Chapter 10
APPROXIMATIONS FOR ANALOG FILTERS
10.1 Introduction, 10.2 Realizability
10.3 to 10.7 Butterworth, Chebyshev, Inverse-Chebyshev,
Elliptic, and Bessel-Thomson Approximations
c 2005- by Andreas Antoniou
Copyright Victoria, BC, Canada
Email: aantoniou@ieee.org
October 28, 2010
Frame # 1
Slide # 1
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
t As mentioned in previous presentations, the solution of the
approximation problem for recursive filters can be accomplished by
using direct or indirect methods.
Frame # 2
Slide # 2
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
t As mentioned in previous presentations, the solution of the
approximation problem for recursive filters can be accomplished by
using direct or indirect methods.
t In indirect approximation methods, digital filters are designed
indirectly through the use of corresponding analog-filter
approximations.
Frame # 2
Slide # 3
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
t As mentioned in previous presentations, the solution of the
approximation problem for recursive filters can be accomplished by
using direct or indirect methods.
t In indirect approximation methods, digital filters are designed
indirectly through the use of corresponding analog-filter
approximations.
t Several analog-filter approximations have been proposed in the past,
as follows:
–
–
–
–
–
Frame # 2
Slide # 4
Butterworth,
Chebyshev,
Inverse-Chebyshev,
elliptic, and
Bessel-Thomson approximations.
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
t As mentioned in previous presentations, the solution of the
approximation problem for recursive filters can be accomplished by
using direct or indirect methods.
t In indirect approximation methods, digital filters are designed
indirectly through the use of corresponding analog-filter
approximations.
t Several analog-filter approximations have been proposed in the past,
as follows:
–
–
–
–
–
Butterworth,
Chebyshev,
Inverse-Chebyshev,
elliptic, and
Bessel-Thomson approximations.
t This presentation deals with the basics of these approximations.
Frame # 2
Slide # 5
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An analog filter such as the one shown below can be represented by
the equation
Vo (s)
N(s)
= H(s) =
Vi (s)
D(s)
where
L
R1
2
C2
vi(t)
Frame # 3
Slide # 6
C1
A. Antoniou
C3
vo(t)
R2
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An analog filter such as the one shown below can be represented by
the equation
Vo (s)
N(s)
= H(s) =
Vi (s)
D(s)
where
– Vi (s) is the Laplace transform of the input voltage vi (t),
L
R1
2
C2
vi(t)
Frame # 3
Slide # 7
C1
A. Antoniou
C3
vo(t)
R2
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An analog filter such as the one shown below can be represented by
the equation
Vo (s)
N(s)
= H(s) =
Vi (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),
L
R1
2
C2
vi(t)
Frame # 3
Slide # 8
C1
A. Antoniou
C3
vo(t)
R2
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An analog filter such as the one shown below can be represented by
the equation
Vo (s)
N(s)
= H(s) =
Vi (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,
L
R1
2
C2
vi(t)
Frame # 3
Slide # 9
C1
A. Antoniou
C3
vo(t)
R2
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An analog filter such as the one shown below can be represented by
the equation
Vo (s)
N(s)
= H(s) =
Vi (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.
L
R1
2
C2
vi(t)
Frame # 3
Slide # 10
C1
A. Antoniou
C3
vo(t)
R2
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t The loss (or attenuation) is defined as
Vi (jω) 2
|Vi (jω)|2
1
1
=
L(ω ) =
=
= 10 log
2
2
|Vo (jω)|
Vo (jω)
|H(jω)|
H(jω)H(−jω)
2
Hence the loss in dB is given by
A(ω) = 10 log L(ω 2 ) = 10 log
1
|H(jω)|2
= −20 log |H(jω)|
In effect, the loss in dB is the negative of the gain in dB.
Frame # 4
Slide # 11
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t The loss (or attenuation) is defined as
Vi (jω) 2
|Vi (jω)|2
1
1
=
L(ω ) =
=
= 10 log
2
2
|Vo (jω)|
Vo (jω)
|H(jω)|
H(jω)H(−jω)
2
Hence the loss in dB is given by
A(ω) = 10 log L(ω 2 ) = 10 log
1
|H(jω)|2
= −20 log |H(jω)|
In effect, the loss in dB is the negative of the gain in dB.
t As a function of ω, A(ω) is said to be the loss characteristic.
Frame # 4
Slide # 12
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t The phase shift and group delay of analog filters are defined
just as in digital filters, namely, the phase shift is the phase
angle of the frequency response and the group delay is the
negative of the derivative of the phase angle with respect to
ω, i.e.,
θ(ω) = arg H(jω)
Frame # 5
Slide # 13
A. Antoniou
and
τ (ω) = −
d θ(ω)
dω
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t The phase shift and group delay of analog filters are defined
just as in digital filters, namely, the phase shift is the phase
angle of the frequency response and the group delay is the
negative of the derivative of the phase angle with respect to
ω, i.e.,
θ(ω) = arg H(jω)
and
τ (ω) = −
d θ(ω)
dω
t As functions of ω, θ(ω) and τ (ω) are the phase response and
delay characteristic, respectively.
Frame # 5
Slide # 14
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As was shown earlier, the loss can be expressed as
L(ω 2 ) =
Frame # 6
Slide # 15
A. Antoniou
1
H(jω)H(−jω)
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As was shown earlier, the loss can be expressed as
L(ω 2 ) =
1
H(jω)H(−jω)
t If we replace ω by s/j in L(ω 2 ), we get the so-called loss
function
L(−s 2 ) =
Frame # 6
Slide # 16
A. Antoniou
D(s)D(−s)
N(s)N(−s)
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As was shown earlier, the loss can be expressed as
L(ω 2 ) =
1
H(jω)H(−jω)
t If we replace ω by s/j in L(ω 2 ), we get the so-called loss
function
L(−s 2 ) =
D(s)D(−s)
N(s)N(−s)
t Thus if the transfer function of an analog filter is known, its
loss function can be readily deduced.
