Changing filter stage order can change the intermediate budget while leaving the final product the same when the same ratios are multiplied. This page treats the ratios as ideal scalar responses; real stage order can also change loading, headroom, noise, phase, nonlinear clipping, and measurement uncertainty.
Independent stage ratios commute for the final product, but not for the middle budget. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
One order has a twenty-unit intermediate budget
The 4/5 stage first gives intermediate budget 20 and final budget 4.
25⋅54⋅51=4
The reversed order has a five-unit intermediate budget
The final budget is still 4, but the intermediate budget is only 5.