1. Introduction
MSK is widely used in VLF/LF communication systems due to its advantages, such as having a continuous phase, high spectral efficiency, a narrow bandwidth, and strong anti-interference [
1]. In VLF/LF communication systems, signals are corrupted primarily by atmospheric noise. Atmospheric noise is usually non-Gaussian and impulsive [
2,
3,
4,
5]. Under impulsive noise, conventional signal processing algorithms based on Gaussian noise typically perform poorly. Therefore, designing a signal processing method with excellent performance under impulsive noise has important practical significance [
6].
The symmetric alpha-stable (
) distribution can well describe the generation mechanism of impulsive noise, so atmospheric noise, which is a type of impulsive noise, can be modeled by the
distribution [
7]. In order to solve the problem of performance degradation in signal processing methods based on Gaussian noise under impulsive noise, people limit the amplitude of the received signals to suppress impulsive noise. The disadvantage of this method is that it is difficult to find a suitable threshold to limit the amplitude of the received signals, and rough amplitude limitations on the received signals can lead to a loss of useful information, ultimately resulting in poor SER performance [
8]. The commonly used optimal demodulation algorithm for wireless communication signals in the
noise channel is the demodulation algorithm based on the maximum likelihood (ML) branch metric. Except for several special cases, the
distribution usually does not have a closed form probability density function, so the ML algorithm is difficult to implement. Meanwhile, even if an approximate ML branch metric is obtained through numerical calculation, the high computational complexity will make the ML algorithm impractical [
9]. The symmetric Cauchy distribution is a special case of the
distribution and has a closed form probability density function. Therefore, a suboptimal Cauchy detection algorithm has been proposed, but this algorithm cannot adapt to varying degrees of impulse noise [
10]. As an improvement to the Cauchy detection algorithm, the myriad detection algorithm has been proposed, which introduces a tunable linearity parameter matching the parameters of the
distribution to adapt to varying degrees of impulsive noise [
11]. People have considered the memory characteristics of MSK and applied the the myriad detection algorithm to MSK demodulation, proposing a demodulation algorithm based on the myriad branch metric, and the symbol error rate (SER) performance of the algorithm is very close to the ML performance [
12]. However, the above improved algorithms still require noise parameter estimation and usually require a large amount of sampling data and complex estimation algorithms to ensure the accuracy of the estimated noise parameters [
13,
14,
15,
16].
In recent years, neural networks have developed rapidly and have been applied in various fields due to their powerful performance. In the field of communication, neural networks are applied to the demodulation of different modulated signals [
17]. Unlike traditional communication signal processing algorithms that require prior estimation of channel noise parameters, communication signal processing algorithms using neural networks can implicitly extract channel features from received signals through training, thus avoiding complex channel noise parameter estimation. Most neural network demodulators only consider Gaussian noise and do not take into account the vast impulsive noise existing in practice, so these neural network demodulators perform poorly under impulsive noise [
18,
19,
20,
21]. The demodulation scheme proposed in [
21] uses a neural network with control gates for MSK demodulation. Unfortunately, that article does not provide specific simulation details and lacks comparative experiments. Our simulation results indicate that the demodulation scheme proposed in [
21] does not seem to provide good training for neural networks under impulsive noise, ultimately resulting in poor SER performance.
In response to the problems of the above methods, this article proposes a novel scheme for MSK demodulation under impulsive noise. Unlike other neural network-based demodulation schemes that use neural networks to replace the entire demodulation process, our demodulation scheme only uses 1D-CNNs to replace the integrator and decision module in the conventional coherent demodulation scheme. This scheme preserves other operations that are beneficial for neural network training. In addition, we remove the symbol conversion before output generation in the conventional coherent demodulation scheme and add the symbol conversion before MSK modulation to further improve the SER performance of our scheme. To the best of our knowledge, this is the first time that 1D-CNNs have been used to replace the integrator and decision module in conventional coherent demodulation for MSK demodulation. The main contributions of this work are as follows:
We use two 1D-CNNs with the same structure to replace the integrator and decision module of the conventional MSK coherent demodulation, while preserving other operations in conventional coherent demodulation. The simulation results indicate that this approach solves the problem of difficulty in training neural networks under impulsive noise;
Our scheme enables a simple neural network model with fewer parameters and less computational complexity to achieve and surpass the performance of other complex neural network models. A simple neural network model is more conducive to subsequent practical deployment.
