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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
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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
As mentioned in previous presentations, the solution of theapproximation problem for recursive filters can be accomplished byusing director indirectmethods.
In indirect approximation methods, digital filters are designedindirectly through the use of corresponding analog-filterapproximations.
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Introduction
As mentioned in previous presentations, the solution of theapproximation problem for recursive filters can be accomplished byusing director indirectmethods.
In indirect approximation methods, digital filters are designedindirectly through the use of corresponding analog-filterapproximations.
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
As mentioned in previous presentations, the solution of theapproximation problem for recursive filters can be accomplished byusing director indirectmethods.
In indirect approximation methods, digital filters are designedindirectly through the use of corresponding analog-filterapproximations.
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
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
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
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
Vi(s) is the Laplace transform of the input voltage vi(t),
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
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
Vi(s) is the Laplace transform of the input voltage vi(t),
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
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
Vi(s) is the Laplace transform of the input voltage vi(t),
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
As was shown earlier, the loss can be expressed as
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)
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Introduction Contd
As was shown earlier, the loss can be expressed as
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.
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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)]
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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.
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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)
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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)
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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)
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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.
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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.
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.
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Introduction Cont d
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.
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.
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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
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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 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
Classical approximations such as the Butterworth, Chebyshev,inverse-Chebyshev, and elliptic approximations lead to a loss
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.
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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()
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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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In practice, the cutoff frequency of the lowpass filter dependson the application at hand.
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.
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Introduction Contd
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In practice, the cutoff frequency of the lowpass filter dependson the application at hand.
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.
Filter denormalization can be applied through the use of aclass of analog-filter transformations.
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Introduction Contd
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In practice, the cutoff frequency of the lowpass filter dependson the application at hand.
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.
Filter denormalization can be applied through the use of aclass of analog-filter transformations.
These transformations will be discussed in the nextpresentation.
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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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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.
The coefficients must be real.
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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.
The coefficients must be real.
This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.
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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.
The coefficients must be real.
This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.
The degree of the numerator polynomial must be equal to or lessthan the degree of the denominator polynomial.
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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.
The coefficients must be real.
This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.
The degree of the numerator polynomial must be equal to or lessthan the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal systemwhich would not be realizable as a real-time analog filter.
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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.
The coefficients must be real.
This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.
The degree of the numerator polynomial must be equal to or lessthan the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal systemwhich would not be realizable as a real-time analog filter.
The poles must be in the left halfs plane.
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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.
The coefficients must be real.
This requirement is imposed by the fact that inductances,capacitances, and resistances are required to be real quantities.
The degree of the numerator polynomial must be equal to or lessthan the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal systemwhich would not be realizable as a real-time analog filter.
The poles must be in the left halfs plane.
Otherwise, the transfer function would represent an unstable system.
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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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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
Typical loss characteristics.
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
Typical loss characteristics.
Available independent parameters.
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
Typical loss characteristics.
Available independent parameters.
Formula for the loss as a function of the independent
parameters.
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
Typical loss characteristics.
Available independent parameters.
Formula for the loss as a function of the independent
parameters. Minimum filter order to achieved prescribed specifications.
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In the slides that follow, the basic features of the variousclassical analog-filter approximations will be presented such as:
The underlying assumptions in the derivation.
Typical loss characteristics.
Available independent parameters.
Formula for the loss as a function of the independent
parameters. Minimum filter order to achieved prescribed specifications.
Formulas for the parameters of the transfer function (e.g.,zeros, poles, coefficients, multiplier constant).
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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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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.
For an n-order approximation, L(2) is assumed to be maximallyflatat zero frequency.
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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.
For an n-order approximation, L(2) is assumed to be maximallyflatat zero frequency.
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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The Butterworth approximation is derived on the assumption thatthe loss function L(s2) is a polynomial. Since
lims
L(s2) = limL(2) =a0+ a22 + +a2n2n in such a case, a lowpass approximation is obtained.
For an n-order approximation, L(2) is assumed to be maximallyflatat zero frequency.
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)
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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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The zeros of the loss function are the poles of the transferfunction and their negatives.
For stability, the poles of the transfer function must belocated in the left-halfs plane.
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The zeros of the loss function are the poles of the transferfunction and their negatives.
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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Butterworth Approximation Contd
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.
Frame # 21 Slide # 62 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
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.
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.
Frame # 21 Slide # 63 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
T i ll i ti th i d filt 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.
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.
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
Frame # 21 Slide # 64 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
T picall in practice the req ired filter order is nkno n
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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.
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.
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.)
Frame # 21 Slide # 65 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
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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.
Frame # 22 Slide # 66 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
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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.
As a result, the required specifications will usually be slightlyoversatisfied.
Frame # 22 Slide # 67 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Butterworth Approximation Contd
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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.
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.
Frame # 22 Slide # 68 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
Frame # 23 Slide # 69 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
On the other hand, the stopband loss is very small atfrequencies close to the stopand edge and very large at very
high frequencies.
Frame # 23 Slide # 70 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
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.
Frame # 23 Slide # 71 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
Frame # 24 Slide # 72 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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Chebyshev Approximation Contd
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.
Frame # 24 Slide # 74 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
Frame # 24 Slide # 75 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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
Frame # 25 Slide # 76 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
Frame # 26 Slide # 77 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
The solution of this differential equation gives the loss asL(2) = 1 +2F2()
where 2 = 100.1Ap 1and F() = T4() = cos(4 cos
1 )
Frame # 26 Slide # 78 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
On using all the information that can be extracted from the figureh d ff l f h f
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shown, a differential equation of the form
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.
