A method and system for low complexity demodulation of a multi-exponential CPFSK signal

Abstract

This invention relates to a low-complexity demodulation method and system for multi-exponential CPFSK signals. The method includes the following steps: Based on the characteristics of multi-exponential CPFSK signals, data features are extracted by processing the signals, including in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. A signal dataset is generated based on the data features, modulation index, and labels, and preprocessed. A demodulation model based on a convolutional neural network and a demodulation model based on convolution and attention mechanisms are designed, using data features and modulation index as inputs and the demodulation result as output. A loss function is constructed based on the error between the predicted demodulation result and the labels, and the demodulation model is trained. The trained demodulation model is tested and post-processed. Based on the post-processing result, one of the demodulation models is selected for demodulating the CPFSK signal. Compared with existing technologies, this invention has advantages such as reducing demodulation complexity and improving the real-time performance of the demodulation process.

Description

Technical Field

[0001] This invention relates to the fields of wireless communication and artificial intelligence, and in particular to a low-complexity demodulation method and system for multi-exponential CPFSK signals. Background Technology

[0002] In wireless communication systems, continuous phase modulation (CPFSK) has become an important modulation technique for low-power, interference-resistant scenarios due to its constant envelope characteristics and high spectral efficiency. Multi-exponential CPFSK, by introducing time-varying modulation exponents, further improves the signal's coding gain and spectral efficiency, effectively mitigating the periodic phase jump problem caused by fixed modulation exponents and enhancing its adaptability. However, demodulation of multi-exponential CPFSK signals faces inherent challenges from traditional methods. At the receiver, coherent demodulation requires precise carrier synchronization and time-varying modulation exponent tracking, while channel distortion and Doppler shift significantly degrade its performance. Incoherent demodulation, while robust to phase noise, suffers from bit error rate performance loss. Maximum likelihood sequence detection based on the Viterbi algorithm can theoretically achieve optimal demodulation, but its state space complexity increases rapidly with the number of modulation exponents, and it has strict requirements for receiver synchronization, leading to a surge in computational resource demands. Especially under low signal-to-noise ratio or dynamic channel conditions, the real-time performance and reliability of traditional demodulation methods struggle to meet the stringent requirements of modern communication systems.

[0003] With the rapid development of artificial intelligence and deep learning technologies, neural networks have shown great potential in signal processing and communication. Currently, numerous research projects have applied neural networks to signal demodulation with promising results. Deep learning's powerful nonlinear modeling capabilities in signal processing, especially its strong ability to extract and process complex features from data, provide new insights into overcoming the bottleneck of multi-index CPFSK signal demodulation. Chinese patent CN111404856A discloses a high-order modulation signal demodulation method based on a deep learning network. This method constructs a five-layer deep learning network, trains it using the log-likelihood function, and extracts and demodulates features of the high-order modulation signal through an input layer, multiple intermediate layers, and a Softmax multi-classification layer, reducing the bit error rate and training difficulty. However, the inherent characteristics of multi-index CPFSK signals make it difficult to directly apply general network architectures to their receivers. Specific processing of the multi-index and continuous phase characteristics is required to achieve more ideal results. Summary of the Invention

[0004] The purpose of this invention is to provide a low-complexity demodulation method and system for multi-exponential CPFSK signals. It addresses the two special characteristics of multi-exponential CPFSK signals, namely multiple modulation exponents and phase continuity, thereby enabling the use of deep learning models to improve demodulation efficiency.

[0005] The objective of this invention can be achieved through the following technical solutions: A low-complexity demodulation method for multi-exponential CPFSK signals includes the following steps: Based on the characteristics of multi-exponential CPFSK signals, data features are extracted from the multi-exponential CPFSK signals. The data features include in-phase I-channel sampling, quadrature Q-channel sampling, and first-order phase difference. A signal dataset is generated based on the data features, modulation index, and labels, and then preprocessed. The labels are the original symbol information. Demodulation models based on convolutional neural networks and convolution and attention mechanisms are designed respectively. The demodulation models take data features and modulation index as input and demodulation results as output. A loss function is constructed based on the error between the predicted demodulation results and the labels of the demodulation models, and the demodulation models are trained. The trained demodulation model is tested and post-processed, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, one of the demodulation models is selected to demodulate the CPFSK signal.

