Spectral Density

The spectral density of a discrete three-dimensional data set can be calculated based in the discrete Fourier transformation, which maps the discrete spatial data set to a frequency domain via where the discrete data set is given by the indicator function χ^{(P )}, cf. Eq. (6.6), and the position vector x = [nx ny nz]^T with nx = 1 . . . Nx, ny = 1 . . . Ny and nz = 1 . . . Nz and the total size of the data set given by Nx, Ny and Nz. The coordinates in the frequency domain are kx, ky and kz. The discrete inverse Fourier transformation is then given by

χ^{(P )}(nx, ny, nz) := 1

mapping the data from the frequency domain to the spatial domain. In the discrete Fourier transformation, the data set is considered as periodically extendable. The spectral density is then calculated by

P_SD^{(P )}:= (F^{(P )})^∗F^{(P )} NxNyNz

, (6.14)

where (F^{(P )})^∗denotes the conjugate complex of the Fourier transform F^{(P )}. As mentioned above, the spectral density and auto-correlation function are related, thus an alternative method in order to obtain the spectral density involves computing the Fourier transfor-mation of the autocorrelation function. However, Eq. (6.14) allows for rapid computa-tions of the spectral density, since fast algorithms are available for the computation of Fourier transformations, e.g. the “FFTW”, (“Fastest Fourier Transform in the West”), developed at the Massachusetts Institute of Technology by M. Frigo and S.G. Johnson (www.fftw.org). In contrast to this, computing the autocorrelation function would require more computational effort.

Example of Spectral Density Calculation for Discrete 1D Pulse

As an example, the computation of the spectral density based on the two above mentioned methods is presented for a discrete signal, which is assumed to be repeated periodically.

The signal sequence x has the period N = 6 and the spatial coordinates n, such that

0

Figure 6.2: (a) Sample pulse, (b) real and (c) imaginary part of discrete Fourier transform of sample pulse.

x(n + N) = x(n) is given by

x(n) =

(1 for n = 0, 1, 2 ,

0 for n = 3, 4, 5 , (6.15)

it depicted in Fig. 6.2a. The discrete Fourier transform for a 1D discrete signal sequence is given by

X(k) =

N−1X

n=0

x(n) exp(−i 2πk n/N), (6.16)

which maps the periodic signal into a periodic, discrete frequency spectrum, described by the coordinate k. The discrete Fourier transform X(k) is calculated via Eq. (6.16) and is shown in Fig. 6.2.

The value X(k = 0) is simply X(k = 0) =PN−1

n=0 exp(0) x(n), thus the sum over all entries, and referred to as the zero component. Reversibly, the inverse discrete Fourier transform is defined by

which maps the discrete Fourier transform, X(k), given in the frequency domain back to x(n) in the spatial domain. Furthermore, the autocorrelation function ϕ(s) describes the correlation of the function with itself with respect to a time shift s. The autocorrelation function of x(n) is defined by

ϕ(s) := 1 N

N−1X

n=0

x(n) x(n + s), (6.18)

which is depicted in Fig. 6.3a and where ϕ(s + N) = ϕ(s), thus the autocorrelation function itself is periodic with period N. The computation of the spectral density can be now carried out based on X(k) given by

SD(k) := X(k) X^∗(k)

N (6.19)

where X^∗(k) is the conjugate complex of X(k). Alternatively, the spectral density can be defined based on the discrete Fourier transform of the autocorrelation function, hence

SD(k) :=

N−1X

s=0

ϕ(s) exp(−i 2πk s/N). (6.20)

The spectral density is depicted in Fig. 6.3b, where the entry zero entry SD(0) is apriori known as (X(0))²/N.

In the discrete spectral density, the frequency coordinates kx, ky and kz and the coordi-nates in the spatial domain nx, ny and nz are related by ki = 2 π ni/Ni where i = x, y, z.

Subsequently, the number of spatial data points Nx,y,z and the number of data points in the frequency domain is the same. The entry in the zero component is deleted from the spectral density and a normalization with respect to the largest entry in the spectral den-sity is computed afterwards. In order to compare the spectral denden-sity of structures with different sizes, i.e. different number of data points Nx,y,z, a rebinning procedure to adjust the resolution must be applied. Thereby, the data set of fine resolution is transferred to a data set of coarse resolution computing mean values of sections of the original set. This is straight forward if the data set size of fine resolution is an integer multiple of the set size of coarse resolution. Then, the mean value of all entries of the set of fine resolution being associated to an entry in the coarsely resolved set is computed. The rebinning procedure in this case is illustrated in Fig. 6.4a. If the size of the finely resolved set is not an integer multiple of the coarsely resolved set’s size, the rebinning must consider the affiliation of one entry in the one grid to two cells in the other grid. Then, the computation of the mean value of the associated entries in the finely resolved set considers a weighting factor which describes the amount of the entry in the currently considered coarse entry. This rebinning case is illustrated in Fig. 6.4b. Rebinning procedures, also called binning procedures, are often found in digital photography. The spectral density is then reordered as indicated in Fig. 6.5. The reordering enables to choose a reduced center section of the reordered spectral density later on to be compared in the optimization process. This section then contains the relevant entries of the spectral density as the values typically abate with increasing distance from the zero component.

The ability of the spectral density to describe periodicity properties is shown with an ex-ample of an artificial, periodic microstructure in 2D, see Fig. 6.6. It shows the microstruc-ture, which is composed of periodically arranged unitcells, one of them is also shown in the

0

Figure 6.3: (a) Autocorrelation function and (b) spectral density of sample pulse.

same figure. When the spectral densities of the two structures are compared, considering the rebinning and reordering operations described above, the similarity of the spectral densities is obvious. This example emphasizes the ability of the spectral density to detect similarities in the microstructural morphology especially in periodic structures.