Frame # 26 Slide # 79 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
Frame # 27 Slide # 80 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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
Frame # 28 Slide # 81 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
Frame # 29 Slide # 82 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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.
Frame # 29 Slide # 83 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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)
Frame # 30 Slide # 84 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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
Frame # 31 Slide # 85 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Chebyshev Approximation Contd
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
Chebyshev Approximation Contd
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
Frame # 33 Slide # 87 A Antoniou Digital Signal Processing Secs 10 1-10 7
Chebyshev Approximation Contd
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.
Frame # 33 Slide # 88 A Antoniou Digital Signal Processing Secs 10 1-10 7
Chebyshev Approximation Contd
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
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 actual minimum stopband loss can be calculated as
Aa =A(a) = 10 log L(2a) = 10 log[1 +
2
T2n (a)]
Frame # 33 Slide # 89 A Antoniou Digital Signal Processing Secs 10 1-10 7
Chebyshev Approximation Contd
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
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 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.
Frame # 33 Slide # 90 A Antoniou Digital Signal Processing Secs 10 1-10 7
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)
Frame # 35 Slide # 93 A Antoniou Digital Signal Processing Secs 10 1 10 7
Inverse-Chebyshev Approximation Contd
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.
Frame # 36 Slide # 94 A Antoniou Digital Signal Processing Secs 10 1 10 7
Inverse-Chebyshev Approximation Contd
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
for evennand D0(s) =
s 1
p0for odd n
1 for even n
Frame # 37 Slide # 95 A Antoniou Digital Signal Processing Secs 10 1 10 7
Inverse-Chebyshev Approximation Contd
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
for evennand D0(s) =
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.
Frame # 37 Slide # 96 A Antoniou Digital Signal Processing Secs 10 1 10 7
Inverse-Chebyshev Approximation Contd
= 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
Frame # 38 Slide # 97 A Antoniou Digital Signal Processing Secs 10 1 10 7
Inverse-Chebyshev Approximation Contd
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
Inverse-Chebyshev Approximation Contd
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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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.
Frame # 39 Slide # 99 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Inverse-Chebyshev Approximation Contd
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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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.
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
Frame # 39 Slide # 100 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Inverse-Chebyshev Approximation Contd
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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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.
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.
Frame # 39 Slide # 101 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
Frame # 40 Slide # 102 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation
The Chebyshev approximation yields a much better passbandand the inverse-Chebyshev approximation yields a much better
b d h h B h i i
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stopband than the Butterworth approximation.
A filter with improved passband and stopbandcan beobtained by using the elliptic approximation.
Frame # 40 Slide # 103 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation
The Chebyshev approximation yields a much better passbandand the inverse-Chebyshev approximation yields a much better
b d h h B h i i
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stopband than the Butterworth approximation.
A filter with improved passband and stopbandcan beobtained by using the elliptic approximation.
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.
Frame # 40 Slide # 104 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation
The Chebyshev approximation yields a much better passbandand the inverse-Chebyshev approximation yields a much better
t b d th th B tt th i ti
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stopband than the Butterworth approximation.
A filter with improved passband and stopbandcan beobtained by using the elliptic approximation.
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.
Frame # 40 Slide # 105 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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
Frame # 41 Slide # 106 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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.
Frame # 42 Slide # 107 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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.
On the basis of the assumed structure of the losscharacteristic, a differential equation is derived, as in the caseof the Chebyshev approximation.
Frame # 42 Slide # 108 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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.
On the basis of the assumed structure of the losscharacteristic, a differential equation is derived, as in the caseof the Chebyshev approximation.
After considerable mathematical complexity, the differentialequation obtained is solved through the use of ellipticfunctions, and the parameters of the transfer function arededuced.
Frame # 42 Slide # 109 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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.
On the basis of the assumed structure of the losscharacteristic, a differential equation is derived, as in the caseof the Chebyshev approximation.
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.
Frame # 42 Slide # 110 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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.
Frame # 43 Slide # 111 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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Elliptic Approximation Contd
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
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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.
Frame # 44 Slide # 113 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
Frame # 45 Slide # 114 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
Frame # 46 Slide # 115 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
Frame # 47 Slide # 116 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
Because of the fact that the filter order is rounded upto thenext integer, the minimum stopband loss is usuallyoversatisfied.
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Frame # 48 Slide # 117 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
Frame # 48 Slide # 118 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
The loss of an elliptic filter is usually calculated by using thetransfer function, i.e.,
A() = 20log 1
|H(j)|
Frame # 48 Slide # 119 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
Elliptic Approximation Contd
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
The loss of an elliptic filter is usually calculated by using thetransfer function, i.e.,
A() = 20log 1
|H(j)|(See textbook for details.)
Frame # 48 Slide # 120 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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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Frame # 49 Slide # 121 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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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.
Frame # 49 Slide # 123 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
The last approximation in Chap. 10, namely, the Bessel-Thomsonapproximation, is derived on the assumption that the group delay ismaximally flatat zero frequency.
Frame # 49 Slide # 124 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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.
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.
Frame # 49 Slide # 125 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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)!
Frame # 50 Slide # 126 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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)!
The group-delay is 1 s. An arbitrary delay can be obtained byreplacing s by0s where 0 is a constant.
Frame # 50 Slide # 127 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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)!
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.
Frame # 50 Slide # 128 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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
Frame # 51 Slide # 129 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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
Frame # 52 Slide # 130 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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This slide concludes the presentation.
Thank you for your attention.
Frame # 53 Slide # 131 A. Antoniou Digital Signal Processing Secs. 10.1-10.7
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