I Introduction
In the past decades, limited feedback techniques have been intensively investigated for realizing closedloop communications in frequencydivision duplex (FDD) systems [11, 2, 17, 10, 6, 5]
. A typical limited feedback procedure is divided into performing channel estimation and feeding back the information. Channel state information (CSI) is first estimated at the receiver
[1], and then is mapped to the quantized values, e.g., the precoding matrix index (PMI) [11]. The quantized CSI is sent back to the transmitter through finiterate feedback channels with the aid of a channel codebook [17].A joint design of the channel estimation and quantization is necessary for identifying the optimal limited feedback systems, since the CSI estimation error is closely related to the codebook design. Nevertheless, due to nonconvex and combinatorial nature of problem formulations, most works on the limited feedback systems [11, 2, 9] have studied the CSI estimation procedure and the quantization separately by assuming an ideal scenario where perfect CSI is available at the receiver. Another problem is a complexity issue on the channel estimation. It is expected that the estimation overhead grows as the training time increases.
Recently, a deep learning (DL) method has been applied for tackling limited feedback systems design problems [16, 15, 12, 4]. [16]
utilized sparsity of the massive multipleinput multipleoutput (MIMO) channel using a convolutional neural network (CNN), and showed improved performance with lower complexity than other baseline schemes. A feedback method in a timevarying channel was addressed in
[15], which adopted a recurrent neural network (RNN) to capture timevarying characteristic of the channel. The authors in
[12] deployed an integrated neural network composed of different neural network structures in both transmitter and receiver. In [4], feedback delay and error were considered in neural network designs. However, these works assumed perfect CSI and did not consider channel estimation. Therefore, a practical feedback system design is needed in the DL approach which takes the channel estimation process into account.In this paper, we consider a pointtopoint MIMO system where a multiantenna receiver sends the quantized CSI back to a multiantenna transmitter via a feedback channel. To this end, pilot sequences are first conveyed from the transmitter so that the receiver can extract useful features of the CSI. Then, we jointly design DL based limited feedback systems which include CSI prediction as well as codebook optimization. Two individual DNNs are implemented at the receiver and transmitter. The receiver DNN accepts the pilotaided received signal as an input and is designed to output bipolar vectors as a quantized representation of the CSI. Such a DNN structure abstracts the channel estimation by directly extracting the binary feature from the received signal which contains the channel information. Also, the transmitter DNN is developed for calculating the beamforming vector for data transmission using the feedback information from the receiver [11]. We jointly train the DNNs at the transmitter and the receiver in an endtoend manner so that the overall DLbased limited feedback scheme can learn efficient feedback rules by exploiting statistics of wireless channels. Consequently, compared to existing DL methods in [16] where perfect CSI is assumed at the receiver, our proposed design is applicable to a more practical scenario with no exact CSI.
A major challenge for such a DNN training stems from the vector quantization operation at the receiver DNN whose gradient becomes zero for all input range. For this reason, gradient decent (GD) based DL libraries such as Tensorflow cannot be straightforwardly applied to the training task of the proposed DNN approach. It should be noted that the quantization process has not been included in the conventional DL studies
[16, 15]as it is not easy to train DNNs with binary constraints. To address this issue, we employ the concept of a stochastic binarization layer and gradient estimation techniques
[14]. As a result, the endtoend training of the proposed DNNbased limited feedback system is made possible with stateoftheart DL libraries. The numerical results verify that the proposed DL method improves the symbol error rate (SER) performance by 1 dB over conventional schemes with reduced computational complexity.Ii System Model
Iia Limited Feedback Systems
We consider a FDD MIMO system where a transmitter with antennas conveys the symbol to a receiver with antennas through the quasistatic frequencyflat fading channel . In data transmission, denoting as the beamforming vector, the received signal is written by
(1) 
where the precoded signal is given by , stands for the symbol transmission energy, and is the additive Gaussian noise.
To achieve closedloop communication, the receiver first estimates the channel matrix using the standard pilotbased channel estimation methods [1]. Defining () as the th pilot sequence, the received signal is given by , where accounts for the Gaussian noise vector. By stacking the received signals for into the matrix , we have
(2) 
where represents the energy for the pilot, and and are denoted as and respectively. The pilot matrix
is determined as the normalized discrete Fourier transform (DFT) matrix where
is defined as . From (2), the receiver can obtain the estimation of the CSI by adopting the linear minimum mean square error (LMMSE) estimation [1].With the estimated CSI at hand, the receiver identifies the PMI by selecting a codeword in a predesigned codebook of size , where stands for the number of feedback bits and () with denotes the th candidate for the PMI. The receiver chooses the optimal PMI which maximizes the effective channel gain evaluated over as [11, 17]
(3) 
Here, the optimal codebook maximizing the average effective channel gain can be determined by the Lloyd algorithm [17]. The receiver informs the index to the transmitter through the feedback channel. Hence, the transmitter readily recovers the PMI based on the codebook, and the corresponding PMI is utilized as a beamforming vector for the data transmission over the channel .
