B. NINNESS, H. HJALMARSSON, AND F. GUSTAFSSON, Generalised Fourier and Toeplitz results for rational orthonormal bases, SIAM Journal on Control and Optimization, 37 (1999), pp. 429--460.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....T [f(#) f(#) 41) 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 Number of Receive Elements Scale Factor Fig. 2. Capacity of channel with increasing numbers of receivers and different values of k where we have used a well known identity for Toeplitz matrices [11]. For # = # we have equal scattering angles for both noise and signal and we may write: H T H # T # 1 = T [f(#) f(#) T [0] I and so (32) simplifies to: C = NR log 2 1 K w (43) for a constant K. This provides linear growth in terms of the number of receive elements. For the ....

B. Ninness, H. Hjalmarsson, and Fredrik Gustafsson, "Generalised fourier and toeplitz results for rational orthonormal bases," SIAM Journal on Control and Optimization, vol. 37, no. 2, pp. 439--460, 1997.

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B. NINNESS, H. HJALMARSSON, AND F. GUSTAFSSON, Generalised Fourier and Toeplitz results for rational orthonormal bases, SIAM Journal on Control and Optimization, 37 (1999), pp. 429--460.

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B. Ninness, H. Hjalmarsson, and F. Gustafsson, Generalised Fourier and Toeplitz results for rational orthonormal bases, to appear in SIAM J. Control and Optimization, (1997).

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B. Ninness, H. Hjalmarsson, and F. Gustafsson, Generalised Fourier and Toeplitz results for rational orthonormal bases, Technical Report EE9740, Department of Electrical and Computer Engineering, University of Newcastle, Australia. To appear in SIAM Journal on Control and Optimization, (1997).

No context found.

B. NINNESS, H. HJALMARSSON, AND F. GUSTAFSSON, Generalised Fourier and Toeplitz results for rational orthonormal bases, Technical Report EE9740, Department of Electrical and Computer Engineering, University of Newcastle, Australia. To appear in SIAM Journal on Control and Optimization, (1997).

No context found.

B. Ninness, H. Hjalmarsson, and F. Gustafsson, Generalised Fourier and Toeplitz results for rational orthonormal bases, Technical Report EE9740, Department of Electrical and Computer Engineering, University of Newcastle, Australia. To appear in SIAM Journal on Control and Optimization, (1998).

No context found.

B. Ninness, H. Hjalmarsson, and F. Gustafsson. Generalised Fourier and Toeplitz results for rational orthonormal bases. Tech. Rep. EE9740, Dept. of Elec. Eng. Uni. Newcastle, Australia. Submitted to SIAM J. Cont. and Optim., 1997. Available

....improved (dashed line) approximation in figure 1 is to absorb the fixed poles into the model structure, but still in an orthonormally parameterised way. In this case, certain new results on generalised Fourier convergence and the algebraic properties of generalised Toeplitz matrices are employed [6] to provide an approximation that is improved since it involves generalised Fourier reconstruction of a function that is invariant to the choice of fixed poles, and hence has fixed smoothness. To be more specific, the strategy employed here involves replacing the model structure (13) with the ....

....0 J Gamma1 is also maintained. The above basis functions fBk (z)g are orthonormal in the sense that hBn ; Bm i 1 2 Z Gamma Bn(e j )Bm (e j ) d = ffi(n Gamma m) where here ffi( Delta) is the Kronecker delta. In this case, the idea of Toeplitz matrices is generalised in [6] to one in which a matrix Mn(f) is defined by a spectral density f( as [Mn (f ) m; 1 2 Z Gamma Bm (e j )B (e j )f( d in which case, taking the RLS case as an example, it is possible to show [5] that Pi t 1=2Mn (oe 2 = Phi u ) Continuing by substituting Gamma n (q) ....

[Article contains additional citation context not shown here]

B. Ninness, H. Hjalmarsson, and F. Gustafsson. Generalised Fourier and Toeplitz results for rational orthonormal bases. Tech. Rep. EE9740, Dept. of Elec. Eng. Uni. Newcastle, Australia. Submitted to SIAM J. Cont. and Optim., 1997. Available via http://www.ee.newcastle.edu.au/users/staff/brett/

....condition numbers of R and R OE compare for the more commonly encountered coloured input case. A key result in this context is that purely by virtue of the orthonormality in the structure (5) an upper bound on the conditioning of R OE may be guaranteed for any Phi u by virtue of the fact that [11, 10] ( R) denotes the set of eigenvalues of the matrix R. min 2[ Gamma ; Phi u ( R OE ) max 2[ Gamma ; Phi u ( 13) No such bounds are available for the matrix R corresponding to the general (non orthonormal) structure (1) This suggests that the numerical conditioning ....