Frame # 6
Slide # 17
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t If
M
(s − zi )
N(s)
= Ni =1
H(s) =
D(s)
i =1 (s − pi )
then
L(−s 2 ) =
Frame # 7
Slide # 18
N
(s − pi ) N
D(s)D(−s)
i =1 (−s − pi )
= iM=1
M
N(s)N(−s)
i =1 (s − zi )
i =1 (−s − zi )
N
N
(s − pi ) i =1 [s − (−pi )]
= (−1)N−M iM=1
M
i =1 (s − zi )
i =1 [s − (−zi )]
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t If
M
(s − zi )
N(s)
= Ni =1
H(s) =
D(s)
i =1 (s − pi )
then
L(−s 2 ) =
t Therefore,
N
(s − pi ) N
D(s)D(−s)
i =1 (−s − pi )
= iM=1
M
N(s)N(−s)
i =1 (s − zi )
i =1 (−s − zi )
N
N
(s − pi ) i =1 [s − (−pi )]
= (−1)N−M iM=1
M
i =1 (s − zi )
i =1 [s − (−zi )]
– the zeros of the loss function are the poles of of the transfer
function and their negatives, and
– the poles of the loss function are the zeros of the transfer
function and their negatives.
Frame # 7
Slide # 19
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t Zero-pole plots for transfer function and loss function:
jω
H(s)
L (−s2)
jω
2
2
s plane
s plane
σ
σ
2
2
Frame # 8
Slide # 20
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An ideal lowpass filter is one that will pass only low-frequency
components. Its loss characteristic assumes the form shown in the
figure.
A(ω)
ωc
ω
(a)
Frame # 9
Slide # 21
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t An ideal lowpass filter is one that will pass only low-frequency
components. Its loss characteristic assumes the form shown in the
figure.
– The frequency range 0 to ωc is the passband.
– The frequency range ωc to ∞ is the stopband.
– The boundary between the passband and stopband, namely,
ωc , is the cutoff frequency.
A(ω)
ωc
ω
(a)
Frame # 9
Slide # 22
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As in digital filters, the approximation step for the design of
analog filters is the process of obtaining a realizable transfer
function that would satisfy certain desirable specifications.
Frame # 10
Slide # 23
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As in digital filters, the approximation step for the design of
analog filters is the process of obtaining a realizable transfer
function that would satisfy certain desirable specifications.
t In the classical solutions of the approximation problem, an
ideal normalized lowpass loss characteristic is assumed with a
cutoff frequency of order unity, i.e., ωc ≈ 1.
Frame # 10
Slide # 24
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t As in digital filters, the approximation step for the design of
analog filters is the process of obtaining a realizable transfer
function that would satisfy certain desirable specifications.
t In the classical solutions of the approximation problem, an
ideal normalized lowpass loss characteristic is assumed with a
cutoff frequency of order unity, i.e., ωc ≈ 1.
t A set of formulas are then derived that yield the zeros and
poles or coefficients of the transfer function for a specified
filter order.
Frame # 10
Slide # 25
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t Classical approximations such as the Butterworth, Chebyshev,
inverse-Chebyshev, and elliptic approximations lead to a loss
characteristic where
Frame # 11
Slide # 26
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t 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 Ap dB over the frequency
range 0 to ωp ;
Frame # 11
Slide # 27
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t 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 Ap dB over the frequency
range 0 to ωp ;
– the loss is equal to or greater than Aa dB over the frequency
range ωa to ∞.
Frame # 11
Slide # 28
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t 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 Ap dB over the frequency
range 0 to ωp ;
– the loss is equal to or greater than Aa dB over the frequency
range ωa to ∞.
t Parameters ωp and ωa are the passband and stoband edges,
Ap is the maximum passband loss (or attenuation), and Aa is
the minimum stopband loss (or attenuation), respectively.
Frame # 11
Slide # 29
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t The quality of an approximation depends on the values of Ap
and Aa for a given filter order, i.e., a lower Ap and a larger Aa
correspond to a better filter.
A(ω)
Aa
Ap
ω
ωp
Frame # 12
Slide # 30
A. Antoniou
ωc
ωa
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t In practice, the cutoff frequency of the lowpass filter depends
on the application at hand.
Frame # 13
Slide # 31
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t In practice, the cutoff frequency of the lowpass filter depends
on the application at hand.
Furthermore, other types of filters are often required such as
highpass, bandpass, and bandstop filters.
Frame # 13
Slide # 32
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t In practice, the cutoff frequency of the lowpass filter depends
on the application at hand.
Furthermore, other types of filters are often required such as
highpass, bandpass, and bandstop filters.
t Approximations for arbitrary lowpass, highpass, bandpass, and
bandstop filters can be obtained through a process known as
denormalization.
Frame # 13
Slide # 33
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t In practice, the cutoff frequency of the lowpass filter depends
on the application at hand.
Furthermore, other types of filters are often required such as
highpass, bandpass, and bandstop filters.
t Approximations for arbitrary lowpass, highpass, bandpass, and
bandstop filters can be obtained through a process known as
denormalization.
t Filter denormalization can be applied through the use of a
class of analog-filter transformations.
Frame # 13
Slide # 34
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Introduction
Cont’d
t In practice, the cutoff frequency of the lowpass filter depends
on the application at hand.
Furthermore, other types of filters are often required such as
highpass, bandpass, and bandstop filters.
t Approximations for arbitrary lowpass, highpass, bandpass, and
bandstop filters can be obtained through a process known as
denormalization.
t Filter denormalization can be applied through the use of a
class of analog-filter transformations.
t These transformations will be discussed in the next
presentation.
