Analog Filters: Network Functions
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Analog Filters: Network Functions
Franco Maloberti
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Franco Maloberti Analog Filters: Network Functions
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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
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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(2) and that for obtaining Q(s) are the same
H(s)H ( s) H ( j ) 2 s22
A( 2 )B( 2)
2 s2
H(s) P(s)Q(s)
P(s)P( s) A( 2) 2 s2 and Q(s)Q( s) B( 2)
2 s2
H (j) 2 H(s)H ( s)sj
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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(2) are
Quadrant symmetry
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General Procedure (iii)
In A(2) replace 2 by -s2 Factor A(-s2) and determine zeros Split pair of real zeros and complex mirrored conjugateExample
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
P(s)P( s) A( 2 ) 2 s2
A( s 2 ) (s 2)(s 2)(s 2 2s 5)(s2 2s 5)(s 2 6)
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General Procedure (iv)
EXAMPLE
H (j) 2 2 3
2 ( 4 6 2 25)
H(s)H ( s) s 2 3
(s 6 6s 4 25)
H(s)H ( s) (s 3)(s 3)
s2 (s12 j)(s1 2 j )(s 1 2 j )(s 1 2 j)one
NO
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Butterworth Network Functions
Remember that
therefore:
The zeros of Q are obtained by
Therefore
Bn j 2
11 2n
Bn s Bn s 11 ( s 2)n
1 ( s2)n ( s 2)n 1 e j( 2 k )
s2 ej2k1n
or s2 e
j2k1n
sk e
j2k12n
2
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Butterworth Network Functions
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Chebyshev Network Functions
Remember that
Therefore
The zeros of Q are obtained by
Let
CHn j 2
112Cn
2(); 2 s 2
CHn s CHn s 1
Q(s)Q( s)
11 2Cn
2( js)
Cn js cos n cos 1( js) j
cos 1 ( js) u jv jscos(u jv) cosu cosh v j sinu sinhv
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Chebyshev Network Functions
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Chebyshev Network Functions (ii)
Equation
Becomes
Equating real and imaginary parts
cos n cos 1 ( js) j
cos n(u jv) cos nu cosh nv j sinnusinhnv j
cosnu cosh nv 0; sinnusinhnv 1
For a real v this is > 1
cosnu 0
u (2k 1)2n
sinnu 1
v 1n
sinh 1 1
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Chebyshev Network Functions (iii)
Remember that
Therefore
The real and the imaginary part of k are such that
Zeros lie on an ellipse.
sk k j k sin(2k 1)
2n
sinh v j cos
(2k 1)2n
cosh v
u(2k 1)
2n
v 1n
sinh 1 1
jscos(u jv) cos u coshv j sin usinhv sinu sinhv j cos u cosh v
k2
sinh2 v
k2
cosh2 v1
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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
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Elliptic Filter
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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
h(t) (t )
H(s) e s
H(s) 1e s
1sinh(s ) cosh(s )
sinh(s )
1 cosh(s )sinh(s )
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Bessel-Thomson Filter Function (ii)
Find an approximation of in the form
And set
Approximations of
Example
cosh xsinh x
M (s)N (s)
H(s) K
M (s) N (s)
cosh x 1 s2
2! s
4
4! s
6
6!
sinh x s s3
3! s
5
4! s
7
7!
H3(s) 15
s3 6s 2 15s15
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Bessel-Thomson Filter
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Different Filter Comparison
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Different Filter Comparison
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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
H(s) (s si)
i
(s si)
i
j si j si
phasej sij si
2tan 1 ( i )
( i); si i ji
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Delay Response
Examples
s 1s1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
3.5
4Delay Response
s2 2 s 1s2 2 s 1