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 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. the i-th row of H(z). If the greatest common divisor (GCD) of 1 3z2 {H (z), ..., H (z)} is 1, then the Bezout identity problem has 1 N H 2z1 +1 0 a solution [21]. We can use the Euclidean algorithm to find Example 1: Is (z1,z2) = invertible? 3 z1 the GCD and also a set of polynomials {G (z), ..., G (z)} 1 N 3z2 5 [22] such that We can use the software Singular[26] to implement the above N result. Gj (z)Hj(z)=1. 1: >ring R=0,(z(1),z(2)),dp; . Xj=1 R is a ring with 2 variables; dp specifies degree reverse However, the univariate GCD criterion and Euclidean algo- lexicographical ordering rithm fail for multivariate polynomials. But the multivariate 2: >matrix H[4][2]; membership problem can be solved by using Gr¨obner bases 3: >H=1,3*z(2),2*z(1)+1,0,3,z(1),3*z(2),5; [6], [23]. Briefly, the theory of Gr¨obner bases implies that any 4: >print(H); set of generators of an ideal or module has a unique reduced 5: 1,3*z(2), Grobner¨ basis for a given ordering. This basis is obtained by 6: 2*z(1)+1,0, using Buchberger’s algorithm [24]. 7: 3,z(1), In particular, suppose {b (z), ..., b (z)} is a Gr¨obner ba- 1 n 8: 3*z(2),5 sis of a C z -submodule r z r z generated by [ ] h 1( ), ...., N ( )i 9: >module S=transpose(H); S P . is the module r1(z), ..., rN (z) [25] where ri(z) belongs to C[z] . Then generated by rows of H(z1,z2) there exists an n × N transformation matrix {W (z)} such ij 10: >option(redSB); . Computes a reduced standard that N basis in any standard basis computation bi(z)= Wij (z)rj(z). (2) 11: >print(std(S)); . Returns the reduced Groebner Xj=1 basis by using above option 1,0, Buchberger’s algorithm is implemented in most computer 12: 0,1 algebra software systems, such as Singular, Macauley2, Maple, 13: H and Mathematica, and hence the computation of Gr¨obner bases By Proposition 1, we know that (z1,z2) is invertible. is available in these systems. The results from algebraic geometry and Gr¨obner bases deal only with polynomial matrices. To be applicable for systems B. Criteria for Left Invertibility with general FIR filters, not just causal or anticausal filters, we need to extend the results from polynomial matrices to Laurent To conclude a general fact between the Gr¨obner bases and polynomial matrices. One method is to multiply both sides of invertibility of a polynomial matrix, we have the following (1) with a monomial of high enough degree. Thus H(z) is proposition. This generalizes Proposition 2 from [15]. Laurent polynomial left invertible if and only if there exist a H z Proposition 1: Suppose ( ) is an N × P polynomial P × N polynomial matrix Gˆ (z) such that matrix. Let S = hh1(z), ...., hN (z)i be the C[z]-submodule P k of C[z] generated by the rows hi(z) of H(z). Then H(z) Gˆ (z)H(z)= z IP (4) is invertible if and only if the reduced Gr¨obner basis of S is k {e } where e is the i-th row of the P × P identity for some integer vector . But finding a suitable integer vector i i=1,...,P i k matrix. might require an extensive search. However, by generalizing Proof: Suppose H(z) is invertible. Then there exist Theorem 2 from [15], we have a simple algorithm to determine whether the given Laurent polynomial matrix is invertible or G(z) = (gij (z)) such that not1. G(z)H(z)= I. Proposition 2: Suppose H(z) is an N × P Laurent poly- Then nomial matrix. Consider the (N + P ) × P matrix N m 0 z H(z) ei = gij (z)hj (z) (3) H (z, w)= (5) diag(1 − z z ...z w) Xj=1 1 2 M for i = 1, ..., P . According to the definition of a Gr¨obner 1Theorem 2 from [15] can be proved directly by using the ”Rabinowitch basis [6, p.121], {ei}i=1,..,P is a Gr¨obner basis of S. It is a trick”. See also [27]. 