2

d„ fT (t)fT (t + „ )e2πiν„ .

S(ν) = lim dt (12.77)

T ’∞ T ’∞

0

On the other hand, (12.76) can be written as

T T

2

dt fT (t)e’2πiνt fT (t )e2πiνt

S(ν) = lim dt

T ’∞ T 0 0

6 The de¬nition of the transform di¬ers slightly from the de¬nitions given in Section 12.1 on

page 315, where 2πν is taken as reciprocal variable rather than ν. Because of this the factors

√

2π disappear.

In fact, we do require that the integral from 0 to ∞ of the correlation function exists; this is

7

not the case when constant or periodic components are present.

328 Fourier transforms

T ’t

T

2

d„ fT (t)fT (t + „ )e2πiν„ .

= lim dt (12.78)

T ’∞ T ’t

0

Now, for large T , t “almost always” exceeds the time over which the corre-

lation function is non-zero. Therefore the integral over „ “almost always”

includes the full correlation function, so that the limits can be taken as ±∞.

In the limit of T ’ ∞ this is exact.

Combination of (12.73) and (12.76) shows that the autocorrelation funct-

ion can be obtained from the inverse FT of the squared frequency ampli-

tudes:

1 ∞—

C(„ ) = 2 lim FT (ν)FT (ν) cos(2πν„ ) dν

T ’∞ T 0

1∞—

FT (ν)FT (ν)e’2πiν„ dν.

= lim (12.79)

T ’∞ T ’∞

Note that the sign in the exponent is irrelevant because of the symmetry of

F —F .

The discrete periodic case is similar. Given a time series fn , periodic on

[0, N ), we obtain the following exact relations:

per

(i) Autocorrelation function Ck de¬ned in (12.70).

(ii) Fourier transform Fm of time series is de¬ned by (12.66).

(iii) Spectral density from Fm :

2—

per

Sm = F Fm . (12.80)

Nm

(iv) Autocorrelation function from inverse FFT of spectral density:

N ’1

N/2’1

1

Sm e’2πimk/N . (12.81)

per per per

Ck = Sm cos(2πmk/N ) =

2N

m=0 m=0

(v) Spectral density from FFT of autocorrelation function:

N ’1

N/2’1

per per

per

Ck e2πimk/N .

Sm =4 Ck cos(2πmk/N ) =2 (12.82)

k=0 k=0

When the data are samples of a continuous time series taken at intervals

per

”t during a total time span T = N ”t, the autocorrelation coe¬cients Ck

provide an estimate for the continuous autocorrelation function C(k”t) of

(3.37). If zero-padding has been applied, the estimate is improved by scaling

C(k”t) with N/(N ’ k) (see (12.70) on page 326). Without zero-padding,

per

Ck provides an estimate for a mixture of C(k”t) and C(T ’ k”t), as

12.8 Autocorrelation and spectral density from FFT 329

given by (12.71). The spectral resolution is ”ν = 1/T and the coe¬cients

Sm represent the “power” in a frequency range ”ν. The continuous power

density per unit of frequency S(ν), given by (12.72) and (12.76), equals

Sm /”ν = T Sm .

If the time series is longer than can be handled with FFT, one may break

up the series into a number of shorter sections, determine the spectral den-

sity of each and average the spectral densities over all sections. One may

then proceed with step (iv). This procedure has the advantage that a good

error estimate can be made based on the variance of the set of spectra ob-

tained from the sections. Spectra obtained from one data series will be noisy

and “ if the data length N greatly exceeds the correlation length of the data

(as it should) “ too ¬nely grained. One may then apply a smoothing proce-

dure to the spectral data. Beware that this causes a subjective alteration of

the spectrum! This can be done by convolution with a local spread function,

e.g., a Gaussian function,8 but the simplest way is to multiply the autocor-

relation function with a window function which reduces the noisy tail of the

correlation function. The smoothed spectral density is then recovered by

FFT of the windowed autocorrelation function. The e¬ect is a convolution

with the FT of the window function. If a Gaussian window is used, the

smoothing is Gaussian as well. See the following example.

In Fig. 12.4 an example is given of the determination of a smoothed spec-

tral density from a given time series. The example concerns an MD sim-

ulation at 300 K of the copper-containing protein azurin in water and the

question was asked which vibration frequencies are contained in the ¬‚uc-

tuation of the distance between the copper atom and the sulphur atom of

a cysteine, one of the copper ligands. Such frequencies can be compared

to experimental resonance-Raman spectra. A time slice of 20 ps with a

resolution of 2 fs (10 000 data points) was considered (Fig. 12.4a). It was

Fourier-transformed (”ν = 1012 /20 Hz = 50 GHz) and its power spectrum,

which had no signi¬cant components above a range of 400 points (20 THz),

computed and plotted in Fig. 12.4b. The complete power spectrum was

subsequently inversely Fourier transformed to the autocorrelation function

(not shown), which was multiplied by a Gaussian window function with

8 See Press et al. (1992) for a discussion of optimal smoothing.

330 Fourier transforms

Cu-S distance (nm)

0.24

0.23

0.22

0.21

4 8 12 16 20

a time (ps)

spectral intensity (a.u.)

spectral intensity (a.u.)

8

100

c

b 7

80 6

5

60

4

40 3

2

20