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#### Linear Interpolation Error Bound

Let h(t) denote the lowpass filter impulse response, and assume it is twice continuously differentiable for all t. By Taylor's theorem [#!Goldstein!#, p. 119], we have (6)

for some , where denotes the time derivative of h(t) evaluated at t=t0, and is the second derivative at t0.

The linear interpolation error is (7)

where , , , and is the interpolated value given by (8)

where and . Thus t0 and t1 are successive time instants for which samples of h(t) are available, and is the linear interpolation factor.

By definition, e(t0)=e(t1)=0. That is, the interpolation error is zero at the known samples. Let te denote any point at which |e(t)| reaches a maximum over the interval (t0,t1). Then we have Without loss of generality, assume . (Otherwise, replace t0 with t1 in the following.) Since both h(t) and are twice differentiable for all , then so is e(t), and therefore e'(te)=0. Expressing e(t0)=0 as a Taylor expansion of e(t) about t=te, we obtain for some . Solving for e(te) gives Defining where the maximum is taken over , and noting that , we obtain the upper bound2 (9)

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