Minimum entropy deconvolution and simplicity: A noniterative algorithm

Minimum entropy deconvolution (MED) is a technique developed by Wiggins (1978) with the purpose of separating the components of a signal, as the convolution model of a smooth wavelet with a series of impulses. The advantage of this method, as compared with traditional methods, is that it obviates strong hypotheses over the components, which require only the simplicity of the output. The degree of simplicity is measured with the Varimax norm for factor analysis. An iterative algorithm for computation of the filter is derived from this norm, having as an outstanding characteristic its stability in presence of noise. Geometrical analysis of the Varimax norm suggests the definition of a new criterion for simplicity: the D norm. In case of multiple inputs, the D norm is obtained through modification of the kurtosis norm. One of the most outstanding characteristics of the new criterion, by comparison with the Varimax norm, is that a noniterative algorithm for computation of the deconvolution filter can be derived from the D norm. This is significant because the standard MED algorithm frequently requires in each iteration the inversion of an autocorrelation matrix whose order is the length of the filter, while the new algorithm derived from the D norm requires the inversion of a single matrix. On the other hand, results of numerical tests, performed jointly with Graciela A. Canziani’, show that the new algorithm produces outputs of greater simplicity than those produced by the traditional MED algorithm. These considerations imply that the D criterion yields a new computational method for minimum entropy deconvolution. A section of numerical examples is included, where the results of an extensive simulation study with synthetic data are analyzed. The numerical computations show in all cases a remarkable improvement resulting from use of the D norm. The properties of stability in the presence of noise are preserved as shown in the examples. In the case of a single input, the relation between the D norm and the spiking filter is analyzed (Appendix B). ‘Institute Argentino de Matemitica, Consejo National de Investigaciones Cientificas y Tknicas, Viamonte 1636, 1055 Buenos Aires, and Departamento de Matemitica, Universidad de Buenos Aires, Pabell6n I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.