A natural transfer function space for linear discrete time-invariant and scale-invariant systems Daniel Alpay, Mamadou Mboup To cite this version: Daniel Alpay, Mamadou Mboup. A natural transfer function space for linear discrete time-invariant and scale-invariant systems. International Workshop on Multidimensional (nD) Systems, 2009- NDS 2009 (Invited session), Jun 2009, Thessaloniki, Greece. pp.1-4, 2009, <http://ieeexplore.ieee.org/servlet/opac?punumber=5174547>. <10.1109/NDS.2009.5196173>. <inria-00428994> HAL Id: inria-00428994 https://hal.inria.fr/inria-00428994 Submitted on 4 Nov 2009 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. A natural transfer function space for linear discrete time-invariant and scale-invariant systems Daniel Alpay* Department of mathematics Ben Gurion University of the Negev, Israel dany@cs.bgu.ac.il Abstract— In a previous work, we have defined the scale shift for a discrete-time signal and introduced a family of linear scale-invariant systems in connection with characterautomorphic Hardy spaces. In this paper, we prove a BeurlingLax theorem for such Hardy spaces of order 2. We also study an interpolation problem in these spaces, as a first step towards a finite dimensional implementation of a scale invariant system. Our approach uses a characterization of character-automorphic Hardy spaces of order 2 in terms of classical de Branges Rovnyak spaces. I. INTRODUCTION The self-similarity property is widely studied in the literature in the framework of stochastic process theory [1], [2], [3], and in the framework of systems theory [4], [5], [6]. In stochastic process theory the property is seen as a weighted form of stationarity in scale while in the systems theory approach, it is interpreted as a scale invariance. The scale shift, defined for a signal x(t) by the operator α 7→ x(αt) thus plays a central rôle in the definition of self-similarity. Though simple and straightforward in the continuous-time domain, this operator is not well defined for discrete-time signal. In this paper, we use the definition given in [7]. Therein, a family of linear discrete both timeand scale-invariant systems is introduced in connection with character-automorphic Hardy spaces. In this paper, we prove a Beurling-Lax theorem for such Hardy spaces of order 2. We also study an interpolation problem in these spaces, as a first step towards a finite dimensional implementation of a scale invariant system. Our approach uses a characterization of character-automorphic Hardy spaces of order 2 in terms of classical de Branges Rovnyak spaces [8]. A. Scale shift for discrete-time signals Let f ∈ L1 (R+ ) (that is, a continuous time signal), with Laplace transform F (s), ℜ(s) > 0. As it is well known, for √ every α = 1/β > 0, the Laplace transform of f (βt) is αF (αs). Therefore, time scaling has the same form both in the time and frequency domains. This remark is the starting point to define the scaling operator for discrete-time signals. Let θ be given such that |θ| < π2 . Consider the Möbius transformation eiθ − s , Gθ (s) = −iθ e +s *Earl Katz Chair in Algebraic System Theory **EPI ALIEN, INRIA Mamadou Mboup** UFR mathématiques et informatique - CRIP5 Université Paris Descartes 45, rue des Saints-Pères - 75270 Paris cedex 06 Mamadou.Mboup@mi.parisdescartes.fr which maps conformally the open right half-plane C+ onto the open unit disk D = {z ∈ C : |z| < 1}. Then, the scale shift Sα : s 7→ Sα (s) = αs, α>0 translates in the unit disc, into the hyperbolic transformation [7] (1) γ{α} = Gθ ◦ Sα ◦ G−1 θ . Any transformation of this form maps the unit open disc (resp. the unit circle) into itself. Conversely, the following lemma is true. Lemma 1.1: For each hyperbolic transformation γ(z) = γ1 z + γ2 , γ2z + γ1 there exist αγ > 0, θγ with |θγ | < π 2 (2) and ξγ such that iξγ z). eiξγ γ(z) = (Gθγ ◦ Sαγ ◦ G−1 θγ )(e (3) In particular, αγ is given by the multiplier of the transformation Proof: We assume that γ is normalized such that |γ1 |2 − 2 |γ2 | = 1. Since γ is hyperbolic, we have the relation [9] γ(z) − ξ1 z − ξ1 =K γ(z) − ξ2 z − ξ2 (4) λγ − λγ eiξγ γ(z) λγ − λγ eiξγ z =K , iξ γ 1 + e γ(z) 1 + eiξγ z (5) √ [ℜ(γ1 )]2 −1+iℑ(γ1 ) λ λ where ξ1 = = γγ2 and ξ2 = − γγ2 are the γ2 two fixed points of γ. The positive constant K is called the 1 multiplier of the transformation [9] and is given by K + K = 2 4[ℜ(γ1 )] −2. Noting that |ξ1 | = |ξ2 | = 1, one may rearrange (4) to obtain λ where eiξγ = γγ2 . Dividing both sides of this equality by λ |λγ |, we get (3) by setting eiθγ = |λγγ | and αγ = K. The set of all linear transformations as in (2) forms a group that we denote by Γ. The lemma then shows that the action of Γ on D is equivalent to the scale operator on C+ . Therefore, we define below the discrete-time frequency domain scale shift by the action of the (hyperbolic) group of automorphisms of D. Given a discrete-time {xn }n≥0 , P∞ signal n consider its Z transform X(z) = x z , which we n n=0 assume convergent in a neighborhood of the origin. The scale α = αγ shift of the sequence {xn }n≥0 is the sequence {xn (γ)}n≥0 defined via the equation Xγ (z) = X 1 X (γ(z)) = xn (γ)z n . γ2z + γ1 (6) n≥0 It is useful to note that this operator also makes sense for vector-valued functions. B. Scale-invariant systems A wide class of causal discrete time-invariant linear systems can be given in terms of convolution in the form yn = n X hn−m xm , n = 0, 1, . . . , (7) m=0 where {hn } is the impulse response and where the input sequence {xm } and output sequence {ym } are requested to belong to some pre-assigned sequences spaces. P∞ The Z transform of the sequence {hn }, that is H(z) = n=0 z n hn , is called the transfer function of the system, and there are deep relationships between properties of H(z) and of the system; see [10] for a survey. In particular, it is well known that if the system is asymptotically stable, then H(z) belongs to the classical Hardy space of order 2. In this paper, we are interested in linear discrete-time systems which, in addition to the time invariance, are also invariant under a scale shift. Scale-invariance is defined [7] similarly to time-invariance: a scale shift in the input sequence induces the same scale shift on the corresponding output sequence. In terms of the Z transforms, this would directly mean that for all γ ∈ Γ, Yγ (z) = H(γ(z))Xγ (z) = H(z)Xγ (z). (8) Therefore, scale-invariance implies that the transfer function of the system be Γ-periodic. Now a function f satisfying f ◦ γ = f , for all γ ∈ Γ is said to be automorphic with respect to Γ. This makes sense only for discrete groups (see [9]). In the following, we will be interested in the characterautomorphic Hardy spaces of order 2. These are the natural transfer function spaces for the LTI and scale-invariant systems. In a previous study, see [8], we have given a characterization of these spaces in terms of associated classical de Branges Rovnyak spaces. In section II, we use this approach to prove a Beurling-Lax theorem and we study interpolation in these Hardy spaces in section III. Leech’s theorem and the characterization of de Branges Rovnyak spaces given in [11, Theorem 3.1.2, p. 85] play an important role in the arguments. In the remaining of the paper, the complex conjugation of is denoted par ∗ and no longer by . II. A NATURAL TRANSFER FUNCTION SPACE A. Definitions Now on, we discretise the scale axis and consider that Γ is a Fuchsian group of Widom type (with no elliptic element) b its dual [12]. We denote by z its uniformizing map and by Γ group, i.e. the group of unimodular characters. Recall that a character α is a function defined on Γ and satisfying: |α(γ)| = 1 and α(γ ◦ ϕ) = α(γ)α(ϕ), ∀γ, ϕ ∈ Γ. b A function