Civil Engineering Reference
In-Depth Information
the other profile values displaced by a constant lag [67]. The power spectrum is the frequency domain
counterpart of the autocorrelation function [67].
The autocorrelation function and power spectrum have been widely applied to qualitatively distinguish
among different profiles and to monitor signals in manufacturin'g processes [25, 43, 72, 73]. A large
number of the applications of the spectral-based methods can be found in the fault monitoring and
detection literature. The field of fault monitoring and detection concentrates on finding suitable signal
processing methods to extract relevant fault-indicating features in manufacturing, such as the errors on
manufactured surfaces. The traditional method of feature extraction relies on the use of the Fourier
transform (FT) to decompose a given signal into its different frequency components [8, 20, 28, 37, 41,
64, 74]. The power spectrum analysis based on the Fourier transform is well understood, and is a
commonly available tool in commercial signal analyzers. However, the Fourier spectrum method has
many limitations that result in an inaccurate picture of the signal decomposition [56, 75], especially
when nonstationary features are present. Any difference in the nature of the data function is smoothed
out because the total signal is encompassed under the integral sign, creating an averaging effect [75].
Since the power spectrum can fail to provide an accurate picture of the correct frequency decomposition,
other methods are commonly used in conjunction with the power spectrum, namely time-domain
representations from common random process analysis methods, such as correlation functions, time-domain
averages, etc. [37]. A time-domain version of the power spectrum, namely the cepstrum, which is the
spectrum of the logarithm of the spectrum of the signal [37, 64], is also sometimes proposed as an aid
to Fourier transform. Its main characteristic is that any periodically occurring peaks in the spectrum will
appear as a single peak in the cepstrum, thus easily identifiable [53]. Finally, the coherence function, a
dimensionless version of the power spectrum, is also used by some researchers [15, 79]. However, these
methods do not always help in providing a more accurate picture of the fault condition [74]. Furthermore,
they add to the large number of tools proposed for use on the production floor, which makes their
acceptance in practice more difficult.
Advanced Mathematical Transforms as Alternatives to the Fourier Transform
The theory of random process analysis is now well established in the field of surface characterization,
but it has yet to make an impact in industry [74]. As discussed above, this is due to several factors: the
difficulty in interpreting the results; the tendency to obtain misleading or incomplete results due to
time-varying or localized surface effects; and the inability to identify when and where exactly the errors
take place.
To overcome the above-mentioned shortcomings of the Fourier transform, especially in representing
time-varying signals, methods that parameterize time, in addition to frequency, are researched as alter-
natives to the common random process analysis methods. In general, the aim of signal analysis is to
extract relevant information from a signal by transforming it [48]. The Fourier transform is such a tool,
resulting in a decomposition into different frequency components based on sines and cosines as basis
vectors [8, 28, 74]. However, there exist many other transforms that researchers have explored with the
purpose of providing a clearer picture of the signal signature.
The most popular alternative of the Fourier transform is a windowed version of the power spectrum,
the short-time Fourier transform (STFT) [74, 77]. The STFT gives a picture of the frequency decompo-
sition of the signal in different time windows, and hence presumably can analyze time-varying signals.
However, stationarity still has to be assumed within the short time window, which causes a problem with
the frequency/time resolution tradeoff [75].
Other attractive alternatives to the Fourier transform are the Wigner-Ville transform, the wavelet
transform, and the higher-order spectral transform. The Wigner-Ville transform has been previously
used in analyzing manufacturing signals, including surface signals [46, 49, 75]. However, this trans-
formation typically introduces redundant features when dealing with multi-component signals. This
redundancy tends to obscure the significant fault features. The wavelet transform has been used in a
number of signal processing applications, including a few manufacturing applications, but there are no
applications to surface profiles [22, 30]. Wavelet transforms provide an exciting tool which eliminates
