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Orders 1 to 5 on a fractional delay of 0.4 samples

Figure [*] shows the amplitude responses of Lagrange interpolation, orders 1 through 5, for the case of implementing an interpolated delay line of length $ 5.4$ 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.

Figure 4.13: Amplitude responses, Lagrange interpolation, orders 1 to 5, for an interpolated delay of $ 5.4$ samples. From the bottom-right corner along the right edge, the curves represent orders 1,3,5,4,2.
\includegraphics[width=0.9\twidth]{eps/tlagrange-1-to-5-ar-c}

Figure 4.14: Phase delays, Lagrange interpolation, orders 1 to 5, for an interpolated delay of $ 5.4$ samples. From bottom to top, the curves represent orders 2,4,1,3,5.
\includegraphics[width=0.9\twidth]{eps/tlagrange-1-to-5-pd-c}

In the phase-delay plots of Fig.[*], all cases are exact at frequency zero. At half the sampling rate they all give 5 samples of delay.

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.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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