IDSD: Iterative Deep-Learning-Based Signal Decomposition

Iris Huijben; Joël Karel; Ralf Peeters; Pietro Bonizzi

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IDSD: Iterative Deep-Learning-Based Signal Decomposition

Iris Huijben, Joël Karel, Ralf Peeters, Pietro Bonizzi

arXiv:2608.27332v1Aug 27, 2026eess.SP

TL;DR

Univariate signal decomposition is ill-posed, while classical methods impose fixed priors or require the component count in advance. IDSD replaces these constraints with learned priors and iterative frequency-domain masking. It extracts varied components adaptively and outperformed baselines on synthetic and real data.

Problem
Univariate decomposition is ill-posed, and classical approaches constrain components with fixed priors or require the number of components beforehand.
Method
IDSD learns data-driven component priors and iteratively predicts frequency-domain masks using a structured U-Net, extracting components until a stopping criterion is met.
Results
IDSD outperformed baselines on synthetic, tidal-gauge, and physiological measurements, including when trained only on synthetic data.
Takeaways & Limitations
One trained model can extract a flexible number of component types without requiring ground-truth components in every target domain.

Abstract

from arXiv · show

Data-driven signal decomposition methods decompose a signal into its underlying components in a flexible and adaptive way, taking into account the signal characteristics. Here we focus on univariate signals, whose decomposition is ill-posed. Classical univariate approaches -- like variational mode decomposition -- constrain the solution space (e.g. through narrowband priors), and often require the number of components to be known in advance. These assumption, however, limit the algorithm's usage in certain real-life applications. Instead, we exploit the flexibility of neural networks to replace fixed (narrowband) priors with data-driven priors. Our model, called Iterative Deep-Learning-Based Signal Decomposition (IDSD), iteratively extracts an adaptive number of various types of components from a signal, with no restrictions on the bandwidth of a component. We show superior performance of IDSD both in a controlled setup with synthetic data, and on two real datasets concerning tidal waves and physiological measurements.

I. INTRODUCTION

IDSD addresses the ill-posedness of univariate signal decomposition by replacing fixed priors with data-driven neural-network priors. It extracts diverse components adaptively without requiring their number during training.

I. INTRODUCTION: Classical methods such as VMD impose narrowband constraints and cannot simultaneously extract broadband and narrowband components.
I. INTRODUCTION: Deep learning provides data-driven priors, enabling more flexible decomposition than methods based on fixed priors.
I. INTRODUCTION: IDSD supports narrowband, broadband, and intermittent components while learning from synthetic signals when ground-truth components are unavailable.
I. INTRODUCTION: IDSD predicts frequency-domain masks, constraining each component spectrum to a subset of the input spectrum.
I. INTRODUCTION: During deployment, one trained model iteratively extracts an adaptive number of components, with model size independent of that number.
II. PROBLEM DEFINITION: Univariate signal decomposition is under-determined, so the true components cannot be obtained analytically from the input alone.

IV. THE IDSD ALGORITHM

IDSD iteratively decomposes a signal by predicting frequency-domain masks on successive residual spectra. The masked spectra are transformed back to time domain, residuals are updated, and the process ends with the residual component.

IDSD approximates each component with a point estimate because the underlying conditional inference problem is intractable in high dimensions.
The pipeline applies a discrete Fourier transform, retains positive frequencies, and uses fθ to predict a soft frequency-domain mask.
The mask is post-processed and applied element-wise to the positive-sided complex residual spectrum.
Each masked positive spectrum is completed on the negative-frequency axis and transformed back to time domain as the extracted component.
The next residual spectrum is computed after extraction, and the procedure repeats until the final residual is converted into the noise component.
The resulting decomposition is exact: the extracted components sum to the input signal, while masking keeps each component spectrum within the input spectrum.

B. Architecture of neural network fθ

IDSD uses a one-dimensional U-Net to transform spectral magnitudes into soft masks of matching dimensionality. Its parameterized encoder-decoder architecture and deployment loop produce extracted components and a final residual.

The neural network fθ transforms a spectrum’s magnitude into a soft mask with values between 0 and 1 and the same dimensionality.
The U-Net encoder progressively downsamples inputs, while the decoder upsamples a latent representation back to the input dimensionality through skip connections.
Deployment initializes the residual with the signal spectrum, repeatedly extracts masked components, checks stopping criteria, and reconstructs components with the inverse Fourier transform.
Each encoder block uses two ReLU-activated 1D convolutions and max pooling, while each decoder block uses transpose convolution followed by two ReLU-activated convolutions.
The architecture is parameterized by B, L, P, and C, which affect trainable parameters and the effective receptive field.