The simulation results show that under impulsive noise, our demodulation scheme has better SER performance than the conventional coherent demodulation scheme and the demodulation algorithm based on the myriad branch metric. The SER performance of the demodulation algorithm based on the myriad branch metric can closely approach the ML performance under impulsive noise. In addition, our scheme does not require complex impulsive noise parameter estimation.
4. Simulation Results
In the simulation, the symbol period and carrier frequency of the MSK signals are set to be
s and
Hz, respectively. Meanwhile, we set the sampling frequency of the receiver to be
KHz. For MSK signals under impulsive noise,
is given in [
11] and the specific expression is as follows:
where
is the average power of the MSK signals and
.
In order to evaluate the performance and to verify the superiority of our proposed scheme, we simulate and generate datasets with different under varying degrees of impulsive noise to test our demodulation scheme and other schemes. We simulate and generate training sets under strong impulsive noise , medium impulsive noise , and weak impulsive noise , with being 0 db, 4 db, and 8 db, respectively. Similarly, We simulate and generate testing sets under strong impulsive noise , medium impulsive noise , and weak impulsive noise , with being 0 db, 2 db, 4 db, 6 db, and 8 db, respectively. The training set in each case includes 100,000 randomly generated symbols and their corresponding received noisy signals, while the test set in each case includes 10,000 randomly generated symbols and their corresponding received noisy signals.
We use the training sets generated under the nine types of impulsive noise mentioned above to form a total training set to train the neural network in our proposed demodulation scheme and the neural networks in other demodulation schemes. After the training is completed, we use the testing sets generated under the fifteen types of impulsive noise mentioned above to test the trained neural networks separately. Compared to the training sets, the testing sets have data under six additional impulsive noise conditions. We can simulate and generate data under different impulsive noise conditions to train neural networks. But, when applying the trained neural network to practical applications, we may encounter data under other impulsive noise conditions that are not within the training sets. The composition of the testing sets can simulate this situation in the practical application very well. The detailed parameters during training are shown in
Table 1.
Figure 5 shows the improvement in SER performance caused by adjusting the symbol conversion from the position before MSK demodulation output to the position before MSK modulation, where
A represents the SER performance of our proposed scheme under different impulsive noise after adjusting the position of the symbol conversion, and
B represents the SER performance of our proposed scheme under different impulsive noise without adjusting the position of the symbol conversion. The improvement in SER performance caused by this adjustment is easy to explain. Assuming the SER of the demodulation unit before the symbol conversion is
, when two adjacent input symbols only have one error symbol, the output of symbol conversion will be incorrect. Therefore, after the symbol conversion, the SER of the entire demodulation output is
. When
is very small,
. When
is very large,
. It can be seen that the symbol conversion before MSK demodulation output will deteriorate the SER performance. Therefore, adjusting the symbol conversion to the position before MSK modulation can improve the SER performance.
The demodulation scheme in [
20] and the demodulation scheme in [
21] are similar to the demodulation scheme shown in
Figure 2. The neural network model of the demodulation scheme in [
20] is the long short-term memory (LSTM) model, which has been proven to achieve excellent performance in demodulation problems under Gaussian noise. The neural network in [
21] has two additional layers of GatedNet Layers behind two one-dimensional convolutional layers, called GCNN. Due to the lack of specific detailed parameters for neural networks in [
21], we can only choose the neural network with a similar structure and good performance in simulation results as its implementation. The final implementation of GCNN is equivalent to adding two GatedNet Layers with four channels after the two one-dimensional convolutional layers of our neural network. The comparison of parameters and floating point operations (FLOPs) between our neural network model and two other neural network models is shown in
Table 2. FLOPs can be understood as being computationally complex. From the table, it can be seen that our neural network has fewer parameters and less computational complexity.