Frame # 13
Slide # 35
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
Frame # 14
Slide # 36
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
Frame # 14
Slide # 37
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
This requirement is imposed by the fact that inductances,
capacitances, and resistances are required to be real quantities.
Frame # 14
Slide # 38
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
This requirement is imposed by the fact that inductances,
capacitances, and resistances are required to be real quantities.
t The degree of the numerator polynomial must be equal to or less
than the degree of the denominator polynomial.
Frame # 14
Slide # 39
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
This requirement is imposed by the fact that inductances,
capacitances, and resistances are required to be real quantities.
t The degree of the numerator polynomial must be equal to or less
than the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal system
which would not be realizable as a real-time analog filter.
Frame # 14
Slide # 40
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
This requirement is imposed by the fact that inductances,
capacitances, and resistances are required to be real quantities.
t The degree of the numerator polynomial must be equal to or less
than the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal system
which would not be realizable as a real-time analog filter.
t The poles must be in the left half s plane.
Frame # 14
Slide # 41
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Realizability Constraints
t Realizability contraints are constraints that must be satisfied by a
transfer function in order to be realizable in terms of an analog-filter
network.
t The coefficients must be real.
This requirement is imposed by the fact that inductances,
capacitances, and resistances are required to be real quantities.
t The degree of the numerator polynomial must be equal to or less
than the degree of the denominator polynomial.
Otherwise, the transfer function would represent a noncausal system
which would not be realizable as a real-time analog filter.
t The poles must be in the left half s plane.
Otherwise, the transfer function would represent an unstable system.
Frame # 14
Slide # 42
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical analog-filter approximations will be presented such as:
Frame # 15
Slide # 43
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical analog-filter approximations will be presented such as:
– The underlying assumptions in the derivation.
Frame # 15
Slide # 44
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical analog-filter approximations will be presented such as:
– The underlying assumptions in the derivation.
– Typical loss characteristics.
Frame # 15
Slide # 45
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical analog-filter approximations will be presented such as:
– The underlying assumptions in the derivation.
– Typical loss characteristics.
– Available independent parameters.
Frame # 15
Slide # 46
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical 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.
Frame # 15
Slide # 47
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical 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.
Frame # 15
Slide # 48
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Classical Analog-Filter Approximations
t In the slides that follow, the basic features of the various
classical 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).
Frame # 15
Slide # 49
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation
t The Butterworth approximation is derived on the assumption that
the loss function L(−s 2 ) is a polynomial. Since
lim L(−s 2 ) = lim L(ω 2 ) = a0 + a2 ω 2 + · · · + a2n ω 2n → ∞
s→∞
ω→∞
in such a case, a lowpass approximation is obtained.
Frame # 16
Slide # 50
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation
t The Butterworth approximation is derived on the assumption that
the loss function L(−s 2 ) is a polynomial. Since
lim L(−s 2 ) = lim L(ω 2 ) = a0 + a2 ω 2 + · · · + a2n ω 2n → ∞
s→∞
ω→∞
in such a case, a lowpass approximation is obtained.
t For an n-order approximation, L(ω 2 ) is assumed to be maximally
flat at zero frequency.
Frame # 16
Slide # 51
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation
t The Butterworth approximation is derived on the assumption that
the loss function L(−s 2 ) is a polynomial. Since
lim L(−s 2 ) = lim L(ω 2 ) = a0 + a2 ω 2 + · · · + a2n ω 2n → ∞
s→∞
ω→∞
in such a case, a lowpass approximation is obtained.
t For an n-order approximation, L(ω 2 ) is assumed to be maximally
flat at zero frequency.
This is achieved by letting
d k L(x) = 0 for k ≤ n − 1
dx k x=0
where x = ω 2 , i.e., n derivatives of the loss are set to zero at zero
frequency.
L(0) = 1,
Frame # 16
Slide # 52
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation
t The Butterworth approximation is derived on the assumption that
the loss function L(−s 2 ) is a polynomial. Since
lim L(−s 2 ) = lim L(ω 2 ) = a0 + a2 ω 2 + · · · + a2n ω 2n → ∞
s→∞
ω→∞
in such a case, a lowpass approximation is obtained.
t For an n-order approximation, L(ω 2 ) is assumed to be maximally
flat at zero frequency.
This is achieved by letting
d k L(x) = 0 for k ≤ n − 1
dx k x=0
where x = ω 2 , i.e., n derivatives of the loss are set to zero at zero
frequency.
L(0) = 1,
t Assuming that L(1) = 2, the loss function in dB can be expressed as
L(ω 2 ) = 1 + ω 2n
Frame # 16
Slide # 53
and
A. Antoniou
A(ω) = 10 log(1 + ω 2n )
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t Typical loss characteristics:
30
25
A(ω), dB
20
n=9
15
n=6
10
n=3
5
0
0
1.0
0.5
1.5
ω, rad/s
Frame # 17
Slide # 54
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t The loss function for the normalized Butterworth
approximation (3-dB frequency at 1 rad/s) is given by
L(−s 2 ) = 1 + (−s 2 )n =
where
Frame # 18
Slide # 55
zi =
2n
(s − zi )
i =1
e j(2i −1)π/2n
for even n
e j(i −1)π/n
for odd n
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t The loss function for the normalized Butterworth
approximation (3-dB frequency at 1 rad/s) is given by
L(−s 2 ) = 1 + (−s 2 )n =
where
zi =
2n
(s − zi )
i =1
e j(2i −1)π/2n
for even n
e j(i −1)π/n
for odd n
t Since |zk | = 1, the zeros of L(−s 2 ) are located on the unit
circle |s| = 1.
