The ramp and step signals

Ramp and step are important signals for understanding some interesting properties of the impulse signal, which will be described in the next section. The step signal is zero for negative input and one for positive. Thus, it is defined by

$$\mathrm{step}(x) = \left\{ \begin{array}{ll} 1 & \textrm{if $x \geq 0$}\\ 0 & \textrm{if $x < 0$}\\ \end{array} \right.$$ (3.1)

The Step signal can be interpreted as a ramp of width zero, when the ramp is defined as follows

$$\mathrm{ramp}(x,w) = \left\{ \begin{array}{ll} 1 & \textrm... ...eq x < w$}\\ 0 & \textrm{if $x < 0$}\\ \end{array} \right.$$ (3.2)

As can easily be seen, if w tends to zero the ramp becomes a step signal. This relationship is important because we cannot differentiate a step signal because it is discontinuous. However, the ramp signal is continuous and therefore differentiable, its differentiation is a box function (see Fig. 3.1).

Figure 3.1: On the left the step is function depicted, in the middle the ramp, and its derivative (the box function) on the right. The smaller w gets the higher and narrower the corresponding box function gets (until it reaches the impulse signal in the limit).