Analog Lowpass Filter Specifications Analog Lowpass Filter Specifications

Analog Lowpass Filter Analog Lowpass Filter
Specifications Specifications
• Typical magnitude response H a ( jΩ) of an • In the passband, defined by 0 ≤ Ω ≤ Ω p , we
analog lowpass filter may be given as require
indicated below 1 − δ p ≤ H a ( jΩ ) ≤ 1 + δ p , Ω ≤ Ω p
i.e., H a ( jΩ) approximates unity within an
error of ± δ p
• In the stopband, defined by Ω s ≤ Ω ≤ ∞ , we
require
H a ( jΩ ) ≤ δ s , Ω s ≤ Ω ≤ ∞
i.e., H a ( jΩ) approximates zero within an
1 2 error of δ s
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra
Analog Lowpass Filter Analog Lowpass Filter

Specifications Specifications
• Ω p - passband edge frequency • Magnitude specifications may alternately be
• Ω s - stopband edge frequency given in a normalized form as indicated
• δ p - peak ripple value in the passband below
• δ s - peak ripple value in the stopband
• Peak passband ripple
α p = − 20 log10 (1 − δ p ) dB
• Minimum stopband attenuation
α s = − 20 log10 (δ s ) dB
3 4
Analog Lowpass Filter

Analog Lowpass Filter Design
Specifications
• Two additional parameters are defined -
• Here, the maximum value of the magnitude
in the passband assumed to be unity Ωp
(1) Transition ratio k =
Ωs
• 1 / 1 + ε 2 - Maximum passband deviation,
given by the minimum value of the For a lowpass filter k < 1
magnitude in the passband
ε
(2) Discrimination parameter k1 =
A2 − 1
• 1 - Maximum stopband magnitude Usually k1 << 1
A
5 6
1
Butterworth Approximation
Butterworth Approximation
• The magnitude-square response of an N-th
order analog lowpass Butterworth filter
is given by • Gain in dB is G (Ω) = 10 log10 H a ( jΩ) 2
1
H a ( jΩ ) =
2
1 + (Ω / Ω c ) 2 N • As G (0) = 0 and
• First 2 N − 1 derivatives of H a ( jΩ) at Ω = 0
2
G (Ωc ) = 10 log10 (0.5) = −3.0103 ≅ −3 dB
are equal to zero Ω c is called the 3-dB cutoff frequency
• The Butterworth lowpass filter thus is said
to have a maximally-flat magnitude at Ω = 0
7 8
Butterworth Approximation Butterworth Approximation

• Typical magnitude responses with Ωc = 1 • Two parameters completely characterizing a
Butterworth Filter
Butterworth lowpass filter are Ωc and N
1
N=2 • These are determined from the specified
N=4
0.8 N = 10 bandedges Ω p and Ω s , and minimum
Magnitude
0.6 passband magnitude 1 / 1 + ε 2, and

0.4
maximum stopband ripple 1 / A
0.2
0
0 1 2 3
Ω
9 10

• Since order N must be an integer, value
• Ωc and N are thus determined from
2
obtained is rounded up to the next highest
H a ( jΩ p ) = 1 = 1 integer
1 + (Ω p / Ω c ) 2 N 1 + ε 2
• This value of N is used next to determine Ωc
H a ( jΩ s ) =
2 1 = 1 by satisfying either the stopband edge or the
1 + ( Ω s / Ω c ) 2 N A2
passband edge specification exactly
• Solving the above we get • If the stopband edge specification is
log [( A2 − 1) / ε 2 ] log10 (1 / k1 ) satisfied, then the passband edge
N = 1 ⋅ 10 = specification is exceeded providing a safety
2 log10 (Ω s / Ω p ) log10 (1 / k )
margin
11 12
2
• Example - Determine the lowest order of a
• Transfer function of an analog Butterworth Butterworth lowpass filter with a 1-dB cutoff
frequency at 1 kHz and a minimum attenuation of 40
lowpass filter is given by dB at 5 kHz
ΩcN ΩcN
H a ( s) = C = N = • Now
DN ( s ) s + ∑lN=−01d s l ∏lN=1 ( s − pl )
l 10 log10 ⎛⎜ 1 2 ⎞⎟ = −1
⎝1 + ε ⎠
where which yields ε 2 = 0.25895
pl = Ωc e j[π ( N + 2l−1) / 2 N ] , 1 ≤ l ≤ N and
• Denominator DN (s ) is known as the 10 log10 ⎛⎜ 12 ⎞⎟ = − 40
Butterworth polynomial of order N ⎝A ⎠
which yields A2 = 10,000
13 14
Butterworth Approximation Chebyshev Approximation

