THE ALGEBRAIC STRUCTURE IN SIGNAL PROCESSING: TIME AND SPACE

Markus Puschel¨ and Jose´ M. F. Moura Electrical and Computer Engineering Carnegie Mellon University Email: {pueschel,moura}@ece.cmu.edu

ABSTRACT visualizations are undirected. We thus call them space models. Fi- nally, we briefly discuss the extension of our approach to 2-D SP The assumptions underlying linear signal processing (SP) produce for separable and nonseparable time and space models. more structure than vector spaces. We capture this structure by de- Organization. We introduce the concept of algebra and mod- scribing the space of filters as an algebra and the space of signals as ule in Section 2. In Section 3, we motivate why linear SP is al- the associated module. We formulate an algebraic approach to SP gebraic in nature and introduce the concept of a signal model that that is axiomatically based on the concept of a signal model. Sig- underlies our algebraic approach. In Sections 4 and 5, we show the nal models for time are visualized as directed graphs. We construct signal models for infinite and finite SP in time and space. Section 6 corresponding models for undirected graphs, which we hence call briefly discusses the extension to separable and nonseparable 2-D space models, and show that, in particular, the 16 DCTs and DSTs SP. Finally, section 7 concludes the paper. are Fourier transforms for these finite space models. Finally, we discuss the extension of our theory to separable and nonseparable 2-D SP. 2. BACKGROUND: ALGEBRAS AND MODULES In this section, we introduce two concepts from abstract algebra: 1. INTRODUCTION Algebra and module. These are needed to identify the algebraic structure in linear signal processing. Both are vector spaces, but Linear signal processing (henceforth simply called SP) is a well- possess additional structure. In Section 3, we will explain that in developed theory and a comprehensive set of tools for processing linear signal processing filter spaces are algebras and signal spaces 1-D time signals, continuous or discrete, of infinite or finite du- are modules. For every given pair of algebra and module, there is ration. Each of these four cases has specific notions of filtering, an associated notion of Fourier transform that we introduce. We Fourier transform, and other key concepts needed in SP. focus on regular modules (the algebra and the module are the same Many other transforms are used in SP, most prominently the set but with different properties) of polynomial algebras, which are class of trigonometric transforms that include the discrete cosine particularly relevant in signal processing, as we will explain later. and sine transforms (DCTs and DSTs). These compute a spec- Algebra (filter space). An algebra A is a vector space that trum of sorts, so conceptually they are Fourier transforms; but, is also a ring, i.e., it permits multiplication and the distributivity questions remain: for which type of signals? what are their infi- law holds. Examples include the field of complex numbers C, nite counterparts? and, just as for the DFT, what are the associ- the space of polynomials in one variable C[x], and the space of ated notions of filtering, convolution, and spectrum? The current Laurent series in x with absolute summable coefficient sequences. derivation of the DCTs based on random processes, [1], or their in- Another example, particularly important in SP, is introduced next. terpretation as Karhunen-Loeve` transforms of random fields, [2], Let p(x) ∈ C[x] be a fixed polynomial of degree deg(p) = n, do not answer these questions. then the set of polynomials In 2-D, SP is usually developed in a separable way, which as- sumes the signal resides on a rectangular lattice. For many ap- C[x]/p(x) = {q(x) ∈ C[x] | deg(q) < deg(p)} (1) plications, this is appropriate. Mersereau et al. [3] provide some of degree smaller than n with addition and multiplication