....key feature of the orthonormal parameterisation (5) is that associated with it is a covariance matrix with numerical conditioning guaranteed by the bounds min 2[ Gamma ; Phi u ( R OE ) max 2[ Gamma ; Phi u ( 38) A natural question to consider is how tight these bounds are. In [10], this was addressed by a strategy of analysis that is asymptotic in p. Specifically, define M OE , lim p 1 R OE . In this case, M OE is an operator 2 2 , so that the eigenvalues of the finite dimensional matrix R, generalize to the continuous spectrum (R1 ) of the operator M OE defined as ....

[Article contains additional citation context not shown here]

B. Ninness, H. Hjalmarsson, and F. Gustafsson, Generalised Fourier and Toeplitz results for rational orthonormal bases, Technical Report EE9740, Department of Electrical and Computer Engineering, University of Newcastle, Australia. Submitted to SIAM Journal on Control and Optimization, (1997). Model Structure and Numerical Properties 22

.... the space spfBw ; w 2 Wg as X n = sp ae 1 1 Gamma k z ; k = 0; 1; 2 Delta Delta Delta ; n Gamma 1 oe = sp fB k (z) k = 0; 1; 2; Delta Delta Delta ; n Gamma 1g RATIONAL BASIS FUNCTIONS FOR ROBUST ESTIMATION 4 where the functions fB k (z)g, which have been considered in detail in [45, 46], are defined by B 0 (z) p 1 Gamma j 0 j 2 = 1 Gamma 0 z) and B k (z) p 1 Gamma j k j 2 1 Gamma k z k Gamma1 Y m=0 z Gamma m 1 Gamma m z ; k = 1; 2; Delta Delta Delta (6) They are orthonormal in H 2 (T) with respect to the inner product hB n ; Bm i = 1 2 Z ....

.... earlier work in a stochastic setting, where it has also been argued that the main utility of orthonormal model structures for system identification is not as an implementational tool (since simpler structures span the same space X n and hence provide identical estimates) but as an analysis tool [45, 46]. Having studied these basic approximation properties, robust estimation using the minimax scheme proposed by [40] and [51, 53] is investigated. Conditions for robust convergence, and explicit quantification of estimation error are derived for each of the spaces A(D) H p (T) and 1 and for both ....

[Article contains additional citation context not shown here]

B. Ninness and H. Hjalmarsson. Generalised Fourier and Toeplitz Results for Rational Orthonormal Bases. Technical Report EE9740, Dept. Elec. and Comp. Eng. Uni. Newcastle, Australia. Submitted to SIAM J. Contr. Optimization, 1997.

....terms of an orthonormal basis. This approach is of greatest utility when accurate system descriptions are achieved with only a small number of basis functions. In recognition of this, there has been much work over the past several decades [M, B, E, R, S1] and, with renewed interest, more recently [WM, W2, W1, HVB, WP, NHG, BGS, NG] on the construction, analysis and application of rational orthonormal bases suitable for providing linear system characterisations. In a system theoretic context, the applications of these orthonormal basis ideas have been manifold, but nevertheless have concentrated mainly on the discrete time ....

....lies in H p ( Pi) 1 p 1) The choice of basis functions on the other hand depends on the class of systems. This subject will not be pursued here. The remainder of this section will be consumed with the extension of Theorem 5. 1 to the discrete time orthonormal basis functions studied in [NG, AN1, NHG] defined on D [ T by B n (z) 4 = p 1 Gamma j n j 2 1 Gamma n z OE n Gamma1 (z) OE n (z) 4 = n Y k=1 z Gamma k 1 Gamma k z ; OE 0 (z) 4 = 1: 19) These basis functions were considered in [AN1] for the purpose of robust estimation. In particular, it was shown that model ....

B. Ninness, H. Hjalmarsson, and F. Gustafsson. Generalised Fourier and Toeplitz results for rational orthonormal bases. SIAM J. Control and Optimization, 37:429--460, 1998.

