SPECTRAL CHARACTERISTICS OF TWO PARAMETER FIFTH DEGREE POLYNOMIAL CONVOLUTION KERNEL

In this paper, the spectral characteristic of a polynomial two parameter convolutional fifth - order interpolation kernel is determined. The spectral characteristic is determined as follows. First, the kernel is decomposed into components. After that, the spectral characteristics of each kernel component were calculated using the Fourier transform. Finally, the spectral characteristic of the interpolation two parameter kernel using a combination of the spectral components of the kernels and the kernel parameters, α and β, is formed. Through a numerical example and a graphical representation of the spectral characteristics of the one parameter and two parameter kernels, greater similarity of the spectral characteristics of the 2 P kernel, relative to the ideal box characteristic, is shown.


INTRODUCTION
Theoretical  analyzes of convolutional interpolation have shown that, for interpolation of band limited signals, the interpolation kernel sin(x) / x (in the notation sinc) should be applied (Keys, 1981). The sinc interpolation kernel is called the ideal interpolation kernel. he boundaries of the sinc interpolation kernel are -∞ ≤ x ≤ + ∞ (Meijering & Unser, 2003). The spectral characteristic of the sinc interpolation kernel is a rectangular or, in some notations, box function. From a practical point of view, it is not possible to implement a kernel with boundaries -∞ ≤ x ≤ + ∞. The solution to this problem was found in the truncated sinc kernel, so that the kernel becomes a finite length L. The process of truncated the kernel to finite length is called windowizing (Dodgson, 1997). Truncated distinguishes the spectral characteristic of the interpolation kernel from the ideal spectral characteristic, ie from the box characteristic. The spectral characteristics of the truncated kernel, sincw, have: a) ripple in the passband and stopband, and b) finite slope in the transition band.
In order to reduce the numerical complexity of the interpolation kernel, and thus reduce the interpolation time, approximation of the sinc kernel with low-degree polynomial functions is performed. Reducing of the interpolation time is especially important for convolutional interpolation in realtime systems. A zero-degree polynomial kernel performs nearest-neighbor interpolation (Dodgson, 1997). The nearestneighbor interpolation has a very high computational speed. However, a large interpolation error is generated (Rukundo & Maharaj, 2014). * Corresponding author: bojan.prlincevic@akademijakm.edu.rs A linear, first-degree interpolation kernel is described in (Rifman, 1973). A quadratic, second-degree interpolation kernel is described in (Dodgson, 1997) and (Deng, 2010). A cubic, third-degree interpolation kernel is described in (Keys, 1981). Detailed numerical analysis of the interpolation error, which is presented in great detail in (Keys, 1981), showed that the interpolation error, when the cubic kernel was applied, is smaller than the interpolation errors when the nearest-neighbor and linear interpolation kernels were applied. Thus, precision, as one of the parameters for estimating the quality of interpolation, is increased.
A cubic kernel with interpolation parameter α is shown in (Keys, 1981). Later, in the scientific literature, the one parametric interpolation kernel from (Keys, 1981), in honor of the author Roberts B. Keys, the 1P Keys interpolation kernel, was named. By adjusting of the parameter α it is possible to minimize of the interpolation error in various applications (image interpolation, audio signal interpolation, etc.) The process of changing the kernel parameter for customization is called parameter optimization. Changing the kernel parameter α affects, among other things, to the ripple of the spectral characteristic. Reduction of ripples was achieved by eliminating members of the Taylor series that predominantly influence on the ripple (Park & Schowengerdt, 1982). The analysis presented in (Meijering et al., 1999) indicates that α = -0.5 is the optimal value of the kernel parameter for reducing the ripple of the spectral characteristic in the passband and stopband. Convolution interpolation realized by the parameterized cubic kernel is called PCC (Parametric Cubic Convolution) interpolation.
Increasing of the interpolation precision of the cubic interpolation kernel by constructing a two parameter (α, β) interpolation kernel (2P), was achieved (Hanssen & Bamler, 1999). The two parameter interpolation kernel is based on the additional parameterization of the 1P Keys kernel. In the scientific literature, this kernel is called the 2P Keys kernel. Optimization of the 2P Keys kernel parameters in the estimation of the fundamental frequency, F 0 , of the speech signal was determined in (Milivojević & Brodić, 2013) and (Milivojević et al., 2017). As a measure of the error estimate of the fundamental frequency MSE was used.
One parameter fifth-degree interpolation kernel, length L= 8, described in (Meijering et al., 1999). In order to increase of the precision of interpolation, the construction of a two parameter fifth-degree interpolation kernel was performed (Savić et al., 2021). The 2P fifth-degree interpolation kernel is constructed by the extended parameterization of the 1P fifthdegree interpolation kernel. The optimal parameters of the 2P fifth-degree interpolation kernel, when interpolating the audio signal, (α opt = 0.1, β opt = 0.0571), were determined using the experiment (Savić et al., 2022). The precision of the interpolation at 2P (MSE = 1.325310 -6 ) is higher than the precision of the interpolation at 1P kernel (MSE= 1.809610 -6 ).
In the previously cited papers, the analytical form of the spectral characteristic of the 2P fifth degre polynomial kernel, has not been determined. In this paper, the spectral characteristic of the 2P kernel, whose parameterization was done in (Savić et al., 2021), was calculated. The spectral characteristic of the 2P kernel is done as follows. It is first done by decomposing the 2P kernel into components. Then, by applying the Fourier transform to each kernel component, the spectral characteristics of each component were determined. In this way, the spectral components of the kernel are determined. Finally, taking into account all spectral components as well as the kernel parameters α and β, the spectral characteristic of the 2P kernel was determined. With the knowledge of the analytical form of the spectral characteristic, it is possible to change its shape by changing the kernel parameters, in order to bring its shape closer to the ideal box characteristic. In addition, the precision of interpolation in Digital Image, Audio and Speech processing can be affected by changing the kernel parameters, and thus minimize the interpolation error. This optimizes the kernel parameters. In this way, the scope of application of parametric convolutional kernels is increased.
Further organization of this paper is as follows. Section II describes: ideal kernel, 1P fifth order kernel, and 2P fifth order interpolation kernel. Section III describes the process of determining the spectral characteristic of the kernel. Section IV is the Conclusion

