Determine the lowest order K and the cutoff frequency ωc of a lowpass Butterworth filter that satisfies the the following speci- fications. When possible, choose ωc to provide sensible passband and...

Determine the lowest order K and the cutoff frequency ωc of a lowpass Butterworth filter that satisfies the the following speci- fications. When possible, choose ωc to provide sensible passband and stopband buffer zones.

(a) α_p
≤ 0.5 dB, α_s
≥ 20 dB, ω_p
≥ 100 rad/s, and ω_s
≤ 200 rad/s.

(b) δ_p
≤ 0.0115, δ_s
≤ 10−3, ω_p
≥ 1000 rad/s, and ω_s
≤ 2000 rad/s.

(c) The gain at 3ω_c
is required to be no greater than −50 dB.