• The magnitude-square response of an N-th
• Therefore 1= A2 − 1 = 196.51334
k1 ε order analog lowpass Type 1 Chebyshev filter
is given by
and 1 = Ωs = 5 1
H a ( s) =
2
k Ωp
1 + ε TN (Ω / Ω p )
2 2
• Hence where TN (Ω) is the Chebyshev polynomial

log (1 / k1 ) of order N:
N = 10 = 3.2811
log10 (1 / k ) ⎧ cos( N cos −1 Ω), Ω ≤1
TN (Ω) = ⎨ −1
• We choose N = 4 ⎩cosh( N cosh Ω), Ω >1
15 16
Chebyshev Approximation Chebyshev Approximation

• If at Ω = Ω s the magnitude is equal to 1/A,
• Typical magnitude response plots of the then
analog lowpass Type 1 Chebyshev filter are 1
H a ( jΩ s ) = = 12
2
shown below 1 + ε TN (Ω s / Ω p ) A
2 2
Type 1 Chebyshev Filter
1
N=2
N=3
• Solving the above we get
cosh −1 ( A2 − 1 / ε ) cosh −1 (1 / k1 )
0.8 N=8
Magnitude
0.6 N= =
0.4 cosh −1 (Ω s / Ω p ) cosh −1 (1 / k )
0.2
• Order N is chosen as the nearest integer
0
0 1
Ω
2 3
greater than or equal to the above value
17 18
3
Chebyshev Approximation Chebyshev Approximation
• The magnitude-square response of an N-th • Typical magnitude response plots of the
order analog lowpass Type 2 Chebyshev analog lowpass Type 2 Chebyshev filter are
(also called inverse Chebyshev) filter is shown below
given by Type 2 Chebyshev Filter
1 N=3
H a ( jΩ ) =
2 1 N=5
⎡T (Ω / Ω p ) ⎤
2 0.8 N=7
Magnitude
1+ ε 2 ⎢ N s ⎥
0.6
⎣ TN (Ω s / Ω) ⎦ 0.4
0.2
where TN (Ω) is the Chebyshev polynomial 0
0 1 2 3
of order N Ω
19 20
Chebyshev Approximation Elliptic Approximation

• The order N of the Type 2 Chebyshev filter • The square-magnitude response of an
is determined from given ε , Ω s , and A elliptic lowpass filter is given by
using 1
H a ( jΩ ) =
2
cosh −1 ( A2 − 1 / ε ) cosh −1 (1 / k1 ) 1 + ε RN ( Ω / Ω p )
2 2
N= =
cosh −1 (Ω s / Ω p ) cosh −1 (1 / k )
where RN (Ω) is a rational function of order
• Example - Determine the lowest order of a N satisfying RN (1 / Ω) = 1 / RN (Ω) , with the
Chebyshev lowpass filter with a 1-dB cutoff
frequency at 1 kHz and a minimum attenuation of roots of its numerator lying in the interval
40 dB at 5 kHz - 0 < Ω < 1 and the roots of its denominator
cosh −1 (1 / k1 ) lying in the interval 1 < Ω < ∞
N= = 2.6059
21 cosh −1 (1 / k ) 22
Elliptic Approximation Elliptic Approximation

