Transfer Generative Adversarial Networks (T-GAN)-based Terahertz Channel Modeling

The channel impulse response (CIR) can be represented as

$$h(\tau)=\sum_{l=0}^{L-1}\alpha_{l}e^{j\phi_{l}}\delta(\tau-\tau_{l}),$$

(1)

where $\tau_{l}$ denotes the delay of the $l^{th}$ multi-path components (MPCs), $L$ denotes the number of MPC, $\alpha_{l}$ refers to the path gain and $\phi_{l}$ represents the phase of the corresponding MPC. To characterize the channel, PDP is an important feature, which indicates the dispersion of power over the time delay, specifically, the received power with respect to the delay in a multi-path channel. It can be extracted from the channel impulse response by

$$P(\tau)=|h(\tau)|^{2},$$

(2)

Then, the channel modeling problem is exploited by learning the distribution of PDPs denoted by $p_{r}$, which is difficult to be analytically represented. Instead, the distribution $p_{r}$ can be captured by generating fake PDP samples with distribution $p_{g}$, such that the generated distribution $p_{g}$ of PDPs can match the actual distribution $p_{r}$.

The GAN can be utilized to learn the distribution of PDPs denoted by $p_{r}$, with the framework depicted in Fig 1. The GAN consists of two sub-networks, namely, generator and discriminator. The generator is aimed at generating fake samples $G(z)$ to fool the discriminator, in which $z$ is the noise sample, by mapping the input noise distribution $p_{z}(z)$ to the generated distribution $p_{g}=p(G(z))$. The discriminator tries to distinguish between real samples $x$ from $p_{r}$ and fake samples $G(z)$ from $p_{g}$, and the output of the discriminator $D(x)$ and $D(G(z))$ can be treated as the probability of being a real sample. The two networks are trained in an adversarial manner, which can be considered as a two-player zero-sum minimax game. Specifically, the training objective can be represented by

$$\mathop{\min}\limits_{G}\mathop{\max}\limits_{D}\mathbb{E}_{\boldsymbol{x}\sim p_{r}}[\log D(\boldsymbol{x})]+\mathbb{E}_{\boldsymbol{z}\sim p_{z}}[\log(1-D(G(\boldsymbol{z})))],$$

(3)

The discriminator is designed to distinguish between generated samples and real samples. The numbers of neurons for the five dense layers in the discriminator are 512, 256, 128, 64, 1, respectively. The activation function chosen for the first 4 layers is the LeakyReLU function introduced before. The activation function for the output layer is linear activation function, which is decided by the objective function of WGAN-GP introduced in (4).

The framework for the proposed T-GAN is depicted in Fig. 3, in which the transfer learning is conducted between the measurement and 3GPP TR 38.901 model [16]. The measured PDPs denote the PDPs extracted from measurement with a small size, while the simulated PDPs refer to the PDPs simulated by the 3GPP model, which is implemented with the extracted statistics from measurement. Then, the proposed GAN and T-GAN with the same network structure, are trained on the simulated PDPs and measured PDPs, respectively, to capture the distribution of PDPs. Since the size of measured PDPs is quite small for the training of T-GAN, which can cause the difficulty of converging or the over-fitting problem, the transfer learning is exploited to tackle these problems.

To describe the transfer learning formally, a domain denoted by $\mathcal{D}$ consists of a feature space $\mathcal{X}$ and a marginal probability distribution $\mathcal{P}(\mathbf{X})$ defined on $\mathbf{X}=\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{N}\}\in\mathcal{X}$, where $N$ is the number of feature vectors in $\mathbf{X}$. As depicted in Fig. 3, the target domain $\mathcal{D}_{t}$ and source domain $\mathcal{D}_{s}$ are defined on measurement and 3GPP model, respectively. The feature spaces for the two domains are both constructed by PDPs, with different marginal probability distributions defined on measured PDPs $\mathbf{X_{t}}$ and simulated PDPs $\mathbf{X_{s}}$.

