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SEMI MF1811-0704 © SEMI 2003, 2004 13 5.5 Periodogram Estimators o f the Rms Profile Roughness and S lope 5.5.1 The periodogram est i mator of the rm s profile height, , q R ) is as follows: ∑ + = = ) 2 / ( 1 2 1 ) ( 1 N…

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5.3 RMS Slope
5.3.1 There are two different estimators for the rms
slope, ∆
q
— one expressed in configuration space, and
the other in frequency space, as follows:
∑
−
=
−+=∆
1
1
2
2
)()1([
1
)Config(
N
n
q
ndZndZ
ND
)))
(29)
and
2
)2/(1
1
1
2
)1(2
)(
1
)Freq(
−π
=∆
∑
+
=
ND
m
mS
ND
N
m
q
)
)
(30)
NOTE 8: The magnitudes of these two estimates are
generally different since the first treats the profile as a
collection of straight-line segments connecting the
measurement points, while the second connects them with a
bandwidth-limited interpolation curve and involves a smaller
bias error.
NOTE 9: The notes for the rms roughness estimators just
made generally apply to slope estimates as well. One
difference is that the significant bandwidth effects on the
slope occur principally at the HFL rather than the LFL as is
the case for rms roughness measurements.
5.4 Periodogram Estimators of the Profile Power
Spectral Density
5.4.1 Form for 1-dimensional Power Spectral Density
for height-measuring profilometers is as follows:
.
)()(FFT2
)(
2
1
N
mKmD
mS
⋅⋅
=
)
(31)
The spatial frequency is evaluated at the discrete
valuesof m, with m = 1, 2, … , [1 + (N/2)]:
.
1
ND
m
f
x
−
= (32)
5.4.2 The symbol FFT in Equation 31 stands for
discrete Fourier transform, which is always evaluated
using some version of the fast Fourier transform as
follows:
∑
=
⋅⋅−−π=
N
n
ndZnWNmnim
1
)()(]/)1)(1(2exp[)(FFT
)
(33)
where:
W(n) = window function (see Section 5.6) and
K(m) = book-keeping factor equal to ½ for m = 1 o
r
m = 1 + (N/2) and equal to 1 otherwise.
NOTE 10: Equation 33 applies for the conventional case of
even N. Different forms apply for odd N.
NOTE 11: The case m = 1 corresponds to the zero-frequency
or dc component of the detrended surface profile and m = 2
corresponds to the spatial frequency 1/(ND), which is
essentially the reciprocal of the trace length, ( N–1)D. On the
opposite extreme, the frequency corresponding to m = 1 + N/2
is the Nyquist frequency, 1/(2D). The extreme range of
surface wavelengths included in the measurement is therefore
1/(ND) < f < 1/(2D). In other words, the extreme LFL =
1/(ND), the extreme HFL = 1/(2D), and the dynamic range of
the measurement is N/2.
NOTE 12: A convenient and readable reference to the FFT
and its evaluation is Chapter 12 in Numerical Recipes by
Press, Flannery, Teukolsky, and Vetterling (see Section 7.1).
NOTE 13: The Brookhaven National Laboratory Report (see
Section 7.2) contains further background information on these
procedures along with a computer program and numerical
examples. (The BASIC routines used there involve different
forms for the quantities M
P
appearing in the expressions for
the detrending polynomials than those discussed in this guide,
although the numerical values of the detrending polynomials
are identical in both cases.)
NOTE 14: The periodogram estimator just given is not the
only method of estimating the power spectral density from a
set of profile data, but it is the most direct and common
method. It is sufficient for general use, and is a necessary
first step to be taken before adding embellishments such as
post-processing or considering more complicated estimators.
This guide does not exclude the use of post-processing or
alternative methods of analysis, but does require that the basic
periodogram estimates just described be included in the
discussion for comparative purposes.
5.4.3 Form for 1-dimensional Power Spectral Density
for slope-measuring profilometers is as follows:
N
mKmD
ND
m
mS
)()(FFT2
)1(2
)(
2
2
1
⋅⋅
⋅
−π
=
−
)
(34)
where:
∑
=
−−π
⋅⋅=
′
N
n
N
mni
ndMnWem
1
)1)(1(2
)()()(TFF
)
and )(ndM
)
is the value of the profile slope measure-
ments detrended using either the least-squares piston or
the piston-plus-tilt expressions (see Section 5.1).
