1. Analog Filters: Network Functions
Franco Maloberti

2. Analog Filters: Network Functions
Introduction
Magnitude characteristic
Network function
Realizability
Can be implemented with real-world components
No poles in the right half-plane
Instability:
goes in the non-linear region of operation of the active or passive components
Self destruct
Franco Maloberti
Analog Filters: Network Functions

3. Analog Filters: Network Functions
General Procedure
The approximation phase determines the magnitude characteristics
This step determines the network function H(s)
Assume that
The procedure to obtain P(s) for a given A(w2) and that for obtaining Q(s) are the same
Franco Maloberti
Analog Filters: Network Functions

4. General Procedure (ii)
P(s) is a polynomial with real coefficients
Zeros of P(s) are real or conjugate pairs
Zeros of P(-s) are the negative of the zeros of P(s)
Zeros of A(w2) are
Quadrant symmetry
Franco Maloberti
Analog Filters: Network Functions

5. General Procedure (iii)
In A(w2) replace jw by -s2
Factor A(-s2) and determine zeros
Split pair of real zeros and complex mirrored conjugate
Example
Four possible choices, but …. B(s) must be Hurwitz, for a the choice depends on minimum-phase requirements
The polynomial A(s) [or B(s)] results
Franco Maloberti
Analog Filters: Network Functions

6. General Procedure (iv)
EXAMPLE
one NO
Franco Maloberti
Analog Filters: Network Functions

7. Analog Filters: Network Functions
Use of Matlab
% Specify coefficient vector
% a=w^6+3*w^4+12*w^2+100
a=[ ]
% Obtain zero roots
b= roots(a)
% Plot the zeros
zplane(b)
% Form the polynomial
x1=input('first zero is # ')
x2=input('second zero is # ')
x3=input('third zero is # ')
c= poly([b(x1) b(x2) b(x3)])
Franco Maloberti
Analog Filters: Network Functions

8. Butterworth Network Functions
Remember that
therefore:
The zeros of Q are obtained by
Therefore
Franco Maloberti
Analog Filters: Network Functions

9. Butterworth NF with Matlab
»ButterNet
order of the filter 5
n = 5
a =
b = i i i i i i
1.0000 i i
c =
Result with n=5
m-file
clear all;
n=input('order of the filter ')
zerocoeff=2*n-1;
lastcoeff=(-1)^n;
a=[1 zeros(1,zerocoeff) lastcoeff]
b=roots(a)
c=poly([b(1:n)])
Franco Maloberti
Analog Filters: Network Functions

10. Butterworth NF with Matlab (ii)
BUTTAP Butterworth analog lowpass filter prototype.
[Z,P,K] = BUTTAP(N) returns the zeros, poles, and gain
for an N-th order normalized prototype Butterworth analog
lowpass filter. The resulting filter has N poles around
the unit circle in the left half plane, and no zeros.
clear all;
n=input('order of the filter ')
[z p k] =buttap(n)
zplane(p)
c=poly(p)
Franco Maloberti
Analog Filters: Network Functions

11. Chebyshev Network Functions
Remember that
Therefore
The zeros of Q are obtained by
Let
Franco Maloberti
Analog Filters: Network Functions

12. Chebyshev Network Functions (ii)
Equation
Becomes
Equating real and imaginary parts
For a real v this is > 1
Franco Maloberti
Analog Filters: Network Functions

13. Chebyshev Network Functions (iii)
Remember that
Therefore
The real and the imaginary part of wk are such that
Zeros lie on an ellipse.
Franco Maloberti
Analog Filters: Network Functions

14. Chebyshev NF with Matlab
CHEB1AP Chebyshev type I analog lowpass filter prototype.
[Z,P,K] = CHEB1AP(N,Rp) returns the zeros, poles, and gain
of an N-th order normalized prototype type I Chebyshev analog
lowpass filter with Rp decibels of ripple in the passband.
Type I Chebyshev filters are maximally flat in the stopband.
%CHEBYNET
clear all;
N=input('order of Chebyshev ')
Rp=input('ripple in the pb (dB) ')
[z,p,k]=cheb1ap(N,Rp)
figure
zplane(p)
e=poly(p) k
0.1 dB
Franco Maloberti
Analog Filters: Network Functions

15. NF for Elliptic Filters
Obtained without obtaining the prior magnitude characteristics
Based on the use of the Conformal transformation
Mapping of points in one complex plane onto another complex plain (angular relationships are preserved)
Mapping of the entire s-plane onto a rectangle in the p-plane
sn is the Jacobian elliptic sine function
Derivation complex and out of the scope of the Course
Design with the help of Matlab
Franco Maloberti
Analog Filters: Network Functions

16. Elliptic NF with Matlab
ELLIPAP Elliptic analog lowpass filter prototype.
[Z,P,K] = ELLIPAP(N,Rp,Rs) returns the zeros, poles, and gain
of an N-th order normalized prototype elliptic analog lowpass
filter with Rp decibels of ripple in the passband and a
stopband Rs decibels down.
%ElliptNet
clear all;
N=input('order of the Elliptic ')
Rp=input('ripple in the pb (dB) ')
Rs=input('stopband attenuation (dB) ')
[z,p,k]=ellipap(N,Rp,Rs)
figure
zplane(z,p)
num=poly(z)
den=poly(p) k
N=4
Rp=1dB
Rs=25dB
Franco Maloberti
Analog Filters: Network Functions

17. Elliptic NF with Matlab (ii)
[n1 n2]=size(num);
[n3 n4]=size(den);
xmax = input('what is the max plotted freq? ');
npoints=500;
w0=linspace(0,xmax,npoints);
p1=0;
for m=1:npoints
w=w0(m);
for j=1:n2
p1=p1+num(j)*(i*w)^(n2-j);
end
numer=abs(p1);
for j=1:n4
p1=p1+den(j)*(i*w)^(n4-j);
denom=abs(p1);
H(m)=k*numer/denom;
figure
plot(w0,H)
Estimate the
Module response
Franco Maloberti
Analog Filters: Network Functions

18. Elliptic NF with Matlab (iii)
»ElliptResp
order of the Elliptic 4
N = 4
ripple in the pb (dB) 1
stopband attenuation (dB) 20
z = i i i i
p = i i i i
k =
what is the max plotted freq? 10
Franco Maloberti
Analog Filters: Network Functions

19. Bessel-Thomson Filter Function
Useful when the phase response is important
Video applications require a constant group delay in the pass band
Design target: maximally flat delay
Storch procedure
Franco Maloberti
Analog Filters: Network Functions

20. Bessel-Thomson Filter Function (ii)
Find an approximation of in the form
And set
Approximations of
Example
Franco Maloberti
Analog Filters: Network Functions

21. Analog Filters: Network Functions
Delay Equalizer
It is a filter cascaded to a filter able to achieve a given magnitude response for changing the phase response
It does not disturb the magnitude response
Made by all-pass filter
The magnitude response is 1 since
Moreover
Franco Maloberti
Analog Filters: Network Functions

22. Analog Filters: Network Functions
Examples
Franco Maloberti
Analog Filters: Network Functions

23. Analog Filters: Network Functions
Examples
Franco Maloberti
Analog Filters: Network Functions