Linear MIMO Precoders Design for Finite Alphabet Inputs via Model-Free Training

In the proposed method, we regard the precoder and the receiver as two independent parameterized functions to optimize: (i) the precoder is presented by the function $f_{\boldsymbol{\theta}_{P}}^{(P)}(\mathbf{x})=\boldsymbol{\theta}_{P}\mathbf{x}$, where $\boldsymbol{\theta}_{P}=\begin{bmatrix}\mathrm{Re}(\mathbf{G})&-\mathrm{Im}(\mathbf{G})\\ \mathrm{Im}(\mathbf{G})&\mathrm{Re}(\mathbf{G})\end{bmatrix}$ is the parameter matrix of the precoder; (ii) the receiver is implemented as $f_{\boldsymbol{\theta}_{R}}^{(R)}:\mathbb{C}^{N_{t}}\rightarrow\{\mathbf{p}\in\mathbb{R}_{+}^{|\mathcal{M}|}|\sum_{i}^{|\mathcal{M}|}p_{i}=1\}$, where $\boldsymbol{\theta}_{R}$ is the parameter vector of receivers, and $\mathbf{p}$ is the probability vector of the transmitted information. Since the NN implementation is limited to the range of real numbers, in this paper, the real and imaginary parts of complex signals involved need to be reshaped into real number vectors before further processing.

Considering the channel follows a conditional probability distribution w.r.t. the channel input, the loss function of the system can be expressed as:

where $\bar{\mathbf{x}}=f_{\boldsymbol{\theta}_{P}}^{(P)}(\mathbf{x})$ represents the precoded signal, $\mathbf{y}\in\mathbb{R}^{2N_{t}\times 1}$ denotes the received signal, as well as the channel output, and $l(f_{\boldsymbol{\theta}_{R}}^{(R)}(\mathbf{y}^{(i)}),\mathbf{x}^{(i)})$ is the sample loss function defined as the categorical cross-entropy (CE) [9, 7] between the $i$th input signal $\mathbf{x}^{(i)}$ and the $i$th received signal $\mathbf{y}^{(i)}$, $S$ is the batch size of training samples. The approximation in (4) means that we can use the sample mean to estimate the mathematical expectation.

Then, the corresponding optimization problem is formulated as:

When Softmax is adopted as the activation function of the output layer, inspired by [7], then we have

Since $\mathcal{I}(\mathbf{x}|\mathbf{y})=\mathcal{H}(\mathbf{x})-\mathcal{H}(\mathbf{x}|\mathbf{y})$, the problem in (5) is nearly the same as the problem in (2), i.e., the precoder obtained by training the AE can also maximize the mutual information.

The training process requires the derivative of the loss function, i.e.,$\nabla_{(\boldsymbol{\theta}_{R},\boldsymbol{\theta}_{P})}\mathcal{L}=[(\nabla_{\boldsymbol{\theta}_{R}}\mathcal{L})^{\mathsf{T}},(\nabla_{\boldsymbol{\theta}_{P}}\mathcal{L})^{\mathsf{T}}]^{\mathsf{T}}$. To enable the training for the proposed framework in the absence of the channel model, inspired by [9], we present an alternating training method with the exact gradient and the approximated gradient, respectively, as shown in Algorithm 1. It should be noted that since the loss function in this algorithm is defined as a categorical CE function, the transmitted information should be in the form of one-hot and then mapped to corresponding discrete constellation points, as shown in Fig. 1. We adopt the Adam optimizer [13] for training.

According to (4), the gradient of $\mathcal{L}$ w.r.t. $\boldsymbol{\theta}_{R}$ can be expressed as:

It can be seen that there is no need to acquire the channel model, as the calculation process only needs to sample the received signal.

Algorithm 2 is the detailed algorithm for training the receiver. First, the signal source randomly generates a batch of one-hot transmitted information $\mathbf{M}$, which is a $|\mathcal{M}|$-by-$S$ matrix and then mapped to the finite alphabet input signals $\mathbf{X}$. Next, the signals are multiplied by $\boldsymbol{\theta}_{P}$ to obtain the precoded signals $\bar{\mathbf{X}}$, and then $\bar{\mathbf{X}}$ enters the channel for transmission. The receiver acquires the channel output $\mathbf{Y}$. After $\mathbf{Y}$ is fed to the receiver network, a batch of probability matrix $\hat{\mathbf{M}}\in\mathbb{R}^{|\mathcal{M}|\times S}$ over the transmitted information are given for the calculating loss $\mathcal{L}$. Finally, $\boldsymbol{\theta}_{R}$ can perform a one-step update using $\nabla_{\boldsymbol{\theta}_{R}}\mathcal{L}$ by (7). As the channel model is unknown, lines 4, 5 in Algorithm 2 only carry out forward propagation and do not need to record the gradient.