NOTE 15: This estimate of the profile power spectrum is the
power spectrum of the profile slope divided by (2πf
m
)
2
. The
prime on the FFT on the left denotes that it involves slope
rather than height data.
NOTE 16: The case m = 1 corresponds to zero spatial fre-
quency and must be excluded in the use of the above
expressions.
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5.5 Periodogram Estimators of the Rms Profile
Roughness and Slope
5.5.1 The periodogram estimator of the rms profile
height, ,
q
R
)
is as follows:
∑
+
=
=
)2/(1
2
1
)(
1
N
m
q
mS
ND
R
)
)
(35)
This quantity has the same dimension as the original
height measurements and is independent of the
magnitude and dimensions of the sampling interval, D.
5.5.2 The corresponding estimator for the rms profile
slope, ,
q
∆
)
is as follows:
∑∑
+
=
+
=
−π=
′
=∆
)2/(1
2
1
2
3
)2/(1
2
1
)()1(
)(
1
2)(
1
N
m
N
m
q
mSm
ND
mS
ND
))
)
(36)
In evaluating these quantities the height measurements,
Z, and the sampling interval, D, must be expressed in
the same length units. Although
q
∆
)
is in units of
radians, its magnitude scales as 1/D.
5.6 Window Functions
5.6.1 Window functions appear in a wide variety of
signal-processing applications, with different shapes
and normalizations. Although this guide recommends
the use of the Hann or Blackman window, other forms
are included for comparison. All are normalized so that
∑
=
=
N
n
nW
N
1
2
1)(
1
in order to preserve the magnitudes of
average values of the mean-square profile statistics.
5.6.2 Particular forms are:
5.6.2.1 Rectangular or Daniell window:
1)( =nW (37)
5.6.2.2 Hann, or Raised Cosine window:
−π
−=
N
n
nW
)1(2
cos124
1728
2
)(
(38)
5.6.2.3 Hamming window:
−π
−⋅=
N
n
nW
)1(2
cos2327
1987
2
)(
(39)
5.6.2.4 Blackman window:
−π
+
−π
−⋅=
N
n
N
n
nW
)1(4
cos4
)1(2
cos2521
1523
2
)(
(40)
NOTE 17: The choice of window functions is of minor
importance for randomly-rough surfaces as long as it
smoothes the data at the ends of the data record. The
rectangular or Daniell window does not do this, but is useful
for numerical checking.
NOTE 18: In the case of profiles with a smooth PSD, the
principal effect the window shape is to change the fine-scale
fluctuations in the periodogram estimate without changing its
ensemble-average value, except, perhaps, near the LFL.
NOTE 19: In the case of profiles involving periodicities, the
window shape can change the shape of the sharp lines in the
PSD, albeit without changing their areas. The choice of the
window shape then involves a trade-off between line width
and smoothness. The raised Hann or Blackman windows are
recommended for general use.
NOTE 20: If the estimation routines are applied to
deterministic profiles, such as individual steps, pits, or bumps,
a data window must still be used to minimize effects of the
finite data record, but the object should be placed in the center
of the profile where the window function is relatively flat.
5.7 Zero Padding
5.7.1 The fastest FFT routines require the total number
of data points to be a power of two, such as N = 2
10
=
1024. If the number of measured points, N, is not a
power of two but lies between 2
a
and 2
b
, the power-of-
two routines can be used by dropping N−2
a
points from
one end of the original data set, or by adding 2
b
−N
zeros and replacing N in the routines everywhere by 2
b
.
5.7.2 The first method is wasteful of data, while the
second uses the full set of measured data but requires
that the estimated PSD be renormalized by multiplying
it by the factor 2
b
/N.
5.8 Averaging of Statistical Quantities:
5.8.1 Power spectral density functions, the mean-
square roughness, and slope values estimated from a
number of individual profiles that have the same
statistical properties can each be averaged together to
obtain composite results. In the case of homogeneously
and isotropically rough surfaces the profiles can lie in
any position and direction on the surface under test. In
the case of homogeneously but anisotropically rough
surfaces they can lie anywhere on the surface but must
lie parallel with each other, preferably perpendicular to
the surface axis. Averaging data lowers the errors
associated with individual measurements.
