How are you,

I hope you can help me with a problem.
I'm designing a second-order Chebyshev filter: Ap = 3dB (ripple), N = 2, ε = 0.997628 , Gain = 1dB and TF is:

H(s) = Ho / (s^2 + 0.6449s + 0.7079)

As N=2 then

H(s=0) = 1 / √ (1+ε^2) then

Ho = 0.5012.

The TF of Low Pass Filter is:

LPF(s) = w^2 / (s^2 + w*s/Q + w^2),

if I compare H(s) and LPF(s), then Ho should be equal to w^2, but Ho = 0.5012 and w^2 = 0.7079.

Will I be doing some wrong calculation?

 

Your equation for the transfer function of a generic LPF assumes a DC response that is unity gain:

\( LPF(j0) = {w^2 \over {(j0)^2 + j0 \; w/Q + w^2}} = {w^2 \over w^2} = 1 \)

However, for a second-order Chebyshev filter, the DC response is -Ap, which is -3 dB in your design. The LPF(s) response should be adjusted for this DC gain. The general LPF transfer function is then

\( LPF(s) = {10^{-3/20} w^2 \over {s^2 + sw/Q + w^2}} \)

and your coefficients should then line up.