C. Training phase of neural network fθ

IDSD is trained once on synthetically generated mixtures whose components are known, enabling supervised learning without real signals containing ground-truth decompositions. The training distribution is assumed to approximate the test distribution, and the model is trained primarily on the dominant component.

IDSD assumes the synthetic training distribution ptrain approximates the test-time distribution ptest and generates mixtures containing varied component types.
Synthetic components vary in amplitude and frequency to create pure sinusoidal, amplitude-modulated, frequency-modulated, and intermittent signals.
The number of components is sampled as K ∼U{1,...,5}, with every signal containing one broadband FM component and additional components selected from specified types.
Known synthetic components permit supervised training by minimizing the MSE between the dominant ground-truth component and the first extracted component.
The dominant ground-truth component is defined as the component with the highest frequency-domain peak, independently of generation order.
Training only on the dominant component is assumed sufficient to learn mappings that also predict high-likelihood components at later extraction steps.

D. Input spectrum pre-processing

IDSD preprocesses the input by converting it to a suitably sized positive-frequency spectrum and then shaping neural-network mask outputs before applying them to the residual spectrum. Temperature controls mask steepness, while post-processing produces a normalized contiguous mask.

IDSD applies an ˜N-point DFT and uses the positive side of the spectrum, choosing ˜N so the frequency-bin count is compatible with the U-Net down-sampling factor P L.
The final sigmoid maps each neural-network output element to a soft-mask value between 0 and 1.
Lower temperature τ produces steeper mask boundaries, whereas higher τ facilitates overlapping boundaries between nearby masks.
Post-processing subtracts the mask minimum, retains only the region covering the dominant residual-spectrum peak, and divides by the maximum.
The resulting mask contains one contiguous region of adjacent frequency bins with values between 0 and 1.

F. Stopping criterion

IDSD stops extraction adaptively by monitoring residual energy rather than requiring a preset component count. Its masking guarantees make residual energy non-increasing, supporting this criterion, while a second tolerance test prevents extracting negligible components.

Extraction stops when residual Fourier-domain energy falls below ζ of the original signal’s energy, with the remaining energy captured in the final residual component.The default threshold is ζ = 1%.
A second criterion stops extraction when updated residual energy fails to decrease by more than tolerance κ, avoiding further extraction of noise or zero-components.The default tolerance is κ = 1%.
Because the post-processed mask values lie between 0 and 1, extracting a new component cannot increase residual-spectrum energy.
Parseval’s theorem and Fourier-spectrum symmetry imply that residual time-domain energy also cannot increase after an additional extraction.
IDSD outperforms all baselines on test sets with K = 2 and K = 5 components, with similar performance across sets and random seeds.
When K is known, users can take the first K extracted components because IDSD extraction does not depend on setting K.

V. EXPERIMENTS ON SIMULATED DATA

On simulated composite signals, IDSD is evaluated against multiple decomposition baselines using normalized component error and qualitative reconstructions. It accurately handles mixed component types, including narrowband, broadband, amplitude-modulated, frequency-modulated, and intermittent components.

10,000 test signals were generated for each of the K = 2 and K = 5 component settings, with AWGN of σ = 0.1.
Quantitative analyses: IDSD outperforms all baselines on both simulated test sets, with performance largely unaffected by the randomization seed.
Qualitative analyses: IDSD predicts masks from residual spectra and reconstructs extracted components in the time domain at each iteration.
Qualitative analyses: IDSD accurately extracts both narrowband and broadband components, whereas VMD, SVMD, and SSD require noise filtering and EWT splits the broadband component.
Qualitative analyses: For a low-frequency plus AM-FM signal, IDSD performs on par with EWT, VMD, and SVMD, although VMD and SVMD require optimal α values and EWT and VMD require known K.

D. Additional analyses

Additional analyses examine IDSD’s architectural structure and compare U-Net mask prediction with alternative model architectures and direct time-domain component prediction. The U-Net frequency-domain mask model performs best among the tested alternatives.

Model design: IDSD combines a U-Net neighborhood prior with frequency-domain mask prediction, ensuring generated component spectra are subsets of the input spectrum.
Model design: MLP mask predictors perform far worse than IDSD’s U-Net despite having more trainable parameters.
Model design: The direct time-domain variant predicts a component from the residual and updates the residual by subtraction.

2) Effect of model size:

The additional experiments assess model size, training-data component counts, loss placement, and residual-energy convergence. A variable-component training strategy supports deployment with different component counts, while residual energy decreases across sequential extraction steps.

