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 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 PM 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 ◦ Since F ( 0 ) is a closed set, is also in F ( 0 ). Thus, −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 0 and N P . Let V ( mi ) n ≥ N { } 3 0 0 0 500 := Z P mi(Z)=0 for all i =1, ..., where mi { ∈ | P
} is a maximal minor of H(Z) with some ordering and H(Z) 1 0 0 500 500 M=2 P 2 0 0 0 500 is the homogenization of H(z) of degree k. Then V ( mi ) is { } 3 0 0 0 0 empty if and only if ht mi = M +1. Therefore if V ( mi ) is empty, then N P h Mi. In other words, if N P{ 0, then an N P By previous discussion, F (H(z)) = 0 for all Laurent poly- polynomial M-variate− matrix H(z) of degree at most×k is nomial left invertible polynomial matrix H(z). This shows generically Laurent polynomial left noninvertible. that if N P 0, then an N P − H z × trix H(z). If N < P , then every polynomial matrix is left polynomial M-variate matrix ( ) of degree at most k is generically polynomial left noninvertible. noninvertible. Now consider H(z) is invertible. Let cij be a coefficient for the constant term of h (z) where H(z) = ij Proof Similar proof from Theorem 3. (hij (z)). Define a function F1 such that

H(z) cij . 2.4. Simulation and Applications 7→ Y i ,...,N j ,...,P =1 =1 From Table 1, we used a random polynomial matrix genera- If hij (z1, ..., zN−P +1, 0, ..., 0) = 0 for some i, j, then it im- tor to generate polynomial matrices with each entry of degree plies cij = 0. This shows that F (H(z)) = 0 in (1). If less than or equal to 4 and the random coefficients are from 1 hij (z1, ..., zN−P +1, 0, ..., 0) = 0 for all i, j, then H(z1, ..., to 100. In each value of N, P and M, we ran 500 samples to 6 zN−P +1, 0, ..., 0) is also invertible because there exists Lau- test the inversibility. We found out that they agreed with our rent polynomial matrix G(z) such that G(z)H(z) = I and theorems. These theorems lead to some applications. For im- G(z1, ..., zN−P +1, 0, ..., 0) is well-defined. We can now as- age deconvolution from multiple FIR blur filters, Harikumar sume that M = N P +1. Define ti(Z) to be the same and Bresler in [6] show that perfect reconstruction is almost − i-th surely, when there are at least three channels. Since image as Theorem 2. Let t(i) = t (z , ..., 0 , ..., z ). Define j j 0 M is two dimension (i.e. M = 2) and the downsampling rate (i) (i) T θi to be a function such that H(z) (t , ..., t ) for is just one (i.e. P = 1), by Theorem 3, we know that the 7→ 1 M i = 0, ..., M. By Lemma 2 and Lemma 3 and the fact that perfect reconstruction is almost surely if number of channels t(i)(Z), ..., t(i)(Z) is the subset of the set of maximal mi- is greater than two (i.e. N 3). Therefore Harikumar and { 1 M } ≥ nors of H(Z), it implies that θi(H(z)) have a nonzero com- Bresler’s image deconvolution is a special case of our main mon zero for some i = 0, ..., M. By the property of the re- theorem. Another application is that we can have an alterna- sultant shown in Theorem 1, we know that given any Lau- tive approach in designing multidimensional filter banks. We rent polynomial left invertible polynomial matrix H(z), so can freely design the analysis side first such that it satisfies the condition (i.e. N P M). Then , by Theorem 3 and (i.e. N P M), then we can almost surely find a per- Lemma 1, we can almost− surely≥ find a perfect reconstruction fect reconstruction− ≥ inverse for the synthesis polyphase matrix. inverse for the synthesis polyphase matrix. These