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Order 5 over a range of fractional delays

Figures 4.17 and 4.18 show amplitude response and phase delay, respectively, for 5th-order Lagrange interpolation, evaluated over a range of requested delays between $ 2$ and $ 3$ samples in steps of $ 0.1$ samples. Note that the vertical scale in Fig.4.17 spans $ 100$ dB while that in Fig.4.15 needed less than $ 9$ dB, again due to the constrained zero at half the sampling rate for odd-order interpolators at the half-sample point.

Figure 4.17: Amplitude responses, Lagrange interpolation, order 5, for the range of requested delays $ [2.0 : 0.1 : 3.0]$ , with $ 2.495$ and $ 2.505$ included as well (see next plot for why).
\includegraphics[width=0.9\twidth]{eps/tlagrange-5-ar}

Figure 4.18: Phase delays, Lagrange interpolation, order 5, for the range of requested delays $ [2.0 : 0.1 : 3.0]$ , with $ 2.495$ and $ 2.505$ included as well.
\includegraphics[width=0.9\twidth]{eps/tlagrange-5-pd}

Notice in Fig.4.18 how suddenly the phase-delay curves near 2.5 samples delay jump to an integer number of samples as a function of frequency near half the sample rate. The curve for $ 2.495$ samples swings down to 2 samples delay, while the curve for $ 2.505$ samples goes up to 3 samples delay at half the sample rate. Since the gain is zero at half the sample rate when the requested delay is $ 2.5$ samples, the phase delay may be considered to be exactly $ 2.5$ samples at all frequencies in that special case.


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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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