Frame # 18
Slide # 56
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t Zero-pole plots for loss function:
s plane
s plane
n=6
jIm s
jIm s
n=5
Re s
Re s
(b)
(a)
Frame # 19
Slide # 57
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t The zeros of the loss function are the poles of the transfer
function and their negatives.
Frame # 20
Slide # 58
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t The zeros of the loss function are the poles of the transfer
function and their negatives.
t For stability, the poles of the transfer function must be
located in the left-half s plane.
Frame # 20
Slide # 59
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t The zeros of the loss function are the poles of the transfer
function and their negatives.
t For stability, the poles of the transfer function must be
located in the left-half s plane.
Therefore, they are identical with the zeros of the loss
function located in the left-half s plane.
Frame # 20
Slide # 60
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t Typically in practice, the required filter order is unknown.
Frame # 21
Slide # 61
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t 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 Cont’d
t 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 Let us assume that we need a normalized Butterworth filter with a
maximum passband loss Ap , minimum stopband loss Aa , passband
edge ωp , and stopband edge ωa .
Frame # 21
Slide # 63
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t 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 Let us assume that we need a normalized Butterworth filter with a
maximum passband loss Ap , minimum stopband loss Aa , passband
edge ωp , and stopband edge ωa .
The minimum filter order that will satisfy the required specifications
must be large enough to satisfy both of the following inequalities:
n≥
Frame # 21
Slide # 64
[− log (100.1Ap − 1)]
(−2 log ωp )
A. Antoniou
and n ≥
log (100.1Aa − 1)
2 log ωa
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
t 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 Let us assume that we need a normalized Butterworth filter with a
maximum passband loss Ap , minimum stopband loss Aa , passband
edge ωp , and stopband edge ωa .
The minimum filter order that will satisfy the required specifications
must be large enough to satisfy both of the following inequalities:
n≥
[− log (100.1Ap − 1)]
(−2 log ωp )
and n ≥
log (100.1Aa − 1)
2 log ωa
(See textbook for derivations and examples.)
Frame # 21
Slide # 65
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
···
n≥
[− log (100.1Ap − 1)]
(−2 log ωp )
and n ≥
log (100.1Aa − 1)
2 log ωa
t The right-hand sides in the above inequalities will normally yield a
mixed number but since the filter order must be an integer, the
value obtained must be rounded up to the next integer.
Frame # 22
Slide # 66
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
···
n≥
[− log (100.1Ap − 1)]
(−2 log ωp )
and n ≥
log (100.1Aa − 1)
2 log ωa
t The right-hand sides in the above inequalities will normally yield a
mixed number but since the filter order must be an integer, the
value obtained must be rounded up to the next integer.
As a result, the required specifications will usually be slightly
oversatisfied.
Frame # 22
Slide # 67
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Butterworth Approximation Cont’d
···
n≥
[− log (100.1Ap − 1)]
(−2 log ωp )
and n ≥
log (100.1Aa − 1)
2 log ωa
t The right-hand sides in the above inequalities will normally yield a
mixed number but since the filter order must be an integer, the
value obtained must be rounded up to the next integer.
As a result, the required specifications will usually be slightly
oversatisfied.
t Once the required filter order is determined, the actual maximum
passband loss and minimum stopband loss can be calculated as
Ap = A(ωp ) = 10 log(1 + ωp2n ) and
Aa = A(ωa ) = 10 log(1 + ωa2n )
respectively.
Frame # 22
Slide # 68
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
t In the Butterworth approximation, the loss is an increasing
monotonic function of ω, and as a result the passband loss is
very small at low frequencies and very large at frequencies
close to the bandpass edge.
Frame # 23
Slide # 69
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
t In the Butterworth approximation, the loss is an increasing
monotonic function of ω, and as a result the passband loss is
very small at low frequencies and very large at frequencies
close to the bandpass edge.
On the other hand, the stopband loss is very small at
frequencies 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
t In the Butterworth approximation, the loss is an increasing
monotonic function of ω, and as a result the passband loss is
very small at low frequencies and very large at frequencies
close to the bandpass edge.
On the other hand, the stopband loss is very small at
frequencies close to the stopand edge and very large at very
high frequencies.
t 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
Cont’d
t As in the Butterworth approximation, the loss function in the
Chebyshev 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
Chebyshev Approximation
Cont’d
t As in the Butterworth approximation, the loss function in the
Chebyshev approximation is assumed to be a polynomial in s,
which would assure a lowpass characteristic.
t The derivation of the Chebyshev approximation is based on
the assumption that the passband loss oscillates between zero
and a specified maximum loss Ap .
Frame # 24
Slide # 73
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t As in the Butterworth approximation, the loss function in the
Chebyshev approximation is assumed to be a polynomial in s,
which would assure a lowpass characteristic.
t The derivation of the Chebyshev approximation is based on
the assumption that the passband loss oscillates between zero
and a specified maximum loss Ap .
On the basis of this assumption, a differential equation is
constructed 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
Cont’d
t As in the Butterworth approximation, the loss function in the
Chebyshev approximation is assumed to be a polynomial in s,
which would assure a lowpass characteristic.
t The derivation of the Chebyshev approximation is based on
the assumption that the passband loss oscillates between zero
and a specified maximum loss Ap .
On the basis of this assumption, a differential equation is
constructed whose solution gives the zeros of the loss function.
t Then by neglecting the zeros of the loss function in the
right-half s plane, the poles of the transfer function can be
obtained.
