So I'm solving some problems on filters and came up with an apparent contradiction that I would like to get clear. Considering a low-pass biquadratic section:


So in the first problem that this appeared I was expected to calculate the group-delay at pole frequency.

I considered the phase being given by (because in this exercise the gain is negative)


and the group delay calculated with $$\tau_g(\omega)=-\frac{d\phi}{d\omega}$$

I come up with $$\tau_g(\omega)=\frac{Q_0\omega_0(\omega_0^2+\omega^2)}{Q_0^2(\omega_0^2-\omega^2)^2+\omega_0^2\omega^2}$$

Finally at \$ \omega_0 \$: $$\tau_g(\omega_0)=\frac{2Q_0}{\omega_0}$$


Now off with the second exercise, I have to consider the same biquadratic section but now it is a Bessel filter with constant group delay \$\tau_0 \$. Now the author calculates the group delay as a function of the pole frequency and the quality factor and comes up with.


I have no idea where this expression comes from and why it contradicts my answer of the previous exercise. Is this because this a Bessel filter now? How do we arrive with this new expression? What are the math steps? Thank you in advance

  • \$\begingroup\$ One might expect an error as group delay at resonance increases with Q \$\endgroup\$ Commented Mar 1, 2021 at 0:28
  • \$\begingroup\$ When the pole-Q is high, the phase function has a steep slope at the pole frequency - that means: Group delay is proportional to the pole-Q. The last equation must be wrong. \$\endgroup\$
    – LvW
    Commented Mar 1, 2021 at 8:28
  • 2
    \$\begingroup\$ If it helps, see this. \$\endgroup\$ Commented Mar 1, 2021 at 15:11
  • \$\begingroup\$ @aconcernedcitizen it did, thank you! I upvoted your answer there too. \$\endgroup\$ Commented Mar 1, 2021 at 22:40

1 Answer 1


Ok, I figured it out! It was a mistake on my part because I assumed that


While in fact


Now it makes sense.

  • \$\begingroup\$ You can select your own answer, it will help future searches to find an accepted solution to this kind of problem. \$\endgroup\$ Commented Mar 2, 2021 at 10:09
  • \$\begingroup\$ @aconcernedcitizen yes I know but I have to wait 24 hours to do it :) 10 more to go! \$\endgroup\$ Commented Mar 2, 2021 at 12:20

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