modulo answers for more general lattices. We would like, for example, to p is called a polynomial algebra. In C[x]/p(x), p(x) = 0 holds. explore generic questions including what are the DCT equivalents As a vector space, C[x]/p(x) has dimension deg(p) = n. We on nonseparable lattices. write elements of A as h, the common letter used for filters in SP. In this paper, we give answers to these questions in an overview Module (signal space). Assume that an algebra A has been on recent work [4, 5]. Our approach is in two steps. First, we chosen. An A-module, M is a vector space that permits an oper- identify the algebraic structure inherent to SP that goes beyond ation of the A on M, written as “·” (multiplication), such that the vector spaces to algebras and modules. We argue that polynomial distributivity law holds (as well as some other properties that we algebras are key concepts in SP. Second, we formulate a general omit). Formally, for h ∈ A, s, s , s ∈M, framework for SP, based on the notion of a signal model that we 1 2 define as a triple of an algebra of filters, a module of signals, and h · s ∈M, h · (s1 + s2) = h · s1 + h · s2. a mapping Φ that generalizes the z-transform. Once a model is given, other concepts such as the Fourier transform are automati- For a given algebra, one choice of module is the regular mod- cally defined. We describe the signal models for infinite and finite ule M = A with the operation of A on M being the multiplica- time SP and visualize them as directed graphs. Then we identify tion in A. Almost all modules considered in this paper are regular. the models underlying the DCTs and DSTs and show that their We will focus on algebras and modules that are C-vector spaces in this paper, but the theory extends to other basefields. We denote This work was supported by NSF through award 0310941. signals in M as s. Fourier transform. Assume that an algebra A and an A- • / • / • / • / • / • module M are given. Then the associated Fourier transform (if x−2 x−1 x0 x1 x2 x3 it exists) is a linear mapping1 that decomposes M into a direct Fig. 1. Visualization of the infinite discrete time model (4). sum of smallest submodules invariant under A. We omit a formal definition, as we are only interested in one special case discussed next. as explained below. Once a signal model is chosen, filtering is C Let A = [x]/p(x) and let M = A be the regular mod- well-defined (the operation of A on M) and the main ingredients ule. Further, assume p(x) is of degree n and separable, i.e., it has for SP follow: spectrum, Fourier transform (see Section 2), fre- pairwise distinct zeros α = (α0,...,αn−1). Then, the associated quency response, as well as others [4]. Fourier transform is given by the Chinese remainder theorem as Example. Consider the space of finite energy sequences s ∈ V = ℓ2(Z). A signal model for V is given by ∆ : C[x]/p(x) → C[x]/(x − α0) ⊕ . . . ⊕ C[x]/(x − αn−1), s(x) 7→ (s(α0), . . . , s(αn−1)). ℓ 1 Z A = {Pℓ∈Z hℓx | (hℓ) ∈ ℓ ( )}, Since ∆ is a linear mapping, we can express it as a matrix w.r.t. cho- ℓ 2 Z (4) M = {Pℓ∈Z hℓx | (hℓ) ∈ ℓ ( )}, sen bases. Let b = (p0, . . . , pn−1) be a basis of M, and choose s ℓ 0 Φ = V →M, 7→ s = s(x) = Pℓ∈Z sℓx . (x ) as basis in each C[x]/(x − αk). Then ∆ corresponds to The mapping Φ is the ordinary z-transform3. For this reason we P = [p (α )] , (2) b,α ℓ k 0≤k,ℓscalar multiplication = amplification by α ∈ C; and focus on providing intuition about them. A detailed exposition addition = parallel connection) but is even an algebra A, since se- is found in [4, 5]. quential connection of filters, i.e., multiplication of filters, is per- missible. The set of signals is also a vector space in linear SP. Fur- ther, filters operate on this space, i.e., filtering a signal produces a 4. 