Triangular Pulse as Convolution of Two Rectangular Pulses

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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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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