analog filter design techniques

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Chapter 10

APPROXIMATIONS FOR ANALOG FILTERS

10.1 Introduction, 10.2 Realizability10.3 to 10.7 Butterworth, Chebyshev, Inverse-Chebyshev,

Elliptic, and Bessel-Thomson Approximations

Copyright c2005- by Andreas AntoniouVictoria, BC, Canada

Email: [email protected]

October 28, 2010

Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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Introduction

As mentioned in previous presentations, the solution of theapproximation problem for recursive filters can be accomplished byusing director indirectmethods.

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Introduction


In indirect approximation methods, digital filters are designedindirectly through the use of corresponding analog-filterapproximations.

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Introduction



Several analog-filter approximations have been proposed in the past,as follows:

Butterworth, Chebyshev,

Inverse-Chebyshev, elliptic, and Bessel-Thomson approximations.

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Introduction



Several analog-filter approximations have been proposed in the past,as follows:

Butterworth, Chebyshev,

Inverse-Chebyshev, elliptic, and Bessel-Thomson approximations.

This presentation deals with the basics of these approximations.

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Introduction Contd

An analog filter such as the one shown below can be represented bythe equation

Vo(s)Vi(s)

=H(s) = N(s)D(s)

where

R2

R1

L2

C2

C1 C3vi(t)vo(t)

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Introduction Contd


Vo(s)Vi(s)

=H(s) = N(s)D(s)

where

Vi(s) is the Laplace transform of the input voltage vi(t),

R2

R1

L2

C2

C1 C3vi(t)vo(t)

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Introduction Contd


Vo(s)Vi(s)

=H(s) = N(s)D(s)

where


Vo(s) is the Laplace transform of the output voltage vo(t),

R2

R1

L2

C2

C1 C3vi(t)vo(t)

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Introduction Contd


Vo(s)Vi(s)

=H(s) = N(s)D(s)

where


Vo(s) is the Laplace transform of the output voltage vo(t), H(s) is the transfer function,

R2

R1

L2

C2

C1 C3vi(t)vo(t)

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Introduction Contd


Vo(s)Vi(s)

=H(s) = N(s)D(s)

where


Vo(s) is the Laplace transform of the output voltage vo(t), H(s) is the transfer function,

N(s) and D(s) are polynomials in complex variable s.

R2

R1

L2

C2

C1 C3vi(t)vo(t)

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Introduction Contd

The loss (or attenuation) is defined as

L(2) =|Vi(j)|2|Vo(j)|2 =

Vi(j)Vo(j)2 = 1|H(j)|2 = 10 log 1H(j)H(j)

Hence the loss in dB is given by

A() = 10log L(2) = 10 log 1

|H(j)|2=20log |H(j)|

In effect, the loss in dB is the negative of the gain in dB.

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Introduction Contd

The loss (or attenuation) is defined as

L(2) =|Vi(j)|2|Vo(j)|2 =

Vi(j)Vo(j)2 = 1|H(j)|2 = 10 log 1H(j)H(j)

Hence the loss in dB is given by

A() = 10log L(2) = 10 log 1

|H(j)|2=20log |H(j)|

In effect, the loss in dB is the negative of the gain in dB.

As a function of, A() is said to be the loss characteristic.

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Introduction Contd

The phase shift and group delay of analog filters are defined

just as in digital filters, namely, the phase shift is the phaseangle of the frequency response and the group delay is thenegative of the derivative of the phase angle with respect to, i.e.,

() = arg H(j) and () = d()

d

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Introduction Contd

The phase shift and group delay of analog filters are defined

just as in digital filters, namely, the phase shift is the phaseangle of the frequency response and the group delay is thenegative of the derivative of the phase angle with respect to, i.e.,

() = arg H(j) and () = d()

d

As functions of, () and () are the phase response anddelay characteristic, respectively.

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Introduction Contd

As was shown earlier, the loss can be expressed as

L(2) = 1H(j)H(j)

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Introduction Contd


L(2) = 1H(j)H(j)

If we replace bys/j in L(2), we get the so-called lossfunction

L(s2) = D(s)D(s)N(s)N(s)


I d i
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Introduction Contd


L(2) = 1H(j)H(j)

If we replace bys/j in L(2), we get the so-called lossfunction

L(s2) = D(s)D(s)N(s)N(s)

Thus if the transfer function of an analog filter is known, itsloss function can be readily deduced.


