Points to Note

• We saw that in OLA with time varying modifications and $R=1$ (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 $R$ is the window length $M$ divided by a small integer like $2$ -$10$ , 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 $50$ % overlap, ($R=2$ ), 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 $h_m$ may vary with time, but its length is limited to that of the window $w$ .

• 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 $w$ , while OLA filters are band-limited by $w$ (another dual relation).

• Recall for comparison that each frame in the OLA method is filtered according to

  $$Y_m = X_m \cdot H_m = [X*W_m] \cdot H_m \;\longleftrightarrow\; \underbrace{[x \cdot w_m]}_{x_m} * h_m$$ (10.34)

  
  
  where $w_m$ denotes $\hbox{\sc Shift}_{mR}(w)$ .
• 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)

For more details, see [9].