Computer Sound Synthesis 2 MUS_TECH 335 Selected Topics

Physical Modeling Synthesis

Karplus-Strong Karplus-Strong string synthesis loops a short noise burst through a filtered delay line to simulate the sound of a hammered or plucked string or some types of percussion. This technique using a feedback loop like a comb filter, especially like those used in reverberation simulation with a low-pass filter in the loop. 1.A short excitation waveform (generally of length L samples), often a burst of noise, is output and simultaneously fed back into a delay line L samples long. 2.The output of the delay line is fed through a low-pass filter, usually a first order one-pole filter. 3.The filtered output is simultaneously mixed back into the output and fed back into the delay line.

Karplus-Strong f f f time Transformation of the noise spectrum through time. The output gradually approaches a pure harmonic waveform.

Karplus-Strong The fundamental frequency is the reciprocal of the total delay, that is, the delay buffer (plus the group delay of the low-pass filter). The delay needed for a particular frequency is the sample rate divided by the frequency. If one uses a delay with no interpolation, the only frequencies available are integer divisions of the sampling rate--- certainly not good enough for equal temperament. Using linear interpolation would create a huge problem because it would add its own low-pass filtering, therefore higher-order interpolation is required.

Karplus-Strong Advantages: Efficient Realistic Every note different Disadvantages: Pitch depends on sampling rate and delay line interpolation

String Simulation Actual vibrations of string and air columns can be modeled with combined masses and springs--- either creating longitudinal waves or transverse waves, as with a string.

String Simulation In simulated longitudinal waves, there is string mass and the string tension that exerts a force to restore the displaced string toward its point of equilibrium.

String Simulation When the string is plucked, displacement waves travel up and down the string. At the fixed ends the waves reflects back with reversed polarity. time

Waveguides Digital waveguides are efficient computational models for physical media through which acoustic waves propagate. A basic one-dimensional digital waveguide models a string with with a rigid termination on one end (left) and a frequency-dependent attenuating filter at the other (right). A lossless digital waveguide realizes the discrete form of the one dimensional wave equation as the superposition of a right-going wave and a left-going wave.

Waveguides A general string waveguide can be related to the physical system as: z -m Bridge termination Finger board termination -b

Waveguides Losses incurred throughout the medium are generally consolidated so that they can be calculated once at the termination of a delay line, rather than many times throughout. The filter, R(z), captures the frequency- dependent losses and dispersion in the medium and can include non- linear elements. In the simplest possible realization, the filter can be replaced by multiplication with a coefficient less than 1. One way of creating the output of the waveguide is to add together the state of the on-going left and right delays, thereby representing the string. Alternatively, the output can be taken at either end.

A pluck or strike can be simulated by inserting a transient into the delay buffers. Thus we see that the Karplus-Strong model is a simplification of a waveguide. Waveguides

Although waveguides such as acoustic tubes may be thought of as three-dimensional, because their lengths are often much greater than their cross-sectional area, it is reasonable and computationally efficient to model them as one dimensional waveguides. Membranes, as used in drums, may be modeled using two-dimensional waveguide meshes, and reverberation in three dimensional spaces may be modeled using three- dimensional meshes.

Waveguides Making a physically modeled instrument consists of connecting a series of these waveguide sections. Note that each waveguide section has two signals going into it, and two signals going out. These two sets of signals model the waves traveling in one direction, and the waves traveling in the opposite direction. Each section contains two summers, six attenuators, and two delay lines. The delay lines models the time it takes for the wave to propagate from one end of the section to the other, in either direction. The attenuators with values g and -g model the losses that may occur in the waveguide. It these losses are frequency-dependent, they are modeled as filters.

Waveguides The other attenuators (with values of r, -r, 1+r and 1-r) model the scattering of the signal at the junction between two sections - part of the signal is transmitted across the junction, while the rest is reflected back in the opposite direction. These attenuations are also usually frequency dependant, that is, the relative proportion, r, of signal that is reflected at a junction depends on frequency. Thus these attenuators are often implemented with filters. Typically the reflection coefficient r has a lowpass characteristic (and therefore 1-r has a highpass characteristic), modeling the usual physical situation that high frequencies are transmitted preferentially, and low frequencies are reflected preferentially.