Power Spectral Density

The Power Spectral Density (PSD) decomposes the surface profile into spatial frequencies.  The PSD is useful for studying the strengths of various periodic components in the surface profile and for comparing these with the strength of the broad spectrum due to the random components.  It is given analytically by

PSD(F)=limit from L to infinity of 1 / L * square of abs(integral from L to 0 of (z(x)*exp(-j*2*pi*F*x*dx))

or in digitized form

PSD(F) = 1/(N*A)*square of abs(sum from i=1 to N of delta z(j)*exp(-i*2*pi*k*j/N))

PSD : Power Spectral Density

delta : the lateral point spacing of the digitized data points

L: The total length of the profile L = N 

F: the set of spatial frequencies


References


T. V. Vorburger, J. Raja,  Surface Finish Metrology Tutorial, NISTIR 89-4088 (National Institute of Standards and Technology, Gaithersburg, MD, 1990)

C. Temperton, Self-sorting mixed-radix fast fourier transforms, Journal of Computational Physics, 52(1):1-23, 1983

E. Marx, I. Malik, Y. E. Strausser, T. Bristow, M. Poduje, J. Stover,  Power Spectral Densities: A Multiple Technique Study of Different Si Wafer Surfaces, J. Vac. Sci. Technol, 20: 31-41, 2002