Frame # 24
Slide # 75
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t In the case of a fourth-order Chebyshev filter the passband
loss is assumed to be zero at ω = Ω01 , Ω02 and equal to Ap
at ω = 0, Ω̂1 , 1 as shown in the figure:
2.0
1.8
1.6
A(ω), dB
1.4
1.2
1.0
0.8
0.6
Ap
0.4
0.2
0
0
0.2
0.4
Ω01
Frame # 25
Slide # 76
A. Antoniou
0.6
0.8
ω, rad/s
ˆ
Ω
1
1.0
1.2
Ω02 ωp
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t On using all the information that can be extracted from the figure
shown, a differential equation of the form
dF(ω)
dω
2
=
M4 [1 − F 2 (ω)]
1 − ω2
can be constructed.
Frame # 26
Slide # 77
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t On using all the information that can be extracted from the figure
shown, a differential equation of the form
dF(ω)
dω
2
=
M4 [1 − F 2 (ω)]
1 − ω2
can be constructed.
t The solution of this differential equation gives the loss as
L(ω 2 ) = 1 + ε2 F 2 (ω)
where
and
Frame # 26
Slide # 78
ε2 = 100.1Ap − 1
F (ω) = T4 (ω) = cos(4 cos−1 ω)
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t On using all the information that can be extracted from the figure
shown, a differential equation of the form
dF(ω)
dω
2
=
M4 [1 − F 2 (ω)]
1 − ω2
can be constructed.
t The solution of this differential equation gives the loss as
L(ω 2 ) = 1 + ε2 F 2 (ω)
where
and
ε2 = 100.1Ap − 1
F (ω) = T4 (ω) = cos(4 cos−1 ω)
t The function cos(4 cos−1 ω) is actually a polynomial known as the
4th-order Chebyshev polynomial.
Frame # 26
Slide # 79
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t Similarly, for an nth-order Chebyshev approximation, one can
show that
A(ω) = 10 log L(ω 2 ) = 10 log[1 + ε2 Tn2 (ω)]
where
and
ε2 = 100.1Ap − 1
cos(n cos−1 ω)
Tn (ω) =
cosh(n cosh−1 ω)
for |ω| ≤ 1
for |ω| > 1
is the nth-order Chebyshev polynomial.
Frame # 27
Slide # 80
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t Typical loss characteristics for Chebyshev approximation:
30
n=7
Loss, dB
20
n=4
Loss, dB
10
1.0
0
0
0.4
0.8
ω, rad/s
1.2
1.6
0
(a)
Frame # 28
Slide # 81
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The zeros of the loss function for a normalized nth-order Chebyshev
approximation (ωp = 1 rad/s) are given by si = σi + jωi where
1
(2i − 1)π
−1 1
sinh
σi = ± sinh
sin
n
ε
2n
1
1
(2i − 1)π
sinh−1
ωi = cosh
cos
n
ε
2n
for i = 1, 2, . . . , n.
Frame # 29
Slide # 82
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The zeros of the loss function for a normalized nth-order Chebyshev
approximation (ωp = 1 rad/s) are given by si = σi + jωi where
1
(2i − 1)π
−1 1
sinh
σi = ± sinh
sin
n
ε
2n
1
1
(2i − 1)π
sinh−1
ωi = cosh
cos
n
ε
2n
for i = 1, 2, . . . , n.
t From these equations, we note that
σi2
ωi2
+
=1
sinh2 u
cosh2 u
where u =
1
1
sinh−1
n
ε
i.e., the zeros of L(−s 2 ) are located on an ellipse.
Frame # 29
Slide # 83
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t Typical zero-pole plots for Chebyshev approximation:
(a) n = 5 Ap = 1 dB; (b) n = 6 Ap = 1 dB.
1.2
1.2
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
jIm s
1.0
0
0
−0.2
jIm s
−0.2
−0.4
−0.6
−0.4
−0.6
−0.8
−0.8
−1.0
−1.0
−1.2
−0.5
Frame # 30
Slide # 84
0
Re s
(a)
0.5
A. Antoniou
−1.2
−0.5
0
Re s
(b)
Digital Signal Processing – Secs. 10.1-10.7
0.5
Chebyshev Approximation
Cont’d
t An nth-order normalized Chevyshev transfer function with a
passband edge ωp = 1 rad/s and a maximum passband loss of Ap
dB can be determined as follows:
H0
(s
− pi )(s − pi∗ )
i
H0
r
=
D0 (s) i [s 2 − 2Re (pi )s + |pi |2 ]
HN (s) =
where
⎧
⎨ n−1
2
r=
⎩n
2
Frame # 31
Slide # 85
D0 (s)
r
for odd n
and
for even n
A. Antoniou
D0 (s) =
s − p0
1
for odd n
for even n
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The poles and multiplier constant, H0 , can be calculated by using
the following formulas in sequence:
ε = 100.1Ap − 1
1
1
sinh−1
p0 = σ(n+1)/2 with σ(n+1)/2 = − sinh
n
ε
for i = 1, 2, . . . , r
pi = σi + jωi
1
(2i − 1)π
−1 1
sinh
where σi = − sinh
sin
n
ε
2n
1
1
(2i − 1)π
sinh−1
ωi = cosh
cos
n
ε
2n
r
for odd n
−p0 i =1 |pi |2
H0 =
r
−0.05Ap
2
for even n
10
i =1 |pi |
Frame # 32
Slide # 86
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
cosh−1 D
100.1Aa − 1
n≥
where
D
=
100.1Ap − 1
cosh−1 ωa
Frame # 33
Slide # 87
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
cosh−1 D
100.1Aa − 1
n≥
where
D
=
100.1Ap − 1
cosh−1 ωa
t As in the Butterworth approximation, the value at the right-hand
side of the inequality must be rounded up to the next integer. As a
result, the minimum stopband loss will usually be slightly
oversatisfied.
Frame # 33
Slide # 88
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
cosh−1 D
100.1Aa − 1
n≥
where
D
=
100.1Ap − 1
cosh−1 ωa
t As in the Butterworth approximation, the value at the right-hand
side of the inequality must be rounded up to the next integer. As a
result, the minimum stopband loss will usually be slightly
oversatisfied.
