Error in the Absence of Interpolation

For comparison purposes, we derive the error incurred when no interpolation of the filter table is performed. In this case, assuming rounding to the nearest table entry, we have

$$\begin{eqnarray*} t & = & t_0+\eta,\qquad \left\vert\eta\right\vert\leq {1\over 2} \\ \hat{h}(t) & = & h(t_0) \\ e(t) & = & h(t) - h(t_0) \\ & = & \eta h^\prime (t_0) + {1\over 2} \eta^2h^{\prime\prime}(t_0+\lambda \eta) \\ \left\vert e(t)\right\vert & \leq &{M_1\over2} + {M_2\over8}, \end{eqnarray*}$$

where $M_1 \isdef \max_{t} \vert h^\prime (t)\vert$. For the ideal lowpass, we have

$$h^\prime (t) = -{1\over\omega_L}\int_0^{\omega_L}\omega\sin(\omega t) d\omega = {\omega_L t\cos(\omega_L t) - \sin(\omega_L t)\over \omega_L t^2} .$$

Note that $h^\prime (L)=-1/L$ and $\vert h^\prime (t)\vert < \omega_L/2 = \pi/2L$. Thus M₁=a/L where $1\leq a <\pi/2$. The no-interpolation error bound is then

$$\left\vert h^\prime (t)\right\vert \leq {a\over2L} + {\pi^2\over24L^2} < {0.7854\over L} + {0.4113\over L^2}.$$