Consider now applying a time varying modification.
refers to the tap of the FIR filter at time .
Hence, the result is the convolution of with the windowed .
- We saw that in OLA with time varying modifications and
``sliding'' DFT), the window served as a lowpass filter on
each individual tap of the FIR filter being implemented.
- In the more typical case in which
is the window length
divided by a small integer like
, we may think of the window as
specifying a type of cross-fade from the LTI filter for one
frame to the LTI filter for the next frame.
- Using a Bartlett (triangular) window with
), the sequence of FIR filters used is obtained simply by
linearly interpolating the LTI filter for one frame to the LTI
filter for the next.
- In FBS, there is no limitation on how fast the filter
may vary with time,
but its length is limited to that of the window
- In OLA, there is no limit on length (just add more zero-padding), but
the filter taps are band-limited to the spectral width of the window.
- FBS filters are time-limited by
, while OLA
filters are band-limited by
(another dual relation).
- Recall for comparison that each frame in the OLA method is filtered
where denotes .
- Time-varying FBS filters are instantly in ``steady state''
- FBS filters must be changed very slowly to avoid clicks and pops (discontinuity distortion is likely when the filter changes)
FBS Fixed Modifications