2017 25th European Signal Processing Conference (EUSIPCO) Characterization of Discrete Linear Shift-Invariant Systems Michael Clausen and Frank Kurth Fraunhofer FKIE Communication Systems 53343 Wachtberg Germany Email: [email protected], [email protected] Abstract—Linear time-invariant (LTI) systems are of funda- i.e., C(A) = C[A]. Note that in this case, the centralizer is a mental importance in classical digital signal processing. LTI sys- commutative algebra. If however A is the N × N unit matrix, tems are linear operators commuting with the time-shift operator. then C(A) = CN×N , which is non-commutative for N ≥ 2. For N-periodic discrete time series the time-shift operator is a circulant N × N permutation matrix. Sandryhaila and Moura Thus centralizers can be non-commutative. In fact, we prove developed a linear discrete signal processing framework and in Section IV, that the centralizer of A is commutative if and corresponding tools for datasets arising from social, biological, only if all eigenspaces of A are one-dimensional. Thus if we and physical networks. In their framework, the circulant permu- are in the degenerate case, then not all LSI systems w.r.t. A N ×N A tation matrix is replaced by a network-specific matrix , are of the form h(A), for a polynomial h. Hence the argument called a shift matrix, and the linear shift-invariant (LSI) systems are all N × N matrices H over C commuting with the shift [1] (p. 1647, following Theorem 2) reducing graph filters for matrix: HA = AH. Sandryhaila and Moura described all those the degenerate case to filters in the non-degenerate case does H for the non-degenerate case, in which all eigenspaces of A not hold. are one-dimensional. Then the authors reduced the degenerate In this paper, we give a complete description of LSI sys- case to the non-degenerate one. As we show in this paper this tems by describing the centralizer of arbitrary complex-valued reduction does, however, not generally hold, leaving open one gap in the proposed argument. In this paper we are able to N × N matrices A. We start in Section II with two examples close this gap and propose a complete characterization of all illustrating the degenerate case. To keep this paper to some (i.e., degenerate and non-degenerate) LSI systems. Finally, we extent self-contained, we introduce in Section III notation describe the corresponding spectral decompositions. required in the following and discuss basic tools: associative algebras and their morphisms, minimal and characteristic poly- I. INTRODUCTION nomials, upper triangular Toeplitz matrices, Jordan canonical Linear time-invariant (LTI) systems are linear operators form, and Hermite interpolation. After compiling these tools, commuting with the time-shift operator. Such systems are of we present in Section IV the characterization of centralizers. fundamental importance in classical digital signal processing. Finally, in Section V we describe the spectral decompositions For N-periodic discrete time series the time-shift operator is of the signal space CN w.r.t. C[A] and C(A). the circulant permutation matrix corresponding to the cyclic II. ILLUSTRATING THE DEGENERATE CASE permutation (0, 1,...,N − 1). In [1], Sandryhaila and Moura There are a number of network-specific matrices A with developed a linear discrete signal processing framework and repeated eigenvalues. Here are two examples. corresponding tools for datasets arising from social, biologi- cal, and physical networks. In their framework, the circulant Example 1. Figure 1 shows the Petersen graph. permutation matrix is replaced by a network-specific matrix bc CN×N bc A ∈ , the shift matrix, and the linear shift-invariant bc bc bc bc (LSI) systems are all matrices in the centralizer of A: bc bc C(A) := {H ∈ CN×N | HA = AH}. bc bc Fig. 1. Peterson graph This centralizer has an additional algebraic structure, briefly N×N reviewed in Subsection III-A: C(A) is a subalgebra of C . The characteristic polynomial pA(X) := det(XI − A) of its Moreover, C(A) contains the algebra C[A] of all polynomial adjacency matrix A equals (X − 1)5 · (X + 2)4 · (X − 3). N×N expressions in A. Thus C[A] ⊆ C(A) ⊆ C . Thus λ1 = 1 is a 5-fold eigenvalue, whereas λ2 = −2 is a 4- In [1], the authors provide an explicit description of the fold eigenvalue