TOR-GAN: A Transformer-Based OFDM Signals Reconstruction GAN

Abstract

Reconstruction techniques for communication signals represent a significant research focus within the field of signal processing. To overcome the difficulty and low precision in reconstructing OFDM signals, we introduce a signal reconstruction technique called TOR-GAN (Transformer-Based OFDM Signal Reconstruction GAN). Reconstructing IQ sequences using a CNN and RNN presents challenges in capturing the correlations between two signals. To tackle this issue, the VIT (vision in transformer) approach was introduced into the discriminator network. The IQ signal is treated as a single-channel, two-dimensional image, divided into blocks of 2 × 2 pixels, with absolute position embedding added. The generator network maps the input noise to the same dimension as the IQ signal dimension × embedding vector dimension and adds two identical position embedding data points to the network learning. In the transformer network, prob sparse attention is employed as a replacement for multi-head attention to tackle the issue of high computational complexity. Finally, combined with the MLP structure, the transformer-based generator and discriminator are designed. The signal similarity evaluation index was constructed, and experiments showed that the reconstructed signal under QPSK and BPSK modulation had good reconstruction quality in the time-domain waveform, constellation diagram, and spectrogram at a high SNR. Compared with other reconstruction algorithms, the proposed algorithm improved the quality of the reconstructed signal while reducing the complexity of the algorithm.

1. Introduction

Signal reconstruction is a research focus in the field of signal processing [1], where the primary function of this technology is to generate digital signals that closely resemble enemy signals, enabling subtle evasion of the physical layer authentication employed by enemy devices and realizing effective deception and interference. For unknown radio communication signals, there are primarily two approaches for their reconstruction: the conventional reconstruction method and the method based on deep learning. The conventional method mainly adopts the technical route of “feature identification, parameter extraction, hardware generated signal” for signal reconstruction. Among them, the feature identification and parameter extraction of the captured signal have emerged as the key issues of the signal reconstruction scheme. To tackle this issue, blind parameter estimation and modulation recognition have become crucial solutions. However, faced with complex communication protocols, conventional blind parameter estimation methods are difficult to effectively reconstruct [2].

In recent years, research based on deep learning has become an active topic in the field of signal reconstruction, with numerous scholars proposing signal reconstruction methods based on deep learning.

In 2020, Yang Hongjie et al. [3] used BEGAN to realize the signal reconstruction of single-carrier BPSK and 8PSK and validated the quality of the reconstructed signal by comparing the symbol rate, spectral characteristics, constellation diagram, and other features of the reconstructed signal with the real signal. However, this scheme encountered challenges in realizing signal reconstruction for a larger number of sampling points. In the same year, Zhang Xiong et al. [4] spliced the WCDMA signals generated by multiple groups of GANs to obtain the imitation WCDMA downlink signal. They further obtained the interference signal by scrambling it and modulated its transmission to the target terminal to achieve interference on the target terminal. The experiment demonstrated that under low-SNR conditions, the symbol error rate of the terminal after being interfered with by the interference signal was more than 10% higher compared with noise interference alone. However, WCDMA is mainly modulated and transmitted by a single carrier.

In 2021, Zhao Fan et al. [5] used a simple GAN to generate BPSK-, QPSK-, 16QAM-, and 2FSK-modulated communication signals, which demonstrated the generalization of the network, but did not give the similarity evaluation index between the reconstructed signal and the real signal nor verify the modulation characteristics of the reconstructed signal through a constellation diagram analysis. In the same year, Ye et al. [6] used a CGAN to simulate the channel characteristics of an AWGN channel, Rayleigh channel, and frequency-selective channel and used the received pilot signal as conditional information so that the CGAN generator could generate the signal after passing through a specific instantaneous channel, but the system could only transmit single-carrier QAM symbols with a single modulation style.

In 2022, FENG Qi et al. [7] proposed DRAGAN, which is a reconstructed network based on CNN architecture, for the reconstruction of complex signals with specific frame structures. The experimental results demonstrated that the reconstructed signals effectively captured the modulation style, symbol rate, and frequency bandwidth of the sample signals. However, this approach solely relies on mean and standard deviation to assess the quality of reconstructed signals without establishing a comprehensive similarity analysis framework.

In summary, although GAN-based methods have achieved good results in reconstructing single-carrier communication signals, in the face of multi-carrier OFDM communication signals with complex protocols, these methods are difficult to apply to the reconstruction of OFDM signals due to their noise-like characteristics in the time domain [8].

For OFDM modulation signals, the existing conventional blind estimation algorithms mainly estimate the modulation mode [9], carrier frequency [10], chip time width [11], cycle prefix length [12], etc. However, blind parameter estimation of OFDM signals in a complex electromagnetic environment still has limitations, such as sensitivity to noise and fuzzy estimation [13], complicated estimation steps, high computational complexity, and large parameter estimation errors [14]. Consequently, conventional blind estimation schemes are difficult to directly apply to OFDM signal reconstruction.

