1 Identification of unknown elements
Every analysis of the perturbation network presumes that each element in the operator is modeled. Some are not: a component may be known only through measurement, or a flame’s transfer function may be the very quantity one wishes to determine. Identification — equivalently de-embedding — recovers such an unknown element’s dynamic response from a measured or simulated network response, given a model of the rest of the network. It is the inverse of the forward acoustic problem: rather than assembling \mathbf{A}(\omega) from known parts to predict a response, one takes a response and a known reference operator and solves for the missing part.
The construction rests on a single structural fact — that an unknown element enters the operator as a low-rank update of a known reference — which makes the recovery a small, well-conditioned linear problem rather than a full inversion.
1.1 The de-embedding principle
The unknown element enters the perturbation operator as a low-rank modification of a reference operator \mathbf{A}_0(\omega), defined as the network with the unknown set to a passive default — an identity transfer matrix for an unknown two-port, or a silenced feedback for an unknown source. Writing the modification in low-rank factors, the two cases are given as:
\text{two-port:}\quad \mathbf{A}(\mathbf{X}) = \mathbf{A}_0 - \mathbf{P}\,(\mathbf{X} - \mathbf{I})\,\mathbf{Q}^{\top}, \qquad \text{source:}\quad \mathbf{A}(\mathbf{G}) = \mathbf{A}_0 + \mathbf{P}\,\operatorname{diag}(\mathbf{G})\,\mathbf{Q}^{\top},
where \mathbf{X} is the unknown transfer matrix and \mathbf{G} the unknown source gains, \mathbf{P} selects the element’s overwritten rows, and \mathbf{Q} carries the upstream-face characteristic map. Because the update is low-rank, the Woodbury identity (Hager 1989) recovers the unknown from the network response while re-using a single factorization of \mathbf{A}_0 per frequency, which serves the whole network whether it is a simple cascade or a branched topology — the branch simply folds into \mathbf{A}_0. Intuitively, all the acoustics of the known part are captured once in \mathbf{A}_0^{-1}, and the measurement then only has to pin down the small block the unknown occupies.
1.2 Recovering a two-port transfer matrix
The measurement is taken at two stations that straddle the unknown — an upstream edge a and a downstream edge b — where a set of excitations produces port-wave matrices \mathbf{W}_a and \mathbf{W}_b, and the measured relation imposes \mathbf{W}_b = \mathbf{M}_{\text{meas}}\,\mathbf{W}_a. Applying \mathbf{A}_0^{-1} to the excitations and projecting onto the two stations expresses each port response as its reference value plus a contribution linear in the unknown, so the measured relation becomes a linear system for the unknown, given at the readable level as:
\mathbf{K}_c \,\mathbf{m} = \mathbf{r}, \qquad \mathbf{K}_c = \mathbf{G}_b - \mathbf{M}_{\text{meas}}\,\mathbf{G}_a, \qquad \mathbf{X} = \mathbf{I} + \mathbf{D}(\mathbf{m}),
where \mathbf{G}_a and \mathbf{G}_b are the reference port responses to the low-rank probe, \mathbf{r} the reference measurement residual, \mathbf{m} the intermediate the Woodbury update solves for, and \mathbf{X} the recovered transfer matrix reconstructed from it. The system is solved in the least-squares sense, one small solve per frequency on top of the single \mathbf{A}_0 factorization, and the same procedure applies unchanged to a branched three-terminal network (tests: test_identify_transfer_matrix_cascade, test_identify_transfer_matrix_branched_3_terminal).
1.3 Recovering a flame transfer function
When the unknown is a flame’s response rather than a full two-port, the same de-embedding recovers the transfer-function gains of the source terms instead of a matrix block. This is the rank-structured special case in which the unknown is a handful of scalar gains — one per reference quantity, such as a velocity and a pressure sensitivity — so the recovery solves a small system for those gains, given as:
\mathbf{A}_{\text{sys}}\,\mathbf{G} = -\mathbf{R}_0,
where \mathbf{G} collects the per-term gains and \mathbf{R}_0 is the reference measurement residual. Because there are far fewer unknowns than in a full matrix, a single measurement can separate several distinct sensitivities — for instance the velocity and pressure responses of the same flame — provided the excitation makes their reference fluctuations sufficiently independent (tests: test_identify_single_input_ftf, test_identify_multi_input_ftf).
