# If the input of the system depends on the future outputs then is the system non-causal?

While I'm aware of the fact that causality implies that the present output is only dependent on present and past inputs, something that is bugging me is what if the input is dependent on future outputs. Would the system be causal or anti-causal in that case? I look at this with the assumption that since input depends on the future outputs, then in that case, the outputs only ever depend on the past and present inputs and thus should be causal. I would love some insight into this. Take, for example: $$x[n]=y[n-1]-2.5y[n]+y[n+1]$$ where $$x[n]$$ & $$y[n]$$ are inputs and outputs of an LTI system, respectively.

The equation

$$x[n]=y[n-1]-\frac52y[n]+y[n+1]\tag{1}$$

is just a way of representing the input-output relation of a linear time-invariant discrete-time system. It does not mean that the input depends on the output. The input is always given and the system behaves in such a way that the output satisfies $$(1)$$.

Note that from $$(1)$$ we cannot determine whether or not the corresponding system is causal. You could rewrite $$(1)$$ as

$$y[n+1]=x[n]-y[n-1]+\frac52y[n]\tag{2}$$

which would suggest that the system is causal, because it seems that the output at time $$n+1$$ only depends on past values of the input and output. However, we could also rewrite $$(1)$$ as

$$y[n-1]=x[n]+\frac52y[n]-y[n+1]\tag{3}$$

which would suggest that the output only depends on future values of the input and output. Finally, another way of rewriting $$(1)$$ is

$$y[n]=\frac25\big(y[n-1]+y[n+1]-x[n]\big)\tag{4}$$

which seems to imply that the output depends on current, past and future values.

From the difference equation $$(1)$$ we simply cannot tell if the system is causal or not. Eq. $$(1)$$ represents three different systems, as expressed by Eqs $$(2)$$, $$(3)$$ and $$(4)$$: one causal, one acausal (left-sided), and one non-causal (two-sided). In order to completely describe the system, we generally need more information than just the difference equation.