Relation of Interpolation Error to Quantization Error

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Relation of Interpolation Error to Quantization Error

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

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

where [画像:$b_i\in\{0,1\}$] 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 [画像:$h_q(t_i)=h(t_i)+\epsilon_i$], where [画像:$\vert\epsilon_i\vert\leq2^{-n_c}$], 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
[画像:\begin{displaymath} \left\vert e_q(t)\right\vert \leq \frac{M_2}{8} + 2^{-n_c}, \end{displaymath}]

which, in the case of [画像:$h(t)=\mbox{sinc}(t/L)$], can be written
[画像:\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 [画像:$\vert e_q(t)\vert < 1.5\cdot 2^{-n_c}$].

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