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The preceding analysis can be extended to the case of multiple
sinusoids in white noise [120]. When the sinusoids are
well resolved, i.e., when window-transform side lobes are
negligible at the spacings present in the signal, the optimal
estimator reduces to finding multiple interpolated peaks in the
spectrum.
One exact special case is when the sinusoid frequencies
coincide with the ``DFT frequencies''
, for
. In this special case, each sinusoidal peak sits
atop a zero crossing in the window transform associated with every
other peak.
To enhance the ``isolation'' among multiple sinusoidal peaks, it is
natural to use a window function which minimizes side lobes. However,
this is not optimal for short data records since valuable data are
``down-weighted'' in the analysis. Fundamentally, there is a
trade-off between peak estimation error due to overlapping side lobes
and that due to widening the main lobe. In a practical sinusoidal
modeling system, not all sinusoidal peaks are recovered from the
data--only the ``loudest'' peaks are measured. Therefore, in such
systems, it is reasonable to assure (by choice of window) that the
side-lobe level is well below the ``cut-off level'' in dB for the
sinusoidal peaks. This prevents side lobes from being comparable in
magnitude to sinusoidal peaks, while keeping the main lobes narrow as
possible.
When multiple sinusoids are close together such that the associated
main lobes overlap, the maximum likelihood estimator calls for a
nonlinear optimization. Conceptually, one must search over the
possible superpositions of the window transform at various relative
amplitudes, phases, and spacings, in order to best ``explain'' the
observed data.
Since the number of sinusoids present is usually not known, the number
can be estimated by means of hypothesis testing in a Bayesian
framework [21]. The ``null hypothesis'' can be ``no
sinusoids,'' meaning ``just white noise.''
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