I assume that (which I call the `signal') for a single, ergodic trajectory has been generated and stored at a fine enough time-resolution to capture all the desired features. This can be done by evaluating along the trajectory, and subtracting the average value (estimated, or often analytically known). We need , where is the upper frequency limit desired for the band profile.
The auto-correlation spectrum of this single signal is , and if the signal exists over all times then is a stochastic quantity uncorrelated in . The microcanonical average (I drop the energy subscript) is reached by averaging over initial conditions of trajectories. It is that is the desired `band profile'. However, is also the local mean of , so can be more easily estimated by smoothing . This smoothing (convolution with Gaussian of width ) is performed using the Fast Fourier Transform (FFT) . Since computationally the signal must be of necessarily finite length , correlations in -space arise in on the scale . Therefore the smoothing operation corresponds to averaging independent samples, and a fractional error of results for the band profile estimate. This estimation error appears as multiplicative noise with frequency correlation scale . Typical parameters were , , giving about 3% RMS estimation error.
There is another `failure mode' of this estimation procedure: There are only independent samples in the finite sample length, therefore only independent values can appear in . If then the errors may exceed that quoted above, because samples are no longer independent. This failure mode only arises when is desired at large , which forces to shrink, and therefore to also shrink if the memory requirements are not to grow. In this case, true averaging over trajectories is the only remedy.
Now I describe how to find
I will always use the Fourier Transform (FT) convention
All FFTs were implemented on a Compaq XP1000 (21264 667MHz Alpha processor) running C++ and the Compaq Extended Math Library (CXML). Using this setup, an 8 million () point double-precision FFT takes about 5 seconds.