results also have potential applications in multidimen- sional signal reconstruction from multichannel filtering and 2.5. Fast Computation of Left Inverse sampling. Another application is that we can improve the Laurent poly- 4. REFERENCES nomial inverse algorithm [11]. Since if N P M and H z H z − ≥ ( ) is a polynomial matrix, then ( ) is generically Lau- [1] P. P. Vaidyannthan, Multirate Systems and Filter Banks. rent polynomial left invertible by theorem 3. However, at the Prentice-Hall, 1993. same time, the H(z) is generically polynomial left invertible by Theorem 2. Therefore instead of apply the Laurent Poly- [2] R. Rajagopal and L. C. Potter, “Multivariate MIMO FIR nomial Inverse Algorithm in [11], we should simply apply the inverses,” IEEE Trans. Image Proc., vol. 12, pp. 458– Polynomial Inverse Algorithm in [11] which is less expensive 465, Apr. 2003. in term of time and storage. For a convenience sake, we de- note our Laurent Polynomial Inverse Algorithm 2 in [11] to [3] J. Zhou and M. N. Do, “Multidimensional oversampled be LPIA and denote our Polynomial Inverse Algorithm 1 in filter banks,” in Proc. of SPIE Conference on Wavelet [11] to be PIA. Applications in Signal and Image Processing XI, San Diego, USA, Jul. 2005. Algorithm 1 (Faster Version) The computational algorithm for a Laurent polynomial left inverse matrix. [4] D. Cox, J. Little, and D. O’Shea, Ideal, Varieties, and Input: N P Laurent polynomial matrix H(z) with M vari- Algorithms. Springer-Verlag, 1996. ables. × [5] R. C. Gunning, Analytic Function of Several Complex Output: P N Laurent polynomial matrix G(z), if it exists. Variables. Prentice-Hall, 1965. 1. Multiply×H(z) by a common monomial zl such that zlH(z) are polynomial matrix. [6] G. Harikumar and Y. Bresler, “Perfect blind restoration l 2. Call PIA with the input z H(z). of images blurred by multiple filters: theory and ef- −l 3. If the output of PIA is J(z), then output z J(z). ficient algorithms,” IEEE Trans On Image Processing, 4. Otherwise call LPIA. vol. 8, no. 2, pp. 202–219, Feb. 1999. Example 1 Compare the processing time between LPIA and [7] R. Rajagopal and L. C. Potter, “Multi-channel multi- Algorithm 1. Let H(z1,z2) variate equalizer computation,” Multidimensional Sys-

−1 2 −1 tems and Signal Processing, vol. 14, pp. 105–108, Jan. 4z1 7z1 z2 + 2 + 10z1 −1 2003.  1 + 10z1 10z1 + 3z2  = −1 −1 7z1 + 9z2 + 10z1 z2 + 10z1 0 [8] F. Macaulay, The Algebraic Theory of Modular Systems.  z −1z2 z −1 z −1z2   8 1 2 + 10 + 4 1 6 1 2  Cambridge University Press, 1916. be a Laurent polynomial matrix. Then we found out that the [9] J. Zhou, “Multidimensional multirate systems: charac- run time of LPIA and Algorithm 1 is 0.23 sec and 0.06 sec re- terization. design, and applications,” Ph.D. dissertation, spectively for using a desktop PC. This agrees that Algorithm University of Illinois at Urbana-Champaign, 2005. 1 is faster than LPIA in this example. [10] E. Fornasini and M. E. Valcher, “Multidimensional sys- tems with finite support behaviors: signal structure, gen- 3. CONCLUSION eration, and detection,” SIAM Journal on Control and We shows that there is a sharp phase transition on the invert- Optimization, vol. 36, no. 2, pp. 760 – 779, Mar. 1998. ibility depending on the size and dimension of a given Lau- [11] K. L. Law, R. M. Fossum, and M. N. Do, rent polynomial matrix. Specifically when N P M, the − ≥ “Invertibility of multidimensional FIR fil- N P polynomial (resp. : Laurent polynomial) of M-variate ter banks and mimo systems,” IEEE Trans. matrix× is generically invertible; whereas when N P