[0006] The method for extracting the data features is as follows: The in-phase and quadrature portions of the CPFSK received signal are sampled to obtain in-phase I-channel and quadrature Q-channel sampled data, respectively. The phase of the complex received signal is then differentially divided and passed through a three-point moving average filter padded with zeros to obtain the first-order phase difference. The expression for the first-order phase difference is as follows: ; in, and These are the phases of the previous symbol period and the current symbol period, respectively. It is a first-order phase difference.

[0007] The signal dataset is divided into a training set, a validation set, and a test set. When generating the training set and the validation set, within a set signal-to-noise ratio range, a signal-to-noise ratio is randomly selected, and noise is added to all individual data. When generating the test set, a separate dataset is constructed for each signal-to-noise ratio with a set test step size.

[0008] The demodulation model based on convolutional neural networks adopts a dual-path architecture that parallelizes temporal feature extraction and modulation index embedding. The processing results of the dual-path architecture are merged and then input into a fully connected layer for decoding. Specifically, the temporal feature extraction part encodes the input data features using the input embedding layer, and then extracts features through convolutional and pooling layers, corresponding to the matched filtering and phase detection parts in the maximum likelihood sequence detection process. Simultaneously, the modulation index embedding part uses the modulation index embedding layer to obtain the input of the modulation index sequence in parallel to provide prior information on how the transmitter uses the modulation index. After the two feature paths are merged, the output is decoded through a fully connected layer, corresponding to the Viterbi decoding part in the maximum likelihood sequence detection process.

[0009] The processing procedure of the demodulation model based on convolutional neural networks is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. After the convolutional layers, batch normalization and LeakyReLU activation functions with the same output dimension are used. Then, pooling layers are used to downsample and reduce complexity. After processing by the modulation index embedding layer, a 32-dimensional modulation index sequence is obtained. The modulation exponential sequence is merged with the features obtained after pooling layer processing. Then, through three fully connected layers, the original merged 160-dimensional output is first increased to 256 dimensions, and then decreased to 128 dimensions. The ReLU activation function and Dropout are used after the first two fully connected layers. Finally, the 1-dimensional demodulation result, i.e. the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

[0010] The demodulation model based on convolution and attention mechanisms introduces a Transformer encoder. After the input embedding, position encoding is performed first, and then features are extracted in parallel using convolutional layers and Transformer encoder layers. Cross attention is then introduced to fuse the two sets of features, which are then merged with the modulation index embedding information from another path and decoded by a fully connected layer before being output.

[0011] The processing procedure of the demodulation model based on convolution and attention mechanisms is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. Global phase information is provided through positional encoding to obtain the input features. The input features are processed in parallel using two 128-dimensional convolutional layers in the CNN part and a 128-dimensional Transformer encoder layer in the Transformer part. The outputs of the two are fused through a cross-attention mechanism to obtain a 128-dimensional fused vector. After the modulation index is processed by the modulation index embedding layer, a 128-dimensional modulation index sequence is obtained. This sequence is then merged with the 128-dimensional fusion vector to obtain a 256-dimensional vector. By using two fully connected layers, the 256-dimensional vector is first reduced to 128 dimensions. Then, the 1-dimensional demodulation result, i.e., the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

[0012] The expression for the loss function is: ; in, Indicates the number of samples. Indicates the first The true value of each sample, i.e., the raw bits in the label of the signal dataset; Indicates the first The predicted value of a sample is the predicted distribution probability of the bit sequence output by the prediction model.

[0013] The bit error rate is calculated as follows: ; in, The number of signs indicating errors. This indicates the total number of symbols transmitted.

[0014] A low-complexity demodulation system for multi-exponential CPFSK signals includes: Dataset generation module: Based on the characteristics of multi-exponential CPFSK signals, the module processes the multi-exponential CPFSK signals to extract data features, including co-channel I-channel sampling, quadrature Q-channel sampling, and first-order phase difference. Based on the data features, modulation index, and labels, the module generates a signal dataset and performs preprocessing. The labels are the original symbol information. Model building and training module: Demodulation model based on convolutional neural network and demodulation model based on convolution and attention mechanism are designed respectively. The demodulation model takes data features and modulation index as input and demodulation result as output. The loss function is constructed according to the error between the predicted demodulation result and the label of the demodulation model, and the demodulation model is trained. Model selection and demodulation module: Tests the trained demodulation model and performs post-processing, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, it selects one of the demodulation models to demodulate the CPFSK signal.

