Signal processing very short question?

Question

We have $Y(k)=x(k-1)+ kx(k-5)+x(k)^4$ .I have to find the impulse response for the function

So I know that $G(z) = \frac{Y(z)}{X(z)}$ but how do I relate that to this?

$Y(z)=(z^{-1}) + k(z^{-5})+ z^4$

Answer 1

The system with input $x(k)$ and output $y(k) = x(k-1)+kx(k-5)+x(k)^4$ is not a linear system, so the usual transfer function methods don't apply.

However, we can still find the impulse response, which is the output $y(k)$ when the input is $x(k) = \delta(k) = \begin{cases}1 & \text{if} \ k = 0 \\ 0 & \text{if} \ k \neq 0\end{cases}$.

Since $x(k) = 0$ for every value of $k$ except $0$, $y(k) = x(k-1)+kx(k-5)+x(k)^4 = 0$ for every value of $k$ except $1$, $5$, and $0$.

Can you evaluate $y(1)$, $y(5)$, and $y(0)$?