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) [161].
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
given
.
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.