$L^1$-Norm Convergence Properties of Correlogram Spectral Estimates

This paper establishes the following results concerning the estimation of the power spectrum of a single, deterministic, infinitely long signal. a) If
S

_x
is the signal's power spectral density, correlogram spectral estimates obtained from increasingly longer signal segments tend to
S

<sub>x

*</sub>

?

_x
/2
p
in the
L

¹
-norm, where ? is the Fourier transform of the window used to generate the estimates. b) The
L

¹
-norm of
S

_x
-
S

<sub>x

*</sub>

?

_x
/2
p
can be made arbitrarily small by an appropriate choice of window. It is further shown that the accuracy of the spectral estimates generated by a given window is related to a newly introduced function, termed the windowing error kernel and that this function yields bounds on the asymptotic error of the estimates. As an example, correlogram spectral estimates are used to analyze spectral regrowth in an amplifier.