J. O. Coleman

ceetoeplitz.pdf (188K)
ceetoeplitz.ps (233K)
J. O. Coleman
A Simple FIR-Filter Interpretation of the Extreme Eigenvalues of a Toeplitz Autocorrelation Matrix
The convergence of LMS adaptive algorithms is typically limited by the eigenvalue spread of a Toeplitz autocorrelation matrix with elements from the central portion of an autocorrelation function. If that autocorrelation function describes a random process input to an FIR filter, the ratio of the filter output power to that obtained in response to a unit-power white input varies, as the filter response is changed, across the closed interval from the minimum eigenvalue to the maximum eigenvalue of the autocorrelation matrix. This simple fact permits important relationships between these extreme eigenvalues and the spectrum at the filter input to be understood easily and without resort to the classic asymptotic approximation with a cyclic matrix. In particular, (1) a pure line spectrum with fewer distinct lines than the matrix order leads to a singular matrix; (2) the spectral minimum/maximum is a lower/upper bound on the minimum/maximum eigenvalue; and (3) those bounds are approached asymptotically with increasing matrix order (the classic result). Further, filter-optimization experience may offer the system designer some intuition for the variation of extreme eigenvalues with matrix order and key spectral parameters.
Computers & Electrical Engineering, February 2000, volume 26, number 2, pp. 141-149. (Publication status)