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Identification of $dq$-Asymmetric Impedances as Complex Transfer Functions Using a Single Arbitrary Excitation

Mohamed Abdalmoaty, Zheran Zeng, Dongsheng Yang, Florian Dörfler

arXiv:2608.29740v1eess.SPeess.SY

TL;DR

Identifying dq-asymmetric grid impedances from one measurement is underdetermined and complicated by transient effects. This paper introduces a single-record, non-parametric frequency-domain method and recovers complex and real transfer functions across a wide band at 1 Hz resolution.

  • Problem

    Single-record identification is underdetermined because asymmetric impedances contain twice as many unknown responses as available complex data equations, while short records retain transient errors.

  • Method

    The method jointly estimates impedance responses, leakage, and transients by fitting each spectral line with a local low-order rational model.

  • Results

    Both complex transfer functions and all four real dq responses were recovered on HIL over a wide band from one one-second record at 1 Hz resolution.

  • Takeaways & Limitations

    The method supports single-record non-parametric identification of dq-asymmetric grid impedances, including near-fundamental coupling features.

  • Takeaways & Limitations

    The method assumes the grid remains sufficiently unchanged during the record, whereas real grids may not hold a constant operating point for long.

Abstract

from arXiv · show

Cross-coupling between the $dq$ coordinates makes the identification of asymmetric grid impedances a challenging problem, particularly near the fundamental frequency where the asymmetric coupling is strongest. Existing schemes usually handle it either by perturbing the two coordinates sequentially, which lengthens the measurement, or by using a time-domain method with a global parametric model whose order must be tuned. This paper develops a single-shot active non-parametric frequency-domain method that avoids both. The equivalent impedance is parameterized by a pair of single-input single-output complex transfer functions. Each spectral line is fitted with a local rational model; the leakage and transient contributions are estimated, so that neither periodic steady-state excitation nor repeated excitation cycles are required. We give the exact finite-time discrete Fourier transform relation for the conjugate-coupled complex-signal model, and analyse the distortion that a stationary-frame filter placed ahead of the Park transform imposes on the identified pair. The method is validated on a controller hardware-in-the-loop platform against an analytically derived small-signal model, for a symmetric grid and for the same grid with an added grid-following converter that renders it asymmetric. Both complex transfer functions and all four real transfer functions of the $dq$ impedance are recovered over a wide band from a single one-second record of a random excitation, at 1 Hz resolution.

I. Introduction … A. Small-signal model

The paper motivates data-driven impedance identification for poorly documented, operating-point-dependent converter-dominated grids and develops a single-record active method for dq-asymmetric models. Its small-signal formulation represents the impedance through direct and conjugate-coupling complex transfer functions, with asymmetry arising from actively controlled devices.

  • I. Introduction: Converter-dominated grids have faster, less documented, operating-point-dependent dynamics, while proprietary models limit analytical characterization.Subsystem interactions can vary with operating point, and manufacturers may not disclose converter models.
  • I. Introduction: Impedance measurements provide local dynamics without requiring protected model information and support stability, harmonic, design, diagnosis, and control applications.Impedance-based criteria are used for small-signal stability assessment because converter-grid interactions can degrade power quality or trigger instabilities.
  • A. Identification methods and their limitations: Existing identification methods trade frequency coverage, measurement duration, excitation amplitude, or protection risk, while dq methods can also be biased by PLL dynamics.Passive methods are sparse in frequency; impulse excitation creates a brief large transient; low-frequency classical measurements may require several periods; and sequential perturbations lengthen measurement.
  • B. Contributions: The proposed active method identifies dq-asymmetric grid models from one random excitation record, without sequential injections or periodic steady-state perturbation.The excitation is applied once and the response is captured in one triggered acquisition.
  • B. Contributions: An exact finite-record DFT relation incorporates transient and leakage contributions, enabling local rational estimation for arbitrary non-periodic data.The record length is determined by the desired frequency resolution and signal-to-noise ratio, not by excitation periodicity.
  • A. Small-signal model: The identified quantity is the dynamic small-signal Thévenin equivalent impedance reconstructed from terminal three-phase voltage and current samples at a point of common coupling.The model makes no assumptions about the devices behind the PCC and assumes balanced operation for the Park transformation.
  • A. Small-signal model: The dq impedance is represented by two complex SISO transfer functions: a direct term multiplying i(t) and a coupling term multiplying i*(t).The conjugate channel captures the non-C-linear behavior associated with dq asymmetry.
  • A. Small-signal model: For symmetric grids, the coupling transfer function vanishes; asymmetric grids retain it, doubling unknowns relative to one measurement set and requiring additional structure for identification.The asymmetry arises naturally from actively controlled devices and their control loops.

