Statistical hypothesis testing for the shape of impulse response function

The problem of estimation of unknown response function of a time-invariant continuous linear system is considered. Integral sample input–output cross-correlogram is taken as an estimator of the response function. The inputs are supposed to be zero-mean stationary Gaussian process. A criterion on the shape of impulse response function is given. For this purpose, we apply a theory of square–Gaussian random processes and estimate the probability that supremum of square–Gaussian process exceeds the level specified by some function.