CT unit impulse, step and ramp relationship

Unit impulse $\delta(t)$

Zero for all $t \neq 0$, with unit area: $\int_{-\infty}^{\infty}\delta(t)\,dt = 1$. It is the limit of a pulse of width $\Delta$ and height $1/\Delta$ as $\Delta \to 0$. Sifting property: $\int x(t)\delta(t-t_0)dt = x(t_0)$. Used to model sudden shocks and to define the impulse response.

Unit step $u(t)$

$$u(t) = \begin{cases} 1, & t \ge 0 \\ 0, & t < 0 \end{cases}$$

Represents switching on a constant (e.g. closing a DC switch at $t=0$).

Unit ramp $r(t)$

$$r(t) = t\,u(t) = \begin{cases} t, & t \ge 0 \\ 0, & t < 0 \end{cases}$$

Increases linearly with slope 1 from $t = 0$.

 delta(t)        u(t)            r(t)
   ^ area 1     1 +-------          /
   |              |               /
   |              |             /  slope 1
 --+------ t    --+------- t  --+------- t
   0              0             0

Relationship

Each is the integral of the previous one:

$$u(t) = \int_{-\infty}^{t}\delta(\tau)\,d\tau, \qquad r(t) = \int_{-\infty}^{t}u(\tau)\,d\tau$$

and each is the derivative of the next one:

$$\delta(t) = \frac{du(t)}{dt}, \qquad u(t) = \frac{dr(t)}{dt}, \qquad \delta(t) = \frac{d^2 r(t)}{dt^2}$$

Impulse → (integrate) → step → (integrate) → ramp; differentiation goes the other way.