I d i
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Introduction Contd

If

H(s) = N(s)

D(s) =M

i=1(s

zi)Ni=1(s pi)

then

L(

s2

) =

D(s)D(

s)

N(s)N(s) = N

i=1(s

pi)Ni=1(

s

pi)

Mi=1(s zi)

Mi=1(s zi)

= (1)NMN

i=1(s pi)N

i=1[s (pi)]Mi=1(s zi)

Mi=1[s (zi)]


I d i
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Introduction Contd

If

H(s) = N(s)

D(s) =M

i=1(s

zi)Ni=1(s pi)

then

L(

s2) = D(s)D(

s)

N(s)N(s) = Ni=1(s

pi)Ni=1(s

pi)

Mi=1(s zi)Mi=1(s zi)

= (1)NMN

i=1(s pi)N

i=1[s (pi)]Mi=1(s zi)

Mi=1[s (zi)]

Therefore,

the zeros of the loss function are the poles of of the transferfunction and their negatives, and

the poles of the loss function are the zeros of the transferfunction and their negatives.


I t d ti
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Introduction Contd

Zero-pole plots for transfer function and loss function:

splane splane

H(s)

2

2

2

2

j j L (s2)


I t d ti C
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Introduction Contd

An ideal lowpassfilter is one that will pass only low-frequencycomponents. Its loss characteristic assumes the form shown in the

figure.

c

A()

(a)


Introduction C d
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Introduction Contd

An ideal lowpassfilter is one that will pass only low-frequencycomponents. Its loss characteristic assumes the form shown in the

figure.

The frequency range 0 to c is the passband.

The frequency range c tois the stopband. The boundary between the passband and stopband, namely,

c, is the cutoff frequency.

c

A()

(a)


Introduction C td
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Introduction Contd

As in digital filters, the approximation step for the design ofanalog filters is the process of obtaining a realizable transferfunction that would satisfy certain desirable specifications.


Introduction C td
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Introduction Contd


In the classical solutions of the approximation problem, anideal normalized lowpass loss characteristic is assumed with acutoff frequency of order unity, i.e., c

1.


Introduction Contd
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Introduction Cont d


In the classical solutions of the approximation problem, anideal normalized lowpass loss characteristic is assumed with acutoff frequency of order unity, i.e., c

1.

A set of formulas are then derived that yield the zeros andpoles or coefficientsof the transfer function for a specifiedfilter order.


Introduction Contd
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Introduction Cont d

Classical approximations such as the Butterworth, Chebyshev,inverse-Chebyshev, and elliptic approximations lead to a loss

characteristic where


Introduction Contd
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Introduction Cont d


characteristic where the loss is equal to or less than ApdB over the frequency

range 0 to p;

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Introduction Contd


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Introduction Cont d


characteristic where the loss is equal to or less than ApdB over the frequency

range 0 to p;

the loss is equal to or greater than Aa dB over the frequencyrangea to

.

Parameters p and a are the passband and stoband edges,Ap is the maximum passband loss (or attenuation), and Aa isthe minimum stopband loss (or attenuation), respectively.


Introduction Contd
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Introduction Cont d

The quality of an approximation depends on the values ofApand Aa for a given filter order, i.e., a lower Apand a larger Aacorrespond to a better filter.

ap

Ap

Aa

c

A()


Introduction Contd
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Introduction Cont d

In practice, the cutoff frequency of the lowpass filter dependson the application at hand.

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Introduction Contd


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Furthermore, other types of filters are often required such ashighpass, bandpass, and bandstopfilters.

Approximations for arbitrary lowpass, highpass, bandpass, and

bandstop filters can be obtained through a process known asdenormalization.


Introduction Contd
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Filter denormalization can be applied through the use of aclass of analog-filter transformations.


Introduction Contd
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Filter denormalization can be applied through the use of aclass of analog-filter transformations.

These transformations will be discussed in the nextpresentation.


Realizability Constraints
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Realizability contraintsare constraints that must be satisfied by atransfer function in order to be realizable in terms of an analog-filternetwork.


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The coefficients must be real.


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This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.


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The degree of the numerator polynomial must be equal to or lessthan the degree of the denominator polynomial.


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Otherwise, the transfer function would represent a noncausal systemwhich would not be realizable as a real-time analog filter.


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The poles must be in the left halfs plane.


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The poles must be in the left halfs plane.

Otherwise, the transfer function would represent an unstable system.


