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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.
\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 $ 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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