filters.ppt

8/21/2019 filters.ppt

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Mrs. LINI MATHEWAssociate Professor

Electrical Engineering Department

NITTTR, Chandigarh

FILTERS


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Four types of mathematical functions toachieve the required approximations in theresponse characteristics of filters (Recursive

Type)

Butterworth Approximation Function

Chebyshev Approximation Function

Elliptical Approximation Function

Bessels Approximation Function

TYPES OF FILTERS


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The transfer function of a first order LPF(RC) is

Squared Magnitude Form

represents aButterworth polynomial

of order n

BUTTERWORTH APPROXIMATION

( )cjCRj)(H

+1

1=

+1

1=

( )2

+1

1=

2

c

)(H

( ) n

c

)(H2

+1

1=


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nthorder filter n stages are cascaded

The cut off becomes sharper as the order of thefilter is increased.

The pass band gain remains more or less constantand flat up to the cut off frequency.

H() in dB

BUTTERWORTH APPROXIMATION

( )

( )[ ]nc1-10log

nc

log)(Hlog

2+=

2+1

1

20=20


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To design a filter the following information isrequired:

(i) Pass band gain required, H1()

(ii) Frequency upto which the pass band gainmust remain more or less constant, 1

(iii) Amount of attenuation required, H2()

(iv) Frequency from which the attenuation muststart, 2

DESIGN PROCEDURE-BUTTERWORTH FILTER


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Step I

At = 1 At = 2

Solve these two eqns to get n and C

Step II Determination of the poles of H(s)Poles of the Butterworth polynomial lie on a circlewhose radius is C

Number of Butterworth poles = 2n


( ) nc)(H 2+11

=

( ) nc)(H 2

1+11=1

( ) nc)(H 2

2+11=2


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Angle between the poles, = 360/2n

Location of the poles (i) If n is even, then thelocation of the first pole is at /2 from thex-axis in the counter clockwise direction.

Location of the subsequent poles are /2 + ,/2 + 2, /2 + 3, ......

(ii) If n is odd, then the location of the first

pole is on the x-axis.Location of the subsequent poles are , 2,....

with angle measured in the counter clockwise

direction.



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Step III Determination of the valid poles ofH(s)

Poles that lie on the left half of s-plane alone

are stable poles.

Poles that lie in between 90oand 270oalone arevalid poles.

If is the angle of a valid pole wrt x-axis,then the pole and its conjugate are located at

C(cos j sin )



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Step IV To find the expression for H(s)

where a+jb and a-jb are the poles and isthe damping factor.

Step V Determination of the filter componentsR & C

Step VI Determination of the amplifierelements R1, R2, R3, R4etc.


2+2+2

2

=+++

2

=

cscs

c)jb-as)(jbas(

c)s(H


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To design a desired digital filter, first developthe expression for H(s) based on the proceduresin the analog filter design, and then modify it inan appropriate fashion to suit the digital domain.

IMPULSE INVARIANT DESIGN

Convert H(s) to h(t) by inverse Laplace transformConvert h(t) to H(z) by direct Z transformUsing H(z) construct the required digital filter

BILINEAR TRANSFORMATIONConvert H(s) to H(z) directly by using theexpression

DESIGN OF DIGITAL IIR FILTERS

)1-z(T

)-1z-(

s +1

12=


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For the design of digital filters, the idea

of sampling has to be incorporated.Step I Normalization of frequencies

Step II Determination of n and C

Step III Determination of valid poles B1,B2, ....

DESIGN OF BUTTERWORTH IIR FILTERS

( ) n

c

)(H2

+1

1=

sf

f12=1

sf

f22=2


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Step IV Finding the expression for H(s) in theanalog domain

Taking the inverse Laplace transform

h(t) = L-1

[H(s)] and H(z) = Z-1

[h(t)]

DESIGN OF BUTTERWORTH IIR FILTERS

)jb-as)(jbas()s(H c

+++

=

2


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Two types:

(i)Type I (regular) Chebyshev Filter

Passband containing ripples, and stopbandcontaining no ripples

(ii) Type II (inverse) Chebyshev Filter

Passband containing ripples, and stopbandalso containing ripples

CHEBYSHEV DIGITAL IIR FILTERS

22

222

+1

=

n

n

C

C)(H


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Chebyshev design makes use of the Chebyshev

polynomial

where is the amount of ripple in themagnitude

Cnis the Chebyshev constant

if /c1

DESIGN OF CHEBYSHEV LOW PASS FILTERS

22

2

+1

1=

nC)(H

]coshncosh[Cc

-1n

=

]cosncos[Cc

-1n

= if /

c1


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For the Chebyshev design

1=

cSpecifications of the desired Chebyshev LPF:

(i) Pass band gain required, H1()

(ii) Frequency upto which the pass band gainmust remain more or less steady, 1=c

(iii) Amount of attenuation required, H2

()

(iv) Frequency from which the attenuation muststart, 2



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Step I Determination of nusing the Chebyshev Polynomial & Chebyshevconstant

Step II Determination of poles of H(s) as perthe following rules

(i) From the given specifications, determine thevalid Butterworth poles ie. B1 = a+jb B2 = c+jd....

