 FILES:

ceetoeplitz.pdf (188K)
ceetoeplitz.ps (233K)
 AUTHORS:
 J. O. Coleman
 TITLE:

A Simple FIRFilter Interpretation
of the Extreme Eigenvalues
of a Toeplitz Autocorrelation Matrix
 ABSTRACT:

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
unitpower 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, filteroptimization experience may offer the
system designer some intuition for the variation of
extreme eigenvalues with matrix order and key spectral
parameters.
 PUBLISHED:

Computers & Electrical Engineering, February 2000, volume
26, number 2, pp. 141149. (Publication
status)