1-D TIME MODELS signal, making the signal space an A-module M, where A is the chosen set of filters. Infinite time. In (4), we presented the signal model adopted for Signal model. Signals do not arise as elements of modules, infinite discrete time. The property of modeling time is best ex- ℓ but as sequences s of numbers. We treat the discrete case, where plained by the operation of the shift x on the basis (x )ℓ∈Z of the ℓ ℓ+1 signals arise as infinite or finite sequences over an index domain I. signal module M: x · x = x . The operation is graphically Examples include infinite signals s ∈ ℓ2(Z) ≤ CZ, or finite sig- displayed in Fig. 1. We call the graph a visualization of the signal nals in s ∈ Cn. In a natural way, one can impose a vector space model (4). The graph is directed, an inherent property of time. structure on these sequences; in contrast, there may be many pos- Finite time. For finite time, the model usually adopted is sible choices of filter space A and filtering operation. To assign C n such a choice, we introduce the concept of signal model.2 A = [x]/(x − 1), M = C[x]/(xn − 1), (5) Definition 1 (Signal model) Let V ≤ CI be a vector space. We Cn s ℓ call the triple (A, M, Φ) a signal model for V , if A is an algebra, Φ = →M, 7→ s = s(x) = P0≤ℓ

1More specifically an A-module homomorphism. 3We set x = z−1 for notational convenience later when considering 2The definition can be stated without explicitly giving the basis, but it more general models. is more intuitive for the purpose of this paper in this form. 4Convolving two ℓ2 sequences does in general not yield an ℓ2 sequence. • x / • / • • / • / • • y % • o / • o / • o / • 0 1 2 n−3 n−2 n−1 9 x x x x x x T0 T1 T2 T3 T4 Fig. 2. Visualization of the finite discrete time model (5). • o / • o / • o / • o / • U0 U1 U2 U3 U4 Table 1. Four types of Chebyshev polynomials. The range for the $ • o / • o / • o / • o / • zeros is 0 ≤ k

Fig. 3. The space shift x · pℓ. which yields again a series in Cℓ, or a signal. The four models are visualized in Fig. 4. The graphs are undi- Infinite case. To construct a space model, we choose the sym- rected (since these are space models) and the left boundary con- metric shift operator x shown in Fig. 3. We now consider the poly- dition (third column in Table 1) is expressed through redirecting the arrow that would go to , based on the boundary condition nomials pℓ. Fig. 3 implies xpℓ = (pℓ−1 + pℓ+1)/2 or C−1 (second column in Table 1). pℓ+1 = 2xpℓ − pℓ−1, (6) Finite space. We introduce the finite space models using an example. We start with the infinite V -transform (7), which im- which is exactly the recurrence for the Chebyshev polynomials [7]. plies a left boundary condition of V−1 = V0. We cut the ba- The exact form is determined by the choice of p0 and p1 as initial sis to length n: b = (V0,...,Vn−1) and introduce (choose) the polynomials of degree 0 and 1. Then pℓ is a polynomial of de- mirrored boundary condition on the right side: V = V − or gree ℓ. Running (6) in the other direction yields also polynomials n n 1 V − V − = 2(x − 1)U − = 0 (the 2 can be omitted). This for the negative indices. Thus, only the polynomials with positive n n 1 n 1 yields the signal model (for V = Cn) subscripts are linearly independent: The model is for rightsided se- quences and has a left boundary. The left side (negative subscript) A = C[x]/(x − 1)Un−1, is the signal extension. We normalize p0 = 1 and choose p1 such that the resulting signal extension is simple, i.e., has a symmetry. M = C[x]/(x − 1)Un−1, (8) This happens exactly in the four cases shown in Table 1 (see the Cn s Φ = →M, 7→ s = s(x) = Pℓ∈N sℓVℓ. symmetry column).6 In summary, we get four infinite space models for rightsided The model is visualized in Fig. 5. sequences, which we collectively state as follows. To ensure con- The Fourier transform (2) for this model is given by vergence in all cases, we require V = ℓ1(N). cos k(ℓ+1/2)π/n Pb,α = , 1 N  cos kπ/(2n) 0≤k,ℓ