In the limited feedback system, the receiver operation can be characterized as a mapping which extracts the integer from the received signal . Similarly, the operation at the transmitter is generally represented by a function which calculates the beamforming vector from the feedback information . Thus, an endtoend limited feedback procedure can be written as . An optimization task for the limited feedback scheme, which maximizes the effective channel gain over an arbitrarily distributed and in (2), can be written by
(4)  
(5) 
Problem (4) handles an endtoend optimization of the overall limited feedback process including the PMI extraction and the beamforming vector computation . The CSI estimation is abstracted in (4), since the receiver does not explicitly predict the CSI but obtains the quantization index containing implicit information of .
Existing codebook designs [11, 17] developed for quantizing the estimated CSI cannot be straightforwardly applied to (4), as a nontrivial feature is required to extract from the pilotaided received signal . In addition, it is difficult to solve (4) through traditional optimization methods, since closedform expressions for the optimization functions and as well as the objective are not available for an arbitrary distributed channel . Hence, there is no general optimization approach for obtaining an efficient solution to (4). To tackle this difficulty, we present a datadriven solution for the limited feedback systems through DL techniques.
IiB Basics of DNN
Denoting as the dimension of the th hidden layer, the th hidden layer output in a fullyconnected DNN with layers is expressed as
(6) 
where an elementwise function
is defined as the activation function, and
andare a weight matrix and a bias vector of hidden layer
, respectively. Then, the overall DNN operation can be represented by a mapping , which is composed of consecutive calculation in (6). The parameter set of the DNN is a collection of the weight matrices and the bias vectors, i.e., . The objective of the DNN training is to find the parameter minimizing a cost function, which mathematically describes the target of a DL task. Stateoftheart DL libraries such as Tensorflow depend on the gradient descent method and its variant for iteratively updating the DNN parameter .Iii Limited Feedback Systems Based on DL
We propose a DL framework for the limited feedback systems illustrated in Fig. 1. We employ two individual DNNs implemented at the receiver and the transmitter whose mapping is defined as and , respectively, each of which approximates the unknown mappings and of the overall limited feedback scheme.^{1}^{1}1The approximation accuracy of DNNs has been mathematically demonstrated both for continuousvalued functions [3] and discrete mappings [13]. We design the receiver DNN such that it produces a bipolar vector of length whose element is either or . The vector is regarded as an equivalent representation of index for the feedback information. The transmitter DNN constructs the unitnorm beamforming vector maximizing the effective channel gain in (4). In the following, we detail the operations of the proposed DL approach.
Iiia Receiver
At the receiver, the received signal matrix is first converted into the real vector representation where stands for the vectorization of , and and denote real and imaginary parts, respectively.^{2}^{2}2Tensorflow does not support complex number calculations. It is then followed by a receiver DNN parameterized by that accepts the received signal in (2) as an input and yields the bipolar vector for the feedback information. We construct the receiver DNN with
fullyconnected hidden layers equipped with a rectified linear unit (ReLU) activation
[8], i.e., for .The dimension of the output layer is fixed to for generating different feedback values. To obtain the bipolar output, the activation at the output layer should be carefully chosen, since the quantization operation typically has zero gradient for all input range. This poses a vanishing gradient issue in the DNN training strategy where GDbased DL optimizers, e.g., the Adam algorithm [7], fail to get updated and converge to a poor solution. To tackle this problem, we employ a stochastic binarization layer [14] which consists of two sequential activations. First, is adopted to force the output of the receiver DNN within a range .
Next, a stochastic activation , which adds a quantization noise to the output of , is chosen to mimic the quantization procedure as
(7) 
where a distribution of the noise is determined as
(8) 
It is obvious that regardless of the value of , the output of the stochastic binarization becomes either or . The setup in (8) leads to the zeromean property for the quantization noise , making
an unbiased estimator for the soft value
, i.e., .Due to the probabilistic operation, the forward propagation (7) of the stochastic binarization layer has no closedform gradient expression which is not applicable to the GDbased DL libraries. This can be solved by the gradient estimation technique [14] where the gradient with respect to is approximated as , i.e., the gradient over the quantization noise . Thanks to the fact , the gradient is calculated as
(9) 
The approximation in (9) is valid when the DNN experiences a large number of the quantization noise , i.e., when the training set is sufficiently large. Note that the gradient estimation (9
) is only performed for the training where the gradient is computed via the backpropagation algorithm
[8]. The forward propagation is produced with the stochastic binarization (7).IiiB Transmitter
At the transmitter, we employ a DNN which consists of hidden layers with the ReLU activations. The output layer of the transmitter DNN is implemented with the normalization activation to produce a unitnorm beamforming vector. We denote the output of the transmitter DNN as . Finally, the complex beamforming vector is readily attained from converting the output .
