The Step Response is the response of an LTI system to a unit step function. In other words, the input to the system is simply the unit step function: x(t) = u(t).
This is equivalent to simply integrating the input from the infinite past up to time t.
Example 1
Superposition (or DivideandConquer):
We can directly apply superposition to find the output of LTI systems if x(t) can be expressed as a linear combination of basis functions Φ_{k}(t).
The Φ_{k}(t) are some convenient set of functions, for example unit impulses, unit step functions, or complex exponentials.
Example 2 If an input is written as:

using superposition, we can write its output as:
where
Ψ_{k}(t) = S[Φ_{k}(t)] = h(t) * Φ_{k}(t) = Φ_{k}(t) * h(t)
Again, using superposition, we can write:
Example 3
Find the output of the system where
x(t) = 2u(t)  u(t  1) 
u(t  2) and
h(t) = e^{at} [ u(t)  u(t  2) ].
Convolution is commutative, associative, and distributive. Keeping this in mind may simplify some convolutions for you.
w(t) = x(t) * h_{1}(t), y(t) = w(t) * h_{2}(t)
= [ x(t) * h_{1}(t) ] * h_{2}(t)
= x(t) * [h_{1}(t) * h_{2}(t)], by associativity of convolution
Therefore the impulse response h(t) for this overall system is h_{1}(t) * h_{2}(t).
We can change the order in which the convolutions are performed due to commutativity. For a cascade of M systems there are M! possible system orderings.
Parallel systems is a large area of research today.