Results on Biorthonormal Filter Banks
CALIFORNIA INST OF TECH PASADENA
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For a maximally decimated nonuniform filter bank, the perfect reconstruction PR property is equivalent to biorthonormality. We start from this result and derive a number of properties of PR filter banks. For example, no two integer decimators in a biorthonormal system can be coprime moreover if all analysis and synthesis filters have unit energy, then perfect reconstruction is equivalent to orthonormality. We also generalize the Nyquist and power complementary properties of orthonormal filter banks, for the biorthonormal case. We then show that whenever the decimation ratios are such that biorthonormality is possible with rational filters, it is in particular possible to obtain orthonormality with rational filters. This is done by developing an orthonormalization procedure. While reminiscent of the Gram-Schmidt approach, the procedure converges in a finite number of steps and furthermore preserves the filter-bank Eke form of the basis functions. We then modify the orthonormalization procedure for the application of subband decorrelation. It will be demonstrated that mere decorrelation of subband signals does not necessarily optimize the coding gain of a system. Finally we consider the problem of alias cancellation, and obtain a generalization of a previously known necessary condition called compatibility.
- Statistics and Probability