Analog Filters: The Approximation
Franco Maloberti
Introduction

The first step of the normalized filter design is to
find a magnitude characteristics, |H(jw)|, that fulfills
the specifications

Handling |H(jw)| it is difficult. It is better to deal with
|H(jw)|2.
H( jw )  H(s)H(s) s jw  H( jw )H * ( jw )
2

|H(jw)|2 is a rationale function in w2

Franco Maloberti
Analog Filters: The Approximation
2
Example

Consider the transfer function
s 2
H(s)  3
s  2s2  2s  3
(s  2)(s  2)
s2  4
H(s)H(s)  3
 6
2
3
2
(s  2s  2s  3)(s  2s  2s  3) s  8s2  9
w2  4
H( jw )  6
w  8w 2  9
2
Much easier to deal than

w2  4
H( jw) 
w 6  8w 2  9
2
Franco Maloberti
Analog Filters: The Approximation
3
Methodology

The approximation phase is based on the following:
2
 Find a real rationale function in w capable to
met the magnitude specifications (pass-band
and stop-band)
 Live with the resulting phase response
 Correct (if necessary) the phase response (or
delay) with a cascaded network
 Work on low-pass characteristics (we will se
shortly how to achieve high-pass, band-pass, …
from a low-pass TF)
Franco Maloberti
Analog Filters: The Approximation
4

Generic Low-Pass TF

A generic form of a low-pass TF is:
A0
H( jw ) 
1 F(w 2 )
2
F(w 2 )  1
for 0  w  w p
F(w 2 )  1
for
w  w p
In the pass-band H(w)=A0
Franco Maloberti
A0
Analog Filters: The Approximation
wp
5
Butterworth Characteristics

A very simple form for F(w2) is
F(w 2 )  w 2n
A0
H( jw ) 
1 w 2n
2

n is the order of the Butterworth filter
 Properties:

H( jw ) Max  H( j0)  1
H( j1) 
Franco Maloberti
1
(3 dB or half  power)
2
Analog Filters: The Approximation
6
Maximally Flat

The magnitude response of the Butterworth filter is
said to be maximally flat.
H( jw)  (1 w 2n )1/2

If we take the first 2n-1 derivatives of H(jw), they
are all zero at w=0
1

0.9
0.8
0.7
0.6
n=8
w=0,…,2
0.5
0.4
0.3
0.2
0.1
0
Franco Maloberti
0
10
20
30
Analog Filters: The Approximation
40
50
60
70
80
90
100
7
Butterworth Plots; |H(jw)|2
clear all;
omega=linspace(0,2,100);
for k=1:6
n=k;
for i=1:100
Butter(i,k)=1/(1+omega(i)^(2*n));
end
end
plot(omega,Butter);
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
All curves cross 0.5
For w=1
Franco Maloberti
0.1
0
0
0.2
0.4
0.6
0.8
Analog Filters: The Approximation
1
1.2
1.4
1.6
1.8
2
8
Butterworth Plots


For w>>1 |H(iw)|2 ≈ 1/(w2n)
H(jw)|dB = -20 n log w
0
clear all;
omega=linspace(0,100,100);
for k=1:4
n=k;
for i=1:100
Butter(i,k)=1/(1+omega(i)^(2*n));
ButdB(i,k) = 20* log10(Butter(i,k));
end
end
semilogx(omega,ButdB);
Franco Maloberti
-100
-200
-300
-400
-500
-600
-700 0
10
Analog Filters: The Approximation
10
1
10
2
9
Design of Butterworth filters

Specifications
dB
s
p
|H(jw)|2
 s  10log(1 w s2n )
 p  10log(1 w 2n
p )
w s2n  10
w
2n
p
s
/10
 p /10
 10
1
1
 s /10
w 
10
1
s

w 
  10 p /10 1
 p 
2n
Franco Maloberti
wp
ws
The values of s, p
and the ratio ws/wp
determine the order
n of the filter.
Analog Filters: The Approximation
10
Chebyshev Characteristics

The Chebishev filters are based on the Chebishev
polynomials
Cn (w)  cos(n cos1 w )


n is the order
It is a polynomial!

cos  w
  cos1 w
Cn (w )  cos(n )
Franco Maloberti
Analog Filters: The Approximation
11
Chebyshev Polynomials
Cn (w )  cos(n )
Cn 1(w )  cos(n   )  cosn cos  sin n sin
Cn1(w )  cos(n   )  cosn cos  sin n sin 
Cn 1(w )  Cn1 (w )  2cosn cos
Cn (w )
w
Cn 1 (w )  2wCn (w )  Cn1(w )

Franco Maloberti
Analog Filters: The Approximation
12
Chebyshev Polynomials
C0 (w )  cos0  1
C1(w )  cos(cos1 w )  w
C2 (w )  2w 2 1
C3 (w )  2(2w 2 1)w  w  4w 3  3w
C4 (w )  2(4w 3  3w )w  (2w 2 1)  8w 4  8w 2  1
Cn (w )  2n1w n 


