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

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

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