Random uncertainty of variance of finite length measurement signals
The finite length of a towing tank induces a random uncertainty in the mean value of a measurement signal. This uncertainty is not negligible if the signal contains low frequency components with large amplitudes, such as the towing force on a ship model. Ideally, the random uncertainty of towing test results is determined by repeat or reproduction tests. Though scientifically correct, in commercial research this is not always viable. An alternative is to obtain an estimate of the uncertainty of the mean value by signal analysis of a single measurement. An analytical solution for the standard deviation of the mean of a band pass noise process is derived. Depending on the bandwidth and lowest frequency of the process, the standard deviation of the mean decays either linearly or with the square root of the inverse of the measurement time as long as the measurement time exceeds a certain threshold. This agrees with the solution obtained from sample statistics for respectively a special class of correlated samples and fully uncorrelated samples. Two methods to estimate the random uncertainty of the mean from a single finite length sample record are derived. One method uses the calculation of the autocovariance function. The other uses the division of a signal into equally-sized segments. Both methods are verified using the analytical solution for a band pass noise process. Finally, the methods are applied to the towing force signal of a ship model.