Contents Introduction
Providing a fast and stable communication link for unmanned aerial vehicles (UAVs) has become essential since most UAVs are used for critical applications such as emergency medical help, disaster management, remote sensing, payload delivery, surveillance, and mapping operations, etc. [1], [8]. As the UAVs keep moving at a speed of 100-500 km/h, establishing a fast, reliable, and stable communication link is a challenging problem in a rapidly changing wireless environment for 5G and beyond applications, especially due to the Doppler effect. At the same time, they are affected by different kinds of noises in channels, which cause substantial interference to the data link [2], [3], [4], resulting in distortion of the transmitted signal, control parameters, image, etc. Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation scheme that offers high bandwidth efficiency and reduces the effect of noise distortions and inter-symbol interference (ISI) into the channel [5], [6]. In OFDM, the carrier signals are made orthogonal to each other, and multiple carriers divide the input data among themselves. At high speeds, subcarrier orthogonality breaks down, leading to degraded OFDM performance. Conventional OFDM is insufficient to handle these conditions effectively [7]. An orthogonal time frequency space (OTFS) modulation is a widely adopted technique to handle the above challenges of high-speed motion between transmitter and receiver [9], [10], [11]. OTFS modulates and demodulates signals in the delay-doppler domain [12], [13], [14]. The time-frequency doubly selective channel can be represented as a sparse channel characterized by multiple taps in the delay-doppler domain. Channel estimation (CE) in this domain efficiently covers the entire signal block. The frequency-domain signal can be transformed into the delay domain using the inverse fast fourier transform (IFFT), whereas signals placed in the delay-doppler domain are converted using the inverse symplectic fast fourier transform (ISFFT). Nevertheless, practical OTFS implementations continue to encounter difficulties in achieving time-frequency dual orthogonality with transceiver filters and frequently depend on rectangular filters. OTFS signals are also affected by inter-carrier interference (ICI) and ISI in high-speed applications [14], [15].
Machine learning (ML) based methods have recently been applied to solve many challenging problems related to communication systems. Information and communication technology, in combination with drones, has demonstrated potential applications in surveillance and monitoring [16]. The UAVs can be combined with internet of things (IoT) devices to increase the coverage and capacity of mobile networks in remote or hazardous areas [17]. The applications of deep learning (DL) for CE in OFDM systems are widely explored [18], [19], [20]. DL methods have also been explored for CE in OTFS systems [26], [41], [42], [45]. Generative adversarial networks (GANs) have recently gained considerable popularity and are seen as a promising technique across various fields, including channel estimation in communication systems [21], [22], [23], [43], [44], [46]. The GANs can generalize well to different environments as they learn to model the distribution of the channel data rather than relying on strict mathematical models and improve the accuracy and robustness of the CE process.
This paper presents a novel GAN architecture for CE in an OTFS-based free space communication channel for UAVs. The proposed GAN architectures comprise a U-Net-based generator and a PatchGAN discriminator for adversarial training of the model. The model performance has been compared with the baseline approaches in terms of bit error rate (BER), outage probability (OP), and Normalized mean squared error (NMSE). The performance of the model has also been evaluated for quadrature phase shift keying (QPSK), 16-quadrature amplitude keying (QAM), and 64-QAM modulations along with time complexity analysis.
A. Related Work
A study on various existing methods for secure wireless communication systems for UAVs is summarised in [25]. The authors have explored blockchain technology for enhancing data security and user privacy in UAV networks. In [24], the authors have explored the line-of-sight (LoS) sensing-based channel estimation technique for UAV-based OFDM systems. The authors derived the LoS and non-line-of-sight signal from the received signal based on threshold parameters. Later, the conventional LS estimation was applied to the LoS signal to improve the channel estimation. The effects of Doppler shifts and ICI on OFDM-based communication for UAVs are presented in [7]. The authors have studied the effects of Doppler shifts and ICI on OFDM-based communication systems for UAVs. The authors have presented the BER analysis and concluded that the above effects are well tolerated at low signal-to-noise ratio (SNR) values (<30 dB) and lower speeds (<400 km/h) of UAVs. However, at higher speeds, the system is affected by Doppler effects are more prominent, and the OFDM modulation is not suitable for such systems. A new 2D modulation scheme, OTFS, is explored in detail for high mobility scenarios in fifth-generation and beyond application in [9] and [10].
