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Order 4 over a range of fractional delays

Figures 4.15 and 4.16 show amplitude response and phase delay, respectively, for 4th-order Lagrange interpolation evaluated over a range of requested delays from $ 1.5$ to $ 2.5$ samples in increments of $ 0.1$ samples. The amplitude response is ideal (flat at 0 dB for all frequencies) when the requested delay is $ 2$ samples (as it is for any integer delay), while there is maximum high-frequency attenuation when the fractional delay is half a sample. In general, the closer the requested delay is to an integer, the flatter the amplitude response of the Lagrange interpolator.

Figure 4.15: Amplitude responses, Lagrange interpolation, order 4, for the range of requested delays $ [1.5 : 0.1 : 2.5]$ , with $ 2.499$ thrown in as well (see next plot for why). From bottom to top, ignoring the almost invisible split in the bottom curve, the curves represent requested delays $ 1.5, 1.6, 1.7, 1.8, 1.9$ , and $ 2.0$ . Then, because the curve for requested delay $ 2+\eta $ is the same as the curve for delay $ 2-\eta $ , for $ \vert\eta \vert<1/2$ , the same curves, from top to bottom, represent requested delays $ 2.0, 2.1, 2.2, 2.3, 2.4$ and $ 2.5$ (which is nearly indistinguishable from $ 2.499$ ).
\includegraphics[width=0.9\twidth]{eps/tlagrange-4-ar}

Figure 4.16: Phase delays, Lagrange interpolation, order 4, for the range of requested delays $ [1.5 : 0.1 : 2.5]$ , and additionally $ 2.499$ .
\includegraphics[width=0.9\twidth]{eps/tlagrange-4-pd}

Note in Fig.4.16 how the phase-delay jumps discontinuously, as a function of delay, when approaching the desired delay of $ 2.5$ samples from below: The top curve in Fig.4.16 corresponds to a requested delay of 2.5 samples, while the next curve below corresponds to 2.499 samples. The two curves roughly coincide at low frequencies (being exact at dc), but diverge to separate integer limits at half the sampling rate. Thus, the ``capture range'' of the integer 2 at half the sampling rate is numerically suggested to be the half-open interval $ [1.5,2.5)$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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