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- Let
be the length of the DFT (or the number of frequency
bins).
is no longer constrained to be the same as
.
is the downsampling factor on each branch
is the length of each polyphase filter,
.
- The length of the lowpass prototype filter
- Case 1 : critically sampled, no overlap
, the filter bank is critically sampled
, the polyphase filters are simply scalar multiplies
- Equivalent to a block transform
- Perfect reconstruction only if
- Case 2 : oversampled OLA of 50% overlap
-
, the filter bank output is oversampled by 2
, the polyphase filters are simply scalar multiplies
- PR requires that the prototype lowpass filter (window) has
constant overlap add in time:
- Case 3 : critically sampled OLA of 50% overlap
, the filter bank is critically sampled
, the polyphase filters are two-tap FIR filters
, the lowpass protoype filter
is twice as
long as the transform length,
- A transform slightly different than the DFT matrix is
needed for perfect reconstruction
- The same form as the Princen-Bradley filter bank, where
and
- Considered a Lapped Transform
- Case 4 : critically sampled OLA of 8:1 overlap
- Similar to Case 3, but
and thus
.
- Transform matrix is close to that of Case 3.
- The same form as the MPEG layer I,II filter bank
- Considered an Extended Lapped Transform
- How do we generate the new transform matrices in Cases 3 & 4?
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Download JFB.pdf
Download JFB_2up.pdf
Download JFB_4up.pdf
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