IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems Ka L. Law, Robert M. Fossum, Member, IEEE, and Minh N. Do, Senior Member, IEEE, Abstract—We study the invertibility of M-variate Laurent original signal. Using the polyphase representation in the z- polynomial N × P matrices. Such matrices represent multi- domain [1], [2], we can represent the analysis part as an dimensional systems in various settings such as filter banks, N × P matrix H(z) (shown in Fig.1(b)) with entries in a multiple-input multiple-output systems, and multirate systems. −1 −1 Laurent polynomial ring C[z1,z2, ..., zM ,z1 , ..., zM ]. In Given an N × P Laurent polynomial matrix H(z1, ..., zM ) of degree at most k, we want to find a P × N Laurent polynomial this case M is the dimension of signals, N is the number left inverse matrix G(z) of H(z) such that G(z)H(z) = I. of channels in the filter bank, and P is the sampling factor We provide computable conditions to test the invertibility and at each channel. An application of this setting may arise in propose algorithms to find a particular inverse. multichannel acquisition. In such an application we collect The main result of this paper is to prove that H(z) is X z generically invertible when N −P ≥ M; whereas when N −P < data about an unknown multidimensional signal ( ) as M, then H(z) is generically noninvertible. As a result, we output of the analysis part in Fig. 1(a). The acquisition system propose an algorithm to find a particular inverse of a Laurent (filters Hi(z) and sampling matrix D) is fixed and known polynomial matrix that is faster than current algorithms known beforehand. The objective is to reconstruct X(z) with a to us. synthesis part G(z). The existence of a synthesis part becomes Index Terms— Left Invertibility, Perfect Reconstruction, a purely mathematical question. Grobner¨ Bases, Multidimensional Multirate Systems, Generic Therefore, our first problem is to consider whether there Property. exists a P × N matrix G(z) over a Laurent polynomial ring −1 −1 C[z1,z2, ..., zM ,z1 , ..., zM ] for which G(z)H(z)= IP where I is the P × P identity matrix. I. INTRODUCTION P One dimensional perfect reconstruction finite impulse re- During the last two decades, one dimensional multirate sponse (FIR) filter banks have been investigated in several systems in digital signal processing have been thoroughly studies [3], [4], [5]. The Euclidean algorithm plays a key role developed. Due to the high demand of multidimensional in the matrix inverse problem for one dimensional perfect processing including image and video processing, volumetric reconstruction FIR filter banks [4] since it can be used to data analysis, and spectroscopic imaging, multidimensional find the GCD of a family of polynomials. For multivariate multirate systems require more extensive study. Perfect re- polynomials, there is a GCD (since the ring is a unique construction, which guarantees that an original input can be factorization domain) but the GCD is not necessarily a linear perfectly reconstructed from the outputs, is one key property combination of the polynomials. The theory of Gr¨obner bases of a multidimensional multirate system. has been introduced to compute with multivariate polynomials In a multidimensional multirate system, a digital signal [6], [7] and the theory is widely used in multidimensional is split into several channels and processed with different signal processing [8], [9], [10], [11], [12], [13]. Methods using sampling rates. The most popular multirate systems are filter Gr¨obner bases techniques for testing the invertibility of and banks shown in Fig. 1(a). In the analysis part, a digital input for computing a particular inverse of an N × 1 multivariate signal is filtered and then downsampled, generating multiple polynomial matrix H(z) were proposed in [14], [15]. For an outputs at the lower rates. In the synthesis part, the multiple N × P multivariate polynomial matrix H(z) where P > 1, outputs are upsampled and then filtered to reconstruct the adjoint matrix methods are employed in [14], [16]. Park in [17] provides a method to find the inverse of a Laurent Copyright (c) 2008 IEEE. Personal use of this material is permitted. polynomial matrix G(z). His method involves transforming However, permission to use this material for any other purposes must be Laurent polynomials into polynomials by multiplying by a obtained from the IEEE by sending a request to [email protected]. K. L. Law was with the Department of Mathematics and the Coor- series of elementary matrices. In this paper, we offer a simpler dinated Science Laboratory, University of Illinois at Urbana-Champaign, and more direct algorithm to compute a particular Laurent Urbana IL 