WDF State Space Interpretation

Digital filters can be expressed in state-space form as

$$\underline{x}(n+1) = A\, \underline{x}(n) + B\, \underline{u}(n)$$

by simply enumerating all delay elements as state variables $\underline{x}^T(n) = [x_1(n), x_2(n),\ldots, x_N(n)]$ , and finding the state transition matrix $A$ by inspection. Any inputs are collected in $\underline{u}(n)$ and determine the $B$ matrix.

• For WDFs, the $A$ matrix is a scattering matrix
• The $A$ matrix is orthogonal (lossless) for reactive elements (masses, springs)
• The state variables are all sampled traveling waves
• Physical state variables (bilinear transformed) are obtainable by summing (capacitors, springs) or subtracting (inductors, masses) the input and output of the unit delays:
  $$y_k(n) = x_k(n) \pm x_k(n-1)$$

• In comparison to other state-space models, WDF state-space form has top numerical properties due to its lossless scattering formulation