Step 3: Analysis Algorithms

V-Listen provides three curve-fitting algorithms for extracting modal parameters from measured FRF data. All three are poly-reference, frequency-domain methods capable of handling closely spaced modes and high modal density.

Algorithm Overview

Algorithm	Full Name	Approach	Best For
PolyProMax	Polyreference Pro Maximum	Standard LSCF + PolyMAX-Plus 3-criterion stabilization (freq + damp + shape MAC, median) + LSCE cross-check	General-purpose; recommended default (drift-free)
LSCF	Least Squares Complex Frequency	Z-domain poly-reference fitting; two selectable backends for computation style	Dense modal data; high numerical precision
LSCE	Least Squares Complex Exponential	Time-domain companion matrix eigenvalue decomposition	Well-separated modes, impulse response data

PolyProMax (Recommended)

PolyProMax is V-Listen's flagship modal analysis algorithm. It combines the strengths of LSCF and LSCE through a cross-validation strategy that automatically identifies and retains only physically meaningful poles.

How It Works

LSCF core: Fit a common-denominator polynomial to the FRF matrix in the z-domain (poly-reference). PolyProMax uses the Standard per-order-independent backend, so pole frequencies do not drift with model order.
PolyMAX-Plus stabilization: Cluster poles across model orders using a three-criterion test — frequency + damping + participation-vector MAC — and take the cluster median frequency. This separates physical modes from mathematical poles and keeps frequencies stable.
LSCE cross-validation: Independently extract poles from impulse responses (inverse FFT of the FRFs) and mark modes confirmed in both domains.
MAC orthogonality filter: Remove near-duplicate modes via the Modal Assurance Criterion.

The result is a drift-free physical-mode set: mode frequencies stay stable to a fraction of a Hz regardless of the chosen maximum model order.

H(ω) = Σ_r [ A_r / (jω - λ_r) + A_r* / (jω - λ_r*) ]

Partial fraction model of the FRF matrix, where λ_r are poles and A_r are residue matrices.

Stabilization & Cross-Validation Tolerances

Parameter	Default	Description
Frequency tolerance	1.0%	Max relative frequency change for a pole to be "stable" across model orders (PolyMAX-Plus criterion 1).
Damping tolerance	5%	Max relative damping change across model orders (criterion 2).
Participation-vector MAC	0.98	Min mode-shape MAC across model orders — the third PolyMAX-Plus criterion that rejects mathematical poles.
Min stable count	5	Minimum number of consecutive orders a pole must remain stable to be accepted.
LSCE cross-validation MAC	0.90	MAC for matching / duplicate removal in the LSCE cross-check.

LSCF (Standalone)

The Least Squares Complex Frequency method operates directly on the FRF data in the z-domain without converting to impulse responses.

Theory

LSCF models the FRF matrix as a right matrix-fraction description:

[H(z)] = [N(z)] [D(z)]^-1

where [D(z)] = Σ_k=0^p [α_k] z^k is the common-denominator polynomial matrix of order p, and [N(z)] is the numerator polynomial matrix.

The denominator coefficients are found by minimizing a linearized least-squares cost function. Poles are then extracted as eigenvalues of the companion matrix formed from the [α_k] coefficients.

Advantages

Numerically stable for high model orders (up to 100+).
Handles closely spaced and heavily damped modes well.
Clean stabilization diagrams with fewer computational poles than time-domain methods.

Backends

The LSCF engine offers two poly-reference backends:

TimeEfficient (default) — optimized for computational speed. Note: per-order pole frequencies shift slightly with the chosen maximum model order.
Standard — each model order builds its equations independently, so pole frequencies do not drift with the maximum model order. This is the textbook PolyMAX behaviour and the backend that PolyProMax builds on.

LSCE (Standalone)

The Least Squares Complex Exponential method works on impulse response functions (IRFs) obtained by inverse FFT of the measured FRFs.

Theory

The IRF is modeled as a sum of complex exponentials:

h(t) = Σ_r=1^2N C_r e^{λ_r t}

A block Toeplitz matrix is formed from the IRF samples. The eigenvalue decomposition of the companion matrix yields the system poles λ_r, from which natural frequencies and damping ratios are computed:

f_n = |λ_r| / (2π) , ζ = -Re(λ_r) / |λ_r|

Advantages

Intuitive connection to physical impulse response.
Well-suited for impact testing data.
Lower computational cost for small-to-medium model orders.

Caution: LSCE may produce noisy stabilization diagrams at high model orders due to the sensitivity of the companion matrix to numerical errors. Consider using PolyProMax or LSCF for high-order analysis.

Model Order Selection

The model order determines how many poles the algorithm attempts to fit. Too low an order misses physical modes; too high an order introduces computational (mathematical) poles.

Manual Selection

Set the minimum and maximum model order. The algorithm runs for all orders in this range and displays poles on the stabilization diagram. Recommended starting values:

Parameter	Default	Guideline
Min Order	2	At least 2 × number of expected modes.
Max Order	60	Increase if modes are missing; decrease if diagram is cluttered.
Order Step	2	Step of 2 is standard (poles appear in conjugate pairs).

Auto AI Mode

Enable Auto AI to let V-Listen automatically determine the optimal model order range based on the data characteristics:

The algorithm analyzes the FRF Sum to estimate the number of visible resonances.
It sets the model order range to 2× to 4× the estimated mode count.
After computation, it pre-selects stable poles using a clustering algorithm.

Tip: Auto AI mode is a good starting point for unfamiliar datasets. Review the auto-selected poles on the stabilization diagram and adjust manually if needed.

Frequency Range Setting

Restrict the analysis to a frequency range of interest to improve computational efficiency and pole quality:

Parameter	Default	Description
Lower Frequency	0 Hz	Start of the analysis band. Set above any DC offset artifacts.
Upper Frequency	Nyquist	End of the analysis band. Set below the noise floor cutoff.

Narrowing the frequency range reduces the number of poles the algorithm must fit and produces cleaner stabilization diagrams. For broadband datasets, consider splitting the analysis into sub-bands.

FRF Sum Modes

The FRF Sum is displayed as a background overlay on the stabilization diagram to help identify resonance frequencies visually. V-Listen offers three summation modes:

Mode	Formula	Use Case
Sum	Σ \|H_ij(f)\|	Default. Shows overall response magnitude. Peaks indicate resonances.
RMS Average	√( Σ \|H_ij(f)\|² / N )	Normalizes by channel count. Better for comparing datasets with different numbers of FRFs.
Max Envelope	max( \|H_ij(f)\| )	Shows the maximum response across all channels. Ensures no resonance is hidden by averaging.

Running the Analysis

Select the algorithm from the Algorithm dropdown.
Set the frequency range using the Lower and Upper frequency fields.
Set the model order range or enable Auto AI.
Click Run (or press F5).
The stabilization diagram appears when computation completes. See Stabilization Diagram for interpretation.

Performance: Computation time depends on the number of FRFs, frequency resolution, and model order range. For large datasets (100+ FRFs), expect 10–60 seconds. A progress bar is displayed during computation.

Step 3: Analysis Algorithms

Algorithm Overview

PolyProMax (Recommended)

How It Works

Stabilization & Cross-Validation Tolerances

LSCF (Standalone)

Theory

Advantages

Backends

LSCE (Standalone)

Theory

Advantages

Model Order Selection

Manual Selection

Auto AI Mode

Frequency Range Setting

FRF Sum Modes

Running the Analysis

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