In [11], the authors propose a channel estimation technique for OTFS massive multiple-input multiple-output (MIMO) systems based on a 3D-structured orthogonal matching pursuit algorithm. In OTFS systems, modulation and demodulation of signals are performed in the delayed Doppler domain. The authors in [14] introduced a pilot-aided channel estimation scheme for OTFS. In this technique, the pilot, guard, and data symbols are strategically arranged in the delay-Doppler plane to prevent interference between the pilot and data symbols at the receiver. the proposed technique has shown improvements in the CE with channels having multipath channel delays and Doppler shifts. As trajectory planning plays an important role in urban mobility and disaster management scenarios, much work has been done recently for this. An OTFS-based UAV communication system is proposed in [27] for trajectory planning/optimization at high speeds. The authors have compared the OFDM and OTFS techniques for UAVs moving at high speed (120 km/h) and medium Doppler effects. The improvement in trajectory planning is demonstrated by applying the proposed methods. In [28], the authors have discussed a novel localization approach using OTFS modulation and leveraging on physical random access channel (PRACH) preambles for UAVs moving up to 500 km/h. The proposed OFTS-superimposed PRACH-based technique has outperformed the conventional PRACH-based localization and better root mean square error performances.
Preliminary findings for wireless channel assessment and signal identification in OFDM systems using DL are presented in [29]. Here, the authors have utilized the DL method to estimate wireless channels using an end-to-end approach. The suggested approach is different from conventional OFDM systems that first asses the channel state information (CSI) specifically and then estimate the transmitted symbols with the help of the assessed CSI. The authors in [30] have put forward a DL-based channel assessment technique for multiple-cell interference-limited massive MIMO communication systems based on [31], in which multiple single-antenna users are represented by base stations fitted with a large number of antennas. The suggested estimator utilizes a specifically developed deep neural network based on the architecture of the deep image prior network [32] to denoise the received MIMO signal, followed by a traditional LS estimator. Here, the authors also assume that there is no mobility between the transmitter and receiver. A feed-forward denoising convolutional neural network (CNN) for image denoising can also be extended for signal denoising [34]. Here, progress in deep architecture is assessed in new learning methods and regularisation techniques. Residual learning and batch normalization are utilized to accelerate the learning process and reduce the noise effects. In [39], the authors have proposed an ML-based technique to increase the throughput of a software-defined network controller for better UAV communications without compromising on data security. An improvement in OP performance in fast Rayleigh fading MIMO-OFDM channels has been demonstrated in [33].
A DL based on the air communication system has been presented in [35]. The authors have developed a wireless communications system based on neural networks (NN). A detailed comparison of block error rate performance has been demonstrated for the “learned” system and a practical baseline system. The proposed model provides competitive performance near 1 dB, even without rigorous hyperparameter tuning. The authors in [36] have proposed a DL-based long short-term memory NN to estimate channel state information for the OFDM system. The proposed channel estimator can provide better results compared to LS and MMSE estimators even when fewer pilot signals are used. A study on communication techniques for multiple UAVs using a single ground control station (GCS) has been explained in [37]. The UAVs can communicate with GCS directly or through the relay. The authors have evaluated the performance of a multiple-carrier relay-based UAV network over a generalized fading wireless communication channel by calculating OP. In [38], the authors have proposed a relay-based multi-hop radio frequency link for a UAV-operated communications system. The model is also capable of determining the optimal height for flying UAVs to ensure reliable communication links. The initial report on DL/NN-based CE in OTFS systems is presented in [41] and [45]. In [42], the authors have presented a special Transformer NN to improve the performance of OTFS-based CE. The simulation results presented have outperformed the conventional CE methods and other DL-based CE methods for OTFS. The authors in [48] have introduced the concept of the internet of vehicles which studies the integration of communication technology and artificial intelligence for high-mobility vehicles. The authors proposed residual channel attention network-based OTFS signal detection which outperforms conventional DL-based OTFS detection in terms of BER.
GAN is an emerging deep-learning framework that can be used to improve the performance of CE [43], [44]. In [23], the authors have presented the advantages of GAN by improving the accuracy of CE without increasing the length of training sequences. GAN has been effectively used to reduce the number of pilot signals by 70% by retaining the performances of conventional estimators such as LS and MMSE [22]. In [21], the authors have proposed a conditional generative adversarial network (cGAN) for CE for a massive MIMO system. The performance of the proposed model is compared with existing DL methods. The cGAN-based model outperformed the based line DL methods in terms of robustness. The authors in [47] have proposed a GAN-based CE for OTFS systems. The performance of the model has been compared with the conventional CE methods. A concept on distributed GAN for SWARM of UAVs is presented in [40]. The authors have utilized a separate GAN for each air-to-ground channel estimation case. The proposed method has improved the UAV downlink communications by 10% compared with baseline CE methods used in real-time.
The comparison highlighting the key contributions of the proposed work vis-à-vis the state of the art for GAN-based CE in high mobility scenarios is given in Table 1. Based on the above literature and comparison highlighted in Table 1, it can be seen that the different DL-based models were used for OTFS systems, and GAN-based models have been explored for high-speed time-varying UAV channels for improving the CE. Hence, it can be concluded that the researchers have made notable contributions to the above two concepts. However, no studies have yet reported on GAN-based CE for OTFS systems in high-speed, time-varying channels for UAVs, which has been proposed and briefly discussed in this work.
