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The Triangular Pulse as a Convolution of Two Rectangular Pulses

The 2-sample wide triangular pulse $ h_l(t)$ (Eq. (I.5)) can be expressed as a convolution of the one-sample rectangular pulse with itself.

Figure I.14: The width $ T$ rectangular pulse.
\begin{figure}\input fig/rectpulse.pstex_t
\end{figure}

The one-sample rectangular pulse is shown in Fig. I.14 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 $ u(t)$ 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..
$

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

$\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 $ u(t+T/2)-u(t-T/2)$ for $ p_T(t)$, it seen more quickly via graphical convolution.


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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