Figure shows the amplitude responses of Lagrange interpolation, orders 1 through 5, for the case of implementing an interpolated delay line of length samples. In all cases the interpolator follows a delay line of appropriate length so that the interpolator coefficients operate over their central one-sample interval. Figure shows the corresponding phase delays. As discussed in §4.2.10, the amplitude response of every odd-order case is constrained to be zero at half the sampling rate when the delay is half-way between integers, which this example is near. As a result, the curves for the two even-order interpolators lie above the three odd-order interpolators at high frequencies in Fig.. It is also interesting to note that the 4th-order interpolator, while showing a wider ``pass band,'' exhibits more attenuation near half the sampling rate than the 2nd-order interpolator.
Note that all three odd-order phase delay curves look generally better in Fig. than both of the even-order phase delays. Recall from Fig. that the two even-order amplitude responses outperformed all three odd-order cases. This illustrates a basic trade-off between gain accuracy and delay accuracy. The even-order interpolators show generally less attenuation at high frequencies (because they are not constrained to approach a gain of zero at half the sampling rate for a half-sample delay), but they pay for that with a relatively inferior phase-delay performance at high frequencies.
Order 4 over a range of fractional delays