Franco Maloberti
Analog Filters: The Approximation
13
Chebyshev Polynomials Features


If w varies in the range -1 +1
 Cn(w)=cos(n+) limited to the range -1 +1
If |w| is larger than 1
-1 w) increases monotonically
 Cn(w)=cosh(n cosh
with wn.
15
10
clear all;
n=7
xmax=1+5/n^2;
w=linspace(-xmax,xmax,200);
for i=1:200
C(i)=cos(n*acos(w(i)));
end
plot(w,C);
5
0
-5
-10
-15
-1.5
Franco Maloberti
-1
-0.5
Analog Filters: The Approximation
0
0.5
1
1.5
14
Chebyshev low-pass

Choose a small number e and let

F(w 2 )  e 2Cn2 (w )
the low-pass transfer function is
H( jw ) 
2


1
1 e 2Cn2 (w )

Normalize wp = 1
Maximum attenuation in the pass-band

1
2
H( jw ) min 
1 e 2
 s  10log(1 e 2 )
Franco Maloberti
Analog Filters: The Approximation
15
Chebyshev low-pass Plots
1
0.9
0.8
0.7
0.6
clear all;
n = input('what is the order? ');
eps = input('what is the value of epsilon? ');
xmax=2;
w=linspace(0,xmax,200);
for i=1:200
C(i)=cos(n*acos(w(i)));
H(i)=1/(1+(eps*C(i))^2);
end
plot(w,H);
0.5
Order 4
Eps=0.4
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
1
0.9
0.8
0.7
0.6
0.5
Order 7
Eps=0.2
0.4
0.3
0.2
0.1
0
Franco Maloberti
0
0.2
0.4
0.6
Analog Filters: The Approximation
0.8
16
Chebyshev Filter Design


Start from specifications
 p 10log(1 e 2 )


|H(jw)|2
Determine e
 p /10

dB
s
p
e  10
1
wp
ws
The order of the filter must be such that
 s 10log[1 e2Cn2 (ws / w p ))]
Remember that

Cn (w s / w p )  cosh(cosh1(w s / w p ))
n
Franco Maloberti
cosh1[Cn (w s / w p )]
cosh1 (w s / w p )
Analog Filters: The Approximation
17
Other Chebyshev Responses
Define another function

Ha ( jw) 1 H( jw)
2

2
Use in the Matlab program
H(i)=1-1/(1+(eps*C(i))^2);

1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
Franco Maloberti
1
1.2
1.4
1.6
1.8
2
0
0
0.2
0.4
Analog Filters: The Approximation
0.6
0.8
1
1.2
1.4
1.6
1.8
2
18
Inverse Chebyshev High-pass
H b ( jw ) 
2
1
1 e 2Cn2 (1 / w )
1
clear all;

n = input('what is the order? ');
eps = input('what is epsilon? ');
xmax=2;
w=linspace(0,xmax,200);
for i=1:200
C(i)=cos(n*acos((1/w(i))));
H(i)=1/(1+(eps*C(i))^2);
end
plot(w,H);
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Franco Maloberti
0
0.2
0.4
0.6
Analog Filters: The Approximation
0.8
1
1.2
1.4
1.6
1.8
2
19
Inverse Chebyshev Low-pass
H c ( jw )  1 H b ( jw )  1
2
2
1
1 e 2Cn2 (1 / w )
1
0.9

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Franco Maloberti
0
0.5
1
1.5
2
2.5
3
Analog Filters: The Approximation
3.5
4
20
Elliptic-Function


Butterworth and Chebyshev don’t have finite zeros
 All-pole functions
Finite zeros help in the pass-band stop-band
transition (see inverse Chebyshev)
Hc ( jw ) 
2

P( jw )
2
Q( jw )
2
Elliptic functions are a special category of zeropole transfer functions

Franco Maloberti
Analog Filters: The Approximation
21
Elliptic-Function (ii)

Define RN(w)
(w12  w 2 )(w 22  w 2 ) (w N2  w 2 )
RN (w ) 
(1 w12w 2 )(1 w 22w 2 ) (1 w N2 w 2 )
or


w (w12  w 2 )(w 22  w 2 ) (w N2  w 2 )
RN (w ) 
(1 w12w 2 )(1 w 22w 2 ) (1 w N2 w 2 )
Poles and zeros are reciprocal of one another
2
  If w1, w2,… are in the range 0 ≤ w ≤1 |R(w)| is
zero at w1, w2,… and infinite at 1/w1, 1/w2,…
Franco Maloberti
Analog Filters: The Approximation
22
Elliptic-Function (iii)

The elliptic characteristics is
H( jw ) 
2
1
1 e 2 RN2 (w )