The actual minimum stopband loss can be calculated as
Aa = A(ωa ) = 10 log L(ωa2 ) = 10 log[1 + ε2 Tn2 (ωa )]
Frame # 33
Slide # 89
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
cosh−1 D
100.1Aa − 1
n≥
where
D
=
100.1Ap − 1
cosh−1 ωa
t As in the Butterworth approximation, the value at the right-hand
side of the inequality must be rounded up to the next integer. As a
result, the minimum stopband loss will usually be slightly
oversatisfied.
The actual minimum stopband loss can be calculated as
Aa = A(ωa ) = 10 log L(ωa2 ) = 10 log[1 + ε2 Tn2 (ωa )]
t In the Chebyshev approximation, the actual maximum passband loss
will be exactly as specified,i.e., Ap .
Frame # 33
Slide # 90
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
t The inverse-Chebyshev approximation is closely related to the
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
Inverse-Chebyshev Approximation
t The inverse-Chebyshev approximation is closely related to the
Chebyshev approximation, as may be expected, and it is
actually derived from the Chebyshev approximation.
t The passband loss in the inverse-Chebyshev is very similar to
that of the Butterworth approximation, i.e., it is an increasing
monotonic function of ω, while the stopband loss oscillates
between infinity and a prescribed minimum loss Aa .
Frame # 34
Slide # 92
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t Typical loss characteristics for inverse-Chebyshev
approximation:
40
Loss, dB
60
n=4
Loss, dB
20
1.0
n=7
0
0
0.2
0.4
1.0
ω, rad/s
2.0
4.0
0
10.0
(b)
Frame # 35
Slide # 93
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The loss for the inverse-Chebyshev approximation is given by
A(ω) = 10 log 1 +
where
δ2 =
1
δ2 Tn2 (1/ω)
1
−1
and the stopband extends from ω = 1 to ∞.
Frame # 36
Slide # 94
100.1Aa
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The normalized transfer function for a specified order, n, stopband
edge of ωa = 1 rad/s, and minimum stopband loss, Aa , is given by
r
HN (s) =
H0 (s − 1/zi )(s − 1/zi∗ )
D0 (s)
(s − 1/pi )(s − 1/pi∗ )
i =1
r
s 2 + |z1i |2
H0 =
1
D0 (s)
2
i =1 s − 2Re
pi s +
where
n−1
r =
Frame # 37
Slide # 95
2
n
2
for odd n
for even n
and
A. Antoniou
D0 (s) =
s−
1
1
p0
1
|pi |2
for odd n
for even n
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The normalized transfer function for a specified order, n, stopband
edge of ωa = 1 rad/s, and minimum stopband loss, Aa , is given by
r
HN (s) =
H0 (s − 1/zi )(s − 1/zi∗ )
D0 (s)
(s − 1/pi )(s − 1/pi∗ )
i =1
r
s 2 + |z1i |2
H0 =
1
D0 (s)
2
i =1 s − 2Re
pi s +
where
n−1
r =
2
n
2
for odd n
for even n
and
D0 (s) =
s−
1
1
p0
1
|pi |2
for odd n
for even n
t The parameters of the transfer function can be calculated by using
the formulas in the next slide.
Frame # 37
Slide # 96
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
δ = √
1
,
100.1Aa − 1
p0 = σ(n+1)/2
Cont’d
(2i − 1)π
for
2n
1
sinh−1
= − sinh
n
zi = j cos
with σ(n+1)/2
pi = σi + jωi
1, 2, . . . , r
1
δ
for 1, 2, . . . , r
with
and
Frame # 38
Slide # 97
1
(2i − 1)π
−1 1
sinh
σi = − sinh
sin
n
δ
2n
1
1
(2i
−
1)π
sinh−1
ωi = cosh
cos
n
δ
2n
⎧
2
r
|zi |
1
⎪
for odd n
⎨ −p0 i =1 |pi |2
H0 =
⎪
⎩r |zi |2
for even n
i =1 |pi |2
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
100.1Aa − 1
cosh−1 D
where D = 0.1Ap
n≥
−1
10
−1
cosh (1/ωp )
Frame # 39
Slide # 98
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
100.1Aa − 1
cosh−1 D
where D = 0.1Ap
n≥
−1
10
−1
cosh (1/ωp )
t The value of the right-hand side of the above inequality is rarely an
integer and, therefore, it must be rounded up to the next integer.
This will cause the actual maximum passband loss to be slightly
overspecified.
Frame # 39
Slide # 99
A. Antoniou
Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
100.1Aa − 1
cosh−1 D
where D = 0.1Ap
n≥
−1
10
−1
cosh (1/ωp )
t The value of the right-hand side of the above inequality is rarely an
integer and, therefore, it must be rounded up to the next integer.
This will cause the actual maximum passband loss to be slightly
overspecified.
The actual maximum passband loss can be calculated as
1
1
Ap = A(ωp ) = 10 log 1 + 2 2
where δ 2 = 0.1Aa
δ Tn (1/ωp )
10
−1
Frame # 39
Slide # 100
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Inverse-Chebyshev Approximation
Cont’d
t The minimum filter order required to achieve a maximum passband
loss of Ap and a minimum stopband loss of Aa must be large
enough to satisfy the inequality
√
100.1Aa − 1
cosh−1 D
where D = 0.1Ap
n≥
−1
10
−1
cosh (1/ωp )
t The value of the right-hand side of the above inequality is rarely an
integer and, therefore, it must be rounded up to the next integer.
This will cause the actual maximum passband loss to be slightly
overspecified.