The SER performance comparison between our scheme and two other neural network-based demodulation schemes after training and testing is shown in
Figure 6, where Scheme 1 represents the scheme in [
20], and Scheme 2 represents the scheme in [
21]. It can be seen that the SER performance of our proposed demodulation scheme is much better than that of the scheme proposed in [
20] and that of the scheme proposed in [
21] under different impulsive noise. At the same time, the neural network model in our proposed scheme has fewer parameters and less computational complexity than two other neural network models. In addition, The simulation results indicate that demodulation schemes similar to the scheme shown in
Figure 2 cannot effectively train the neural network under impulsive noise, ultimately resulting in poor SER performance.
We believe that the reason for the significant difference in SER performance between these three neural network-based demodulation schemes is that the two schemes similar to the demodulation scheme shown in
Figure 2 use neural networks to fit the more complex entire demodulation process, while our scheme only uses neural networks to perform the decision function throughout the entire demodulation process. Relatively speaking, the decision function is simpler and our scheme is more explanatory. In addition, due to preserving other operations before the integrator and decision module, our model is equivalent to introducing some prior information that is more conducive to neural network training.
In order to better demonstrate the superiority of our proposed scheme, we replace the neural network model of our proposed scheme with the two other neural network models. The SER performance of the two other neural network models and our neural network model under our scheme after training and testing is shown in
Figure 7. It can be seen that under our proposed scheme, LSTM and GCNN are well trained and the SER performance of the two other neural network models is significantly improved. In addition, it can also be seen that under our demodulation scheme, the SER performance of our model is close to or even better than that of the two other neural network models, indicating that the proposed demodulation scheme enables the simple neural network model with fewer parameters and less computational complexity to achieve and surpass the performance of the complex neural network model. And, in practical applications, fewer parameters and less computational complexity mean less computational overhead, which will be beneficial for the practical deployment of the model.
We also compare the SER performance of the proposed demodulation scheme with the performance of the demodulation algorithm based on the myriad branch metric and the performance of the conventional coherent demodulation scheme. The performance of the demodulation algorithm based on the myriad branch metric is very close to the ML performance. For a more fair comparison, like our proposed scheme, we remove the symbol conversion operation before output generation and add the symbol conversion operation before modulation in the conventional coherent demodulation scheme. This adjustment to the conventional coherent demodulation scheme will eliminate the impact of symbol conversion and ensure that the improvement in SER performance of the proposed scheme mainly comes from the use of neural networks. In the process of implementing the demodulation algorithm based on the myriad branch metric, we directly use the correct noise parameters to avoid the performance degradation of the demodulation algorithm based on the myriad branch metric caused by noise parameter estimation. This means the that the performance of the demodulation algorithm based on the myriad branch metric being implemented is the best it can achieve. The simulation results are shown in
Figure 8. It can be seen that the SER performance of our scheme is superior to the performance of the demodulation algorithm based on the myriad branch metric and the performance of the conventional coherent demodulation scheme. And, it can also be seen from the figure that as the parameter decreases, the SER performance of our scheme and the performance of the demodulation algorithm based on myriad branch metric improve. The fundamental reason for this phenomenon is that the smaller the parameter, the more energy of impulsive noise is concentrated on the impulses, which is more conducive to suppressing the impact of impulse noise [
24].
In addition, our scheme does not require complex noise parameter calculations. Although it takes time to train the model, it is a one-time complexity. The demodulation algorithm based on the myriad branch metric requires a large amount of data and complex estimation methods to estimate noise parameters. And, as the symbol sequence increases, the computational complexity of the demodulation algorithm based on the myriad branch metric during demodulation will gradually increase. Since the input of the neural network in our proposed scheme is a fixed-length sampling data sequence, the computational complexity of our proposed scheme during demodulation will not increase with the increase in the symbol sequence.