Causality, data use, and signal roles

This page fixes the vocabulary used throughout the examples. The package contains runtime filters, batch estimators, offline diagnostics, and finite-block transforms. They can all be useful, but they make different promises about what data are available when an output or prediction is formed.

Causal one-step prediction

In the prediction examples, causal means that a prediction for the current sample or vector is made before the current value is observed. For a scalar signal,

\[\widehat y[n] = g\bigl(y[n-1], y[n-2], \ldots\bigr).\]

For a multichannel signal \(y[n]\in\mathbb{R}^c\), causal MIMO prediction allows cross-channel history but still forbids future samples:

\[\widehat y[n] = g\bigl(y[n-1], y[n-2], \ldots\bigr), \qquad y[n-k]\in\mathbb{R}^c,\ k\ge 1.\]

Equivalently, a vector AR predictor has

\[\widehat y[n] = -\sum_{k=1}^p A_k y[n-k], \qquad A_k\in\mathbb{R}^{c\times c}.\]

The matrices \(A_k\) may mix channels, so channel 0 at time n may use channel 2 at time n-1. It may not use channel 2 at time n or any future time unless the task is explicitly noncausal/offline.

Causal filtering

For input-output filtering, causal means the output at time n uses only the current input, previous inputs, and previous state:

\[y[n] = \sum_{i=0}^{q} b_i x[n-i] - \sum_{j=1}^{p} a_j y[n-j].\]

With the impulse-response convention

\[y[n] = \sum_{k\in\mathbb{Z}} h[k] x[n-k],\]

causality means \(h[k]=0\) for \(k<0\), because negative k would use future input samples \(x[n+|k|]\).

The scalar lattice filters, lattice-ladder filters, streaming block processor, state-space simulator, online MIMO lattice predictor, and online matrix-lattice all-pass runtime are causal in this sense.

Batch estimation versus causal use

An estimator may use a full finite training record and still produce a model that is later used causally. This is the case for Burg, Levinson-Durbin, block Levinson-Durbin, and finite-Hankel/model-reduction routines.

The distinction is:

Step	Data-use promise
Coefficient estimation	May use a complete training record, covariance sequence, impulse response, or Markov tensor.
Held-out prediction/filtering	Uses the fitted coefficients with no lookahead on the held-out stream.
Frequency-response or Hankel diagnostics	Offline analysis of a finite object; useful for inspection, not a streaming runtime by itself.

In docs language, batch or offline describes the estimation/diagnostic step. Causal describes the runtime or prediction step.

Noncausal and finite-block transforms

Some examples intentionally use finite blocks or frequency-domain multiplication:

\[Y[\ell] = H(e^{j\omega_\ell}) X[\ell].\]

This is a circular/block operation unless embedded in an overlap-add or state-space runtime. It is useful for checking unitarity, adjoint identities, perfect-reconstruction diagnostics, and compression of frequency-dependent responses. It should not be read as a sample-by-sample streaming filter unless the page explicitly uses a streaming runtime such as OnlineMatrixLatticeAllPass.

For all-pass systems there is also an important inverse distinction. The forward all-pass can be causal and stable. Its time-domain impulse response has

\[y[n] = \sum_{k\ge 0} H_k x[n-k].\]

The finite-record adjoint used for reconstruction diagnostics is

\[x_{adj}[n] = \sum_{k\ge 0} H_k^H y[n+k].\]

That adjoint is time-domain but noncausal as an online inverse because it needs future transformed samples. The docs therefore separate causal forward runtime from finite-record adjoint/perfect-reconstruction diagnostic.

Transductive, inductive, and held-out language

The examples avoid using future samples of the held-out test record when they claim online prediction. A useful vocabulary is:

Term	Meaning in these docs
Online causal	Output or prediction is produced left-to-right with no future samples from the same stream.
Inductive evaluation	Fit coefficients on a training record, then run causally on a separate held-out record.
Transductive/offline evaluation	Use the whole record being evaluated to choose coefficients or a transform before reporting results on that same record.
Block/circular	Process a complete finite block, usually through an FFT or dense matrix operator.

Transductive experiments can be valid diagnostics, especially for response compression or finite-block unitary layers, but they are not the same claim as online prediction on a held-out stream.

Single-signal prediction versus two-signal filtering

Prediction uses one observed signal. At time n the model predicts the current sample from its own past:

\[\widehat y[n] = g(y[n-1], y[n-2], \ldots), \qquad e[n]=y[n]-\widehat y[n].\]

Echo cancellation and system identification use two signals with different physical roles. A reference or far-end signal \(x[n]\) drives an unknown path, and the microphone or desired signal \(m[n]\) contains the echo plus other content:

\[\widehat e_{echo}[n] = h_\theta(x)[n], \qquad r[n] = m[n] - \widehat e_{echo}[n].\]

A causal echo-path filter may use current and past reference samples \(x[n],x[n-1],\ldots\). It should not use future reference samples or future microphone samples. In real double-talk conditions, near-end speech and nonstationarity add control logic beyond the package’s synthetic metric demos; those examples are therefore presented as controlled DSP tests, not production echo cancellers.

Package examples by data-use type

The detailed MIMO verification map in MIMO verification map turns this vocabulary into concrete checks for the multichannel examples.

Example family	Data-use status
Scalar lattice IIR, lattice-ladder, streaming block processing	Causal filtering of one input stream.
Adaptive lattice/NLMS/RLS examples	Causal adaptive updates from the current input/desired pair after output and error are formed.
Burg, Levinson, block Levinson, AR spectral estimation	Batch coefficient estimation; fitted AR predictors are causal once coefficients are fixed.
MIMOLatticePredictor examples	Causal online vector prediction after a batch or precomputed coefficient fit.
OnlineMatrixLatticeAllPass examples	Causal forward multichannel all-pass filtering.
Paraunitary, ML-style unitary convolution, adjoint reconstruction demos	Forward analysis now uses streaming matrix-lattice runtimes where applicable; reconstruction checks use finite-record noncausal time-domain adjoints.
Finite-Hankel, Nehari/AAK-style, and response-compression pages	Offline finite-dimensional analysis or model selection; resulting reduced models may be causal when implemented as state-space/IIR systems.