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

  • 2
    $\begingroup$ 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 '21 at 23:40
  • 1
    $\begingroup$ @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 '21 at 19:44
  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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.