(ii) Determine the factor k from


( )11

= 1-sinh

nk


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(iii) Find tanh(k) and cosh(k)

(iv) Multiply the real part (a,c,...) of the poleswith tanh(k). This is called normalisation. Thusthe normalised (modified) Butterworth poles

are B1 = a tanh(k) +jb,B2 = c tanh(k) +jd

(v) Now, denormalise the poles by multiplying with

cosh(k) to get the desired Chebyshev poles.The Chebyshev poles are

C1= {cosh(k) [a tanh(k) +jb]}

C2= {cosh(k) [c tanh(k) +jd]}



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Step III Determination of transfer function

H(s)

where P={|C1||C4|}{|C2||C3|}


( )( )( )( )4321 C-sC-sC-sC-s

P)s(H =


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A condition to be satisfied before using bilineartransformation

Step I Normalization of frequenciesStep II Conversion of digital frequencies to

analog frequencies

Step III Determination of n and cStep IV Determination of the valid poles and

formulating H(s) and H(z)

DIGITAL FILTER DESIGN USING BILINEARTRANSFORMATION

)(tanT

22

=


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DESIGN OF FILTERS USINGWINDOW FUNCTIONS

There are infinite number of coefficients in aFourier Series Expansion.

The abrupt termination of the Fourier series

coefficients to a finite value produces sharptransients or introduce ripples in the frequencyresponse characteristic H(). This is due to thenon-uniform convergence of the Fourier Series at

a discontinuity.The oscillatory behaviour near the band edge ofthe filter is called Gibbs Phenomenon


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DESIGN OF FILTERS USINGWINDOW FUNCTIONS

Window functions are mathematical functions thatare designed to have tapering characteristics

When the impulse response h(n), derived from

the given transfer function H(), is multiplied byan appropriate window function w(n), we get amodified impulse function h(n), which showsgradually decreasing filter coefficients.

These filter coefficients will ensure the absenceof the Gibbs Phenomenon from the operatingregions of the filter.


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Low pass filter

designed withrectangularwindow

Low pass filter

designed with

Hamming window


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Low pass filter

designed with

Blackman window

Low pass filter

designed with

Kaiser window


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INDOW FUNCTIONS FILTER DESIG


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INDOW FUNCTIONS FILTER DESIG


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DESIGN OF DIGITAL FIR FILTERS

FIR filters are designed by assuming that themagnitude of transfer function H() is unity.

The phase of H() is not taken as unity

|H(

)|=1 and H(

) = ejn

here is constant, hence FIR filters are calledconstant phase filters

We do not use any type of frequencytransformation techniques, in the design of FIRfilters


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DESIGN OF DIGITAL FIR FILTERSUSING THE FOURIER SERIES

METHODStep I Normalisation of cut off frequency

Step II Fixing the transfer function to be used

Step III Determining the impulse response ofthe filter h(n) = sinc(n/2)

s

cc f

f2=

elsewhere0//2-e)H(

elsewhere0

//2-)(H

nj- 0

=

2=

=

21=


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DESIGN OF DIGITAL FIR FILTERSUSING THE FOURIER SERIES

METHODStep IV Determining the coefficients of the

impulse response sequence

ie. h(0), h(1), h(2), ....

Step V Determining the transfer functionback from the impulse response

sequence ie.H(z) is determinedStep VI Implementation of the filter


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FREQUENCY SAMPLING (FOURIERTRANSFORMATION) METHOD

In this method, we make use of thetheory of DFT for determining the filtercoefficients

The impulse response h(n) is determinedusing IDFT

N is the order of the filter & L=N+1 isthe length of the filter

0=

1+2

1+

1=

N

k

N/knje)k(H

N

)n(h


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FREQUENCY SAMPLING (FOURIERTRANSFORMATION) METHOD

We assume the transfer functionH(k)=1

H(k) is a periodic function with samplevalues at k=0,1,2,....N. These samplevalues are to be determined first fromthe transfer characteristics of thefilter

After this h(n) should be computed

Take the z transform H(z)

FREQUENCY RESPONSE


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FREQUENCY RESPONSECHARACTERISTICS OF LPFFourier transformation of a signal produces

infinite frequency bands in the positive andnegative frequency axes

In frequency sampling method, the period

for the computation of h(n) is considered tobe 0 to 2

E G EP E


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Step I Normalisation of frequency

Step II Fixing the transfer function to be used

Step III Determination of the locations andamplitudes of the samples

We have to convert the limits from the factorsrelated to (0 to 2) to numbers related to k.ie. interval between adjacent samples =2/N+1

samples are located at k = 2k/N+1

s

cc f

f2=

)k(H 1=

DESIGN STEPS IN THEFREQUENCY SAMPLING METHOD

DE GN EP N HE


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Step IV Determination of the values ofimpulse response

Step V Determination of the transferfunction H(z)

DESIGN STEPS IN THEFREQUENCY SAMPLING METHOD


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THANKS

filters.ppt

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