**2.2.2 Frequency domain characterization**

Although wood surface description is the main subject here, the potential advantages of frequency analysis has been investigated for metal surface measurement as well as road

Most surface quality analysis including three-dimensional analysis has been traditionally based upon the surface tracing or surface profile. Most analysis of the surface profile generated by the stylus system has been evaluated using time domain parameters such as height deviations and asperity spacing or wavelength. Whitehouse (1982) gives a brief history of the development of surface quality evaluation techniques and the confusion that has developed due to new developments in measurement technology, lack of coordinated efforts between countries, changes in manufacturing processes resulting in different surface textures for a given part, and economic considerations affecting instrumentation development. King and Spedding (1983) discussed three categories of approaches that have

Statistical characterization of the surface profile (magnitude of surface irregularities,

The figure below (figure 3) shows a common example of time domain measurements. The measurements include a measure of the average roughness, Rq (second moment, root mean square), a measure of "extremes" Rtm, a measure of whether the surface defects are above or below the average surface, a measure of skewness, Rsk (third moment), and a measure of the

Sample Length (l)

Although wood surface description is the main subject here, the potential advantages of frequency analysis has been investigated for metal surface measurement as well as road

Rmax

**Rsk** = (1/**Rq**

**Rku** = (1/**Rq**

3 )(1/N)Zi 3

4 )(1/N)Zi 4

**Rq** = [Z1 2 + Z2 2 + Z3 2 + Zi 2 + ... ZN 2 /N]0.5

**Rtm** = mean value of the **Rmax** of five consecutive sample lengths (l)

**2.2.1 Time domain characterization** 

been used to characterize a surface:

etc.)

Surface Height Z (



0

Z1

Z2 Zi

20

in or

m)

 Characterization by process specification (sawing, milling, etc.) Characterization according to function (intended use of workpiece)

shape of the surface defects, Rku kurtosis (4th moment).

<sup>40</sup> Evaluation Length (L)

Fig. 3. Definitions of surface descriptors

**2.2.2 Frequency domain characterization** 

Distance (inch or mm) 0.0 0.5 1.0 1.5 2.0 2.5

Mean or Reference Line

surface measurement. The use of standard surface descriptors based on time domain analysis is sufficient for some applications; however it does not provide information as to the periodicity of the surface characteristics or the nature or cause of the defects. Frequency is most often expressed as cycles per second known as Hertz (Hz). However, frequency can also be expressed spatially such as cycles per unit length (cycles per inch).

As stated by Brock (1983), in the field of signal processing and analysis as applied to sound and vibration problems, the transformation of the signal from the time domain to the frequency domain is very common due to the ease with which the signal can be analyzed and characterized. Although this approach is not common in the field of surface quality analysis, the same benefits can be realized. The main advantage of frequency analysis is that it can reveal the dominant frequency components contained in the transducer signal. Ber and Braun (1968) showed that the frequency spectra resulting from the measurements on surfaces obtained by turning, grinding, and lapping are dissimilar. Raja and Radhakrishnan (1979) separated the roughness from the waviness component on a surface by using fast Fourier transform techniques. Staufert (1979) also used frequency domain analysis to separate periodic components from random components in the surface. In the literature an industry that has tended to use the power spectrum for surface quality analysis is that of road surface evaluation. In an article by Bruscella, Rouillard, and Sek (1999) a laser based optical profilometer was used to obtain a surface profile of the road. Both the time and frequency domains were analyzed.

Work by Lemaster (1997b) has addressed the use of the frequency spectrum of the surface profile to detect "periodicity" within a surface profile. This approach is suitable because a surface profile is often composed of both random and periodic components. Under ideal cutter conditions, the tool produces evenly spaced cutter marks which occur periodically. In cases where the tool is not concentric, out of balance, or the workpiece is not properly held, the marks are unevenly spaced and vary in depth. More random defects often result from the detachment of material from the workpiece. The utility of simple frequency analysis is demonstrated, for several idealized (simulated) examples of surface quality issues relevant to wood machining is discussed below.

