Delay-Line Interpolation Summary

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Delay/Signal Interpolation

Summary of Windowed Sinc Interpolation

Time-Varying Delay Effects

The 'Cardinal sine function' is defined as sin(pi x)/(pi x) — Click for https://ccrma.stanford.edu/~jos/mdft/Sinc_Function.html

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

Click for https://ccrma.stanford.edu/~jos/Interpolation/Welcome.html

Click for http://www.circuitsage.com/filter.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Bandlimited interpolation is the ideal type of interpolation for properly sampled signals. — Click for https://ccrma.stanford.edu/~jos/resample/

Lagrange interpolation is interpolation based on passing a polynomial through the given sample-points. — Click for https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html

The group delay of an LTI filter is minus the phase response divided by frequency — Click for https://ccrma.stanford.edu/~jos/filters/Group_Delay.html

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

Allpass bandlimited interpolation is ideal with respect to amplitude, but generally has some delay error at high frequencies (usually inaudible). — Click for https://ccrma.stanford.edu/~jos/pasp/Thiran_Allpass_Interpolators.html

In digital signal processing, FIR is an acronym for Finite Impulse Response — Click for https://ccrma.stanford.edu/~jos/filters/FIR_Digital_Filters.html

Click for https://ccrma.stanford.edu/~jos/Interpolation/Welcome.html

A feedback signal creates a loop. That is, the term 'feedback' refers to when an output signal is 'fed back' to an input of the same system. — Click for http://en.wikipedia.org/wiki/Feedback

Click for https://mathworld.wolfram.com/LinearCombination.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

Click for https://ccrma.stanford.edu/~jos/pasp/Linear_Interpolation.html

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

Click for https://ccrma.stanford.edu/~jos/filters/Allpass_Filters.html

Click for https://ccrma.stanford.edu/~jos/pasp/Delay_Line_Interpolation.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Delay/Signal Interpolation

Summary of Windowed Sinc Interpolation

Time-Varying Delay Effects

Delay-Line Interpolation Summary

The most commonly used closed-form methods for delay-line interpolation may be summarized by the following table:

$\begin{displaymath} \begin{array}{c} \mbox{Order}\\ \begin{array}{\vert\vert r\vert\vert c\vert c\vert c\vert c\vert} \hline \hline & 1 & N & \mbox{Large $N$} & \infty \\ \hline \hline \mbox{FIR} & \mbox{Linear} & \mbox{Lagrange} & \mbox{Windowed Sinc} & \\ \hline \mbox{IIR} & \mbox{Allpass}_1 & \mbox{Thiran} & & \mbox{Sinc} \\ \hline \end{array}\end{array}\end{displaymath}$

In the first column, we have linear and first-order allpass interpolation, as discussed above in sections §4.1.1 and §4.1.2, respectively. These are the least-expensive methods computationally, and they find wide use, especially in audio applications with large sampling rates. While linear and first-order allpass interpolation cost the same, only linear interpolation offers ``random access mode''. That is, since the interpolated signal is a (finite) linear combination of known samples, the signal can be evaluated at any arbitrary time within the range of known samples -- the interpolation can ``jump around'' as desired. On the other hand, allpass interpolation has no gain error, so it may preferred inside a feedback loop to provide slowly varying fractional delay filtering.

In the second column -- th-order interpolation -- we list Lagrange for the FIR case (top) and Thiran for the IIR case (bottom), and these are introduced in §4.2 and §4.3, respectively. Lagrange and Thiran interpolators, properly implemented, enjoy the following advantages:

Gain bounded by 1 at all frequencies
Coefficients known in closed form as a function of desired delay
Maximally flat at low frequencies:

-

Lagrange: maximally flat frequency response at dc

-

Thiran: maximally flat group delay at dc

In the high-order FIR case, one should also consider ``windowed sinc'' interpolation (introduced in §4.4) as an alternative to Lagrange interpolation. In fact, as discussed in §4.2.16, Lagrange interpolation is a special case of windowed-sinc interpolation in which a scaled binomial window is used. By choosing different windows, optimalities other than ``maximally flat at dc'' can be achieved.

In the most general th-order case, the interpolation-filter impulse response may be designed to achieve any optimality objective, such as Chebyshev optimality (Fig.4.11). That is, design a digital filter (FIR or IIR) that approximates

$\displaystyle H_\Delta\left(e^{j\omega T}\right) = e^{-j\omega\Delta T},\quad \Delta = \hbox{Desired delay in samples}$

optimally in some sense, with coefficients tabulated over a range of $\Delta$ samples (and interpolated on lookup). Tabulated filter-designs of this nature, while generally giving the best interpolation quality, are not included in the above table because the filter-coefficients are not known in closed form.

FIR interpolators have the advantage that they can be used in ``random access'' mode. IIR interpolators, on the other hand, require a sequential stream of input samples and produce a sequential stream of interpolated signal samples (typically implementing a fractional delay). In IIR fractional-delay filters, the fractional delay must change slowly relative to the IIR duration.

Finally, we note in the last column of the above table that if ``order is no object'' ( $N\to\infty$ ), then the ideal bandlimited-interpolator impulse-response is simply a sampled sinc function, as discussed in §4.4.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``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

Delay-Line 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