According to (4), the gradient of $\mathcal{L}$ w.r.t. $\boldsymbol{\theta}_{P}$ is:

Since the channel model $p(\mathbf{y}|\bar{\mathbf{x}})$ is unknown and $\nabla_{\bar{\mathbf{x}}}p(\mathbf{y}|\bar{\mathbf{x}})$ cannot be calculated, inspired by [9], we resort to another approach that relax the channel input $\bar{\mathbf{x}}$ into a random variable $\tilde{\mathbf{x}}$, which follows a distribution of $\pi_{\bar{\mathbf{x}}}=\delta(\tilde{\mathbf{x}}-\bar{\mathbf{x}})$, where $\delta$ refers to the delta distribution. The position of the relaxation operation in the algorithm process is shown in the dotted box in Fig. 1. Then, the relaxed system loss can be expressed as:

Besides, $\nabla_{\boldsymbol{\theta}_{P}}\mathcal{L}$ can be approximated by the following expression:

where $\hat{\pi}_{\bar{\mathbf{x}},\sigma_{\pi}}$ represents the approximated probability distribution function of $\pi_{\bar{\mathbf{x}}}$, whose variance is $\sigma_{\pi}^{2}$, since $\pi_{\bar{\mathbf{x}}}$ is non-differentiable. As such, the function requiring the derivative of $\boldsymbol{\theta}_{P}$ is subtly transformed from the unknown channel model $p(\mathbf{y}|\mathbf{x})$ to the known distribution $\hat{\pi}$. Through the estimated gradient in (10), we can complete the training process of precoders without the need of the channel model.

Algorithm 3 is the detailed algorithm for training the precoder. The signal source randomly generates the transmitted information $\mathbf{M}$ and maps them to the input signals $\mathbf{X}$. After precoding, $\bar{\mathbf{X}}$ can be obtained. For the approximated gradient calculation, it is necessary to relax $\bar{\mathbf{X}}$ into random variables $\tilde{\mathbf{X}}$, and then sends the random variables to the channel for transmission to obtain the received signals $\bar{\mathbf{Y}}$. The receiver then establishes the probability matrix of the received signals over the transmitted information and then calculates the per-sample loss $l$. However, at this time, $l$ does not directly calculate the gradient of $\boldsymbol{\theta}_{P}$, as shown in (10), and $\hat{\pi}_{\bar{\mathbf{x}},\sigma_{\pi}}$ should be used to calculate the approximated gradient. Then, $\boldsymbol{\theta}_{P}$ can perform a step update through $\nabla_{\boldsymbol{\theta}_{P}}\mathcal{\hat{L}}$, which still needs to satisfy the power constraints of the precoders. In order to ensure that $\nabla_{\boldsymbol{\theta}_{P}}\mathcal{\hat{L}}$ can be estimated and receiver parameters are not affected, lines 6, 7 in Algorithm 3 only carry out forward propagation and do not need to record the gradient. In other words, the channel in this algorithm is only used to observe its input and the corresponding output, and the complete channel model itself does not participate in the precoders design process.

