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Biorthogonal Wavelet Pairs: biorNr.Nd
More about biorthogonal wavelets can be found in [Dau92] pages 259, 262-285 and in [Coh92]. By typing waveinfo('bior') at the MATLAB command prompt, you can obtain a survey of the main properties of this family, as well as information about Nr and Nd orders and associated filter lengths.
Figure 6-15: Biorthogonal Wavelets bior2.4 on the Left and bior4.4 on the Right
The new family extends the wavelet family. It is well known in the subband filtering community that symmetry and exact reconstruction are incompatible (except for the Haar wavelet) if the same FIR filters are used for reconstruction and decomposition. Two wavelets, instead of just one, are introduced:
In addition, the wavelets 
are related by duality in the following sense:
as soon as 
or 
and even
as soon as 
It becomes apparent, as Cohen pointed out in his thesis, that "the useful properties for analysis (e.g., oscillations, zero moments) can be concentrated on the 
function whereas the interesting properties for synthesis (regularity) are assigned to the 
function. The separation of these two tasks proves very useful" (see [Coh92] page 110).
, 
can have very different regularity properties (see [Dau92] page 269).
The 
, 
, 
, and 
functions are zero outside of a segment.
The calculation algorithms are maintained, and thus very simple.
The filters associated with m0 and 
can be symmetrical. The functions used in the calculations are easier to build numerically than those used in the usual wavelets.
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, is used in the analysis, and the coefficients of a signal s are
, is used in the synthesis