Multiple Operator Deconvolution with Additive Noise; The Envelope Operator.

The methods for multiple operator deconvolution of Berenstein, Taylor, and Yger are examined for the case of the addition of a noise signal after each of the multiple convolutions and preceding the deconvolutions. It is shown that for strongly coprime multiple operators there is an obvious choice for optimal deconvolvers. The case of m strongly coprime, parallel convolvers with m independent noise sources is compared to m identical, parallel convolvers with m independent, identically distributed noise sources. A performance criterion is defined. The performance for selected collections of strongly coprime convolvers is shown to be at least as good as that for the corresponding collection of an equal number of identical, parallel convolvers. That is, there is no penalty for the additional frequency response available with deconvolution, at least for the noncompactly supported optimal deconvolvers. Qualitative methods are developed to characterize the properties of strongly coprime configurations. These methods enable the description of circumstances in which it is advantageous to use strongly coprime multiple detectors of large support