Chapt 4 Design of Digital Filters_FIR

7/31/2019 Chapt 4 Design of Digital Filters_FIR

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2/2/2012Watumull Institute of Technology, Worli

Prepared by Chandrashekhar Padole, DSP-BE Computer, Mumbai Uni 1

Chapt 04

Design of Digital Filters ( FIR)B.E. Comps, Mumbai Uni

PrePrepared by Chandrashekhar Padole

Lecturer

Watumull Institute of Tech , Worli


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Chapt 04 Design of Digital Filters

Design of FIR filters

Design of IIR filters from analog filters

Frequency transformation

Design of digital filters based on least-squares method

Digital filters from analog filters

Properties of FIR filters

Design of FIR filters using windows

Comparison of IIR and FIR filters

Linear phase filters


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What is filter ?

Basic building block of DSP:Input is given to filter and output of the system would be

signal obtained from input and filter's impulse response.

FilterInput Output

[ 1 1]Input Output

[ 1 -1]Input Output

Integrator ( Lowpass filter)

Differentiator(High pass filter)


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Contd..

Different impulse response different characteristics

Characteristic of the filter Frequency domain characteristics

FrequencyCharacteristic

PhaseCharacteristic


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Frequency Characteristic

H(w)

W

Frequency characteristic plot between frequencies and magnitude

For given system if particular frequency is passed thorough the system,its magnitude at output can be obtained from corresponding magnitudefrom frequency characteristicNote: most of the time input wont be single pure sinusoidal/frequency but

would be a set of frequencies


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Phase Characteristic

(w)

W

Phase characteristic plot between frequencies and phase shift

For given system if particular frequency is passed thorough the system,its delay/phase shift at output can be obtained from corresponding phsefrom phase characteristicNote: most of the time input wont be single pure sinusoidal/frequency butwould be a set of frequencies


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Ideal Characteristics

Frequency Phase

Sharp cutoffLinear

phase

Sharp cutoff is requirement foralmost all the applications

Non-linear phase characteristicdistorts the signal.Linear phase characteristic has

constant group delay.


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Effect of Linear Phase Characteristic

Let us pass signal , made up of summing frequency 1 Hz, 2Hz & 3Hz, through filters with different phase characteristics


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Contd..


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Contd..


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Contd..


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Contd..


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Contd..


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Challenge ..


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Verify .


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M file used in MATLAB for this simulation

clear all;close all;figure,

subplot(4,1,1);plot(sin(2*pi*1*[0:0.01:3]));title(' Inoput Signal Composition');subplot(4,1,2);

plot(sin(2*pi*2*[0:0.01:3]));subplot(4,1,3);plot(sin(2*pi*3*[0:0.01:3]));

subplot(4,1,4);plot(sum([sin(2*pi*1*[0:0.01:3]);sin(2*pi*2*[0:0.01:3]);sin(2*pi*3*[0:0.01:3])]));

figure,subplot(4,1,1);plot(sin(2*pi*1*[0:0.01:3]+pi/8));title(' Filtered Signal with phase characteristic pi/8');subplot(4,1,2);plot(sin(2*pi*2*[0:0.01:3]+pi/8));subplot(4,1,3);plot(sin(2*pi*3*[0:0.01:3]+pi/8));

subplot(4,1,4);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/8);sin(2*pi*2*[0:0.01:3]+pi/8);sin(2*pi*3*[0:0.01:3]+pi/8)]));


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Contd..

figure,subplot(4,1,1);plot(sin(2*pi*1*[0:0.01:3]+pi/4));title(' Filtered Signal with phase characteristic pi/4');subplot(4,1,2);plot(sin(2*pi*2*[0:0.01:3]+pi/4));subplot(4,1,3);

plot(sin(2*pi*3*[0:0.01:3]+pi/4));


figure,subplot(4,1,1);

plot(sin(2*pi*1*[0:0.01:3]+pi/2));title(' Filtered Signal with phase characteristic pi/2');subplot(4,1,2);plot(sin(2*pi*2*[0:0.01:3]+pi/2));subplot(4,1,3);plot(sin(2*pi*3*[0:0.01:3]+pi/2));

subplot(4,1,4);

plot(sum([sin(2*pi*1*[0:0.01:3]+pi/2);sin(2*pi*2*[0:0.01:3]+pi/2);sin(2*pi*3*[0:0.01:3]+pi/2)]));


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Contd..