of A. A is symmetric, hence diagonalizable. centralizer of A in the non-degenerate case in which the Thus the dimension of C[A] equals the number of distinct eigenspace of every eigenvalue of A is one-dimensional. More eigenvalues of A, which is 3, whereas the centralizer C(A) is precisely, they showed that in this case the centralizer of A isomorphic to the non-commutative algebra C5×5 ⊕ C4×4 ⊕ coincides with the algebra of all polynomial expressions in A, C1×1, which is of dimension 42. ♦ ISBN 978-0-9928626-7-1 © EURASIP 2017 366 2017 25th European Signal Processing Conference (EUSIPCO) Example 2. As a second example we consider the adjacency B. Upper Triangular Toeplitz Matrices matrix A of a clique with nodes. Here, m×n = (ai,j ) N ai,i = 0 Recall that a matrix T = (ti,j ) ∈ C is a Toeplitz matrix, and ai,j = 1, for all i and all j 6= i. Figure 2 illustrates the if each descending diagonal from left to right is constant, i.e., case N = 6. ti,j = ti+1,j+1, whenever both sides are defined. Thus every m × n Toeplitz matrix is completely specified by the entries in its leftmost column and its topmost row. We are mainly interested in upper triangular Toeplitz matrices. Such matrices are specified by its topmost row. For a = (a0, . , ad−1) ∈ Fig. 2. Clique for N = 6. d d×d C define T d(a) = T d(a0, . , ad−1) = (ti,j ) ∈ C by t := 0, if i > j, and t := a , if i ≤ j. Thus A is a circulant matrix. (Recall that an N × N matrix of i,j i,j j−i the form (aj−i mod N )i,j is called circulant.) It is well- a0 a1 a2 . ad−2 ad−1 pq a0 a1 a2 ad−2 known that the N × N DFT matrix F = (ω )0≤p,q<N , .. .. .. ω := exp(2πi/N), performs a simultaneous diagonalization of T (a) = . d all circulant N × N matrices A. More precisely, if a denotes a0 a1 a2 a0 a1 the leftmost column of A, then a0 −1 ∆ F AF = diag(F a) =: . Td denotes the set of all d × d upper triangular Toeplitz In our case, a = (0, 1,..., 1)⊤ ∈ CN . A straightforward matrices. calculation shows that F a ⊤. In = (N − 1, −1,..., −1) Lemma 3. Td is a commutative C-algebra. More precisely, particular, is a simple eigenvalue, whereas is an d N − 1 −1 if ej denotes the jth unit vector in C , then the matrices (N − 1)-fold eigenvalue of A. Let’s compare C[∆] ≃ C[A] T d(e0),..., T d(ed−1) are a basis of Td and for 0 ≤ i, j < d, and C(∆) ≃ C(A) for ∆ := diag(N − 1, −1,..., −1). If C ∆ T d(ei+j ) if i + j < d, h ∈ [X], then h( ) = diag(h(N − 1), h(−1), . , h(−1)). T d(ei) · T d(ej) = This shows that C[∆] and hence C[A] is 2-dimensional, 0 otherwise. C ∆ d in contrast to the centralizer ( ) which is equal to the In general, if a, b, c ∈ C and T d(a) · T d(b) = T d(c), then space C1×1 ⊕ C(N−1)×(N−1). Thus the centralizer C(A) is k ck = j=0 aj bk−j, for k ∈ [0, d − 1]. of dimension 1 + (N − 1)2. ♦ ProofP. Follows by a straightforward computation. Both examples show that in the degenerate case the central- Thus the multiplication of upper triangular Toeplitz matrices izer of A can be much larger than the space of all polynomial in T boils down to truncated polynomial convolution. Via expressions in A. d FFT, this multiplication can be performed with O(d log d) III. PRELIMINARIES arithmetic operations. Below, we need the following gener- A. Review of Associative Algebras alization of Td. An associative C-algebra A is a vector space over C with Definition 4. For positive integers d, e let Td,e denote the an associative multiplication and a unit element 1A. Moreover, set of all d × e matrices T = (ti,j )0≤i<d,0≤j<e, such that addition, scalar multiplication, and multiplication must be ti,j = ti+1,j+1, whenever both sides are defined. Moreover, compatible. The space ti,j is zero if j − i < max(0, e − d). n j Note that T = T . C[X] := hjX | n ≥ 0; h0, . , hn ∈ C d,d d nXj=0 o Example 5. Let a0, . , a4 ∈ C. Then typical elements in T3,5 N×N of all univariate polynomials or the space C of all N ×N resp. T5,3 look as follows: matrices are examples of C-algebras. A subalgebra B of A is a linear subspace of A closed under multiplication a0 a1 a2 0 0 a a a 0 a a and containing 1A. For example, the space of all upper 2 3 4 0 1 triangular N × N matrices is a subalgebra of CN×N . An 0 0 0 a2 a3 resp. 0 0 a0 . algebra morphism T : A → B is a C-linear map that is also 0 0 0 0 a2 0 0 0 0 0 0 multiplicative and maps 1A to 1B.