Moreover, given a domain $\mathcal{D}(\mathcal{X},P(\mathbf{X}))$, a task denoted by $\mathcal{T}$ is defined by a label space $\mathcal{L}$ and a predictive function $f(\cdot)$, and the predictive function is learned from the pairs $(\mathbf{x}_{n},l_{n})$ with $\mathbf{x}_{n}\in\mathbf{X}$ and $l_{n}\in\mathcal{L}$. In the target domain $\mathcal{D}_{t}$ and source domain $\mathcal{D}_{s}$, the tasks are the same to capture the distribution of PDPs, and the label space is $\mathcal{L}=\{0,1\}$ representing whether the PDP sample is generated by the proposed GAN or from the training dataset. The T-GAN and GAN serve as the predictive functions $f_{t}$ and $f_{s}$. Then, transfer learning is aimed at learning the function $f_{t}$ in target domain $\mathcal{D}_{t}$ with the knowledge of $\mathcal{T}_{s}$ in source domain $\mathcal{D}_{s}$, i.e., transferring the knowledge stored in GAN trained on simulated PDPs to T-GAN trained on the measured PDPs.

The method of fine-tuning [13] is adopted for the transfer learning. The T-GAN is initialized with the weights of the GAN trained on the simulated PDPs, and is then fine-tuned on the measured PDPs with small size. It is worth noting that both the generator and discriminator in the GAN are transferred, which can yield the better performance in generating high quality samples and fast convergence, compared with transferring only the generator or the discriminator [13].

With transfer learning, the performance of T-GAN can be largely enhanced. Specifically, the channel statistics extracted for 3GPP method are captured by the proposed GAN trained on simulated PDPs, which are further transferred to T-GAN. Moreover, T-GAN can learn the features of PDPs that are not captured by 3GPP method, directly from measurement, which further improves the performance of T-GAN in modeling the distribution of PDPs.

The dataset is collected from the measurement campaign in [14]. which is conducted in an indoor corridor scenario at 306-321 GHz with 400 ns maximum delay, as depicted in Fig. 4. With the measurement data, the PDPs can be extracted to characterize the channel in the 21 receiver points. Since the sample frequency interval is relatively small, as 2.5 MHz, the measured PDPs are very long, including 6001 sample points, which results in extraordinary computation and time consumption to train the GANs. To address this problem, we only use the measured channel transfer functions in the frequency band from 314 to 315 GHz, based on which the PDPs can be shorten to 401 sample points.

The PDPs of the 21 measured channels make up the measured dataset. In addition to the measured dataset, the dataset of simulated PDPs can be generated by 3GPP model with the extracted statistics from the measurement, which consists of 10000 channels. Compared to the measured dataset, the simulated dataset has larger data size with the channel statistics embedded. Moreover, the PDPs in two datasets are normalized into the range of [0, 1] by the min-max normalization method.

The training procedure of the GAN network is explained in detail as follows. Firstly, the input noise vector $z$ of size 100 is generated by the multivariate normal distribution, which can provide the capabilities to transform into the desired distribution. The gradient penalty parameter $\lambda$ in (4) is set as 10, which works well in the training process. Moreover, the stochastic gradient descent (SGD) optimizer is applied for the generator network, and the adaptive moment estimation (Adam) optimizer is chosen for the discriminator network. In addition, the learning rates of the two optimizers are both set as 0.0002 to stabilize the training.

All the experimental results are implemented on a PC with AMD Ryzen Threadripper 3990X @ 2.19 GHz and four Nvidia GeForce RTX 3090 Ti GPUs. In addition, the training of GAN network is carried out in the Pytorch framework.

The proposed GAN is first trained on the simulated dataset, and is then fine-tuned on the measured dataset with transfer learning to develop the T-GAN. The numbers of epochs for training the proposed GAN and T-GAN are both set as 10000. A epoch is defined as a complete training cycle through the training dataset, in which the generator and discriminator are iteratively trained for once. To demonstrate the benefits of transfer learning, the GAN is also trained on the measured dataset without transfer learning for comparison. The loss of generator denoted by G_loss and loss of discriminator denoted by D_loss are shown in the Fig. 5, in which the TG_loss and TD_loss correspond to the losses for T-GAN. For the simulated dataset, it is clear that the generator and discriminator reach the equilibrium in the end. For the measured dataset, the loss of T-GAN is close to the loss for the simulated dataset except for some small fluctuations. The fluctuations are due to the small size of the measured dataset. By comparison, the training is not stable for the GAN network without transfer leaning. There is large fluctuation in the discriminator loss, and the absolute values of G_loss and D_loss are quite large compared to the losses for the simulated dataset. The comparison demonstrates the benefits of the transfer learning in the training of GAN network, which enables T-GAN to converge with a small training dataset. Moreover, it takes only 4000 epochs for T-GAN to converge, compared to 6000 epochs for GAN trained on the simulated dataset. The training time of T-GAN on the measured dataset is also small, which is only 114 seconds compared to 7 hours for GAN trained on the simulated dataset. From these results, it is clear that the transfer learning technique can improve the convergence rate of T-GAN, and reduce the training overhead with the knowledge from the pre-trained model.

