butterworth and chebyshev filter

1. butterworth and chebyshev filter

1) How to derive expression (3.11) for chebyshev filter ?

2) Why there is a factor of 2 in gi expression (3.24) and n expression (3.25) for butterworth filter ?

•

2. Re: butterworth and chebyshev filter

Originally Posted by promach
1) How to derive expression (3.11) for chebyshev filter ?
Definition of Chebyshev polynomials.

Originally Posted by promach
2) Why there is a factor of 2 in gi expression (3.24) and n expression (3.25) for butterworth filter ?
(3.24) is due to complex conjugate pair of poles.

(3.25) is from (3.7).

•

3. Re: butterworth and chebyshev filter

(3.24) is due to complex conjugate pair of poles.
I understand that butterworth function uses complex conjugate poles, but that does not justify the use of a factor of 2 in gi expression.

•

4. Re: butterworth and chebyshev filter

Originally Posted by promach
I understand that butterworth function uses complex conjugate poles,
but that does not justify the use of a factor of 2 in gi expression.
Consider coefficients of polynomials.
Calculate S11 from S21.
Then calculate Zin.
Then evaluate g-parameters by continuation fraction expansion of Zin.

All zeros of S11 are located at origin for Butterworth filter.
So you can get Both S11 and Zin as close form.

--[[ ]]--