.... which may be expressed as K n (z; n Gamma1 X k=1 B k ( B k (z) z; 2 C: 28) It derives its name from the property that for any f 2 X n and 2 C such that 62 f 0 ; Delta Delta Delta ; n Gamma1 g then f( hf(z) K n (z; i (29) and by virtue of which K n (z; is unique [35]. In x 6 it will be shown how the property (29) can be exploited in order to quantify bias error, but to do so a more explicit expression for (28) is required. This has been derived in [35] where in analogy with equivalent formulae in the study of orthogonal polynomials [11, 41] it is termed a ....

.... Delta ; n Gamma1 g then f( hf(z) K n (z; i (29) and by virtue of which K n (z; is unique [35] In x 6 it will be shown how the property (29) can be exploited in order to quantify bias error, but to do so a more explicit expression for (28) is required. This has been derived in [35] where in analogy with equivalent formulae in the study of orthogonal polynomials [11, 41] it is termed a Christoffel Darboux formula: K n (z; 1 Gamma n ( n (z) z Gamma 1 ; z; 2 D: 30) In this expression, n (z) is defined as the Blaschke product like quantity 0 (z) 1; ....

[Article contains additional citation context not shown here]

B. NINNESS, H. HJALMARSSON, AND F. GUSTAFSSON, Generalised Fourier and Toeplitz results for rational orthonormal bases, Technical Report EE9740, Department of Electrical and Computer Engineering, University of Newcastle, Australia. To appear in SIAM Journal on Control and Optimization, (1997).

....improved (dashed line) approximation in figure 1 is to absorb the fixed poles into the model structure, but still in an orthonormally parameterised way. In this case, certain new results on generalised Fourier convergence and the algebraic properties of generalised Toeplitz matrices are employed [6] to provide an approximation that is improved since it involves generalised Fourier reconstruction of a function that is invariant to the choice of fixed poles, and hence has fixed smoothness. To be more specific, the strategy employed here involves replacing the model structure (13) with the ....

.... Sigma = J Gamma1 Sigma 0 J GammaT is also maintained. The above basis functions fBk (z)g are orthonormal in the sense that hBn ; Bm i 1 2 Z Gamma Bn(e j )Bm (e j ) d = ae 1 ; m = n 0 ; m 6= n In this case, the idea of Toeplitz matrices is generalised in [6] to one in which a matrix Mn(f) is defined by a spectral density f( as [Mn (f ) m; 1 2 Z Gamma Bm (e j )B (e j )f( d is considered, in which case, taking the RLS case as an example, it is possible to show [5] that Pi t 1=2Mn ( oe 2 = Phi u ) Continuing by ....

[Article contains additional citation context not shown here]

B. Ninness, H. Hjalmarsson, and F. Gustafsson. Generalised Fourier and Toeplitz results for rational orthonormal bases. Tech. Rep. EE9740, Dept. of Elec. Eng. Uni. Newcastle, Australia. Submitted to SIAM J. Cont. and Optim., 1997. Available via http://www.ee.newcastle.edu.au

....AND F. GUSTAFFSON SpanfB0 ; B1 ; Delta Delta Delta ; Bn Gamma1 g and which may be expressed as Kn (z; n Gamma1 X k=1 Bk ( Bk (z) z; 2 C: 27) It derives its name from that property that for any f 2 Xn ; 2 D f( hf(z) Kn (z; i (28) and by virtue of which, it is unique [24]. In x VI it will be shown how the property (28) can be exploited in order to quantify bias error, but to do so a more explicit formulation of (27) is required which has been derived in [24] where in analogy with equivalent formulae in the study of orthogonal polynomials [8] 28] it is termed a ....

.... property that for any f 2 Xn ; 2 D f( hf(z) Kn (z; i (28) and by virtue of which, it is unique [24] In x VI it will be shown how the property (28) can be exploited in order to quantify bias error, but to do so a more explicit formulation of (27) is required which has been derived in [24] where in analogy with equivalent formulae in the study of orthogonal polynomials [8] 28] it is termed a Christoffel Darboux formula: Kn(z; 1 Gamma n ( n(z) z Gamma 1 ; z; 2 D: 29) In this expression, n(z) is defined as the Blaschke product like quantity n (z) n Gamma1 ....

[Article contains additional citation context not shown here]

B. Ninness, H. Hjalmarsson, and F. Gustafsson, Generalised Fourier and Toeplitz results for rational orthonormal bases, Submitted to SIAM Journal on Control and Optimization, (1997).

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