IEEE SIGNAL PROCESSING LETTERS: SPL.SP.4.1 THEORY, ALGORITHMS, AND SYSTEMS 0 Designing Commutative Cascades of Multidimensional Upsamplers and Downsamplers Brian L. Evans, Member, IEEE Abstract In multiple dimensions, the cascade of an upsampler by L and a downsampler by L commutes if and only if the integer matrices L and M are right coprime and LM = ML. This pap er presents algorithms to design L and M that yield commutative upsampler/dowsampler cascades. We prove that commutativity is p ossible if the 1 Jordan canonical form of the rational resampling matrix R = LM is equivalent to the Smith-McMillan form of R. A necessary condition for this equivalence is that R has an eigendecomp osition and the eigenvalues are rational. B. L. Evans is with the Department of Electrical and Computer Engineering, The UniversityofTexas at Austin, Austin, TX 78712-1084, USA. E-mail: [email protected], Web: http://www.ece.utexas.edu/~b evans, Phone: 512 232-1457, Fax: 512 471-5907. This work was sp onsored in part by NSF CAREER Award under Grant MIP-9702707. July 31, 1997 DRAFT IEEE SIGNAL PROCESSING LETTERS: SPL.SP.4.1 THEORY, ALGORITHMS, AND SYSTEMS 1 I. Introduction 1 Resampling systems scale the sampling rate by a rational factor R = L=M = LM , or 1 equivalently decimate by H = M=L = L M [1], by essentially upsampling by L, ltering, and downsampling by M . In converting compact disc data sampled at 44.1 kHz to digital audio tap e 48000 Hz 160 data sampled at 48 kHz, R = = . Because we can always factor R into coprime 44100 Hz 147 integers L and M , we can always commute the upsampler and downsampler which leads to ecient p olyphase structures of the resampling system. In multiple dimensions, resampling is describ ed by a rational matrix R. Multidimensional resampling systems are essentially a 1 cascade of an upsampler by L, a lter, and a downsampler by M, such that R = LM and L and M are non-singular integer matrices. Although it is rare that a cascade of an upsampler and downsampler commutes in multiple dimensions, we can nonetheless always nd p olyphase structures for multidimensional resampling systems. Polyphase structures exist b ecause we can always factor R into right coprime L and M which is known as relaxed commutativity [2]. Commutativity of a multidimensional upsampler and downsampler in cascade [2], [3], [4], [5] o ccurs when 1: L and M are right coprime, and 1 2: LM = ML Many approaches exist for checking whether or not a cascade commutes, given values of L and M. Techniques exist for generating L and M that are right coprime by decomp osing R into its Smith-McMillan form [6]. Finding algorithms to generate cascades that satisfy b oth commutativity conditions is an op en problem [7]. These algorithms play a role in rearranging [4] and scheduling [8] multidimensional multirate systems. In the pap er, we develop two algorithms to design L and M that yield commutative cascades: 1. generate L and M, and 1 2. factor a desired R into LM 1 ^ ^ ^ ^ if L and M are given, use R = L M The key to the new algorithms is that we can satisfy b oth commutativity conditions. In partic- 1 ular, we can satisfy the condition that LM = ML if the Jordan canonical form of R = LM is equivalent to a Smith-McMillan form. We show that a necessary condition for this equivalence is that R has an eigendecomp osition and the eigenvalues are rational. Once we satisfy LM = ML using our approach, we automatically satisfy the other commutativity condition. July 31, 1997 DRAFT IEEE SIGNAL PROCESSING LETTERS: SPL.SP.4.1 THEORY, ALGORITHMS, AND SYSTEMS 2 II. Background In multiple dimensions, downsampling is describ ed by a non-singular integer downsampling matrix M. On average, a downsampler outputs one sample for every group of j det Mj input samples. Its resp onse x [n] to input x[n]is d x [n]=x[Mn] 2 d Upsampling is describ ed by a non-singular integer upsampling matrix L. It outputs j det Lj samples for every input sample. Its resp onse x [n] to input x[n]is u 8 > 1 1 < x[L n] if L n 2R I x [n]= 3 u > : 0 otherwise An upsampler/downsampler cascade commutes if L and M meet the conditions in 1. The Smith form decomp osition is useful for testing [4] and generating [6] L and M to meet the right coprime condition. The Smith form decomp osition of a non-singular integer matrix S is S = U V 4 S S S where is a diagonal integer matrix and U and V are unimo dular integer matrices a S S S unimo dular matrix has a determinantof1. The Smith form decomp osition decouples a linear op eration and is not unique. When applied to non-singular rational matrix, it is called the Smith- McMillan form decomp osition, and has the same form as 4 but is a diagonal rational matrix. The Smith form decomp osition always exists for non-singular integer and rational matrices [9]. 