STFT with Modifications
FBS Fixed Modifications
Consider applying a fixed (time-invariant) filter to each before resynthesizing the signal:(10.28) |
where, is the sampled frequency response of a filter with impulse response
(10.29) |
Let's examine the result this has on the signal in the time domain:
(10.30) |
while OLA gives (for )
(10.31) |
- 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.
Time Varying Modifications in FBS
Consider now applying a time varying modification.(10.32) |
where
(10.33) |
refers to the tap of the FIR filter at time .
Points to Note
- We saw that in OLA with time varying modifications and (a ``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 % overlap, ( ), 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
according to
(10.34)
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)
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STFT Summary and Conclusions
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Downsampled STFT Filter Banks