GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR MULTIRATE SYSTEMS AND FILTER BANKS Minh N. Do Ka L. Law, Robert M. Fossum Department of Electrical Department of Mathematics and Computer Engineering University of Illinois at Urbana-Champaign University of Illinois at Urbana-Champaign Urbana IL 61801 Urbana IL 61801 ABSTRACT polyphase matrix G(z). The existence of a synthesis part becomes a purely mathematical question. The perfect recon- We study the invertibility of M-variate polynomial (respec- struction condition holds if and only if G(z)H(z) = IP tively : Laurent polynomial) matrices of size N by P . Such where IP is the P P identity matrix. matrices represent multidimensional systems in various set- Then it is a natural× question to ask: When does the system tings including filter banks, multiple-input multiple-outputsys- have a high probability of the existence of an inverse? Ra- tems, and multirate systems. The main result of this paper is jagopal and Potter [2] and Zhou and Do [3] have investigated H z to prove that when N P M, then ( ) is generically this question and made several conjectures. We investigate invertible; whereas when− ≥ , then H z is gener- N P <M ( ) the systems by varying M, N and P . In the experiments, we ically noninvertible. As a result,− we can have an alternative found that when M N P , the existence of an inverse is approach in design of the multidimensional systems. “almost surely”. On− the other≥ hand, when M N < P , the − Index Terms— Generic Invertible, Left Invertibility, Per- nonexistence of an inverse is “almost surely”. To precisely fect Reconstruction, Multirate Systems, Generic Property. study this inverse existence problem, we employ the concept of “hold generically” [4]. 1. INTRODUCTION 2. GENERIC INVERTIBILITY During the last two decades, one dimensional multirate sys- 2.1. Generic Property tems in digital signal processing were thoroughly developed. In recent years, due to the high demand in multidimensional In [3], Zhou and Do made the following conjectures. processing including image and video processing, volumetric Conjecture 1 Suppose H z is an -variate poly- data analysis and spectroscopic imaging, multidimensional ( ) N P M nomial (resp. : Laurent polynomial) matrix× with . If multirate systems have been studied more extensively. One N P , then it is “almost surely” polynomial≥ (resp. : key property of a multidimensional multirate system is its per- N P M Laurent− ≥ polynomial) left invertible. Otherwise, it is “almost fect reconstruction, which guarantees that an original input surely” polynomial (resp. : Laurent polynomial) left nonin- can be perfectly reconstructed from the outputs. vertible. In a multidimensional multirate system, a digital signal is split into several channels and processed with different sam- However, Zhou and Do did not give a precise definition of pling rates. The most popular multirate systems are filter “almost surely”. In order to have the appropriate language, banks. Using the polyphase representation in the z-domain we employ the concept of “hold generically”. [1], we can represent the analysis part as an matrix N P Definition 1 (Generic) [4] A property is said to hold gener- H(z) with entries in a Laurent polynomial ring C×[z ,z , ..., 1 2 ically for polynomials f , .., f of degree at most k , ..., k if z ,z −1, ..., z −1]. Here M is the dimension of signals, 1 n 1 n M 1 M there is a nonzero polynomial F in the coefficients of the f N is the number of channels in the filter bank, and P is the i such that the property holds for f , ..., f whenever the poly- sampling factor at each channel. An application of this set- 1 n nomial F (f , ..., f ) is nonvanishing. ting may arise in multichannel acquisition. Here we collect 1 n data about unknown multidimensional signal X(z) as out- Lemma 1 If a property of polynomials of degree at most k1, put of the analysis part. The acquisition system (filters Hi(z) ..., kn in m variables is generic, then the coefficient space of and sampling matrix D) is fixed and known beforehand. The polynomials whose polynomials failed to satisfy the properC ty objective is to reconstruct X(z) with an P N synthesis is measure zero and nowhere dense. × Proof By the definition of hold generically, there exists a and N P , then the P P maximal minors of H(z) have ≥ × nonzero polynomial F in the coefficients of the fi such that a common zero. Suppose (˜z1/z˜0, z˜2/z˜0, ..., z˜M /z˜0) is a so- the property fails to satisfy for f , ..., fn for which the polyno- lution of the maximal minors of H(z) where z˜ = 0. Then 1 0 6 mial F (f1, ..., fn) is vanishing. Let Ri bethesetof M-variate (˜z0, z˜1, z˜2, ..., z˜M ) is a nonzero solution of maximal minors polynomialsof degree less than or equal to ki. By Gunningin of H(Z). Since t1, ..., tM+1 is a part of the subset of the n { H Z} H z [5, p.9], λl( (f1, ..., fn) i=1 Ri F (f1, ..., fn)=0 )= set of maximal minors of ( ), this implies that φ( ( )) { 1 ∈ Qn | } 0 where l = k +m +...