A signal that does not start before t=0 is a causal signal i.e. x(t)=0, ∀ t<0.
A system is causal or non-anticipatory if the output at any time to depends only on the values of the input at the present time and in the past.
Note: All memoryless systems are causal systems since the output responds only to the current value of the input.
A signal that starts before t=0 is a non-causal signal.
A signal that ends before t=0 is an anti-causal signal i.e. x(t)=0, ∀ t>0.
Note: Any signal x(t) that does not contain any singularities (a delta function or its derivative) at t=0 can be written as the sum of a causal part x+(t) and anti-causal part x−(t) i.e.
x(t) = x+(t) + x−(t)
For example, x(t) = e−at can be written as
x(t) = e−at u(t) + e−at u(−t)
Note: Multiplying the signal by the unit step ensure that the resulting signal is causal.