Time Varying Modifications in FBS
Consider now applying a time varying modification.
![]() |
(10.32) |
where
![]() |
(10.33) |



![\begin{eqnarray*}
y(m) &=& \frac{1}{N} \sum_{k=0}^{N-1} Y_m(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{k=0}^{N-1} X_m(\omega_k)H_m(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right\} H_m(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_k(m-n)} \\
&=& \sum_{n=-\infty}^\infty x(n) [ w(n-m) h_m(m-n)] \\
&=& \sum_{n=-\infty}^\infty x(n) [\tilde{w}(m-n)h_m(m-n)] \\
&=& (x*[\tilde{w} \cdot h_m])(m) \\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1711.png)
Hence, the result is the convolution of
with the windowed
.
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)
wheredenotes
.
- 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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Useful Preprocessing
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FBS Fixed Modifications