The actual maximum passband loss can be calculated as
1
1
Ap = A(ωp ) = 10 log 1 + 2 2
where δ 2 = 0.1Aa
δ Tn (1/ωp )
10
−1
t 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
t The Chebyshev approximation yields a much better passband
and the inverse-Chebyshev approximation yields a much better
stopband than the Butterworth approximation.
Frame # 40
Slide # 102
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
t The Chebyshev approximation yields a much better passband
and the inverse-Chebyshev approximation yields a much better
stopband than the Butterworth approximation.
t A filter with improved passband and stopband can be
obtained by using the elliptic approximation.
Frame # 40
Slide # 103
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
t The Chebyshev approximation yields a much better passband
and the inverse-Chebyshev approximation yields a much better
stopband than the Butterworth approximation.
t A filter with improved passband and stopband can be
obtained by using the elliptic approximation.
t The elliptic approximation is more efficient that all the other
analog-filter approximations in that the transition between
passband and stopband is steeper for a given approximation order.
Frame # 40
Slide # 104
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
t The Chebyshev approximation yields a much better passband
and the inverse-Chebyshev approximation yields a much better
stopband than the Butterworth approximation.
t A filter with improved passband and stopband can be
obtained by using the elliptic approximation.
t The elliptic approximation is more efficient that all the other
analog-filter approximations in that the transition between
passband and stopband is steeper for a given approximation order.
In fact, this is the optimal approximation for a given piecewise
constant approximation.
Frame # 40
Slide # 105
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
A(ω)
t Loss characteristic for a 5th-order elliptic approximation:
Aa
Ap
Ωˆ 1
ω
Ω01
Ωˆ
Ω∞1
Ωˇ 2
2
Ω02
ωp
Frame # 41
Slide # 106
ˇ
Ω
1
Ω∞2
ωa
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t The passband loss is assumed to oscillate between zero and a
prescribed maximum Ap and the stopband loss is assumed to
oscillate between infinity and a prescribed minimum Aa .
Frame # 42
Slide # 107
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t The passband loss is assumed to oscillate between zero and a
prescribed maximum Ap and the stopband loss is assumed to
oscillate between infinity and a prescribed minimum Aa .
t On the basis of the assumed structure of the loss
characteristic, a differential equation is derived, as in the case
of the Chebyshev approximation.
Frame # 42
Slide # 108
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t The passband loss is assumed to oscillate between zero and a
prescribed maximum Ap and the stopband loss is assumed to
oscillate between infinity and a prescribed minimum Aa .
t On the basis of the assumed structure of the loss
characteristic, a differential equation is derived, as in the case
of the Chebyshev approximation.
t After considerable mathematical complexity, the differential
equation obtained is solved through the use of elliptic
functions, and the parameters of the transfer function are
deduced.
Frame # 42
Slide # 109
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t The passband loss is assumed to oscillate between zero and a
prescribed maximum Ap and the stopband loss is assumed to
oscillate between infinity and a prescribed minimum Aa .
t On the basis of the assumed structure of the loss
characteristic, a differential equation is derived, as in the case
of the Chebyshev approximation.
t After considerable mathematical complexity, the differential
equation obtained is solved through the use of elliptic
functions, and the parameters of the transfer function are
deduced.
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
Cont’d
t The passband and stopband edges and cutoff frequency of a
normalized elliptic approximation are defined as follows:
ωp =
√
k,
1
ωa = √ ,
k
ωc =
√
ωa ωp = 1
Constants k and k1 given by
ωp
k=
ωa
and k1 =
100.1Ap − 1
100.1Aa − 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
Elliptic Approximation
Cont’d
t A normalized elliptic lowpass filter with a selectivity factor k,
√
√
passband edge ωp = k, stopband edge ωa = 1/ k, a maximum
passband loss of Ap dB, and a minimum stopband loss equal to or
in excess of Aa dB has a transfer function of the form
r
s 2 + a0i
H0 HN (s) =
D0 (s)
s 2 + b1i s + b0i
⎧
⎨
where
r =
⎩
and
Frame # 44
Slide # 112
D0 (s) =
i =1
n−1
2
for odd n
n
2
for even n
s + σ0
for odd n
1
for even n
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t A normalized elliptic lowpass filter with a selectivity factor k,
√
√
passband edge ωp = k, stopband edge ωa = 1/ k, a maximum
passband loss of Ap dB, and a minimum stopband loss equal to or
in excess of Aa dB has a transfer function of the form
r
s 2 + a0i
H0 HN (s) =
D0 (s)
s 2 + b1i s + b0i
⎧
⎨
where
r =
⎩
and
D0 (s) =
i =1
n−1
2
for odd n
n
2
for even n
s + σ0
for odd n
1
for even n
t The parameters of the transfer function can be obtained by using
the 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
k =
Cont’d
1 − k2
√ 1 1 − k
√
q0 =
2 1 + k
q = q0 + 2q05 + 15q09 + 150q013
100.1Aa − 1
D = 0.1Ap
10
−1
log 16D
(round up to the next integer)
n≥
log(1/q)
100.05Ap + 1
1
ln 0.05Ap
Λ=
−1
2n 10
2q 1/4 ∞ (−1)m q m(m+1) sinh[(2m + 1)Λ] m=0
σ0 = m q m2 cosh 2mΛ
1+2 ∞
(−1)
m=1
Frame # 45
Slide # 114
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
W =
Ωi =
1+
2q 1/4
1+
Frame # 46
Slide # 115
σ2
1+ 0
k
∞
where
Vi =
kσ02
Cont’d
1−
m m(m+1) sin (2m+1)πμ
m=0 (−1) q
n
∞
2 m=1 (−1)m q m2 cos 2mπμ
n
μ=
kΩ2i
⎧
⎨i
for odd n
⎩i −
1
2
Ω2
1− i
k
i = 1, 2, . . . , r
for even n
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
a0i =
1
Ω2i
b0i =
(σ0 Vi )2 + (Ωi W )2
2
1 + σ02 Ω2i
b1i =
2σ0 Vi
1 + σ02 Ω2i
H0 =
Frame # 47
Slide # 116
Cont’d
r
σ0 i =1
b0i
a0i
10−0.05Ap
r
b0i
i =1 a0i
for odd n
for even n
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t Because of the fact that the filter order is rounded up to the
next integer, the minimum stopband loss is usually
oversatisfied.
