Synthesizing Modes

• Though the mass-spring-damper approximation may appear overly simplistic, more complex vibrating systems can often be reasonably analyzed in terms of a set of independent exponentially decaying sinusoids.

• An exponentially decaying sinusoid can be represented as
    $$x(t) = e^{-\alpha t} A \cos(2 \pi f_c t + \phi),$$ (18)
  where $\alpha = 1 / \tau$ is a decay constant, $\tau$ is the time it takes for the signal to decay by a factor $e$, $A$ is the amplitude, $f_c$ is the natural radian frequency and $\phi$ is a phase offset.

• A decaying sinusoidal component will correspond to a peak, or resonance, in its spectrum and the width of the spectral peak will be directly proportional to the sinusoidal decay constant.

• Modal synthesis involves the use of second-order digital resonators to synthesize sounds that have a relatively small number of distinct, decaying resonances or modes.

• A second-order IIR filter can be designed to produce a single frequency magnitude peak at a specified frequency by setting its feedback coefficients as:
  • $a_1 = -2 r \cos( 2 \pi f T_s )$
  • $a_2 = r^2$

  where $r$ is a value between 0 – 1.0 that sets the pole radius and $T_s$ is the sample period.

• This filter can also be designed to have unity magnitude gain at the resonance peak with:
  • $b_0 = (1 - r^2) / 2$
  • $b_1 = 0$
  • $b_2 = -b_0$

• When “impulsed,” a digital resonator produces a decaying sinusoid, as shown in Fig. 16 below.

  Figure 16: A digital resonance filter: impulse response (top) and frequency magnitude response (bottom).

  

• A “bank” of resonance filters can be combined in parallel to simulate all the resonant modes of a system. Each resonator should have its own amplitude, center frequency, and rate of decay. The filter parameters are typically tuned and calibrated from measured recordings, and then excited with an appropriate “dry hit” (and/or residual) signal, as illustrated in Fig. 17.

  Figure 17: A traditional modal synthesis block diagram.

  

• Any object that exhibits just a few decaying strong modes and is excited by striking or plucking is a good candidate for model synthesis.

• A simple modal synthesis example (using arbitrary mode parameters) is provided in the Matlab script modal.m.