Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Quantized Interpolation Error Bound

The quantized interpolation factor and its complement are representable as

\begin{eqnarray*}
\eta_q&=&\eta + \nu \\
\overline{\eta }_q &=& \overline{\eta }- \nu
\end{eqnarray*}

where, since $\eta,\overline{\eta }$ are unsigned, $\vert\nu\vert\leq 2^{-({n_\eta }+1)}$. The interpolated coefficient look-up then gives

\begin{eqnarray*}
\hat{h}_{qq}(t) &=& (\overline{\eta }-\nu)[h(t_0)+\epsilon_0] + (\eta+\nu)[h(t_1)+\epsilon_1] \\
&=& \hat{h}(t) + \overline{\eta }\epsilon_0 + \eta\epsilon_1 + \nu[h(t_1)-h(t_0)],
\end{eqnarray*}

where second-order errors $\nu\epsilon_0$ and $\nu\epsilon_1$ are dropped. Since $\vert h(t_1)-h(t_0)\vert\leq M_1$, we obtain the error bound

\begin{displaymath}
\left\vert e_{qq}(t)\right\vert\leq 2^{-n_c} + 2^{-({n_\eta }+1)}M_1 + {1\over8}M_2.
\protect
\end{displaymath} (14)

The three terms in Eq.$\,$(14) are caused by coefficient quantization, interpolation quantization, and linear-approximation error, respectively.


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download resample.pdf
[How to cite and copy this work]  [Comment on this page via email]

``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
CCRMA