B. Motivation and Contributions
The applications of OTFS-based communication systems for high-speed scenarios are well-documented in [9], [10], [11], [12], [13], and [14]. DL methods have also been explored for CE in OTFS systems [41], [42], [45]. An overview of the application of ML in UAV communication systems is presented in [17]. GANs are becoming increasingly popular and are considered a promising technique across a variety of scenarios, including channel estimation in communication systems [21], [22], [23], [43], [44]. The GANs can be modeled to learn based on channel data rather than relying on strict mathematical models. They can generalize well to the different channel environments and improve the accuracy and robustness of the CE process. The above characteristics of the GAN can be very useful in estimating the received signal in dynamic communication scenarios required for UAVs. To the best of our knowledge, GAN-based channel estimation in OTFS systems for UAV applications has not yet been explored in the literature. This motivates the authors to fill the above gap by exploring the application of the GAN for OTFS-based communication systems for UAVs. Based on the above motivation, the main contribution of this work can be highlighted as follows:
This work proposes a GAN-based CE model for an OTFS-based communication system for UAVs. A modified U-Net model is used as the Generator to train the data set and the PatchGAN discriminator is used for identifying the real vs fake estimate.
The U-Net-based generator and the PatchGAN discriminator were appropriately modified for the system under consideration and to handle the synthetically generated dataset, making the proposed model more suitable for real-time applications.
The proposed model is compared with the three baseline approaches i.e. con-conventional LS estimator, ML-based CE estimator, and CNN-based estimator in terms of BER, OP, and NMSE for different velocities.
Further, the proposed model is evaluated for three different modulation schemes, i.e., QPSK, 16-QAM, and 64-QAM, and the results for BER and OP analysis are compared with the baseline estimators. Moreover, the complexity and processing time of the proposed model have been examined.
The remaining paper is organized as follows: the system model and problem formulation are explained in Section II. The proposed model, along with baseline approaches, is detailed in Section III. Section IV gives the details on the results and discussions, and the concluding remarks are given in Section V.
System Model and Problem Formulation
The block diagram of the overall system model is represented in Fig. 1. The ground station or transmitter is stationary. It can be a hand-held radio transmitter or a laptop/computer running the mission planning software. The air unit or receiver is installed on the UAV, which is moving at high speeds. The transmitter uses the OTFS modulation technique to send the commands from the ground station. The radio signals are passed through the UAV channel (free space) and received by the moving receiver at the UAV. The characteristics of UAV communication channels are shaped by their unique operating environments and mobility patterns. Some key aspects of the UAV channel are LoS dominance, high mobility and Doppler effect, altitude dependency, path losses, and weather effects. The current study assumes favorable weather conditions without any rain or crosswinds and LoS operations at a constant altitude. The received signal is distorted due to multi-path delays caused by various obstacles in the medium and the Doppler effect caused by the high-speed motion of the UAV. The conventional CE estimation with OTFS still faces challenges and struggles to achieve time-frequency dual orthogonality with transceiver filters, often resorting to rectangular filters. Further, OTFS signals are also affected by inter-carrier and inter-symbol interference in high-speed applications [14], [15]. This leads to the motivation for intervention of other advanced ML/DL techniques.
A. Orthogonal Time Frequency Space Model
OTFS is a 2D modulation scheme which is a relatively new technique designed to improve the performance of wireless communication systems, particularly in high-mobility environments where traditional 1D modulation schemes like OFDM might struggle due to Doppler effects. OTFS operates in the delay-Doppler domain, making it more robust against time and frequency dispersions. The block diagram of the OTFS model is explained in [42]. The OTFS model consists of a modulation and demodulation block. The modulation block is having 3 major components, i.e., input data mapping, inverse symplectic finite fourier transform, and heisenberg transform. The demodulation block consists of the wigner transform, symplectic finite fourier transform (SFFT), and equalization and demapping blocks. The different blocks are as defined below:
1) OTFS Modulation
Delay-Doppler domain input: In OTFS, the data symbols are mapped onto a 2D grid in the delay-Doppler domain. Let
$x[k, l]$ Inverse SFFT (ISFFT): The delay-Doppler domain symbols are transformed into the time-frequency domain symbols
$X[n, m]$ \begin{equation*} X[n, m] = \frac {1}{\sqrt {NM}} \sum _{k=0}^{N-1} \sum _{l=0}^{M-1} x[k, l] e^{j2\pi \left ({{\frac {nk}{N} - \frac {ml}{M}}}\right )}, \tag {1}\end{equation*} View Source
\begin{equation*} X[n, m] = \frac {1}{\sqrt {NM}} \sum _{k=0}^{N-1} \sum _{l=0}^{M-1} x[k, l] e^{j2\pi \left ({{\frac {nk}{N} - \frac {ml}{M}}}\right )}, \tag {1}\end{equation*}
Heisenberg transform: The time-frequency domain symbols
$X[n, m]$ \begin{equation*} s(t) = \sum _{n=0}^{N-1} \sum _{m=0}^{M-1} X[n, m] g_{tx}(t - nT) e^{j2\pi m \Delta f (t - nT)}, \tag {2}\end{equation*} View Source\begin{equation*} s(t) = \sum _{n=0}^{N-1} \sum _{m=0}^{M-1} X[n, m] g_{tx}(t - nT) e^{j2\pi m \Delta f (t - nT)}, \tag {2}\end{equation*}
$g_{tx}(t)$ $\Delta f$
2) OTFS Demodulation
Wigner transform: The received time-domain signal
$r(t)$ \begin{equation*} Y[n, m] = \int r(t) g_{rx}^{*}(t - nT) e^{-j2\pi m \Delta f (t - nT)} dt, \tag {3}\end{equation*} View Source\begin{equation*} Y[n, m] = \int r(t) g_{rx}^{*}(t - nT) e^{-j2\pi m \Delta f (t - nT)} dt, \tag {3}\end{equation*}
$g_{rx}(t)$ SFFT: The time-frequency domain symbols
$Y[n, m]$ \begin{equation*} y[k, l] = \frac {1}{\sqrt {NM}} \sum _{n=0}^{N-1} \sum _{m=0}^{M-1} Y[n, m] e^{-j2\pi \left ({{\frac {nk}{N} - \frac {ml}{M}}}\right )}. \tag {4}\end{equation*} View Source\begin{equation*} y[k, l] = \frac {1}{\sqrt {NM}} \sum _{n=0}^{N-1} \sum _{m=0}^{M-1} Y[n, m] e^{-j2\pi \left ({{\frac {nk}{N} - \frac {ml}{M}}}\right )}. \tag {4}\end{equation*}
Equalization and demapping: Equalization in the delay-Doppler domain is performed using estimated channel response
$h[k, l]$ $x[k, l]$ $Y[k, l]$
B. Generative Adversarial Network Model
GANs are a type of deep learning model tailored for generative tasks. They consist of two neural networks—the generator and the discriminator—that are trained concurrently in a competitive process.
1) GAN Components
Generator: The generator G takes a noise vector z, sampled from a prior distribution
$p_{z}(z)$ $G(z)$ Discriminator: The discriminator D receives either a real data sample x or a generated sample
$G(z)$ $D(x)$ $D(G(z))$
2) Training Process
Objective functions: The discriminator aims to maximize the following objective:
\begin{align*} \max _{D} \mathbb {E}_{x \sim p_{data}(x)}[\log D(x)] + \mathbb {E}_{z \sim p_{z}(z)}[\log (1 - D(G(z)))]. \tag {5}\end{align*} View Source\begin{align*} \max _{D} \mathbb {E}_{x \sim p_{data}(x)}[\log D(x)] + \mathbb {E}_{z \sim p_{z}(z)}[\log (1 - D(G(z)))]. \tag {5}\end{align*}
\begin{equation*} \min _{G} \mathbb {E}_{z \sim p_{z}(z)}[\log (1 - D(G(z)))]. \tag {6}\end{equation*} View Source\begin{equation*} \min _{G} \mathbb {E}_{z \sim p_{z}(z)}[\log (1 - D(G(z)))]. \tag {6}\end{equation*}
\begin{equation*} \max _{G} \mathbb {E}_{z \sim p_{z}(z)}[\log D(G(z))]. \tag {7}\end{equation*} View Source\begin{equation*} \max _{G} \mathbb {E}_{z \sim p_{z}(z)}[\log D(G(z))]. \tag {7}\end{equation*}
Optimization: The optimization of G and D is performed using stochastic gradient descent (SGD) or its variants, such as the Adam optimizer.
GANs are powerful generative models that have been used in a variety of applications, including image synthesis, video generation, and data augmentation. The adversarial training process results in the creation of highly realistic data samples.
Proposed Model
The conventional OTFS model uses decision-based estimators for CE in high-speed applications. The proposed model utilizes a data-driven GAN-based approach for CE at the OTFS receiver end in place of traditional estimators such as LS/MMSE estimators. In traditional CE estimators, it is difficult to learn the channel parameters quickly in high-speed scenarios using conventional pilot modes. However, data-driven DL models overcome this problem by automatically learning the channel parameters more effectively without depending on a pilot signal. The overall architecture of the proposed model is given in Fig. 2. The input signal is modulated using the OTFS technique and transmitted through the free space channel. The transmitted signal is affected by noise and multi-path distortion while traveling through the channel. As the receiver on the drone is moving, the received signal also gets affected by the Doppler shift. The received signal is further demodulated using an OTFS demodulator and fed to the GAN block for the final CE process. The GAN block consists of a modified U-Net model as the generator and a PatchGAN discriminator.
