Research on downlink channel state information prediction technique for 5G system based on deep neural network

Consider a simple single-user MIMO [23–24] (SU-MIMO) system where the base station is equipped with N_t transmit antenna and the user is equipped with N_r receive antennas. The channel matrix H ∈ C^N_r×^N_t of a massive MIMO channel is defined as: 1H=[ h1,1h1,2⋯h1,Nth2,1h2,2⋯h2,Nt⋮⋮⋮hNr,1hNr,2⋯hNr,Nt ] h_i,j denotes the channel gain coefficient between the i nd antenna at the base station and the j rd antenna at the user.

Let the transmitted signal from the antenna array at the base station be x =[x₁,x₂,⋯,x_Nt]^T. The transmitted signal arrives at the receiving end through an additive Gaussian white noise (AWGN) [25] channel. The signal y ∈ C^N_r×1 received by the user can be expressed as: 2y=Hx+ω

where y = [y₁, y₂,⋯,y_Nt]^T, y_i denote the received signals at the user’s i rd antenna port; ω ∈ C^N_r×1 denotes the additive Gaussian white noise of the wireless channel that satisfies E{ωω*} = N₀I_Nr, (·)* denotes the conjugate transpose of the matrix, and N₀ is the power spectral density of the noise.

Massive MIMO technology dramatically increases the channel capacity of a wireless communication system by concentrating the energy of radio waves into a smaller spatial area using a massive antenna array. The channel capacity of a communication system is defined as the maximum mutual information of the transmitted and received signals, so the channel capacity of a massive MIMO system can be expressed as: 3C=maxf(x)l(x;y)

The mutual information of transmit vector x and receive vector y is defined as: 4I(x;y)=H(y)−H(y∣x)

Since emission vector x and noise vector ω are statistically independent, according to Eq. (2): 5I(x;y)=H(y)−H(ω)

According to the power-limited maximum entropy theorem, the receiving vector y is maximized when H(y) obeys a zero-mean cyclic symmetric Gaussian distribution, and then the transmitting vector x must also obey a zero-mean cyclic pairwise Gaussian distribution. Let the covariance matrices of x and y be K_x and K_y, respectively. Under the above assumptions, H(y) and H (ω), respectively: 6H(y)=log2det(πeKy) 7H(z)=log2det(πeN0INr) where N₀ denotes the noise power. Thus, the mutual information of transmit vector x and receive vector y can be expressed as: 8I(x;y)=log2det(INτ+HKxH*N0)

Limit the maximum transmit energy of the antenna array to P_t. From the above derivation, the MIMO channel capacity is obtained: 9C=maxTr(Kx)≤Ptlog2det(INr+HKxH*N0)

When the base station does not hold CSI, the transmit power is divided equally among the antennas, and the channel capacity of the MIMO system at this time is: 10Cwithout-CSI =log2det(INr+HH*N0)

The above derivation shows that the MIMO system can utilize space-division multiplexing to improve the spectral efficiency when there is no base station that does not have knowledge of the CSI, which is called open-loop MIMO. The channel capacity mentioned above can be achieved by using the hierarchical space-time coding (BLAST) proposed by Bell Labs. When the CSI is known to the base station, the MIMO at this point is called closed-loop MIMO. Using singular value decomposition (SVD), the original channel can be decomposed into n_min parallel subchannels, n_min = min(N_t,_Nτ), as shown in the following equation: 11H=UΛV* where U ∈ C^N_τ×N_τ and V ∈ C^N_τ×N_τ denote the left singular and right singular matrices, respectively, both of which are You matrices. Λ ∈ C^N_r×N_t is the diagonal matrix and the elements on the diagonal are the singular values of the channel matrix H. Matrix H has a total of n_min non-zero singular values and is denoted as {λ_i},i = 1…n_min. Let the power assigned to the subchannels by the base station be {P},i = 1…n_min, then the signal-to-noise ratio of each subchannel is: 12SNRi=Piλi2N0 i=1,2,…,nmin

