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

Relation of Interpolation Error to Quantization Error

If $h(t)\in[-1,1-2^{-n_c}]$ is approximated by hq(t) which is represented in two's complement fixed-point arithmetic, then

\begin{displaymath}
h_q(t_0) = -b_0 + \sum_{i=1}^{n_c-1} b_i 2^{-i},
\end{displaymath}

where $b_i\in\{0,1\}$ is the ith bit, and the worst-case rounding error is

\begin{displaymath}
\left\vert h(t)-h_q(t)\right\vert \leq 2^{-n_c}.
\end{displaymath}

Letting $h_q(t_i)=h(t_i)+\epsilon_i$, where $\vert\epsilon_i\vert\leq2^{-n_c}$, the interpolated look-up becomes

\begin{displaymath}
\hat{h}_q(t_0+\eta) = \overline{\eta }h_q(t_0) + \eta h_q(t_1)
= \hat{h}(t_0+\eta) + \overline{\eta }\epsilon_0 + \eta\epsilon_1.
\end{displaymath}

Thus the error in the interpolated lookup between quantized filter coefficients is bounded by

\begin{displaymath}
\left\vert e_q(t)\right\vert \leq \frac{M_2}{8} + 2^{-n_c},
\end{displaymath}

which, in the case of $h(t)=\mbox{sinc}(t/L)$, can be written

\begin{displaymath}
\left\vert e_q(t)\right\vert < {0.412\over L^2} + 2^{-n_c}
= 0.412 \cdot 2^{-2n_l} + 2^{-n_c}.
\end{displaymath}

If L=2nc/2, then $\vert e_q(t)\vert < 1.5\cdot 2^{-n_c}$.


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