Due to the inherent characteristics of OFDM signals, deep learning algorithms are unable to directly learn noise-like waveforms in OFDM. An existing method based on deep learning converts signals into time-frequency patterns under known communication protocols, allowing for the extraction of pixel information from graphs [15]. However, this approach requires excessive computational resources and necessitates signal conversion into an image for training purposes, while also heavily relying on prior knowledge of the OFDM signal.

In recent years, deep learning technology has gained widespread adoption in various domains, including image processing and natural language processing. GANs exhibit excellent feature extraction capabilities and find extensive applications in time series data analysis, such as anomaly detection [16], sequence prediction [17], and data augmentation [18]. For OFDM signals, a GAN learns the timing characteristics of IQ signals. However, due to the high complexity of OFDM signals, existing radio signal reconstruction networks based on a single carrier cannot be directly applied to reconstructing OFDM signals. Therefore, this study’s main contributions are as follows:

• The reconstruction model of the OFDM signal was established, and a novel sequential data reconstruction algorithm for OFDM signals based on the transformer was proposed.
• The multi-carrier OFDM signal was reconstructed and the similarity evaluation index system was established. The experimental results show that the reconstructed signal had high similarity to the time domain waveform, constellation, and spectrogram of the original OFDM signal, effectively capturing its distinctive features.
• In comparison with the alternative reconstruction algorithm, our proposed approach achieved reduced algorithmic complexity while enhancing the quality of the reconstructed signal.

2. Communication Signal Dataset Based on OFDM Modulation

For OFDM communication signals, no open channel dataset resources were found; therefore, this study chose a simulation dataset for the subsequent OFDM signal reconstruction.

2.1. OFDM Signal Simulation

The OFDM communication system was constructed using MATLAB [19], and the system flowchart is illustrated in Figure 1. The input of the system is the transmission stream encoded by the source, and then through modulation mapping; the insertion of the pilot; serial to parallel; IFFT; insert cyclic prefix; channel transmission; remove cyclic prefix; FFT; channel equalization; parallel to serial; demodulation reflection; and finally, through the channel decoding to recover the bit stream.

2.2. Establishment of Dataset

In order to simulate signal interception in a real communication countermeasure environment, the signal $y(t)$ directly transmitted through the channel was selected to construct the dataset. For the directly intercepted signal $y(t)$, we first needed to carry out the pre-processing steps, such as symbol timing synchronization, frequency offset estimation, and frequency offset correction.

First, for the symbol timing synchronization, it was necessary to estimate the parameters, such as carrier frequency and chip time width of the signal, and turn the signal from a carrier signal into a baseband signal. Then, the modulation parameters of OFDM signals needed to be blindly estimated, such as the effective length of the OFDM symbol, the full length of the OFDM symbol, and the length of the cycle prefix. After estimating the modulation parameters, subsequent signal synchronization was carried out [11]. The synchronization involved the symbol timing synchronization first, finding the starting point of the signal, and then carrying out the carrier frequency synchronization of the signal.

After the synchronization was completed, for the forward frequency offset estimation and correction, because the cyclic prefix is the copy of the signal tail, the data of the two parts were the same, and thus, the phase difference between the two parts could be used to solve the frequency offset. By calculating the phase difference between the cyclic prefix and its repeating part, the offset value of carrier frequency was calculated according to the phase difference to rectify the frequency deviation [20].

After eliminating these effects on $y(t)$, we then constructed the dataset of the OFDM we needed. In an actual situation, if the signal SNR is too high, it will be too ideal, while if the SNR is too low, the features in the signal will be completely covered by noise. Moreover, to ensure the fidelity of signal reconstruction, two modulation styles of QPSK and BPSK were selected, and the SNR was 10 dB, 15 dB, and 20 dB. There were 2000 signals of each modulation style under each SNR and a total of 12,000 signal data. The dataset information is shown in

Table 1. Dataset information.

Details
Modulation Mode	QPSK, BPSK
Sampling number	256
SNR range	[10 dB, 15 dB, 20 dB]
Signal bandwidth	2160 kHz
Signal sample size	12,000
Channel	AWGN

.

3. Transformer-Based Generative Adversarial Networks

3.1. Generative Adversarial Networks

Since it was proposed by Goodfellow et al. [21] in 2014, GAN has quickly become a hot topic in the research field. Its fundamental architecture comprises two primary components: a discriminator and a generator. The two collaborate in resolving an adversarial minimax problem through an iterative training procedure. The discriminator’s primary task is to discern whether input data are real or generated by a generator. Simultaneously, the generator endeavors to generate a data distribution that closely approximates the distribution of authentic data. Through this adversarial mechanism, the generator learns how to generate data similar to the real data and finally reaches a Nash equilibrium. At this point, it is difficult for the discriminator to distinguish whether the data are from a real sample or generated data.