Thus, according to Shannon’s Second Theorem, the system and capacity expressed in Eq. (9) can again be written as: 13CSVD=∑i=1nmin log(1+Piλi2N0)

The allocation of power according to the water-flooding algorithm (WF) allows the system and capacity of the above equation to be maximized. The optimal power allocation is expressed as: 14Pi*=(μ−N0λi2)+ where (·)* is an identifier only and does not denote a conjugate transpose; μ is a constant, and the value of μ is required to satisfy the constraints on the total power, i.e.,; ∑i=1nminPi*=Pt and (x)⁺ is defined as: 15(x)+={ x,x≥00,x<0

From Eq. (14), it can be seen that the principle of the water injection algorithm is to allocate more transmit power to the subchannels with good channel quality.

The analytic expression shows that the sum rate of the MIMOBC channel is a K -dimensional capacity region under the total power constraint and consists of the set of reachable rates of K users, denoted as: 16R(π,Q1,Q2,…,QK)=[ Rπ(1),Rπ(2),…,Rπ(K) ]T where π = {π(1),π(2)…,π(K)} denotes an ordered arrangement for K users; {Q₁,Q₂,…,Q_k} denotes the covariance matrix of each user’s transmit signal, which is required to satisfy the total transmit power limit of the base station Σk=1KTr(Qk)≤Ptr . Based on the dirty paper coding (DPC) principle, the reachable rate of user π (k) is given and denoted as: 17Rπ(k)=log2| N0I+Hπ(k)(∑j≥kQπ(j))Hπ(k)H/N0 || N0I+Hπ(k)(∑j>kQπ(j))Hπ(k)H/N0 |k=1⋯K

Thus, the capacity of the MIMOBC channel can be obtained by optimizing the ordering of the users and the covariance matrix of the transmitted signals as shown in the following equation: 18CBC=maxπ,Q1,⋯QK∑k=1KRπ(k) 19s.t. ∑k=1KTr(Qk)≤Ptr

It can be seen that obtaining accurate downlink CSI is extremely important for massive MIMO systems.

Neuron is the most basic unit in deep learning and all neural networks are neurons combined in different ways. The basic structure of a neuron is shown in Figure 1. A complete neuron consists of a linear model and an activation function. For the linear model, it is assumed that the input sample X is a N -dimensional column vector, W is the weight parameter matrix of the sample, the number of rows of the matrix is the same as that of X, and the number of columns can be adjusted according to the need. The column parameters in W characterize the importance of each element in X, and b represents the bias term. The input sample X can be linearly calculated to obtain the linear relationship between each feature in sample X and the output.

Figure 1.

Basic structure of neurons

For the activation function f (·), it is generally a nonlinear function, which is usually located after the linear model of neurons, and serves two purposes: one is to provide nonlinear factors for the network, and the other is to map the function according to different task objectives. Common activation functions are Sigmoid, linear rectifier function (ReLU), tanh, etc., of which the mathematical formula of ReLU is as follows: 20ReLU(x)=max(0,x)

ReLU rectifies all negative values in input x to 0 and leaves the rest of the values unchanged, and is commonly used in the field of image processing, located after convolutional computation. The next subsection describes the CSI feedback framework based on deep learning.

Neurons can be combined in different ways to form different network structures. Common neural network structures are described below. 1)

Convolutional Neural Network

The structure of the convolutional neural network is shown in Fig. 2. The blue part indicates a 4 × 4 single-channel input feature map, whose dashed part is the padding value (Padding), which is generally zero-padding, so the input size of the convolutional layer is 5 × 5, which is denoted as C_in : the gray part indicates a 3 × 3 single-channel convolutional kernel, which is denoted as W_kcmd ; the green part indicates a 4 × 4 single-channel output feature map, which is denoted as C_ost. The computation rule is: each element in C_ost is obtained by multiplying and summing the corresponding elements of W_kret and C_m, and W_kermel is shifted one element at a time during the computation. This process is called convolutional computation and is represented by the following equation: 21Cout =Wkarad *Cin where * denotes the convolution calculation. When there are more than one channel in W_kσxl and C_in, the calculation process is the same, only the convolution calculation is performed for each channel in turn, and the number of channels in its output C_out is the same as the number of channels in W_kσmcl.

