# Random Frequency Division Multiplexing
## 3 Random Frequency Division Multiplexing
### DNN Detection Method  
After the receiver filter and CP removal, (4) can be written as  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{v}=\mathbf{H}\mathbf{A}\mathbf{F}_{\mathbf{N}}^{ \mathbf{H}}\mathbf{s}+\mathbf{v}. \tag{6}\]  
Since the matrix \(\mathbf{A}\) can compress the signal which will require fewer subcarriers or the number of antenna for transmitting the same data, and the original \(\mathbf{s}\) is not sparse, these lead to the challenge of RFDM: developing a low-complexity detector with acceptable accuracy. This detector consists of two parts: (1) Estimate \(\mathbf{\hat{x}}\) according to the channel information as \(\mathbf{\hat{x}}=\mathbf{H}^{-1}\mathbf{y}=\mathbf{x}+\mathbf{H}^{-1}\mathbf{v}\). (2) Reconstruct the original signal \(\mathbf{\hat{s}}\) based on the DNN, as shown in Figure 1.  
DNN, as one of the well-known algorithms of DL, can automatically learn and extract features from raw data, effectively model complex nonlinear relationships, and powerful computational capabilities to efficiently train on large datasets, achieving better generalization performance and enhancing the understanding and representation of data.  
Figure 1: A RFDM-based multicarrier communication system.  
Generally, a DNN consists of \(C\) layers: an input layer, \(C-2\) hidden layers, and an output layer. Given an input vector \(\mathbf{d}\), the output of every layer can be written as follows:  
\[\mathbf{d}_{c}=f(\mathbf{W}_{c}\cdot\mathbf{d}_{c-1}+\mathbf{b}_{c}), \tag{7}\]  
where \(\mathbf{d}_{c-1}\) represents the input of the \(c\)-th layer, \(\mathbf{d}_{c}\) denotes its output, and \(\mathbf{W}_{c}\) is the weight matrix associated with the \((c-1)\)-th and \(c\)-th layers, which can be continuously updated during the training process using the backpropagation algorithm to minimize the loss function, allowing the network to better adapt to the training data. The weight matrices of different layers can learn various levels of features, assisting the network in more complex pattern recognition. \(\mathbf{b}_{c}\) is the bias vector and \(f(\cdot)\) is the activation function.  
Consequently, the output of DNN can be mathematically expressed as:  
\[\hat{\mathbf{s}}=F(\hat{\mathbf{x}};\Theta)=f^{(C)}(f^{(C-1)}(\cdots f^{(1)}(\hat{\mathbf{ x}}))), \tag{8}\]  
where \(\Theta\) represents the learned parametric set and \(\hat{\mathbf{x}}\) is the input of the network. The parameter set \(\Theta\) is optimized by reducing the loss function defined as the distance between the prediction and the regression vector. We define \(T\) as the size of the dataset. The pair \(\{\mathbf{x}^{(t)},\mathbf{s}^{(t)}\}_{t=1}^{T}\) is utilized to train \(\Theta\) to minimize the loss function. The mean squared error (MSE) is adopted as the loss function to express the distance between the transmitted vector \(\mathbf{x}\) and the output of DNN \(\mathbf{s}\) as:  
\[\mathcal{L}(\mathbf{x},\mathbf{s})=\sum_{t=1}^{T}(\mathbf{x}^{(t)}-\mathbf{A}\mathbf{F}^{\mathbf{H }}_{\mathbf{N}}\mathbf{s}^{(t)})^{2}. \tag{9}\]  
Although increasing the number of hidden layers, neurons, and samples increases the training time, this DNN model can be trained offline in advance. Once the parameters \(\Theta\) of DNN model are obtained, the testing process is very fast. Then, the DNN-based network can predict \(\hat{\mathbf{s}}\) from \(\hat{\mathbf{x}}\).  
It is important to note that the dataset collection process and DNN training are performed without affecting the classical communication system operation. Hence, it is feasible to collect a large dataset for capturing the dynamics in the environment because it does not interfere with the classical system operation.