What is the cause of slow roll off of windowed filters

Question

Exactly why using the of windowing function, such as Hamming or Blackman, causes a slower transition to stop band? Everywhere it is mentioned just as an obvious fact that this happens but I haven't found a text that mentions what is the cause.

An example for what I mean:

• An FIR low-pass filter designed with cut off frequency 1 Hz without a window function reaches stop band at little over this frequency (1.04 Hz or so).
• When window function is applied, the filter reaches the stop band at over 1.2 Hz

Also, is there a way to compensate for this? For example, if I really want the cut off frequency to be 1 Hz and use window function, should I design the filter to have cut off say 0.9 Hz?

Answer 1

A (non-full-width-rectangular) window attenuates data at its edges, thus narrowing the width (decreasing the 1st moment) of the bulk hump of the input. Decreasing width in the time domain increases uncertainty in the frequency domain, which is represented by a filter response which has a shallower transition band stretching farther from the center frequency.

One way to roughly compensate is to start with a wider data window (longer FIR), which when "narrowed" by a non-rectangular window function will constitute roughly the same bulk "width" as before the widening.