I have the following transfer function in the time domain:
$h\left(t\right) = B x\left(t\right) + a x^2\left(t\right) + b x^3\left(t\right)$.
My simulation is in the frequency domain, so I would like to have a transfer function of the form $H\left(\omega\right)$.
I didn't find any suitable Fourier/Laplace identities so I tried substituting as follows $\theta_1\left(t\right) = x\left(t\right)$, $\theta_2\left(t\right) = x^2\left(t\right)$ and $\theta_3\left(t\right) = x^3\left(t\right)$. The resultant frequency domain function
$H\left(\omega\right) = B \Theta_1 \left(\omega\right) + a \Theta_2 \left(\omega\right) + b \Theta_3 \left(\omega\right)$,
wasn't of much use to me because I only have $\Theta_1\left(\omega\right)$ and I don't know how $\Theta_2$ and $\Theta_3$ are related to $\Theta_1$.
I would like my transfer function to depend only on $\Theta_1$, which is the variable that I know.
How can I go about solving this problem?