The objective optimization function of a GAN is formulated as follows:

$$\underset{G}{\min }\underset{D}{\max }V\left(D,G\right)=E_{x\sim p_{data}\left(x\right)}\left[\log \left(D\left(x\right)\right)\right]+E_{z\sim p_z\left(z\right)}\left[\log \left(1-D\left(G\left(z\right)\right)\right)\right]$$

(1)

where $z$ is the input following the random distribution $p_z(z)$; $V(D,G)$ is the combined loss function of the GAN; $G(z)$ represents the data generated by G; and $D(x)$ and $D(G(z))$ represent the probability that D gives the correct discrimination for the real data and the generated data, respectively.

3.2. Transformer-Based GAN

In 2017, Ashish Vaswani et al. [22] proposed the transformer model architecture, which relies on its self-attention mechanism to capture the global relationships within data while transforming input features into output features. This approach has achieved excellent performance in the field of NLP (natural language processing).

Inspired by the transformer, numerous researchers have proposed various GAN models based on transformers for tasks such as image generation, image translation, image super-resolution, video synthesis, text-to-image generation, and timing signal generation in recent years. In 2021, Jiang [23] first designed a GAN based on a pure transformer, which constructs a generator network by upsampling input noise multiple times and achieves an FID score of 9.26 on the CIFAR-10 dataset.

In 2022, Google introduced the VITGAN (Training GANs with Vision Transformers) for the first time [24]. Inspired by the VIT and BERT structures, it divides the image into patch_size × patch_size pixel blocks and sends them to a transformer-based discriminator for training. Furthermore, the noise $z$ is transformed into a latent vector $w$ within the generator and utilized as an input for network training. After generating pixel blocks, the image is spliced to form a complete image, exhibiting comparable performance to that of the CNN model.

In the same year, Li [25] pioneered the application of a transformer in generating time-series data. Taking inspiration from the VIT, the transformer treats multiple one-dimensional time-series signals as multi-channel images and inputs them into a network similar to a Trans-GAN [23] for training, which also achieves good outcomes.

In recent years, the application of a transformer in wireless communication has witnessed a significant surge. In 2021, Yang Li et al. [26] employed the transformer network for device activity detection. Compared with the state-of-the-art covariance method, the transformer network not only achieves superior performance in activity detection but also reduces the computational time. In 2022, Hao Jiang et al. [27] utilized the transformer for channel prediction and proposed a parallel channel prediction scheme based on this architecture. Furthermore. In 2023, Zhiwei Zhang et al. [28] applied the transformer to network traffic prediction and achieved good accuracy by incorporating the CBAM attention mechanism into their model. Additionally, Dexin Zhang et al. [29] introduced an OFDM signal-detection system based on the transformer framework that leverages orthogonal frequency division multiplexing and index modulation techniques, outperforming existing DL detectors with slightly increased complexity.

With the continuous application of a transformer in the wireless communication field, we innovated to apply it in the field of communication signal reconstruction.

4. TOR-GAN: Transformer-Based Communication Signal Reconstruction Generative Adversarial Network

In this part, the signal reconstruction model is proposed according to the characteristics of the OFDM signal, followed by a description of the network structure for signal reconstruction. Subsequently, a loss function is presented that establishes a comprehensive OFDM signal reconstruction system.

4.1. Signal Reconstruction Model

Since the time domain signal of baseband OFDM is superimposed by the time domain sequences transmitted over several orthogonal subcarriers, according to the central limit theorem, the time domain signal of OFDM follows the Gaussian distribution with zero mean [8] and has characteristics similar to Gaussian noise, which is difficult to extract features from using neural networks. OFDM frequency domain signals contain more obvious features and are more conducive to the learning of neural networks. Therefore, according to the modulation process of OFDM, we established a reconstruction procedure to acquire its frequency domain information.

First, based on the prior knowledge of the cyclic prefix length, OFDM symbol boundary, and FFT size, the OFDM signal is preprocessed by S/P, the cyclic prefix is removed, an FFT is used, the signal with obvious feature information is sent to the TOR-GAN network for reconstruction, the reconstructed frequency domain signal is carried out using IFFT, the cyclic prefix is inserted, and P/S is used to obtain the reconstructed OFDM signal. The signal reconstruction model is illustrated in Figure 2.

