WAVELETS AND RECURSIVE FILTER BANKS

Recent work has shown that perfect reconstruction filter banks can be used to derive continuous-time bases of wavelets; the case of finite i mpulse response filters, which lead to compactly supported wavelets, h as been examined in detail. In this paper we show that infinite impuls e response filters lead to more general wavelets of infinite support. We give a complete constructive method which yields all orthogonal two channel filter banks, where the filters have rational transfer functi ons, and show how these can be used to generate orthonormal wavelet ba ses. A family of orthonormal wavelets, which shares with those of Daub echies the property of having a maximum number of disappearing moments , is shown to be generated by the halfband Butterworth filters. When t here is an odd number of zeros at pi we show that closed forms for the filters are available without need for factorization. A still larger class of orthonormal wavelet bases having the same moment properties i s presented, and contains the Daubechies and Butterworth filters as th e boundary cases. We then show that it is possible to have both linear phase and orthogonality in the infinite impulse response case, and gi ve a constructive method. We show how compactly supported bases may be orthogonalized, and construct bases for the spline function spaces. T hese are alternatives to those of Battle and Lemarie, but have the adv antage of being based on filter banks where the filters have rational transfer functions and are thus realizable. Design examples are presen ted throughout.