# Identify whether to have unique output in this ARMA system

my friends and I have no idea whether to diagnose to have the specific output from the formula below: The question is that Assume we are given a specific input signal x[k]x[k]. Explain and justify whether or not there is a unique output y[k] satisfying this difference equation.

• I don't see $x[k]$ (for any value of $k$) in the equation. This means that the signal $y$ is not created by a signal put into a filter. There may be a signal that satisfies that equation, but it does not depend on any input signal. I suspect that there is a typographical error. I suspect that one of the sums on the right-hand side should involve $x[k-i]$ or $x[k-j]$. – Joe Mack Jul 15 '20 at 17:49
As I mentioned in my comment, I think there is a typographical error in the equation. I think it is supposed to be $$\begin{equation} y[k] = \sum_{i=1}^{M}a_iy[k-i] + \sum_{j=1}^{N}b_jx[k-j]. \end{equation}$$ This is an example of a linear constant-coefficient difference equation. These are introduced very early in most textbooks about digital signal processing. For example, they are introduced in Section 2.5 (page 33) in the 1989 edition of Discrete-Time Signal Processing by Oppenheim and Schafer.
Given an input signal $$x$$, there will not be a unique solution signal $$y$$. This is because a specific solution requires specifying the input signal as well as initial conditions for $$y$$. Different initial conditions lead to different solutions.