4 IEEE TRANSACTIONS ON SIGNAL PROCESSING where m ∈ NM is such that zmH(z) is a polynomial matrix, These algorithms use Gr¨obner bases and are based on Propo- w is a new variable, and diag(y) is a P × P diagonal matrix sition 1 and Proposition 2. with element y on the diagonal. Then H(z) is Laurent poly- 0 nomial left invertible if and only if H (z, w) is a polynomial Algorithm 1 Particular Polynomial Inverse left invertible matrix. The computational algorithm for a polynomial left inverse Proof: If H(z) is Laurent polynomial left invertible, matrix zmH z then ( ) is also Laurent polynomial left invertible. Then Input: N × P polynomial matrix H(z) over C[z1, ..., zM ] there exists a polynomial matrix G(z) = (gij (z)) satisfying Output: P × N polynomial matrix G(z), if it exists k m0 ZM (4). Among these , pick one for which ∈ + is the 1: compute the reduced Gr¨obner basis of {h (z), ..., h (z)} m0 1 N least integer vector. Let m0 be the maximal entry of = where hi(z) is a row of H(z) and the associated trans- {m1, ..., mM }. If m0 = 0, then H(z) is polynomial left z 0 formation matrix {Wij ( )} as defined in (2) invertible and so is H z . Otherwise is positive. Now ( , w) m0 2: if the reduced Gr¨obner basis is {ei}i=1,..,P , then output let (Wij (z)) m −m m0 M 0 k z 3: else there is no solution w k=1 zk gij ( ), i = 1,...,P ; j = 1,...,N; 0 − z m0Q 1 M k k 4: end if gij ( , w)= k=0 ( l=1 zl )w , if i = j − N; 0P, Q otherwise. G0 z 0 z Let ( , w) = (gij ( , w)) be the corresponding P ×(P +N) Algorithm 2 Particular Laurent Polynomial Inverse matrix. Then by a straightforward computation, we can con- The computational algorithm for a Laurent polynomial left 0 0 clude that G (z, w) is a polynomial left inverse of H (z, w). inverse matrix 0 Now suppose H (z, w) is polynomial left invertible. There Input: N × P Laurent polynomial matrix H(z) with M 0 0 0 exists G (z, w) such that G (z, w)H (z, w) = I with variables G0 z 0 z G z ( , w)= gij ( , w) . Set Output: P × N Laurent polynomial matrix ( ), if it exists m 1: multiply H(z) by a common monomial z such that M 0 G z z−m 0 z −1 H (z, w) is polynomial matrix from Proposition 2 ( ) = ( gij ( , zk ))i=1,...,P ; j=1,...,N . 0 2: call Algorithm 1 with input H (z, w) kY=1 0 3: if the output of Algorithm 1 is G (z, w), then output Then we have G(z)H(z) = I and G(z) is a Laurent −m 0 M −1 z (G (z, z ))i ,...,P j ,...,N polynomial matrix. Hence H(z) is Laurent polynomial left ij k=1 k =1 ; =1 4: else there is noQ solution invertible. 5: end if H z z1 z1 Example 2: Is ( ) = 2 2 invertible? z2 +3 z2 +1 Clearly it is not polynomial invertible because the determinant Example 3: Find an inverse of H(z1,z2) = is zero when z1 is zero. To verify that the matrix is Laurent 1 3z2 polynomial left invertible, we need to introduce a new vari- 2z1 +1 0 H H0 z . By Example 1, we know that (z1,z2) able and the ( , w) from (5) and test the invertibility of 3 z1 H0 z ( , w). 3z2 5 1: >ring R=0,(z(1),z(2),w),dp; is invertible. 2: >matrix H’[4][2]; To calculate a left inverse of polynomial matrix, we have the 3: >H’=z(1),z(1),z(2)ˆ2+3,z(2)ˆ2+1, following: 1-z(1)*z(2)*w,0,0,1-z(1)*z(2)*w; 1: >matrix U[2][2]=unitmat(2); . U is the 2 × 2 4: >print(H’); identity matrix 5: z(1),z(1), 2: >matrix G[2][4]; 6: z(2)ˆ2+3,z(2)ˆ2+1, 3: >G=transpose(lift(transpose(H),U)); . lift 7: -z(1)*z(2)*w+1,0, is function that returns a transformation matrix L where T 8: 0,-z(1)*z(2)*w+1 U = H ∗ L 9: >module S=transpose(H’); 4: >print(G); 10: >option(redSB); 5: 2/179z(1),18/179z(2)-1/179,-6/179z(2)+ 11: >print(std(S)); 60/179, -12/179z(1), 12: 1,0, 6: 12/179z(1),3/895z(2)-6/179,-36/179z(2)+ 13: 0,1 2/179, -2/895z(1)+1/5 This implies that H(z) is Laurent polynomial left invertible. 7: >print(G*H); 8: 1,0, 0,1 III.PROPOSEDALGORITHMS 9: Thus G(z ,z ) is a left inverse of H(z ,z ). A. Computation of Left Inverses 1 2 1 2 Example 4: Find an inverse of H(z) = In this section we introduce two new algorithms to generate z1 z1 H z 2 2 . By Example 2, we know that ( ) is an inverse matrix of a given matrix if the matrix is invertible. z2 +3 z2 +1 LAW, FOSSUM, AND DO: GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR FILTER BANKS AND MIMO SYSTEMS 5 Laurent polynomial left invertible. To calculate a left inverse First we make a simple observation. Suppose g,h are using Singular: elements in some ring for which gh =1 while hg 6=1. Then 1: >matrix U[2][2]=unitmat(2); hghg = hg and (1 − hg)h = 0. If a is any element, then 2: >matrix G’[2][4]; a(1 − hg)h =0 and hence 3: >G’=transpose(lift(transpose(H’),U)); (g + a(1 − hg))h =1. 