Normalization and 78 dB/90 dB attenuation numerical

Significance of normalization and denormalization

Normalization means designing the filter with convenient reference values, usually a cut-off (or half-power) frequency of $\omega_o = 1$ rad/s and a termination of $R = 1\ \Omega$. Denormalization (scaling) converts the normalized element values to the actual frequency and impedance level required.

Importance:

• Filter tables (Butterworth, Chebyshev, Bessel poles and ladder element values) are published only in normalized form; one table serves every frequency and impedance level.
• Calculations use simple numbers like 1, 1.414, 2 instead of values like $10^{-9}$ and $10^{4}$, so errors are fewer.
• Designs can be compared and reused: a 1 rad/s prototype becomes a 1 kHz or 1 MHz filter by simple scaling.
• Scaling lets the designer choose practical element values (e.g. capacitors in nF–μF, resistors in kΩ) without changing the shape of the response.

The scaling relations are $R' = k_m R$, $L' = k_m L/k_f$, $C' = C/(k_m k_f)$.

Numerical

Attenuation in dB is defined as

$$\alpha = 20\log_{10}\frac{V_{in}}{V_{out}}\ \text{dB} \;\Rightarrow\; V_{out} = V_{in}\times10^{-\alpha/20}$$

At $f$ = 30 kHz, $\alpha$ = 90 dB and $V_{in}$ = 1 V:

$$\begin{aligned} V_{out} &= 1 \times 10^{-90/20} = 10^{-4.5} \\ &= 3.162\times10^{-5}\ \text{V} \end{aligned}$$

Answer: output amplitude at 30 kHz ≈ 31.6 μV.

(The 78 dB at 20 kHz applies only to a 20 kHz input; a 1 V signal at 20 kHz would come out as $10^{-78/20} = 126\ \mu$V.)