Much work has been conducted on using wavelets in filtering or de-composing the surface profile. The category of interest here is the use of wavelets to separate these surface components. Much of the work discussing wavelets as applied to surface roughness are based on analyzing the gray scales of an image of the surface which is beyond the scope of this chapter and will not be discussed here but the reader is referred to Fricout et. al. (2002) for one discussion of this approach. Other works discussing wavelets and surface texture consists of multi-resolution decomposition of the surfaces including separating the error of form, waviness, roughness, and localized defects. Work by Khawaja (2011) demonstrated the insensitivity of the shape of the wavelet in its ability to decompose the components of a surface trace and obtain a standard roughness descriptor. While these works are very important in the complete understanding of surface texture analysis, it was not the main thrust of the topic in this chapter. In fact, the work by Lemaster (2004) found that this use of wavelets did indeed provide a means of removing the form of the surface texture that, in many cases, yielded superior filtering than the traditional phase correct Gaussian filter.

The Use of the Wavelet Transform to Extract

windows (Figure 5).

windows.

**3. Basic joint time / frequency analysis 3.1 The Short Time Fourier Transform (STFT)** 

varied; i.e. rectangular, Hanning, cosine taper, etc.

a manufacturing environment.

perfect reconstruction of the original signal g(t) is still possible.

Additional Information on Surface Quality from Optical Profilometers 107

The FT is very versatile, but is inadequate when one is interested in the "local" (in time or space) frequency content of the signal. A transform method that can analyze non-stationary signals

An obvious method, following on from the FT, is to analyze the time (space) signal over 0-T seconds in a train of shorter intervals such as 0-T/4, T/4-T/2, T/2-3T/4, 3T/4-T, known as

where the frequency information changes with time is required for this type of analysis.

Fig. 5. Short Time Fourier Transform (STFT) with moving non-overlapping rectangular

The individual windows, being only of length T/4 in this case, mean that the lowest frequency fL will be only one-quarter of the full 0-T window value. This method, first described by Gabor (1956), is known as the **Short Time Fourier Transform** (STFT), (see Goswani and Chan (1999) and Qian (2002) for a full discussion). Today, the individual transforms are usually performed using the FFT algorithm where the window shape can be

**0 T/4 T/2 3T/4 T**

**t**

Note in an STFT, as in the FT, the size of the window is fixed but the frequency of the sinusoids that are compared to the signal varies as does the number of oscillations. A small window is unable to detect low frequencies which are too large for the window. If too large a window is used then information about a brief change will be lost. This implies prior knowledge of the signal's characteristics and will become an important criterion for choosing the analysis method. An additional advantage of the non-overlapping STFT is that

A more recent, but slower, method known as the **adaptive Gabor spectrogram** was developed by Qian and Chen (1994) where the time and frequency resolutions are defined by one parameter. Unlike the classical Gabor expansion, where the time and frequency resolutions are fixed, the time and frequency resolutions of the adaptive Gabor expansion can be adjusted optimally. This method while, it would be acceptable for "off-line" surface measurements was not investigated further in this research because of the slower computational times and the desire to have an efficient method that could be used on-line in

#### **2.2.3 Shortcomings of simple time and frequency analysis**

One of the main objectives of developing a surface quality evaluation system was to be able to detect variations in surface quality from time to time which actually may be viewed as discontinuities. Besides detecting if a random or periodic component exists it is also important to determine if the defect is consistent (stationary) or if it changes with time (non-stationary). This can occur in practice from such things as a failure in the feed system or variation in thicknesses of a board being planed. The problem in defining a non-stationary surface is linked to the time frame being observed. A sanding ridge can be considered non-stationary when only a small sample distance is considered (one board), however, if the ridge occurs over numerous boards and all boards are included in the analysis, then the ridge can be considered stationary as far as the process is concerned. Traditional time and frequency analyses cannot distinguish between stationary and non-stationary surfaces. The following section illustrates this shortcoming and discusses some recent developments in **joint time-frequency analysis (JTFA)** that may overcome these shortcomings in surface quality assessment. Figure 4 illustrates the difficulty or shortcomings of traditional frequency analysis. Two significantly different surface profiles can result in similar frequency spectra.

These two examples show the weakness of traditional frequency analysis in the current descriptions of wood surface applications. Though both signals have a similar frequency spectrum, one signal is non-stationary (top – left) where the other one (lower – left) is stationary. This illustrates a need for a more advanced form of frequency analysis.

Fig. 4. Two types of signals that have similar frequency spectra
