Passband, transition band, roll-off and design steps

Terms

 |T|(dB)
   0 |~~~~~~~~~~~~~~.
     |  passband     \   transition band
     |                \  (roll-off slope)
 -As |- - - - - - - - -\_______________
     |                  :   stopband
     +-------------+----+--------------> w
                   wp   ws

• Passband: the range of frequencies passed with attenuation not more than $\alpha_{max}$ (LPF: $0$ to $\omega_p$).
• Stopband: the range attenuated by at least $\alpha_{min}$ (LPF: $\omega \ge \omega_s$).
• Transition band: the region between $\omega_p$ and $\omega_s$, where attenuation rises from $\alpha_{max}$ to $\alpha_{min}$. An ideal filter has zero transition width.
• Roll-off: the rate at which the gain falls beyond the cut-off, in dB/decade or dB/octave. An $n$th-order all-pole LPF rolls off at $20n$ dB/decade ($6n$ dB/octave); a steeper roll-off needs a higher order.
• Bandwidth: the width of the passband: $\omega_p$ (or $\omega_{3dB}$) for LPF, $\omega_2 - \omega_1$ between half-power points for BPF.

Basic steps in filter design

1. Specifications: state the filter type (LP, HP, BP, BS) and $\alpha_{max}, \alpha_{min}, \omega_p, \omega_s$, gain, impedance levels.
2. Normalization: convert to a normalized low-pass prototype ($\omega_p$ = 1 rad/s, 1 Ω), using frequency transformation for HP/BP/BS.
3. Approximation: choose a response (Butterworth, Chebyshev, inverse Chebyshev, elliptic, Bessel), find the order $n$ and the transfer function $T(s)$ that meets the specifications.
4. Realization (synthesis): build a circuit for $T(s)$: a passive LC ladder, or cascaded active biquads (Sallen–Key, MFB, Tow–Thomas), or switched-capacitor sections.
5. Denormalization (scaling): apply frequency and magnitude scaling (and the inverse frequency transformation) to get practical element values.
6. Study of non-idealities: check sensitivity to element tolerances, op-amp limits and temperature; simulate.
7. Construction and testing: build, measure and tune the filter.