4.2. Signal Reconstruction Network Structure

The extraction of time dimension information is a crucial aspect in the reconstruction of a communication signal, particularly for OFDM signal sequences that typically exhibit wide transmission bandwidths. In comparison with conventional radio communication, higher sampling rates are necessitated to accurately capture these frequency components, resulting in a significant increase in sampling points within the temporal domain. However, the conventional temporal processing network is susceptible to the issues of vanishing or exploding gradients when handling lengthy sequential data.

When processing time-series data, the transformer model can capture dependencies at different locations in the series through the self-attention mechanism, which is not constrained by temporal order and enables better modeling of long-range dependencies. Consequently, we constructed the generator and discriminator with the transformer model as their core to enhance information capture in the temporal dimension.

4.2.1. Discriminator Network

The construction of the discriminator, as depicted in Figure 3, was initially inspired by VIT [24]. Considering that IQ data represents two mutually orthogonal time-domain sampling points, the amplitude of I is closely correlated with Q. Moreover, the original signal’s amplitude is the square root of the sum of squares between the I and Q amplitudes. Consequently, we treated it as a single channel × two-channel sampling sequence × N sampling points for image data input into the discriminator network. The specific parameters of the encoder network are presented in

Table 2. Network structure of embedding.

Types	Size/Step	Output Size
2D convolution	2 × 2/2	128 × embed_size
Positional_encoding	-	128 × embed_size

. The input signal is divided into N//2 2 × 2-pixel blocks using a 2 × 2 Conv2d with an output size of 128 × embed_size. The transformer method is then utilized to apply the positional_encoding module for absolute position embedding, resulting in a final output of 128 × embed_size.

The encoded signal is then transmitted to the core network for processing based on the transformer architecture.

Table 3. Network structure of discriminator_transformer.

Types	Parameters	Output Size
BatchNorm 1D	Embed_size	128 × embed_size
Prob attention	Num_heads = 16	128 × embed_size
Dropout	Probability = 0.3	128 × embed_size
Layer normalization	Embed_size	128 × embed_size
Linear	Expansion = 4	128 × 4 × embed_size
GELU	-	128 × 4 × embed_size
Linear	Expansion = 4	128 × embed_size
Dropout	Probability = 0.3	128 × embed_size

shows the specific parameters of the core network. First, batch normalization is performed to normalize the data, followed by prob sparse self-attention to learn positional relationships between each sequence block. After layer normalization, the MLP module processes and combines representations generated by attention. The MLP module consists of two fully connected layers with an expansion factor of 4 and utilizes the GELU activation function. Finally, a dropout and residual connection are applied respectively to output the results from self-attention and MLP.

Finally, after the layer normalization in the discriminant module, it is sent to the full connection layer to determine the authenticity of the signal.

Table 4. Network structure of discriminant module.

Types	Paraments	Output Size
Layer normalization	Embed_size	128 × embed_size
Linear	-	128 × 1

shows the specific parameters of the discriminant module.

4.2.2. Generator Network

The generator network, as depicted in Figure 4, is inspired by TTS-GAN [25].

Table 5. Network structure of embedding.

Types	Paraments	Output Size
Linear	Laten_dim	2 × 256 × embed_size
Positionl_encoding	-	2 × 256 × embed_size

shows the specific parameters of the generator coding network. First, the input size is (batch_size, laten_dim), which follows Gaussian distribution noise $z$. Subsequently, this input noise is mapped to the same dimension as that of the IQ signal dimension × embedding vector dimension through a fully connected layer. The same absolute position embedding as the discriminator is subsequently applied to embed identical positional information in both IQ signals, ensuring the correlation between IQ sequences during subsequent learning.

Subsequently, it is forwarded to the transformer-based core network for processing.

Table 6. Network structure of generator_transformer.

Types	Paraments	Output Size
BatchNorm 1D	Embed_size	512 × embed_size
Prob attention	Num_heads = 16	512 × embed_size
Layer normalization	Embed_size	128 × embed_size
Dropout	Probability = 0.3	512 × embed_size
Linear	Expansion = 4	512 × 4 × embed_size
GELU	-	512 × 4 × embed_size
Linear	Expansion = 4	512 × 4 × embed_size
Dropout	Probability = 0.3	512 × embed_size
2D convolution	Kernel_size = 1	2 × 256

presents its specific parameters, wherein the core network structure resembles that of the discriminator network. Finally, Conv2d is applied to reduce the dimensionality of the embedding vector, resulting in two IQ data outputs.

4.3. Loss Function

The definition of the loss function is crucial to the performance of the model, and the loss function of the discriminator $d\_loss$ comprises three components. The negative average value is initially employed to optimize the discriminant outcome of the real signal $D(real)$, thereby enabling the discriminator to accurately identify it as genuine data. Subsequently, the positive average value is utilized to minimize the discriminant result $D(G(z))$ of the reconstructed signal, which facilitates the discriminator in distinguishing it as a false signal. Finally, a gradient penalty term is incorporated to restrict the gradient of the discriminator and enhance model stability.

