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# General system--windowed estimation of correlation spectrum

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 given . I will always use the Fourier Transform (FT) convention      (B.1)

For this general real signal , the Wiener-Khinchin Theorem (easy to prove ) equates the power spectrum to the FT of the auto-correlation, (B.2)

I choose to be a windowed version of the signal, (B.3)

where and has finite characteristic width . This windowing function defines the maximum trajectory length needed. Therefore the auto-correlation is related to the desired by (B.4)

where the last step involved identifying as a -dependent weighting function whose -integral is , giving a generalization of the top-hat' weighting function of e.g. Eq.(2.30). Applying the Convolution Theorem and dividing by , (B.5) is a narrowly peaked function where the normalization is . Therefore we have the LHS as our estimate for . Structure below frequency scale (the -function width) is lost. I typically used the Gaussian window , for which . The maximum signal length needed to evaluate the windowed signal was typically chosen as , beyond which is negligible. Because dies to zero very smoothly, the windowed signal of length can now be used as a periodic discretely-sampled array, and the FFT used to find .

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.   Next: Considerations in a billiard Up: Appendix B: Numerical evaluation Previous: Trajectories the issue of
Alex Barnett 2001-10-03