Impulse response $h[n]$ of a non linear I/O equation

my question concerns an assignment that was given to us at school. We have been given a list of I/O equations and it is required, for each of them, to find the impulse response $$h[n]$$ and the ratio between the variance of the input noise and the variance of the output one, namely $$\frac{\sigma_{y_v}^2}{\sigma_v^2}$$ , assuming that the output consists of the signal $$y[n]$$ and a superimposed noise $$y_v[n]$$, i.e:

$$y[n] + y_v[n]$$

Same for the input, which is:

$$x[n] + v[n], \quad \text{where v[n] is white noise, that generates y_v[n]}$$

This list also includes the following equation:

$$y[n]=x[n]^2$$ My first reaction was that $$h[n]$$ in this case does not exist since this equation does not describe a LTI (Linear Time Invariant) System. Furthermore, if we consider the noise at the input, we can not say that the output is simply equal to $$y[n]+y_v[n]$$, because of the non linearity of the latter. We would instead have:

$$\big(x[n] + v[n]\big)^2 = x[n]^2 + v[n]^2 + 2x[n]v[n] = y[n] + y_v[n] + 2x[n]v[n]$$

My question is whether my answer is complete or there is a more formal way of answering at a mathematical level.

• The system $y[n] = x^2[n]$ is nonlinear, and does not have an impulse reponse $h[n]$, you are right. Aug 30, 2021 at 23:40
2. Your suggested approach to finding the statistics of $$x$$ won't work, because the definition is circular. There are methods for finding the probability density function of a function of another random variable.