Astronomy and Astrophysics – Astrophysics
Scientific paper
Jan 1981
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1981a%26a....93..269k&link_type=abstract
Astronomy and Astrophysics, Vol. 93, P. 269, 1981
Astronomy and Astrophysics
Astrophysics
6
Scientific paper
The maximum entropy method (MEM) of spectral analysis is examined from a somewhat novel viewpoint.
A spectral function being necessarily positive, its (discrete) Fourier transform is an autocorrelation sequence. We show that associated with an sequence there is a sequence of unit vectors whose scalar products give the auto-correlations. Specifying the autocorrelation up to some finite lag imposes constraints on the vector sequence and thus on unspecified auto- correlation terms. Each unspecified term lies within a convex connected area in the complex plane. The first lies within a circle; we give expressions for its centre and radius. Any choice within this "circle of constraint" is consistent with positivity, but choosing the centre maximizes the area available to the next unspecified term, which is a circle of the same radius. Sequentially setting each term to the centre of its circle maximizes the area available for each subsequent term; this procedure also yields the MEM spectral estimate.
The MEM is shown to be a type of model fitting; the model consists of a number of asymmetric "peaks" equal to the number of autocorrelation terms originally specified. Since each peak requires two complex numbers to specify it, the individual characteristics of the peaks are not independently adjustable. This can lead to undesirable results, for example spurious splitting of peaks.
If the circles of constraint contract to points the peaks reduce to delta functions. Measuring error drives the auto-correlations towards this fully constrained limit, and thus tends to enhance the "peakiness" of the spectrum.
These and other features of the MEM are substantiated by numerical examples.
Komesaroff M. M.
Narayan Ramesh
Nityananda Rajaram
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