B. Grid-connected converter … III. Impedance Identification Method

The paper uses a grid-connected converter to inject a wideband arbitrary excitation, records synchronized voltage and current measurements, and identifies the conjugate-coupled impedance functions non-parametrically from a single record’s DFT.

  • B. Grid-connected converter: A grid-connected voltage-sourced converter with an LCL filter provides the excitation, using dq-frame current control, PLL synchronization, and PWM voltage-reference realization.The DC link is ideal, while outer power loops are present but omitted from the figure.
  • B. Grid-connected converter: A zero-mean random binary sequence is superimposed on converter voltage references to achieve higher excitation bandwidth than injection at the current reference.The sequence is aperiodic and may be truncated at any length.
  • B. Grid-connected converter: Any sufficiently exciting signal can be used, provided its spectrum phase varies widely over frequency, allowing energy shaping within a chosen amplitude and record length.The excitation-spectrum design is outside the stated scope.
  • C. Measurement setup: Voltage and current recordings begin when excitation is applied and require anti-alias filtering, synchronized channels, relative calibration, and sufficiently fast uniform sampling.Calibration accounts for transducer responses, while the measurement setup also includes switching harmonics and measurement noise.
  • D. Choice of reference frame: Park transformation requires the grid fundamental-voltage phase, commonly obtained from a PLL whose dynamics can distort measurements unless explicitly modeled.At constant grid frequency, dq-frame impedance is linearly connected to modified-sequence and stationary-frame alternatives.
  • III. Impedance Identification Method: From a single arbitrary record, the method constructs a non-parametric estimator of G+(jω) and G−(jω) on a uniform frequency grid.The data set consists of paired sampled voltage and current values, and the estimator maps these data to the two complex transfer functions.
  • III. Impedance Identification Method: The identification computes N-point DFTs efficiently with FFT algorithms and accounts for complex-signal conjugacy and reversal rather than Hermitian symmetry.The spectra are periodic with period N, and the DFT of the conjugated signal uses the reversed modulo-N index.

A. Relation between voltage and current DFT spectra · B. Local parametric modeling

The method derives an exact finite-record DFT relation for conjugate-coupled complex signals and estimates the transfer functions separately at each spectral line. Local low-order rational models use neighboring lines to estimate transient effects and resolve the underdetermined single-record problem without assuming a global fixed-order model.

  • A. Relation between voltage and current DFT spectra: The exact DFT relation holds without periodic data or approximation for finite observations of stable, causal G+ and G−.It applies at every spectral line of the recorded interval.
  • A. Relation between voltage and current DFT spectra: The transient term captures spectral leakage, truncation-induced aliasing, and unforced responses caused by mismatched initial and final conditions.Its contribution decays at rate O(N^-1/2).
  • A. Relation between voltage and current DFT spectra: The complex transfer functions G+ and G− combine the same four real dq transfer functions, with G− = 0 for a dq-symmetric system.The reversed and conjugated spectrum appears in the second channel of the complex-signal relation.
  • A. Relation between voltage and current DFT spectra: The single-record problem is underdetermined because N complex equations must identify 2N complex responses, while short records retain transient distortion.Waiting for transients to decay or using a second perturbation would lengthen measurement time.
  • A. Relation between voltage and current DFT spectra: The method estimates the transient term together with the complex transfer functions and obtains missing equations from neighboring spectral lines through local parametric modeling.This avoids suppressing transients or waiting for them to decay.
  • B. Local parametric modeling: At each frequency ωk, G+, G−, and T are approximated over a short interval by low-order models, with every line contributing an equation to the estimate at ωk.Different-frequency estimates are linked only through the raw data, preserving non-parametric estimation.
  • B. Local parametric modeling: A common denominator models nearby poles, and the parameterization contains 4R + 3 unknown complex parameters before extracting the center-frequency estimates.The denominator is normalized to unity at the interval center by setting a0(k) = 1.
  • B. Local parametric modeling: Multiplying the local spectral relation by the common denominator makes it linear, enabling least-squares fitting separately over [ωk−ℓ, ωk+ℓ].Unlike a global model, each local model is fitted at one spectral line and does not impose one order across the whole band.

