Title :
Eigenvalue method for approximating cycle slip rates in PLLs
Author_Institution :
Dept. of Electr. & Comput. Eng., Alabama Univ., Huntsville, AL, USA
Abstract :
The small eigenvalues of a PLL´s Fokker-Planck operator are useful for approximating the rates at which noise-induced cycle slips occur. A method is presented for numerically approximating these small eigenvalues. An a finite dimensional subspace of 2πm-periodic functions (m is a user-supplied problem-dependent integer), the eigenvalue problem is reformulated (in an approximate sense) by a finite dimensional system of differential equations with 2π-periodic coefficients. The unknown eigenvalue appears as a parameter of this system; it must be selected so that the system has a particular periodic solution. Next, a method is discussed for numerically approximating the periodic solutions and small eigenvalues. The method utilizes as a constraint the known scale factor change that the eigenfunctions must experience from one 2π interval to the next. This scale factor is shown to be a multiplier of the system of differential equations. Finally, the method is applied to a simple example
Keywords :
approximation theory; differential equations; eigenvalues and eigenfunctions; phase locked loops; Fokker-Planck operator; PLL; cycle slip rates approximation; differential equations; eigenfunctions; eigenvalue method; eigenvalue problem; finite dimensional subspace; finite dimensional system; noise induced cycle slips; periodic coefficients; periodic functions; periodic solutions; scale factor; system parameter; Boundary conditions; Density functional theory; Eigenvalues and eigenfunctions; Equations; H infinity control; Phase locked loops; Phase noise; Shape control; Steady-state;
Conference_Titel :
System Theory, 1997., Proceedings of the Twenty-Ninth Southeastern Symposium on
Conference_Location :
Cookeville, TN
Print_ISBN :
0-8186-7873-9
DOI :
10.1109/SSST.1997.581578