Classical Analog-Filter Approximations
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:


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The underlying assumptions in the derivation.


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Typical loss characteristics.


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Available independent parameters.


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Formula for the loss as a function of the independent

parameters.


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parameters. Minimum filter order to achieved prescribed specifications.


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parameters. Minimum filter order to achieved prescribed specifications.

Formulas for the parameters of the transfer function (e.g.,zeros, poles, coefficients, multiplier constant).


Butterworth Approximation
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The Butterworth approximation is derived on the assumption thatthe loss function L(s2) is a polynomial. Since

lims

L(s2) = lim

L(2) =a0+ a22 + +a2n2n in such a case, a lowpass approximation is obtained.


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lims

L(s2) = lim


For an n-order approximation, L(2) is assumed to be maximallyflatat zero frequency.


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lims

L(s2) = lim



This is achieved by letting

L(0) = 1, dkL(x)

dxk

x=0

= 0 for k n 1where x=2, i.e., n derivatives of the loss are set to zero at zero

frequency.


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lims

L(s2) = limL(2) =a0+ a22 + +a2n2n in such a case, a lowpass approximation is obtained.


This is achieved by letting

L(0) = 1, dkL(x)

dxk

x=0

= 0 for k n 1where x=2, i.e., n derivatives of the loss are set to zero at zero

frequency. Assuming that L(1) = 2, the loss function in dB can be expressed as

L(2) = 1 +2n and A() = 10 log(1 +2n)


Butterworth Approximation Contd
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Typical loss characteristics:

0 0.5 1.0 1.5

10

5

0

15

20

25

30

A(),dB

, rad/s

n = 3

n = 6

n = 9




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The loss function for the normalized Butterworthapproximation (3-dB frequency at 1 rad/s) is given by

L(s2) = 1 + (s2)n =2ni=1

(s zi)

where zi =ej(2i1)/2n for even n

ej(i1)/n for odd n


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The loss function for the normalized Butterworthapproximation (3-dB frequency at 1 rad/s) is given by

L(s2) = 1 + (s2)n =2ni=1

(s zi)

where zi =ej(2i1)/2n for even n

ej(i1)/n for odd n

Since|zk| = 1, the zeros ofL(s2) are located on the unitcircle|s| = 1.


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Zero-pole plots for loss function:

splanen = 6splanen = 5

(a) (b)

Res

jIm

s

Res

jIm

s


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The zeros of the loss function are the poles of the transferfunction and their negatives.


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For stability, the poles of the transfer function must belocated in the left-halfs plane.


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For stability, the poles of the transfer function must belocated in the left-halfs plane.

Therefore, they are identical with the zeros of the lossfunction located in the left-halfs plane.

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T i ll i i h i d fil d i k


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Typically in practice, the required filter order is unknown.

For Butterworth, Chebyshev, inverse-Chebyshev, and elliptic filters,it can by easily deduced if the required specifications are known.



T i ll i i h i d fil d i k
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Let us assume that we need a normalized Butterworth filter with amaximum passband loss Ap, minimum stopband loss Aa, passbandedge p, and stopband edge a.



T i ll i ti th i d filt d i k
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The minimum filter order that will satisfy the required specificationsmust be large enough to satisfy both of the following inequalities:

n [ log(100.1Ap 1)]

(

2log p)

and n log(100.1Aa 1)

2log a



T picall in practice the req ired filter order is nkno n
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For Butterworth, Chebyshev, inverse-Chebyshev, and elliptic filters,

it can by easily deduced if the required specifications are known.


The minimum filter order that will satisfy the required specificationsmust be large enough to satisfy both of the following inequalities:

n [ log(100.1Ap 1)]

(

2log p)

and n log(100.1Aa 1)

2log a

(See textbook for derivations and examples.)


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n

[ log(100.1Ap 1)]

(2log p) and n

log(100.1Aa 1)

2log a

The right-hand sides in the above inequalities will normally yield amixed number but since the filter order mustbe an integer, thevalue obtained must be rounded upto the next integer.


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n

[ log(100.1Ap 1)]

(2log p) and n

log(100.1Aa 1)

2log a


As a result, the required specifications will usually be slightlyoversatisfied.


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n

[ log(100.1Ap 1)]

(2log p) and n

log(100.1Aa 1)

2log a


As a result, the required specifications will usually be slightlyoversatisfied.

