Multiple Force-Pulse Synthesis

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

A lowpass filter rejects high frequencies and does not affect low frequencies. — Click for http://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_I.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

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

A tapped delay line is like a water pipe having a series of holes along its side where water is 'tapped out'. — Click for https://ccrma.stanford.edu/~jos/pasp/Tapped_Delay_Line_TDL.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

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

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

Multiple Force-Pulse Synthesis

One method of generating multiple force pulses as shown in Fig.9.31 is to sum several single-pulse filters, as shown in Fig.9.33. Furthermore, the three input impulses can be generated using a single impulse into a tapped delay line (§2.5). The sum of the three filter outputs gives the desired superposition of three hammer-string force pulses. As the collision velocity

increases, the output pulses become taller and thinner, showing less overlap. The filters LPF1-LPF3 can be given

as side information, or the input impulse amplitudes can be set to

, or the like.

**Figure 9.33:** Multiple hammer-string interaction force-pulse as the superposition of impulse-responses of a parallel lowpass-filter (LPF) bank.
$\includegraphics[width=\twidth]{eps/pianoForceFilterPulses}$

Multiple Force-Pulse Synthesis

``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