Wiener series

The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience.

The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Itô expansion (1951) which has a different form but is entirely equivalent to it.

The Wiener series should not be confused with the Wiener filter, which is another algorithm developed by Norbert Wiener used in signal processing.

Wiener G-functional expressions

Given a system with an input/output pair ${\displaystyle (x(t),y(t))}$ where the input is white noise with zero mean value and power A, we can write the output of the system as sum of a series of Wiener G-functionals ${\displaystyle y(n)=\sum _p(G_{p}x)(n)}$

In the following the expressions of the G-functionals up to the fifth order will be given:

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${\displaystyle (G_{0}x)(n)=k_0=E\left\{y(n)\right\};}$

${\displaystyle (G_{1}x)(n)=\sum _{{\tau }_1=0}^{N_1-1}{k}_1({\tau }_1)x(n-{\tau }_1);}$

${\displaystyle (G_{2}x)(n)=\sum _{{\tau }_1,{\tau }_2=0}^{N_2-1}{k}_2({\tau }_1,{\tau }_2)x(n-{\tau }_1)x(n-{\tau }_2)-A\sum _{{\tau }_1=0}^{N_2-1}{k}_2({\tau }_1,{\tau }_1);}$

${\displaystyle (G_{3}x)(n)=\sum _{{\tau }_1,\dots ,{\tau }_3=0}^{N_3-1}{k}_3({\tau }_1,{\tau }_2,{\tau }_3)x(n-{\tau }_1)x(n-{\tau }_2)x(n-{\tau }_3)-3A\sum _{{\tau }_1=0}^{N_3-1}\sum _{{\tau }_2=0}^{N_3-1}{k}_3({\tau }_1,{\tau }_2,{\tau }_2)x(n-{\tau }_1);}$

${\displaystyle \begin{array}{cc}(G_{4}x)(n)=& \sum _{{\tau }_1,\dots ,{\tau }_4=0}^{N_4-1}{k}_4({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4)x(n-{\tau }_1)x(n-{\tau }_2)x(n-{\tau }_3)x(n-{\tau }_4)+\\ & -6A\sum _{{\tau }_1,{\tau }_2=0}^{N_4-1}\sum _{{\tau }_3=0}^{N_4-1}{k}_4({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_3)x(n-{\tau }_1)x(n-{\tau }_2)+3A^2\sum _{{\tau }_1,{\tau }_2=0}^{N_4-1}{k}_4({\tau }_1,{\tau }_1,{\tau }_2,{\tau }_2);\end{array}}$

${\displaystyle \begin{array}{cc}(G_{5}x)(n)=& \sum _{{\tau }_1,\dots ,{\tau }_5=0}^{N_5-1}{k}_5({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4,{\tau }_5)x(n-{\tau }_1)x(n-{\tau }_2)x(n-{\tau }_3)x(n-{\tau }_4)x(n-{\tau }_5)+\\ & -10A\sum _{{\tau }_1,\dots ,{\tau }_3=0}^{N_5-1}\sum _{{\tau }_4=0}^{N_5-1}{k}_5({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4,{\tau }_4)x(n-{\tau }_1)x(n-{\tau }_2)x(n-{\tau }_3)\\ & +15A^2\sum _{{\tau }_1=0}^{N_5-1}\sum _{{\tau }_2,{\tau }_3=0}^{N_5-1}{k}_5({\tau }_1,{\tau }_2,{\tau }_2,{\tau }_3,{\tau }_3)x(n-{\tau }_1).\end{array}}$