# A Novel Measurement Standard for Surface Roughness on Involute Gears
## 2 Methods for the Analysis of Surface Roughness on Involutes
### Filtration  
In standardised procedures of geometrical product specification, geometrical irregularities or form deviations contained in a scale-limited area or profile are referred to as surface texture (see [16] Clause 3.1.2). The ISO standard [16] specifying the statistical parameters for characterizing roughness defines the term primary profile, in short P-profile, as the residuals of a profile trace on the surface of a geometrical object obtained by removing the form. Furthermore, it defines the waviness profile, W-profile, as a scale-limited profile obtained by removing small scale lateral components and the roughness profile, R-profile, as a scale-limited profile obtained by removing large scale lateral components. The removal and the separation of small and large scale lateral components, respectively, are performed by applying a filter with specified cut off wavelength \(\lambda_{c}\), which, in roughness metrology, is also referred to as nesting index. There exist a variety of filter algorithms specified in a series of ISO standards. The commonly used filter is a Gaussian filter specified in ISO standard 16610-2 [17]. Let \(z^{P}(x)\) be the primary profile, then the waviness profile is obtained by a linear low pass filter either by convolution of the following:  
\[z^{W}(x)=\int s(x^{\prime})\,z^{P}(x\,-\,x^{\prime})\,\mathrm{d}x^{\prime} \tag{7}\]  
with a convolution kernel  
\[s(x^{\prime})=\frac{1}{a\lambda_{c}}e^{-\pi\left(\frac{x^{\prime}}{a\lambda_{ c}}\right)^{2}} \tag{8}\]or equivalently in the Fourier domain with the following transfer function. \[H(\lambda)=2^{-(\frac{\lambda_{c}}{\lambda})^{2}}=e^{-\pi(\alpha\frac{\lambda_{c }}{\lambda})^{2}} \tag{9}\]  
\(\alpha=\sqrt{\frac{\ln(2)}{\pi}}\) and \(\lambda\) are the spatial wavelengths. The transfer function \(H\) is the low pass filter, while \(1-H(\lambda)\) is the high pass delivering the roughness profile, R-profile and \(z^{\text{R}}\). Equivalently, the R-profile is given by the P-profile by removing the W-profile: \(z^{\text{R}}=z^{\text{P}}(x)-z^{\text{W}}\). In order to minimize filtration errors of signals of finite size, the borders need to be dealt with appropriately. The usage of any kind of windowing function suppresses information and amplitude content within the border regions. However, using a combination of the Gaussian filter with a Savitzky-Golay filter minimizes such effects that the roughness metrology refers to as end effects. Savitzky-Golay filters employ regression polynomials to locally follow a signal, and the local range it acts on depends on the width of the regression polynomial. The degree of the regression polynomial determines the shape it follows. A straight line follows the slope of the signal within the covered range, a parabola of the curvature and so forth. To minimize end effects, the width of the range covered by the regression polynomial is equal to the cut off wavelength \(\lambda_{c}\), and the degree shall comply by following the signal slope at the border regions. A shape retainment expressed by following the primary profile's curvature is not required because it is assumed that the curved shape lies inside the form associated when fitting the involute. The Savitzky-Golay polynomial chosen is, therefore, the straight line, i.e., the order of the polynomial is \(p=1\). The filter is called a Gaussian regression filter, as developed in [18; 19], and subsequently standardised as ISO 16610-28 [20]. For a discrete set of data points \((x_{k},z_{k})\) representing a profile and with \(z^{\text{W}}\) denoting the waviness profile and \(\mathbf{z}^{\text{P}}\), which is the vector of height values of the primary profile, the Gaussian regression for order \(p=1\) is given by the following convolution:  
\[z_{k}^{\text{W}}=(1\;0)\left(\mathbf{X}_{k}^{T}\;\mathbf{S}_{k}\;\mathbf{X}_{k }\right)^{-1}\;\mathbf{X}_{k}^{T}\;\mathbf{S}_{k}\;\mathbf{z}^{\text{P}} \tag{10}\]  
with \(\mathbf{X}_{k}^{T}\;\mathbf{S}_{k}\;\mathbf{X}_{k}=\)  
\[\left(