On Factorization of a Subclass of 2-D Digital FIR Lossless Matrices for 2-D QMF Bank Applications

The role of one-dimensional (I-D) digital FIR lossless matrices in the design of FIR perfect reconstruction QMF banks has been explored in several recent articles. Structures which can realize the complete family of FIR lossless transfer matrices have also been developed, with QMF application in mind. For the case of 2-D QMF banks, the same concept of lossless polyphase matrix has been used to obtain perfect reconstruction. However, the problem of finding a structure to cover all 2-D FIR lossless matrices of a given degree has not been solved. In this letter we make some progress in this direction. We obtain a structure which completely covers a well-defined subclass of 2-D digital FIR lossless matrices. The design of maximally decimated digital filter banks with perfect reconstruction has been addressed by a number of authors in recent years [1]-[3] along with 2-D extensions [4]-[6]. This letter is related to the method reported in [3] and the 2-D version reported in [6]. The notations we shall use are the same as those in [3] and [6]. Consider the M-channel maximally decimated QMF bank used in [3]. Let each analysis filter H , ( z ) be represented in its polyphase form as in [3] so that we can define the M X M polyphase matrix E ( z ) = [Ekn(z)], which completely characterizes the analysis filters. The perfect reconstruction property in [3] was based on the losslessness property of E ( z ) . As a remainder, the FIR matrix E ( z ) is said to be lossless -if it satisfies the paraunitary property & z ) E ( z ) = I where E ( z ) A E $ ( z ' ) (subscript * stands for coefficient conjugation only). On the unit circle, this property reduces to the unitariness of ,!$eJ"'). In [7], [8] structures for FIR lossless transfer matrices were developed based on the discrete-time lossless lemma (DTL Lemma [9]) which is a result about the state-space manifestation of losslessness. This lemma [9] merely says that E ( z ) is lossless if is unitary. Because of the FIR nature of E ( z ) , this lemma leads to a structure which was exploited in [8] to optimize attenuation characteristics of the analysis filters H,(z). When the parameters of the structure are being optimized, the perfect reconstruction property remains intact because the losslessness of E ( z ) is structurally guaranteed. 2-D QMF banks [4]-[6] find applications in subband coding of images. The results in [3] on perfect reconstruction has been extended to the 2-D case in [6] (for rectangular decimation) and ip [lo] (for general decimation matrices). The use of 2-D FIR lossless polyphase matrices in obtaining perfect reconstruction has been recognized in [6], [lo]. However, unlike in the 1-D case, it has not been possible in the 2-D case to obtain a complete structural characterization of M x M FIR lossless transfer matrices. The main difficulty is because there is no 2-D equivalent of the lossless lemma. In other words, given a 2-D lossless system, we are not guaranteed to find a structure whose system matrix is unitary. For the 1-D case, a second alternative form of structures were introduced in [ l l ] which does not require a state-space formulation. This form also completely covers all M X M FIR causal lossless systems and has minimum number of parameters (and delays) as in [8]. The advantages of this characterization are summarized in [ 111. (A full journal article is scheduled to appear [13]). According to the 1-D results in [ll], any M X M causal FIR lossless matrix E ( z ) of McMillan degree K can be realized io the form E ( z ) =VK(z)VK-1(z ) " . V 1 ( Z ) H " (2) where Ho is a constant M x M unitary matrix and V, ( z ) are M X M FIR lossless matrices of McMillan degree one, of the special form V,( z ) = I U,U,t + u,,uLz-' (3) where U , are unit-norm column vectors. The parameters of this characterization (i,e., H, and U,,) have been optimized in [ l l ] to obtain analysis filters H , ( z ) with good attenuation characteristics. Note that any matrix of the form (3) where U , has unit norm can be verified [ l l ] to be lossless with determinant equal to 2 l . For the 2-D case a causal FIR E ( z l , z,) is lossless if E(ZI ,Z*)E(ZI ,Z , ) = I (4) for all z I , z 2 . However a factorization analogous to (2) has not been established. Based on the fact that U , has unit-norm, it is easy to prove that in the 2-D case [ I U ~ U ~ + u , , u ~ z ~ " ~ z ~ ~ ~ ] is lossless for arbitrary integers n 1 , n 2 > 0. And it is possible to cascade sections of this type to obtain 2-D non-separable FIR lossless systems of arbitrary degree (see [6] for further examples of 2-D FIR lossless systems). But none of these will result in a general structure that can realize arbitrary 2-D FIR lossless systems. Manuscript received April 1 1 , 1989; revised June 29, 1989. This work was supported in part by the National Science Foundation under Grant DCI 8552579 and under Grant MIP 8604456. This letter was recommended by In this correspondence we shall restrict our attention to a particular Of 2-D lossless systems Of the form Associate Editor T. R. Vaidyanathan. 1 K The authors are with the Department of Electrical Engineering, California ~ ( z , , z , ) = ek , ,k , z ;k lz ;k2 , K > 0. (5) k l = O k , = O Institute of Technology, CA 91 125. IEEE Log Number 8930742. 0098-4094/90/0600-0852$01.00