The overall objective of the discriminator is to maximize the discriminant outcomes for real signals, minimize the discriminant outcomes for reconstructed signals, and enhance model stability through the inclusion of a gradient penalty term. The loss function of the discriminator can be formulated as follows:

$$d\_loss=\max \{-E[D(real)]+E[D(G(z))]+\alpha gp\}$$

(2)

The loss function $g\_loss$ of the generator consists of two components. First, the negative mean is utilized to maximize the discriminant output $D(G(z))$ to enhance the realism of the reconstructed signal. Second, a feature loss term is introduced to capture the differences in features between reconstructions. This term is obtained by calculating $E\left[\left|G(z_2)-G(z_1)\right|\right]$, which represents the difference between reconstructed signals, and $E\left(\left|z_2-z_1\right|\right)$, which represents the difference between random vectors, and dividing them by a constant $eps$. Ultimately, the objective of the generator is to maximize discriminant outputs for reconstructed signals while promoting diversity through the feature loss term. The expression for the generator’s loss function can be formulated as follows:

$$loss1=\min \{-E[D(G(z))]+\frac{1}{\frac{E[\left|G(z_2)-G(z_1)\right|]}{E(\left|z_2-z_1\right|)}+eps}\}$$

(3)

The subsequent section introduces the generator’s constellation diagram loss, which can be determined by evaluating the cluster count of the reconstructed signal and comparing it with that of the real signal. Here, $N$ represents the number of clusters in total, $N_{G(z)}$ denotes the number of clusters in the reconstructed signal, and $N_{real}$ signifies the number of clusters in reality. The similarity between these two cluster numbers is calculated using $y_{sim}=1-\frac{\left|N_{G(z)}-N_{real}\right|}{N_{real}}$. Finally, MSELoss is used to calculate the final constellation diagram loss, and the formula is shown as follows:

$$loss2=\min \{\frac{1}{N}{\displaystyle \sum _{k=1}^N{(y_{sim}-y_{real})}^2}\}$$

(4)

The loss of the final generator is determined by both the antagonistic loss and the constellation diagram loss. Therefore, the final loss function is formulated as a weighted sum of these two components. The expression of the loss function is presented below:

$$g\_loss=\lambda loss1+(1-\lambda )loss2$$

(5)

5. Experiment and Analysis

The proposed method was applied to reconstruct communication signals, and the training details and parameters are described in this section. To validate the effectiveness of the proposed method, we compared it with existing networks and conducted a similarity analysis. The hardware and software environment for this experiment is presented in

Table 7. Hardware and software environments.

Environment	Technical Parameters
OS	Windows 10
CPU	Intel Xeon Silver 4212R
GPU	NVIDIA GeForce 4090
Memory	128 G
Python	Python 3.8.8
Pytorch	Pytorch 1.8.1

.

5.1. Training Details and Parameters

The dataset comprised 12,000 signals, consisting of 6000 QPSK and 6000 BPSK modulations. We input the dataset into the network model while adhering to the specific training parameters outlined in

Table 8. Network model training parameters.

Model Hyperparameters
$\alpha$	10
$\lambda$	0.7
$eps$	$e^{-5}$
Generator embed_dim	160
Discriminator embed_dim	384
Batch size	128
Epochs	100
Learning rate	0.001
${\beta }_1,{\beta }_2$	0.9, 0.999
Optimizer	Adam

.

The generator input is a random vector sampled from a Gaussian distribution (batch size × 200). Both the generator and discriminator had an initial learning rate of 0.001. A linear decay learning rate function was employed to gradually reduce the learning rate to 0 during the final stages of training. Throughout the training process, iterative optimization was performed using the loss function, and upon completion of the model training, the final weight models for both the generator and discriminator were saved.

5.2. Experimental Results

In order to evaluate the generator performance of the trained TOR-GAN, this section presents the results of evaluating the quality of reconstructed signals by visual comparison methods of time domain waveform, constellation diagram, spectrogram, and OFDM sequence waveform to make a visual comparison of signal similarity and using MSE, MAE, and EVM as signal similarity evaluation indexes to evaluate the quality of the reconstructed signals.

5.2.1. QPSK Modulation Signal Reconstruction

This section first presents the results of the reconstructed signal analysis of the QPSK modulation style.