4: >print(G’); 5: -1/2*z(2)ˆ3*w-1/2*z(2)*w,1/2*z(1)*z(2)*w,Thus if hg 6= 1, then we can find infinitely many left 1,0, inverses to h. With this in mind we have the following general 6: 1/2*z(2)ˆ3*w+3/2*z(2)*w,-1/2*z(1)*z(2)*w,statements. 0,1 Lemma 1 (Zhou): [19], [29] Suppose H(z) is an N × P 7: >print(G’*H’); polynomial matrix and G˜ (z) is a P × N polynomial matrix 8: 1,0, such that G˜ (z)H(z) = I. Then G(z) is an polynomial 9: 0,1 inverse matrix of H(z) if and only if G(z) can be written as According to Algorithm 2, G(z) = 1 −1 2 1 −1 1 G(z)= G˜ (z)+ A(z)(I − H(z)G˜ (z)) (6) − 2 z1 z2 − 2 z1 2 H z 1 −1 2 3 −1 1 is a left inverse of ( ). 2 z1 z2 + 2 z1 − 2 where A(z) is an arbitrary P × N polynomial matrix. Rajagopal and Potter explore the computation of the syn- Theorem 1 (Park): [30] Suppose H(z) is an N × P poly- thesis part of an M-variate perfect reconstruction FIR filter. nomial matrix and G˜ (z) is a P × N polynomial matrix such Their algorithm [14], [16] first computes every maximal minor that G˜ (z)H(z) = I. Let h , h , ..., h be row vectors of H z 1 2 N of ( ) and the corresponding adjoint matrix. Then it uses H(z). Then G(z) is an polynomial inverse matrix of H(z) them to compute an inverse of H(z). The size of the set G z N if and only if ( ) can be written as of maximal minors is P , which could be large if N − P ˜ is large. When this is the case, we find in practice that the G(z)= G(z)+ A(z)Syz(h1, h2, ..., hN ) (7) algorithm is extremely slow. In order to avoid the problem A z that the computation of maximal minors poses, our Algorithm where ( ) is an arbitrary polynomial matrix and Syz is the h h h 2 computes an inverse directly by using the computation of the syzygy [6] of { 1, 2, ..., N }. reduced Gr¨obner bases for modules. Park [17] also presents an Remark 1: Both of these theorems hold when polynomial algorithm. To find the inverse of Laurant polynomial matrices, is replaced by Laurent polynomial. I H z G˜ z H z Park transforms the Laurent polynomial matrix into a polyno- Remark 2: Note that since ( − ( ) ( )) ( )=0, the I H z G˜ z mial matrix by multiplying by a series of elementary matrices. element ( − ( ) ( )) is a syzygy of {h1,...,hN }. Our approach simply transforms Laurent polynomials into Zhou’s method provides a simple characterization of in- polynomials by multiplying by a large enough monomial. verses which is easy to implement. Park’s method is more Therefore our approach is simpler and provides a closed form complicated. However the matrix size of the free parameter A z formula to compute an inverse. ( ) in Lemma 1 is P ×N, while the smallest possible matrix A z When one designs a filter bank, one would like to estimate size of ( ) in Theorem 1 is P × (N − P ) in theory. Though the degree of the entries in the inverse matrices. Caniglia et al. syzygy provided by Singular does not necessary attain this A z [28] propose an upper bound on the degree of N ×N invertible optimal size, the matrix size for ( ) obtained in Park’s method is in general smaller than Zhou’s method. IP matrix K(z) such that K(z)H(z) = and the degree z z +1 0 1 1 K z z2 + z1 z1 bound of deg( ( )) is optimal in order. Example 5: Let be H(z)= . Find the H z 3 z1 +2 Proposition 3: [28] Assume that ( ) is an N ×P invert- ible matrix in M variables. Let deg(H(z)) be the maximum z1 z2 A of the degrees of the entries of H(z) and let d = deg(H(z))+ size of (z1,z2) from Theorem 1 using Singular. 1. Then there exists an N × N invertible matrix K(z) such 1: >ring R=0,(z(1),z(2)),dp; that 2: >matrix H[4][2]; I >H=z(1),z(1)+1,z(2)+z(1),z(1),3,z(1)+2, K(z)H(z)= P 3: 0 z(1),z(2); 4: >option(redSB); and deg(K(z)) is (P d)O(M). 