TIB parametrization of signal processing

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Publication MaxEnt 2014
OAI : oai:www.see.asso.fr:9603:11321


TIB parametrization of signal processing


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        <identifier identifierType="DOI">10.23723/9603/11321</identifier><creators><creator><creatorName>Xiao Yu</creatorName></creator></creators><titles>
            <title>TIB parametrization of signal processing</title></titles>
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	    <date dateType="Created">Sat 30 Aug 2014</date>
	    <date dateType="Updated">Mon 2 Oct 2017</date>
            <date dateType="Submitted">Thu 22 Mar 2018</date>
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            <description descriptionType="Abstract"></description>

TIB parametrization of signal processing Xiao Yu 15th May 2014 We describe a generalization of the TIB representation of a time invariant linear system from scalar (SISO) to vector (MIMO) case. Several advantageous features of the scalar case, such as numerical stability, fast state update and ef- …cient dimensionality reduction, are also extended to the matrix case. Consider a d-dimensional innovations model, z (t + 1) = Az (t) + Bx (t) ; (1) y (t) = Cz (t) + x (t) ; (2) with y (t) being a sequence of d-dimensional measurement vectors and z (t) being n-dimensional state vectors. By appropriate choice of state-space coordinates we choose A to be a lower triangular, and input balanced: AA +BB = I: As a special case of orthogonal …lter, input balance ensures good numerical perform- ance, and the triangular realization has high precision location of eigenvalues and a band matrix fraction representation which a¤ords fast algorithms and a very e¢ cient model reduction algorithm for both SISO and MIMO cases is also known for the TIB parametrization. According to Douglas-Shaprio-Shields factorization, any strictly proper ra- tional stable function can be factored into an unstable part and rational lossless part, where for the TIB realization the unstable part is related to C and the lossless part is associated with A and B. Once the lossless part has been de- termined, the transfer functions have convenient geometry. The Schur algorithm constructs scalar rational lossless functions, the extension to the MIMO case is the Schur tangential algorithm, Bi (1=wi) ui = vi; (3) where Bi is a sequence of rational lossless functions with degree/McMillan degree i. When vi = 0, then wi are the poles of the system, and Bi are the Blaschke- Potapov factorization in MIMO case (and Blaschke factors in the SISO case). The TIB pair (A; B) is uniquely determined by the poles of the system for SISO, whereas for MIMO, the null vectors ui are also required to determine the lossless part. The reduction algorithm we propose is an hybrid one, which obtains the lossless part by minimizing the Hankel (H1 ) norm and the unstable part by minimizing H2 norm. The lossless part (A; B) can be determined from a partial 1 SVD of the Hankel matrix given by Hij = fi+j 1; i; j = 0; 1; : : : (4) where ffig are the impulse responses. C can be therefore obtained by a well conditioned least squares regression. In the SISO case, the choice of model parameters as the power series coe¢ - cients of the logarithm of the transfer function results in the Fisher information matrix is an identity matrix, providing a Euclidean statistical (Hilbert) mani- fold. Denoting log f (z) = a0 + a1z + a2z2 + ; since f and log f has the same singularity, we can apply our reduction algorithm on the alternative Hankel matrix Aij = ai+j 1; i; j = 0; 1; : : : (5) and then recover the parameters after exponentiation: Mullhaupt and Choi proved in the SISO case, the information geometry is determined by the un- stable part of the transfer function, which our algorithm can extend to the MIMO case. References [1] A.P.Mullhaupt, K.S.Riedel, Fast Identi…cation of Stable Innovation Filtersg [2] A.P.Mullhaupt, K.S.Riedel, Band Matrix Representation of Triangular Input Balanced Form [3] A.P.Mullhaupt, K.S.Riedel, Low Grade matrices and matrix fraction repres- entations [4] A.P.Mullhaupt, K.S.Riedel, Exponential Condition Number of Solutions Dis- crete Lyapunov Equation [5] B.Hanzon, M.Olivi, R.M.Peeters, Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm [6] R.G.Douglas, H.S.Shapiro, A.L.Shields, Cyclic vectors and invariant sub- spaces for the backward shift operator [7] J.Marmorat, M.Olivi, B.Hanzon, R.M.Peeters, Matrix rational H2 approx- imation: a state-space approach using Schur parameters 2