5This differs from current practice in which the distinction between All possible choices of the left boundary conditions in Fig. 7 time and space is one of 1-D versus 2-D. with their mirrored versions for the right side yield a total of 16 6See [5] for a more rigorous explanation of “simple” signal extension. finite space models, corresponding to the 16 types of DCTs and •O Table 2. Signal models associated with the 16 DCTs and DSTs. • C Cn − Cn−2 Cn Cn − Cn−1 Cn + Cn−1 O •o • / • T DCT-1 DCT-3 DCT-5 DCT-7 2 (x − 1)Un−2 Tn (x − 1)Wn−1 (x + 1)Vn−1 • / •  U DST-3 DST-1 DST-7 DST-5 • Tn Un Vn Wn (a) 2-D time (b) 2-D space V DCT-6 DCT-8 DCT-2 DCT-4

(x − 1)Wn−1 Vn (x − 1)Un−1 2Tn Fig. 6. Shifts for the separable 2-D time (directed) and 2-D space W DST-8 DST-6 DST-4 DST-2 (undirected) models. Both yield rectangular lattices. • • (x + 1)Vn−1 Wn 2Tn (x + 1)Un−1 Y33 E ••_? ? 33 ??  3 ??  3 ?  •o • / • • o •? / •  ?? DSTs. An overview is provided in Table 2. For a given DCT or  ??  ? DST, let p(x) be the polynomial below it, and let C ∈ {T,U,V,W } Õ  •• be the type of Chebyshev polynomial given in the first column. • • Then the associated model is given by A = M = C[x]/p(x), and (a) 2-D space (hexagonal lattice) (b) 2-D space (quincunx lattice) s , which we call a finite -transform. The Φ : 7→ P0≤ℓalgorithms can be derived easily by manipu- bijective mapping (a generalization of the z-transform). Instanti- lating the polynomial algebra rather than the transform matrix [8]. ations of the signal model give rise to infinite and finite discrete time and discrete space linear signal processing models with the corresponding linear transforms and other basic concepts. 6. 2-D TIME AND SPACE MODELS 8. REFERENCES The introduced algebraic approach to 1-D SP is readily extended to higher-dimensional SP, and can be used to derive new SP schemes [1] K. R. Rao and P. Yip, Discrete Cosine Transform: Algo- for nonseparable SP. We briefly discuss the 2-D case. rithms, Advantages, Applications, Academic Press, 1990. Separable 2-D signal models. Usually, 2-D SP is done in a [2] J. M. F. Moura and M. G. S. Bruno, “DCT/DST and Gauss- separable way, which assumes the signal resides on a rectangular Markov fields: Conditions for equivalence,” IEEE Trans. on lattice with two shift operators operating orthogonally. In the time Signal Processing, vol. 46, no. 9, pp. 2571–2574, 1998. case (directed, Fig. 6(a)), the signal model consists of Laurent se- [3] D. E. Dudgeon and R. M. Mersereau, Multidimensional Dig- ries in two variables (infinite case) or of the polynomial algebra n n k ℓ ital Signal Processing, Prentice-Hall, 1984. C[x, y]/hx − 1, y − 1i (finite case) with basis (x y )k,ℓ. In the space case (undirected, Fig. 6(b)), the signal model consists [4] M.Puschel¨ and J. M. F. Moura, “Algebraic theory of signal analogously of series (infinite case) or polynomials (finite case) processing: Foundation and 1-D time,” submitted for publi- cation. in Ck(x)Cℓ(y). In the finite case, the polynomials are elements of polynomial algebras in two variables. For example, applying [5] M.Puschel¨ and J. M. F. Moura, “Algebraic theory of signal a 2-D DCT-2 to an n × n image (as done in JPEG image com- processing: 1-D space,” submitted for publication. pression), assumes the space model A = M = C[x, y]/h(x − [6] H. J. Nussbaumer, Fast Fourier Transformation and Convo- , s . 1)Un−1(x), (y − 1)Un−1(y)i Φ : 7→ P0≤k,ℓalgorithm. See [9, 10, 11] for more details. [9] M. Puschel¨ and M. Rotteler,¨ “The Discrete Triangle Trans- form,” in Proc. ICASSP, 2004. [10] M. Puschel¨ and M. Rotteler,¨ “Cooley-Tukey FFT like fast 7. CONCLUSION algorithms for the discrete triangle transform,” in Proc. 11th IEEE DSP Workshop, 2004. The paper presents an algebraic theory of signal processing in [11] M. Puschel¨ and M. Rotteler,¨ “Fourier transform for the spa- which the basic building block is the signal model: a triple of a tial quincunx lattice,” in Proc. ICIP, 2005. space of filters (an algebra), a space of signals (a module), and a