Appendix 2


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The synthesis of the video filter


The filter specifications represent the input of the synthesis procedure. First we have to compute the transfer function of the filter and after that to realise this transfer as a LC-ladder. The synthesis procedure is based on Darlington synthesis procedure for partial and total removal of the poles from ¥ and simultaneous realisation of the zeros for z21 and the poles for z11 and z21 from the [Z] matrix of the filter. The specifications are given in the table below.

 

w p

2 p 5.5MHz

w c

2 p 14.8MHz

a max

3dB

a min

31dB

w c/w p

2.69

Table A2.1: Video filter specifications

The ripple of the filter is found from the value of amin:

(A2.1)

Hence we can compute the order of the filter as:

(A2.2)

Therefore the chosen order is n=3. The zeros of the transmission can be determined from the order of the transmission:

(A2.3)

The roots of the numerator in the transfer are the zeros of the transmission. Now we can compute the numerator of the transfer from (A2.3).

(A2.4)

The poles of the transfer Ymp are given by reciprocals of YK where:

(A2.5)

Finally, the normalized transfer of the filter is:

(A2.6)

After frequency scaling to get the passband edge, the new transfer becomes:

(A2.7)

For H(s) we have to find the LC-ladder with 1W termination. The reflection coefficient at the input is found from:

(A2.8)

The poles of r (s) are the poles of H(s) and the zeros can be chosen arbitrary with extra condition that zeros are complex conjugates. The input impedance in the LC ladder when termination is 1W is a function of r (s):

(A2.9)

We can define four functions related to the even and odd parts of numerator and denominator:

(A2.10)

Accordingly we have now the elements of the Z matrix associated to the ladder:

(A2.11)

The Darlington synthesis procedure is based on the total and partial removal of the poles from ¥ and simultaneous realisation of the zeros for z21 and the poles for z11 and z21. Fig.A2.1 shows the final result of the synthesis.

Fig.A2.1: The LP prototype after scaling

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