On Computational Aspects of Tchebichef Polynomials for Higher Polynomial Order

Tchebichef polynomials (TPs) and their moments are widely used in signal processing due to their remarkable performance in signal analysis, feature extraction, and compression capability. The common problem of the TP is that the coefficients computation is prone to numerical instabilities when the polynomial order becomes large. In this paper, a new algorithm is proposed to compute the TP coefficients (TPCs) for higher polynomial order by combining two existing recurrence algorithms: the three-term recurrence relations in the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-direction and <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>-direction. First, the TPCs are computed for <inline-formula> <tex-math notation="LaTeX">$x,n=0,1, {\dots },({N}/{2})-1$ </tex-math></inline-formula> using the recurrence in the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>-direction. Second, the TPCs for <inline-formula> <tex-math notation="LaTeX">$x=0,1, {\dots },({N}/{2})-1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n=({N}/{2}), ({N}/{2})+1, {\dots },N-1$ </tex-math></inline-formula> based on <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> directions are calculated. Finally, the symmetry condition is applied to calculate the rest of the coefficients for <inline-formula> <tex-math notation="LaTeX">$x=({N}/{2}),({N}/{2})+1, {\dots },N-1$ </tex-math></inline-formula>. In addition to the ability of the proposed algorithm to reduce the numerical propagation errors, it also accelerates the computational speed of the TPCs. The performance of the proposed algorithm was compared to that of existing algorithms for the reconstruction of speech and image signals taken from different databases. The performance of the TPCs computed by the proposed algorithm was also compared with the performance of the discrete cosine transform coefficients for speech compression systems. Different types of speech quality measures were used for evaluation. According to the results of the comparative analysis, the proposed algorithm makes the computation of the TP superior to that of conventional recurrence algorithms when the polynomial order is large.