**8. Spectral analysis**

140 Woven Fabrics

Waviness

am

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) leads to the increase of PC. Lowest values of PC are around *ym* equal to the 0.5. This behavior is "inverse" to the behavior of MAD and SD. The PC parameter is useful in tribological applications. The lower the slope the smaller will be the friction and wear. Also, the reflectance property of a surface

The *MAD* and *PC* characteristics for all slices for cord fabric (see fig. 11) are shown in

a) b)

For the characterization of hand, it will be probably the best to use waviness *PC*. The characteristics of slope and curvature can be computed for the case of fractal surfaces from

In the case of MAD, a systematic trend is visible. The variation of PC is nearly random.

Fig. 24. The a) MAD and b) PC values for all slices of cord fabric (fig. 11)

power spectral density, autocorrelation function or variogram.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1


ym

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 23. Influence of *am* and *ym* on the PC

increases in the case of small PC.

the fig. 24.

The primary tool for evaluation of periodicities is expressing of signal *R(d)* by the Fourier series of sine and cosine wave. It is known that periodic function given by equally spaced values *Ri, i = 0, ..., N – 1* can be generally expressed in the form of Fourier series at Fourier frequencies *fj = j/N, 1 ≤ j ≤ [N/2]*. If *N* is odd with *N = 2m + 1*, the Fourier series has form (Quinn Hannan, 2001)

$$R\_i = a\_0 + \sum\_{k=1}^{m} \left( a\_k \cos(o\_k \ i) + b\_k \sin(o\_k \ i) \right) \quad i = 0, \ldots N - 1 \tag{18}$$

where 2 2 / 1,.. *k k f k Nk m k = 1, ..., m* are angular frequencies. The eqn. (17) is for known frequencies harmonic linear regression model with *2m + 1* parameters (intercept and *2m* sinusoids amplitudes at the *m* Fourier frequencies). The sinusoid with the *j-th* Fourier frequency completes exactly *j* cycles in the span of the data. Due to selection of Fourier frequencies all regressors (*sin(.)* and *cos(.)* terms) are mutually orthogonal, so that standard least-squares method leads to estimates 0*a R* and

$$a\_k = \frac{2\sum\_{i=0}^{N-1} R\_i \cdot \cos(o\_k \cdot i)}{N} \quad b\_k = \frac{2\sum\_{i=0}^{N-1} R\_i \cdot \sin(o\_k \cdot i)}{N} \quad k = 1, \ldots, m\tag{19}$$

Basic statistical characteristic in the frequency domain is power spectral density PSD defined as Fourier transform of covariance function.

The simple estimator of power spectral density is called periodogram. The periodogram of an equally spaced series *Ri, i = 0, ..., N – 1* is defined by equation

$$I(o\nu) = \frac{1}{N} \left(\sum\_{i=0}^{N-1} R\_i \cdot \cos(o\nu \text{ i})\right)^2 + \frac{1}{N} \left(\sum\_{i=0}^{N-1} R\_i \cdot \sin(o\nu \text{ i})\right)^2 \tag{20}$$

and can be expressed in the alternate form

$$I(o\_k) = \frac{N}{4}(a\_k^2 + b\_k^2) \quad k = 1 \dots m \tag{21}$$

For rough SHV from fig. 4 periodogram is shown in fig. 25.

The periodogram ordinates correspond to analysis of variance decomposition into *m*  orthogonal terms with *2* degrees of freedom each because,

$$\sum\_{k=1}^{m} I(o\_k) = 0.5 \sum\_{i=0}^{N-1} \left( R\_i - \overline{R} \right)^2 \tag{22}$$

The normalized periodogram with ordinates

Woven Fabrics Surface Quantification 143

The well-known trigonometric identity *cos (t- s) = (cos t)(cos s) + (sin t)(sin s)* allows to write

cos( ) sin( ) cos( ) *k k k k k kk a tb t A t*

The influence of teeth profile parameters *am* and *ym* on the amplitude *A1* of most important

Fourier amplitude 1

am

<sup>1</sup> of most important Fourier term are shown on the fig. 28.

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) leads to the decrease of the amplitude *A1*. Highest values of the amplitude *A1* are around *ym* equal to the 0.5 (similar behaviour as in the case of MAD). The influence of teeth profile parameters *am* and *ym* on

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) have the small

The periodogram is unbiased only in case of Gaussian noise. The variance of periodogram

<sup>2</sup> sin( ) ( ( )) ( ) 1 sin( ) *<sup>N</sup> DI I*

 

*N*

<sup>1</sup> The influence of *ym* on the phase

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

 

 

*b a* . Coefficients *Ak* creates amplitude spectrum and

(24)

0.1

<sup>1</sup> is more important with

(25)

2

0.2

0.3

0.4

0.5

*<sup>k</sup>* creates phase spectrum.

each paired sinusoid term as

ym

0.1

the phase

influence on the phase

minimum at *ym* = 0.5.

