On Measuring the Root-Mean-Square Value of a Finite Record Length Periodic Waveform

An analysis of the uncertainty in measuring the root-mean-square, rms or R_q, value of a periodic waveform which results from the use of a finite record length is presented. Even though the results of the analysis are somewhat as expected, i.e., that the uncertainty is inversely proportional to the number of periods in the record, the explicit relationship between the magnitude of the uncertainty and properties of the waveform does not appear to be available in the literature. The paper first presents an introductory example in terms of the reasonably well known case of bandwidth limited Gaussian waveform to introduce definitions. Following this is an analysis of the periodic waveform using the same approach. It is shown that for a large number of periods, n, in the record length, the normalized three standard deviation of the rms value is given by 3/(8πn).