Effect of model size: Increasing trainable parameters has little effect for K = 2, but slightly improves performance for the more difficult K = 5 test set.
Training strategy: Training with a varying number of components up to K = 5 performs at least as well as specialized models on both K = 2 and K = 5 test sets.
Training strategy: A model trained with varying component counts can extract any number of K components without retraining for each deployment setting.
Training strategy: Training with MSE loss on only the dominant component slightly outperforms loss applied to all components and makes training faster and more stable.
Residual-energy convergence: Residual energy decreases over eight extraction steps, becoming negligible for most K = 5 test signals after five components.

VI. EXPERIMENTS ON REAL DATA

On tidal-gauge and physiological data, IDSD decomposes mixed signals into meaningful components and supports denoising under added noise. It extracts both tidal and tsunami components and preserves physiologically relevant PPG structure and timing.

A. Tidal waves: IDSD extracted narrowband tidal and broadband tsunami components with the lowest average error among the compared methods.Errors were 0.180, 0.172, and 0.171 across three seeds; the best baseline, EWT, achieved ϵ=0.283.
A. Tidal waves: Without finetuning, IDSD still outperformed all baselines for extracting the tidal and tsunami components, although relative errors increased.Relative errors increased from (ϵ(1), ϵ(2)) = (0.226, 0.118) to (ϵ(1), ϵ(2)) = (0.323, 0.169).
B. Physiological data: IDSD identified the fundamental heart-rate peak at 1.6 Hz and its harmonics in noisy PPG spectra.The example used AWGN with N(0, 0.3), corresponding to 96 beats per minute.
B. Physiological data: IDSD denoising retained PPG waveform shape and cardiac-cycle consistency better than bandpass filtering on noisy signals.The resulting waveform also retained a downward-slope deflection associated with the dicrotic notch and diastolic peak.
B. Physiological data: IDSD-derived peak intervals best resembled ECG RR-interval distributions, measured by KL divergence across subjects and noise levels.KL divergence was lower for IDSD than for bandpass filtering as noise increased; lower values indicate greater similarity.

VII. DISCUSSION

IDSD uses learned frequency-domain structure and iterative extraction to decompose signals with flexible component counts and types. Experiments show strong performance across synthetic and real data, while performance can depend on signal length and phase-overlap conditions.

Properties: IDSD learns data-driven component priors rather than relying on fixed narrowband assumptions.Its frequency-domain masks and built-in structure constrain extracted spectra to the input signal.
Results: IDSD outperformed baselines on synthetic, tidal-gauge, and physiological measurements, including cases trained only on synthetic data.Fine-tuning with one ground-truth tidal component further improved performance on the tidal-gauge measurement.
Implications: IDSD’s complete Fourier dictionary makes the method sensor- and application-agnostic, supporting signals whose components do not follow narrowband assumptions.The conclusion identifies unknown component counts and non-narrowband components as key use cases.
Properties: One trained IDSD model extracts a flexible number of components through repeated deployment until an adaptive stopping criterion is met.The model size and training process do not depend on the eventual number of extracted components.
Limitations: Large differences between training and test signal lengths can produce overly narrow bandpass masks because spectral resolution changes.Suggested mitigations include matching synthetic training lengths to deployment signals, using Welch’s estimator, or increasing τ above 0.5.
Limitations: Phase errors from overlapping spectra are a known issue for time-frequency mask prediction, although they appeared marginal for the signals tested here.The paper identifies phase-error mitigation from speech separation as a direction for future improvement.

APPENDIX A RECEPTIVE FIELD OF U-NET

The appendix analyzes the U-Net encoder’s effective receptive field and relates it to the frequency-bin resolution used for synthetic signals. It also includes comparative relative-error distributions and an IRCNN+ benchmark.

Receptive field: Equation (20) is used to analyze the encoder’s effective receptive field across different values of B, P, and L.The resulting configurations are reported in table V.
Architecture: The U-Net encoder contains 3B + 2 layers, comprising convolutional blocks, pooling layers, and a bottleneck block.Convolutional layers use kernel size L and stride 1, while pooling uses kernel size and stride P.
Receptive field: 513 positive-spectrum frequency bins motivate choosing an effective receptive field close to 513 for the synthetic 1024-sample signals.The signals have one-second duration and sampling frequency fs = 1024 Hz.
Benchmark: With kernel size 32, the reported IRCNN+ average MAE is 0.0338 and RMSE is 0.0627, while the implementation obtains 0.0304 and 0.0439.The implementation’s improvement may reflect a learning-rate scheduler and training until convergence.
Results: For relative errors, IDSD performs best on set AB, while all baselines perform worse on sets AC and AD containing FM and intermittent components.VMD and SVMD degrade on set AB when α(max) is lower.