Statistical Errors in Measurements on Random Time Functions

The statistical errors in time‐average measurements of various properties of (stationary) random time functions are analyzed for their dependence on the averaging interval, the method of averaging, and the statistics of the particular time function under observation. The specific averaging processes considered here are as follows: (1) The continuous time average $(\frac{1}{T}{\int }_0^T\mathrm{Z(t)dt)}$ , (2) The summation (by discrete sampling techniques) $(\frac{1}{N}{\sum }_{\text{n=1}}^N{\mathrm{Z(nT}}_0\mathrm{))}$ , and (3) The smoothing accomplished by various low‐pass filters (for example, a dc meter). Here Z(t) may represent the output of a linear or nonlinear device. As a particular example, the measurement of the power in a random noise voltage of band width ω_F is analyzed in detail. The expected root‐mean‐square error approaches asymptotically the value (K/ω_FT)^½, where K is some numerical constant, depending on the device in question. For (1) and (2) T is the total observation time while for (3) it is proportional to a relevant time constant of the low‐pass filter. A short discussion of the relative merits of correlators and spectrum analyzers is included, together with a brief treatment of the distribution function of the error in the measurement.