1.4 Isentropic versus full identification
The identification runs in one of two modes, distinguished by whether the convected entropy wave is retained, and the choice is not cosmetic. The full mode carries all three waves (f, g, h) and retains the entropy content, so the recovered response includes the coupling by which composition and temperature fluctuations generate sound. The isentropic mode pins the entropy (and composition waves) amplitude to zero and recovers a purely acoustic two-port, which is the classical flame transfer matrix and the right object when the measurement itself is acoustic.
An important remark is that the isentropic mode misses the indirect combustion noise: the entropy-to-acoustic conversion that occurs when an entropy spot reaches a compact nozzle — the coupling R_s of the perturbation network — is invisible to an analysis that has set the entropy wave to zero. Accordingly, an identification aimed at the acoustic two-port of a classical flame is run isentropically, while one that must account for entropy-generated (indirect) noise is run in the full mode (tests: test_identify_acoustic_only_isentropic, test_isentropic_analysis_misses_the_indirect_noise).
1.5 Identifiability and its diagnostic
Recovery is a linear inverse problem, and like any inverse problem it can be well or ill posed depending on how informative the measurement is. The diagnostic is the per-frequency condition number \operatorname{cond}(\cdot) of the identification system, reported alongside the recovered response. A large condition number at a frequency flags that the excitations there are too collinear to separate the unknown — for a multi-input flame, that the reference fluctuations of the several terms are nearly parallel, so their gains cannot be resolved independently — and the recovered value at such a frequency is not to be trusted. The recovery degrades gracefully rather than failing catastrophically as the conditioning worsens, so the diagnostic is the honest reading of where the identification is reliable (tests: test_identify_noise_degrades_gracefully, test_identify_multi_input_ftf).
Two interpretive points deserve to be stated plainly. First, transfer-matrix identification can target an interior element that also carries active flame feedback. In the reference operator \mathbf{A}_0, that node’s dynamic source is set acoustically silent and its perturbation acoustics are replaced by an identity transfer matrix, while the mean-flow jump is left unchanged. The recovered matrix therefore includes any active feedback folded into the element’s linear two-port, not its passive acoustics alone. Second, a marker attached to an element declares what is unknown; until identification runs, that element is assembled with a passive default (an identity transfer matrix or a silent source). The edges a and b supplied to the identification routine declare where the measured network response is read out; they need not coincide with the marked element’s own faces, and the two choices should not be conflated.
1.6 From data to the complex plane
The identification returns a real-frequency table of the recovered response and, by default, its analytic continuation. The raw table lives only on the real axis, which suffices for a forced response and for the real-axis Nyquist stability driver of analyses, both of which evaluate only there. The stability eigensolver, however, searches the complex plane for the roots of \det\mathbf{A}(\omega) and therefore needs an analytic function, so the table is continued by one of the two fits of dynamic sources — the impulse-response fit when the recovered response dies out in finite time (a flame, a compact element), which is the recommended route, or the barycentric-rational (AAA) fit (Nakatsukasa et al. 2018) when it carries a genuine resonance; by default the identification applies the rational fit, since it needs no memory-length input. The same continued object then drives both the real-axis Nyquist analysis and the complex-plane eigensolver, and the conversions between transfer and scattering forms preserve its analyticity (test: test_identify_transfer_matrix_continuation_is_analytic).
Identification thus closes the loop between measurement and model: a measured or simulated network response becomes a network-ready element — a transfer matrix or a flame transfer function — continued into the complex plane and ready to be placed back into the operator for the stability and response analyses of analyses.