figure,subplot(4,1,1);plot(sin(2*pi*1*[0:0.01:3]+pi/8));title(' Filtered Signal with phase characteristic w(linear phase with group delay 1)');subplot(4,1,2);plot(sin(2*pi*2*[0:0.01:3]+pi/4));subplot(4,1,3);

plot(sin(2*pi*3*[0:0.01:3]+pi/2));


figure,subplot(4,1,1);

plot(sin(2*pi*1*[0:0.01:3]+pi/4));title(' Filtered Signal with phase characteristic 2w(linear phase with group delay 2)');subplot(4,1,2);plot(sin(2*pi*2*[0:0.01:3]+pi/2));subplot(4,1,3);plot(sin(2*pi*3*[0:0.01:3]+pi/1));

subplot(4,1,4);

plot(sum([sin(2*pi*1*[0:0.01:3]+pi/4);sin(2*pi*2*[0:0.01:3]+pi/2);sin(2*pi*3*[0:0.01:3]+pi/1)]));


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Chapt 04

figure,

subplot(6,1,1);plot(sum([sin(2*pi*1*[0:0.01:3]);sin(2*pi*2*[0:0.01:3]);sin(2*pi*3*[0:0.01:3])]));title('I am Input Signal , which of followings looks like me ?');

subplot(6,1,2);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/8);sin(2*pi*2*[0:0.01:3]+pi/8);sin(2*pi*3*[0:0.01:3]+pi/8)]));%title(' Filtered Signal with phase characteristic pi/8');



subplot(6,1,5);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/8);sin(2*pi*2*[0:0.01:3]+pi/4);sin(2*pi*3*[0:0.01:3]+pi/2)]));%title(' Filtered Signal with phase characteristic w(linear phase with group delay 1)');

subplot(6,1,6);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/4);sin(2*pi*2*[0:0.01:3]+pi/2);sin(2*pi*3*[0:0.01:3]+pi/1)]));%title(' Filtered Signal with phase characteristic 2w(linear phase with group delay 2)');


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Chapt 04

figure,

subplot(6,1,1);plot(sum([sin(2*pi*1*[0:0.01:3]);sin(2*pi*2*[0:0.01:3]);sin(2*pi*3*[0:0.01:3])]));title(' Input Signal ');

subplot(6,1,2);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/8);sin(2*pi*2*[0:0.01:3]+pi/8);sin(2*pi*3*[0:0.01:3]+pi/8)]));title(' Filtered Signal with phase characteristic pi/8');




title(' Filtered Signal with phase characteristic w(linear phase with group delay 1)');

subplot(6,1,6);plot(sum([sin(2*pi*1*[0:0.01:3]+pi/4);sin(2*pi*2*[0:0.01:3]+pi/2);sin(2*pi*3*[0:0.01:3]+pi/1)]));title(' Filtered Signal with phase characteristic 2w(linear phase with group delay 2)');


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Linear Phase Conclusion

If filter doesn't have linear phase characteristic( constant groupdelay) then signal shape gets distorts.e.g. First 3 filtered outputs

If filter has linear phase characteristic( constant group delay) thensignal shape would be preserved.e.g. Last 2 filtered outputs

Linear phase can be obtained easily by FIR filter by having

symmetric/anti-symmetric property in its impulse response.


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FIR Filter

FIR filter has finite number of samples in its impulse response

Advantages:

The question of stability and realizability never arise for FIR filters ( it isalways stable and realizable) It gives linear phase relationship with frequency , which can beachieved by having symmetric or anti-symmetric impulse response of thefilter

Disadvantages :

Long sequences for h(n) are generally required to adequatelyapproximate sharp cut-off filters

Hence , higher computation complexityThe delay of linear phase FIR filters need not always be an integer

number of samples . This non-integral delays can lead to problems insome signal processing applications


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Characteristic of FIR filter

Let { h(n)} be a causal finite durations sequence of length N( 0 to N-1).Its z-transform

=

=

1

0

)()(N

n

nznhzH --------(1)

Its Fourier transform,

=

=

1

0

)()(N

n

jwnjwenheH

Which is periodic in frequency with period of 2

)()()2( mjj

eHeH +

= ......3,2,1,0 =mfor

--------(2)

Consider h(n) be real and its magnitude and phase ,

)()()(

jjwjweeHeH =


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Contd..

