Low-pass to low-pass transformation Provided the prototype low pass filter G(s) with unit band width (i.e., cut-off frequency = 1 rad/sec.) and unity gain at Ω = 0 (i.e., |G(j₀)| = 1), the transformation s → s/Ω_c provides us a new filter, H(s), with cut-off frequency of Wc. The filter H(s) is shown by

H(s) = G(s)]_{s→ s /} Ωc

The critical frequency Ω_r of the filter G(s) is changed to W¢r of the filter H(s), provided by Ω´_c = Ω_r Ω_c. Both G(s) and H(s) are low pass filters.

More usually, the transformation

s → s Ω_c/Ω´c

transforms a low pass filter with a cut-off (or critical) frequency Ω_c to a low pass filter with a cut-off (or critical) frequency Ω´_c .

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