# Causal Systems

1. Sep 23, 2007

### Tom McCurdy

We have been going over causal systems and I am still having trouble determining what defines a system to be causal.

I was told that if the input is anything besides x(a*t) where a=1 then the system is non causal. I can kind of see this, but it is still a bit blurry for me. I also was wondering if that would still apply if you removed t directly from the input equation...

say like if you had $$y(t) = \int_{-\infty}^{t}x(5{\tau}) d\tau$$

then is this automatically not causal because of the the 5 coefficient on the inside of x()

2. Sep 24, 2007

### antonantal

That's not true.

The system defined by $$y(t) = x(t-1)$$ is causal although $$x(t-1)$$ is something else than $$x(t)$$.

The general definition for a causal system (linear or non-linear, time-invariant or time-variant) is:

Given 2 input signals $$x_1(t)$$ and $$x_2(t)$$ such that $$x_1(t) = x_2(t)$$ for any $$t < t_0$$, the system is causal if the output signals $$y_1(t) = y_2(t)$$ for any $$t < t_0$$

If the system is linear then if we apply a signal $$x(t) = x_1(t) - x_2(t)$$ the output should be $$y(t) = y_1(t) - y_2(t)$$, so the condition for the system to be causal (in the case of linear systems) reduces to:

if $$x(t) = 0$$ for $$t<t_0$$ then $$y(t) = 0$$ for $$t<t_0$$

If the system is linear and time invariant, the condition for causality reduces to:

$$h(t)=0$$ for $$t<0$$

So depending on the kind of system and your known data you should check one of these conditions.

In the case of $$y(t) = \int_{-\infty}^{t}x(5{\tau}) d\tau$$ I know that it's linear because it's defined by an integral which is a linear operation so I will check the second condition.
I pick an instant $$t_0$$ at which the output will be $$y(t_0) = \int_{-\infty}^{t_0}x(5{\tau}) d\tau$$

So we see that the output depends on values of $$x(t)$$ till $$5t_0$$ but
we know that $$x(t) = 0$$ only for $$t<t_0$$ and thus the output will not be 0 for any $$t<t_0$$ which means that the system is not causal.

Last edited: Sep 25, 2007
3. Sep 24, 2007

### rbj

i sorta like the Wikipedia definition of causal system (i had a hand in it before they kicked me out of Wikipedia):

i think you can conclude that for an LTI system, causality is equivalent to the impulse response h(t) being zero for all t < 0. t0 is not a parameter of the impulse response. the impulse response is the LTI to a unit impulse applied at t=0.

Last edited: Sep 24, 2007
4. Sep 25, 2007

### antonantal

You're right of course. I just copied the latex expression above it and forgot to replace t0 with 0 as well. I'll edit it now. Thanks!