Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Khaled A Al-Utaibi; Sadiq H Abdulhussain; Basheera M Mahmmod; Marwah Abdulrazzaq Naser; Muntadher Alsabah; Sadiq M Sait

doi:10.3390/e23091162

Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Entropy (Basel). 2021 Sep 3;23(9):1162. doi: 10.3390/e23091162.

Authors

Khaled A Al-Utaibi¹, Sadiq H Abdulhussain², Basheera M Mahmmod², Marwah Abdulrazzaq Naser³, Muntadher Alsabah⁴, Sadiq M Sait⁵

Affiliations

¹ Department of Computer Engineering, University of Ha'il, Ha'il 682507, Saudi Arabia.
² Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq.
³ Department of Architectural Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq.
⁴ Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 4ET, UK.
⁵ Department of Computer Engineering, Interdisciplinary Research Center for Intelligent Secure Systems, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia.

Abstract

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

Keywords: Krawtchouk moments; computation cost; discrete Krawtchouk polynomials; energy compaction; propagation error.