5: >matrix S=transpose(syz(transpose(H))); This suggests that the maximum degree of the entries of the . where syz computes the syzygy P × N inverse matrix G(z) is also less than or equal to 6: > print(S); (P d)O(M). 7: S[1,1],S[1,2],z(2)ˆ2+z(1),z(1)- z(2)-3, 8: S[2,1],z(1)ˆ2-z(1)-3,z(1)*z(2)+z(1)+z(2), B. Characterization of Inverses 0, Algorithms 1 and 2 do not guarantee that the inverse would 9: S[3,1],S[3,2],z(2)ˆ3-z(1)-z(2),-z(2)ˆ2- be well behaved. In this section, we refer to some results that z(1)-4*z(2) characterize the set of all inverses. Once we have a particular where S[i, j] is some long polynomial expression. Thus the inverse, we can parametrize the set of all inverses. required free parameter A(z1,z2) in Theorem 1 is a 2 × 3 6 IEEE TRANSACTIONS ON SIGNAL PROCESSING 2 matrix. It is not the optimal matrix size, namely 2 × 2. But Suppose F (f) is nonzero (i.e. c2(c1 − 4c2c0) 6= 0). Then A 2 the size of (z1,z2) in Zhou’s method is 2 × 4. Therefore c2 6= 0 and c1 − 4c2c0 6= 0. So f has two distinct solutions. 2 applying Park’s method using Singular would lead to smaller Therefore by the above definition, f(x)= c2x +c1x+c0 has size of A(z) in this case. two distinct solutions generically. In the set of all inverses, an optimal set of synthesis filters Lemma 3: If a property of polynomials of degree at most can be obtained according to some design criteria [11], [15], k1, ..., kn in M variables is generic, then the coefficient space [18]. C of polynomials whose polynomials failed to satisfy the property is measure zero and nowhere dense. IV. GENERIC INVERTIBILITY Proof: By the definition of hold generically, there exists A. Lebesgue Measure and Generic Property a nonzero polynomial F in the coefficients of the fi such that the property fails to satisfy for f , ..., f for which the When designing filter banks, an important question is how 1 n polynomial F (f , ..., f ) is vanishing. Let R be the set of likely it is that the synthesis part of the perfect reconstruction 1 n i M-variate polynomials of degree less than or equal to k . By filter banks exists. If it does not exist, then in general we are i lemma 2, not able to reconstruct the original signal. n In [19], Zhou and Do made the following conjecture. λ ({(f , ..., f ) ∈ R | F (f , ..., f )=0})=0 Conjecture 1: Suppose H(z) is an M-variate N × P l 1 n i 1 n iY=1 polynomial (resp.: Laurent polynomial) matrix with N ≥ P . k1+M kn+M If N − P ≥ M, then it is “almost surely” polynomial where l = M + ... + M is the dimension of the (resp.: Laurent polynomial) left invertible. Otherwise, it is coefficient space. Thus, the coefficient space C of polynomials “almost surely” polynomial’ (resp.: Laurent polynomial) left whose polynomials failed to satisfy the property is measure noninvertible. zero. To show the set is nowhere dense, it is equivalent to Rajagopal and Potter made another conjecture related to show that the closure of the set contains no open set. Suppose “almost surely” invertible in their paper [14]. it contains an open ball B() with some radius > 0. − − Corollary 6 in [14]: Suppose H(z) is an N × PM-variate Since F 1({0}) is a closed set, C is also in F 1({0}). N Thus, F −1({0}) contains the open ball B(). However, this polynomial matrix with N > P . If P > M, then it is −1 “almost surely” invertible. contradicts the fact that F ({0}) is measure zero. Therefore, Unfortunately, Corollary 6 in [14] is not correct. Please refer the coefficient space of polynomials whose polynomials failed to Zhou’s thesis [29] for more details. to satisfy the property is nowhere dense. Suppose the conjecture posed by Zhou and Do is true. If The immediate consequence is that if f1, ..., fn are drawn we design filter banks such that N − P ≥ M, then “almost independently from a probability distribution with respect to surely” there exists a synthesis part of the filter banks which the Lebesgue measure, the property of f1, ..., fn holds with is able to reconstruct the original signal perfectly. probability one. Furthermore, suppose f˜0, ..., f˜n satisfies the However, Zhou and Do did not give a precise definition