In Figure 5, the red line represents the real part of the signal, and the blue line represents the imaginary part of the signal. It can be observed that at an SNR of 20 dB, the reconstructed signal of QPSK modulation exhibited a high resemblance to the time domain waveform of the original signal, eight OFDM symbol numbers can be clearly distinguished, and the pilot signal position on each OFDM symbol can be learned. It can be seen from the constellation diagram that the constellation diagram of the reconstructed signal was consistent with the modulation characteristics of the original signal. The distribution of the DC subcarriers around the sample points and in the center of the constellation diagram was similar to the original distribution, and the pilot signal at (1,0) was also learned. After the IFFT of the generated signal and insertion of a cyclic prefix, the OFDM reconstructed signal spectrogram was obtained, as shown in Figure 5b. It can be observed that the reconstructed spectrogram basically restored the original spectrum structure, with obvious characteristics of the number of subcarriers, and the corresponding pilot signals were also restored at the seventh and fourteenth subcarriers.

Figure 6 shows the recovered OFDM waveform compared with the actual OFDM signal waveform on the real and imaginary parts. In the figure, the blue curve is the real waveform and the red curve is the reconstructed waveform. It can be seen that the OFDM time-domain waveform presents the characteristics of noise, but the real and imaginary waveforms of the reconstructed signal had a high coincidence degree with the real signal.

It can be observed from Figure 7 that when SNR = 15 dB, the reconstructed signal and the original signal also had a high resemblance in time-domain waveform, and the OFDM symbol number and pilot signal position feature were obvious, but the pilot amplitude feature on the fifth symbol was drowned out by the signal, and there was an abnormal part in the sixth and seventh symbols.

As can be observed from the constellation diagram, the reconstructed signal accords with the modulation characteristics of the original signal, but due to the reduction in the SNR, some noise points appeared in the constellation diagram, causing discrete pilot signal characteristics at (1,0). Compared with SNR = 20 dB, the subcarrier characteristic information of the reconstructed signal spectrogram in Figure 7b is more chaotic, but the corresponding pilot signal can still be identified. It can be observed in Figure 8 that a certain degree of coincidence was maintained between the OFDM reconstructed signal and the real signal.

The time-domain waveform diagram of the reconstructed signal in Figure 9 exhibits an increased presence of noise points and a chaotic waveform when SNR = 12 dB, indicating the influence of noise. Additionally, the constellation diagram became discrete; however, it was still possible to distinguish symbol number and modulation characteristics while maintaining relatively complete pilot frequency information. In the spectrogram of the reconstructed signal in Figure 9b, it is difficult to distinguish the number of subcarriers at this time, and noise signals similar to pilot signals appear in the spectrogram.

It can be seen from Figure 10 that the reconstructed signal waveform has a high coincidence degree with the real waveform.

5.2.2. BPSK Modulation Signal Reconstruction

In this section, the results of the reconstructed signal quality evaluation are given by visual analysis of each characteristic map of the signal for the reconstruction signal of the BPSK modulation pattern.

The reconstructed signal of the BPSK modulation style exhibited a strong resemblance to the time-domain waveform diagram of the original signal when SNR = 20 dB, as depicted in Figure 11. Additionally, the number of eight OFDM symbols was obvious. The constellation diagram of the reconstructed signal also demonstrates consistency with the modulation characteristics of the original signal, while exhibiting a similar distribution of sample points and DC subcarriers, as observed in the original distribution. However, due to the BPSK modulation, the pilot signal became submerged amidst other signals; nevertheless, it was still discernible from the constellation diagram at coordinates (1,0).

In Figure 11b, it can be observed that the reconstructed signal spectrogram basically restored the structure of the original spectrogram, and the characteristics of the subcarrier number and pilot frequency were obvious. In the OFDM signal waveform diagram in Figure 12, it can be observed that the reconstructed signal and the real signal also maintained a certain degree of coincidence, but the two were still approximate to the noise distribution.

It can be observed from Figure 13 that when SNR = 15 dB, the time-domain waveform had a high resemblance and obvious symbolic features. The constellation diagram revealed the emergence of noise points in both the real and reconstructed signals as the noise levels increased, yet the reconstructed signal still preserved the modulation characteristics inherent to the original signal. In the reconstructed signal spectrogram depicted in Figure 13b, it can be observed that the number of subcarriers was difficult to identify, but the pilot characteristics were still learned. In the OFDM signal waveform diagram shown in Figure 14, it can be noted that a certain level of correspondence was maintained between the reconstructed signal and the actual signal.

The time-domain waveform similarity between the reconstructed signal and the original signal was significantly reduced when SNR = 12 dB, as depicted in Figure 15. However, the number of symbols could still be recognized. The distribution of the reconstructed signal and original signal in the constellation diagram appeared disordered due to noise interference, yet it still maintained the characteristic features of BPSK modulation. In Figure 15b, subcarriers could not be identified in the reconstructed signal spectrogram, and the pilot frequency was drowned by noise. In the OFDM signal waveform diagram shown in Figure 16, the reconstructed waveforms of the real part and the imaginary part maintained a certain coincidence degree.