A. U-Net Generator
The U-Net model mainly consists of an encoder block or contracting path and a decoder block or expansive path. The basic framework for the proposed generator model is defined in Table 2. In the modified U-Net network, the input layer is a received signal of
Input layer: Let
$ X $ $ H \times W $ \begin{equation*} X \in \mathbb {R}^{H \times W} \tag {8}\end{equation*} View Source\begin{equation*} X \in \mathbb {R}^{H \times W} \tag {8}\end{equation*}
Encoder (contracting path): The encoder applies convolutional layers followed by max-pooling.
Convolutional layers at level
$ l $ \begin{equation*} F_{l} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{l-1})))) \tag {9}\end{equation*} View Source\begin{equation*} F_{l} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{l-1})))) \tag {9}\end{equation*}
$ \sigma $ $ F_{l} $ $ l $ Max-pooling:
\begin{equation*} F_{l} = \text {MaxPool}(F_{l}) \tag {10}\end{equation*} View Source\begin{equation*} F_{l} = \text {MaxPool}(F_{l}) \tag {10}\end{equation*}
Bottleneck (bridge): At the bottom of the U-shape:
\begin{equation*} F_{B} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{L})))) \tag {11}\end{equation*} View Source\begin{equation*} F_{B} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{L})))) \tag {11}\end{equation*}
$ F_{B} $ Decoder (expansive path): The decoder upscales the feature maps and concatenates them with corresponding encoder features.
Upsampling:
\begin{equation*} F_{l+1} = \text {UpConv}(F_{l}) \tag {12}\end{equation*} View Source\begin{equation*} F_{l+1} = \text {UpConv}(F_{l}) \tag {12}\end{equation*}
Concatenation:
\begin{equation*} F_{l} = \text {Concat}(F_{l+1}, F_{L-l}) \tag {13}\end{equation*} View Source\begin{equation*} F_{l} = \text {Concat}(F_{l+1}, F_{L-l}) \tag {13}\end{equation*}
Convolutional layers after concatenation:
\begin{equation*} F_{l} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{l})))) \tag {14}\end{equation*} View Source\begin{equation*} F_{l} = \sigma (\text {Conv}_{2}(\sigma (\text {Conv}_{1}(F_{l})))) \tag {14}\end{equation*}
Output layer: The final layer produces the predicted segmentation map:
\begin{equation*} Y_{\text {pred}} = \text {Conv}_{1 \times 1}(F_{0}) \tag {15}\end{equation*} View Source\begin{equation*} Y_{\text {pred}} = \text {Conv}_{1 \times 1}(F_{0}) \tag {15}\end{equation*}
B. PatchGAN Discriminator
The PatchGAN discriminator is a type of discriminator used in GANs, particularly in image-to-image translation tasks in the Pix2Pix model. Unlike traditional discriminators that classify entire images/datasets as real or fake, the PatchGAN discriminator classifies individual patches of the image/dataset, thereby focusing on local details. PatchGAN has fewer parameters compared to a discriminator that operates on the entire dataset, making it computationally efficient. The basic framework for the proposed discriminator model is defined in Table 3. The PatchGAN discriminator utilized in this work has four convolution layers. Each layer performs a 2D convolution using a
Given an input dataset \begin{equation*} D(I) = \{D_{ij}(P_{ij}(I))\}_{i,j} \tag {16}\end{equation*}
The discriminator loss is computed as:\begin{align*} {\mathcal {L}}_{D} = -\sum _{i,j} \left [{{ \log D_{ij}(P_{ij}(I_{\text {real}})) + \log (1 - D_{ij}(P_{ij}(I_{\text {fake}}))) }}\right ] \tag {17}\end{align*}
The generator’s objective is to fool the discriminator:\begin{equation*} {\mathcal {L}}_{G} = -\sum _{i,j} \log D_{ij}(P_{ij}(I_{\text {fake}})) \tag {18}\end{equation*}
The overall algorithm of the proposed GAN model is presented in Algorithm 1, and notations used are given in Table 4. The training data set is synthetically generated using Matlab. Both the generator and the discriminator are initialized for the training. The model has been trained for 10 epochs with a batch size of 16.