Figure 2.

Convolutional neural network structure

Figure 3.

Full connection layer network structure

The CSI feedback method for designing massive MIMO systems using deep learning techniques does not require precise analysis of channel data features and clear modeling. Therefore, it can be applied to a variety of channel scenarios and greatly simplifies the complexity and difficulty of signal processing. The deep learning-based CSI feedback method is shown in Fig. 4. In the offline training phase, the CSI matrix data of the massive MIMO system needs to be collected in advance and fed into the network for learning and training, so that the network parameters can fit the corresponding channel data features. In the online prediction phase, using the already trained network model for CSI reconstruction can greatly reduce the time complexity and design difficulty of the system. Depending on the focus and direction of problem solving, the neural network model may have a large gap.

Figure 4.

CSI feedback model based on deep learning

The classification results of CSI feedback methods based on existing deep learning are shown in Figure 5. At present, for the CSI feedback methods based on white coding, the use of encoder + decoder structure of the implementation of the form of more, and the reconstruction accuracy is higher: a single decoder of the network, although the complexity of the network is lower, but the reconstruction accuracy is lower, and can not meet the actual needs: for the CSI feedback methods of adaptive quantizer, its main purpose is to he characterized CSI with lower quantization bit number, in order to reduce the feedback process required for the parametric number: the depth-expanded CSI feedback method focuses on the modification of certain high-performance compressed-aware reconstruction algorithms, using neural networks to replace all or part of the parameters in the algorithms in order to improve the reconstruction accuracy.

Figure 5.

Deep learning CSI feedback method classification

Assuming that in an FDD massive MIMO system, there is M ≫ 1 antenna at the base station (using a uniform antenna array), 1 receiving antenna at the user equipment, and P transmission paths between the base station and the user equipment, the channel function can be expressed as follows: 23h(f)=∑p=1pαpe−j2πfτp+jϕpa(θp) where f is the carrier frequency, and α_p, ϕ_p, τ_p, θ_p are the fading factor, phase offset, time delay, and angle of arrival direction for the p rd path, respectively.

The fading factor α_p is determined by the following variables: the distance between the base station and the user equipment, the transmission antenna and reception antenna gains, the carrier frequency, and the scattering environment. Phase ϕ_p depends on the scattering material and the angle of incidence of the wave on the scatterer. The time delay τ_p depends on the distance the signal travels along the p path.

a(θ_p) is an array vector and can be expressed as: 24a(θp)=[1,e−jχsinθp,⋯e−jχ(M−1)sinθp]T where χ = 2πdf / c, d is the antenna spacing and c is the speed of light. The DOA observed by the base station is confined to a certain area, i.e: 25θp∈[θ−Δθ/2,θ+Δθ/2]

Define h(f_U) and h(f_D) as the uplink channel function and the downlink channel function from the user equipment to the base station, where f_U and f_D are the uplink frequency and the downlink frequency, respectively. For FDD systems, h(f_D) cannot be simply obtained from h(f_U). However, there is an intrinsic connection between the uplink CSI and the downlink CSI because the downlink and the uplink are located in the same propagation environment with the same physical path, and the spatial propagation characteristics of the wireless channel are basically unchanged within a certain bandwidth range.

Next, this paper will follow the definition of an uplink to downlink mapping function and prove its existence. From Eq. (23), the channel function h(f) is completely determined by parameters such as α_p, ϕ_p, τ_p, P, Δθ, θ while α_p, ϕ_p, τ_p, P,Δθ it depends on the average DOAθ and the distance between the base station and the user equipment D and other communication environment factors. Based on this, the following location-channel mapping is given: 26Φf:{(D,θ)}→{h(f)} where {(D,θ)} and {h(f)} are the domain of definition and the domain of values of mapping Φ_f, respectively.

