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A natural transfer function space for linear discrete time-invariant and scale-invariant systems

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A natural transfer function space for linear discrete time-invariant and scale-invariant systems
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  A natural transfer function space for linear discretetime-invariant and scale-invariant systems Daniel Alpay, Mamadou Mboup To cite this version: Daniel Alpay, Mamadou Mboup. A natural transfer function space for linear dis-crete time-invariant and scale-invariant systems. International Workshop on Mul-tidimensional (nD) Systems, 2009- NDS 2009 (Invited session), Jun 2009, Thessa-loniki, Greece. pp.1-4, 2009,  < http://ieeexplore.ieee.org/servlet/opac?punumber=5174547 > . < 10.1109/NDS.2009.5196173 > .  < inria-00428994 > HAL Id: inria-00428994https://hal.inria.fr/inria-00428994 Submitted on 4 Nov 2009 HAL  is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire  HAL , estdestin´ee au d´epˆot et `a la diffusion de documentsscientifiques de niveau recherche, publi´es ou non,´emanant des ´etablissements d’enseignement et derecherche fran¸cais ou ´etrangers, des laboratoirespublics ou priv´es.  A natural transfer function space for linear discrete time-invariant andscale-invariant systems Daniel Alpay*Department of mathematicsBen Gurion University of the Negev, Israel dany@cs.bgu.ac.il Mamadou Mboup**UFR math´ematiques et informatique - CRIP5Universit´e Paris Descartes45, rue des Saints-P`eres - 75270 Paris cedex 06 Mamadou.Mboup@mi.parisdescartes.fr  Abstract —In a previous work, we have defined the scaleshift for a discrete-time signal and introduced a family of linear scale-invariant systems 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 studyan interpolation problem in these spaces, as a first step towardsa finite dimensional implementation of a scale invariant system.Our approach uses a characterization of character-automorphicHardy spaces of order 2 in terms of classical de BrangesRovnyak spaces. I. INTRODUCTIONThe self-similarity property is widely studied in the lit-erature 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 aweighted form of stationarity in scale while in the systemstheory approach, it is interpreted as a scale invariance.The scale shift, defined for a signal  x ( t )  by the operator α  →  x ( αt )  thus plays a central rˆole in the definition of self-similarity. Though simple and straightforward in thecontinuous-time domain, this operator is not well definedfor discrete-time signal. In this paper, we use the definitiongiven in [7]. Therein, a family of linear discrete both time-and scale-invariant systems is introduced in connection withcharacter-automorphic Hardy spaces. In this paper, we provea Beurling-Lax theorem for such Hardy spaces of order 2.We also study an interpolation problem in these spaces, asa first step towards a finite dimensional implementation of ascale invariant system. Our approach uses a characterizationof character-automorphic Hardy spaces of order 2 in termsof classical de Branges Rovnyak spaces [8].  