C. Identifiability and excitation

Identifiability is guaranteed exactly when the local regression matrix has full column rank, requiring both sufficient frequency-interval width and sufficiently rich local spectra. Random-binary-sequence excitation supports this spectral richness through irregular phase variation across frequency lines.

  • Identifiability: The solution is unique if and only if Φ_k has full column rank, imposing conditions on interval width and local spectra.These are a counting condition and a data-excitation condition.
  • Identifiability: The local model contains 4R + 3 complex parameters and 2ℓ + 1 complex data equations, yielding a necessary counting condition.
  • Excitation: Rank deficiency occurs when the local current spectrum and its mirror satisfy a polynomial relation of degree at most R over the interval.This is the frequency-domain counterpart of identification with unknown initial conditions.
  • Excitation: Sufficient excitation means that no nontrivial degree-at-most-R polynomial description makes input-related columns indistinguishable from those corresponding to T.The precise condition and proof are given in Appendix C.
  • Excitation: RBS excitation has randomly varying phase in [−π, π] across spectral lines, making the required polynomial relation unlikely for the closed-loop current spectrum.A degree-R polynomial is fixed by R + 1 values, while the remaining 2ℓ−R lines would also need to satisfy the same relation.

The RBS excitation: … E. Extracting the four real TFs

The method excludes the uninformative zero-frequency dq line, requiring sufficiently wide local intervals after its removal. It also provides special-case estimators and maps the identified complex transfer functions to the four real dq transfer functions, with distinct error-propagation effects.

  • The RBS excitation:: The dq-frame line at ω = 0 corresponds to the fundamental ωg in abc coordinates and is not excited in the data.At this frequency, the DFT line is its own mirror, so the direct and conjugate regressors coincide.
  • The estimate at ωg:: Because mean removal makes V0 and I0 vanish, the zero-frequency regression row carries no channel information and forces Ck’s constant coefficient to zero.This biases the transient term.
  • The estimate at ωg:: At the minimum interval width ℓ = 2R + 1, removing the zero-frequency row leaves fewer equations than parameters, so wider intervals are required.With ℓ = 4R + 2, the interval has 8R + 5 lines against 4R + 3 parameters, making the loss immaterial.
  • D. Special cases: For periodically repeated excitation with an integer number of steady-state periods, T(jω) = 0 and the Ck columns can be removed, but local modeling remains necessary.A single measurement still provides fewer data equations than unknowns.
  • D. Special cases: If dq symmetry is known, the model reduces to Vk = G(jωk)Ik + T(jωk), allowing the B−k columns to be removed while leakage still requires local modeling.When symmetry and periodic steady-state conditions both hold, G(jωk) = Vk/Ik is available, but only under steady-state measurements.
  • E. Extracting the four real TFs: The symmetric complex transfer function maps to real components through Gd(s) = 0.5(G(s) + G∗(s)) and Gq(s) = −0.5j(G(s) − G∗(s)).Here, G∗(s) denotes the conjugate reflection [G(s∗)]∗.
  • E. Extracting the four real TFs: For the asymmetric case, inverting the model uses estimates at positive and mirrored negative-frequency lines to construct the real dq impedance transfer functions.The mirrored frequency is represented by ω̄k = ω(N−k)N for even N.
  • E. Extracting the four real TFs: Mapping the complex pair to Zg changes error propagation: a zero of G+ generally is not a zero of Zdd because the mirrored contribution fills the antiresonance.Consequently, an error at a G+ notch can remain small in absolute terms while entering a Zg element with non-small magnitude.

IV. Measurement Chain and Frame Alignment … B. Distortion due to pre-Park filtering

The section explains how real acquisition filters interact with Park-frame signals and derives the resulting pre-Park distortion and correction conditions. It shows that filter placement and measurement-chain mismatch affect the identified dq transfer functions.

