Analog Filters: Network Functions Franco Maloberti.

Analog Filters: Network Functions

Franco Maloberti

Franco Maloberti Analog Filters: Network Functions

2

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


3

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 s2

2 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


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(2) are

Quadrant symmetry


5

General Procedure (iii)

In A(2) replace 2 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

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)


6

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


7

Butterworth Network Functions

Remember that

therefore:

The zeros of Q are obtained by

Therefore

Bn j 2

1

1 2n

Bn s Bn s 1

1 ( s 2 )n

1 ( s2 )n ( s 2)n 1 e j( 2 k )

s2 ej2k1n

or s2 e

j2k1n

sk e

j2k12n

2


8

Butterworth Network Functions


9

Chebyshev Network Functions

Remember that

Therefore

The zeros of Q are obtained by

Let

CHn j 2

1

12Cn2 ()

; 2 s 2

CHn s CHn s 1

Q(s)Q( s)

1

1 2Cn2( js)

Cn js cos n cos 1 ( js) j

cos 1 ( js) u jv jscos(u jv) cosu cosh v j sinu sinhv


10

Chebyshev Network Functions


11

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 1

nsinh 1 1


12

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 1

nsinh 1 1

jscos(u jv) cos u coshv j sin usinhv sinu sinhv j cos u cosh v

k2

sinh2 v

k2

cosh2 v1


13

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


14

Elliptic Filter


15

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) 1

e s 1

sinh(s ) cosh(s )

sinh(s )

1 cosh(s )

sinh(s )


16

Bessel-Thomson Filter Function (ii)

Find an approximation of in the form

And set

Approximations of

Example

cosh x

sinh 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


17

Bessel-Thomson Filter


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Different Filter Comparison


19

Different Filter Comparison


20

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


21

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 1

s1

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 1

s2 2 s 1

Analog Filters: Network Functions Franco Maloberti.

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