Subband adaptive filtering decimation constraints for oversampled nonuniform filterbanks

Mean square error analysis of nonuniform subband adaptive filters for system modeling

“…The cross-correlation (and degenerate auto-correlation) vectors for the th band polyphase input signals will be given by (12) where , and .…”

“…Using a complex filterbank that only reconstructs the positive frequencies, as suggested in [12], twice the real part of the output signal will reconstruct the negative spectrum. Thus, using the above definitions, we can conclude that (41) since the index selects the th polyphase component of the synthesis filter.…”

“…By making use of Theorem 2, we can assume that the subband cross-correlations will be small. This implies that we can approximate (60) by (61) If a complex filterbank is used as suggested in [12], one must take twice the real part of to reconstruct the negative frequency spectrum. This gives the spectral density of the final error for index (62) where…”

“…The filterbanks are complex and cover only the positive half of the spectral domain (the negative frequency half is redundant for real signals). The subband filters are constructed by forming a 40-subband uniform complex-modulated pseudo-QMF "constituent" filterbank of which neighboring subbands are added together to form the nonuniform and uniform filterbanks [12] shown in Fig. 6.…”

“…(The analysis also holds for real filters, with a few exceptions, which are noted.) Complex filterbanks are recommended for implementation, since real filterbanks can be shown to require more complexity due to the necessary constraints placed on the subband decimation factors to avoid significant inband aliasing [12]. When a complex filterbank is used with a real input signal, as in this paper, the filterbank is assumed to only reconstruct the positive half of the frequency spectrum, since the appropriate real signal can then be recovered by simply taking twice the real part of the filterbank output upon synthesis [12].…”