αp, αs, ωp, ωs and element scaling

$\alpha_p$, $\alpha_s$, $\omega_p$, $\omega_s$

Attenuation (loss) is $\alpha(\omega) = -20\log_{10}|T(j\omega)|$ dB, so a large $\alpha$ means a small output.

 alpha(dB)
     |                     ______________
amin |- - - - - - - - - - |
     |                   /
     |                  /  transition
amax |~~~~~~~~~~~~~~~~ /
   0 +---------------+---+---------------> w
       passband      wp  ws   stopband

• $\alpha_p$ ($\alpha_{max}$): the maximum attenuation allowed anywhere in the passband (e.g. 0.5 dB). It sets the allowed ripple or droop.
• $\alpha_s$ ($\alpha_{min}$): the minimum attenuation required everywhere in the stopband (e.g. 40 dB).
• $\omega_p$: passband edge frequency, the last frequency at which $\alpha \le \alpha_{max}$.
• $\omega_s$: stopband edge frequency, from which $\alpha \ge \alpha_{min}$.

The smaller $\alpha_p$, the larger $\alpha_s$ and the narrower the transition band $\omega_s/\omega_p$, the higher the filter order required.

Scaling

Scaling changes element values so that the response moves to another frequency or impedance level without changing its shape. It converts normalized designs (1 rad/s, 1 Ω) into practical filters.

Magnitude (impedance) scaling

Every impedance in the network is multiplied by $k_m$, at the same frequency. A voltage transfer function is a ratio of impedances, so it does not change.

$$\begin{aligned} R' &= k_m R \\ sL' = k_m (sL) &\Rightarrow L' = k_m L \\ \frac{1}{sC'} = k_m \frac{1}{sC} &\Rightarrow C' = \frac{C}{k_m} \end{aligned}$$

Frequency scaling

The response at the old frequency $\omega$ must appear at the new frequency $k_f\omega$, where $k_f = \omega_{new}/\omega_{old}$. So each element must have at $k_f\omega$ the same impedance that the old element had at $\omega$:

$$\begin{aligned} j(k_f\omega)L' = j\omega L &\Rightarrow L' = \frac{L}{k_f} \\ \frac{1}{j(k_f\omega)C'} = \frac{1}{j\omega C} &\Rightarrow C' = \frac{C}{k_f} \\ R' &= R \quad (\text{frequency independent}) \end{aligned}$$

The new transfer function is $T'(s) = T(s/k_f)$: the response keeps its shape but is shifted along the frequency axis.

Combined scaling

Applying both:

$$R' = k_m R, \qquad L' = \frac{k_m}{k_f}L, \qquad C' = \frac{C}{k_m k_f}$$