Characteristics of Butterworth approximation

The Butterworth (maximally flat) low-pass response has magnitude

$$|T(j\omega)|^2 = \frac{1}{1 + \varepsilon^2\omega^{2n}}$$

(normalized so that $\omega_p$ = 1 rad/s; with $\varepsilon = 1$ the half-power frequency is 1 rad/s). Its characteristics are:

• Maximally flat at $\omega = 0$: the first $2n-1$ derivatives of $|T(j\omega)|^2$ are zero at $\omega = 0$, so the passband is as flat as possible.
• Monotonic: no ripple in the passband or the stopband; the gain falls continuously with frequency.
• $|T(j0)| = 1$; at $\omega = 1$ (when $\varepsilon = 1$) $|T| = 1/\sqrt{2}$ (−3 dB) for every order.
• Roll-off of $20n$ dB/decade far into the stopband; the transition is wider than Chebyshev or elliptic for the same $n$.
• All-pole function: all transmission zeros are at $\omega = \infty$.
• Poles lie on a circle of radius $\omega_o = \varepsilon^{-1/n}$ in the left-half s-plane, equally spaced by $\pi/n$, symmetric about the real axis.
• Phase is fairly linear (better than Chebyshev), with a moderate delay peak near the cut-off.
• As $n \to \infty$ the response approaches the ideal brick-wall.

 |T|
  1 |----.__
    |       `.   n = 2
0.707 - - - - :\.  n = 4
    |         : \ \.
    |         :  `. `-._
    +---------+---------`----> w
              1