A B-wavelet-based noise-reduction algorithm.pdf

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5, MAY 1996 1219 approximation of S1/4(e3”) (one obvious way of doing this is by autoregresive (AR) modeling of S1/4(eJW)). If S(eJa) = 0 on some interval, it can be shown that both pre and postfilters can be chosen to be zero on the same interval, so that there are no stability problems. The AR modeling approach not only insures stability of the pre and postfilters, but it also offers a computationally very efficient way of obtaining rational approximations of optimal pre and postfilters. In order to obtain a minimum phase stable approximation of S- ’ /4(eJ”) , all we have to do is compute ,,/m (using the fast Fourier transform, for example), and then use Levinson’s recursion to find a polynomial approximation of S-1/4(eJ”). In. EXAMPLES Example 3.1-DCT Filter Bank with Prejiltering: The above de- veloped technique will be applied to a very simple PU FB. Let {P~(z) Q k ( z ) } be a DCT FB, i.e., the one in which the polyphase matrix E(z) is the DCT IV matrix [Il l . The DCT filters have poor attenuations. Fig. 3 shows [ l /S(e3w)]1/4, the test function chosen for this example (dotted curve). The solid curve is its second-order rational approximation (i.e., Pa ( z ) is a second-order filter). The input PSD function S ( e J w ) was the lowpass AR(5) model of speech [7]. Fig. 4 shows the coding gain for different FB’s. We can see that even prefilter alone (without any FB) gives some coding gain (see [7], Ch. 7). The coding gain changes only slightly if the ideal prefilter [l/S( eJw) ]1 ’4 is approximated by a second-order rational filter. Notice that the coding gain of PPU FB approximately halves the gap (on a dB scale) between the coding gain achieved with the PU FB and the prediction gain bound on the coding gain given by (2.15). The next example is striking in the sense that a finite-order FB performs better than a brick-wall FB. Example 3.2-Tree-Structured Filter Bank with P