Time Series Generation and Transformation—The Role of Phase

Abstract

Random signals with some desired power spectrum may be generated by a variety of methods, including summation of sinusoids, convolution of white noise with a kernel function or integration of a differential equation driven by white noise. These continuous models of the spectrum have discrete time equivalents, the moving average (MA) and autoregressive moving average (ARMA), respectively. This paper relates the various techniques and discusses a problem which appears in all of them. Each method involves an amplitude function, the square root of the desired power spectrum and some phase behavior which is unknown and must therefore be assumed. Results from the time series literature presented here provide a rational basis for the choice of phase. In the convolution method, the resulting kernel function is the shortest possible, consistent with the given spectrum and hence minimizes the computational cost of the subsequent signal generation. The corresponding differential equation is the simplest possible, consistent with the form of the amplitude curve. No properties of the time series are affected by the assumed phase spectrum, but phase is critical to the synthesis of correlated signals such as velocity at different points in a random wave field. The results are illustrated by a nondimensional fourth-order differential equation model of the Pierson-Moskowitz spectrum.