[0015] Compared with the prior art, the present invention has the following beneficial effects: (1) Based on the characteristics of multi-exponential CPFSK signals, this invention specifically uses the modulation index embedding layer in the network structure and a single one-dimensional modulation index input in the input data to solve the challenge brought by multi-exponential modulation signals with time-varying modulation index to the demodulation model. At the same time, it utilizes the continuous phase characteristics of CPFSK signals and adds a separate first-order phase difference dimension on the basis of the commonly used in-phase / orthogonal two-way sampling of input, so that the model can directly extract phase change information. Furthermore, it introduces Transformer encoder and position encoding in the modulation model based on convolution and attention mechanism to enhance this point.

[0016] (2) This invention proposes a demodulation model based on a convolutional neural network and a demodulation model based on convolution and attention mechanisms. These two models, which balance demodulation complexity and demodulation bit error rate, can adapt to environments with higher requirements for bit error rate and higher requirements for complexity, respectively. Compared with the traditional Viterbi decoding scheme, both models have significant reductions in complexity, advantages in real-time demodulation, and only a small performance loss compared to the theoretically optimal demodulation bit error rate. Attached Figure Description

[0017] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the demodulation model based on a convolutional neural network in this invention; Figure 3 This is a schematic diagram of the demodulation model based on convolution and attention mechanisms in this invention; Figure 4 This is a comparison chart of bit error rate performance in one embodiment. Detailed Implementation

[0018] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0019] This embodiment provides a low-complexity demodulation method for multi-exponential CPFSK signals, such as... Figure 1 As shown, it includes the following steps: S1. Based on the characteristics of multi-exponential CPFSK signals, the multi-exponential CPFSK signals are processed to extract data features. Based on the data features, modulation index, and label, a signal dataset is generated and preprocessed.

[0020] The data features include in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference, and the label is the original symbol information.

[0021] The expression for the multi-exponential CPFSK transmit signal is: ; in, For symbolic energy, For symbol period, It is an infinitely long sequence of unrelated binary symbols. For time, For carrier frequency, For phase, This is the initial phase. In this embodiment, A simulation experiment was conducted with a sequence length of 1024, and the sign was a random value between +1 and -1.

[0022] Under multiple modulation indices, the expression for the phase is: ; in, It is the first The modulation index used in each symbol period It is the first Binary symbols in a symbol period, This represents the instantaneous frequency pulse function.

[0023] As can be seen from the phase expression, the phase changes continuously with time. For the case of multiple modulation indices, it is usually assumed that the modulation indices used are... The modulation indices repeat sequentially in a period, using only one specific modulation index within the same symbol period. Generally, the modulation indices are selected from a set of indices with a common denominator. The rational numbers chosen from, that is: ; in, and All are small integers, and , In practical applications, The value generally does not exceed 16.

[0024] This embodiment uses the modulation index as the selection factor. Let's take an example to illustrate. Then, under simulation conditions, the following condition is satisfied: .

[0025] For instantaneous frequency pulse function , its in It is always positive within the range and satisfies the normalization condition. The expression is: ; The signal to be sent is obtained accordingly.

[0026] Using an additive white Gaussian noise channel as the communication channel, the received signal can be expressed as: ; in, It is a power spectral density with a mean of 0 and a one-sided power spectral density of A Gaussian random process.

[0027] The in-phase and quadrature portions of the CPFSK received signal are sampled to obtain in-phase (I-channel) and quadrature (Q-channel) sampled data, respectively, serving as the basic two-dimensional data of the received signal. Considering the phase continuity of the multi-exponential CPFSK signal, the characteristics of the received signal cannot be efficiently and accurately reflected by in-phase and quadrature sampling alone. Therefore, the phase of the complex received signal is further differentiated by first order and passed through a three-point moving average filter padded with zeros to obtain the first-order phase difference, which serves as the third dimension of the data. This ensures that the neural network can accurately learn the characteristics of continuous phase changes. The expression for the first-order phase difference is: ; in, and These are the phases of the previous symbol period and the current symbol period, respectively. This is a first-order phase difference. Adding a phase difference allows the data to explicitly include information about phase changes centrally, and also mitigates the effects of phase changes. Potential problems arising from periodicity.