1 ^ ^ ^ ^ Given a rational resampling matrix R, or given L and M and forming R = L M , we 1 can always factor R = LM such that L and M are right coprime using the Smith form decomp osition of R [6]: R = UV 1 = U V L M 5 1 1 = U V L M 1 = LM 1 is a diagonal rational matrix, = , and are diagonal integer matrices, and L M L M 1 L = U and M = V 6 L M 1 Since V is unimo dular, V is unimo dular. So, L and M are always right coprime provided that l each rational numb er on the diagonal of is reduced to where l and m are coprime integers. m July 31, 1997 DRAFT IEEE SIGNAL PROCESSING LETTERS: SPL.SP.4.1 THEORY, ALGORITHMS, AND SYSTEMS 3 III. Designing Commutative Cascades In this section, we determine the conditions for which L and M satisfy the commutativity conditions given by 1. Since L and M are non-singular, the condition LM = ML holds when L and M have the same eigenvector matrix denoted by U [10] 1 1 L = U U and M = U U 7 L M b ecause 1 1 U U LM = U U M L 1 = U U L M 1 = U U M L = ML Notice that L and M are coprime when and are coprime. Forming the rational matrix L M 1 1 R = LM = UU 8 we develop an algorithm to generate L and M, and another to determine when and how to factor a given R matrix into L and M, that satisfy the commutativity conditions. Since a non-singular matrix do es not always have an eigendecomp osition, suchas 3 2 1 1 0 7 6 7 6 0 1 1 5 4 0 0 1 which cannot b e diagonalized, we use the Jordan canonical form which always exists for a non- singular matrix [10]. The Jordan canonical form decomp oses a non-singular matrix R into 1 SJS . When J is a diagonal matrix, the Jordan canonical form is an eigendecomp osition of R in which the diagonal elements of J are the eigenvalues and the columns of S are the eigenvectors corresp onding to the eigenvalues [10]. In our case, R is a rational resampling matrix. Theorem 1 states that we can always nd a rational eigenvector matrix for a rational matrix when the eigenvalues are rational. Theorem 2 de nes equivalence b etween the eigendecomp osi- tion and Smith form decomp osition. Theorem 3 determines how to compute 7 from R when Theorem 2 applies. Theorem 1: Given a square non-singular rational matrix R with Jordan canonical form 1 ^ ^ R = SJS , if J is a diagonal rational matrix, then we can always nd an alternative 1 Jordan canonical form R = SJS in which S is a rational matrix. July 31, 1997 DRAFT IEEE SIGNAL PROCESSING LETTERS: SPL.SP.4.1 THEORY, ALGORITHMS, AND SYSTEMS 4 Pro of: If J is diagonal, then the Jordan canonical form of R is an eigendecomp osition of R. ^ ^ ^ De ne s to b e the ith column vector eigenvector of S. Therefore, s satis es i i ^ ^ R s = s i i i where = J , i.e., the ith eigenvalue. An eigenvector is unique only up to a scale factor, so i ii ^ s = s where is a scalar. Since R is a rational matrix and is a rational numb er, we can i i i i ^ always compute by taking the gcd of the non-zero elements of s to yield a rational eigenvector i 1 ^ s = s . Then, we construct S by setting its ith column vector to b e equal to s for all i. QED i i i 1 Theorem 2: An eigendecomposition of a non-singular rational matrix R = SJS is 1 equivalent to the Smith-McMil lan decomposition of R = UV if the eigenvalues of R given by the diagonal elements of J are rational and there exists a rational diagonal matrix D for which U = SD is a unimodular integer matrix. Pro of: Let D b e a diagonal matrix. Generalize the eigendecomp osition of R as 1 1 1 1 R = SJS = SJDD S =SD JSD since the pro duct of diagonal matrices commutes JD = DJ. Equating the general eigende- comp osition and the Smith-McMillan decomp osition, we get J = and U = SD Since is a diagonal rational matrix, J is a diagonal rational matrix, which means that R has rational eigenvalues.