Once the required filter order is determined, the actual maximumpassband loss and minimum stopband loss can be calculated as

Ap=A(p) = 10 log(1 + 2np ) and Aa =A(a) = 10 log(1 +

2na )

respectively.


Chebyshev Approximation

In the Butterworth approximation the loss is an increasing
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In the Butterworth approximation, the loss is an increasingmonotonic function of, and as a result the passband loss is

very small at low frequencies and very large at frequenciesclose to the bandpass edge.



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On the other hand, the stopband loss is very small atfrequencies close to the stopand edge and very large at very

high frequencies.



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On the other hand, the stopband loss is very small atfrequencies close to the stopand edge and very large at very

high frequencies. A more balanced characteristic with respect to the passband

can be achieved by employing the Chebyshev approximation.


Chebyshev Approximation Contd

As in the Butterworth approximation the loss function in the
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As in the Butterworth approximation, the loss function in theChebyshev approximation is assumed to be a polynomial in s,

which would assure a lowpass characteristic.

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As in the Butterworth approximation, the loss function in the


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s t e utte o t app o at o , t e oss u ct o t eChebyshev approximation is assumed to be a polynomial in s,

which would assure a lowpass characteristic. The derivation of the Chebyshev approximation is based on

the assumption that the passband loss oscillates between zeroand a specified maximum loss Ap.

On the basis of this assumption, a differential equation isconstructed whose solution gives the zeros of the loss function.



As in the Butterworth approximation, the loss function in the
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pp ,Chebyshev approximation is assumed to be a polynomial in s,

which would assure a lowpass characteristic. The derivation of the Chebyshev approximation is based on

the assumption that the passband loss oscillates between zeroand a specified maximum loss Ap.

On the basis of this assumption, a differential equation isconstructed whose solution gives the zeros of the loss function.

Then by neglecting the zeros of the loss function in theright-halfsplane, the poles of the transfer function can beobtained.



In the case of a fourth-order Chebyshev filter the passband
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y ploss is assumed to be zero at = 01, 02 and equal to Ap

at = 0, 1, 1 as shown in the figure:

0 0.2 0.4 0.6 0.8 1.0 1.20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

A(),dB

, rad/s

1

Ap

01 02

p



On using all the information that can be extracted from the figure
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shown, a differential equation of the form

dF()

d

2

= M4[1 F2()]

1 2

can be constructed.



On using all the information that can be extracted from the figure
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dF()

d

2

= M4[1 F2()]

1 2

can be constructed.

The solution of this differential equation gives the loss asL(2) = 1 +2F2()

where 2 = 100.1Ap 1and F() = T4() = cos(4 cos

1 )



On using all the information that can be extracted from the figureh d ff l f h f
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dF()

d

2

= M4[1 F2()]

1 2

can be constructed.

The solution of this differential equation gives the loss asL(2) = 1 +2F2()

where 2 = 100.1Ap 1and F() = T4() = cos(4 cos

1 )

The function cos(4 cos1 ) is actually a polynomial known as the4th-order Chebyshevpolynomial.



Similarly, for an nth-order Chebyshev approximation, one can
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show that

A() = 10log L(2) = 10 log[1 +2T2n ()]

where 2 = 100.1Ap 1

and Tn() =

cos(n cos1 ) for|| 1cosh(n cosh1 ) for|| >1

is the nth-orderChebyshev polynomial.



Typical loss characteristics for Chebyshev approximation:
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1.0

0 00

Loss,

dB

1.6

, rad/s

Loss,

dB

(a)

1.20.80.4

10

20

30

n= 4

n= 7



The zeros of the loss function for a normalized nth-order Chebyshevi ti ( 1 d/ ) i b + j h
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approximation (p= 1 rad/s) are given by si=i+ji where

i = sinh

1n

sinh1 1

sin(2i 1)

2n

i = cosh

1

nsinh1

1

cos

(2i 1)2n

for i= 1, 2, . . . , n.



The zeros of the loss function for a normalized nth-order Chebyshevapproximation ( 1 rad/s) are given by s + j where
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approximation (p= 1 rad/s) are given by si=i+ji where

i = sinh

1n

sinh1 1

sin(2i 1)

2n

i = cosh

1

nsinh1

1

cos

(2i 1)2n

for i= 1, 2, . . . , n. From these equations, we note that

2isinh2 u

+ 2i

cosh2 u= 1 where u=

1

nsinh1

1

i.e., the zeros ofL(s2) are located on an ellipse.



