How to convert my transfer function to the frequency domain

Question

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?

Answer 1

Since this is not an LTI system, we cannot have $H(\omega) = K\Theta_1(\omega)$, where $K$ is a scalar complex number. But as OP mentioned in the comment he has knowledge of $\Theta_1(\omega)$, we can compute $\Theta_2(\omega)$ and $\Theta_3(\omega)$. As $x^2(t)=x(t)\times x(t)$, $$ \Theta_2(\omega)= \Theta_1(\omega)*\Theta_1(\omega) =\int_{-\infty}^{+\infty} \Theta_1(\alpha)\Theta_1(\omega-\alpha)d\alpha $$ where $*$ is the convolution operation. Similarly, $x^3(t)=x(t)\times x(t)\times x(t)$ $$ \Theta_3(\omega)= \Theta_1(\omega)*\Theta_1(\omega)*\Theta_1(\omega)=\Theta_1(\omega)*\Theta_2(\omega) $$ So $H(\omega)=B\Theta_1(\omega)+a\Theta_2(\omega)+b\Theta_3(\omega)$, where $\Theta_2$ and $\Theta_3$ are computed from $\Theta_1$ above.