5.2.3. Model Comparison and Similarity Evaluation

The number of parameters and time complexity of TOR-GAN were compared with other existing signal reconstruction networks, as shown in

Table 9. Comparison of model parameters.

Model Name	Generator		Discriminator
	Parameters/B	Time Complexity/FLOPs	Parameters/B	Time Complexity/FLOPs
TOR-GAN (prob attention)	1.67 × 107	2.57 × 108	4.21 × 106	1.53 × 108
TOR-GAN (multi-attention)	1.69 × 107	3.45 × 108	5.37 × 106	1.82 × 108
Pattern–constellation dual GAN	1.65 × 107	1.78 × 109	2.04 × 107	2.34 × 108
DRAGAN	1.43 × 106	8.53 × 107	6.99 × 107	1.93 × 109

.

First, the TOR-GAN based on the prob attention mechanism proposed in this paper was compared with the TOR-GAN based on the multi-attention mechanism in the original transformer. It can be observed that the parameter count in the generator of prob attention-based TOR-GAN was essentially consistent with that of the multi-attention-based TOR-GAN, and the time complexity decreased by about 25%. Additionally, there was a reduction of approximately 20% in the parameter count of the discriminator, and the time complexity was reduced by about 16%.

In comparison with the pattern–constellation dual GAN [15], which transforms OFDM signals into images for learning, the proposed TOR-GAN reconstruction model in this paper had a similar number of generator parameters, and the generator time complexity was reduced by one order of magnitude. At the same time, the parameters of the discriminator were reduced by about 80%, and the time complexity of the discriminator was reduced by about 35%. In comparison with the recently proposed single-carrier signal reconstruction DRAGAN [7], the number of discriminator parameters and time complexity were significantly reduced.

Continuing the similarity analysis between the reconstructed signal and the real signal, we selected two time-series data similarity evaluation indexes, namely, MSE (mean squared error) and MAE (mean absolute error). Then, a signal similarity evaluation index EVM (error vector magnitude) was also selected, which is an indicator to measure the performance of the digital communication system and was used to evaluate the difference between the sent signal and the received signal. EVM was used to indicate the degree of deviation between the received signal and the ideal reference signal. It is usually expressed as a percentage or in dB, and the smaller the value, the closer the received signal is to the ideal signal.

As shown in

Table 10. Reconstruction similarity evaluation.

Model Name	Similarity Analysis	BPSK			QPSK
		10 dB	15 dB	20 dB	10 dB	15 dB	20 dB
TOR-GAN (prob attention)	MSE	0.4197	0.3265	0.1339	0.4714	0.3101	0.2096
	MAE	0.4124	0.2701	0.1279	0.4683	0.3115	0.1593
	EVM	1.3008	0.9663	0.7429	1.3314	1.0969	0.8248
TOR-GAN (multi-attention)	MSE	0.3970	0.3083	0.1460	0.4307	0.3086	0.1981
	MAE	0.3413	0.2849	0.1284	0.4142	0.2981	0.1660
	EVM	1.1395	0.9588	0.7911	1.2415	0.9935	0.8303
DRAGAN	MSE	0.7051	0.5319	0.4197	0.6424	0.5740	0.4200
	MAE	0.6376	0.4945	0.4086	0.5359	0.5542	0.3447
	EVM	2.7872	1.5969	1.2761	2.4858	1.3715	1.1097

, all three indexes exhibited comparable performance to TOR-GAN (prob attention) and TOR-GAN (multi-attention). In the BPSK modulation scheme, TOR-GAN (prob attention) outperformed TOR-GAN (multi-attention) in the three metrics when SNR = 20 dB. Additionally, at SNR = 15 dB, the MAE value was also superior. Moreover, under the QPSK modulation style, TOR-GAN (prob attention) exhibited better performance in terms of the MAE and EVM indexes. Compared with DRAGAN, the proposed TOR-GAN demonstrated enhanced capability in learning OFDM signals and achieved significant improvements across all indicators.

5.3. Network Generalization Ability Verification

The RML2016.10a dataset was selected and the signal waveform was reconstructed to assess the generalization and robustness of the proposed network, as no other publicly available OFDM signal datasets were found for verification purposes.

5.3.1. Dataset Introduction

The RML2016.10a dataset was specifically designed for wireless signal recognition, and we tried to use it for signal reconstruction research. It encompasses 11 modulation modes and 20 center frequencies, comprising a total of 220,000 signal samples with each combination of modulation mode and center frequency having 2000 samples. Each sample represents an IQ sample with a duration of 128 microseconds and includes a label indicating its corresponding modulation mode and center frequency.

Table 11. Dataset information.

Details
Modulation mode	8PSK, BPSK, CPFSK, GFSK, PAM4, 16QAM, 64QAM, QPSK
Sampling number	128
SNR range	[−8 dB, −6 dB, …, 10 dB]
Signal sampling frequency	1MHz
Signal sample size	80,000

contains the details of this dataset.

