Three methods are introduced for generating realizations of time histories with a specified auto-spectral density while controlling the kurtosis. One of the methods also allows the skewness to be specified. A second method allows large excursions (that produce large kurtosis) to be randomly distributed or almost periodic. In addition, the second method allows the average number of large excursions per unit of time to be specified. All the methods are variations of the inverse Welch method. The shape of the discrete Fourier magnitude is specified for a frame of data, thus controlling the shape of the auto-spectral density. The phase of the frame of data or the magnitude of the amplitude spectrum is modified to control the kurtosis or the skewness. The frames of data are multiplied by a window and overlapped and added to produce the realization.

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