Fig. 2 shows the network structure of the algorithm proposed in this paper, where Fig. 2 is the precoding part and Fig. 2 is the receiver part. As shown in the figure, the precoding matrix $\mathbf{G}$ is implemented by a linear layer without bias, whose parameters are $\boldsymbol{\theta}_{P}$. The network structure of the receiver is mainly composed of two parts: the first one is the “equalization” network, which consists of two linear layers, and the activation function between is the $tanh$ function; the second one is the decision network, which is composed of multiple linear layers, between which the activation function is ReLU, and Softmax at the output layer. It should be noted that since the proposed method is used in the case of unknown channel models, the receiver may show poor performance [9] without prior information. Therefore, in order to improve the accuracy of the receiver, the “equalization” network is added to extract part of implicit prior channel information $\hat{\mathbf{H}}\in\mathbb{C}^{N_{t}\times S}$ from the received signals. Then, it calculates the product of $\frac{1}{\|\hat{\mathbf{H}}\|_{2}^{2}}\begin{bmatrix}\mathrm{Re}(\hat{\mathbf{H}})&\mathrm{Im}(\hat{\mathbf{H}})\\ -\mathrm{Im}(\hat{\mathbf{H}})&\mathrm{Re}(\hat{\mathbf{H}})\end{bmatrix}$ and the received signal $\mathbf{y}$, similar to the channel equalization in traditional communication systems. Next, the received signals after “equalization” can be exploited to establish the probability matrix $\hat{\mathbf{M}}$ over the input information through the decision network, and finally the loss function and the corresponding gradient can be calculated to train the whole network. It should be noted that $\hat{\mathbf{H}}$ is just an intermediate variable of the receiver, which does not correspond strictly to the true channel model. Instead, it can be regarded as the implicit channel information, explaining the reason why such a structure can improve the performance of the receiver. The method itself does not need to assume the channel model in advance.

As described in Section III-A, when Softmax and the categorical CE are adopted, the sizes of layers in the AE grow exponentially with the number of $N_{t}$, which is very expensive in large systems. Therefore, inspired by [7], we use $N_{t}\log_{2}(M)$ bits to represent one training sample. Then, the loss function and the activation function at the last layer of the receiver will be adjusted to binary CE function and Sigmoid accordingly. Such a choice may make the process of training AE equivalent to maximizing the lower bound of mutual information [7]. Even so, the proposed method is easier to generalize to more complex scenarios, such as MIMO Multiple Access Channels (MAC) with finite discrete inputs, MIMO orthogonal frequency-division multiplexing (MIMO-OFDM) systems, etc. The accurate channel models of these systems are complex, while the assumptions of the channel models are relatively simple in the traditional research methods, or the channel model contains some non-differentiable components (such as preamble insertion in MIMO-OFDM), so the simulation results may differ greatly from the actual performance. The proposed method without the knowledge of an accurate channel model would have inherent advantages in these systems and may alleviate this problem to some extent.

For example, in a $K$-user MIMO MAC communication system, considering the signal model:

where $\mathbf{H}_{i}\in\mathbb{C}^{N_{r}\times N_{t}}$ represents the complex channel matrix between the $i$th transmitter and the receiver; $\mathbf{G}_{i}\in\mathbb{C}^{N_{t}\times N_{t}}$ is each user’s precoding matrix; $\mathbf{x}=[\mathbf{x}_{1}^{\mathsf{T}},\mathbf{x}_{2}^{\mathsf{T}},\cdots,\mathbf{x}_{K}^{\mathsf{T}}]^{\mathsf{T}}$ contains the signal of all transmitters, assuming $\mathbf{x}_{i}$ of different users are independent from each other; the receiver noise $\mathbf{n}\in\mathbb{C}^{N_{r}\times 1}$, and $\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I})$. Suppose there are $N_{r}$ antennas at the receiver and each user has $N_{t}$ transmit antennas.

According to [14], the boundary of the constellation-constrained capacity region can be characterized by the solution of the sum rate optimization problem. Then, the proposed precoders design methods can be applied by: i) using $N_{t}K\log_{2}(M)$ bits to represent one training target; ii) the parameter $\boldsymbol{\theta}_{P}$ of the precoders is adjusted to Bdiag$(\boldsymbol{\theta}_{P_{1}},\cdots,\boldsymbol{\theta}_{P_{K}})$, where $\boldsymbol{\theta}_{P_{i}}=\begin{bmatrix}\mathrm{Re}(\mathbf{G}_{i})&-\mathrm{Im}(\mathbf{G}_{i})\\ \mathrm{Im}(\mathbf{G}_{i})&\mathrm{Re}(\mathbf{G}_{i})\end{bmatrix}$; iii) the receiver function correspondingly becomes as $f_{\boldsymbol{\theta}_{R}}^{(R)}:\mathbb{R}^{2N_{t}K}\rightarrow\mathbb{R}^{N_{t}K\log_{2}(M)}$. Since the channel model of each user is unknown, the precoder and receiver parts still need the true and estimated gradients for training, respectively. It should be noted that because the optimal precoders of different users depend on each other [14], we should iteratively optimize one user’s precoder at a time with others fixed.