The synthesis of transient processes

This brief presents a procedure for synthesizing a transient process by exciting a causal linear shift-variant system with an orthonormal random sequence. The output process has finite average energy and is described by its average energy density spectrum. This spectrum is equal to the ensemble average of the magnitude-squared of the Fourier transform process associated with the transient process. A prototypical bandpass average energy density spectrum is introduced, and a more general spectrum is obtained by adding together a collection of prototypical spectra each having different values for the center frequency, bandwidth, and peak amplitude. Realizations of the transient process are obtained by a matrix transformation applied to an ensemble of MATLAB-generated random input vectors. The average of the magnitude-squared discrete Fourier transform of the resulting random output vectors approximates the average energy density spectrum of the transient process