the magnitude, we multiply be H0

20logH0 = 5dB which leads H0 = 1.7783

To restore the frequency we replace p by s/1000

122)( 23

0

pppHpH

1

10002

10002

1000

7783.1)( 231000/

ssspHsH sp

9623

9

10102200010.7783.1)(

ssssH

Slide ٢٢Digital Signal Processing

Butterworth filter

If the passband edge is defined for Ap 3 dB (i.e. 1).

The design equation needs to be modified. The formula for calculating the order will become

And the poles are given by

Home Study: Repeat the previous example if Ap = 0.5 dB

s

AsN

log2

110log 10/

Nkep NNkjNk ,...,2,12/)12(/1