From eq (2) , since wnjwne jwn sincos =

symmetricAnti-

symmetric

Magnitude of Fourier transform is symmetric and the phaseis an anti-symmetric function

)()(jwjw

eHeH

=

0)()( =

Consider , we have to have linear phase responsei.e. =)(

where is constant ( phase delay in samples)


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Contd..

We would find , condition/restriction on impulse response such that itwill give linear phase response

Mathematically,

=

=

1

0

)()(N

n

jwnjwenheH

jjweeH

= )( Requirement

=

=

1

0)()(

N

n

jwnjw

enheH

Given

=

=

=

1

0

1

0 )sin()()cos()(

N

n

N

n wnnhjwnnh

=

=

=

1

0

1

01

)cos()(

)sin()(

tanN

n

N

n

wnnh

wnnh


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Since n=0, sin(wn)=0Hence , limit starts

from 1



Contd..

=

=

=

1

0

1

0

)cos()(

)sin()(

)tan(N

n

N

n

wnnh

wnnh

+

=

=

=

1

1

1

1

)cos()()0(

)sin()(

)tan(N

n

N

n

wnnhh

wnnh

There are two possible solutions

1) =0 , h(0) can be arbitrary& h(n)=0 for n0

In this case filter will have order 0 and it would bejust amplifier and not a filter. This is not toouseful result


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Contd..

2) 0

==

=

=

1

0

1

0

)cos()(

)sin()(

)cos(

)sin()tan(

N

n

N

n

nnh

nnh

=

=

=

1

0

1

0

)cos()sin()()sin()cos()(N

n

N

n

nnhnnh

0)}cos()sin()sin()){cos((

1

0=

= nnnh

N

n

0)sin()(1

0

=

=

nnhN

n

0)(sin)(1

0

=

=

nnhN

n


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Contd..

From this equation , unique solution is obtained for the set ofconditions

1) = (N-1)/2i.e. for any value of sequence , there is only one value of phase delay

, which is the condition to obtain linear phase.2) h(n)=h(N-1-n) for 0 n N-1i.e. impulse response sequence must have a special kind of symmetry

for the value of

For linear phase filter , impulse response should be either symmetric oranti-symmetric.As impulse response can be of odd or even length, there are four

possible types of impulse response which will have linear phasecharacteristic.


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Linear Phase Filters Impulse Responses

1) Symmetric and Even N

2) Anti-symmetric and Even N

3) Symmetric and Odd N

4) Anti-symmetric and Odd N


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Symmetric and Even & Odd N

0 1 2 3 4 5 6 7 8 9 10

N=11 ( Odd)= 5

Center ofsymmetry

0 1 2 3 4 5 6 7 8 9

N=10 ( Even)= 4.5

Center ofsymmetry


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Anti-Symmetric and Even & Odd N

0 1 2 3 4 5 6 7 8 9 10

N=11 ( Odd)= 5

Center of

symmetry

0 1 2 3 4 5 6 7 8 9

N=10 ( Even)= 4.5

Center ofsymmetry

P f f Li h h i i f ll f


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Proofs of Linear phase characteristics for all four types

Refer Rabinar and Gold

P iti f i Li h filt


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Position of zeros in Linear phase filters

=

=

1

0

)()(N

n

nznhzHWe have

)1()2()3(

4321

)0()1()2(

)4()3()2()1()0(

++++=

NNN zhzhzh

zhzhzhzhh

Symmetry/ anti-symmetry in impulse response

2/)1()(

=

NzzH [ ]

[ ]

[ ]})2(

)1(

)0({

2/)5(2/)5(

2/)3(2/)3(

2/)1(2/)1(

+

+

+

NN

NN

NN

zzh

zzh

zzh

-----------(1)

C td


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Contd..

If z is replaced by z-1 in eq 1 we get

2/)1(1)(

=

NzzH [ ]

[ ][ ]

}

)2(

)1(

)0({

2/)5(2/)5(

2/)3(2/)3(

2/)1(2/)1(

+

+

+

NN

NN

NN

zzh

zzh

zzh

-----------(2)

Comparing eq 1 & 2 we get

)()()1(1 zHzzH N =

H(z) and H(z

-1

) are identical within adelay of (N-1) samples and multiplier 1(r,-)

(r,)

(1/r,)

(1/r,-)

Id l Filt


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Ideal Filter

Consider ideal low-pass filter characteristic

H(w)

Wc w

=0

1)(H

)(

~

kH2/3

Chapt 4 Design of Digital Filters_FIR

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