• For given Ω p , Ω s , ε , and A, the filter order • Example - Determine the lowest order of a elliptic
can be estimated using lowpass filter with a 1-dB cutoff frequency at 1
2 log10 (4 / k1 ) kHz and a minimum attenuation of 40 dB at 5 kHz
N≅
log10 (1 / ρ ) Note: k = 0.2 and 1 / k1 = 196.5134
where k ' = 1 − k 2 • Substituting these values we get
ρ0 = 0.00255135,
ρ0 = 1 − k '
k '= 0.979796,
2(1 + k ') ρ = 0.0025513525
ρ = ρ0 + 2( ρ0 )5 + 15( ρ0 )9 + 150( ρ0 )13 • and hence N = 2.23308
• Choose N = 3
23 24
4
Analog Lowpass Filter Design
Elliptic Approximation • Example - Design an elliptic lowpass filter
of lowest order with a 1-dB cutoff
• Typical magnitude response plots with Ω p = 1 frequency at 1 kHz and a minimum
are shown below attenuation of 40 dB at 5 kHz
Elliptic Filter
N=3
• Code fragments used
1 N=4 [N, Wn] = ellipord(Wp, Ws, Rp, Rs, ‘s’);
0.8
Magnitude
0.6 [b, a] = ellip(N, Rp, Rs, Wn, ‘s’);

0.4 with Wp = 2*pi*1000;
0.2
Ws = 2*pi*5000;
0
0 1 2 3 Rp = 1;
Ω
25 26 Rs = 40;
Design of Analog Highpass,

Highpass,
Analog Lowpass Filter Design Bandpass and Bandstop Filters
• Steps involved in the design process:
• Gain plot Step 1 - Develop of specifications of a
Lowpass Elliptic Filter prototype analog lowpass filter H LP (s)
0
from specifications of desired analog filter
HD (s ) using a frequency transformation
Gain, dB
-20
Step 2 - Design the prototype analog
-40
lowpass filter
-60 Step 3 - Determine the transfer function HD (s )
0 2000 4000 6000
Frequency, Hz of desired analog filter by applying the
inverse frequency transformation to H LP (s)
27 28
Design of Analog Highpass,

Highpass,
Analog Highpass Filter Design
Bandpass and Bandstop Filters
• Let s denote the Laplace transform variable • Spectral Transformation:
of prototype analog lowpass filter H LP (s) Ω p Ω̂ p
and ŝ denote the Laplace transform s=
sˆ
variable of desired analog filter HD (sˆ)
where Ω p is the passband edge frequency of
• The mapping from s-domain to ŝ -domain is H LP ( s) and Ω̂ p is the passband edge
given by the invertible transformation
frequency of H HP ( sˆ)
s = F (sˆ)
• On the imaginary axis the transformation is
• Then H D ( s
ˆ ) = H LP ( s ) s = F ( sˆ)
Ω pΩ ˆp
H LP ( s ) = HD ( sˆ) sˆ = F −1 ( s ) Ω=−
Ω
ˆ
29 30
5
• Example - Design an analog Butterworth
Ω pΩ ˆp highpass filter with the specifications:
Ω=− Fˆ p = 4 kHz, Fˆs = 1 kHz, α p = 0.1 dB,
Ωˆ
α s = 40 dB
Ω
Stopband − Ω s − Ω p Ω p Ωs Stopband • Choose Ω p = 1
0
Ω Lowpass
Passband • Then 2πFˆ p Fˆ p 4000
Ωs = = = =4
Passband Stopband Passband 2πFˆs Fˆs 1000
−Ω
ˆ p −Ω
ˆs 0 Ω̂ s Ω̂ p
Ω̂ Highpass • Analog lowpass filter specifications: Ω p = 1,
31 32
Ω s = 4 , α p = 0.1 dB, α s = 40 dB
Analog Highpass Filter Design Analog Bandpass Filter

