Features of Modal State-Space Coordinates

• Input and output locations must be known and fixed

• The $i$ th mode of the system is simulated in isolation
  $$\begin{eqnarray*} {\bf x}_n[i] &=& \lambda_i {\bf x}_{n-1}[i] + b^\prime_i u_n\\ y_n &=& {{\bf C}^\prime} {\bf x}_n \end{eqnarray*}$$
  

  Notes:

  • the input gain $b^\prime_i$ specifies how the input signal $u_n$ excites mode $i$
  • the output state-gain-vector ${{\bf C}^\prime}$ specifies how all the modes add up (weighted) to form the output
  • The mode-input gain vector ${\bf B}^\prime = [b^\prime_1, \ldots,b^\prime_N]$ changes if the physical input is moved
  • ${{\bf C}^\prime}$ changes if the physical output is moved
  • The system modes do not change when the input or output are moved
• Dispersion modeling (mode tuning) is much easier than in a finite difference approximation

• Frequency-dependent damping is also much easier

• Extends to higher order models (time and/or space)

• Extends simply to spatially varying media
  (not Fourier based)

• Can also be used for implicit finite difference schemes

• Relatively efficient when modes are inharmonic
  (bells, gongs, mallet percussion)

• When mode frequencies are nearly harmonic (strings, woodwinds, brasses) digital waveguide models are more efficient (we'll take them up later).