4. UNIFORM SAMPLING
4.3 Selection of sampling length
4.3.3 Sampling length determination
It is known that periodic signals have a DFT expressed by equation (4.13) and most structured surfaces fall into this type of signal. Equation (4.23) gives a necessary condition to avoid spectral leakage in DFT/FFT computation; but this is an insufficient condition. Improper selection of the sampling length would produce a leaked spectrum which cannot be correctly sampled with any frequency sampling resolution. An illustrative example can be seen in Figure 4.9.
With a proper sampling length, the sampled spectrum can be the same as the original (see Figure 4.10 for example). As described by equation (4.13), a periodic surface
2 2
( ) ( )
f x L with period T 2 has a discrete spectrum only at the positions
2
101
has a sinc like spectrum with zero values at 2
{k k l k/ , }. Therefore, it is possible that the original discrete frequency spectrum can be retained if
,
lZT Z , (4.24)
where Z is usually taken as Z10 with reference to the stability of statistics [69]. However, leakage cannot essentially be avoided in this way. The only benefit is that such an integer period sampling produces a frequency sampling result only at the un- leaked frequency positions (see Figure 4.10 for a schematic).
The sampling technique which asks that sampling length is an integral multiple of the signal period is called integral period sampling. A coherent sampling technique has been proposed to eliminate spectral leakage [19, 30], which is originated from the integral period sampling. The coherent sampling states that
signal sampling
f M
f N . (4.25)
where fsignal and fsampling are the input signal frequency and sampling frequency, M
is the number of sampled cycles and N is the number of total sampling points (M N, ). The equation (4.25) indicates that the sampling length is an integral multiple of the signal period, i.e.
sampling signal
lN T M T . (4.26)
In this case, there is no spectral leakage when a rectangular window is used [19]. The effect of retaining the original discrete frequency spectrum is presented in Figure 4.10. In comparison with Figure 4.9, where the sampling length is a non-integral multiple of the period of original signal, the integral period sampling shows an excellent performance regarding the avoidance of spectral leakage.
102 (a)
(b)
(c)
(d)
Figure 4.9. Spectral leakage caused by non-integer period sampling (a) Partial discrete frequency spectrum of a periodic signal; (b) the spectrum of a sampling window with 2.5 times period length; (c) thus the spectrum is leaked because of the sampling truncation based
on equation (4.16); (d) The frequency sampling result (red dots) based on DFT/FFT computation using equation (4.23).
Figure 4.10. The effect of integral period sampling. (a) Partial discrete frequency spectrum of a periodic signal; (b) The spectrum of a sampling window with three times period length; (c)
The spectrum is leaked because of the sampling truncation based on equation
Comparison of the leaked spectrum (blue curve) and the frequency sampling result (red dots) based on FFT/DFT computation using equation
103 (a)
(b)
(c)
(d)
. The effect of integral period sampling. (a) Partial discrete frequency spectrum of a periodic signal; (b) The spectrum of a sampling window with three times period length; (c)
The spectrum is leaked because of the sampling truncation based on equation
Comparison of the leaked spectrum (blue curve) and the frequency sampling result (red dots) based on FFT/DFT computation using equation (4.23).
. The effect of integral period sampling. (a) Partial discrete frequency spectrum of a periodic signal; (b) The spectrum of a sampling window with three times period length; (c)
The spectrum is leaked because of the sampling truncation based on equation (4.16); (d) Comparison of the leaked spectrum (blue curve) and the frequency sampling result (red dots)
Integral period sampling also shows excellent performance in nois Taking a noisy sinusoidal profile as an example:
( ) cos(2 /10) ( )
x n n r n
in which
r n( )
is a normally distributed random noise sampling length l1=100 andin Figure 4.11. Another example of analyzing a Figure 4.12. The surface is simulated as
( , ) cos(2
/ 5)cos(2
/ 5)
( , )
f x y
=
x
y
+r x y
where
r∼N(1,0.125)
. Sampling areascalculated. It is observed that the sampling length period of the original surface,
Figure 4.11. Power spectrum of the noised sinusoidal profile expressed by equation with the sampling length (a
104
sampling also shows excellent performance in noisy signal analysis. sinusoidal profile as an example:
( ) cos(2 /10) ( ) x n =
π
n +r n ,is a normally distributed random noise
r∼N(1,0.125)
and l2 =128, the corresponding power spectrum is presented . Another example of analyzing a noisy sinusoidal surface is shown in
is simulated as
( , ) cos(2
/ 5)cos(2
/ 5)
( , )
f x y
=
πx
πy
+r x y
,Sampling areas of 50 µm × 50 µm and 52 µm ×
calculated. It is observed that the sampling length, which is an integral multiple of the , presents an accurate spectrum, without spectral leakage.
(a)
(b)
. Power spectrum of the noised sinusoidal profile expressed by equation with the sampling length (a) 100 points and (b) 128 points.
signal analysis.
(4.27)
(1,0.125)
. Adopting , the corresponding power spectrum is presentedsinusoidal surface is shown in
(4.28)
× 52 µm are integral multiple of the presents an accurate spectrum, without spectral leakage.
105 (a)
(b)
Figure 4.12. Power spectrum of the noised sinusoidal surface expressed by equation (4.28) with sampling length (a) 50 µm × 50 µm (b) 52 µm × 52 µm.
106
(a) N = 100, = 0 (good) (b) N = 101, = 0.1(good)
(c) N = 102, = 0.2 (slightly blurred) (d) N = 103, = 0.3 (slightly blurred)
(e) N = 104, = 0.4 (blurred) (f) N = 105, = 0.5 (blurred)
(g) N = 106, = -0.4 (blurred) (h) N = 107, = -0.3 (blurred)
(i) N = 108, = -0.2 (slightly blurred) (j) N = 109, = -0.1 (slightly blurred)
(k) N = 110, = 0 (good)
Figure 4.13. Comparison of the DFT spectrums with the sampling length from 100 to 110, i.e. M from ten to eleven, for the signal expressed by equation (4.27).
107
In practice, an exact integer-period sampling is usually difficult to manipulate, i.e. M in equation (4.24) is usually not an exact integer, because the period of a real signal is usually unknown or has a discrepancy from an expected value due to manufacturing error. However, the spectral leakage can still be minimised by controlling Z close to an integer. For example, if test the DFT computation of the signal (4.27) using sampling lengths from 100 (Z = 10) to 110 (Z = 11), the spectrums can sequentially be obtained in Figure 4.13. It can be observed that the spectra gradually become distorted as Z becomes further away from its nearest integer. Therefore, the following conclusion for the determination of sampling length/area for practical measurement can be obtained
( ) ,
l Z T Z , (4.29)
where Z is usually taken as ten or more, has not been decided at the moment but the evidence in Figure 4.13 indicates that [0,0.2) is acceptable. More research on this topic should validate this conclusion.