# Anti-causal systems

If Anti-causal systems are defined as those whose output depends solely upon future inputs.(Is this definition correct as I understand)

So i see that $y[n] = x[n+2]$ ; is anticausal system

How is a time reversing system such as:

$$y(t) = x(-t)$$

anticausal?

e.g. $y(2) = x(-2)$ it infact depends upon past.

Looking for some explanation.

• i would call it "acausal" not "anticausal". – robert bristow-johnson Oct 16 '18 at 0:41

$y(t) = x(-t)$ is not anticausal, but it is acausal. For negative values of $t$, $y(t)$ is not causal, while for positive $t$, it is causal, since, for example, $y(2) = x(-2)$, but $y(-2) = x(2)$.

• Thats what I thought, so this Wikipedia entry en.wikipedia.org/wiki/Causal_system needs correction then. – goldenmean Jan 20 '12 at 15:00
• I would agree with you. – Jason R Jan 20 '12 at 15:16
• Yes, the Wikipedia entry is not correct. Note that Wikipedia says that some authors distinguish between non-causal or acausal and anti-causal systems. My personal preference is to not join in this hair-splitting. – Dilip Sarwate Jan 20 '12 at 16:11
• @DilipSarwate: It is not hair splitting, its plain incorrect, Even if we go by the sentence on Wikipedia which you are referring to which says "Note that some authors have defined an anticausal system as one that depends solely on future and present input values or, more simply, as a system that does not depend on past input values" – goldenmean Jan 20 '12 at 17:14
• What I meant by hairsplitting is that I prefer to not join the some authors in creating a subcategory of anticausal systems in the broadly defined set of noncausal systems. For those who do choose to define anticausal systems, we are in agreement that the Wikipedia example of $y(t) = x(-t)$ as a anti-causal system is plain incorrect. – Dilip Sarwate Jan 20 '12 at 17:30

If Anti-causal systems are defined as those whose output depends solely upon future inputs. (Is this definition correct as I understand)

How particular are you about the words solely and future in your definition?

A causal system is one with the property that the output $y$ at every time $t$ depends only on the current and past inputs, that is, the values of the input $x$ at times in $(-\infty, t]$, or more formally,

For each $t$, $-\infty < t < \infty$, $y(t)$ is a function of $\{x(\tau) \colon \tau \in (-\infty, t]\}$ only and does not depend on any $x(\tau^\prime), \tau^\prime > t$.

If anticausal is taken to mean not causal, then the complement of the definition of causal is not what you have written. In this sense of "non-causal = not causal",

• a non-causal system is defined as one for which there is at least one time instant $t_0$ (and a $\tau > t_0$) such that the output $y(t_0)$ depends on the value of $x(\tau)$, a future input.

• Note that $y(t_0)$ might depend on past inputs as well as on the future input $x(\tau)$.

• Note that there may be other time instants $t_1$ for which $y(t_1)$ depends only on past and curent $\{x(t) \colon t \in (-\infty, t_1]\}$ and on no future inputs. All it takes for a system to be bad-mouthed as non-causal is one bad apple time instant $t_0$ for which causality is violated.

On the other hand, your definition of anti-causal is that for every instant $t$, $y(t)$ depends solely on future inputs $\{x(\tau) \colon \tau \in (t,\infty)\}$; even the current input is excluded. So, the time-reversal system $y(t) = x(-t)$ is not an anti-causal system by your definition. Trivially, $y(0)$ equals current input $x(0)$, and if you chill a bit and amend your definition to say "current and future inputs", then, as Jason R has already pointed out, $y(2)$ depends on a past input $x(-2)$ and so the system is not anti-causal as per your amended definition either. In fact, there is a huge class of systems that would be classified as non-causal as per the definition of non-causal given here that do not meet your definition of anti-causality; the time-reversal system is just one example of a system that is not anti-causal.

Time reversal system is not "non-causal" or "not causal", rather it is not time invariant system. This may cause the confusion.