Algorithm 1 Algorithm for the Proposed GAN Model
Training set of input signals
Trained generator model G
Initialize:
Initialize U-Net generator G
Initialize PatchGAN discriminator D
Define optimizers for G and D (Adam with learning rate
Define loss functions:
Discriminator loss:\begin{equation*} {\mathcal {L}}_{D} = \mathbb {E}[\log D(I_{target})] + \mathbb {E}[\log (1 - D(G(I_{real})))]\end{equation*}
Generator loss:\begin{equation*} {\mathcal {L}}_{G} = \mathbb {E}[\log (1 - D(G(I_{real})))] + \lambda \mathbb {E}[\|I_{target} - G(I_{real})\|_{1}]\end{equation*}
for epoch
Shuffle training set
for batch
Sample a batch of
Generate fake images
Update discriminator:
Compute discriminator loss
Update D using gradient descent:
Update generator:
Compute generator loss
Update G using gradient descent:
end for
end for
return: Trained generator G
C. Data Set Generation
The input sequence is generated randomly, which is further encoded and modulated with a QPSK/16-QAM/64-QAM modulation scheme. The modulated signal is superimposed on sub-carriers to generate the OTFS symbol. The dimension of the OTFS symbol generated is
D. Baseline Approaches
The following baseline approaches are used to compare the performance of the proposed model:
Least squares estimation: In the first approach, we consider a conventional least squares (LS) estimation method for CE. The received signal is first converted into a delay-Doppler domain from the time-frequency domain using an OTFS demodulator, and then the CE is done using the LS estimator. The received signal model for LS estimation can be expressed as:
\begin{equation*} R(\tau , \nu ) = h(\tau , \nu ) * P(\tau , \nu ) + N(\tau , \nu ) \tag {19}\end{equation*} View Source\begin{equation*} R(\tau , \nu ) = h(\tau , \nu ) * P(\tau , \nu ) + N(\tau , \nu ) \tag {19}\end{equation*}
$R(\tau , \nu)$ $h(\tau , \nu)$ $P(\tau , \nu)$ $N(\tau , \nu)$ $h(\tau , \nu)$ \begin{equation*} \hat {h}(\tau , \nu ) = \arg \min _{h(\tau , \nu )} \| R(\tau , \nu ) - h(\tau , \nu ) * P(\tau , \nu ) \|_{2}^{2} \tag {20}\end{equation*} View Source\begin{equation*} \hat {h}(\tau , \nu ) = \arg \min _{h(\tau , \nu )} \| R(\tau , \nu ) - h(\tau , \nu ) * P(\tau , \nu ) \|_{2}^{2} \tag {20}\end{equation*}
\begin{equation*} \hat {h}(\tau , \nu ) = \frac {P^{H}(\tau , \nu ) R(\tau , \nu )}{|P(\tau , \nu )|^{2}} \tag {21}\end{equation*} View Source\begin{equation*} \hat {h}(\tau , \nu ) = \frac {P^{H}(\tau , \nu ) R(\tau , \nu )}{|P(\tau , \nu )|^{2}} \tag {21}\end{equation*}
$P^{H}(\tau , \nu)$ $|P(\tau , \nu)|^{2} $ $\hat {h}(\tau , \nu)$ Machine learning: SVM is a powerful supervised ML algorithm primarily used for classification tasks, though it can also be adapted for regression. SVM is well-regarded for its ability to handle high-dimensional data and for finding the optimal boundary between different classes. In this approach, the demodulated signal is fed to the ML-based model for CE. The SVM model is implemented using Python’s sci-kit-learn library. A systematic grid search varying the kernels and regularisation parameters has yielded the best performance using the polynomial degree 3 kernels and regularisation parameter 10.
Convolutional neural network: A CNN is a type of deep learning model specifically designed to process data with a grid-like topology. A CNN-based U-Net model has been considered to compare the performance with the proposed GAN model. The U-Net model comprises an encoder block, a bottleneck layer, and a decoder block. The encoder and decoder blocks have four layers each. In the encoder block, each layer performs operations such as convolution, BN, Max-pooling, etc. In the decoder layer, operations such as up-sampling, concatenation, convolution, etc, are performed. The final output layer produces the predicted signal.
Results and Discussions
A GAN-based model is proposed for a high-speed UAV communication system. The standard simulation parameters used have been given in Table 5. The proposed model is compared with three baseline approaches, i.e., 1. OTFS with conventional LS estimation, 2. OTFS with an ML-based approach, and 3. OTFS with DL-based approach.
