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Nonlinear System Identification

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The Nonlinear System Identification module provides tools for characterizing nonlinearities in audio systems using the Parallel Hammerstein Model (PHM). By utilizing Synchronized Swept Sine (SSS) excitation, the system extracts isolated kernels representing the linear fundamental response ($h_1$) and higher-order harmonic distortions ($h_2$ through $h_5$).

Architecture and Data Flow

The identification process is split between high-precision offline analysis (for detailed modeling) and a real-time engine (for live monitoring).

Nonlinear Analysis Pipeline

The identification follows a multi-step DSP pipeline:

  1. Excitation Generation: A phase-synchronized exponential sine sweep is generated according to Novak et al. (2015) constraints src/core/nonlinear_analyzer_core.py:111-168.
  2. Synchronous Capture: The signal is played and recorded across multiple amplitude steps (typically 5-10) src/gui/widgets/nonlinear_analyzer.py:161-167.
  3. Clock Drift & Latency Compensation: If input/output devices differ, the system performs sub-sample peak detection and windowed sinc resampling to correct for clock drift src/core/nonlinear_analyzer_core.py:9-108.
  4. Deconvolution: The recorded signal is deconvolved with the excitation signal in the frequency domain using Tikhonov regularization src/core/nonlinear_analyzer_core.py:171-192.
  5. Kernel Extraction: Harmonic responses are isolated and converted into Parallel Hammerstein kernels via Chebyshev inversion src/core/nonlinear_analyzer_core.py:194-250.

Nonlinear Identification Logic

The following diagram illustrates the relationship between the GUI, the core analysis logic, and the real-time engine.

System Identification Component Map

graph TD
    subgraph "GUI Layer"
        NAW["NonlinearAnalyzerWidget"]
        RVW["ResponseViewerWidget"]
    end

    subgraph "Analysis Core"
        NAC["nonlinear_analyzer_core.py"]
        HM["hammerstein_model.py"]
        RSSS["RealtimeSSSEngine"]
    end

    subgraph "Audio Backend"
        AE["AudioEngine"]
        PRS["PlayRecSession"]
    end

    NAW -->|Triggers| NAC
    NAW -->|Uses| PRS
    PRS -->|Interacts| AE
    NAC -->|Saves/Loads| HM
    HM -->|Shared Cache| RVW
    RSSS -->|Real-time Stats| NAW
Loading

Sources: src/gui/widgets/nonlinear_analyzer.py:156-193, src/core/nonlinear_analyzer_core.py:1-6, src/core/realtime_sss_core.py:16-41, src/core/hammerstein_model.py:85-112

Core Components

RealtimeSSSEngine

The RealtimeSSSEngine provides low-latency harmonic extraction. It uses a local least-squares (LS) extractor to estimate harmonic coefficients $h_1 \dots h_5$ block-by-block during a sweep src/core/realtime_sss_core.py:16-31.

Feature Implementation Detail
Phase Trajectory Calculated using Novak's constraints for perfect phase sync src/core/realtime_sss_core.py:73-109.
LS Extractor Fits complex harmonics to a sliding history of signal samples src/core/realtime_sss_core.py:190-210.
Decimation Optimized by processing history in decimated chunks to save CPU src/core/realtime_sss_core.py:220-250.

Sources: src/core/realtime_sss_core.py:16-250

Parallel Hammerstein Model (PHM) Extraction

The process_amplitude_responses function is the primary entry point for extracting the full model. It takes responses at different amplitudes and solves the Chebyshev inversion matrix src/core/nonlinear_analyzer_core.py:194-250.

  • Amplitude Steps: Multiple sweeps at different levels allow the system to differentiate between power-series components (e.g., $x^2$ vs $x^4$) that both contribute to the 2nd harmonic src/gui/widgets/nonlinear_analyzer.py:167.
  • Chebyshev Inversion: This mathematical transform ensures that the extracted kernels $h_n$ are independent of amplitude, allowing for accurate simulation at any input level src/core/nonlinear_analyzer_core.py:210-240.

Sources: src/core/nonlinear_analyzer_core.py:194-250, src/gui/widgets/nonlinear_analyzer.py:156-170

Visualization and Simulation

Response Viewer

The Response Viewer module consumes the output of the Nonlinear Analyzer. It visualizes the 1D kernels and generates 2D "Distortion Maps" src/gui/widgets/response_viewer.py:48-61.

Data Flow: Identification to Visualization

sequenceDiagram
    participant NA as NonlinearAnalyzer
    participant HM as HammersteinModel (Cache)
    participant RV as ResponseViewer
    
    NA->>NA: Run SSS Sweeps (Amp 1..N)
    NA->>NA: Extract Kernels (h1..h5)
    NA->>HM: set_active_model(data)
    Note over HM: Data stored in _ActiveModelCache
    RV->>HM: get_active_model()
    RV->>RV: Generate 2D THD Map
    RV->>RV: Simulate Tone Response
Loading

Sources: src/core/hammerstein_model.py:85-112, src/gui/widgets/response_viewer.py:119-122

2D Harmonic Mapping

The ResponseViewerWidget can generate a 2D surface representing THD or specific harmonics ($h_2 \dots h_5$) across both frequency and amplitude axes src/gui/widgets/response_viewer.py:152-185. This is achieved by interpolating the complex kernels and calculating the predicted response for a virtual input signal src/core/hammerstein_model.py:115-138.

Visualization Mode Description
Bode Magnitude Frequency response of individual kernels in dB src/gui/widgets/nonlinear_analyzer.py:57-61.
Impulse Response Time-domain view of kernels $h_1 \dots h_5$ src/gui/widgets/nonlinear_analyzer.py:66-69.
THD Map 2D color map of Total Harmonic Distortion vs. Frequency and Level src/gui/widgets/response_viewer.py:157-160.

Sources: src/gui/widgets/nonlinear_analyzer.py:53-70, src/gui/widgets/response_viewer.py:152-185

Technical Implementation Details

Clock Drift Compensation

When the input and output clocks are not synchronized (e.g., two different USB devices), the sweep will "smear," causing phase errors. The system detects this by comparing the measured duration to the expected duration src/core/nonlinear_analyzer_core.py:72-108.

  • Function: sinc_resample
  • Method: Kaiser-windowed sinc interpolation applied in chunks to manage memory footprint src/core/nonlinear_analyzer_core.py:81-102.

Sub-sample Alignment

To ensure accurate phase for high-order harmonics, the impulse response peak must be found with precision higher than one sample src/core/nonlinear_analyzer_core.py:9-56.

  • Function: find_subsample_peak
  • Method: Performs 100x FFT-based upsampling around the integer peak to find the true fractional maximum src/core/nonlinear_analyzer_core.py:37-50.

Sources: src/core/nonlinear_analyzer_core.py:9-108

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