[0028] When constructing the dataset, to accommodate the data learning and processing patterns of the neural network model, the samples are divided into three parts: data features, labels, and modulation index. The data features part contains all the data generated for each single symbol, namely the sampling of the I / Q signals and the first-order phase differential signal, totaling three data streams. Noise can be added according to a set signal-to-noise ratio when the dataset is generated and accessed. The label part contains the original symbol information, used for comparison after the model learns the data and outputs the demodulation results to evaluate the model's demodulation accuracy. The modulation index part contains the modulation index used for each symbol data modulation.

[0029] Simultaneously, the training, validation, and test datasets need to be appropriately divided. When generating the training and validation datasets, noise is randomly added to all individual data points with a signal-to-noise ratio (SNR) ranging from 0dB to 12dB, and the datasets are randomly divided into training and validation datasets in an 8:2 ratio. This ensures the reliability of the training and validation losses, which can be observed in real-time during training, and prevents data leakage from affecting experimental results. When generating the test dataset, datasets are generated for each SNR condition according to the needs of simulation testing and comparison. These datasets are then read and demodulated separately during simulation testing and comparison, with no fewer than 1000 test samples for each SNR condition.

[0030] S2. Based on the special characteristics of multi-index CPFSK signals, including the cyclic use of multiple modulation exponents and continuous signal phase, demodulation models based on convolutional neural networks and demodulation models based on convolution and attention mechanisms are designed respectively. The demodulation models take data features and modulation exponents as inputs and demodulation results as outputs. A loss function is constructed based on the error between the predicted demodulation results and the labels of the demodulation models, and the demodulation models are trained.

[0031] This step proposes two demodulation models, which offer a trade-off between demodulation error rate and complexity, and can be adapted to environments with higher requirements for complexity or error rate, respectively.

[0032] The demodulation model based on convolutional neural networks adopts a dual-path architecture that parallelizes temporal feature extraction and modulation index embedding. The processing results of the two paths are merged and then input into a fully connected layer for decoding. In the temporal feature extraction part, the input data features are encoded using the input embedding layer and then extracted through convolutional and pooling layers, corresponding to the matched filtering and phase detection parts in the maximum likelihood sequence detection process. At the same time, the modulation index embedding part uses the modulation index embedding layer in parallel to obtain the input of the modulation index sequence to provide prior information on how the transmitter uses the modulation index. After the two feature paths are merged, the output is decoded through a fully connected layer, corresponding to the Viterbi decoding part in the maximum likelihood sequence detection process.

[0033] like Figure 2 As shown, the processing procedure of the demodulation model based on convolutional neural networks is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. After the convolutional layers, batch normalization and LeakyReLU activation functions with the same output dimension are used. Then, pooling layers are used to downsample and reduce complexity. After processing by the modulation index embedding layer, a 32-dimensional modulation index sequence is obtained. The modulation exponent sequence is merged with the features obtained after pooling layer processing. Similar to the Viterbi decoding process, the output is first increased from the original 160-dimensional output to 256-dimensional output through three fully connected layers, and then reduced to 128-dimensional output. The ReLU activation function and Dropout are used after the first two fully connected layers. Finally, the 1-dimensional demodulation result, i.e. the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

[0034] The demodulation model based on convolution and attention mechanisms introduces a Transformer encoder. After the input embedding, position encoding is performed first, and then features are extracted in parallel using convolutional layers and Transformer encoder layers. Cross attention is then introduced to fuse the two sets of features, which are then merged with the modulation index embedding information from another path and decoded by a fully connected layer before being output.

[0035] like Figure 3 As shown, the processing procedure of the demodulation model based on convolution and attention mechanisms is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. Global phase information is provided through positional encoding to obtain the input features. The input features are processed in parallel using two 128-dimensional convolutional layers in the CNN part and a 128-dimensional Transformer encoder layer in the Transformer part. The outputs of the two are fused through a cross-attention mechanism to obtain a 128-dimensional fused vector. After the modulation index is processed by the modulation index embedding layer, a 128-dimensional modulation index sequence is obtained. This sequence is then merged with the 128-dimensional fusion vector to obtain a 256-dimensional vector. By using two fully connected layers, the 256-dimensional vector is first reduced to 128 dimensions. Then, the 1-dimensional demodulation result, i.e., the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

[0036] The difference between the results obtained during demodulation model training and the corresponding labels in the dataset is quantified using the binary cross-entropy loss function, expressed as: ; in, Indicates the number of samples. Indicates the first The true value of each sample, i.e., the raw bits in the label of the signal dataset; Indicates the first The predicted value of a sample is the predicted distribution probability of the bit sequence output by the prediction model.

