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Score-based Source Separation with Applications to Digital Communication Signals

Tejas Jayashankar, Gary C. F. Lee, Alejandro Lancho, Amir Weiss, Yury Polyanskiy, Gregory W. Wornell

TL;DR

This work tackles single-channel source separation for discrete, structured sources in RF-like signals by formulating a Bayesian MAP objective that leverages independently trained diffusion priors. It introduces alpha-RGS, a gradient-based inference scheme that combines an alpha-posterior with randomized Gaussian smoothing across multiple noise levels to transform a non-differentiable discrete prior into a differentiable objective using diffusion-model scores. The authors show analytically that the method concentrates on the modes of the underlying discrete distributions and demonstrate substantial BER and MSE gains over classical and learning-based baselines in RF interference scenarios, including strong-interference regimes. By treating diffusion priors as general-purpose priors and connecting to Score Distillation Sampling, the approach offers a versatile, data-driven framework for RF interference mitigation and other inverse problems in communications, with potential applicability to multi-source scenarios and beyond.

Abstract

We propose a new method for separating superimposed sources using diffusion-based generative models. Our method relies only on separately trained statistical priors of independent sources to establish a new objective function guided by maximum a posteriori estimation with an $α$-posterior, across multiple levels of Gaussian smoothing. Motivated by applications in radio-frequency (RF) systems, we are interested in sources with underlying discrete nature and the recovery of encoded bits from a signal of interest, as measured by the bit error rate (BER). Experimental results with RF mixtures demonstrate that our method results in a BER reduction of 95% over classical and existing learning-based methods. Our analysis demonstrates that our proposed method yields solutions that asymptotically approach the modes of an underlying discrete distribution. Furthermore, our method can be viewed as a multi-source extension to the recently proposed score distillation sampling scheme, shedding additional light on its use beyond conditional sampling. The project webpage is available at https://alpha-rgs.github.io

Score-based Source Separation with Applications to Digital Communication Signals

TL;DR

This work tackles single-channel source separation for discrete, structured sources in RF-like signals by formulating a Bayesian MAP objective that leverages independently trained diffusion priors. It introduces alpha-RGS, a gradient-based inference scheme that combines an alpha-posterior with randomized Gaussian smoothing across multiple noise levels to transform a non-differentiable discrete prior into a differentiable objective using diffusion-model scores. The authors show analytically that the method concentrates on the modes of the underlying discrete distributions and demonstrate substantial BER and MSE gains over classical and learning-based baselines in RF interference scenarios, including strong-interference regimes. By treating diffusion priors as general-purpose priors and connecting to Score Distillation Sampling, the approach offers a versatile, data-driven framework for RF interference mitigation and other inverse problems in communications, with potential applicability to multi-source scenarios and beyond.

Abstract

We propose a new method for separating superimposed sources using diffusion-based generative models. Our method relies only on separately trained statistical priors of independent sources to establish a new objective function guided by maximum a posteriori estimation with an -posterior, across multiple levels of Gaussian smoothing. Motivated by applications in radio-frequency (RF) systems, we are interested in sources with underlying discrete nature and the recovery of encoded bits from a signal of interest, as measured by the bit error rate (BER). Experimental results with RF mixtures demonstrate that our method results in a BER reduction of 95% over classical and existing learning-based methods. Our analysis demonstrates that our proposed method yields solutions that asymptotically approach the modes of an underlying discrete distribution. Furthermore, our method can be viewed as a multi-source extension to the recently proposed score distillation sampling scheme, shedding additional light on its use beyond conditional sampling. The project webpage is available at https://alpha-rgs.github.io
Paper Structure (82 sections, 6 theorems, 62 equations, 18 figures, 1 table, 4 algorithms)

This paper contains 82 sections, 6 theorems, 62 equations, 18 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Broader Impact.
  3. Contributions.
  4. Paper Organization.
  5. Prerequisites
  6. Finite Alphabet Signal Processing
  7. Diffusion Models
  8. A Bayesian Framework for Source Separation
  9. Maximum a Posteriori (MAP) Source Separation
  10. Signal statistical structure.
  11. MAP problem formulation.
  12. Proposed Method (a-RGS)
  13. The Smoothing Model and a-posterior Generalized Bayes'
  14. Surrogate distribution.
  15. Gaussian Smoothing Model.
  16. ...and 67 more sections

Key Result

Proposition 1

Let $s(t)$ and $b(t)$ be two sources following eq:signal processing equation with underlying symbols $c^s_p$ and $c^b_p$ respectively. Assume that the symbols are obtained as, where $f:\mathcal{U}^L \rightarrow \mathbb{C}$ and $h:\mathcal{V}^L\rightarrow \mathbb{C}$ are mappings from a sequence of length $L$ over the discrete alphabets $\mathcal{U}$ and $\mathcal{V}$ respectivelyIn digital commun

Figures (18)

  • Figure 1: Left: Two discrete sources, with infinitesimal additive noise, superimposed to produce a joint distribution with 8 equiprobable modes. An observed mixture $\mathbf{y}$, imposes a linear constraint in this space. Middle: Extending vanilla MAP ($\omega=1$) to multiple noise levels still has a relatively large local minima. Right: By using $\omega=\kappa^2$, we are able to accentuate the correct mode and smooth the landscape even further. Colored curves correspond to \ref{['eq:single noise level loss expectation']} evaluated with $T=1$ and $t=u$.
  • Figure 2: The single-carrier digital modulation pipeline with an intelligent decoder that performs a pre-processing stage of source separation. Illustrated is an example with a QPSK constellation and a root-raised cosine (RRC) pulse shaping function.
  • Figure 3: Left: Source separation results for a mixture with RRC-QPSK SOI and OFDM (QPSK) interference. All curves are obtained by averaging 400 different mixtures per SIR and using $\omega=\kappa^2$. Our proposed method significantly outperforms traditional and learning-based baselines (BASIS) in terms of BER and MSE across all noise levels. Middle: Similar comparisons only in BER for mixtures with OFDM (BPSK) and CommSignal2 interference, respectively. BER is slightly higher for the CommSignal2 mixture since it contains background noise that is amplified for large $\kappa$. The black dotted line in the bottom figure denotes the (presumed) BER lower bound assuming the background noise is an additive white Gaussian noise. Right: BER and MSE versus $\omega/\kappa^2$ for different SIR levels. Clearly a good choice in this setting, in the sense of minimum BER and MSE, is $\omega=\kappa^2$.
  • Figure 4: Simulating \ref{['eq:proposed loss single term']} on a BPSK source. The loss at individual timestamps is visualized in addition to the total loss. The minima are at the modes of the BPSK source distribution, $-1$ and $+1$. Larger noise levels allow for exploration between modes and smaller noise levels sharpen the mode-seeking behavior.
  • Figure 5: Left: A GMM source with two equiprobable modes at $-1$ and $-2$. Two smaller modes are present at $+2$ and $+4$. Right: The minima of \ref{['eq:proposed loss single term']} lie (approximately) at the modes of the source distribution (black curve). Colored curves correspond to \ref{['eq:proposed loss single term']} evaluated with $T=1$.
  • ...and 13 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5