Filtering relationships and signal-processing vocabulary

This page collects basic signal-processing terminology used by the examples. It is intentionally concrete: the goal is to make equalization, echo cancellation, prediction, adaptivity, recursivity, and phase terminology mean specific mathematical objects before readers encounter the lattice and MIMO algorithms.

For data-use language such as causal, batch/offline, transductive, and finite-block/circular, see Causality, data use, and signal roles. This page focuses on the physical roles of the signals once that data-use convention is fixed.

Three filtering relationships

Many adaptive-filter examples have the same algebraic core but different signal roles. A filter produces

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

and an error signal compares that output with a desired signal d[n]:

\[e[n] = d[n] - y[n].\]

The meaning of x, d, y, and e changes by task.

Relationship	Input to adaptive filter	Desired signal / error	What the filter is learning
Equalization	Received distorted signal r[n]	Known training symbol or decision s[n]; e[n]=s[n]-y[n]	An inverse or partial inverse of a channel so that y[n] resembles the transmitted signal.
Echo cancellation	Far-end/reference signal x[n]	Microphone signal m[n]; residual e[n]=m[n]-\hat e_{echo}[n]	The echo path from loudspeaker/reference to microphone, so its estimate can be subtracted.
Prediction	Past samples [x[n-1],x[n-2],\ldots]	Current sample x[n]; innovation e[n]=x[n]-\hat x[n]	A model of temporal structure; the residual is the unpredictable innovation.

The same update law can therefore appear in different applications. What changes is the physical interpretation of the regressor and the error.

Single-signal prediction and two-signal echo cancellation

The prediction examples are single-signal models. The signal being predicted is also the signal whose past is used as the regressor:

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

A causal predictor must form \(\widehat x[n]\) before observing \(x[n]\). Multichannel prediction changes the scalar sample into a vector, but the data-use rule is the same: previous vectors may be used and future vectors may not.

Echo cancellation and system identification use two signals. The adaptive filter receives a reference/far-end signal \(x[n]\); the desired or microphone signal \(m[n]\) is compared with the estimated path output:

\[\widehat d[n] = h_\theta(x)[n], \qquad r[n] = m[n] - \widehat d[n].\]

In echo-cancellation language, \(x[n]\) is the loudspeaker/far-end reference, \(\widehat d[n]\) is the estimated echo, \(m[n]\) is the microphone mixture, and \(r[n]\) is the residual sent downstream. The synthetic ERLE examples use this two-signal metric structure, but they do not implement the control logic, delay estimation, nonlinearities, double-talk handling, or product engineering expected from a production acoustic echo canceller.

Adaptivity

A fixed filter has parameters that do not change with time. An adaptive filter has a parameter vector or state-dependent parameterization \theta_n that is updated from streaming data:

\[y[n] = f_{\theta_n}(x_n), \qquad e[n] = d[n] - y[n], \qquad \theta_{n+1} = \theta_n + \Delta(\theta_n, x_n, d[n]).\]

For FIR LMS, for example,

\[w_{n+1} = w_n + \mu x_n e[n].\]

For normalized LMS,

\[w_{n+1} = w_n + \frac{\mu}{\epsilon + \|x_n\|_2^2} x_n e[n].\]

Even in the FIR case, the learning rate μ is a real design parameter rather than a formality. If it is too small, convergence is slow; if it is too large, the recursion can have large misadjustment or diverge. The safe range depends on the signal power and input covariance, so practical step-size selection is often a tuning problem rather than a one-line formula.

For lattice IIR adaptation, the parameter vector includes numerator/ladder parameters and reflection coefficients. The important distinction is that adaptive recursive filters keep the LMS step-size problem and add a denominator stability problem. Reflection coefficients give a structural guard: in the scalar all-pole case, keeping |k_i| < 1 keeps the recursive denominator stable.

Recursivity

A recursive filter feeds previous outputs or internal states back into the current computation. A direct-form IIR model is recursive because

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

Equivalently, a state-space model is recursive because

\[x_{s}[n+1] = A x_{s}[n] + B u[n], \qquad y[n] = C x_{s}[n] + D u[n].\]

The pole locations of A(z) or the eigenvalues of A control stability. This is why adaptive IIR filtering is more delicate than adaptive FIR filtering: changing denominator or state-recursion parameters changes the stability of the algorithm itself.

Minimum phase and maximum phase

For a causal discrete-time transfer function

\[H(z)=\frac{B(z)}{A(z)},\]

poles determine stability and zeros determine phase/invertibility properties. A stable causal scalar system is minimum phase when its zeros are also inside the unit disk. In that case the inverse filter is causal and stable, up to delay and gain conventions. Minimum-phase systems concentrate energy early in the impulse response and are the easiest case for stable equalization.

A maximum-phase scalar system has zeros outside the unit disk. Its stable inverse would have poles outside the unit disk if implemented causally without additional delay or noncausal processing. Equalizing such a channel is therefore numerically and structurally harder.

A useful scalar split is

\[H(z) = H_{\mathrm{outer}}(z) H_{\mathrm{inner}}(z).\]

The outer factor is the minimum-phase part that carries the invertible amplitude information. The inner/all-pass factor has unit magnitude on the unit circle and carries phase or delay-like structure. In lattice language, all-pass and lossless completions are natural companions to this inner/outer viewpoint.

Long memory and speed

The speed motivation for IIR filters is easiest to see from impulse-response length. A long FIR path stores many taps and computes

\[y[n] = \sum_{m=0}^{L-1} h[m] x[n-m].\]

For large L and millions of samples, this can be expensive even when block or FFT convolution is used. A recursive model can represent some long-memory responses with far fewer parameters. For example, the exponential tail

\[h[m] = (1-r) r^m, \qquad 0 < r < 1,\]

has the one-state recursion

\[y[n] = (1-r) x[n] + r y[n-1].\]

That is the numerical attraction of stable IIR and model-reduction workflows: when a long impulse response has compact recursive structure, the implementation can process very long signals using small state rather than a long tap vector. The package includes a million-sample throughput tutorial that demonstrates this point for an acoustic-like decay while keeping the benchmark local and reproducible.

Why this matters for this package

lattice-dsp focuses on stable recursive coordinates. The practical message is:

• recursive filters are powerful because they represent long memory with few parameters;

• adaptive recursive filters are risky unless stability is parameterized or monitored;

• reflection coefficients provide a stable scalar IIR coordinate system;

• finite Hankel/model-reduction tools help look for compact recursive structure in long impulse-response or Markov-parameter data;

• MIMO and matrix cases replace scalar magnitudes by contraction and block-Hankel diagnostics;

• equalization, echo cancellation, and prediction use similar mathematics but different signal roles and validation metrics.