STFT with Modifications
where, is the sampled frequency response of a filter with impulse response
Let's examine the result this has on the signal in the time domain:
We see that the result is convolved with a windowed version of the impulse response . This is in contrast to the OLA technique where the result gave us a windowed filtered by without the window having any effect on the filter, provided it obeys the COLA constraint and sufficient zero padding is used to avoid time aliasing.
In other words, FBS gives
while OLA gives (for )
- In FBS, the analysis window
smooths the filter frequency response by time-limiting the corresponding impulse response.
- In OLA, the analysis window can only affect scaling.
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)
STFT Summary and Conclusions
Downsampled STFT Filter Banks