Choice of Interpolation Resolution

We now consider the error due to finite precision in the linear interpolation between stored filter coefficients. We will find that the number of bits ${n_\eta }$ in the interpolation factor should be about half the filter coefficient word-length ${n_c}$.

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

$$\begin{eqnarray*} \hat{h}_{qq}(t) &=& (\overline{\eta }-\nu)[h(t_0)+\epsilon _0]... ...rline{\eta }\epsilon _0 + \eta \epsilon _1 + \nu[h(t_1)-h(t_0)], \end{eqnarray*}$$

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

$$\left\vert\tilde{h}_{qq}(t)\right\vert\leq 2^{-{n_c}} + 2^{-({n_\eta }+1)}M_1 + {3\over8}M_2. \protect$$ (31)

The three terms in Eq.$\,$(31) are caused by coefficient quantization, interpolation quantization, and linear-approximation error, respectively.

Ideal Lowpass Filter.

For the ideal lowpass, the error bound is

$$\left\vert\tilde{h}_{qq}(t)\right\vert \leq 2^{-{n_c}} + a 2^{-({n_l}+{n_\eta }+1)} + {\pi^2\over 8} 2^{-{n_l}}.$$ (32)

Let ${n_l}=1+{n_c}/2$ and require that the added error is at most ${1\over2}2^{-{n_c}}$. Then we arrive at the requirement

$${n_\eta }\geq {{n_c}\over2}.$$ (33)