On the identification of the parameters of linear systems using Chebyshev polynomials of the first kind and Chebyshev polynomials orthogonal on a uniform grid

I. I. Sharapudinov ; M. S. Sultanakhmedov ; T. N. Shakh-Emirov ; T. I. Sharapudinov ; M. G. Magomed-Kasumov ; G. G. Akniev ; R. M. Gadzhimirzaev

Daghestan Electronic Mathematical Reports, Tome 2 (2014), pp. 1-32.

Résumé

Linear system in which the input signal $y = y(t)$ and the output $x = x(t)$ are related by the equation $x^{(r)}(t)=\sum_{\nu=0}^{r-1}a_\nu(t)x^{(\nu)}(t)+\sum_{\mu=0}^s b_\mu(t)y^{(\mu)}(t)$ is considered. The goal is to find the unknown variable coefficients $a_\nu(t)$ $(\nu=0,\ldots,r-1)$ and $b_\mu(t)$ $(\mu=0,\ldots,s)$ in case when the signal values are given in the nodes of a uniform grid $\Omega_N=\{t_j=-1+jh\}_{j=0}^{N-1}$, where $h=\frac2{N-1}$. It is assumed that the values of $x(t)$ and $y(t)$ are obtained as a result of experimental observations and are noised. For pretreatment of discrete information we apply «anti-aliasing» based on the use of Chebyshev polynomials orthogonal on a uniform grid $\Omega_N$. On the next step we switched from the original equation to the dual equation by representing of all figuring there functions (including derivatives) in the form of series by Chebyshev polynomials of the first kind $C_n(t)=\cos{(n\arccos{t})}$. The result is a system of linear equations for the Fourier – Chebyshev coefficients of $a_\nu(t)$ and $b_\nu(t)$. Solving this system numerically, we obtain the variable coefficients of the original system of equations, thus completing the solution of the identification problem.

Détail

Keywords: Chebyshev polynomials of the first kind; Chebyshev polynomials orhtogonal on uniform grid; linear systems; signal processing; identification problem.

Bibliographie

[1] Chebyshev P.L., “O nepreryvnykh drobyakh (1855)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 103–126 | MR

[2] Chebyshev P.L., “Ob odnom novom ryade”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 236–238 | MR

[3] Chebyshev P.L., “Ob interpolirovanii po sposobu naimenshikh kvadratov (1859)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 314–334 | MR

[4] Chebyshev P.L., “Ob interpolirovanii (1864)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 357–374 | MR

[5] Chebyshev P.L., “Ob interpolirovanii velichin ravnootstoyaschikh (1875)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 66–87 | MR

[6] Sharapudinov I.I., Mnogochleny, ortogonalnye na diskretnykh setkakh, Izd-vo Dag. gos. ped. un–ta, Makhachkala, 1997

[7] Gasper G., “Positiviti and special function”, Theory and appl. Spec. Funct., 1975, 375–433 | DOI | MR | Zbl

[8] Sharapudinov T.I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Vestnik Dagestanskogo nauchnogo tsentra RAN, 2007, no. 29, 12-23

[9] Sege G., Ortogonalnye mnogochleny, Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moskva, 1962, 500 pp.

[10] Sharapudinov I.I., “O skhodimosti metoda naimenshikh kvadratov”, Matem. zametki, 53:3 (1993), 131–143 | MR | Zbl

[11] Sharapudinov I.I., “Asimptoticheskie svoistva i vesovye otsenki dlya ortogonalnykh mnogochlenov Chebysheva–Khana”, Matem. sb., 182:3 (1991), 408-420 | Zbl

[12] Sharapudinov I.I., “Asimptoticheskie svoistva ortogonalnykh mnogochlenov Khana diskretnoi peremennoi”, Matem. sb., 180:9 (1989), 1259–1277 | Zbl

[13] Sharapudinov I.I., “Ob asimptotike mnogochlenov Chebysheva, ortogonalnykh na konechnoi sisteme tochek”, Vestnik MGU. Seriya 1, 1992, no. 1, 29-35 | MR

[14] Sharapudinov I.I., “Approksimativnye svoistva operatorov $\mathscr Y_{n+2r}(f)$ i ikh diskretnykh analogov”, Matem. zametki, 72:5 (2002), 765–795 | DOI | MR | Zbl

[15] Sharapudinov I.I., “Priblizhenie diskretnykh funktsii i mnogochleny Chebysheva, ortogonalnye na ravnomernoi setke”, Matem. zametki, 67:3 (2000), 460–470 | DOI | MR | Zbl

[16] Sharapudinov I.I., Smeshannye ryady po ortogonalnym polinomam. Teoriya i prilozheniya, DNTs RAN, Makhachkala, 2004, 276 pp.