Franco Maloberti
Analog Filters: The Approximation
23
Plot of the Elliptic Function
-6
3
1.0005
x 10
1
clear all;
n=5;
Stop
eps=8;
xmax=0.91;
om1sq=0.1;
Pass
om2sq=0.4;
om3sq=0.65;
om4sq=0.8;
omega=linspace(0,xmax,200);
for i=1:200
w=omega(i);
Num=w*(om1sq-w*w)*(om2sq-w*w)*(om3sq-w*w)*(om4sq-w*w);
Den=(1-om1sq*w*w)*(1-om2sq*w*w)*(1-om3sq*w*w)*(1-om4sq*w*w);
R(i)=Num/Den;
Ellip(i)=1/(1+(eps*R(i))^2);
end
plot(omega,Ellip);
2.5
0.9995
2
0.999
0.9985
1.5
0.998
1
0.9975
0.997
0.5
0.9965
0.996
Franco Maloberti
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1.1
1.2
Analog Filters: The Approximation
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
24
Determine the order of a Filter
»help buttord
BUTTORD Butterworth filter order selection.
[N, Wn] = BUTTORD(Wp, Ws, Rp, Rs) returns the order N of the lowest
order digital Butterworth filter that loses no more than Rp dB in
the passband and has at least Rs dB of attenuation in the stopband.
Wp and Ws are the passband and stopband edge frequencies, normalized
from 0 to 1 (where 1 corresponds to pi radians). For example,
Lowpass: Wp = .1,
Ws = .2
Highpass: Wp = .2,
Ws = .1
Bandpass: Wp = [.1 .8], Ws = [.2 .7]
Bandstop: Wp = [.2 .7], Ws = [.1 .8]
BUTTORD also returns Wn, the Butterworth natural frequency (or,
the "3 dB frequency") to use with BUTTER to achieve the specifications.
[N, Wn] = BUTTORD(Wp, Ws, Rp, Rs, 's') does the computation for an
analog filter, in which case Wp and Ws are in radians/second.
Franco Maloberti
Analog Filters: The Approximation
25
Verification
Butterorder
clear all;
omega=linspace(0,1.2,200);
n=12;
for i=1:200
Butterw(i)=1/(1+omega(i)^(2*n));
end
plot(omega,Butterw);
Wp=80;
Rp=10*log10(Butterw(Wp+1));
Ws=180;
Rs=10*log10(Butterw(Ws));
[N, Wn] = buttord(Wp, Ws, Rp, Rs, 's')
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
»Butterorder
N = 12
Wn = 166.7598
Franco Maloberti
Analog Filters: The Approximation
26
Order of Chebishev or Elliptic filters
[N, Wn] = CHEB1ORD(Wp, Ws, Rp, Rs, 's') does the computation for an
analog filter, in which case Wp and Ws are in radians/second.
[N, Wn] = ELLIPORD(Wp, Ws, Rp, Rs, 's') does the computation for an
analog filter, in which case Wp and Ws are in radians/second.
Comparing the orders
clear all;
Wp=2*pi;
k=input('normalized stop frequency ')
Ws=k*Wp;
Rp=input('attenuation passband ')
Rs=input('attenuation stopband ')
[Nbutter, Wn] = BUTTORD (Wp, Ws, Rp, Rs, 's')
[NCheb1, Wn] = CHEB1ORD(Wp, Ws, Rp, Rs, 's')
[Nellip, Wn] = ELLIPORD(Wp, Ws, Rp, Rs, 's')
Franco Maloberti
»OrderFilter
normalized stop frequency 1.2
attenuation passband 0.2
attenuation stopband 25
Nbutter = 25
Wn = 6.7203
NCheb1 = 9
Wn = 6.2832
Nellip = 5
Wn = 6.2832
Analog Filters: The Approximation
27
Butter and Cheb with same specs
stop frequency 1.2
attenuation PB 0.2 dB
attenuation SB 25 dB
Nbutter = 25
Wn = 6.7203
NCheb1 = 9
Wn = 6.2832
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Franco Maloberti
0
0.5
1
1.5
Analog Filters: The Approximation
2
2.5
28
Butter and Cheb with same specs (ii)
clear all;
point=2000;
range=2;
omega=linspace(0,range,point)+range/2/point;
n=25;
m=9;
eps=0.186;
k=6.7203/6.2832;
for i=1:point
Butter(i)=1/(1+(omega(i)/k)^(2*n));
C(i)=cos(m*acos(omega(i)));
Cheb(i)=1/(1+(eps*C(i))^2);
end
plot(omega,Butter,omega,Cheb);
Butter(1000)
Cheb(1000)
Butter(1200)
Cheb(1200)
1
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.9
0.9
Franco Maloberti
0.92
0.94
0.96
0.98
Analog Filters: The Approximation
1
1.02
1.04
1.06
1.08
1.1
29