• Code fragments used Design
[N, Wn] = buttord(1, 4, 0.1, 40, ‘s’); • Spectral Transformation
[B, A] = butter(N, Wn, ‘s’); sˆ 2 + Ω
ˆ o2
[num, den] = lp2hp(B, A, 2*pi*4000); s = Ωp
sˆ(Ω
ˆ p2 − Ω ˆ p1 )
• Gain plots
Prototype Lowpass Filter Highpass Filter where Ω p is the passband edge frequency
0 0
of H LP (s ), and Ω
ˆ p1 and Ω
ˆ p 2 are the lower
-20 -20
and upper passband edge frequencies of
Gain, dB
Gain, dB
-40 -40
desired bandpass filter H BP (sˆ)
-60 -60
-80 -80
0 2 4 6 8 10 0 2 4 6 8 10
33 Ω Frequency, kHz 34
Analog Bandpass Filter Analog Bandpass Filter

Design Design
Ω
ˆ o2 − Ω
ˆ2
Ωˆ 2 −Ω
ˆ2 Ω = −Ω p
Ω = −Ω p o Ω ˆ Bw
Ω
ˆ Bw
Stopband − Ω s − Ω p Ω p Ωs
where Bw = Ω ˆ p2 − Ω
ˆ p1 is the width of Stopband
Ω Lowpass
0
passband and Ω̂ o is the passband center Passband
frequency of the bandpass filter

Stopband Passband Stopband Passband Stopband
• Passband edge frequency ± Ω p is mapped ˆ s2 ↓ − Ω
−Ω ˆ ↓ −Ω ˆ s1 ↓ Ω̂ o ↓ Ω
ˆ s1 0 Ω ˆ s2
Ω̂ Bandpass
ˆ p2 o− Ω
into m Ω
ˆ p1 and ± Ωˆ p 2, lower and upper −Ω ˆ p1 Ω
ˆ p1 Ω
ˆ p2
passband edge frequencies

35 36
6
Design Design
• Stopband edge frequency ± Ω s is mapped • Case 1: Ωˆ p1Ω
ˆ p2 > Ω
ˆ s1Ω
ˆ s2
ˆ s1 and ± Ω
into m Ω ˆ s 2 , lower and upper
To make Ω p1Ω p 2 = Ω s1Ω
ˆ ˆ ˆ ˆ s 2 we can either
stopband edge frequencies
increase any one of the stopband edges or
• Also, decrease any one of the passband edges as
Ωˆ o2 = Ω
ˆ p1Ω
ˆ p2 = Ω ˆ s1Ω
ˆ s2 shown below
• If bandedge frequencies do not satisfy the ← Passband ←
above condition, then one of the frequencies
needs to be changed to a new value so that Stopband → → Stopband
Ω̂
the condition is satisfied Ω
ˆ s1 Ω
ˆ p1 Ω
ˆ p2 Ω
ˆ s2
37 38

Design Design
(1) Decrease Ω ˆ p1 to Ωˆ s1Ω
ˆ s2 / Ωˆ p2 • Note: The condition Ω ˆ o2 = Ω
ˆ p1Ω
ˆ p2 = Ω
ˆ s1Ω
ˆ s2
larger passband and shorter can also be satisfied by decreasing Ω ˆ p2
leftmost transition band which is not acceptable as the passband is
(2) Increase Ωˆ s1 to Ωˆ p1Ωˆ p2 / Ωˆ s2 reduced from the desired value
No change in passband and shorter • Alternately, the condition can be satisfied
leftmost transition band by increasing Ω ˆ s 2 which is not acceptable
as the upper stop band is reduced from the
desired value
39 40
Analog Bandpass Filter