of property. Since the coefficient space C of polynomials whose “almost surely”. In order to have the appropriate language, we polynomials failed to satisfy the property is nowhere dense, employ the concept of Lebesgue measure and the concept of there exists an open ball B() around f˜0, ..., f˜n for some > 0 “hold generically”. such that the property is satisfied within the open ball B() . In the 2-dimensional plane, it is obvious that any “simple” This shows that the system with the property is robust [33]. line (i.e. not a locally space filling curve) has zero area. In 3- dimensional space, we also know that any “simple” surface B. Generically Invertible when N − P ≥ M has zero volume. To generalize this property, we have the To prove our main theorem in this section, we need to following lemma. employ the resultant of the polynomials. Lemma 2: [31, p.9] Let f be holomorphic (which means Theorem 2 (Resultant): [7, p.80] If we fix positive degrees infinitely differentiable) in the domain D ⊂ CM , and suppose k0, ..., kn, then there is a unique nonzero polynomial called the f is not identically zero. Then λM ({z ∈ D | f(z)=0})=0 n C k +M resultant RES(k0,...,kn) ∈ [ i {uij } i ] where where λ denote the 2M-dimensional Lebesgue measure. =1 j=1,...,( M ) M the variables u correspond toS the coefficients of i-th poly- Definition 3 (Generic): [32] A property is said to hold ij nomial. Then we have the following property: generically for polynomials f , .., f of degree at most 1 n If F , ..., F ∈ C[x , ..., x ] are homogeneous of degrees k , ..., k if there is a nonzero polynomial F in the coefficients 0 n 0 M 1 n k , ..., k , then F , ..., F have a nontrivial common zero over of the f such that the property holds for f , ..., f whenever 0 n 0 n i 1 n C if and only if RES (F , ..., F )=0. the polynomial F (f , ..., f ) is nonvanishing. (k0,...,kn) 0 n 1 n Now we can translate the first half of Conjecture 1 into the Intuitively, a property of polynomials is generic if it holds following mathematical framework. for “almost all” polynomials. Theorem 3: If N − P ≥ M and k > 0, then an N × P Example 6: [32] The property “f(x)= c x2 +c x+c has 2 1 0 polynomial M-variate matrix H(z) of degree at most k is two distinct solutions” is generic. generically polynomial left invertible. Proof: Let F be a polynomial of the coefficients of f = Proof: The strategy of this proof is to find a nonzero c x2 + c x + c given by 2 1 0 polynomial F such that F (H(z))=0 for every noninvertible 2 H z F = c2(c1 − 4c2c0) matrix ( ) of degree at most k. LAW, FOSSUM, AND DO: GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR FILTER BANKS AND MIMO SYSTEMS 7 Let Z = (z0, ..., zM ). If f(z)= f0(z)+f1(z)+...+fl(z) is Proof: We know that if a polynomial matrix H(z) the decomposition of the polynomial f(z) into sums of forms is Laurent polynomial left noninvertible, then H(z) is also fi(z) of degree i, then the homogenization f(Z) of f(z) of polynomial left noninvertible. According to Theorem 3, this Z k z k−1 z H z degree k is defined to be f( ) = z0 f0( )+ z0 f1( )+ shows that RES(Pk,...,Pk) ◦ φ( ( )) = 0 for all Laurent k−l z h Z H z ... + z0 fl( ). Let i( ) be the ith row of an N × P matrix polynomial left noninvertible polynomial matrix ( ). H(z). Let ti−1(Z) be the determinant of the P ×P submatrix h Z h Z h Z containing i( ), i+1( ), ..., i+P −1( ). Define φ to be a C. Generically Noninvertible when N − P N If N < P , then every polynomial matrix is left noninvert- 1 2 3 4 ible. Thus the statement is true. For example we may arbitrar- 1 0 500 500 500 ily set F =1. Now we assume N ≥ P . Suppose H(z) is in- M=1 P 2 0 0 500 500 vertible. Since there exists a Laurent polynomial matrix G(z) 3 0 0 0 500 4 0 0 0 0 such that G(z)H(z) = I and G(z1, ..., zN−P +1, 1, ..., 1) is H 1 0 0 500 500 well-defined, (z1, ..., zN−P +1, 1, ..., 1) is also Laurent poly- M=2 P 2 0 0 0 500 nomial invertible. We can now assume that M = N − P +1. 