. 2020 Jul 3;10:10968. doi: 10.1038/s41598-020-67364-0

Table 2.

Polynomial expressions in (18-21) for the case of $N = 3$ and Gaussian source with $σ_{0}$ ( $b_{j, j + 1} \neq 0$ for $j \in [0, 2]$ ).

$p o l_{1}$	$π (- 2 ı b_{01} q_{7} π β_{2}^{2} - 2 π (2 π (a_{01} q_{7} - b_{12} q_{12}) β_{2}^{2} + b_{12} b_{23} ı q_{19}) σ_{0}^{2} + b_{01} b_{12} b_{23} q_{11})$
$p o l_{2}$	$2 π (4 β_{1}^{2} π^{2} (a_{01} q_{7} - b_{12} q_{12}) β_{2}^{2} - a_{01} b_{01} b_{12}^{2} b_{23} + 2 b_{12} ı π q_{17}) σ_{0}^{2} + b_{01} (4 β_{1}^{2} ı π^{2} q_{7} β_{2}^{2} + b_{01} b_{12}^{2} b_{23} (- ı) - 2 b_{12} q_{23} π)$
$p o l_{3}$ , $p o l_{4}$	$- 2 b_{01} b_{12} b_{23} π (2 a_{01} π σ_{0}^{2} + b_{01} ı)$ , $π (b_{01} (b_{01} b_{12} q_{12} - 2 ı β_{1}^{2} q_{7} π) - 2 π (2 π (a_{01} q_{7} - b_{12} q_{12}) β_{1}^{2} + a_{01} b_{01} b_{12} ı q_{12}) σ_{0}^{2})$
$p o l_{5}$	$- 4 β_{2}^{2} b_{01}^{2} b_{12}^{3} b_{23} π^{2} q_{20} (b_{12}^{2} b_{01}^{4} + 4 β_{1}^{2} π^{2} q_{13} b_{01}^{2} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} q_{14} b_{01}^{2} + (4 π^{2} q_{28} β_{1}^{2} + a_{01}^{2} b_{01}^{2} b_{12}^{2}) σ_{0}^{2}))$
$p o l_{6}$	$\begin{matrix} (4 b_{01}^{2} π^{2} q_{11}^{2} β_{1}^{4} + b_{01}^{4} b_{12}^{2} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} b_{01}^{2} b_{12}^{2} + (4 π^{2} q_{19}^{2} β_{1}^{4} + a_{01}^{2} b_{01}^{2} b_{12}^{2}) σ_{0}^{2})) \\ \times ((16 β_{1}^{4} π^{4} q_{7}^{2} β_{2}^{4} + b_{01}^{2} b_{12}^{4} b_{23}^{2} + 4 b_{12}^{2} π^{2} q_{8}) b_{01}^{2} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} b_{01}^{2} q_{9} b_{12}^{2} + q_{22} σ_{0}^{2})) \end{matrix}$
$p o l_{7}$ , $p o l_{10}$	$16 β_{1}^{2} β_{2}^{4} b_{01} b_{12} π^{4} (4 a_{01} π^{2} q_{19} σ_{0}^{4} + b_{01}^{2} q_{11}) q_{29}$ , $ı β_{1}^{2} β_{2}^{2} b_{01} b_{12} σ_{0} (b_{01} - 2 ı a_{01} π σ_{0}^{2}) (b_{01} (b_{01} b_{12} - 2 ı β_{1}^{2} q_{11} π) - 2 π q_{18} σ_{0}^{2})$
$p o l_{8}$	$\begin{matrix} - 8 β_{1}^{2} β_{2}^{2} b_{01} b_{12}^{2} b_{23} π^{3} (4 a_{01} π^{2} q_{19} σ_{0}^{4} + b_{01}^{2} q_{11}) (b_{12}^{2} b_{01}^{4} + 4 β_{1}^{2} π^{2} q_{13} b_{01}^{2} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} q_{14} b_{01}^{2} + (4 π^{2} q_{28} β_{1}^{2} \\ + a_{01}^{2} b_{01}^{2} b_{12}^{2}) σ_{0}^{2})) \end{matrix}$