A. Simulation Setup
The sample data and corresponding labels are generated in MATLAB software. Simulations were performed on a workstation with 64 GB of RAM and a 16 GB NVIDIA GTX graphics card for training datasets. However, sample testing could be conducted on a device with lower computational resources, such as a GCS or laptop with 8 GB of RAM and a 2 GB NVIDIA GeForce graphics card. The received signal is demodulated using OTFS, and then channel estimation is performed. LS-based CE is done in MATLAB software, whereas the U-Net-based DL model is implemented in Python using tensor flow. An ML-based SVM regression model is implemented using Python’s scikit-learn library. The proposed GAN model is built in Python using tensor flow. The following parameters are used as performance major parameters for the performance evaluation of the different models under consideration:
1) Bit Error Rate
The BER can be defined as the ratio of bits in error to the total bits received at the receiver.\begin{equation*} BER = {\frac {b_{e}}{{b_{t}}}} \tag {22}\end{equation*}
2) Outage Probability
It is defined as the probability that the received signal SNR falls below a specified threshold value and can be written as:\begin{equation*} P_{out}(\gamma _{t}) = P_{\gamma }(\gamma \lt \gamma _{t}) \tag {23}\end{equation*}
\begin{equation*} P_{out} = \int _{0}^{\gamma _{t}} p_{\gamma }(\gamma ) \,d\gamma \tag {24}\end{equation*}
\begin{equation*} P_{out} = \int _{0}^{\gamma _{t}} {\frac {1}{\bar {\gamma }}}\exp {(-\gamma /\bar {\gamma })} \,d\gamma \tag {25}\end{equation*}
3) Normalized Mean Squared Error
NMSE is a key performance metric used to evaluate the accuracy of the estimated channel coefficients compared to the true channel coefficients in wireless channel estimation. It quantifies the estimation error relative to the actual channel conditions, providing insight into the effectiveness of the channel estimation technique being used. The NMSE for a channel matrix H is given by:\begin{equation*} \text {NMSE} = \frac {\mathbb {E}\left [{{\|\mathbf {H} - \hat {\mathbf {H}}\|_{F}^{2}}}\right ]}{\mathbb {E}\left [{{\|\mathbf {H}\|_{F}^{2}}}\right ]} \tag {26}\end{equation*}
\begin{equation*} \|\mathbf {A}\|_{F} = \sqrt {\sum _{i=1}^{M} \sum _{j=1}^{N} |a_{ij}|^{2}} \tag {27}\end{equation*}
B. Results
The experiments are carried out for OTFS channel estimation by considering the high mobility of the receiver with respect to the transmitter at different velocities. The model’s efficacy is compared for BER, OP, and NMSE for various input SNR values. The results are obtained for the conventional LS channel estimator, ML-based estimator, DL-based estimator, and the suggested GAN-based channel estimator. The proposed model has shown improved performance across the entire range of SNR. The training performance in terms of the model’s accuracy for the real and imaginary (Img) part of the data set is given in Table 6.
1) Bit Error Rate Vs Signal-to-Noise Ratio Analysis
The performance comparison of the proposed model with the baseline models in terms of BER has been plotted in Fig. 3 for receiver moving at 150 and 350 km/h for various SNR values in the range of −5 to 25 dB with the 16-QAM modulation scheme. The BER has been decreasing for an increase in SNR values from −5 dB to 25 dB. At transmitted SNR of 10 dB for receiver moving at 150 km/h (Fig. 3(a)), the BERs for the LS estimator, the SVM-based estimator, the U-Net-based estimator, and the proposed GAN-based estimator are 0.0499, 0.0400, 0.0325, and 0.0214, respectively. This indicates that at transmitted SNR of 10 dB, the proposed estimator has reduced the BER by 57.11%, 46.50%, and 34.15% compared to the LS estimator, the SVM-based estimator, and the U-Net-based estimator, respectively. It is also observed that BER has reduced significantly when SNR is increased from −5 dB to 25 dB in all four models. When the velocity of the receiver is increased to 350 km/h (Fig. 3(b)), the BERs for the LS estimator, the SVM-based estimator, the U-Net-based estimator, and for the proposed GAN-based estimator are 0.1492, 0.0892, 0.0601, and 0.0282, respectively. This indicates that at transmitted SNR of 10 dB, the proposed estimator has reduced the BER by 81.10%, 68.38%, and 53.08% compared to the LS estimator, the SVM-based estimator, and the U-Net-based estimator, respectively. Based on the above values, it can be inferred that the proposed model has reduced the BER for any input SNR value in the given range approximately by 70%, 55%, and 45% compared to the conventional LS estimator, the SVM-based estimator, and the U-Net-based estimator, respectively in high-speed scenarios. It is also clear that due to the Doppler effect, the BER has increased in all the cases as velocity is increased from 150 km/h to 350 km/h.
Performance analysis in terms of BER with receiver moving at a) 150 km/h, b) 350 km/h, considering 16-QAM modulation scheme.
2) Outage Probability Vs Signal-to-Noise Ratio Analysis
The OP vs SNR curves have been plotted in Fig. 4 for receiver moving at 150 and 350 km/h for various SNR values in the range of −5 to 25 dB and threshold SNR (
Performance analysis in terms of OP with receiver moving at a) 150 km/h, b) 350 km/h, considering 16-QAM modulation scheme.
3) Comparative Analysis with Different Modulation Schemes
The comparison plots for different modulations for the proposed estimator in terms of BER and OP have been presented in Fig. 5 and Fig. 6, respectively, for receiver moving at 450 km/h for various SNR values in the range of −5 to 25 dB. The detailed BER comparisons for the conventional LS estimator, SVM-based estimator, U-Net-based estimator, and proposed GAN model for QPSK, 16-QAM, and 64-QAM modulations are presented in Table 7 for receiver moving at 400 km/h. It is evident from Table 7 that for any given SNR value and modulation scheme, BER for the proposed model is less than the BER for the U-Net-based model, which in turn is less than the BER for the SVM-based model, and finally, the BER for LS estimator is the highest. Further, it can be seen for the proposed estimator for any value of SNR, BER for QPSK<BER for 16-QAM<BER for 64-QAM.