3 0 0 0 0 4 0 0 0 0 Define ti(Z) to be the same as in the proof of Theorem 3. Let i-th 1 0 0 0 500 (i) M=3 P 2 0 0 0 0 tj = tj (z0, ..., 0 , ..., zM ). For each i =0, ..., M, define θi to be a function such that 3 0 0 0 0 4 0 0 0 0 H z (i) ˆ (i) (i) T ( ) 7→ (t0 , ..., ti , ..., tM ) TABLE I (i) ˆ (i) INVERSIBILITY TEST FOR A RANDOM POLYNOMIAL MATRIX GENERATOR where ti means that the term ti is omitted from the coordinates. By Remark 3 and the fact that WITH DIFFERENT N , P AND M IN 500 TESTCASES (i) Z ˆ(i) Z (i) Z {t0 ( ), ..., ti ( ), ..., tM ( )} is the subset of the set of maximal minors of H(Z) implies that θi(H(z)) have a nonzero common zero for some i = 0, ..., M. By the property of the resultant shown in Theorem 2, we know generically, we conclude that H(z) of degree at most k is for any Laurent polynomial left invertible polynomial matrix generically polynomial left noninvertible matrix. H z ( ) that Remark 4: Let H(z) be a polynomial matrix. If H(z) is polynomial left invertible, then H(z) is Laurent polynomial RES(Pk,...,Pk)(θi(H(z))) = 0 for some i =0, ..., M. (9) left invertible. But the converse is not true in general. Also if Now let H(z) is Laurent polynomial left noninvertible, then H(z) is M polynomial left noninvertible. Also the converse is not true in F = RES(Pk,...,Pk) ◦ θi. (10) general. iY=0 Example 7: Let (z) be a 1 × 1 matrix. It is not polynomial Then F (H(z))=0 for all Laurent polynomial left invertible left invertible matrix but it is a Laurent polynomial left H z (i) invertible matrix as (z−1)(z)=1. polynomial matrix ( ). The RES(Pk,...,Pk) and tj are polynomials, so is F . Lastly, we need to show F is not a Theorem 6: If N − P This implies z˜M =0. If z˜M =0, then in [33], [38] show that perfect reconstruction is almost surely, when there are at least three channels. Since the image is two i-th P k dimensional (i.e. M = 2) and the downsampling rate is just tM−1(˜z0, ..., 0 , ..., z˜M−1, 0)=z ˜ − =0. M 1 one (i.e. P = 1), by Theorem 4, we know that the perfect Thus z˜M−1 = 0. Continuing the process, we can conclude reconstruction is almost surely if the number of channels z˜0 =z ˜1 = ... =z ˜M =0. This contradicts the assumption that is greater than two (i.e. N ≥ 3). Therefore Harikumar and ˜ Z is nontrivial. So RES(Pk,...,Pk)(θi(T (z))) 6= 0 for all i. Bresler’s image deconvolution is a special case of our main Therefore F is not a zero function. By the definition of hold theorem. LAW, FOSSUM, AND DO: GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR FILTER BANKS AND MIMO SYSTEMS 9 Another application is that we can have an alternative 4: >matrix U[2][2]=unitmat(2); approach in designing multidimensional filter banks. We can 5: >matrix G’[2][6]; freely design the analysis side first such that it satisfies the 6: G’=transpose(lift(transpose(H’),U)); condition (i.e. N −P ≥ M). Then , by Theorem 4 and Lemma 7: //used time: 0.23 sec. . Using a desktop PC 3, we can almost surely find a perfect reconstruction inverse −1G0 −1 −1 Then the left inverse is (z1 (z1,z2,z1 z2 ))i=1,2,j=1,..,4. for the synthesis polyphase matrix. To calculate a Laurent polynomial left inverse using Algorithm 3 in Singular: V. FAST COMPUTATION OF LEFT INVERSES 1: >matrix U[2][2]=unitmat(2); By Theorem 4, we know we should design the filter banks 2: >matrix J[2][4]; such that N − P ≥ M. Suppose N − P ≥ M. Since H(z) 3: J=transpose(lift(transpose(z(1)*H),U)); is a Laurent polynomial matrix, there exists l ∈ NM such that 4: //used time: 0.06 sec. l z H z −1 ( ) is a polynomial matrix and is generically Laurent Then the left inverse is z1 J(z1,z2). polynomial left invertible. However, at the same time, the This agrees that Algorithm 3 is faster than Algorithm 2. zlH(z) is generically polynomial left invertible by Theorem 3. Due to this fact, we can improve our Algorithm 2. VI. CONCLUSION Algorithm 3 Faster Version In this paper we study the inverse problem of a Laurent The computational algorithm for a Laurent polynomial left polynomial matrices. Such matrices arise in FIR filter banks inverse matrix. as polyphase matrices. We use the computation