Ideal interpolation kernel
The ideal interpolation kernel for interpolating band limited signals is the form sinc = sin(x) / x (Keys, 1981). The definition of the sinc kernel is in the interval (-∞, +∞). Therefore, due to its infinite length, the sinc kernel cannot be realized. For this reason, it was necessary to truncated the sinc interpolation kernel to a finite length L (interval [-L / 2, L / 2]). The truncated sinc kernel, sincw, has a spectral characteristic that differs from the box characteristic. The difference is reflected in: a) the appearance of ripples of the spectral characteristics in the passband and stopband and b) finite slope of the spectral characteristics in the transition band (Savić et al., 2021). The ideal interpolation kernel sinc in the range (-20, 20) is shown in Fig. 1a. The spectral characteristics of the truncated kernel sincw for some lengths L (3, 5, 10, 20, + ∞) are shown in Figs. 1 In order to reduce the numerical complexity, the sinc function is approximated by simpler mathematical functions. In the field of the Digital Signal Processing, DSP, and especially in the field of the Digital Image Processing, the approximation of the sinc function with polynomial functions is intensively used. In the field of Digital Image Processing, the interpolation kernels, formed from the third-order polynomials, are very popular. The most significant are the parameterized 1P, 2P and 3P Keys kernels (Hanssen & Bamler, 1999). The need to increase precision of the interpolation has led to the construction of fifth-order polynomial interpolation kernels.
In the further part of this paper, the definition of one parameter and two parameter fifth-order kernels and their parameterization is described. After that, the spectral characteristic of the two parameter kernel was determined by applying the Fourier transform.

One parameter fifth-order kernel
The general form of the convolutional fifth-degree interpolation kernel is (Meijering et al., 1999 (1) The coefficients of the kernel are determined subject to the following conditions: a) r(0) = 1, b) r (x) = 0 for |x| = 1, …, 4; and c) r (l) (x) are continuous for |x| = 0, …, 4. By parameterizing of the kernel, taking into account the previous conditions, form of the one parameter fifth-degree interpolation kernel is determined. The form of the 1P fifthdegree interpolation kernel is: where α is the kernel parameter.

Two parameter fifth-order kernel
In order to increase of the interpolation precision, a two parameter fifth-degree convolutional kernel is created. The process of kernel parameterization is presented in great detail in (Savić et al., 2021). The final form of the parametric 2P kernel is: where α and β are kernel parameters.

SPECTRAL CHARACTERISTICS OF THE 2P FIFTH-ORDER INTERPOLATION KERNEL
The algorithm for determining the spectral characteristic of the fifth-order kernel (Eq. (3)) was implemented in the following steps: a) decomposing the kernel into components, b) determining the spectral characteristics of each kernel component, and c) determining the spectral characteristics of the kernel depending on the parameters α and β.

2P kernel components
The 2P fifth-degree kernel (Eq. (3)) can be written in the form: where r 0 , r 1 and r 2 are kernel components: As an example, the kernel components r 0 , r 1 and r 2 are shown in Fig. 3.a. The ideal interpolation kernel, r sinc , and the 2P fifth degree kernel, r αβ , for α = 0.025 and β = -0.04 (Eq. (4)), on the interval [-5, 5], are shown in Fig. 3 and It can be seen that the spectral characteristic of the 2P kernel, H αβ , in relation to the spectral characteristic of the 1P kernel, H α , has a better similarity with the box characteristic.
Further research will be based on optimizing the spectral characteristics of the 2P kernel in the spectral domain. The optimization will be aimed at minimizing the ripple of the spectral characteristic, and thus, towards greater similarity with the ideal box spectral characteristic.

CONCLUSION
In this paper, the spectral characteristic of two parametric fifth order polynomial interpolation kernels is calculated. By applying the Fourier transform, the spectral characteristic of the interpolation kernel was determined. Due to the finite length of the kernel, the spectral characteristic has ripples in the passband and stopband, as well as finite slopes in the transition band. Therefore, there is a deviation of the spectral characteristic of the 2P kernel from the characteristics of the ideal sinc interpolation kernel, that is, the box characteristic. By adjusting the kernel parameters α and β it is possible to minimize the ripple, and thus increase the similarity of the spectral characteristics of the kernel, with the box characteristic. Examples of minimizing the interpolation error, in Digital Image Processing, as well as in Audio and Speech Processing, are described in the scientific literature. In this way, the optimal values of the kernel parameters are determined. MSE is most often used as a measure of the interpolation precision.
Further research will be aimed at minimizing the ripples of spectral characteristic in the spectral domain. Minimization of the ripple will be carried out by analyzing the effect of the Taylor series members of the spectral characteristics that predominantly affect the ripple.