Impulse response determination using derivatives of correlation function

In this short paper, a new method of system identification is proposed in which the impulse response of a system is determined from a linear combination of the derivatives of the cross-correlation function between an input and the corresponding output of the system. This method is particularly simple when the autocorrelation function
\phi_{xx{(\tau)
of the input is an exponential or a triangular function of the variable \tau. This method merely requires the autocorrelation function to be undifferentiable at some points in the sense of usual functions. Essentially, the method utilizes the δ-functions appearing when
\phi_{xx}(\tau)
is differentiated in the sense of the distribution, or the generalized function proposed by Schwartz. Results of experiments by computer simulation showed good agreement with the theoretical considerations.