Fig. 27. Influence of *am* and *ym* on the *A1*

does not decrease with increasing *N* and has the form

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

coefficients

with 2 2 *A ab k kk* and tan( ) / *k kk*

Fourier term are shown on the fig. 27.

Fig. 25. Periodogram for twill fabric

$$I\{o\_k\} / \sum\_k I\{o\_k\} = A\_k \,/\sum\_i (R\_i - \overline{R})^2 \tag{23}$$

is then simply interpretable. The *k-th* ordinate gives the proportion of the total variation due to sinusoidal oscillation at the *k-th* Fourier frequency, and thus is a partial correlation coefficient *R2*. The so called scree plot is in fact dependence of relative contribution to the total variance from individual Fourier frequencies arranged according to their importance. For rough SHV from fig. 4, scree plot is shown in fig. 26.

Fig. 26. Scree plot for twill fabric

FFT periodogram

0 0.5 1 1.5 2 2.5 3 3.5

angular frequency [-]

<sup>2</sup> ( )/ ( ) / ( ) *k kk i k i I I A RR*

is then simply interpretable. The *k-th* ordinate gives the proportion of the total variation due to sinusoidal oscillation at the *k-th* Fourier frequency, and thus is a partial correlation coefficient *R2*. The so called scree plot is in fact dependence of relative contribution to the total variance from individual Fourier frequencies arranged according to their importance.

Scree plot

0 2 4 6 8 10

index [-]

(23)

 

For rough SHV from fig. 4, scree plot is shown in fig. 26.

0

0

Fig. 26. Scree plot for twill fabric

5

10

15

rel. periodogram [%]

20

25

30

35

Fig. 25. Periodogram for twill fabric

0.05

0.1

0.15

periodogram [-]

0.2

0.25

0.3

0.35

The well-known trigonometric identity *cos (t- s) = (cos t)(cos s) + (sin t)(sin s)* allows to write each paired sinusoid term as

$$a\_k \cos(\alpha\_k \ t) + b\_k \sin(\alpha\_k \ t) = A\_k \cos(\alpha\_k \ t - \phi\_k) \tag{24}$$

with 2 2 *A ab k kk* and tan( ) / *k kk b a* . Coefficients *Ak* creates amplitude spectrum and coefficients *<sup>k</sup>* creates phase spectrum.

The influence of teeth profile parameters *am* and *ym* on the amplitude *A1* of most important Fourier term are shown on the fig. 27.

### Fourier amplitude 1

Fig. 27. Influence of *am* and *ym* on the *A1*

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) leads to the decrease of the amplitude *A1*. Highest values of the amplitude *A1* are around *ym* equal to the 0.5 (similar behaviour as in the case of MAD). The influence of teeth profile parameters *am* and *ym* on the phase <sup>1</sup> of most important Fourier term are shown on the fig. 28.

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) have the small influence on the phase <sup>1</sup> The influence of *ym* on the phase <sup>1</sup> is more important with minimum at *ym* = 0.5.

The periodogram is unbiased only in case of Gaussian noise. The variance of periodogram does not decrease with increasing *N* and has the form

$$D(I(o)) \approx I^2(o) \left[ 1 + \left( \frac{\sin(o \text{ N})}{N \cdot \sin(o)} \right)^2 \right] \tag{25}$$

Woven Fabrics Surface Quantification 145

Waviness component

0 100 200 300 400 500

0 100 200 300 400 500

index

2 22 *g DRF conj DRF T abs DRF T* ( ) ( )/ ( ) /

contains values corresponding to contribution of each frequency to the total variance of *R*. The periodogram and power spectral density are primary tool for evaluation of periodicities.

repeated pattern and height corresponds to the nonuniformity of this pattern.

(27)

*)* is estimator of spectral density function and

graphs is corresponding to the length of

index

Noise component

Fig. 29. SHV component corresponding to waviness for twill fabric

Fig. 30. SHV component corresponding to noise for twill fabric

Frequency of global maximum on the *I g* ( ) or ( )

where conj(.) denotes conjugate vector. The *g(*

Vector *DRF* may be used for creation of power spectral density (PSD)

0

0

0.02

0.04

0.06

SHV trace

0.08

0.1

0.12

0.002 0.004 0.006 0.008 0.01 0.012 0.014

SHV trace

In some cases, it is useful to express Fourier series in the complex exponential form (Quinn Hannan, 2001).

$$R\_i = \sum\_{k=1}^{m} \left( \mathbb{C}\_k \cdot \exp(o\_k \text{ i } j) \right) \text{ i } i = 0 \text{ \dots N - 1} \tag{26}$$

where *j* is imaginary unit and complex coefficients *Ck* have real and imaginary part *C j kk k* Re Im . The values *Ck* creates the complex discrete spectrum.

Fig. 28. Influence of *am* and *ym* on the 1

For discrete data the Fast Fourier Transform (FFT) leads to transformed complex vector *DRF.* The vector *DRF* can be decomposed to the high and low frequency component. After back transformation into original, the SHV part corresponding to noise (high frequencies) and to waviness (low frequencies) can be separated.

For rough SHV from fig. 4 SHV component is corresponding to waviness in fig. 29 and SHV component corresponding to noise is in fig. 30. The number of high frequency components equal to 20 was selected.

Some other techniques for separation of roughness, waviness and form are based on smoothing or digital filtering (Raja et. al, 2002). The smoothing by neural network can be used as well. For estimation of smoothing degree, the minimization of the mean error of prediction is usually applied (Meloun Militký, 2011).

In some cases, it is useful to express Fourier series in the complex exponential form (Quinn

exp( ) 0,.. - 1

Fourie phase 1

0

ym -1.5

0

For discrete data the Fast Fourier Transform (FFT) leads to transformed complex vector *DRF.* The vector *DRF* can be decomposed to the high and low frequency component. After back transformation into original, the SHV part corresponding to noise (high frequencies)

For rough SHV from fig. 4 SHV component is corresponding to waviness in fig. 29 and SHV component corresponding to noise is in fig. 30. The number of high frequency components

Some other techniques for separation of roughness, waviness and form are based on smoothing or digital filtering (Raja et. al, 2002). The smoothing by neural network can be used as well. For estimation of smoothing degree, the minimization of the mean error of

0.2

0.4

1

0.6

0.8

and to waviness (low frequencies) can be separated.

prediction is usually applied (Meloun Militký, 2011).

1 -2

Fig. 28. Influence of *am* and *ym* on the

equal to 20 was selected.

0

2

0.2 0.4 0.6 0.8 1

am



0

0.5

1

(26)

*R C ij i N* 

where *j* is imaginary unit and complex coefficients *Ck* have real and imaginary part

1

*C j kk k* Re Im . The values *Ck* creates the complex discrete spectrum.

*k*

*i kk*

*m*

Hannan, 2001).

Fig. 29. SHV component corresponding to waviness for twill fabric

Fig. 30. SHV component corresponding to noise for twill fabric

Vector *DRF* may be used for creation of power spectral density (PSD)

$$\text{g(o)} = DRF \text{ conj(DRF)} / \text{T}^2 = \text{abs(DRF)}^2 / \text{T}^2 \tag{27}$$

where conj(.) denotes conjugate vector. The *g()* is estimator of spectral density function and contains values corresponding to contribution of each frequency to the total variance of *R*.

The periodogram and power spectral density are primary tool for evaluation of periodicities. Frequency of global maximum on the *I g* ( ) or ( ) graphs is corresponding to the length of repeated pattern and height corresponds to the nonuniformity of this pattern.

Woven Fabrics Surface Quantification 147

The computation of sample autocorrelation directly from definition for large data is tedious.

<sup>1</sup> ( ) ( ) exp( ) <sup>2</sup> *g C t i t dt*

These relations show that characteristics in the space and frequency domain are

ACF

0 50 100 150 200 250 300

The dashed lines in fig. 31 are the approximate 95% confidence limits of the autocorrelation function of an IID process of the same length. Sample autocorrelations lying outside the 95% confidence intervals of an IID process are marked by black circles. The slow decrease of ACF for large lags indicates long-range correlation, which may be due to non-stationarity and/or

In spatial statistics variogram is more frequent (Kulatilake et. al, 1998) which is defined as

The variogram is relatively simpler to calculate and assumes a weaker model of statistical stationarity, than the power spectrum. Several estimators have been suggested for the

( ) 0.5 [ ( ) ( )] *h DRd Rd h* (34)

Lag

 

 

(32)

(33)

0

0 *ACF h S i h d* ( ) ( ) exp( ) 

The spectral density is the Fourier transform of covariance function *C(h)*

For rough SHV from fig. 4 is *ACF* till lag 320 component in the fig. 31.

The ACF is inverse Fourier transform of spectral density.

interchangeable.


Fig. 31. ACF for twill fabric

dynamic non-linearity.

one half variance of differences (*R*(d) - *R*(d+h))

variogram. The traditional estimator is

0

Autocorrelation function

0.5

Spectral density function is therefore generally useful for evaluation of hidden periodicities. The statistical geometry of an isotropic random Gaussian surface could be expressed in the terms of the moment of power spectral function called spectral moments (Zhang Gopalakrishnan, 1996).

$$m\_k = \int\_{a\_{\mathcal{H}}}^{a\_l} \alpha^k \, \mathbf{g}(\alpha) d\alpha \tag{28}$$

The *mo* is equal to the variance oh heights and *m2* is equal to the variance of slopes between bound frequencies. The frequencies *H* and *<sup>L</sup>* are high and low frequency bounds of integration of the spectrogram. These frequency bound can be converted to the wavelength limits. The long wavelength limit is *lH H* 2 / and the short wavelength limit is *l L L* 2 / . For rough SHV from fig. 4, the selected spectral moments have the following values:


Corresponding spectral statistical characteristics are:


The roughness *Rq* = SD (standard deviation) is simply *Rq m* <sup>0</sup> and the density of summits is defined as

$$DS = \frac{m\_4}{m\_2 \text{ é } \pi \sqrt{3}}\tag{29}$$
