A VIRTUAL BANDPASS FILTER IN SUB-BAND ADAPTIVE FILTERS AND CONVERGENCE ANALYSIS

In a sub-band adaptive filter, an i-th analysis filter Fi(z) separates the i-th sub-band Ωi. An unkown system H(z) is identified as H(z)Fi(z) = Fi(z)Ai(zK), ω ∈ Ωi, where Ai(zK) is the i-th sub-band adaptive filter. Fi(z) can be cancelled as Ai(zK) = H(z), ω ∈ Ωi. Furthermore, this can be expressed as Ai(zK) = Bi(z)H(z), where Bi(z) = 1, ω ∈ Ωi, = 0, ω ∈ Ωj, i = j. Bi(z) is not really used, rather it appears equivalently in deriving the ideal sub-band adaptive filters. An impulse response bi(n) of Bi(z) dis-tributes over both negative and positive time domains. Since the impulse response ai(n) of Ai(zK) is a convolution sum of the impulse response h(n) of H(z) and bi(n), it also dis-tributes in both time domains. Thus, the ideal sub-band adaptive filters are noncausal. Two methods for relaxing the noncausal condition are investigated. One of them is to insert time delay in serial to the unknown system in order to shift ai(n) toward the positive time domain. The other is to set the sampling rate in the sub-bands high so as to generate split bands and reduce the ratio of ai(n) in the negative time domain. 1