The Relationship Between the Variance of Energy Estimates and the Uncertainty Parameter

The relationship between the uncertainty (4π times the product of rms duration, D_t, and rms bandwidth, D_f) and the variance of energy estimates of random transients is examined. If a transient can be modeled with the product model, it is shown that for many cases the normalized error in the energy estimate, as measured by the uncertainty, is approximately the same as the normalized error determined from the more exact statistical duration and bandwidth. The uncertainty is a quantitative measure of the expected energy estimate errors. If the transient has a significant random component, a small uncertainty parameter implies a large error in the energy estimate. Attempts to resolve a time/frequency spectrum near the uncertainty limits of a transient with a significant random component will result in large errors in the spectral estimates.