In the experiment, the samples of PDP from measurement, 3GPP method, the proposed GAN and T-GAN are compared as in Fig. 6(a). It is clear that the PDPs are similar to each other, which proves that the proposed GAN and T-GAN can learn the features of PDPs. Moreover, it is observed that PDP of measurement is more complex than PDP of 3GPP method. There are more peaks and fluctuations in the temporal domain. This shows that 3GPP cannot well capture the channel effects embedded in PDP. Comparing PDPs generated by the proposed GAN and T-GAN, the PDP generated by T-GAN is close to measurement, while the PDP generated by the proposed GAN is similar to the 3GPP approach. This is reasonable, since the T-GAN can capture the features of PDP from measurement through transfer learning, while the propose GAN can only learn the features of the simulated PDPs by 3GPP method.

In addition, the average PDPs for these method are plotted in Fig. 6(b). It is clear that T-GAN shows good agreement with measurement, while 3GPP and GAN have large deviations from measurement. The deviations can be measured by root-mean-square error (RMSE), calculated as

$$\mathrm{RMSE}=\sqrt{\frac{1}{N_{\tau}}\sum(P_{m}(i\Delta\tau)-P_{g}(i\Delta\tau))^{2}},$$

(7)

where $N_{\tau}$ denotes the number of sampling points in PDP, $i$ indexs temporal sample points of PDPs, $N_{\tau}$ represents the number of sampling points and $\Delta\tau$ is the sampling interval. Moreover, $P_{m}(i\Delta\tau)$ and $P_{g}(i\Delta\tau)$ are the average power in the $i^{\text{th}}$ sample point of measured PDPs and generated PDPs, respectively. The results of RMSE for 3GPP, the proposed GAN and T-GAN are 4.29 dB, 4.12 dB and -4.82 dB, respectively. The T-GAN improves the performance of RMSE by about 9 dB, compared with other methods, which demonstrates that the T-GAN outperforms the other methods in terms of modeling the average power of PDP. This is attributed to the powerful capability of GAN in modeling the complex distribution, and the benefits of transfer learning in better utilizing the small measurement dataset.

Moreover, to measure the similarity quantitatively, Structure Similarity Index Measure (SSIM) is introduced, which is widely applied to evaluate the quality and similarity of images. The range of SSIM is from 0 to 1, and the value of SSIM is larger when the similarity between images is higher. The PDPs generated by 3GPP method, the proposed GAN and T-GAN are compared with measurement. The cumulative probability functions (CDFs) of SSIM for these method are shown in Fig. 7. It can be observed that the proposed T-GAN can achieve higher SSIM values compared with other methods. More than 40$\%$ of SSIM values are higher than 0.6 for T-GAN, compared to only 20$\%$ for 3GPP and the proposed GAN. This further demonstrates the better performance of T-GAN in modeling the PDPs.

Delay spread characterizes the power dispersion of multi-path components in the temporal domain, which can be calculated as the second central moment of PDPs, by

$$\begin{split}\bar{\tau}&=\frac{\sum_{i=0}^{N_{\tau}}i\Delta\tau P(i\Delta\tau)\Delta\tau}{\sum_{i=0}^{N_{\tau}}P(i\Delta\tau)\Delta\tau},\\ \tau_{rms}&=\sqrt{\frac{\sum_{i=0}^{N_{\tau}}(i\Delta\tau-\bar{\tau})^{2}P(i\Delta\tau)\Delta\tau}{\sum_{i=0}^{N_{\tau}}P(i\Delta\tau)\Delta\tau}},\end{split}$$

(8)

where $\bar{\tau}$ denotes the mean delay weighted by the power, $\tau_{rms}$ refers to the root-mean-square (RMS) delay spread, and $P(i\Delta\tau)$ are the power in the $i^{\text{th}}$ sample point of PDPs.

Then, the CDF plot of delay spread for measurement, 3GPP, the proposed GAN and T-GAN is depicted in Fig. 8. It can be observed that the CDFs of delay spread for 3GPP, the proposed GAN and T-GAN match the measurement well.