+ k +m is the dimension of the co- have a nontrivial common zero. Therefore, by the property of m m efficient space. Thus, the coefficient space of polynomials the resultant shown in Theorem 1, we know RES(Pk,...,Pk) whose polynomials failed to satisfy the propertyC is measure φ(H(z)) = 0 for all noninvertible matrices H(z) of de- ◦ zero. To show the set is nowhere dense, it is equivalent to gree at most k. The RES(Pk,...,Pk) and ti are polynomials, show that the closure of the set contains no open set. Sup- so is RES Pk,...,Pk φ. Last but not least, we need to show ( ) ◦ pose it contains an open ball B(ǫ) with some radius ǫ > 0. RES(Pk,...,Pk) φ is not a zero function. Let −1 −1 ◦ ½ Since F ( 0 ) is a closed set, is also in F ( 0 ). Thus, ¼ −1 { } C { } 1 0 . 0 F ( 0 ) contains the open ball B(ǫ). However, this contra- k − z1 1 . 0 dicts the{ } fact that F 1( 0 ) is measure zero. Therefore, the . { } . coefficient space of polynomials whose polynomials failed to . 1 satisfy the property is nowhere dense. k k .. k zM zM−1 . z1 The immediate consequence is that if f1, ..., fn are drawn in- T z k . ( )= . zM . dependently from a probability distribution with respect to the 0 . . Lebesgue measure, the property of holds with prob- . zk f1, ..., fn . M ˜ ˜ ability one. Furthermore, suppose f0, ..., fn satisfies the prop- 0 . 0 0 erty. Since the coefficient space of polynomials whose poly- . nomials failed to satisfy the propertyC is nowhere dense, there . 0 . 0 0 exists an open ball B(ǫ) around f˜0, ..., f˜n for some ǫ> 0 such that the property is satisfied within the open ball B(ǫ) . This be an N P matrix. Suppose RES φ(T (z)) = × (Pk,...,Pk) ◦ shows that the system with the property is robust [6]. 0. By Theorem 1, we know that ti’s have a nontrivial com- mon zero. i.e. there exists Z˜ a nonzero solution such that Z˜ P k 2.2. Generically Invertible when N P M tM+1( )=˜zM =0. This implies z˜M =0. If z˜M =0, then − ≥ P k t (˜z , z˜ , ..., z˜ − , 0)=z ˜ =0. Thus z˜ − =0. Con- To prove our main theorem in this section, we need to employ M 0 1 M 1 M−1 M 1 tinuing the process, we can conclude z˜ =z ˜ = ... =z ˜ = the resultant of the polynomials. 0 1 M 0. This contradicts the assumption that Z˜ is nontrivial. So Theorem 1 (Resultant) If we fix positive degrees k0, ..., kn, RES(Pk,...,Pk) φ(T (z)) =0. Therefore RES(Pk,...,Pk) φ then there is a unique nonzero polynomial called the resul- is not zero function.◦ By the6 definition of hold generically,◦ we C tant RES(k0,...,kn) [ ui,j ] where the variables ui,j cor- conclude that H(z) of degree at most k is generically poly- ∈ { } respond to the coefficients of i-th polynomial. If F0, ..., Fn nomial left invertible matrix. ∈ C[x0, ..., xn] are homogeneous of degrees k0, ..., kn, then F0, ..., Fn have a nontrivial common zero over C if and only if Theorem 3 If N P M and k > 0, then an N P polynomial M-variate− matrix≥ H(z) of degree at most×k is RES(k0,...,kn)(F0, ..., Fn)=0. generically Laurent polynomial left invertible. Theorem 2 If N P M and k > 0, then an N P − ≥ × polynomial M-variate matrix H(z) of degree at most k is Proof If a polynomial matrix H(z) is Laurent polynomial generically polynomial left invertible. left noninvertible, then H(z) is also polynomial left nonin- Proof The strategy of this proof is to find a nonzero poly- vertible. Accordingto Theorem2, this shows that RES(Pk,...,Pk) H z nomial F such that F (H(z)) = 0 for every noninvertible φ( ( )) = 0 for all Laurent polynomial left noninvertible polynomial◦ matrix H(z). matrix H(z) of degree at most k. Let Z = (z0, ..., zM ). If f(z) = f0(z)+ f1(z)+ ... + fl(z) is the decomposition of the polynomial f(z) into sums of forms fi(z) of degree 2.3. Generically Noninvertible when N P <M Z z − i, then the homogenization f( ) of f( ) of degree k is de- Pn Z k z k−1 z k−l z Projective n-space is the set of equivalence classes of (n+ fined to be f( )= z0 f0( )+ z0 f1( )+ ..