IiiC Training and Implementation
We train the receiver DNN along with the transmitter DNN for the endtoend optimization of the overall limited feedback procedure. To this end, we reformulate the original formulation in (4) as a training task of the DNNs by replacing the receiver and transmitter operations and with the DNNs and , respectively. Thanks to the binarization layer given in Sec. IIIA, the combinatorial constraint in (5) can be removed. We thus obtain the training problem as
(10) 
where the optimization variables now turn out to be the DNN parameters and .
The training task (10) can be tackled via the minibatch stochastic GD (SGD) algorithm [8], which replaces the expectation in (10) with the empirical average over a minibatch set containing several samples of the training data. In our case, the training data set is composed of numerous tuples of CSI and the noise for generating the received signal matrix as the input to the DNNs. Defining as a collection of the DNN parameters, an iterative update rule at the th iteration of the minibatch SGD is given by
(11) 
where represents the DNN parameter computed at the th iteration and is a learning rate.
The DNN training (11) is an offline process, whereas the online computations of the trained DNNs are realized by simple linear matrix multiplications in (6). Once the DNNs are trained, we store the learned parameters and at the memory units of the transmitter and the receiver, respectively, for realtime limited feedback tasks. This can be seen as the sharing of the predesigned PMI codebook in the conventional limited feedback scenarios. Notice that since the trained transmitter DNN leads to an onetoone mapping from the feedback information to the corresponding beamforming vector , a lookup table implementation is possible for the transmitter. Thus, the computational complexity of the proposed DNNbased limited feedback scheme is dominated by the structure of the receiver DNN such as the dimension of each output of layer and the number of hidden layers, and it is expressed as .
Iv Numerical Results
In this section, we present numerical results for evaluating the performance of the proposed DLbased limited feedback schemes. Each element of
follows a zeromean complex Gaussian distribution with covariance
.^{3}^{3}3Thanks to the datadriven training rule in (11), the proposed DL approach can be applied to any channel distribution. Here, the th element of is fixed as [2](12) 
where is the complex correlation coefficient. All the simulation results are averaged over the phase of .
The receiver DNN consists of hidden layers each of which has the dimension , and , while the transmitter DNN employs a reversed structure with respect to the receiver DNN with hidden layers whose output dimension is , and . We randomly generate minibatch samples for each SGD training iteration (11) with , whereas for testing learned DNNs, we evaluate 100,000 independently sampled data. All the simulation is implemented with Tensorflow and Python.
Fig. 2 illustrates the average normalized effective channel gain of (1) with , , as a function of the pilot sequence length for different correlation coefficients
and the signaltonoise ratio (SNR) defined as
wheredenotes the maximum eigenvalue of
. As a reference, we compare the performance of the proposed DL approach with conventional limited feedback systems which adopt the LMMSE channel estimation [1] and the DFT channel codebook. We also examine the performance of the Lloyd algorithm which is known to be the optimal codebook design strategy under the assumption of no estimation error [17]. For all schemes, we set dB for the pilot transmission. From the figure, it is observed that the proposed DL method performs better than baseline schemes, especially when . This implies that the proposed DL approach is more beneficial if the channel training duration is not long enough. It is due to suboptimality which stems from separate optimization of the estimation and feedback process. We can also see that the DLbased limited feedback scheme offers a gain over the Lloyd codebook, which would not be optimal in the presence of the CSI estimation error.We investigate the average symbol error rate (SER) performance in Fig. 3 for the QPSK modulated systems with . It is clear that our proposed method exhibits lower SER compared to baseline schemes. Also, the performance of DL is shown to be more effective when the antennas are highly correlated. It can be shown that proposed schemes offers about an 1 dB gain over the other methods for .
L=2  L=5  L=10  L=20  

MMSE + Lloyd, DFT  0.0625  0.0668  0.0771  0.0981 
Deep Learning  0.0270  0.0286  0.0290  0.0311 
Finally, we compare the average CPU running time in Table I. For fair comparison, the execution time of the baseline methods only includes the online computations, i.e., the channel estimation via the LMMSE and the quantization (3). Since the Lloyd algorithm is an offline procedure, two baselines have the identical complexity. For the DL approach, the realtime complexity is only rely on the receiver DNN. We can see that the DLbased scheme reduces the execution time of the baseline methods by half. This verifies the effectiveness of the proposed DL approach for practical limited feedback designs.
V Conclusion
We have proposed DLbased limited feedback methods for the MIMO systems. The overall limited feedback process has been implemented by two DNNs at the receiver and the transmitter. Both DNNs have been jointly trained to produce efficient quantization and beamforming vectors. Numerical results have demonstrated that the proposed DL approach can improve the performance of conventional limited feedback schemes with reduced complexity.
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