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 p...
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| Vydané v: | IEEE access Ročník 5; s. 2470 - 2478 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2169-3536, 2169-3536 |
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| Abstract | 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 n-direction and x-direction. First, the TPCs are computed for x, n = 0, 1, ... , (N/2) - 1 using the recurrence in the x-direction. Second, the TPCs for x = 0, 1, ... , (N/2) - 1 and n = (N/2), (N/2) + 1, ... , N - 1 based on n and x directions are calculated. Finally, the symmetry condition is applied to calculate the rest of the coefficients for x = (N/2), (N/2) + 1, ... , N - 1. 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. |
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| AbstractList | 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 n-direction and x-direction. First, the TPCs are computed for x, n = 0, 1, ... , (N/2) - 1 using the recurrence in the x-direction. Second, the TPCs for x = 0, 1, ... , (N/2) - 1 and n = (N/2), (N/2) + 1, ... , N - 1 based on n and x directions are calculated. Finally, the symmetry condition is applied to calculate the rest of the coefficients for x = (N/2), (N/2) + 1, ... , N - 1. 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. |
| Author | Al-Haddad, Syed Abdul Rahman Mahmmod, Basheera M. Abdulhussain, Sadiq H. Ramli, Abd Rahman Jassim, Wissam A. |
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| SubjectTerms | Algorithm design and analysis Algorithms Coefficients Compression tests Computation Computers Discrete cosine transform Discrete cosine transforms Electronic mail Feature extraction Image reconstruction Indexes numerical propagations errors Orthogonal polynomials Polynomials recurrence algorithm Signal analysis Signal processing Signal processing algorithms Speech Speech compression Tchebichef polynomials |
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| Title | On Computational Aspects of Tchebichef Polynomials for Higher Polynomial Order |
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