$p o l_{9}$ , $p o l_{12}$	$- 8 β_{2}^{4} b_{01}^{2} b_{12}^{2} π^{3} q_{20} q_{29}$ , $2 β_{2}^{2} b_{12}^{2} π^{2} (- 4 β_{1}^{2} q_{13} π^{2} b_{01}^{2} - 8 β_{1}^{2} π^{2} q_{14} σ_{0}^{2} b_{01}^{2} - 4 (4 π^{4} q_{28} β_{1}^{2} + a_{01}^{2} b_{01}^{2} b_{12}^{2} π^{2}) σ_{0}^{4} - b_{01}^{4} b_{12}^{2})$
$p o l_{11}$	$\begin{matrix} (2 a_{01} π σ_{0}^{2} + b_{01} ı) (2 π q_{15} σ_{0}^{2} + b_{01} (2 π q_{11} β_{1}^{2} + b_{01} b_{12} ı)) \\ \times (b_{01} q_{16} - 2 π (4 β_{1}^{2} π^{2} (a_{01} q_{7} - b_{12} q_{12}) β_{2}^{2} - a_{01} b_{01} b_{12}^{2} b_{23} + 2 b_{12} ı π q_{17}) σ_{0}^{2}) \end{matrix}$
$p o l_{13}$	$(16 β_{1}^{4} π^{4} q_{7}^{2} β_{2}^{4} + b_{01}^{2} b_{12}^{4} b_{23}^{2} + 4 b_{12}^{2} π^{2} q_{8}) b_{01}^{2} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} b_{01}^{2} q_{9} b_{12}^{2} + q_{22} σ_{0}^{2})$
$p o l_{14}$	$\begin{matrix} 4 π^{3} σ_{0}^{2} (σ_{0}^{2} (a_{01}^{2} b_{01}^{2} b_{12}^{4} b_{23}^{2} d_{23} + 16 π^{4} β_{1}^{4} β_{2}^{4} q_{4} (a_{01} q_{7} - b_{12} q_{12}) + 4 π^{2} b_{12}^{2} q_{1}) + 2 β_{1}^{2} b_{01}^{2} b_{12}^{2} (4 π^{2} β_{2}^{2} q_{6} + b_{12}^{2} b_{23}^{2} d_{23})) \\ + π b_{01}^{2} (16 π^{4} β_{1}^{4} β_{2}^{4} q_{5} q_{7} + b_{01}^{2} b_{12}^{4} b_{23}^{2} d_{23} + 4 π^{2} b_{12}^{2} q_{2}) \end{matrix}$
$\begin{matrix} The functions q_{j} for j \in [1, 30] utilized while defining the polynomials are defined as follows: \\ q_{1} \equiv b_{12}^{2} b_{23}^{2} d_{23} β_{1}^{4} - 2 a_{01} b_{12} b_{23}^{2} q_{11} d_{23} β_{1}^{4} + a_{01}^{2} q_{2}, q_{2} \equiv a_{23}^{2} b_{01}^{2} b_{12}^{2} d_{23} β_{2}^{4} - a_{23} b_{01}^{2} b_{12} q_{27} β_{2}^{4} + b_{23} q_{3}, \\ q_{3} \equiv b_{23} d_{23} q_{11}^{2} β_{1}^{4} + 2 β_{2}^{2} b_{01}^{2} b_{23} d_{23} β_{1}^{2} + β_{2}^{4} b_{01}^{2} d_{12} q_{26}, q_{4} \equiv b_{12} (b_{12} - q_{12} d_{23}) - a_{01} b_{12} q_{11} + a_{01} d_{23} q_{7}, \\ q_{5} \equiv - b_{01} b_{23} d_{23} + a_{12} b_{01} q_{25} + b_{12} d_{01} q_{25}, q_{6} \equiv (β_{1}^{2} b_{23}^{2} + β_{2}^{2} q_{12}^{2}) d_{23} - β_{2}^{2} b_{12} q_{12}, q_{7} \equiv - b_{01} b_{23} + a_{12} b_{01} q_{12} + b_{12} d_{01} q_{12}, \\ q_{8} \equiv a_{23}^{2} b_{01}^{2} b_{12}^{2} β_{2}^{4} + 2 a_{23} b_{01}^{2} b_{12} b_{23} d_{12} β_{2}^{4} + b_{23}^{2} q_{24}, q_{9} \equiv 4 π^{2} (β_{1}^{2} b_{23}^{2} + β_{2}^{2} q_{12}^{2}) β_{2}^{2} + b_{12}^{2} b_{23}^{2}, q_{10} \equiv {(b_{12} q_{12} - a_{01} q_{7})}^{2}, \\ q_{11} \equiv a_{12} b_{01} + b_{12} d_{01}, q_{12} \equiv a_{23} b_{12} + b_{23} d_{12}, q_{13} \equiv b_{12}^{2} d_{01}^{2} β_{1}^{2} + 2 a_{12} b_{01} b_{12} d_{01} β_{1}^{2} + (a_{12}^{2} β_{1}^{2} + β_{2}^{2}) b_{01}^{2}, q_{14} \equiv 2 β_{1}^{2} π^{2} β_{2}^{2} + b_{12}^{2}, \\ q_{15} \equiv a_{01} b_{01} b_{12} - 2 ı β_{1}^{2} q_{19} π, q_{16} \equiv - 4 ı β_{1}^{2} q_{7} π^{2} β_{2}^{2} + b_{01} b_{12}^{2} b_{23} ı + 2 b_{12} π q_{23}, q_{17} \equiv a_{01} q_{23} - β_{1}^{2} b_{12} b_{23}, q_{18} \equiv 2 π q_{19} β_{1}^{2} + a_{01} b_{01} b_{12} ı, \\ q_{19} \equiv a_{01} a_{12} b_{01} - b_{12} + a_{01} b_{12} d_{01}, q_{20} \equiv b_{01}^{2} + 4 π^{2} σ_{0}^{2} (β_{1}^{2} + a_{01}^{2} σ_{0}^{2}), q_{21} \equiv b_{12}^{2} b_{23}^{2} β_{1}^{4} - 2 a_{01} b_{12} b_{23}^{2} q_{11} β_{1}^{4} + a_{01}^{2} q_{8}, \\ q_{22} \equiv 16 β_{1}^{4} π^{4} q_{10} β_{2}^{4} + a_{01}^{2} b_{01}^{2} b_{12}^{4} b_{23}^{2} + 4 b_{12}^{2} π^{2} q_{21}, q_{23} \equiv b_{23} q_{11} β_{1}^{2} + a_{23} β_{2}^{2} b_{01} b_{12} + β_{2}^{2} b_{01} b_{23} d_{12}, \\ q_{24} \equiv a_{12}^{2} b_{01}^{2} β_{1}^{4} + b_{12}^{2} d_{01}^{2} β_{1}^{4} + 2 a_{12} b_{01} b_{12} d_{01} β_{1}^{4} + 2 β_{2}^{2} b_{01}^{2} β_{1}^{2} + β_{2}^{4} b_{01}^{2} d_{12}^{2}, q_{25} \equiv b_{23} d_{12} d_{23} + b_{12} (a_{23} d_{23} - 1), \\ q_{26} \equiv b_{23} d_{12} d_{23} - b_{12}, q_{27} \equiv b_{12} - 2 b_{23} d_{12} d_{23}, q_{28} \equiv q_{13} a_{01}^{2} - 2 β_{1}^{2} b_{12} q_{11} a_{01} + β_{1}^{2} b_{12}^{2}, \\ q_{29} \equiv 4 b_{01}^{2} π^{2} q_{11} q_{7} β_{1}^{4} + b_{01}^{4} b_{12}^{2} q_{12} + 4 π^{2} σ_{0}^{2} (2 β_{1}^{2} b_{01}^{2} q_{12} b_{12}^{2} + q_{30} σ_{0}^{2}) and finally q_{30} \equiv 4 π^{2} q_{19} (a_{01} q_{7} - b_{12} q_{12}) β_{1}^{4} + a_{01}^{2} b_{01}^{2} b_{12}^{2} q_{12} . \end{matrix}$