First-Order Allpass Interpolation

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Delay-Line Interpolation

Fractional Delay Filtering by Linear Interpolation

Minimizing First-Order Allpass Transient Response

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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

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_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

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.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

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

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

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

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The difference equation is a recipe for computing samples of the output signal of a digital filter based on samples of the input signal and the filter coefficients. In an Infinite-Impulse-Response (IIR) digital filter, there are typically both feedforward and feedback coefficients in the difference equation. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

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A delay line is used to delay a signal in time. Delay lines for digital signals are typically implemented in software using a circular buffer. — Click for https://ccrma.stanford.edu/~jos/pasp/Delay_Lines.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Delay-Line Interpolation

Fractional Delay Filtering by Linear Interpolation

Minimizing First-Order Allpass Transient Response

First-Order Allpass Interpolation

A delay line interpolated by a first-order allpass filter is drawn in Fig.4.3.

**Figure 4.3:** Allpass-interpolated delay line.
$\includegraphics[width=\twidth]{eps/delayai}$

Intuitively, ramping the coefficients of the allpass gradually ``grows'' or ``hides'' one sample of delay. This tells us how to handle resets when crossing sample boundaries.

The difference equation is

$\begin{eqnarray*} {\hat x}(n-\Delta) \isdef y(n) &=& \eta \cdot x(n) + x(n-1) - \eta \cdot y(n-1) \\ &=& \eta \cdot \left[ x(n) - y(n-1)\right] + x(n-1). \end{eqnarray*}$

Thus, like linear interpolation, first-order allpass interpolation requires only one multiply and two adds per sample of output.

The transfer function is

$\displaystyle H(z) = \frac{\eta + z^{-1}}{1 + \eta z^{-1}}. \protect$

(5.2)

At low frequencies ( $z\to 1$ ), the delay becomes

$\displaystyle \Delta \approx \frac{1-\eta}{1+\eta} \protect$

(5.3)

Figure 4.4 shows the phase delay of the first-order digital allpass filter for a variety of desired delays at dc. Since the amplitude response of any allpass is 1 at all frequencies, there is no need to plot it.

**Figure 4.4:** Allpass-interpolation phase delay for a variety of desired delays (exact at dc).
$\includegraphics[width=\twidth]{eps/allpass1}$

The first-order allpass interpolator is generally controlled by setting its dc delay to the desired delay. Thus, for a given desired delay $\Delta$ , the allpass coefficient is (from Eq.(4.3))

$\displaystyle \eta \approx \frac{1-\Delta}{1+\Delta}$

From Eq.(4.2), we see that the allpass filter's pole is at $z=-\eta$ , and its zero is at $z=-1/\eta$ . A pole-zero diagram for $\Delta =0.1$ is given in Fig.4.5. Thus, zero delay is provided by means of a pole-zero cancellation! Due to inevitable round-off errors, pole-zero cancellations are to be avoided in practice. For this reason and others (discussed below), allpass interpolation is best used to provide a delay range lying wholly above zero, e.g.,

$\displaystyle \Delta\in[0.1,1.1] \;\longleftrightarrow\; \eta\in[-0.05,0.82]$

**Figure 4.5:** Allpass-interpolator pole-zero diagram for $\Delta =0.1$ .
$\includegraphics[width=\twidth]{eps/ap1pz}$

Note that, unlike linear interpolation, allpass interpolation is not suitable for ``random access'' interpolation in which interpolated values may be requested at any arbitrary time in isolation. This is because the allpass is recursive so that it must run for enough samples to reach steady state. However, when the impulse response is reasonably short, as it is for delays near one sample, it can in fact be used in ``random access mode'' by giving it enough samples with which to work.

The STK class implementing allpass-interpolated delay is DelayA.

Subsections

Minimizing First-Order Allpass Transient Response

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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

First-Order Allpass Interpolation

``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