Design
Design
• Case 2: Ωˆ p1Ω
ˆ p2 < Ω
ˆ s1Ωˆ s2
To make Ω ˆ p1Ω
ˆ p2 = Ωˆ s1Ωˆ s 2 we can either
decrease any one of the stopband edges or (1) Increase Ω ˆ p 2 to Ωˆ s1Ω
ˆ s2 / Ω
ˆ p1
increase any one of the passband edges as larger passband and shorter
shown below rightmost transition band
→ Passband (2) Decrease Ω ˆ s 2 to Ω
ˆ p1Ωˆ p2 / Ω
ˆ s1
→
No change in passband and shorter
←
Stopband ← Stopband
Ω̂
rightmost transition band
Ω
ˆ s1 Ω
ˆ p1 Ω
ˆ p2 Ω
ˆ s2
41 42
7
Design Design
• Example - Design an analog elliptic
• Note: The condition Ω ˆ o2 = Ω
ˆ p1Ω
ˆ p2 = Ω
ˆ s1Ω
ˆ s2 bandpass filter with the specifications:
can also be satisfied by increasing Ω ˆ p1 Fˆ p1 = 4 kHz, Fˆ p 2 = 7 kHz, Fˆs1 = 3 kHz
which is not acceptable as the passband is Fs 2 = 8 kHz, α p = 1 dB, α s = 22 dB
ˆ
reduced from the desired value
• Alternately, the condition can be satisfied • Now Fˆ p1Fˆ p 2 = 28 × 106 and Fˆs1Fˆs 2 = 24 × 106
by decreasing Ω ˆ s1 which is not acceptable
• Since Fˆ p1Fˆ p 2 > Fˆs1Fˆs 2 we choose
as the lower stopband is reduced from the
desired value Fˆ p1 = Fˆs1Fˆs 2 / Fˆ p 2 = 3.571428 kHz
43 44

Analog Bandpass Filter Design
Design • Code fragments used
[N, Wn] = ellipord(1, 1.4, 1, 22, ‘s’);
• We choose Ω p = 1 [B, A] = ellip(N, 1, 22, Wn, ‘s’);
• Hence [num, den]
24 − 9 = lp2bp(B, A, 2*pi*4.8989795, 2*pi*25/7);
Ωs = = 1.4 • Gain plot
(25 / 7) × 3 Prototype Lowpass Filter Bandpass Filter
• Analog lowpass filter specifications: Ω p = 1,

0
0
Ω s = 1.4 , α p = 1 dB, α s = 22 dB
Gain, dB
Gain, dB
-20
-20
-40 -40
-60 -60
45 46 0 2 4 6 8 0 5 10 15
Copyright © 2005, S. K. Mitra Ω Frequency, kHz © 2005, S. K. Mitra
Copyright
Analog Bandstop Filter Design Analog Bandstop Filter Design

• Spectral Transformation ΩˆB
ˆ s2 − Ω
sˆ(Ω ˆ s1 ) Ω = Ωs 2 w 2
s = Ωs Ω
ˆ o −Ω ˆ
sˆ + Ω
2 ˆ 2o where Bw = Ω s 2 − Ω s1 is the width of
ˆ ˆ
where Ω s is the stopband edge frequency stopband and Ω̂o is the stopband center
of H LP ( s ) , and Ω
ˆ s1 and Ω
ˆ s 2 are the lower frequency of the bandstop filter
and upper stopband edge frequencies of the • Stopband edge frequency ± Ω s is mapped
desired bandstop filter H BS ( sˆ) into m Ωˆ s1and ± Ω
ˆ s 2 , lower and upper
stopband edge frequencies
47 48
8
• Passband edge frequency ± Ω p is mapped • Also,
into m Ω
ˆ p1 and ± Ω
ˆ p 2 , lower and upper Ω
ˆ o2 = Ω
ˆ p1Ω
ˆ p2 = Ω
ˆ s1Ω
ˆ s2
passband edge frequencies • If bandedge frequencies do not satisfy the
Stopband − Ω s − Ω p Ω p Ωs Stopband above condition, then one of the frequencies
Ω Lowpass
0
Passband
needs to be changed to a new value so that
the condition is satisfied
Passband Stopband Passband Stopband Passband
Ω̂ Bandpass
ˆ p2 ↓ − Ω
−Ω ˆ o ↓ −Ω ˆ p10 Ω
ˆ p1 ↓ Ω̂ o ↓ Ω ˆ p2
−Ω
ˆ s2 −Ω
ˆ s1 Ωˆ s1 Ω
ˆ s2
49 50

