Using the autocorrelation function to characterize time series of voltage measurements

The standard deviation of the mean is the most common basis for specifying the statistical uncertainty of repeated measurements, yet it is often calculated incorrectly. The variance of the mean, var(X), of a time series of correlated measurements of a weakly stationary process is correctly expressed in terms of the autocorrelation function (ACF) at lag k, rho(k). This approach is used to evaluate var(X) for white voltage noise measured at regular time intervals tau(0) through a low pass filter of bandwidth B by four methods: (1) by developing the expression rho(k) = exp(-4B tau(0)k) evaluated by estimating B from the sample spectrum; (2) by noting that rho(k) = phi(k), phi = exp(-4B tau(0)) < 1 is the ACF of a first-order autoregressive process, AR(1), for which var(X) is readily evaluated in terms of phi; (3) by estimating var(X) from the sample ACF, rho(k), using the cut-off lag for an AR(1) process; and (4) by applying the general method recently proposed by Zhang (2006 Metrologia 43 S276-81), to estimate var(X) from rho(k), assuming that the data may be described by a moving average process with a cut-off lag deduced from the rho(k) themselves. The values of var(X) from the four methods are in good agreement. This provides firm support to Zhang's method; this is important because of this method's wide scope of application.