Triangular Pulse as Convolution of Two Rectangular Pulses

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

**Figure 4.8:** The width rectangular pulse.
$\includegraphics{eps/rectpulse}$

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

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

where

is the Heaviside unit step function:

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

with itself produces the two-sample triangular pulse

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

While the result can be verified algebraically by substituting

for

, it seen more quickly via graphical convolution.

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Linear Interpolation Frequency Response
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Frequency Response of Linear Interpolation

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