6 Numerical Test Sequences
6.1 Table 1 presents a set of numerical data for testing
the execution of the users’ implementations of
algorithms discussed in Section 5.
6.2 Although these simulated profile data are in
standard notation, they have been generated by a
SEMI MF1811-0704 © SEMI 2003, 2004 14
random number generator corresponding to a constant
PSD and are not the results of an actual measurement.
In addition, the number of data points has been limited
to N = 32 and the profile heights have been rounded to
digits with magnitudes less than 100 to simplify their
manual input into the users’ programs.
6.3 Actual measured data sets would generally involve
many more data points with height values involving a
larger number of significant digits, and with different
orders of magnitude than those used in this test
sequence.
6.4 In order to provide a means for checking the proper
inclusion of the sampling distance, D, in the spectral-
estimation routines, the value D = 0.1 has been used.
6.5 Tables 2 and 3 give the values of the periodogram
estimates of the profile power spectral density,
,)(
1
mS
)
of the data in Table 1 for the three different types of
detrending described in Section 5.1. The values of
these estimates depend on the data window used. Table
2 uses a rectangular window and Table 3 uses the
Blackman window.
6.6 The dimensions of the power spectral densities in
these tables is length-cubed = (units of Z)
2
·(units of D),
and its magnitude at a given spatial frequency scales as
the sampling interval, D.
6.7 The spatial frequency is given as follows:
ND
m
f
m
1−
= (41)
where:
m = 1 corresponds to the dc or piston part of the
profile, and
m = 1 + N/2 = 17 is the Nyquist frequency in
(units of D
–1
).
6.7.1 Note that since the window function has been
applied after the detrending process, the dc terms do not
necessarily vanish for a non-rectangular window
functions.
6.8 Table 4 gives values of
q
R
)
derived from the
spectra in Table 2 and Table 3 using the expression
given in Table 4.
6.8.1 As mentioned, the unit of
q
R
)
is the same as that
of the height measurement since the magnitude and
dimensions of the sampling interval, D, cancels out in
the evaluation of R
q
.
6.9 The data in Table 2 through Table 4 are adequate
for checking the users' implementation of the estimators
described in Section 5, and further test data are not
included in this guide.
7 Related Documents
7.1 Press, W. H., Flannery, B. P., Teukolsky, S. A., and
Vetterling, W. T., Numerical Recipes – the Art of
Scientific Computing, Cambridge University Press,
Cambridge, 1986.
7.2 Church, E. L., and Takacs, P. Z., “BASIC program
for power spectrum estimation”, Brookhaven National
Laboratory Report BNL No. 49035, May 1993 (revised
May 1994).
7.3 Stover, J. C., Optical Scattering: Measurement and
Analysis, SPIE Press, l995.
7.4 Kay, S. M. Modern Spectral Estimation;Theory
and Application, Prentice Hall, l988.
7.5 Marple, Jr., S. L., Digital Spectral Analysis with
Applications (Prentice Hall, 1987).
7.6 Harris, F. J., “On the Use of Windows for
Harmonic Analysis With the Discrete Fourier
Transform,” Proceedings IEEE 66, 51–83 (1978).
7.7 Oppenheim, A. V., and Schafer, R. W., Digital
Signal Processing, Chapter 11, Prentice Hall, l975.
7.8 Bendat, J. S., and Piersol, A. G., Random Data;
Analysis and Measurement, Chapter 9 (Wiley, l971).
7.9 Church, E. L., Vorburger, T. V., and Wyant, J. C.,
“Direct Comparison of Mechanical and Optical
Measurements of the Finish of Precision Machined and
Optical Surfaces,” Optical Engineering, 24, 388–395
(1985).
7.10 Blackman, R. B., and Tukey, J. W., The
Measurement of Power Spectra (Dover, l959).
8 Keywords
8.1 estimates; estimators; power spectral density; rms
values; root mean square; roughness; slope; surface
roughness; surface slope; surface statistics.