  • IV. Measurement Chain and Frame Alignment: Real acquisition paths apply transducer and anti-alias/decimation dynamics before the Park transform, so filter placement determines their dq-frame behavior.The section derives the effect and its exact correction for this measurement chain.
  • A. A stationary-frame filter seen from the dq frame: A real-coefficient stationary-frame filter becomes a complex dq-frame filter whose response is evaluated at the shifted frequency ωg + ω.Its dq impulse response is h(σ)e^−jωgσ, with frequency response H(j(ωg + ω)).
  • Frame mapping:: The Park demodulation shifts the stationary-frame convolution into a dq-frame convolution with impulse response h(σ)e^−jωgσ.This follows by splitting the demodulation exponential inside the convolution.
  • Frame mapping:: The analysis applies this frame mapping to transducer and anti-aliasing dynamics present in the measurement acquisition before the Park transform.These dynamics are treated as pre-Park transform filters.
  • B. Distortion due to pre-Park filtering: At each spectral line, the estimator’s result depends on the voltage and current acquisition-chain responses evaluated on the corresponding dq sidebands.The chains are represented by real-coefficient responses Cv = SvH and Ci = SiH applied before the Park transform.
  • B. Distortion due to pre-Park filtering: With common anti-alias/decimation filters, dG+ = G+ if and only if Sv = Si; transducer mismatch produces a proportional bias Sv,+/Si,+.The bias can be corrected when the transducers are relatively calibrated.
  • B. Distortion due to pre-Park filtering: Even with matched chains, dG− = DG−, while a flat transducer response across both sidebands reduces D to the response ratio of H alone.The section then addresses mitigation of this distortion.

C. The moving-average decimator … B. Test system configuration

The paper analyzes measurement-chain and frame-alignment effects on dq impedance identification, then validates the method on a hardware-in-the-loop system with passive and converter-interconnected grid cases. The analysis shows that stationary-frame filtering primarily distorts the conjugate-coupled transfer function, whose importance is greatest near the fundamental frequency.

  • C. The moving-average decimator: The moving-average decimator imposes a constant phase on G−, while its sideband magnitude ratio ρ remains close to unity over a narrow identification band.The frequency-dependent linear phases cancel, leaving phase −2ωgτ and magnitude ratio ρ.
  • C. The moving-average decimator: The pre-Park filter leaves the rotationally invariant impedance part unchanged but maps the non-rotationally invariant part through ρR(−2ωgτ).Consequently, Zdd + Zqq and Zqd − Zdq remain clean, whereas individual entries mix G+ with distorted G−.
  • C. The moving-average decimator: The analysis applies to any real-coefficient FIR filter; only the numerical values of τ and ρ change, although the moving-average filter was selected because it was available on the real-time simulator.The paper notes that moving-average filtering is generally not preferred for anti-aliasing because flatter-passband, sharper-transition designs are possible.
  • D. Frame alignment: The dq frame uses a synthetic angle at the nominal grid frequency, avoiding synchronization dynamics but leaving an arbitrary phase offset relative to the PCC-voltage-aligned frame.The offset θ1 is obtained from the settled PLL angle immediately before excitation.
  • V. Experimental Validation: The proposed responses are experimentally assessed on a controller hardware-in-the-loop platform against an analytically derived small-signal model.The test setup uses a ModelingTech MT8020 real-time simulator with a 1 µs real-time step.
  • A. HIL experiment platform: The simulator decimates from 10^6 to 10^4 samples per second with Nf = 100, producing τ = 49.5 µs, phase −1.78° at 50 Hz, and ρ within ±0.03 dB over the evaluation band.The filter operates on stationary-frame signals before the Park transform, so the derived correction applies.
  • A. HIL experiment platform: The estimator uses local order R = 4 and 37 spectral lines to determine 19 complex parameters in each local problem.The same settings are used for both test cases, and the zero-frequency line may be removed when necessary while preserving identifiability and excitation.
  • B. Test system configuration: Case 1 is a passive three-phase grid, whereas Case 2 adds a grid-following voltage-source converter through an LC filter, line, and shunt load.In Case 2, the converter branch connects at PCC2 through the breaker to the main network.

C. Case 1: a dq-symmetric grid · D. Case 2: a dq-asymmetric grid · E. Accuracy

The proposed single-record estimator recovers symmetric and asymmetric dq impedance responses over a wide frequency band, resolving sharp coupling features and achieving near-perfect accuracy in the symmetric case and high accuracy in the asymmetric case.

