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Trapezoidal Rule

The trapezoidal rule is defined by

$\displaystyle \underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + T\, \frac{\dot{\underline{\hat{x}}}(n) + \dot{\underline{\hat{x}}}(n-1)}{2}. \protect$ (8.13)

Thus, the trapezoidal rule is driven by the average of the derivative estimates at times $ n$ and $ n-1$ . The method is implicit in either forward or reverse time.

The trapezoidal rule gets its name from the fact that it approximates an integral by summing the areas of trapezoids. This can be seen by writing Eq.(7.13) as

$\displaystyle \underline{\hat{x}}(n) = \underline{\hat{x}}(n-1) + T\,\dot{\underline{\hat{x}}}(n-1) + \frac{1}{2} T\,\left[\dot{\underline{\hat{x}}}(n) - \dot{\underline{\hat{x}}}(n-1)\right] $

Imagine a plot of $ \dot{\underline{\hat{x}}}(n)$ versus $ n$ , and connect the samples with linear segments to form a sequence of trapezoids whose areas must be summed to yield an approximation to $ \underline{\hat{x}}(n)$ . Then the integral at time $ n$ , $ \underline{\hat{x}}(n)$ , is given by the integral at time $ n-1$ , $ \underline{\hat{x}}(n-1)$ , plus the area of the next rectangle, $ T\,\dot{\underline{\hat{x}}}(n-1)$ , plus the area of the new triangular piece atop the new rectangle, $ T\,[\dot{\underline{\hat{x}}}(n) - \dot{\underline{\hat{x}}}(n-1)]/2$ . In other words, the integral at time $ n$ equals the integral at time $ n-1$ plus the area of the next trapezoid in the sum.

An interesting fact about the trapezoidal rule is that it is equivalent to the bilinear transform in the linear, time-invariant case. Carrying Eq.(7.13) to the frequency domain gives

\begin{eqnarray*} X(z) &=& z^{-1}X(z) + T\, \frac{s X(z) + z^{-1}s X(z)]}{2}\\ \Longleftrightarrow\quad s &=& \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}. \end{eqnarray*}

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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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