• Case 1: Ωˆ p1Ω
ˆ p2 > Ω
ˆ s1Ωˆ s2
• To make Ω ˆ p1Ω
ˆ p2 = Ωˆ s1Ωˆ s 2 we can either
increase any one of the stopband edges or (1) Decrease Ω ˆ p 2 to Ωˆ s1Ωˆ s2 / Ω
ˆ p2
decrease any one of the passband edges as larger high-frequency passband
shown below and shorter rightmost transition band
(2) Increase Ωˆ s 2 to Ω ˆ p1Ωˆ p2 / Ωˆ s2
Passband
No change in passbands and
Passband
shorter rightmost transition band
Stopband
Ω̂
51 Ω
ˆ p1 Ω
ˆ s1 Ω
ˆ s2 Ω
ˆ p2 52

ˆ p1Ω
ˆ p2 = Ωˆ s1Ω
ˆ s2 • Case 1: Ωˆ p1Ω
ˆ p2 < Ω
ˆ s1Ω
ˆ s2
can also be satisfied by decreasing • To make Ω p1Ω p 2 = Ω s1Ω
ˆ ˆ ˆ ˆ s 2 we can either
which is not acceptable as the low- Ω ˆ p1
decrease any one of the stopband edges or
frequency passband is reduced from the increase any one of the passband edges as
desired value shown below
• Alternately, the condition can be satisfied
by increasing Ω ˆ s1 which is not acceptable
Passband Passband
as the stopband is reduced from the desired
value Stopband
Ω̂
53 54 Ω
ˆ p1 Ω
ˆ s1 Ω
ˆ s2 Ω
ˆ p2
9
ˆ p1Ω
ˆ p2 = Ω
ˆ s1Ω
ˆ s2
(1) Increase Ωˆ p1 to Ω
ˆ s1Ω
ˆ s2 / Ω
ˆ p1 can also be satisfied by increasing Ω ˆ p2
larger passband and shorter which is not acceptable as the high-
leftmost transition band frequency passband is decreased from the
desired value
(2) Decrease Ωˆ s1 to Ω
ˆ p1Ω
ˆ p2 / Ωˆ s1
• Alternately, the condition can be satisfied
No change in passbands and by decreasing Ω ˆ s 2 which is not acceptable
shorter leftmost transition band as the stopband is decreased
55 56
10

Analog Lowpass Filter Specifications Analog Lowpass Filter Specifications

Uploaded by

Copyright:

Available Formats

Analog Lowpass Filter Specifications Analog Lowpass Filter Specifications

Uploaded by

Document Information

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Analog Lowpass Filter Specifications Analog Lowpass Filter Specifications

Uploaded by

Copyright:

Available Formats

Analog Lowpass Filter Analog Lowpass Filter

Analog Lowpass Filter Analog Lowpass Filter

Analog Lowpass Filter

Butterworth Approximation Butterworth Approximation

0.6 passband magnitude 1 / 1 + ε 2, and

Butterworth Approximation Butterworth Approximation

Butterworth Approximation Chebyshev Approximation

• Hence where TN (Ω) is the Chebyshev polynomial

Chebyshev Approximation Chebyshev Approximation

Chebyshev Approximation Elliptic Approximation

Elliptic Approximation Elliptic Approximation

0.6 [b, a] = ellip(N, Rp, Rs, Wn, ‘s’);

Design of Analog Highpass,

Design of Analog Highpass,

Analog Highpass Filter Design Analog Bandpass Filter

Analog Bandpass Filter Analog Bandpass Filter

frequency of the bandpass filter

passband edge frequencies

Analog Bandpass Filter Analog Bandpass Filter

Analog Bandpass Filter

Analog Bandpass Filter

• Analog lowpass filter specifications: Ω p = 1,

Analog Bandstop Filter Design Analog Bandstop Filter Design

Analog Bandstop Filter Design Analog Bandstop Filter Design

Analog Bandstop Filter Design Analog Bandstop Filter Design

You might also like