Why roll-off of butterworth is 20 dbdecade?

Answer 1

The roll-off of a Butterworth filter is 20 dB/decade because it is designed to have a maximally flat frequency response in the passband, meaning it exhibits no ripples. The filter's transfer function has poles that are evenly spaced around a semicircle in the s-plane, resulting in a gradual transition from the passband to the stopband. Each pole contributes to the attenuation, and for a first-order Butterworth filter, the roll-off is 20 dB per decade; for higher orders, this increases proportionally to the number of poles, maintaining a smooth roll-off characteristic.