Minimizing First-Order Allpass Transient Response

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Physical Audio Signal Processing

First-Order Allpass Interpolation

Linear Interpolation as Resampling

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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

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

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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Any function that is not linear is nonlinear. There are no general methods for solving nonlinear equations or nonlinear differential equations. This is why it is usually easier, although generally less accurate, to approximate a nonlinear function by a linear function if possible. — Click for http://en.wikipedia.org/wiki/Nonlinear

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

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

Transients are short intervals during which the signal evolves quickly in some nontrivial or relatively unpredictable way (Bello, 2005). In computer music, the short attack portion of a sound is often treated as a transient signal. — Click for http://en.wikipedia.org/wiki/Transient_%28acoustics%29

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Physical Audio Signal Processing

First-Order Allpass Interpolation

Linear Interpolation as Resampling

Minimizing First-Order Allpass Transient Response

In addition to approaching a pole-zero cancellation at , another undesirable artifact appears as $\Delta\to0$ : The transient response also becomes long when the pole at $z=-\eta$ gets close to the unit circle.

A plot of the impulse response for $\Delta =0.1$ is shown in Fig.4.6. We see a lot of ``ringing'' near half the sampling rate. We actually should expect this from the nonlinear-phase distortion which is clearly evident near half the sampling rate in Fig.4.4. We can interpret this phenomenon as the signal components near half the sampling rate being delayed by different amounts than other frequencies, therefore ``sliding out of alignment'' with them.

**Figure 4.6:** Impulse response of the first-order allpass interpolator for $\Delta =0.1$ .
$\includegraphics[width=\twidth]{eps/ap1ir}$

For audio applications, we would like to keep the impulse-response duration short enough to sound ``instantaneous.'' That is, we do not wish to have audible ``ringing'' in the time domain near . For high quality sampling rates, such as larger than kHz, there is no issue of direct audibility, since the ringing is above the range of human hearing. However, it is often convenient, especially for research prototyping, to work at lower sampling rates where is audible. Also, many commercial products use such sampling rates to save costs.

Since the time constant of decay, in samples, of the impulse response of a pole of radius is approximately

$\displaystyle \frac{\tau}{T} \approx \frac{1}{1-R},$

and since a 60-dB decay occurs in about 7 time constants (`` $t_{60}$ '') [454, p. 38], we can limit the pole of the allpass filter to achieve any prescribed specification on maximum impulse-response duration.

For example, suppose 100 ms is chosen as the maximum $t_{60}$ allowed at a sampling rate of . Then applying the above constraints yields $\eta\leq 0.993$ , corresponding to the allowed delay range .

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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

Minimizing First-Order Allpass Transient Response

``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