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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.

Thank you in advance

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    $\begingroup$ The system $y[n] = x^2[n]$ is nonlinear, and does not have an impulse reponse $h[n]$, you are right. $\endgroup$
    – Fat32
    Aug 30, 2021 at 23:40
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    $\begingroup$ @Fat32 every system that has an impulse in its domain has an impulse response. But only for linear or even also time-invariant systems it's guaranteed to be useful for constructing the output from the input. In fact, there are even non-linear systems that can be simplified by finding the impulse response(s). So your comment is misleading at best. $\endgroup$
    – Jazzmaniac
    Aug 31, 2021 at 19:44

1 Answer 1

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  1. You are correct that there is no transfer function, because the system is nonlinear
  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.
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