Two constrained deconvolution methods using spline functions

AbstractThis paper describes two new methods to solve the following estimation problem. Given n1noisy measurements (yi1,i,i=1,..., n1)of the response of a system to a knowninput [A1(t) where tindicates time], and n2noisy measurements yi,2,i=l,..., n2of the response of a system to an unknowninput [A2(t)], obtain an estimate of A2(t) and K(t) (the unit impulse response function of the system) under the model: $$y_{ij} = \int_0^{t_{ij} } {A_j (s)} K(t_{ij} - s)ds + \varepsilon _{ij} $$ wereεi,jare independent identically distributed random variables. Both methods use spline functions to represent the unknown functions, and they automatically select the spline functions representing the unknown input and unit impulse response functions. The first method estimates separately the unit impulse response function and the input, recasting the problem in terms of inequality-constrained linear regression. The second method jointly estimates the unit impulse response function and the input function, recasting the problem in terms of inequality-constrained nonlinear regression. Simulated and real data analysis are reported.