filters - Calculate the Output of Linear Time Invariant System Given it Impulse Response

A filter is defined as $ h \left[ n \right] = \delta \left[ n \right] - \delta \left[ n - 1 \right] $.

Given a signal $ h \left[ n \right] $ defined as:

$$ x \left [ n \right ] = \begin{cases} 1 & \text{ if } x \geq 0 \\ 0 & \text{ if } x < 0 \end{cases} $$

Let $ y \left[ n \right] = \left( x \ast h \right) \left[ n \right] $.
What is the value of $ y \left[ -1 \right], \, y \left[ 0 \right], \, y \left[ 1 \right], \, y \left[ 2 \right] $?

Answer

The Discrete Delta Function, $ \delta \left[ n \right] $ is the identity operator of Linear Time Invariant Systems.

Moreover, since it LTI System we can computer for each element of the filter by itself.
So the first element of the filter, $ \delta \left[ n \right] $, just outputs the signal itself.
The other element $ \delta \left[ n - 1 \right] $ just shifts the input signal.
Since the input is 1 for any $ n \geq 0 $ we subtract 1 from 1 unless it is $ n = 0 $ then we subtract zero from 1.

Hence the solution is 0, 1, 0, 0.