This high-pass-complements-low-pass lesson checks that, for the same first-order index, ideal high-pass and low-pass squared-response fractions add to one. At index 2, the 1/5 low-pass and 4/5 high-pass fractions close exactly only for the stated ideal complementary pair. Real filters include tolerance shifts, loading, parasitic paths, nonideal rolloff, bandwidth limits, finite precision, and measurement uncertainty.
For the same first-order index, high-pass and low-pass squared ratios add to one. 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 high-pass keeps the same fast signal
At index 2, the high-pass squared ratio is 4/5.
∣H∣2=54
The two complementary fractions add to one
At the same index, the low-pass fraction is 1/5 and the high-pass fraction is 4/5.