Frame # 48
Slide # 117
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t Because of the fact that the filter order is rounded up to the
next integer, the minimum stopband loss is usually
oversatisfied.
t The actual minimum stopband loss is given by the following
formula:
Aa = A(ωa ) = 10 log
Frame # 48
Slide # 118
100.1Ap − 1
+
1
16q n
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t Because of the fact that the filter order is rounded up to the
next integer, the minimum stopband loss is usually
oversatisfied.
t The actual minimum stopband loss is given by the following
formula:
Aa = A(ωa ) = 10 log
100.1Ap − 1
+
1
16q n
t The loss of an elliptic filter is usually calculated by using the
transfer function, i.e.,
A(ω) = 20 log
Frame # 48
Slide # 119
1
|H(jω)|
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Elliptic Approximation
Cont’d
t Because of the fact that the filter order is rounded up to the
next integer, the minimum stopband loss is usually
oversatisfied.
t The actual minimum stopband loss is given by the following
formula:
Aa = A(ωa ) = 10 log
100.1Ap − 1
+
1
16q n
t The loss of an elliptic filter is usually calculated by using the
transfer function, i.e.,
A(ω) = 20 log
1
|H(jω)|
(See textbook for details.)
Frame # 48
Slide # 120
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
t Ideally, the group delay of a filter should be constant; equivalently,
the phase shift should be a linear function of frequency to minimize
delay distortion (see Sec. 5.7).
Frame # 49
Slide # 121
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
t Ideally, the group delay of a filter should be constant; equivalently,
the phase shift should be a linear function of frequency to minimize
delay distortion (see Sec. 5.7).
t Since the only objective in the approximations described so far is to
achieve a specific loss characteristic, there is no reason for the phase
characteristic to turn out to be linear.
Frame # 49
Slide # 122
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
t Ideally, the group delay of a filter should be constant; equivalently,
the phase shift should be a linear function of frequency to minimize
delay distortion (see Sec. 5.7).
t Since the only objective in the approximations described so far is to
achieve a specific loss characteristic, there is no reason for the phase
characteristic 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
t Ideally, the group delay of a filter should be constant; equivalently,
the phase shift should be a linear function of frequency to minimize
delay distortion (see Sec. 5.7).
t Since the only objective in the approximations described so far is to
achieve a specific loss characteristic, there is no reason for the phase
characteristic to turn out to be linear.
In fact, it turns out to be highly nonlinear, as one might expect,
particularly in the elliptic approximation.
t The last approximation in Chap. 10, namely, the Bessel-Thomson
approximation, is derived on the assumption that the group delay is
maximally flat at zero frequency.
Frame # 49
Slide # 124
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
t Ideally, the group delay of a filter should be constant; equivalently,
the phase shift should be a linear function of frequency to minimize
delay distortion (see Sec. 5.7).
t Since the only objective in the approximations described so far is to
achieve a specific loss characteristic, there is no reason for the phase
characteristic to turn out to be linear.
In fact, it turns out to be highly nonlinear, as one might expect,
particularly in the elliptic approximation.
t The last approximation in Chap. 10, namely, the Bessel-Thomson
approximation, is derived on the assumption that the group delay is
maximally flat at zero frequency.
t As in the Butterworth and Chebyshev approximations, the loss
function is a polynomial. Hence the Bessel-Thomson approximation
is essentially a lowpass approximation.
Frame # 49
Slide # 125
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
Cont’d
t The transfer function for a normalized Bessel-Thomson
approximation is give by
H(s) = n
b0
i
i =0 bi s
where
Frame # 50
Slide # 126
bi =
=
b0
s n B(1/s)
(2n − i )!
− i )!
2n−i i !(n
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
Cont’d
t The transfer function for a normalized Bessel-Thomson
approximation is give by
H(s) = n
b0
i
i =0 bi s
where
bi =
=
b0
s n B(1/s)
(2n − i )!
− i )!
2n−i i !(n
t The group-delay is 1 s. An arbitrary delay can be obtained by
replacing s by τ0 s where τ0 is a constant.
Frame # 50
Slide # 127
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
Cont’d
t The transfer function for a normalized Bessel-Thomson
approximation is give by
H(s) = n
b0
i
i =0 bi s
where
bi =
=
b0
s n B(1/s)
(2n − i )!
− i )!
2n−i i !(n
t The group-delay is 1 s. An arbitrary delay can be obtained by
replacing s by τ0 s where τ0 is a constant.
t Function B(·) is a Bessel polynomial , and s n B(1/s) can be
shown to have zeros in the left-half s plane, i.e., the
Bessel-Thomson approximation represents stable analog filters.
Frame # 50
Slide # 128
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
Cont’d
t Typical loss characteristics:
30
25
Loss, dB
20
15
n=9
10
n=6
5
n=3
0
0
Frame # 51
Slide # 129
1
2
3
ω, rad/s
4
5
6
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
Bessel-Thomson Approximation
Cont’d
t Typical delay characteristics:
1.2
1.0
n=9
n=6
τ, s
0.8
0.6
n=3
0.4
0.2
0
Frame # 52
Slide # 130
0
1
2
3
ω, rad/s
4
5
6
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7
This slide concludes the presentation.
Thank you for your attention.
Frame # 53
Slide # 131
A. Antoniou Digital Signal Processing – Secs. 10.1-10.7