Approximating Shortened Excitations as Noise

Figure 8.14b suggests that the many
damped modes remaining in the shortened body impulse response may not
be clearly audible as resonances since their decay time is so short.
This is confirmed by listening to shortened and spectrally flattened
body responses which sound somewhere between a click and a noise
burst. These observations suggest that the shortened, flattened, body
response can be replaced by a psychoacoustically equivalent
*noise burst*, as suggested for modeling piano soundboards
[522]. Thus, the signal of
Fig. 8.14b can be synthesized
qualitatively by a white noise generator multiplied by an amplitude
envelope. In this technique, it is helpful to use LP on the residual
signal to flatten it. As a refinement, the noise burst can be
time-varying filtered so that high frequencies decay faster
[522]. Thus, the stringed instrument model may consist of
*noise generator
excitation amplitude-shaping
time-varying lowpass
string model
parametric resonators
multiple outputs.*
In addition, properties of the physical excitation may be
incorporated, such as comb filtering to obtain a virtual pick or
hammer position. Multiple outputs
provide for individual handling of the most important resonant modes
and facilitate simulation of directional radiation characteristics
[514].

It is interesting to note that by using what is ultimately a noise-burst excitation, we have almost come full circle back to the original extended Karplus-Strong algorithm [239,209]. Differences include (1) the amplitude shaping of the excitation noise to follow the envelope of the impulse-response of the highly damped modes of the guitar body (convolved with the physical string excitation), (2) more sophisticated noise filtering which effectively shapes the noise in the frequency domain to follow the frequency response of the highly damped body modes (times the excitation spectrum), and (3) the use of factored-out body resonances which give real resonances such as the main Helmholtz air mode. The present analysis also provides a theoretical explanation as to why use of a noise burst in the Karplus-Strong algorithm is generally considered preferable to a theoretically motivated initial string shape such as the asymmetric triangular wave which leans to the left according to pick position [442, p. 82].

From a psychoacoustical perspective, it may be argued that the
excitation noise burst described above is not perceptually
uniform. The amplitude envelope for the noise burst provides
noise-shaping in the time domain, and the LP filter provides
noise-shaping in the frequency domain, but only at
*uniform resolution* in time and frequency. Due to properties of
hearing, it can be argued that *multi-resolution* noise shaping
should be used. Thus, the LP fit for obtaining the noise-shaping
filter should be carried out over a Bark frequency axis as in
Fig. 8.19b. Since LP operates on the
autocorrelation function, a warped autocorrelation can be computed
simply as the inverse FFT of the warped squared-magnitude spectrum.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University