The precision limits of a waveform processing oscilloscope have been investigated for the situation where the instrument is used to recover a repetitive exponential signal from high levels of noise. In these studies, the oscilloscope was programmed to sample the signal, average it, and then calculate its exponential decay time constant. The measurement errors in the resulting time constants were found to decrease with continued averaging, but not without limit. In fact, we found that there was an optimum number of samples needed to minimize the error for any particular value of the signal‐to‐noise ratio at the input to the oscilloscope. We present here our empirically determined values of those optima, an intercomparison of them with the results of a theoretical model, and a description of the measurement system developed for this work.