4th order Butterworth TF; phase and gain equalization

Normalized 4th-order Butterworth transfer function

With $\varepsilon$ = 1 and $\omega_c$ = 1 rad/s, $|T(j\omega)|^2 = 1/(1 + \omega^8)$. The poles are the left-half-plane roots of $1 + (-s^2)^4 = 0$, on the unit circle:

$$p_k = -\sin\theta_k \pm j\cos\theta_k, \qquad \theta_k = \frac{(2k-1)\pi}{8} = 22.5°,\ 67.5°$$ $$\begin{aligned} T(s) &= \frac{1}{(s^2 + 0.7654s + 1)(s^2 + 1.8478s + 1)} \\ &= \frac{1}{s^4 + 2.6131s^3 + 3.4142s^2 + 2.6131s + 1} \end{aligned}$$

Phase and gain equalization

• Gain (amplitude) equalization: adding a network whose magnitude response is the inverse of the unwanted variation of a system (e.g. a cable or telephone line whose loss rises with frequency), so the overall gain becomes flat over the band. Example: line equalizers, audio graphic equalizers.
• Phase (delay) equalization: adding all-pass networks, $|T| = 1$, whose phase/delay complements that of a filter or channel, so the total phase becomes nearly linear (constant group delay) without changing the magnitude response. Example: equalizing the delay peak of a Chebyshev filter for data transmission.