Modification of Chebyshev Pseudospectral Method to Minimize the Gibbs Oscillatory Behaviour in Resynthesizing Process

Abstract

The Gibbs phenomenon describes oscillations of small or large amplitudes that occur, when a signal with steep gradients or noise components is approximated. Such interruptions can degrade the quality of desired signal. Reduction of such oscillations is an essential task to extract vital information from the desired signal. This paper, therefore, presents the Chebyshev spectral method (CSM) that is combined with two novel concepts to reduce the influence of oscillatory structures. The first notion uses a thresholding approach to estimate true expansion coefficients in a noisy environment, while the second concept introduces a new smoothing function. The basic framework of the proposed concept is to introduce an additional threshold procedure into pre-existing Chebyshev collocation method to handle the fluctuations of noise interferences. Moreover, the CSM is the global-behaviour approximation based on the points of an entire domain, which allows for high-order convergence to be recovered. The method is implemented for sharp gradient-contained function and to a signal that has been distorted by noise. Through computational experiments, efficiency of the proposed method is verified graphically and numerically. Signal-to-noise ratio of 37.4810 dB is achieved with corresponding mean square error about 1.79e−04. The percentage root-mean-square difference (PRD) and maximum error are obtained as 1.3402% and 0.0399, respectively.

Appendices

Appendix

Theorem: A.1: Accuracy of Chebyshev polynomial interpolation on unequally spaced points [38]

Let $\phi$ be a function that belongs to ${\mathcal{H}}$ Hilbert space and $\left\{ {\varsigma_{j}^{{\prime }} s} \right\}$ are the interpolation points that follows the density function given by Eq. (11). Assume $N > 0$, for any $N\epsilon {\mathbb{Z}}$, there exists an interpolation polynomial $P_{N} \phi$ based on Chebyshev polynomials, which coincides with $\phi$ exactly at $\left\{ {\varsigma_{j}^{{\prime }} s} \right\}$. Correspondingly, the electrostatic potential is given as:

$$ z\left( \varsigma \right) = \mathop \int \limits_{ - 1}^{1} \Theta \left( \zeta \right)\ln \left( {\left| {\zeta - \varsigma } \right|} \right)d\zeta . $$

(A.1)

And defines the supremum of potential

$$ z^{\sup } = \underbrace {\sup }_{{\varsigma \epsilon \left[ { - 1,1} \right]}}z\left( \varsigma \right). $$

(A.2)

If $\exists$ is an upper bound such that $z^{{{\text{constt}}}} > z^{\sup }$, in a closed domain $\left\{ {\zeta \epsilon {\mathbb{C}}: z\left( \varsigma \right) \le z^{{{\text{constt}}}} } \right\}, {\text{then}}\,\,\exists$ is a constant $A > 0$, such that for each $N$

$$ \left| {P_{N} \phi \left( \varsigma \right) - \phi \left( \varsigma \right)} \right| \le \frac{A}{{\exp \left( {N\left( {z^{{{\text{constt}}}} - z^{\sup } } \right)} \right)}} , $$

(A.3)

for all $\varsigma \epsilon \left[ { - 1,1} \right].$