of Gr¨obner Input: N × P Laurent polynomial matrix H(z) with M bases to test the invertibility. Then we propose algorithms to variables find a particular left inverse. We note that there is a sharp Output: P × N Laurent polynomial matrix G(z), if it exists phase transition on the invertibility depending on the size and l 1: multiply H(z) by a common monomial z such that dimension of a given Laurent polynomial matrix. Specifically zlH(z) are polynomial matrix when N − P ≥ M, the M-variate N × P polynomial (resp.: l 2: call Algorithm 1 with the input z H(z) Laurent polynomial) matrix is generically invertible; whereas −l 3: if the output of Algorithm 1 is J(z), then output z J(z) when N − P [10] H. Park, T. Kalker, and M. Vetterli, “Gr¨obner bases and multidimen- Ka L. Law was born in Hong Kong in 1979. He sional FIR multirate systems,” Multidimensional Systems and Signal received the B.Sc. degrees major in computer sci- Processing, vol. 8, pp. 11–30, 1997. ence from the University of Melbourne, Melbourne, [11] H. Park, “Optimal design of synthesis filters in multidimensional perfect Australia in 2002 and received the B.Sc. degrees reconstruction FIR filter banks using Gr¨obner bases,” IEEE Trans. Circ. in mathematics from the University of Illinois at and Syst., vol. 49, pp. 843–851, Jun. 2002. Urbana-Champaign in 2003. From September 2003 [12] H. Park and G. Regensburger, Eds., Grobner Bases in Control Theory to December 2008, he studied at the Department of and Signal Processing. Walter de Gruyter, 2007. Mathematics, the University of Illinois at Urbana- [13] L. Zhiping, L. Xu, and N. Bose, “A tutorial on Gr¨obner Bases with Champaign, where he received the Ph.D. degree applications in signals and systems,” IEEE Trans on Circ. and Syst., in Mathematics with Computational Science and vol. 55, no. 1, pp. 445–461, Feb 2008. Engineering Option. [14] R. Rajagopal and L. C. Potter, “Multivariate MIMO FIR inverses,” IEEE He is now with the Department of Communication Systems, Technische Trans. Image Proc., vol. 12, pp. 458–465, Apr. 2003. Universit¨at, Darmstadt 64283, Germany. His research interests include multi- [15] J. Zhou and M. N. Do, “Multidimensional multichannel FIR deconvo- dimensional signal processing for FIR filter banks and MIMO systems; adap- lution using Gr¨obner bases,” IEEE Trans. Image Proc., vol. 15, no. 10, tive beamforming in MIMO communication systems; convex optimization; pp. 2998–3007, Oct. 2006. and applied algebra and algebraic geometry. [16] R. Rajagopal and L. C. Potter, “Multi-channel multi-variate equalizer He was a Research Assistant at the Coordinated Science Laboratory, Uni- computation,” Multidimensional Systems and Signal Processing, vol. 14, versity of Illinois at Urbana-Champaign. He received the Research Assistant pp. 105–108, Jan. 2003. Fellowship by Joint Advising Support Program of Applied Mathematics in [17] H. Park, “A computational theory of Laurent polynomial rings and mul- 2006. tidimensional FIR systems,” Ph.D. dissertation, University of California at Berkeley, 1995. [18] K. L. Law and M. N. Do, “Multidimensional filter bank signal reconstruction from multichannel acquisition,” IEEE Trans. Image Proc., submitted 2008; available at http://www.ifp.uiuc.edu/˜minhdo/publications/. [19] J. Zhou and M. N. Do, “Multidimensional oversampled filter banks.” San Diego, USA: Proc. of SPIE Conference on Wavelet Applications in Robert M. Fossum (M’04) was born in Northfield, Signal and Image Processing XI, Jul. 2005. Minnesota in 1938. He received a B.A degree in [20] H. Royden, Real Analysis, 3rd ed. Prentice Hall, 1988. mathematics and physics from St. Olaf College in [21] C. A. Berenstein, A. V. R. Gay, and A. Yger, Residue Currents and 1959 and a PhD in mathematics from The University Bezout Identities. Birkh¨auser, 1993. of Michigan in 1965. [22] R. E. Blahut, Fast Algorithms for Digital Signal Processing. Addison- From 1964 to 2008 he was on the faculty of the Wesley, 1985. University of Illinois at Urbana-Champaign in the [23] T. Becker and V. Weispfenning, Gr¨obner Bases. Springer-Verlag, 1993. Department of Mathematics, in the Department of [24] B. Buchberger, “Theoretical basis for the reduction of polynomials to Electrical and Computer Engineering, and jointly canonical forms,” ACM SIGSAM Bull, vol. 10, no. 3, pp. 19–29, Aug. with Beckman Institute for Advanced Science and 1976. Technology. He is a Professor Emeritus since May [25] S. Lang, Algebra. Springer-Verlag, 2002. 