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Error in the Absence of Interpolation

For comparison purposes, we derive the error incurred when no interpolation of the filter table is performed. In this case, assuming rounding to the nearest table entry, we have

\begin{eqnarray*}
t & = & t_0+\eta,\qquad \left\vert\eta\right\vert\leq {1\over 2} \\
\hat{h}(t) & = & h(t_0) \\
e(t) & = & h(t) - h(t_0) \\
& = & \eta h^\prime (t_0) + {1\over 2} \eta^2h^{\prime\prime}(t_0+\lambda \eta) \\
\left\vert e(t)\right\vert & \leq &{M_1\over2} + {M_2\over8},
\end{eqnarray*}

where $M_1 \isdef \max_{t} \vert h^\prime (t)\vert$. For the ideal lowpass, we have

\begin{displaymath}
h^\prime (t) = -{1\over\omega_L}\int_0^{\omega_L}\omega\sin(\omega t) d\omega
= {\omega_L t\cos(\omega_L t) - \sin(\omega_L t)\over \omega_L t^2} .
\end{displaymath}

Note that $h^\prime (L)=-1/L$ and $\vert h^\prime (t)\vert < \omega_L/2 = \pi/2L$. Thus M1=a/L where $1\leq a <\pi/2$. The no-interpolation error bound is then

\begin{displaymath}
\left\vert h^\prime (t)\right\vert \leq {a\over2L} + {\pi^2\over24L^2} < {0.7854\over L} +
{0.4113\over L^2}.
\end{displaymath}


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``The Digital Audio Resampling Home Page'', by Julius O. Smith III.
Copyright © 2014-01-10 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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