Abstract

The spectroscope is considered as a communication channel. The message space is the class of all possible spectra, which are essentially sets of randomly spaced delta functions of Gaussian amplitude distribution. Hence the capacity of the channel can be evaluated by Shannon’s formula,

C=12π02dτlog2{1+d4[2ρE2/N0][sindτ/dτ]4[(2-τ)/4]2}

bits per diffraction unit, where d is the slit width, ρ the density of lines, E the intensity with unit slit, N0 the noise power in the band 0⩽τ⩽2, determined by the aperture. This integral rises rapidly to a constant value, indicating that the choice of d need only be based on ease of interpretation.

An equivalent width τC in frequency space is defined, which includes most of the integrand of C, and to which there corresponds an information half-width aC=π/2τC which measures the resolution in the presence of noise. The most suitable experimental conditions are when aC~ac, the Cauchy half-width, which occurs when the noise is about 5 percent of the signal and two readings are taken per slit width.

By the conventional interpretation the amount of information in a spectrum of length Δx is measured by Hartley’s formula as Δx/2aC bits of position information. The ratio of the maximum amount to this is 2aCC, which, in a typical case, amounts to a factor of 4 in resolution if a suitable decoding or filtering procedure could be discovered.

© 1953 Optical Society of America

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Equations (17)

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