2008. [26] G. M. Greuel and G. Pfister, A Singular Introduction to Commutative His research interests are in commutative algebra, algebraic geometry, Algebra. Springer-Verlag, 2002. algebraic groups, and in applications of these mathematical fields to computer [27] D. Eisenbud, Introduction to Commutative Algebra with a View Towards vision, image formation and processing, signal processing. He is a foreign Algebraic Geometry. Springer-Verlag, 1995, vol. 150. member of the Royal Norwegian Society of Sciences and Letters (Det [28] L. Caniglia, G. Corti˜nas, S. Dan´on, J. Heintz, T. Krick, and P. Solern, Kongelig Norske Videnskabers Selskab). “Algorithmic aspects of Suslin’s proof of Serre’s conjecture,” Comput Complexity, vol. 3, pp. 31–55, 1993. [29] J. Zhou, “Multidimensional multirate systems: characterization. design, and applications,” Ph.D. dissertation, University of Illinois at Urbana- Champaign, 2005. [30] H. Park, “Complete parametrization of synthesis in multidimensional Minh N. Do (M’01, SM’07) was born in Vietnam perfect reconstruction FIR systems,” IEEE International Symposium on in 1974. He received the B.Eng. degree in computer Circuits and Systems, vol. 5, pp. 41–44, 1999. engineering from the University of Canberra, Aus- [31] R. C. Gunning, Analytic Function of Several Complex Variables. tralia, in 1997, and the Dr.Sci. degree in commu- Prentice-Hall, 1965. nication systems from the Swiss Federal Institute [32] D. Cox, J. Little, and D. O’Shea, Ideal, Varieties, and Algorithms. of Technology Lausanne (EPFL), Switzerland, in Springer-Verlag, 1996. 2001. [33] G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred Since 2002, he has been on the faculty at the by multiple filters:theory and efficient algorithms,” IEEE Trans. Image University of Illinois at Urbana-Champaign, where Proc. , vol. 8, no. 2, pp. 202–219, Feb 1999. he is currently an Associate Professor in the De- [34] H. Matsumura, Commutative Ring Theory. Cambridge University Press, partment of Electrical and Computer Engineering, 1986. and hold joint appointments with the Coordinated Science Laboratory and [35] R. Hartshorne, Algebraic Geometry. Springer-Verlag, 1977. the Beckman Institute for Advanced Science and Technology. His research [36] F. Macaulay, The Algebraic Theory of Modular Systems. Cambridge interests include image and multi-dimensional signal processing, wavelets and University Press, 1916. multiscale geometric analysis, computational imaging, and visual information [37] E. Fornasini and M. E. Valcher, “Multidimensional systems with finite representation. support behaviors: signal structure, generation, and detection,” SIAM He received a Silver Medal from the 32nd International Mathematical Journal on Control and Optimization, vol. 36, no. 2, pp. 760 – 779, Olympiad in 1991, a University Medal from the University of Canberra in Mar. 1998. 1997, a Doctorate Award from the EPFL in 2001, a CAREER Award from the [38] G. Harikumar and Y. Bresler, “Exact image deconvolution from multiple National Science Foundation in 2003, and a Young Author Best Paper Award FIR blurs,” IEEE Trans. Image Proc., vol. 8, no. 6, pp. 846–862, Jun. from IEEE in 2008. He was named a Beckman Fellow at the Center for 1999. Advanced Study, UIUC, in 2006, and received of a Xerox Award for Faculty Research from the College of Engineering, UIUC, in 2007. He is a member of the IEEE Signal Processing Theory and Methods and Image, Video, and Multidimensional Signal Processing Technical Committees, and an Associate Editor of the IEEE Transactions on Image Processing.