A. Scale shift for discrete-time signals Let  f   ∈ L 1 ( R + )  (that is, a continuous time signal), withLaplace 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 inthe time and frequency domains. This remark is the startingpoint to define the scaling operator for discrete-time signals.Let  θ  be given such that  | θ |  <  π 2 . Consider the M¨obiustransformation G θ ( s ) =  e iθ − se − iθ + s, *Earl Katz Chair in Algebraic System Theory**EPI ALIEN, INRIA which maps conformally the open right half-plane  C +  ontothe open unit disk  D =  { z  ∈ C :  | z |  <  1 } . Then, the scale shift S  α  :  s  →  S  α ( s ) =  αs, α >  0 translates in the unit disc, into the hyperbolic transformation[7] γ  { α }  =  G θ  ◦ S  α ◦ G − 1 θ  .  (1)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 ) =  γ  1 z  + γ  2 γ  2 z  + γ  1 ,  (2)there exist  α γ   >  0 ,  θ γ   with  | θ γ  |  <  π 2  and  ξ  γ   such that e iξ γ γ  ( z ) = ( G θ γ ◦ S  α γ ◦ G − 1 θ γ )( e iξ γ z ) .  (3)In particular,  α γ   is given by the multiplier of the transfor-mation Proof:  We assume that  γ   is normalized such that | γ  1 | 2 −| γ  2 | 2 = 1 . Since  γ   is hyperbolic, we have the relation [9] γ  ( z ) − ξ  1 γ  ( z ) − ξ  2 =  K z − ξ  1 z − ξ  2 (4)where  ξ  1  = √  [ ℜ ( γ  1 )] 2 − 1+ i ℑ ( γ  1 ) γ  2 =  λ γ γ  2 and  ξ  2  =  − λ γ γ  2 are thetwo fixed points of   γ  . The positive constant  K   is called themultiplier of the transformation [9] and is given by  K  +  1 K   =4[ ℜ ( γ  1 )] 2 − 2 . Noting that | ξ  1 |  =  | ξ  2 |  = 1 , one may rearrange(4) to obtain λ γ   − λ γ  e iξ γ γ  ( z )1 + e iξ γ γ  ( z ) =  K λ γ   − λ γ  e iξ γ z 1 + e iξ γ z ,  (5)where  e iξ γ =  λ γ γ  2 . Dividing both sides of this equality by | λ γ  | , we get (3) by setting  e iθ γ =  λ γ | λ γ |  and  α γ   =  K  .The set of all linear transformations as in (2) forms agroup that we denote by  Γ . The lemma then shows thatthe action of   Γ  on  D  is equivalent to the scale operator on C + . Therefore, we define below the discrete-time frequencydomain scale shift by the action of the (hyperbolic) group of   automorphisms of   D . Given a discrete-time signal  { x n } n ≥ 0 ,consider its  Z   transform  X  ( z ) =   ∞ n =0 x n z n , which weassume convergent in a neighborhood of the origin. Thescale  α  =  α γ   shift of the sequence  { x n } n ≥ 0  is the sequence { x n ( γ  ) } n ≥ 0  defined via the equation X  γ  ( z ) = 1 γ  2 z  + γ  1 X   ( γ  ( z )) =  n ≥ 0 x n ( γ  ) z n .  (6)It is useful to note that this operator also makes sense forvector-valued functions.  B. Scale-invariant systems A wide class of causal discrete time-invariant linear sys-tems can be given in terms of convolution in the form y n  = n  m =0 h n − m x m , n  = 0 , 1 ,...,  (7)where  { h n }  is the impulse response and where the inputsequence  { x m }  and output sequence  { y m }  are requestedto belong to some pre-assigned sequences spaces. The  Z  transform of the sequence { h n } , that is  H  ( z ) =  ∞ n =0 z n h n ,is called the  transfer function  of the system, and there aredeep relationships between properties of   H  ( z )  and of thesystem; see [10] for a survey. In particular, it is well knownthat if the system is asymptotically stable, then  H  ( z )  belongsto the classical Hardy space of order 2.In this paper, we are interested in linear discrete-timesystems which, in addition to the time invariance, are alsoinvariant under a scale shift. Scale-invariance is defined[7] similarly to time-invariance: a scale shift in the inputsequence induces the same scale shift on the correspondingoutput sequence. In terms of the  Z   transforms, this woulddirectly mean that for all  γ   ∈  Γ , Y  γ  ( z ) =  H  ( γ  ( z )) X  γ  ( z ) =  H  ( z ) X  γ  ( z ) .  (8)Therefore, scale-invariance implies that the transfer functionof the system be  Γ -periodic. Now a function  f   satisfying f   ◦  γ   =  f  , for all  γ   ∈  Γ  is said to be  automorphic  withrespect to  Γ . This makes sense only for discrete groups (see[9]).In the following, we will be interested in the character-automorphic Hardy spaces of order 2. These are the naturaltransfer function spaces for the LTI and scale-invariantsystems. In a previous study, see [8], we have given a charac-terization of these spaces in terms of associated classical deBranges Rovnyak spaces. In section II, we use this approachto prove a Beurling-Lax theorem and we study interpolationin these Hardy spaces in section III. Leech’s theorem andthe characterization of de Branges Rovnyak spaces given in[11, Theorem 3.1.2, p. 85] play an important role in thearguments.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  isa Fuchsian group of Widom type (with no elliptic element)[12]. We denote by  z  its uniformizing map and by   Γ  its dualgroup,  i.e.  the group of unimodular characters. Recall that acharacter  α  is a function defined on  Γ  and satisfying: | α ( γ  ) |  = 1  and  α ( γ   ◦ ϕ ) =  α ( γ  ) α ( ϕ ) ,  ∀ γ,ϕ  ∈  Γ . A function  f   satisfying  f   ◦ γ   =  α ( γ  ) f,  ∀ γ   ∈  Γ  for  α  ∈   Γ ,is called  character-automorphic  with respect to  Γ . Given acharacter  α  of    Γ , the character-automorphic Hardy space H α 2 ( D )  of order  2  is the space of character-automorphicfunctions which belong to the classical Hardy space  H 2 ( D ) .Its reproducing kernel has been characterized in [13, Lemma4.4.2 p. 387] as follows: k α ( z,ω ) =  c ( α ) k αµ ( z, 0) b ( z )  k α ( ω, 0) ∗ −  k αµ ( ω, 0) b ( ω )  ∗ k α ( z, 0)  z ( z ) −  z ( ω ) ∗ (9)where  c ( α ) =  z (0) b (0) k αµ (0 , 0)  >  0 . In (9),  b  is the Green’s functionof   Γ , and the character associated to the Green’s functionis denoted by  µ . Formula (9) expresses that the kernel is structured  , and of the form of the kernels studied in thepapers [14], [15]. It depends on a  C 1 × 2 -valued function of one variable. Using (9) we proved in [8] that there existsa Schur function  S   α , associated to the de Branges space H ( S   α ) , such that H α 2 ( D ) =  F  ( z ) =  A α ( z )1 − i  z ( z ) f  ( σ ( z )) ; f   ∈ H ( S   α )  (10)where  A α ( z ) =   c ( α )  k αµ ( z, 0) b ( z )  + ik α ( z, 0)  and  σ ( z ) =  1+ i z ( z )1 − i z ( z ) , and with the norm  F   H α 2  ( D )  =   f   H ( S  α ) . We close this subsection with the  Definition  2.1: A causal linear time-invariant system iscalled scale-invariant with respect to  Γ , if its transfer functionis an element of   H α 2 ( D )  for some character  α .Note that such a system is not rational, unless  Γ  is a finitegroup.  B. The shift operator in  H α 2 ( D ) Set  p  ∈ D  and given a function  f   analytic in  D , considerthe operators ( R p f  )( λ ) =  f  ( λ ) − f  ( p ) λ − p . These operators  R p  satisfy the resolvent equation R p − R q  = ( p − q ) R p R q . Let  m ( z ) =  A α ( z )1 − i z ( z ) . The isomorphism F  ( z ) =  m ( z ) f  ( σ ( z ))  (11)  between the de Branges space  H ( S  α )  and the character-automorphic Hardy space allows one to define the followingoperators: R p F   =  m ( z )( R p f  )( σ ( z )) . As we may directly