Typical zero-pole plots for Chebyshev approximation:( ) 5 A 1 dB (b) 6 A 1 dB
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(a) n= 5 Ap= 1 dB; (b) n= 6 Ap= 1 dB.

0.51.2 1.2

0 0.5

1.0

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

Res

jIm

s

(a)

0.5 0 0.5

1.0

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

Res

jIm

s

(b)



An nth-order normalized Chevyshev transfer function with apassband edge = 1 rad/s and a maximum passband loss of A
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passband edgep= 1 rad/s and a maximum passband loss ofApdB can be determined as follows:

HN(s) = H0

D0(s)r

i(s pi)(s pi)=

H0

D0(s)r

i[s2 2Re (pi)s+ |pi|2]

where

r=

n12 for odd n

n2

for evennand D0(s) =

s p0 for odd n1 for even n



The poles and multiplier constant, H0, can be calculated by usingthe following formulas in sequence:
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the following formulas in sequence:

=

100.1Ap 1p0 = (n+1)/2 with (n+1)/2 = sinh

1

nsinh1

1

pi = i+ji for i= 1, 2, . . . , r

where i= sinh

1n

sinh1 1

sin(2i 1)

2n

i = cosh

1

nsinh1

1

cos

(2i 1)2n

H0 =p0ri=1 |pi|2 for odd n

100.05Apr

i=1 |pi|2 for even n

Frame # 32 Slide # 86 A Antoniou Digital Signal Processing Secs 10 1-10 7


The minimum filter order required to achieve a maximum passbandloss of Ap and a minimum stopband loss of Aa must be large
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loss ofApand a minimum stopband loss ofAa must be largeenough to satisfy the inequality

n cosh1

D

cosh1 awhere D=

100.1Aa 1100.1Ap 1



The minimum filter order required to achieve a maximum passbandloss of Ap and a minimum stopband loss of Aa must be large
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loss ofApand a minimum stopband loss ofAa must be largeenough to satisfy the inequality

n cosh1

D

cosh1 awhere D=

100.1Aa 1100.1Ap 1

As in the Butterworth approximation, the value at the right-hand

side of the inequality must be rounded up to the next integer. As aresult, the minimum stopband loss will usually be slightlyoversatisfied.



The minimum filter order required to achieve a maximum passbandloss ofAp and a minimum stopband loss ofAa must be large
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oss o p a d a u stopba d oss o a ust be a geenough to satisfy the inequality

n cosh1

D

cosh1 awhere D=

100.1Aa 1100.1Ap 1



The actual minimum stopband loss can be calculated as

Aa =A(a) = 10 log L(2a) = 10 log[1 +

2

T2n (a)]



The minimum filter order required to achieve a maximum passbandloss ofAp and a minimum stopband loss ofAa must be large


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p p a genough to satisfy the inequality

n cosh1

D

cosh1 awhere D=

100.1Aa 1100.1Ap 1



The actual minimum stopband loss can be calculated as

Aa =A(a) = 10 log L(2a) = 10 log[1 +

2

T2n (a)]

In the Chebyshev approximation, the actual maximum passband losswill be exactly as specified,i.e., Ap.


Inverse-Chebyshev Approximation

The inverse-Chebyshevapproximation is closely related to theChebyshev approximation as may be expected and it is
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Chebyshev approximation, as may be expected, and it is

actually derived from the Chebyshev approximation.

Frame # 34 Slide # 91 A Antoniou Digital Signal Processing Secs 10 1 10 7
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Inverse-Chebyshev Approximation Contd

Typical loss characteristics for inverse-Chebyshevapproximation:


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approximation:

1.0

00

Loss,

dB

0.2 0.4 1.0 2.0 4.0 10.0

, rad/s

20

40

60

n= 4

Loss,d

B

n= 7

0

(b)



The loss for the inverse-Chebyshev approximation is given by
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A() = 10 log

1 +

1

2T2n (1/)

where

2 = 1

100.1Aa

1

and the stopband extends from = 1 to.