[0037] S3 tests the trained demodulation model and performs post-processing, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, one of the demodulation models is selected for demodulating the CPFSK signal.

[0038] The bit error rate is calculated as follows: ; in, The number of signs indicating errors. This indicates the total number of symbols transmitted.

[0039] In the simulation experiment of this embodiment, Python and PyTorch platforms are used to simulate the generation of Multi-h CPFSK transmission sequences and the simulation of their reception sequences after passing through additive white Gaussian noise channel interference.

[0040] In this simulation, the symbol sequence length is 1024, the number of samples in the dataset for each signal-to-noise ratio is 1000, and the number of samples per symbol is 16. Due to the presence of the modulation index embedding layer, the receiver can obtain prior information about how the transmitter uses the modulation index.

[0041] When verifying the performance, a dataset of 1000 samples was generated for each signal-to-noise ratio (SNR) condition within the range of 0 to 8 dB with a step size of 1 dB, and the bit error rate performance comparison curves of each scheme were obtained.

[0042] The bit error rate performance comparison chart is as follows: Figure 4 As shown in the figure, the neural network demodulation scheme suffers only a small performance loss compared to the traditional maximum likelihood sequence detection scheme.

[0043] Under simulation conditions, the running time of the convolutional neural network scheme is approximately 32% of that of the traditional scheme, while the running time of the convolutional and attention mechanism scheme is approximately 58% of that of the traditional scheme. This demonstrates the advantage of the neural network model over the traditional model in terms of demodulation real-time performance.

[0044] This embodiment also provides a low-complexity demodulation system for multi-exponential CPFSK signals, including: Dataset generation module: Based on the characteristics of multi-exponential CPFSK signals, the module processes the multi-exponential CPFSK signals to extract data features, including co-channel I-channel sampling, quadrature Q-channel sampling, and first-order phase difference. Based on the data features, modulation index, and labels, the module generates a signal dataset and performs preprocessing. The labels are the original symbol information. Model building and training module: Demodulation model based on convolutional neural network and demodulation model based on convolution and attention mechanism are designed respectively. The demodulation model takes data features and modulation index as input and demodulation result as output. The loss function is constructed according to the error between the predicted demodulation result and the label of the demodulation model, and the demodulation model is trained. Model selection and demodulation module: Tests the trained demodulation model and performs post-processing, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, it selects one of the demodulation models to demodulate the CPFSK signal.

[0045] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working process of the described module can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.

[0046] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A low-complexity demodulation method for multi-exponential CPFSK signals, characterized in that, Includes the following steps: Based on the characteristics of multi-exponential CPFSK signals, data features are extracted from the multi-exponential CPFSK signals. The data features include in-phase I-channel sampling, quadrature Q-channel sampling, and first-order phase difference. A signal dataset is generated based on the data features, modulation index, and labels, and then preprocessed. The labels are the original symbol information. Demodulation models based on convolutional neural networks and convolution and attention mechanisms are designed respectively. The demodulation models take data features and modulation index as input and demodulation results as output. A loss function is constructed based on the error between the predicted demodulation results and the labels of the demodulation models, and the demodulation models are trained. The trained demodulation model is tested and post-processed, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, one of the demodulation models is selected to demodulate the CPFSK signal.

2. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The method for extracting the data features is as follows: The in-phase and quadrature portions of the CPFSK received signal are sampled to obtain in-phase I-channel and quadrature Q-channel sampled data, respectively. The phase of the complex received signal is then differentially divided and passed through a three-point moving average filter padded with zeros to obtain the first-order phase difference. The expression for the first-order phase difference is as follows: ； in, and These are the phases of the previous symbol period and the current symbol period, respectively. It is a first-order phase difference.

3. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The signal dataset is divided into a training set, a validation set, and a test set. When generating the training set and the validation set, within a set signal-to-noise ratio range, a signal-to-noise ratio is randomly selected, and noise is added to all individual data. When generating the test set, a separate dataset is constructed for each signal-to-noise ratio with a set test step size.

4. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The demodulation model based on convolutional neural networks adopts a dual-path architecture that parallelizes temporal feature extraction and modulation index embedding. The processing results of the dual-path architecture are merged and then input into a fully connected layer for decoding. Specifically, the temporal feature extraction part encodes the input data features using the input embedding layer, and then extracts features through convolutional and pooling layers, corresponding to the matched filtering and phase detection parts in the maximum likelihood sequence detection process. Simultaneously, the modulation index embedding part uses the modulation index embedding layer to obtain the input of the modulation index sequence in parallel to provide prior information on how the transmitter uses the modulation index. After the two feature paths are merged, the output is decoded through a fully connected layer, corresponding to the Viterbi decoding part in the maximum likelihood sequence detection process.

5. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 4, characterized in that, The processing procedure of the demodulation model based on convolutional neural networks is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. After the convolutional layers, batch normalization and LeakyReLU activation functions with the same output dimension are used. Then, pooling layers are used to downsample and reduce complexity. After processing by the modulation index embedding layer, a 32-dimensional modulation index sequence is obtained. The modulation exponential sequence is merged with the features obtained after pooling layer processing. Then, through three fully connected layers, the original merged 160-dimensional output is first increased to 256 dimensions, and then decreased to 128 dimensions. The ReLU activation function and Dropout are used after the first two fully connected layers. Finally, the 1-dimensional demodulation result, i.e. the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

6. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The demodulation model based on convolution and attention mechanisms introduces a Transformer encoder. After the input embedding, position encoding is performed first, and then features are extracted in parallel using convolutional layers and Transformer encoder layers. Cross attention is then introduced to fuse the two sets of features, which are then merged with the modulation index embedding information from another path and decoded by a fully connected layer before being output.

7. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 6, characterized in that, The processing procedure of the demodulation model based on convolution and attention mechanisms is as follows: The data features obtained after the received signal is sampled and calculated enter the input embedding layer and are transformed into an input vector, which includes three dimensions: in-phase I-path sampling, quadrature Q-path sampling, and first-order phase difference. The input vector is processed through two convolutional layers to extract features and increase dimensionality. The original 3-dimensional input is first increased to 64 dimensions, and then to 128 dimensions. Global phase information is provided through positional encoding to obtain the input features. The input features are processed in parallel using two 128-dimensional convolutional layers in the CNN part and a 128-dimensional Transformer encoder layer in the Transformer part. The outputs of the two are fused through a cross-attention mechanism to obtain a 128-dimensional fused vector. After the modulation index is processed by the modulation index embedding layer, a 128-dimensional modulation index sequence is obtained. This sequence is then merged with the 128-dimensional fusion vector to obtain a 256-dimensional vector. By using two fully connected layers, the 256-dimensional vector is first reduced to 128 dimensions. Then, the 1-dimensional demodulation result, i.e., the predicted distribution probability of the original bit sequence, is obtained by the Sigmoid activation function integrated in the last fully connected layer.

8. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The expression for the loss function is: ； in, Indicates the number of samples. Indicates the first The true value of each sample, i.e., the raw bits in the label of the signal dataset; Indicates the first The predicted value of a sample is the predicted distribution probability of the bit sequence output by the prediction model.

9. The low-complexity demodulation method for multi-exponential CPFSK signals according to claim 1, characterized in that, The bit error rate is calculated as follows: ； in, The number of signs indicating errors. This indicates the total number of symbols transmitted.

10. A low-complexity demodulation system for multi-exponential CPFSK signals, characterized in that, include: Dataset generation module: Based on the characteristics of multi-exponential CPFSK signals, the module processes the multi-exponential CPFSK signals to extract data features, including co-channel I-channel sampling, quadrature Q-channel sampling, and first-order phase difference. Based on the data features, modulation index, and labels, the module generates a signal dataset and performs preprocessing. The labels are the original symbol information. Model building and training module: Demodulation model based on convolutional neural network and demodulation model based on convolution and attention mechanism are designed respectively. The demodulation model takes data features and modulation index as input and demodulation result as output. The loss function is constructed according to the error between the predicted demodulation result and the label of the demodulation model, and the demodulation model is trained. Model selection and demodulation module: Tests the trained demodulation model and performs post-processing, including calculating the bit error rate and comparing the demodulation time. Based on the post-processing results, it selects one of the demodulation models to demodulate the CPFSK signal.

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