Retrieving the spectrum of physical radiation from experimental measurements typically involves using a mathematical algorithm to deconvolve the instrument response function from the measured signal. However, in the field of signal processing known as "Source Separation"(SS), which refers to the process of computationally retrieving the separate source components that generate an overlapping signal on the detector, the deconvolution process can become an ill-posed problem and crosstalk complicates the separation of the individual sources. To overcome this problem, we have designed a magnetic spectrometer for inline electron energy spectrum diagnosis and developed an analysis algorithm using techniques applicable to the problem of SS. An unknown polychromatic electron spectrum is calculated by sparse coding using a Gaussian basis function and an L1 regularization algorithm with a sparsity constraint. This technique is verified by using a specially designed magnetic field electron spectrometer. We use Monte Carlo simulations of the detector response to Maxwellian input energy distributions with electron temperatures of 5.0 MeV, 10.0 MeV, and 15.0 MeV to show that the calculated sparse spectrum can reproduce the input spectrum with an optimum energy bin width automatically selected by the L1 regularization. The spectra are reproduced with a high accuracy of less than 4.0% error, without an initial value. The technique is then applied to experimental measurements of intense laser accelerated electron beams from solid targets. Our analysis concept of spectral retrieval and automatic optimization of energy bin width by sparse coding could form the basis of a novel diagnostic method for spectroscopy.
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