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 [523]. 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 [523]. Thus, the stringed instrument model may consist of noise generator $\rightarrow$ excitation amplitude-shaping $\rightarrow$ time-varying lowpass $\rightarrow$ string model $\rightarrow$ parametric resonators $\rightarrow$ 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 [515].

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 [238,208]. 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 [441, 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.