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Feedforward Compensation Algorithm

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youtube-at-vach edited this page Jul 23, 2026 · 1 revision

Feedforward Compensation Algorithm

Relevant source files

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The Feedforward Compensation subsystem implements the Iterative Linear-Inverse Compensated Feedforward (LICFF) algorithm. This system uses a Parallel Hammerstein Model (PHM) identified from the Nonlinear System Identification module to pre-distort audio signals, effectively cancelling both linear frequency response errors and nonlinear harmonic distortions in the playback chain.

LICFF Algorithm Overview

The LICFF algorithm operates by applying a series of inverse kernels to the input signal. Unlike simple linear equalization, LICFF accounts for the interaction between frequency-dependent gain and nonlinear power-series terms. The implementation supports both real-time processing and iterative offline compensation for high-precision laboratory measurements.

Data Flow and Engine Architecture

The LICFFEngine is the primary computational entity, responsible for parsing the PHM model and preparing the inverse filters.

"LICFF Engine Data Flow"

graph TD
    subgraph "Model Parsing"
        A["JSON Model File"] --> B["LICFFEngine.parse_model()"]
        B --> C["Power Series Kernels (q0-q5)"]
    end

    subgraph "Filter Preparation"
        C --> D["LICFFEngine._prepare_buffers_for_length()"]
        D --> E["FFT of Kernels (Q_fft)"]
        D --> F["Tikhonov Regularized Inverse (F_inv_lin)"]
    end

    subgraph "Signal Processing"
        G["Input Signal (u)"] --> H["LICFFEngine.compensate()"]
        F --> H
        E --> H
        H --> I["Pre-distorted Signal (u_comp)"]
    end
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Sources: src/gui/widgets/feedforward_compensator.py:36-73, src/gui/widgets/feedforward_compensator.py:187-200

Implementation Details

Kernel Mapping and Normalization

The engine consumes kernels $h_1$ through $h_5$ extracted via Synchronized Swept Sine (SSS) deconvolution. In the LICFFEngine.parse_model function, these are mapped directly to power-series coefficients $q_p$ src/gui/widgets/feedforward_compensator.py:98-104.

To ensure stability and prevent digital clipping, the engine:

  1. Normalizes kernels based on the peak of the linear response $h_1$ src/gui/widgets/feedforward_compensator.py:125-134.
  2. Thresholds high-order kernels (e.g., $h_4, h_5$) against a noise floor (typically -40 dB to -80 dB relative to $h_1$) to prevent compensating for measurement noise src/gui/widgets/feedforward_compensator.py:107-122.

Inverse Filter Design

The linear inverse filter $F_{inv}$ is calculated in the frequency domain using Tikhonov regularization to prevent extreme boosts at nulls: $$H_{inv}(f) = \frac{H_1^*(f)}{|H_1(f)|^2 + \epsilon}$$ The engine uses solve_eps_in to perform a logarithmic bisection search for an $\epsilon$ value that matches a user-defined maximum boost (e.g., +12 dB) src/gui/widgets/feedforward_compensator.py:160-185.

Iterative Compensation

For nonlinear compensation, the compensate method can run iteratively. In each iteration, it calculates the predicted nonlinear distortion of the current pre-distorted signal and subtracts it from the input src/gui/widgets/feedforward_compensator.py:440-470.

Instability Detection and Stabilization

A critical component of the compensator is ensuring that the pre-distortion does not cause system instability.

"Stability Management and Out-of-Band Handling"

graph TD
    [u_comp_signal] --> [LICFFEngine.compensate()]
    [LICFFEngine.compensate()] --> [Check Out-of-Band Mode]
    
    subgraph "OOB Handling"
        [Check Out-of-Band Mode] -- "bypass_aligned" --> [Apply Bandpass Filter]
        [Check Out-of-Band Mode] -- "full_compensation" --> [Unfiltered Output]
    end

    subgraph "Stabilization Logic"
        [Apply Bandpass Filter] --> [stabilize_poly]
        [stabilize_poly] --> [Polynomial Root Reflection]
        [Polynomial Root Reflection] --> [Stable u_comp]
    end
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Sources: src/gui/widgets/feedforward_compensator.py:59-60, src/gui/widgets/feedforward_compensator.py:201-215

Out-of-Band (OOB) Handling

Since measurement models are only valid within the sweep range (e.g., 20Hz - 20kHz), the engine provides several modes defined in out_of_band_mode:

  • bypass_aligned: Applies a zero-phase bandpass filter matching the model's valid range to the compensated signal, ensuring no attempt is made to correct DC or ultrasonic regions where the model is undefined src/gui/widgets/feedforward_compensator.py:201-225.
  • full_compensation: Allows the inverse filter to operate across the entire Nyquist bandwidth (risky).

Key Classes and Functions

Entity Role Location
LICFFEngine Core DSP implementation of the LICFF algorithm. src/gui/widgets/feedforward_compensator.py:36-36
OfflineFFCompWorker QThread wrapper for batch processing WAV files. src/gui/widgets/feedforward_compensator.py:545-545
FeedforwardCompensator Module-level logic for managing model state. src/gui/widgets/feedforward_compensator.py:657-657
solve_eps_in Bisection search for regularization parameters. src/gui/widgets/feedforward_compensator.py:160-160
forward_model Simulates system output for a given input. src/gui/widgets/feedforward_compensator.py:349-349

Verification and Testing

The algorithm is verified through test_licff_engine_basic and test_compensate_delay_cancellation_nonlinear, which ensure that:

  1. Linear delays in the model are correctly cancelled by the inverse filter tests/logic_verification/gui/widgets/test_feedforward_compensator.py:177-213.
  2. Noise thresholding correctly zeros out insignificant high-order kernels tests/logic_verification/gui/widgets/test_feedforward_compensator.py:138-175.
  3. The iterative process converges for known nonlinearities (e.g., quadratic/cubic systems) tests/logic_verification/gui/widgets/test_feedforward_compensator.py:71-92.

Sources: src/gui/widgets/feedforward_compensator.py:1-600, tests/logic_verification/gui/widgets/test_feedforward_compensator.py:1-231

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