Comparision analysis for the proposed model in terms of BER with receiver moving at 450 km/h, for different modulation schemes.
Comparision analysis for the proposed model in terms of OP with receiver moving at 450 km/h, for different modulation schemes.
Similarly, the detailed OP comparisons for the conventional LS estimator, SVM-based estimator, U-Net-based estimator, and the proposed model for QPSK, 16-QAM, and 64-QAM modulations are presented in Table 8 for receiver moving at 400 km/h and threshold SNR (
4) Normalized Mean Squared Error Analysis
The NMSE performance is compared at different SNR values in Fig. 7. It is apparent from the plot that the CE performance improves as the SNR increases. The primary reason is the significant interference caused by the noise at low SNR values. We have noticed that our proposed model outperforms the baseline approaches even at lower SNR values. For example, at 10 dB SNR, the NMSE, the LS, SVM, and U-Net-based estimator are −11.50, −14.30, and −16.47 dB, respectively, whereas our GAN-based model requires only −19.56 dB to achieve a similar performance. This allows for a reduction in input power overhead. Fig. 8 illustrates the NMSE capability of the proposed model and other baseline models at different speeds. We can infer that the NMSE decreases with the increase in velocity. The primary reason for this is the accelerated rate of change in channel properties at higher velocities, leading to errors in the channel estimation process. Although the NMSE performance declines, the variation is minimal, only around 2-3 dB per 50 km/h, owing to the strong robustness of the OTFS system against high Doppler spread.
5) Complexity Analysis
An algorithm’s complexity and processing power are crucial considerations, particularly for quick systems like UAVs. The complexity of the conventional LS-based CE primarily involves assessing the computational resources needed to estimate channel coefficients. The time complexity of LS is
Further, the training and testing times for the conventional and proposed models are presented in Table 9. The training time of LS-based CE, SVM-based CE, U-Net-based CE, and proposed GANCE are 485.89 seconds (s), 23400 s, 590.52 s, and 1874 s, respectively. Similarly, the testing time is 125 s, 95.8 s, 78 s, and 31.25 s, respectively. It can be observed that the training and testing time for the proposed model are better than the SVM model. The SVM model takes a lot of time for training as it is mostly CPU-bound and doesn’t take advantage of GPU units for parallel processing. The training time of the proposed model is longer compared to LS and U-Net due to increased complexity and adversarial training. However, the testing time of the proposed model has been reduced significantly due to the advantage of PatchGAN discriminator. Despite the additional computational requirements, the benefits of the GAN-based CE model make it a compelling choice.
Conclusion
This paper presented a GAN-based channel estimation method for the OTFS system in high mobility scenarios for 5G and beyond applications. The results of the proposed estimator are compared with the baseline approaches. The proposed model has also been evaluated for three different modulation schemes i.e., QPSK, 16-QAM, and 64-QAM. The proposed model employs a GAN-based approach for CE at the OTFS receiver, replacing conventional estimators like LS and MMSE. The GAN block comprises a modified U-Net model as the generator and a PatchGAN as the discriminator. The GAN model has been trained for a range of velocity operations from 100 to 500 km/h with 16-QAM modulation. The efficacy of the suggested estimator has been assessed by comparing the BER, OP, and NMSE of the estimated signal with the LS estimator, ML-based estimator, and DL-based estimator as defined in subsection baseline approaches. The proposed GAN-based estimation technique provides better BER, OP, and NMSE performance than conventional LS channel estimation, SVM-based CE, and U-Net-based CE. Overall, the model has achieved an average improvement of 70% in BER and 40% in OP compared to the LS estimator. The proposed model has also outperformed the ML-based SVM estimator, showing an average performance improvement of 55% in BER and 30% in OP. The model has shown significant improvement in BER (45%) and OP (20%) compared to the DL-based U-Net model. The performance analysis in terms of BER and OP for the conventional LS estimator, SVM-based estimator, U-Net-based estimator, and the proposed model for QPSK, 16-QAM, and 64-QAM modulations is presented in Table 7 and Table 8, respectively. Further, it is evident that for any given SNR value and modulation scheme, BER and OP for the proposed model is less than the U-Net-based estimator, which in turn is less than the SVM-based estimator, and finally, the LS estimator has the highest BER and OP. The model demonstrated robustness against Doppler effects resulting from high-speed channel variations, with NMSE performance decreasing by only 2-3 dB for every 50 km/h increase in speed. Finally, the time complexity and execution time analysis for the proposed model is done and it is found to be feasible. Therefore, this work concludes that the proposed GAN-based model is suitable for wireless channel estimation in high-speed UAV applications.
In future work, the model can be extended for MIMO-OTFS systems considering the Doppler effect. The GAN-based approach proposed in this article may be further studied using other generator and discriminator techniques. The GAN-based approach may also be tested for inter-UAV communication, SWARM networks, etc.