61801. He is now with the Department of Communica- polynomial inverse. We can then generate all inverses from a tion Systems, Technische Universit¨at, Darmstadt 64283, Germany (email: [email protected]). particular inverse. In this set of inverses, one find an optimal R. M. Fossum was with the Department of Mathematics and the Beckman set of synthesis filters according to some design criteria [11], Institute, University of Illinois at Urbana-Champaign, Urbana IL 61801 [15], [18]. (email: [email protected]). M. N. Do is with the Department of Electrical and Computer En- The second problem is: When is the probability for the gineering, the Coordinated Science Laboratory, and the Beckman Insti- existence of an inverse for a given system high? Rajagopal tute, University of Illinois at Urbana-Champaign, Urbana IL 61801 (email: and Potter [14] and Zhou and Do [19] have investigated [email protected]). The partial results of this paper have been presented in ICASSP’09, Taipei, this problem and made several conjectures. We investigated Taiwan, April 2009. systems by varying M, N and P . In experiments, we found 2 IEEE TRANSACTIONS ON SIGNAL PROCESSING ANALYSIS SYNTHESIS H0 D D G0 H1 D D G1 X + Xˆ HN−1 D D GN−1 (a) ANALYSIS SYNTHESIS −l l z 0 D D z 0 −l l z 1 D D z 1 H z + X ( ) G(z) Xˆ −l l z |D|−1 D D z |D|−1 (b) Fig. 1. Example system represented by a polynomial matrix. (a) A multidimensional N-channel oversampled filter bank: Hi and Gi are analysis and synthesis filters, respectively; D is an M × M sampling matrix with sampling rate P = | det D|≤ N. (b) Polyphase representation: H(z) and G(z) are analysis and synthesis polyphase transformation matrices, respectively; {li} is a basis of the lattice generated by the sampling matrix D. that when M − N ≥ P , an inverse “almost surely” exists. II. MATHEMATICAL CONTEXTS On the other hand, when M − N < P , an inverse “almost surely” does not exist. To make precise the study of this inverse A. (Left) Inverse Polynomial Matrix Problem existence problem, we employ measure theory [20] and the We use boldface letters to denote vectors, or matrices. Let concept of “holds generically” [7]. z be an M-dimensional complex variable z = (z1, ..., zM ) in M M C . For n = (n1, ..., nM ) ∈ Z , we define the monomial zn M ni = i=1 zi . In this paper, we will always assume that N, P , andQ M are positive integers. Definition 1 (Polynomial or Laurent Polynomial Matrix): H z The paper is organized as follows. In Section II, we show An N × P matrix ( ) is said to be a polynomial matrix how to verify the invertibility of a Laurent polynomial matri- (resp.: Laurent polynomial matrix) if every entry is a ces. In Section III, we propose algorithms to find a particular polynomial (resp.: Laurent polynomial). inverse based on the Gr¨obner bases computation. Next, we Definition 2 (Left Invertible): An N ×P polynomial (resp.: H z characterize the set of all inverses. In Section IV, we prove Laurent polynomial) matrix ( ) is said to be polynomial that when N −P ≥ M, then a polynomial matrix of degree at (resp.: Laurent polynomial) left invertible if there exists a P × G z most k is generically polynomial (resp.: Laurent polynomial) N polynomial (resp.: Laurent polynomial) matrix ( ) such left invertible; whereas when N − P <M, then a polynomial that matrix of degree at most k is generically polynomial (resp.: G(z)H(z)= IP . (1) Laurent polynomial) noninvertible. Based on this result, we present a fast algorithm to find a particular inverse in Section Otherwise H(z) is said to be polynomial (resp.: Laurent V. We conclude with a summary in Section VI. polynomial) left noninvertible . LAW, FOSSUM, AND DO: GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR FILTER BANKS AND MIMO SYSTEMS 3 The discussion of polynomial (resp.: Laurent polynomial) reduced Gr¨obner basis since the ei are linearly independent. left invertible can also apply to polynomial (resp.: Laurent By the uniqueness of reduced Gr¨obner basis with respect to a polynomial) right invertible. To avoid repetition, throughout given term order, {ei}i=1,..,P is the reduced Gr¨obner basis of the paper we use the word “invertible” to represent either S. polynomial left invertible or Laurent polynomial left invertible. Suppose on the other hand that the reduced Gr¨obner basis It will be clear in the context whether it is polynomial left of S is {ei}i=1,..,P . Then there exist some {gij(z)} satisfying invertible or Laurent polynomial left invertible. We will also (3). Let G(z) = (gij (z)). Then restrain from using the pedantic “(resp.: Laurent polynomial)” G z H z I when it is understood in the context. ( ) ( )= . C Consider an N × 1 matrix H(z) over [z] where Hi(z) is Thus H(z) is invertible.