High-Pass Then Low-Pass Makes a Window - Electronic Circuits VII & VIII: Frequency Response and Active Filters Step by Step

Cascading a high-pass and a low-pass forms a pass window, where each stage contributes its own squared-ratio share of the budget and the window is their product. With 3 Hz and 12 Hz cutoffs the 6 Hz signal gives 4/5 times 4/5 equals 16/25 exactly for the ideal first-order stages; real filters add component tolerance, loading, phase shift, op-amp bandwidth, source and load impedance, temperature drift, noise, and measurement uncertainty.

A band window is a product of a high-pass and low-pass squared ratio. 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 side keeps this signal

The high-pass cutoff is 3 hertz, so the 6 hertz signal has index 2.

|H|^2={4\over 5}

The low-pass side also keeps this signal

The low-pass cutoff is 12 hertz, so the combined passive ratio is 16/25.

{4\over 5}\cdot {4\over 5}={16\over 25}