Additivity theorem for constrained optimum filters in the presence of colored noise

It is well known that the optimum filter in presence of an additive stationary noise of spectral density N(ω) and an input signal composed by a sum of delta-like components is obtained as the sum of the optimum filters for the single delta components, each weighted by the amplitude of the corresponding delta component. This holds if no constraints are enforced in the filter weighting function (WF). However, this no longer holds if constraints are introduced in the filter WF, such as finite duration and area balance. Recently, it has been observed that if such constraints are enforced the optimum filter seems again to be given by a weighted sum of the optimum WFs for the single delta components of the signal. However, each weight is given in this case by the product of the delta-pulse amplitude with the inverse of the variance associated to the corresponding WF. A rigorous proof for this additivity property will be given, while considering the typical white and non-white noises of a spectrometer of ionizing radiations using semiconductor detectors.