The normalizedtransfer function for a specified order, n, stopbandedge ofa = 1 rad/s, and minimum stopband loss, Aa, is given by
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HN(s) = H0

D0(s)

ri=1

(s 1/zi)(s 1/zi)(s 1/pi)(s 1/pi)

= H0

D0(s)

r

i=1s2 + 1|zi|2

s2 2Re 1pi s+ 1|pi|2where

r =

n1

2 for odd n

n2


s 1

p0for odd n

1 for even n



The normalizedtransfer function for a specified order, n, stopbandedge ofa = 1 rad/s, and minimum stopband loss, Aa, is given by
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HN(s) = H0

D0(s)

ri=1

(s 1/zi)(s 1/zi)(s 1/pi)(s 1/pi)

= H0

D0(s)

ri=1

s2 + 1|zi|2

s2 2Re 1pi s+ 1|pi|2where

r =

n1

2 for odd n

n2


s 1

p0for odd n

1 for even n

The parameters of the transfer function can be calculated by usingthe formulas in the next slide.



= 1100.1Aa 1 , zi =jcos

(2i 1)2n

for 1, 2, . . . , r
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p0 = (n+1)/2 with (n+1)/2 = sinh1nsinh1 1pi = i+ji for 1, 2, . . . , r

with i =

sinh1

n

sinh11

sin(2i 1)

2n

i = cosh

1

nsinh1

1

cos

(2i 1)2n

and H0 =

1p0

r

i=1|zi|

2

|pi|2 for odd n

ri=1

|zi|2

|pi|2 for evenn



The minimum filter order required to achieve a maximum passbandloss ofApand a minimum stopband loss ofAa must be largeenough to satisfy the inequality
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enough to satisfy the inequality

n cosh1 D

cosh1(1/p)where D=

100.1Aa 1100.1Ap 1

Frame # 39 Slide # 98 A Antonio Digital Signal Processing Secs 10 1 10 7


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g y q y

n cosh1 D

cosh1(1/p)where D=

100.1Aa 1100.1Ap 1

The value of the right-hand side of the above inequality is rarely aninteger and, therefore, it must be rounded up to the next integer.

This will cause the actual maximum passband loss to be slightlyoverspecified.



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g y q y

n cosh1 D

cosh1(1/p)where D=

100.1Aa 1100.1Ap 1



The actual maximum passband loss can be calculated as

Ap=A(p) = 10 log

1 +

1

2T2n (1/p)

where

2

=

1

100.1Aa 1



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g y q y

n cosh1 D

cosh1(1/p)where D=

100.1Aa 1100.1Ap 1



The actual maximum passband loss can be calculated as

Ap=A(p) = 10 log

1 +

1

2T2n (1/p)

where

2

=

1

100.1Aa 1 In this approximation, the actual minimum stopband loss will be

exactly as specified, i.e., Aa.


Elliptic Approximation

The Chebyshev approximation yields a much better passbandand the inverse-Chebyshev approximation yields a much better
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stopband than the Butterworth approximation.




b d h h B h i i
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A filter with improved passband and stopbandcan beobtained by using the elliptic approximation.




b d h h B h i i
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The elliptic approximation is more efficient that all the other

analog-filter approximations in that the transition betweenpassband and stopband is steeper for a given approxima-tion order.




t b d th th B tt th i ti
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The elliptic approximation is more efficient that all the other

analog-filter approximations in that the transition betweenpassband and stopband is steeper for a given approxima-tion order.

In fact, this is the optimalapproximation for a given piecewiseconstant approximation.


Elliptic Approximation Contd

Loss characteristic for a 5th-order elliptic approximation:
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A()

Aa

Ap

1

2

1

2

p a

101

202



The passband loss is assumed to oscillate between zero and aprescribed maximum Apand the stopband loss is assumed tooscillate between infinity and a prescribed minimum A
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oscillate between infinity and a prescribed minimum Aa.



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On the basis of the assumed structure of the losscharacteristic, a differential equation is derived, as in the caseof the Chebyshev approximation.



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After considerable mathematical complexity, the differentialequation obtained is solved through the use of ellipticfunctions, and the parameters of the transfer function arededuced.



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After considerable mathematical complexity, the differentialequation obtained is solved through the use of ellipticfunctions, and the parameters of the transfer function arededuced.

The approximation owes its name to the use of elliptic

functions in the derivation.



The passband and stopband edges and cutoff frequency of anormalizedelliptic approximation are defined as follows:
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p= k, a = 1k

, c= ap= 1

Constants k and k1 given by

k= pa

and k1 =

100.1Ap 1100.1Aa 1

1/2

are known as the selectivity and discrimination constants.