5.3.2. Experimental Results

The reconstructed signals used in the validation experiments were BPSK modulated and QPSK modulated, and the quality of these signals was assessed based on the time-domain waveform of IQ signals.

The BPSK and QPSK signals reconstructed by TOR-GAN are depicted in Figure 17, Figure 18, Figure 19 and Figure 20 It can be observed from the waveforms that the proposed network could learn the waveform characteristics of the signals when the SNR was 10 dB and 8 dB and could reconstruct the signal waveform similar to the original signal. The network showed a good generalization ability on the RML2016.10a dataset, but compared with the original signal, the reconstructed waveform was more chaotic and fluctuated more sharply than the original waveform. Further improvement of the network is still needed to improve the reconstruction effect.

The results presented in

Table 12. Reconstruction similarity evaluation.

Similarity Analysis	BPSK		QPSK
	8 dB	10 dB	8 dB	10 dB
MSE	0.0027	0.0021	0.0019	0.0012
MAE	0.0064	0.0023	0.0052	0.0039
EVM	0.5708	0.3809	0.5946	0.4615

demonstrate the excellent signal reconstruction quality achieved by BPSK and QPSK modulation schemes. Specifically, for BPSK modulation, the EVM index reached 0.3809 at an SNR of 10 dB and 0.5708 at an SNR of 8 dB. Similarly, QPSK modulation exhibited a comparable EVM value to that of BPSK modulation. However, due to the small amplitude range (maximum value was approximately 0.015) in this sample, the variations in MSE and MAE values were minimal but still reflected the quality of signal reconstruction to a certain extent.

5.4. Signal Reconstruction in Multipath Channels

In multipath channels, when the transmission time difference between different paths is larger than the CP length of the OFDM signal, multipath interference will occur, resulting in ISI (inter-symbol interference) at the receiving end.

In cooperative communication schemes, the receiver usually performs frequency domain equalization to the received OFDM signal to eliminate ISI to achieve accurate demodulation of the signal. Frequency domain equalization can modify the received signal in the frequency domain according to the result of channel estimation so that the signal on each subcarrier can be restored to the original modulation symbol.

The simulated OFDM signal served as an example, wherein the absence of modulation characteristics in the constellation distribution was observed when transmitted through multi-path channels. However, upon MMSE equalization of the received signal, the constellation distribution map distinctly revealed QPSK modulation characteristics, along with DC subcarrier and pilot distributions in the constellation distribution diagram of Figure 21.

In non-cooperative communication schemes, blind frequency-domain equalization techniques, such as higher-order cumulants [30], can be employed to estimate and compensate for the channel effect by analyzing and modeling the statistical characteristics of the received signal. Following channel estimation to mitigate multipath effects, the signal can be fed into the proposed TOR-GAN network for signal reconstruction.

6. Summary and Prospects

6.1. Summary

In this study, we established a model for reconstructing OFDM signals and proposed an innovative sequential data reconstruction algorithm based on a transformer for OFDM signals. In the proposed TOR-GAN, IQ sequences are treated as single-channel two-dimensional images in the discriminator and partitioned by Conv2d for learning. To ensure correlation between IQ sequences, we added the same absolute position coding to the generator and adopted prob sparse self-attention in both the generator and discriminator. We also proposed a loss function of the constellation map that showed good reconstruction quality in the time-domain waveform, constellation map, and spectrogram while constructing a similarity evaluation index. Compared with other models, our approach improved the reconstruction accuracy while reducing parameter count.

6.2. Prospects for Follow-Up Research Work

The present study successfully achieved the reconstruction of BPSK and QPSK modulation for two commonly employed OFDM signals. However, it should be noted that the dataset and algorithm proposed in this paper solely simulated OFDM signal transmission under AWGN channel conditions, while real-world scenarios may involve more complex channel environments. Furthermore, the proposed OFDM signal reconstruction method in this paper requires implementation within a specific scenario where certain prior knowledge is available; thus, its applicability becomes challenging in blind reconnaissance scenarios.

And for the more complex modulation modes of 16QAM and 64QAM, the current method could only recover the time-domain waveform. However, on the constellation chart, we could not accurately reconstruct the detailed information corresponding to the 16 and 64 constellation points. In order to solve this problem, future research can further optimize the network structure, aiming to restore the time-domain waveform while also ensuring the accuracy of the constellation map. With such improvements, we expected to be able to deal with various modulation modes of OFDM signals more comprehensively.

Considering that prob sparse attention achieves better results than multi-head attention only under partial SNR while reducing the number of parameters, we hope to find a self-attention mechanism that can reduce the number of parameters while maintaining the quality of signal reconstruction.