Triangular Pulse as Convolution of Two Rectangular Pulses

The 2-sample wide triangular pulse $h_l(t)$ (Eq.(4.4)) can be expressed as a convolution of the one-sample rectangular pulse with itself.

Figure 4.8: The width $T$ rectangular pulse.

The one-sample rectangular pulse is shown in Fig.4.8 and may be defined analytically as

$$p_T(t) \isdef u\left(t+\frac{T}{2}\right) - u\left(t-\frac{T}{2}\right),$$

where $u(t)$ is the Heaviside unit step function:

$$u(t) \isdef \left\{\begin{array}{ll} 1, & t\geq 0 \\ [5pt] 0, & t<0 \\ \end{array} \right..$$

Convolving $p_T(t)$ with itself produces the two-sample triangular pulse $h_l(t)$ :

$$h_l(t) = (p_T\ast p_T)(t) \isdef \int_{-\infty}^{\infty} p_T(\tau)p_T(t-\tau)d\tau$$

While the result can be verified algebraically by substituting $u(t+T/2)-u(t-T/2)$ for $p_T(t)$ , it seen more quickly via graphical convolution.