Relevance of analog filters; frequency scaling

Why analog filters are still relevant

Digital filters need an ADC and DAC, and the analog world is always at the input and output, so analog filters remain essential:

• Anti-aliasing and reconstruction: every ADC needs an analog low-pass filter before it (to remove components above $f_s/2$), and every DAC needs a smoothing filter after it. Example: audio codecs, data acquisition cards.
• High frequencies: RF and microwave signals (GHz) in mobile phones, Wi-Fi and radar are filtered with LC, SAW/BAW or cavity filters, because ADCs at such rates are costly or not available.
• Low power and real time: analog filters have no sampling delay and use very little power, useful in hearing aids, wearable sensors and implanted medical devices.
• Power and EMI: mains EMI filters, harmonic filters in power systems and ripple filters in power supplies handle large currents that DSP cannot.
• Dynamic range: a front-end filter removes strong out-of-band interferers before they saturate the ADC (e.g. receiver preselectors, ECG front ends).

Active vs passive filters

Frequency scaling

Frequency scaling moves the frequency response of a filter to a new frequency without changing its shape or impedance level. It is used to convert a normalized (1 rad/s) design to the required cut-off frequency.

Let $k_f = \omega_{new}/\omega_{old}$. The new network must have at $k_f\omega$ the same impedance as the old one at $\omega$:

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

and $T'(s) = T(s/k_f)$. Example: a 1 rad/s prototype with $L$ = 1 H and $C$ = 2 F, scaled to $10^3$ rad/s, becomes $L$ = 1 mH and $C$ = 2 mF.