Relation of Interpolation Error to Quantization Error

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

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

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

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

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

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

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

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

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

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

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