DT unit impulse, step and ramp relationship

Discrete-time unit impulse

$$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}$$

A single sample of height 1 at $n = 0$. It has the sampling property $x[n]\delta[n-k] = x[k]\delta[n-k]$, and any sequence can be written as $x[n] = \sum_k x[k]\delta[n-k]$. The response of a system to $\delta[n]$ is its impulse response $h[n]$.

Discrete-time unit step

$$u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}$$

Samples of height 1 from $n = 0$ onwards. Used to switch signals on at $n = 0$ (make them causal).

Discrete-time unit ramp

$$r[n] = n\,u[n] = \begin{cases} n, & n \ge 0 \\ 0, & n < 0 \end{cases}$$

Sample values $0, 1, 2, 3, \dots$ growing linearly for $n \ge 0$.

 delta[n]         u[n]              r[n]
                                         o
   o          o  o  o  o ..           o  |
   |          |  |  |  |           o  |  |
 --o--> n   --o--o--o--o--> n   o--o--o--o--> n
   0          0  1  2  3        0  1  2  3

Relationship

• Step from impulse (running sum) and impulse from step (first difference):

$$u[n] = \sum_{k=-\infty}^{n}\delta[k], \qquad \delta[n] = u[n] - u[n-1]$$

• Ramp from step and step from ramp:

$$r[n] = \sum_{k=-\infty}^{n-1}u[k] = \sum_{k=-\infty}^{n} u[k-1], \qquad u[n-1] = r[n] - r[n-1]$$

(equivalently $u[n] = r[n+1] - r[n]$).

So summation moves impulse to step to ramp, and the first difference moves back the other way, just as integration and differentiation link $\delta(t)$, $u(t)$ and $r(t)$ in continuous time.