## 3 Conditional Diffusion Model for OFDM-Based Image Reconstruction
### Conditional Diffusion and Reverse Processes  
In order to reconstruct OFDM-based images, the proposed CDM modifies the mean of the Gaussian noise addition step from the existing diffusion process defined in Equation (7) by introducing the interpolation parameter \(v_{t}\). It was assumed that \(v_{t}\) starts with \(v_{0}\approx 0\) at \(t=0\) and increases by gradual steps to reach \(v_{T}\approx 1\) at \(t=T\), so that the average of \(x_{t}\) changes from the original image data \(x_{0}\) to the distorted OFDM-based image \(y\), as shown  
Figure 4: Conditional diffusion and reverse processes for OFDM-based image reconstruction. in Figure 4. This parameter integrates both the original image data \(x_{0}\) and the distorted received signal \(y\) into the conditional diffusion process as follows:  
\[q(x_{t}|x_{0},y)=\mathcal{N}\Big{(}x_{t};(1-v_{t})\sqrt{\bar{\alpha}_{t}}x_{0}+v _{t}\sqrt{\bar{\alpha}_{t}}y,\ \ \sigma_{t}\mathbf{I}\Big{)}, \tag{17}\]  
where \(\sigma_{t}\) denotes the variance of the conditional diffusion process, which is set in such a manner as to generalize the variance of the conventional diffusion process as follows:  
\[\sigma_{t}=(1-\bar{\pi}_{t})-v_{t}^{2}\bar{\pi}_{t}. \tag{18}\]  
The probability distribution of \(q(x_{t}|x_{t-1},y)\) can be derived from the relationship between \(x_{t}\) and \(x_{t-1}\), after defining \(q(x_{t}|x_{0},y)\) and \(q(x_{t-1}|x_{0},y)\) based on Equation (17) as follows:  
\[q(x_{t}|x_{t-1},y)=\mathcal{N}\bigg{(}x_{t};\frac{1-v_{t}}{1-v_{t-1}}\sqrt{ \alpha_{t}}x_{t-1}+\bigg{(}v_{t}-\frac{1-v_{t}}{1-v_{t-1}}v_{t-1}\bigg{)} \sqrt{\bar{\alpha}_{t}}y,\ \sigma_{t|t-1}\mathbf{I}\bigg{)}, \tag{19}\]  
\[\sigma_{t|t-1}=\sigma_{t}-\bigg{(}\frac{1-v_{t}}{1-v_{t-1}}\bigg{)}^{2}\alpha _{t}\sigma_{t-1}. \tag{20}\]  
In the reverse process of the CDM, the conditional denoising process is carried out based on the distorted images resulting from the conditional diffusion process. Based on Equations (17) and (19), \(q(x_{t-1}|x_{t},x_{0},y)\) can be derived through [PERSON] theorem and the Markov chain property as follows [21]:  
\[ \begin{array}{rl}q(x_{t-1}|x_{t},x_{0},y)=&\mathcal{N}(x_{t-1};\frac{1-v_{t }}{1-v_{t-1}}\frac{\sigma_{t-1}}{\sigma_{t}}\sqrt{\alpha_{t}}x_{t}+(1-v_{t}) \sqrt{\bar{\alpha}_{t}}x_{0}+\\ &(v_{t-1}\sigma_{t}-\frac{v_{t}(1-v_{t})}{1-v_{t-1}}x_{t}\sigma_{t-1})\frac{ \sqrt{\bar{\alpha}_{t-1}}}{\sigma_{t}}y,\bar{\sigma}_{t}\mathbf{I}),\\ &\bar{\sigma}_{t}=\frac{\sigma_{t|t-1}\sigma_{t-1}}{\sigma_{t}}.\end{array}  \tag{21}\]  
In order to predict \(x_{t-1}\) from \(x_{t}\) and \(y\) in the conditional reverse process, it is necessary to employ a probability distribution such as \(q(x_{t-1}|x_{t},y)\), which can be defined as:  
\[p_{\theta}(x_{t-1}|x_{t},y)=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},y,t),\ \bar{ \sigma}_{t}), \tag{23}\]  
\[\mu_{\theta}(x_{t},y,t)=c_{x}x_{t}+c_{y}y-\ c_{e}\epsilon_{\theta}(x_{t},y,t), \tag{24}\]  
where \(\mu_{\theta}(x_{t},y,t)\) represents the mean of the probability distribution used to predict \(x_{t-1}\) during the reverse process, and \(\epsilon_{\theta}(x_{t},y,t)\) denotes the estimated noise vector at time step t, conditioned on the current noisy data \(x_{t}\) and the distorted image \(y\). The coefficients \(c_{x}\), \(c_{y}\), and \(c_{e}\) of \(\mu_{\theta}(x_{t},y,t)\) can be derived by combining Equations (17) and (21) as follows:  
\[c_{x}=\frac{1-v_{t}}{1-v_{t-1}}\frac{\sigma_{t-1}}{\sigma_{t}}\sqrt{\alpha_{t }}+(1-v_{t-1})\frac{\sigma_{t|t-1}}{\sigma_{t}}\frac{1}{\sqrt{\alpha_{t}}},\] (variance (LV), as this approach demonstrates better sampling performance. However, the sampling performance seems to be affected by \(\mathcal{L}_{\text{VLB}}\) depending on the adjustment of \(\Sigma_{\theta}(x_{t},t)\) when the diffusion time step is small [22]. In wireless communications, it is important to accurately reconstruct distorted signals, but fast data processing is also critical.