  • C. Case 1: a dq-symmetric grid: In the dq-symmetric grid, G+ overlays the analytical model across [−1, 1] kHz, including the −50 Hz antiresonance where |G+| reaches −20.8 dB.The antiresonance is the image of the abc direct-current point under the shifted dq frequency axis.
  • C. Case 1: a dq-symmetric grid: The estimator identifies both transfer functions without assuming symmetry, with G− at a median −77 dB noise floor versus a 6.0 dB peak of |G+|.The approximately 80 dB separation shows that the absence of coupling is inferred from the data.
  • D. Case 2: a dq-asymmetric grid: Closing the breaker adds a grid-following converter, changes G+, and makes the equivalent impedance asymmetric so that G− becomes nonzero.The asymmetric-grid responses are compared with the analytical model for both complex and four real transfer functions.
  • D. Case 2: a dq-asymmetric grid: In Case 2, the G+ antiresonance is 26.1 dB deep with a 3 dB width of 17.6 Hz, and a new peak appears near 0 Hz.The −50 Hz notch is retained after asymmetry is introduced.
  • D. Case 2: a dq-asymmetric grid: The G− coupling peaks at −16.9 dB near the fundamental, reaches minima of −55.4 dB and −54.5 dB at ∓50 Hz, and is tracked by the estimate across the band.It decays to roughly −56 dB at ±1 kHz, while edge scatter grows as the coupling channel becomes SNR-limited.
  • D. Case 2: a dq-asymmetric grid: All three sharpest features are resolved at 1 Hz, including the G+ −50 Hz notch, the near-fundamental G− peak, and the G− −50 Hz minimum.The near-fundamental peak rises by nearly 20 dB within ±8 Hz of zero despite only a handful of spectral lines spanning it.
  • E. Accuracy: Case 1 fits all four real responses to within rounding of 100% with a 0.4% worst-case relative error, while Case 2 remains above 99.7% with a 6.5% worst-case error.The largest Case 2 error occurs at 1 Hz near the sharp fundamental feature; G+ is recovered essentially exactly in both cases, while G− achieves an 85.57% fit.

F. Local model order · VI. Conclusions

The method remains accurate across local model orders and identifies asymmetric dq impedances from a single arbitrary record without periodic steady-state data or sequential perturbation. The analysis establishes the finite-time DFT relation, filter-induced distortion of G−, and validation against an analytical model on HIL hardware.

  • F. Local model order: Across R = 4, 6, 8 and 10, Fit% changes by less than 0.03 in Case 1 and less than 0.04 in Case 2, while relative H∞ error changes by less than 0.01.The estimates are therefore fairly insensitive to local model order over the tested range.
  • F. Local model order: Local rational modeling shows weaker dependence on order than global parametric methods, whose accuracy depends strongly on the selected order.The reported results use R = 4 with ℓ = 18, while alternative orders adjust ℓ accordingly.
  • F. Local model order: G+ is essentially exact at every tested order, whereas G− in Case 2 is poorer but improves slightly with larger R.G− is three orders of magnitude smaller than G+ around 0 Hz and negligible at higher frequencies, explaining its poorer estimates.
  • VI. Conclusions: The paper develops a non-parametric frequency-domain method that parameterizes the equivalent impedance with two SISO complex transfer functions and jointly estimates transient and leakage contributions at each spectral line.This avoids suppressing those contributions through windowing.
  • VI. Conclusions: The method removes requirements for periodic steady-state data and sequential perturbation, leaving record length governed by desired frequency resolution and SNR.The approach is based on a single arbitrary record rather than repeated coordinate perturbations.
  • VI. Conclusions: The exact finite-time DFT relation for conjugate-coupled complex signals enables joint estimation of transient terms and system responses, alongside established identifiability and excitation conditions.These results provide the analytical basis for fitting responses from finite records.
  • VI. Conclusions: A stationary-frame dynamical filter distorts G− but not G+, even when both channels use identical transducers and filters.The paper specifically identifies anti-aliasing or decimation filtering as an example of this effect.
  • VI. Conclusions: Validation on HIL hardware used an analytically derived small-signal model for a passive dq-symmetric grid and the same grid with a grid-following converter that rendered it asymmetric.The validation therefore covered both symmetric and asymmetric grid conditions.

Appendix A Conjugation and Reversal of the DFT · Appendix B Proof of Theorem 1 · A. The symmetric case

The appendices establish conjugation and index-reversal identities for finite-length complex DFTs, then derive the finite-time frequency-domain relation for a causal symmetric system. The derivation separates initial-condition, final-condition, and aliasing contributions when connecting continuous spectra to sampled DFT coefficients.

  • Appendix A Conjugation and Reversal of the DFT: Conjugating a complex record reverses its DFT index, with the reversed index interpreted modulo N.The identity uses the N-periodicity of the DFT sum to fold −k into the index range 0,…,N−1.
  • Appendix A Conjugation and Reversal of the DFT: The k = 0 line is its own reversal mirror, although its DFT coefficient need not be real for a complex record.At k = 0, the reversal reduces to the coefficient’s complex conjugate relation without implying that I0 is real.
  • Appendix A Conjugation and Reversal of the DFT: For real signals, DFT conjugation becomes Hermitian symmetry and halves the number of independent spectral lines; complex records retain distinct spectra.The distinction is central to the complex-signal treatment used in the paper.
  • Appendix B Proof of Theorem 1: Theorem 1 is proved first for the dq-symmetric case, while the general case follows by applying the result separately to G+ and G−.The proof introduces Fourier and convolution integrals before specializing to the symmetric system.
  • A. The symmetric case: Causality removes the pre-zero contribution from the convolution response over the finite observation interval.The argument assumes a stable, causal G with impulse response g and t ∈ [0, T], where T = NTs.
  • A. The symmetric case: The finite-time spectra include leakage terms from final and initial conditions, respectively.V fin(ω) represents the final-condition contribution, while V init(ω) represents the initial-condition contribution.
  • A. The symmetric case: Sampling and Poisson summation connect continuous spectra to the DFT lines, yielding Vk = G(jωk)Ik + T(jωk).The disturbance term is T(jωk) = V init,k − V fin,k + αk, where αk carries aliasing.

B. The asymmetric case

The asymmetric model combines direct and reversed-conjugated spectral contributions, while transient leakage and aliasing are captured by a low-order local rational term. Identifiability depends on a rank condition for the local regression matrix, with deficiency equivalent to a polynomial relation among the current spectrum, its mirror, and a constant sequence.

  • Spectral relation: The asymmetric spectrum includes a reversed and conjugated contribution because the conjugated signal transforms as [I(−ω)]* on the continuous-frequency representation.On the sampled grid, the reversed frequency −ω_k corresponds to line (N−k)N.
  • Transient modeling: Transient effects comprise endpoint leakage, unforced initial and final responses, and aliasing that repeats the system poles through spectral folding.These contributions share the dynamics of G+ and G− and are smooth over short frequency intervals, enabling joint local modeling.
  • Transient modeling: O(N −1/2) decay follows from the 1/N normalization and bounded endpoint terms.The decay applies to the transient contribution under the stated normalization and boundedness conditions.
  • Rank condition: Φ_k is rank deficient exactly when a nontrivial polynomial relation of degree at most R exists among the local current spectrum, its mirror, and a constant sequence.The corresponding polynomial coefficients include α, β+, β−, and γ, with α(0)=0.

Appendix D The DFT Line at Zero Frequency · Appendix E Proof of Proposition 1

Appendix D shows that the zero-frequency DFT line must be excluded because mean removal forces c0 = 0 and biases leakage estimates. Appendix E establishes that smooth, nonzero acquisition-chain and transducer factors preserve the local model class and input-column independence.

  • Appendix D The DFT Line at Zero Frequency: For N = 8, R = 1, and ℓ = 3, an interval centered on k = 0 has 7 equations for 7 unknowns, including the zero-frequency line.The example uses r ∈ {−3, . . . , 3} and 2ℓ + 1 = 7 data equations for 4R + 3 = 7 unknowns.
  • Appendix D The DFT Line at Zero Frequency: At r = 0, every entry weighted by r vanishes, so the corresponding row reduces to the constant-coefficient relation.The fourth row is the line k = 0, where the r-weighted terms disappear.
  • Appendix D The DFT Line at Zero Frequency: Because mean removal gives V0 = I0 = 0, the zero-frequency row forces c0 = 0 regardless of the data.This constraint affects the constant coefficient of Ck rather than reflecting the measured local behavior.
  • Appendix D The DFT Line at Zero Frequency: Including k = 0 in a local interval biases the leakage-term estimate, whereas removing that row eliminates the constraint.The zero-frequency line counts as one spectral line, and its loss is immaterial for the interval widths used here.
  • Appendix E Proof of Proposition 1: The estimator operates on spectra after the acquisition chains and Park transform, with leakage represented by the second terms in the transformed equations.The conjugate regressor is evaluated at the mirrored line (N − k)N, whose dq frequency is −ωk.
  • Appendix E Proof of Proposition 1: When the relevant factors are nonzero and smooth within each local interval, reparameterization remains in the local model class and preserves input-column linear independence.The stated condition holds for FIR and transducer responses across the band.
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