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SEMI MF1811-0704 © SEMI 2003, 2004 15 Figure 1 Different Forms of the Measurement-Transfer or Instr u men tal-Response Function as a Function of Spatial Frequency, f x . Table 1 Simulated He ight Data ( N = 32) n Z(n) n …

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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.
SEMI MF1811-0704 © SEMI 2003, 2004 15
Figure 1
Different Forms of the Measurement-Transfer or Instrumental-Response Function as a Function of Spatial
Frequency, f
x
.
Table 1 Simulated Height Data (N = 32)
n Z(n) n Z(n) n Z(n) n Z(n)
1 –38 9 –40 17 3 25 –35
2 15 10 45 18 6 26 23
3 36 11 20 19 17 27 4
4 22 12 3 20 20 28 –8
5 29 13 47 21 24 29 –45
6 –43 14 –18 22 16 30 26
7 –1 15 45 23 –5 31 1
8 –5 16 43 24 –17 32 6
Table 2 Periodogram Estimates (m)S
)
for Different
Types of Data Detrending Using a Rectangular
Window
m None Piston Piston+Tilt
Full
Quadratic
1 120.0500 0 0 0
2 205.9506 205.9506 182.5409 14.81781
3 142.1861 142.1861 137.3580 204.3631
4 56.98463 56.98463 44.47469 53.08546
5 147.9039 147.9039 139.6268 126.5831
6 58.60630 58.60630 55.67305 50.13899
7 248.4120 248.4120 264.7417 266.1277
8 152.8321 152.8321 158.6038 154.5160
9 185.1250 185.1250 192.8103 190.0105
10 28.81090 28.81090 26.22179 25.40131
11 28.61438 28.61438 27.90264 28.74841
12 153.1641 153.1641 145.7812 145.2367
13 356.5711 356.5711 366.4113 365.4717
14 199.1965 199.1965 202.6268 203.4514
15 41.53757 41.53757 40.26207 40.50671
16 163.6550 163.6550 168.8428 169.0258
17 16.20000 16.20000 17.87214 17.87215
Table 3 Periodogram Estimates (m)S
)
for Different
Types of Data Detrending Using a Blackman
Window
m None Piston Piston+Tilt
Full
Quadratic
1 312.4632 87.20874 90.11245 9.832211
2 536.6033 268.4217 260.2453 76.63136
3 195.1412 166.9871 162.0176 93.70938
4 1.046170 1.046172 1.041742 1.303189
5 30.14155 30.14156 30.16028 30.25176
6 54.87353 54.87350 54.91950 54.94658
7 188.9816 188.9818 188.8795 188.8293
8 74.45938 74.45934 74.50156 74.51955
9 45.28967 45.28967 45.27541 45.29324
10 59.40473 59.40471 59.41950 59.41210
11 94.24771 94.24768 94.23248 94.23342
12 202.7328 202.7328 202.7405 202.7253
13 289.7414 289.7414 289.7450 289.7637
14 130.1287 130.1287 130.1230 130.1162
15 76.22277 76.22275 76.22527 76.22248
16 62.42836 62.42840 62.42742 62.43071
17 5.585947 5.585948 5.585947 5.584926
SEMI MF1811-0704 © SEMI 2003, 2004 16
Table 4 Values of the Estimates
q
R
)
for Different Types of Detrending
Followed by Different Types of Windowing
Type of Data Detrending
Type of Window
None Piston Piston+Tilt Full Quadratic
Rectangular 26.13517 26.13517 26.05133 25.34362
Hann 25.49031 24.02299 23.91725 22.31003
Hamming 25.53997 24.26072 24.15406 22.58759
Blackman 25.29220 23.38999 23.30221 21.54917
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