It is desirable that the stored filter impulse response be sampled sufficiently densely so that interpolating linearly between samples does not introduce error greater than the quantization error. We will show that this condition is satisfied whenever the filter table contains at least entries per zero-crossing, where is the number of bits allocated to each table entry.
Linear Interpolation Error Bound.
Let denote the lowpass filter impulse response, and assume it is twice continuously differentiable for all . By Taylor's theorem [155, p. 119], we have
for some , where denotes the time derivative of evaluated at , and is the second derivative at .The linear interpolation error is defined as
Expressing as
Application to the Ideal Lowpass Filter.
For the ideal lowpass filter, we have
where , and is the number of table entries per zero-crossing. Note that the rightmost form in Eq. (I.3.4) is simply the inverse Fourier transform of the ideal lowpass-filter frequency response. Twice differentiating with respect to , we obtain from which it follows that the maximum magnitude is Note that this bound is attained at . Substituting Eq. (I.3.4) into Eq. (I.3.4), we obtain the error bound Thus for the ideal lowpass filter sinc, the pointwise error in the interpolated lookup of is bounded by . This means that must be about half the coefficient word-length used for the filter coefficients. For example, if is quantized to bits, must be of the order of . In contrast, we will show that without linear interpolation, must increase proportional to for -bit samples of . In the -bit case, this gives . The use of linear interpolation of the filter coefficients reduces the memory requirements considerably.The error bounds obtained for the ideal lowpass filter are typically accurate also for lowpass filters used in practice. This is because the error bound is a function of , the maximum curvature of the impulse response , and most lowpass designs will have a value of very close to that of the ideal case. The maximum curvature is determined primarily by the bandwidth of the filter since, generalizing equations Eq. (I.3.4) and Eq. (I.3.4),
Relation of Interpolation Error to Quantization Error.
If is approximated by which is represented in two's complement fixed-point arithmetic, then
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
where . For the ideal lowpass, we have