Several symbols in the paper describe different realizations of the same operator language. The following distinctions are used throughout this guide.
| Object | Literal meaning | What it is not |
|---|---|---|
| \(S,\delta_0\) | Sample shift and unit impulse | A finite truncation of every \(\ell^2\) signal |
| \(P_{N,r}f\) | A masked signal on the original lattice | The shorter decimated channel |
| \(f_r[q]\) | The value \(f[Nq+r]\) | An average of neighboring samples |
| \(D\) | One channel-index delay | Necessarily one original-sample delay |
| \(E(D),R(D)\) | Finite Laurent analysis and synthesis matrices | Values at one test frequency |
| \(W_\alpha\) | The addressed Haar analysis operator | An edge detector by itself |
| \(W_\alpha^*W_\alpha\) | A projector on the original signal space | A scalar packet energy |
| \(\Lambda\) | A complete covering antichain of a finite tree | An arbitrary collection of nodes |
| \(j,\widehat j\) | Known and estimated discontinuity packets | Interchangeable hypotheses |
| \(\Gamma,z\) | Reflection coefficient and normalized impedance | Coefficients that can be linearly interchanged |
Three arithmetic conventions
FIR examples use bilateral sequences with finite support and zero extension. Haar examples use vectors of length \(16\), with lengths divisible by the required powers of two. The three-tap smoother and nonlinear neighbor example use periodic indexing. These boundary conventions are stated where they enter. None is silently substituted for another. The polynomial variable \(t\) on Visual 20 indexes coefficients. The delay symbol \(D\) acts on channels. The Smith variable \(z\) is impedance. Keeping them separate prevents a visual resemblance from becoming an invalid identity.
Read this as
A correct formula needs the correct object, product, norm, and boundary convention. A familiar-looking matrix does not supply them automatically.