Even-Order Lagrange Interpolation Summary

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Physical Audio Signal Processing

Lagrange Interpolation

Lagrange Frequency Response Magnitude Bound

Odd-Order Lagrange Interpolation Summary

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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Physical Audio Signal Processing

Lagrange Interpolation

Lagrange Frequency Response Magnitude Bound

Odd-Order Lagrange Interpolation Summary

Even-Order Lagrange Interpolation Summary

We may summarize some characteristics of even-order Lagrange fractional delay filtering as follows:

Two-sample bounded-by-1 delay-range instead of only one-sample
No gain zero at half the sampling rate for the middle delay
No phase-delay discontinuity when crossing the middle delay
Optimal (central) delay range is centered about an integer

To stay within the central, one-sample delay range for even-order Lagrange interpolators, the delay range should be

$\displaystyle \Delta\in\left(\frac{N}{2}-\frac{1}{2},\frac{N}{2}+\frac{1}{2}\right),$

where

is the order of the interpolator.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Even-Order Lagrange Interpolation Summary

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University