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A normalized elliptic lowpass filter with a selectivity factor k,passband edgep=

k, stopband edge a = 1/

k, a maximum

passband loss ofApdB, and a minimum stopband loss equal to or


113/131

in excess ofAa dB has a transfer function of the form

HN(s) = H0

D0(s)

ri=1

s2 +a0is2 +b1is+ b0i

where r =

n12 for odd n

n2 for even n

and D0(s) =

s+0 for odd n

1 for even n

The parameters of the transfer function can be obtained by usingthe formulas in the next three slides in sequence in the order shown.



k =

1 k2

q0 = 1 1

k


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q0 2

1 + kq = q0+ 2q

50 + 15q

90+ 150q

130

D = 100.1Aa 1100.1Ap

1

n log 16Dlog(1/q)

(round up to the next integer)

= 1

2nln

100.05Ap + 1

100.05Ap 1

0 =2q

1/4 m=0(1)

m

qm(m+1)

sinh[(2m+ 1)]1 + 2

m=1(1)mqm2 cosh 2m



W =

1 +k20

1 +

20k
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i =

2q1/4

m=0(1)mqm(m+1) sin (2m+1)n1 + 2

m=1(1)mqm2 cos 2mn

where = i for odd ni 12 for evenn i= 1, 2, . . . , r

Vi =

1 k2

i

1 2i

k



a0i = 1

2i
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b0i = (0Vi)2 + (iW)2

1 +202i

2b1i =

20Vi

1 +20

2i

H0 =

0r

i=1b0ia0i

for odd n

100.05Apri=1

b0ia0i

for even n



Because of the fact that the filter order is rounded upto thenext integer, the minimum stopband loss is usuallyoversatisfied.
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The actual minimum stopband loss is given by the followingformula:

Aa =A(a) = 10log100.1Ap 1

16qn + 1



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Aa =A(a) = 10log100.1Ap 1

16qn + 1

The loss of an elliptic filter is usually calculated by using thetransfer function, i.e.,

A() = 20log 1

|H(j)|



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Aa =A(a) = 10log100.1Ap 1

16qn + 1

The loss of an elliptic filter is usually calculated by using thetransfer function, i.e.,

A() = 20log 1

|H(j)|(See textbook for details.)


Bessel-Thomson Approximation Ideally, the group delay of a filter should be constant; equivalently,

the phase shift should be a linear function of frequency to minimizedelay distortion (see Sec. 5.7).
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Since the only objective in the approximations described so far is toachieve a specific loss characteristic, there is no reason for the phasecharacteristic to turn out to be linear.

In fact, it turns out to be highly nonlinear, as one might expect,

particularly in the elliptic approximation.



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The last approximation in Chap. 10, namely, the Bessel-Thomsonapproximation, is derived on the assumption that the group delay ismaximally flatat zero frequency.



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The last approximation in Chap. 10, namely, the Bessel-Thomsonapproximation, is derived on the assumption that the group delay ismaximally flatat zero frequency.

As in the Butterworth and Chebyshev approximations, the lossfunction is a polynomial. Hence the Bessel-Thomson approximationis essentially a lowpassapproximation.


Bessel-Thomson Approximation Contd

The transfer function for a normalized Bessel-Thomsonapproximation is give by

H( ) b0 b0
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H(s) = b0ni=0 bis

i = b0

snB(1/s)

where bi = (2n i)!2nii!(n

i)!




H( ) b0 b0
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H(s) = b0ni=0 bis

i = b0

snB(1/s)


i)!

The group-delay is 1 s. An arbitrary delay can be obtained byreplacing s by0s where 0 is a constant.




H( ) b0 b0
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H(s) = b0ni=0 bis

i = b0

snB(1/s)


i)!

The group-delay is 1 s. An arbitrary delay can be obtained byreplacing s by0s where 0 is a constant.

Function B() is a Bessel polynomial, and snB(1/s) can beshown to have zeros in the left-halfsplane, i.e., theBessel-Thomson approximation represents stable analog filters.


Bessel-Thomson Approximation Contd Typical loss characteristics:

25

30
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0 1 2 3 4 5 60

5

10

15

20

25

Lo

ss,

dB

, rad/s

n = 3

n = 6

n = 9


Bessel-Thomson Approximation Contd Typical delay characteristics:

1 0

1.2
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0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1.0

,s

, rad/s

n = 3

n = 6

n = 9

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This slide concludes the presentation.

Thank you for your attention.

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analog filter design techniques

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