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Non-Identical Diffusion Models in MIMO-OFDM Channel Generation

Yuzhi Yang, Omar Alhussein, Mérouane Debbah

TL;DR

This work addresses generating high quality MIMO-OFDM channels from uneven initial estimates by introducing a non-identical diffusion model that uses an element-wise time indicator $\bm{t}$. It extends diffusion theory to a vector time with per-element noise progression and develops a two-dimensional time embedding within an MLP-Mixer backbone for channel recovery across $N_a$ antennas and $N_c$ subcarriers. Theoretical results include forward diffusion and non-identical DDIM denoising correctness, and experiments show the method outperforms standard identical diffusion in biased noise scenarios and that training noise patterns and embedding choices critically affect performance. The approach enables reliable CSI generation under non-uniform uncertainty and has practical implications for pilot design and robust channel estimation in wireless systems.

Abstract

We propose a novel diffusion model, termed the non-identical diffusion model, and investigate its application to wireless orthogonal frequency division multiplexing (OFDM) channel generation. Unlike the standard diffusion model that uses a scalar-valued time index to represent the global noise level, we extend this notion to an element-wise time indicator to capture local error variations more accurately. Non-identical diffusion enables us to characterize the reliability of each element (e.g., subcarriers in OFDM) within the noisy input, leading to improved generation results when the initialization is biased. Specifically, we focus on the recovery of wireless multi-input multi-output (MIMO) OFDM channel matrices, where the initial channel estimates exhibit highly uneven reliability across elements due to the pilot scheme. Conventional time embeddings, which assume uniform noise progression, fail to capture such variability across pilot schemes and noise levels. We introduce a matrix that matches the input size to control element-wise noise progression. Following a similar diffusion procedure to existing methods, we show the correctness and effectiveness of the proposed non-identical diffusion scheme both theoretically and numerically. For MIMO-OFDM channel generation, we propose a dimension-wise time embedding strategy. We also develop and evaluate multiple training and generation methods and compare them through numerical experiments.

Non-Identical Diffusion Models in MIMO-OFDM Channel Generation

TL;DR

This work addresses generating high quality MIMO-OFDM channels from uneven initial estimates by introducing a non-identical diffusion model that uses an element-wise time indicator . It extends diffusion theory to a vector time with per-element noise progression and develops a two-dimensional time embedding within an MLP-Mixer backbone for channel recovery across antennas and subcarriers. Theoretical results include forward diffusion and non-identical DDIM denoising correctness, and experiments show the method outperforms standard identical diffusion in biased noise scenarios and that training noise patterns and embedding choices critically affect performance. The approach enables reliable CSI generation under non-uniform uncertainty and has practical implications for pilot design and robust channel estimation in wireless systems.

Abstract

We propose a novel diffusion model, termed the non-identical diffusion model, and investigate its application to wireless orthogonal frequency division multiplexing (OFDM) channel generation. Unlike the standard diffusion model that uses a scalar-valued time index to represent the global noise level, we extend this notion to an element-wise time indicator to capture local error variations more accurately. Non-identical diffusion enables us to characterize the reliability of each element (e.g., subcarriers in OFDM) within the noisy input, leading to improved generation results when the initialization is biased. Specifically, we focus on the recovery of wireless multi-input multi-output (MIMO) OFDM channel matrices, where the initial channel estimates exhibit highly uneven reliability across elements due to the pilot scheme. Conventional time embeddings, which assume uniform noise progression, fail to capture such variability across pilot schemes and noise levels. We introduce a matrix that matches the input size to control element-wise noise progression. Following a similar diffusion procedure to existing methods, we show the correctness and effectiveness of the proposed non-identical diffusion scheme both theoretically and numerically. For MIMO-OFDM channel generation, we propose a dimension-wise time embedding strategy. We also develop and evaluate multiple training and generation methods and compare them through numerical experiments.

Paper Structure

This paper contains 28 sections, 2 theorems, 25 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Works
  4. Contributions
  5. Notations
  6. Paper Organization
  7. Problem Formulation
  8. Non-Identical Diffusion Process
  9. Noise adding process
  10. Denoising process
  11. Realizing Non-Identical Diffusion with NN
  12. Training
  13. Generation
  14. Non-Identical Time Embedding for Multi-Antenna OFDM Channels
  15. Problem Restatement and Backbone NN
  16. ...and 13 more sections

Key Result

Theorem 1

The law of $\bm{H}_t$ in Definition def:1 is the same as that of $\bm{\alpha}_t\circ\bm{H}_0+\bm{\beta}_t\circ\bm{\xi}$, where $\bm{\beta}_t:=\sqrt{\bm{1}-\bm{\alpha}^2_t}$, and $\bm{\xi}\sim \mathcal{N}(\bm{0}, \mathbb{I})$ is independent of $\bm{H}_0$, i.e., the PDF of $\bm{H}_t$ is

Figures (8)

  • Figure 1: A conceptual illustration of non-identical diffusion.
  • Figure 2: The backbone NN structure used in this paper, where ①-④ are potential places for time embedding operations.
  • Figure 3: A conceptual illustration of the proposed two-dimensional time embedding method
  • Figure 4: Training convergence curve with different embedding methods.
  • Figure 5: Training convergence curve with different normalization and dimensional averaging methods.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof