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Definition of three-order cumulant

Three-order cumulant

Diagonal of three-order cumulant

Definition 1

$E\left\{y\left(n\right)y\left(n+{\tau }_1\right)y\left(n+{\tau }_2\right)\right\}$

0

0

Definition 2

$E\left\{y^{\ast }\left(n\right)y\left(n+{\tau }_1\right)y\left(n+{\tau }_2\right)\right\}$

$\begin{array}{l}{A}_1{A}_2{A}_3{\omega }_1{\omega }_2{\omega }_3[\exp j\left({\omega }_1{\tau }_1+{\omega }_2{\tau }_2+\text{π}/2\right)\\ +\exp j\left({\omega }_2{\tau }_1+{\omega }_1{\tau }_2+\text{π}/2\right)]\end{array}$

$2A_1{A}_2{A}_3{\omega }_1{\omega }_2{\omega }_3\left[\exp j\left({\omega }_3\tau +\text{π}/2\right)\right]$

Definition 3

$E\left\{y^{\ast }\left(n\right)y\left(n+{\tau }_1\right)y^{\ast }\left(n+{\tau }_2\right)\right\}$

$\begin{array}{l}{A}_1{A}_2{A}_3{\omega }_1{\omega }_2{\omega }_3[\exp j\left({\omega }_3{\tau }_1-{\omega }_1{\tau }_2-\text{π}/2\right)\\ +\exp j\left({\omega }_3{\tau }_1-{\omega }_2{\tau }_2-\text{π}/2\right)]\end{array}$

$\begin{array}{l}{A}_1{A}_2{A}_3{\omega }_1{\omega }_2{\omega }_3[\exp j\left({\omega }_2\tau -\text{π}/2\right)\\ +\exp j\left({\omega }_1\tau -\text{π}/2\right)]\end{array}$