Discretization

• Closed network defines a set of ODEs in the state variables (solution may not exist or be unique)
• Need a discretization method (for digital filtering, or simulation)

Wave Digital Filters are based on the application of a spectral mapping or bilinear transform:

$$s\rightarrow \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

which takes the $s$ RHP to the $z$ outer disk:

$T$ is the time step and $\frac{1}{T}$ is the sampling frequency.

• Stable causal analog filters mapped to stable causal digital filters. Also: DC $\rightarrow$ DC, $\infty\rightarrow$ Nyquist.
• Discrete ``impedances'' inherit passivity property (now called pseudo-passivity), and are called positive real in the outer disk.

Trapezoid Rule

In the time domain, this bilinear transform is equivalent to applying the trapezoid rule in order to integrate (or differentiate).

• Example: for an inductor

  $$\hat{v} = L\frac{d\hat{i}}{dt}$$

  
  
  Under bilinear transformation:

  $$Z(s) = Ls \rightarrow \frac{2L}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

  
  
  In time domain:

  $$\frac{v(n)+v(n-1)}{2} = L\frac{i(n)-i(n-1)}{T}$$

  
  

• Accurate to order $T^{2}$
• The above difference equation defines a digital $one-port$ , with discrete-time voltage and current $v$ and $i$ .
• Only reactive elements are affected by this transformation (i.e. not resistors, ideal transformers, or Kirchoff's Laws).

Connecting Digital N-ports

• Can connect digital one-ports using Kirchoff's Laws, and still have power conservation. If elements are pseudopassive, then so is a network constructed from such elements.
• Example: parallel $LC$ connection

  
  
  
  
  

  
  
  

• Problem: Delay-free loops.
• Result: Signal-flow diagrams are non-realizable (unless one is willing to perform matrix inversions).

Wave Variables

It is possible to get around these realizability problems by introducing wave variables (1971) borrowed from microwave engineering

Introduce, for any port variables $v$ and $i$ , the quantities

$$a = v+iR$$

$$b = v-iR$$

$R$ is an arbitrary positive constant, called the port resistance. We can also define power-normalized wave variables as

$$a' = \frac{v+iR}{2\sqrt{R}}$$

$$b' = \frac{v-iR}{2\sqrt{R}}$$

The two types of waves are simply related to eachother by

$$a = 2\sqrt{R}a'$$

$$b = 2\sqrt{R}b'$$

Power waves are useful in dealing with time-varying and nonlinear circuit elements.

Example: Wave Digital Inductor

From the trapezoid rule (bilinear transform) we have

$$\frac{v(n)+v(n-1)}{2} = L\frac{i(n)-i(n-1)}{T}$$

Inserting wave variables, we get:

$$a(n+1)+b(n+1)+a(n)+b(n) = \frac{2L}{RT}\left(a(n+1)-b(n+1)-a(n)+b(n)\right)$$

And under the choice $R = \frac{2L}{T}$ , we get

$$b(n) = -a(n-1)$$

A strictly causal input/output relationship. (Same in power-normalized case).

• Energetic interpretation:
  

  $$p_{inst}(n) = v(n)i(n)$$

  $$= a'(n)^{2}-b'(n)^{2}$$

  $$= a'(n)^{2}-a'(n-1)^{2}$$

  
  
• Square of value in delay register has interpretation as stored energy

Wave-Digital Elements

We can derive wave digital equivalents of the standard circuit elements.

Adaptors

Connections between the elements are governed, as before, by Kirchoff's Laws. In terms of wave variables, we have, for a connection of $k$ ports:

$$b_{m} = a_{m} -\frac{2R_{m}}{\sum_{j=1}^{k}R_{j}}\sum_{j=1}^{k}a_{j}, m=1\hdots k {\mbox {\rm Series Connection}}$$

$$b_{m} = -a_{m} +\frac{2}{\sum_{j=1}^{k}G_{j}}\sum_{j=1}^{k}G_{j}a_{j}, m=1\hdots k {\mbox {\rm Parallel Connection}}$$

where $G_{j} = \frac{1}{R_{j}}$ is the conductance at port $j$ . The signal processing block which carries out this operation is called an adaptor

Scattering Matrices

The series and parallel adaptor equations can be written as

$${\bf b} = {\bf Sa}$$

where

$$\begin{array}{rccll} {\mathbf S} &=& & {\mathbf I} - {\mathbf \alpha}^{T}{\bf 1}\hspace{0.5in}&{\mbox {\rm {(Series)}}}\nonumber \\ {\bf S} &=& - & {\mathbf I} + {\mathbf 1}^{T}{\mathbf \beta}\hspace{0.5in}&{\mbox {\rm (Parallel)}}\nonumber \end{array}$$

where ${\bf\alpha} = \frac{2}{\sum_{i=1}^{k}R_{i}}\left(R_{1}\hdots R_{k}\right)$ , ${\bf\beta} = \frac{2}{\sum_{i=1}^{k}G_{i}}\left(G_{1}\hdots G_{k}\right)$ we have also: ${\bf S^{2}=I}$ in either case. For power-normalized waves:

$${\mathbf S'} = {\mathbf I} - {\mathbf \alpha'}^{T}{\bf\alpha'}\hspace{0.5in} {\mbox {\rm { Series}}}$$

$${\bf S'} = -{\mathbf I} + {\mathbf \beta'}^{T}{\mathbf \beta'}\hspace{0.5in} {\mbox {\rm Parallel}}$$

where $\alpha' = \sqrt{\alpha}$ and $\beta' = \sqrt{\beta}$ (over all components). We have ${\bf S'}^{T}{\bf S'} = {\bf S'}{\bf S'}^{T} = {\bf I}$ (orthonormal, unitary).

• Form of the adaptor equation is simple (O(N) adds, multiplies)
• Easy to apply rounding rules so that junction behaves passively, even in finite arithmetic:
  • signals:Extended precision within junction. Magnitude truncation on outputs
  • reflection parameters ( ${\bf\alpha}, {\bf\beta}$ ) may also be truncated without affecting passivity (though accuracy will of course suffer).