check, these operators  R p  also satisfya resolvent equation. Therefore, they can be written in theform R p  = ( T − p ) − 1 . Here  T  is not an operator in general. It is a linear relationand is given by ( T F  )( z ) =  σ ( z ) F  ( z ) + m ( z ) c F  , where  c F   is such that σ ( z ) F  ( z ) + m ( z ) c F   ∈ H α 2 ( D ) . C. A Beurling theorem in  H α 2 ( D ) The classical Beurling theorem (see for instance [16,Th´eor`eme 17.21 p. 330]) gives a characterization of theclosed subspaces  M  of the Hardy space  H 2 ( D )  of the unitdisk   D :  Any such subspace is of the form  M  =  j H 2 ( D )  ,where  j  is an inner function  (the case of vector-valued func-tions was first considered by Lax; see [17] for a discussionand references). The orthogonal complement of   M  is thereproducing kernel Hilbert space with reproducing kernel K  j ( z,w ) = (1  −  j ( z )  j ( w ) ∗ ) / (1  −  zw ∗ ) . If one replaces  j  by a Schur function  s , that is, by a function analyticand contractive in  D , the kernel  k s ( z,w )  is still positivein  D . Its associated reproducing kernel Hilbert space wasdenoted in the preceding subsection by  H ( S   ) . These spacesare called de Branges Rovnyak spaces, and srcinate withthe work [18]. When allowing  S    to be vector-valued, theyhave been fully characterized in [11, Theorem 3.1.2]. Theyare contractively included, but in general not isometricallyincluded, in  H 2 ( D ) . Theorem 2.2:  A Hilbert space  M  is contractively in-cluded in  H α 2 ( D ) , and invariant under  R p , and satisfies theinequality R 0 F   2 M  ≤  F   2 M −| F  (0) | 2 (12)if, and only if, its reproducing kernel is of the form A α ( z ) √  21 −  R  ( σ ( z ))  R  ( σ ( w )) ∗ − i (  z ( z ) −  z ( w ) ∗ ) A α ( w ) ∗ √  2 , where  R   is a vector-valued Schur function such that S   α  =  RR  1 ,  (13)with  R  1  also a vector-valued Schur function.  Remark 1:  The inequality(12) is automatically satisfied if  M  is isometrically included in  H α 2 ( D ) . Proof:  of 2.2 Associated to the space M , there is, by theisomorphism (11), a Hilbert space M α  which is  R p -invariantand satisfying  R 0 f   2 M α ≤  f   2 M α −| f  (0) | 2 .  (14)Using [11, Theorem 3.1.2], we see that the reproducingkernel of   M α  is of the form 1 −  R  ( λ )  R  ( p ) ∗ 1 − λ p ∗  , where  R   is a vector-valued Schur function. Since  M α  iscontractively included in  H ( S   α ) , the kernel  R  ( λ )  R  ( p ) ∗ − S   α ( λ ) S   α ( p ) ∗ 1 − λ p ∗  , is positive. We therefore get the factorization (13) by usingthe 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 reproducingkernel of the space  H α 2 ( D )  is k α ( z,w ) =  A α ( z )1 − i  z ( z )1 − S   α ( σ ( z )) S   α ( σ ( w )) ∗ 1 − σ ( z ) σ ( w ) ∗ A α ( w ) ∗ 1 + i  z ( w ) ∗ =  A α ( z ) √  21 − S   α ( σ ( z )) S   α ( σ ( w )) ∗ − i (  z ( z ) −  z ( w ) ∗ ) A α ( w ) ∗ √  2 . Reversing these different arguments allows one to establishthe converse.III. I NTERPOLATION As we already mention, the elements of the character-automorphic Hardy space  H α 2 ( D )  are not rational functionsunless we consider a finite number of possible scale shifts.Since the corresponding systems are of infinite dimension,a finite dimensional approximation step is necessary beforetheir implementation. This is the motivation of the interpo-lation problem studied in this section.To proceed, denote by F   =  { z  ∈ D  :  | γ  ′ ( z ) |  <  1  for all  γ   ∈  Γ ,γ    =  id }  (15)the  normal fundamental domain  of   Γ  with respect to  0 : thereis no transformation in  Γ , which sends one point of   F   intoanother point of   F  .So, we consider the following interpolation problem: Given  N   complex numbers  F  i  and   N   points  z i  ∈ F ∩ D  , describe the set of all functions  F   ∈ H α 2 ( D )  with  F   H α 2  ( D )  ≤  1  satisfying F  ( z i ) =  F  i , i  = 1 ,...N,  (16)Note that this problem is different from the interpolationproblems considered by Abrahamse in [21] and by Kupinand Yuditskii in [13]. These studies were interested in findingmultipliers having given values at prescibed points while herewe have the constraint that  F   must belong to  H α 2 ( D ) .The function P  α  = (1 − S   α )(1 + S   α )  (17)is analytic and has positive real part in  D . By the Herglotzrepresentation theorem, we can write: P  α  =  ic α  +    2 π 0 e it + λe it − λdσ α ( t ) ,  with  c α  ∈ R  and  dσ α  is a postive measure  [0 , 2 π ) . Defining Q α ( λ )  △ = √  2(1 + S   α ( λ )) = 1 + P  α ( λ ) √  2 where the second equality follows directly from (17), we seethat can also write P  α ( λ ) + P  α ( p ) ∗ 1 − λ p ∗  = Q α ( λ )1 − S   α ( λ ) S   α ( p ) ∗ 1 − λ p ∗  Q α ( λ ) ∗ . (18)Now this equation (18) implies that the operator of mul-tiplication by  Q α ( λ )  is a unitary transformation from thereproducing kernel space  L ( P  α ) , with kernel P  α ( λ ) + P  α ( p ) ∗ 1 − λ p ∗  , λ, p  ∈ D to the space H ( S   α ) . Moreover L ( P  α )  is the set of functionsof the form x ( λ ) =    2 π 0 e it h ( t ) dσ α ( t ) e it − λ ,  (19)where  h  belongs to the closure (subsequently denoted H 2 ( D ,dσ α ) ) of the set of functions  1 / (1 − e it w ∗ )  ( | w |  <  1 )in  L 2 ( dσ α ) . We therefore have the following: Proposition 3.1:  F   ∈ H α 2 ( D )  if, and only if, F  ( z ) =  A α ( z )1 − i  z ( z )1 + S   α ( σ ( z )) √  2    2 π 0 e it h ( t ) dσ α ( t ) e it − σ ( z )  (20)with  h  ∈ H 2 ( dσ α ) .The interpolation problem in the character-automorphicHardy space then reduces to an interpolation problem in L ( P  α ) , or more precisely, to an orthogonal projection in H 2 ( D ,dσ α ) : Problem 1:  Let  λ ℓ  =  σ ( z ℓ )  and x ℓ  = 1 −  z ( z ℓ ) A α ( z ℓ ) √  2 F  ℓ 1 + S   α ( λ ℓ )) , ℓ  = 1 ,...N. Find all  h  ∈ H 2 ( D ,dσ α )  such that    2 π 0 e it h ( t ) dσ α ( t ) e it − λ ℓ =  x ℓ , ℓ  = 1 ,...N. Now this is a classical Hilbert space problem: it admits asolution with minimum norm, corresponding to a function h min  of the form h min ( t ) = N   ℓ =1 c ℓ e it − λ ℓ , and any other solution has the form h min  + h, h  ⊥  h min . 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 modelsfor 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. Wilson, “On the self-similar nature of Ethernet traffic (extended version),”  IEEE/ACM Trans. Networking , pp. 1–15, Feb. 1994.[3] B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions,fractional noises and applications,”  SIAM Review , vol. 10, no. 4, pp.422–437, 1968.[4] M. Mboup, “On the structure of self-similar systems: A Hilbert spaceapproach,” in  Operator Theory: Advances and applications , vol. OT-143, pp. 273–302. Birkh¨auser-Verlag, 2003.[5] B. Yazıcı and R. L. Kashyap, “A class of second-order stationaryself-similar processes for  1 /f   phenomena,”  IEEE Trans. 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