f satisfying f ◦ γ = α(γ)f, ∀γ ∈ Γ for α ∈ Γ, is called character-automorphic with respect to Γ. Given a b the character-automorphic Hardy space character α of Γ, H2α (D) of order 2 is the space of character-automorphic functions which belong to the classical Hardy space H2 (D). Its reproducing kernel has been characterized in [13, Lemma 4.4.2 p. 387] as follows: ∗  αµ k (ω,0) kαµ (z,0) α ∗ k α (z, 0) k (ω, 0) − b(z) b(ω) k α (z, ω) = c(α) z(z) − z(ω)∗ (9) where c(α) = kz(0)b(0) αµ (0,0) > 0. In (9), b is the Green’s function of Γ, and the character associated to the Green’s function is denoted by µ. Formula (9) expresses that the kernel is structured, and of the form of the kernels studied in the papers [14], [15]. It depends on a C1×2 -valued function of one variable. Using (9) we proved in [8] that there exists a Schur function Sα , associated to the de Branges space H(Sα ), such that   Aα (z) α f (σ(z)) ; f ∈ H(Sα ) H2 (D) = F (z) = 1 − iz(z) (10)   αµ p (z,0) + ik α (z, 0) where Aα (z) = c(α) k b(z) and σ(z) = 1+iz(z) 1−iz(z) , and with the norm kF kHα2 (D) = kf kH(Sα ) . We close this subsection with the Definition 2.1: A causal linear time-invariant system is called scale-invariant with respect to Γ, if its transfer function is an element of H2α (D) for some character α. Note that such a system is not rational, unless Γ is a finite group. B. The shift operator in H2α (D) Set p ∈ D and given a function f analytic in D, consider the operators (Rp f )(λ) = f (λ) − f (p) . λ−p These operators Rp satisfy the resolvent equation Rp − Rq = (p − q)Rp Rq . Let m(z) = Aα (z) 1−iz(z) . The isomorphism F (z) = m(z)f (σ(z)) (11) between the de Branges space H(Sα ) and the characterautomorphic Hardy space allows one to define the following operators: Rp F = m(z)(Rp f )(σ(z)). Using [11, Theorem 3.1.2], we see that the reproducing kernel of Mα is of the form As we may directly check, these operators Rp also satisfy a resolvent equation. Therefore, they can be written in the form Rp = (T − p)−1 . where R is a vector-valued Schur function. Since Mα is contractively included in H(Sα ), the kernel Here T is not an operator in general. It is a linear relation and is given by is positive. We therefore get the factorization (13) by using the Leech theorem. See [19, Theorem 2, p. 134] and [20, Example 1, p. 107]. We conclude by using the isomorphism (11) and the fact that (see equation (10)) the reproducing kernel of the space H2α (D) is (TF )(z) = σ(z)F (z) + m(z)cF , where cF is such that σ(z)F (z) + m(z)cF ∈ H2α (D). The classical Beurling theorem (see for instance [16, Théorème 17.21 p. 330]) gives a characterization of the closed subspaces M of the Hardy space H2 (D) of the unit disk D: Any such subspace is of the form M = jH2 (D), where j is an inner function (the case of vector-valued functions was first considered by Lax; see [17] for a discussion and references). The orthogonal complement of M is the reproducing kernel Hilbert space with reproducing kernel Kj (z, w) = (1 − j(z)j(w)∗ )/(1 − zw∗ ). If one replaces j by a Schur function s, that is, by a function analytic and contractive in D, the kernel ks (z, w) is still positive in D. Its associated reproducing kernel Hilbert space was denoted in the preceding subsection by H(S ). These spaces are called de Branges Rovnyak spaces, and originate with the work [18]. When allowing S to be vector-valued, they have been fully characterized in [11, Theorem 3.1.2]. They are contractively included, but in general not isometrically included, in H2 (D). Theorem 2.2: A Hilbert space M is contractively included in H2α (D), and invariant under Rp , and satisfies the inequality kR0 F k2M ≤ kF k2M − |F (0)|2 (12) if, and only if, its reproducing kernel is of the form Aα (z) 1 − Sα (σ(z))Sα (σ(w))∗ Aα (w)∗ 1 − iz(z) 1 − σ(z)σ(w)∗ 1 + iz(w)∗ ∗ Aα (z) 1 − Sα (σ(z))Sα (σ(w)) Aα (w)∗ √ . = √ −i(z(z) − z(w)∗ ) 2 2 Reversing these different arguments allows one to establish the converse. III. I NTERPOLATION As we already mention, the elements of the characterautomorphic Hardy space H2α (D) are not rational functions unless we consider a finite number of possible scale shifts. Since the corresponding systems are of infinite dimension, a finite dimensional approximation step is necessary before their implementation. This is the motivation of the interpolation problem studied in this section. To proceed, denote by F = {z ∈ D : |γ ′ (z)| < 1 for all γ ∈ Γ, γ 6= id} where R is a vector-valued Schur function such that (13) with R1 also a vector-valued Schur function. Remark 1: The inequality(12) is automatically satisfied if M is isometrically included in H2α (D). Proof: of 2.2 Associated to the space M, there is, by the isomorphism (11), a Hilbert space Mα which is Rp -invariant and satisfying (14) (15) the normal fundamental domain of Γ with respect to 0: there is no transformation in Γ, which sends one point of F into another point of F. So, we consider the following interpolation problem: Given N complex numbers Fi and N points zi ∈ F ∩ D, describe the set of all functions F ∈ H2α (D) with kF kHα2 (D) ≤ 1 satisfying F (zi ) = Fi , i = 1, . . . N, Aα (z) 1 − R(σ(z))R(σ(w))∗ Aα (w)∗ √ √ , −i(z(z) − z(w)∗ ) 2 2 kR0 f k2Mα ≤ kf k2Mα − |f (0)|2 . R(λ)R(p)∗ − Sα (λ)Sα (p)∗ , 1 − λp∗ k α (z, w) = C. A Beurling theorem in H2α (D) Sα = RR1 , 1 − R(λ)R(p)∗ , 1 − λp∗ (16) Note that this problem is different from the interpolation problems considered by Abrahamse in [21] and by Kupin and Yuditskii in [13]. These studies were interested in finding multipliers having given values at prescibed points while here we have the constraint that F must belong to H2α (D). The function (1 − Sα ) Pα = (17) (1 + Sα ) is analytic and has positive real part in D. By the Herglotz representation theorem, we can write: Z 2π it e +λ dσα (t), Pα = icα + eit − λ 0 with cα ∈ R and dσα is a postive measure [0, 2π). Defining √ 2 1 + Pα (λ) △ √ = Qα (λ) = (1 + Sα (λ)) 2 where the second equality follows directly from (17), we see that can also write Pα (λ) + Pα (p)∗ 1 − Sα (λ)Sα (p)∗ = Q (λ) Qα (λ)∗ . α 1 − λp∗ 1 − λp∗ (18) Now this equation (18) implies that the operator of multiplication by Qα (λ) is a unitary transformation from the reproducing kernel space L(Pα ), with kernel Pα (λ) + Pα (p)∗ , 1 − λp∗ λ, p ∈ D to the space H(Sα ). Moreover L(Pα ) is the set of functions of the form Z 2π it e h(t)dσα (t) , (19) x(λ) = eit − λ 0 where h belongs to the closure (subsequently denoted H2 (D, dσα )) of the set of functions 1/(1 − eit w∗ ) (|w| < 1) in L2 (dσα ). We therefore have the following: Proposition 3.1: F ∈ H2α (D) if, and only if, Z Aα (z) 1 + Sα (σ(z)) 2π eit h(t)dσα (t) √ F (z) = (20) 1 − iz(z) eit − σ(z) 2 0 with h ∈ H2 (dσα ). The interpolation problem in the character-automorphic Hardy space then reduces to an interpolation problem in L(Pα ), or more precisely, to an orthogonal projection in H2 (D, dσα ): Problem 1: Let λℓ = σ(zℓ ) and √ 2Fℓ 1 − z(zℓ ) , ℓ = 1, . . . N. xℓ = Aα (zℓ ) 1 + Sα (λℓ )) Find all h ∈ H2 (D, dσα ) such that Z 2π it e h(t)dσα (t) = xℓ , ℓ = 1, . . . N. eit − λℓ 0 Now this is a classical Hilbert space problem: it admits a solution with minimum norm, corresponding to a function hmin of the form hmin (t) = N X ℓ=1 cℓ , eit − λℓ and any other solution has the form hmin + h, h ⊥ hmin . We thus deduce the description of all x of the form (19), and hence a description of the functions F by formula (20). R EFERENCES [1] R. Narasimha S. Lee, W. Zhao and R.M. Rao, “Discrete-time models for statistically self-similar signals,” IEEE Trans. Signal Processing, vol. 51, no. 5, pp. 1221–1230, 2003. [2] W. Leland, M. Taqqu, W. Willinger, and D. 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