Filter Ratio Cross Scans

Low-Pass Cutoff Three-Row Scan

cutoff moves the ratio

Holding signal frequency fixed shows how the low-pass squared ratio moves with cutoff. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

At cutoff the low-pass keeps one half

The signal and cutoff are both 6 hertz, so index 1 gives low-pass ratio 1/2. The output budget is 10.

20⋅12=1020\cdot {1\over 2}=10
Low-pass scan rowAt cutoff the squared budget halves.ffcratioinoutlowpass

A higher cutoff keeps more of the same signal

The cutoff rises to 12 hertz while the signal stays 6 hertz. The ratio is 4/5, so output budget is 16.

20⋅45=1620\cdot {4\over 5}=16
Low-pass scan rowA lower normalized index keeps more low-pass budget.ffcratioinoutlowpass

A lower cutoff rejects more of the same signal

The cutoff drops to 3 hertz, making index 2. The ratio is 1/5, so output budget is 4.

20⋅15=420\cdot {1\over 5}=4
Low-pass scan rowThree rows show the low-pass ratio moving with cutoff.ffcratioinoutlowpass