Slow Then Fast Cascade Uses Distinct Cutoffs - Electronic Circuits VII & VIII: Frequency Response and Active Filters Step by Step

In a cascade with distinct cutoffs, the same signal can be below one reference and above another, so each stage contributes a different response ratio. The 6 hertz, 12 hertz, and 3 hertz rows are exact training comparisons; real cascades add tolerance, loading, phase shift, source impedance, noise, and measured cutoff uncertainty.

The same signal can have different normalized indexes at different stages. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The same signal is slow for the first cutoff

Against cutoff 12 hertz, the signal index is 1/2 and the ratio is 4/5.

x_1={6\over 12}={1\over 2}

The same signal is fast for the second cutoff

Against cutoff 3 hertz, the index is 2 and the final product is 4/25.

{4\over 5}\cdot {1\over 5}={4\over 25}