The power of the signal in a given frequency band [ f 1 , f 2 ] , where 0 < f 1 < f 2 , can be calculated by integrating over frequency. Since S x x ( − − f ) = S x x ( f ) , an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used): P bandlimited = 2 ∫ ∫ f 1 f 2 S x x ( f ) d f More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval T is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than 1 / T are not sampled, and results at frequencies which are not an integer multiple of 1 / T are not independent. Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to statistical ensembles of realizations of x ( t ) evaluated over the specified time window.