I recently solved a problem which required that I compute the convolution of an input signal with the impulse response of a filter. I got the resulting discrete signal which is verified to be the correct answer:
$$2\delta[n] + 6\delta[n - 1] + 3\delta[n - 2] + 3\delta[n - 3] + \delta[n - 4] + 4\delta[n - 5] + 2\delta[n - 7]$$
This is apparently equivalent to:
$$2\delta[n] + 6\delta[n - 1] + 3u[n - 2] - 2\delta[n - 4] + \delta[n - 5] - 3u[n - 6] + 2\delta[n - 7]$$
Where $u$ is the unit-step.
I understand that the unit-step is zero when $u[n - k]$ has ($k > n$), and one when ($k \leq n$). But beyond that I can't understand why this representation is identical. How can I to transform my output to match this format, or to verify they are the same?