The inverse mapping of definition Φ_f is: 27Φf−1:{h(f)}→{(D,θ)}

The probability that the inverse mapping Φf−1 holds in a practical massive MIMO communication system is high and approaches 100% as the number of base station antennas increases.

The following mapping relation can be obtained from Eqs. (26) and (27): 28Φf0:{(D,θ)}→{h(fD)} 29Φfc−1:{h(fU)}→{(D,θ)}

The following mapping relationship exists for upstream and downstream links introduced from the above equation: ΨU→D=ΦfD∘ΦfC−1:{ h(fU) }→{ h(fD) } where Φf0∘Φf0−1 denotes the composite mapping of Φ_f0 and Φf0−1 .

The effectiveness of the data-driven scheme will be demonstrated mathematically below. Since Φ_fn and Φfn−1 are continuous mappings and the composition of a continuous mapping is still a continuous mapping, it is always a continuous function for any x ∈ h(f_U), Ψ_U→D(x). According to the literature, if Ψ_U → D(x) satisfies the continuous function condition, for a given small error ε > 0, there always exists a sufficiently large positive constant N that satisfies: 30supx∈H‖ ON(x,w)−ΨU→D(x) ‖≤ε,H={ h(fU) } where O_N(x, w) is the output of a feed-forward neural network, and x, w, N denotes the input data, model weights, and number of hidden layer neurons, respectively. We can train a neural network to predict the downlink CSI based on the uplink CSI, which significantly reduces the overhead required for the process of obtaining CSI in massive MIMO systems.

In order to facilitate modeling and simulation, this paper only considers a simple FDD massive MIMO system, which uses orthogonal frequency division multiplexing (OFDM) [26–27] modulation mode. Assuming that there are N_T transmitting antennas at the base station end and N_R receiving antennas at the user equipment end, and that there are N_S subcarriers in this communication system, and that the data is collected once every N_i time slots, the signal received on the ith subcarrier can be expressed as: 31yi=hixi+ni , i∈[1,Ns] where y_i ∈ C is the signal at the receiving end, x_i ∈ C is the transmit data vector, and n_i ∈ C is the additive Gaussian white noise in the channel. h_i ∈ C^{N_i×N_k×N_i} is the channel matrix between the N_T transmitting antenna and the N_R receiving antenna on the i th subcarrier, and h_i the complex values in the matrix represent the fading gain and phase shift of the subcarriers.

The channel matrix corresponding to the CSI is the set of information for each subcarrier, then the CSI matrix H ∈ C^{N_τ×N_k×N_i×N_j} can be expressed as: 32H=[h1,h2,h3,⋯hNs]H where (·)^H denotes the conjugate transpose.

CsiNet is the first multilayer neural network designed for the problem of downlink CSI prediction in FDD massive MIMO environments. The CsiNet network is implemented by a convolutional neural network structure and is capable of processing channel state information from multiple antennas and outputting high quality channel estimation results.

LeakyReLU is a variant of modified linear unit (ReLU). The mathematical expression of LeakyReLU is shown below: 33f(x)={ x,x≥0αx,x<0

The development of CsiNet network provides a new solution to the downlink CSI feedback problem in massive MIMO systems. Simulation results of CsiNet show that CsiNet network exhibits better prediction performance at all compression rates compared to compressed sensing techniques.

Inspired by the CsiNet decoder network model, this paper proposes a network model for predicting the downlink CSI based on the uplink CSI, which is called the three-dimensional convolutional neural network model [28] (3D-CsiNet), and the overall network structure of 3D-CsiNet is shown in Fig. 6. The design of the convolutional layer, residual network and other parts of this network will be described separately below. 1)

Three-dimensional convolution

In convolutional neural networks, 2D convolution refers to performing convolution operation on 2D data, which is done by sliding a small matrix called convolution kernel over an image or matrix and multiplying the convolution kernel with the localized region of the input, and finally adding all the results to get the output feature map. 3D convolution, on the other hand, refers to the convolution operation on 3D data, which is similar to 2D convolution, but adds a dimension (depth) to the convolution kernel by sliding a 3D convolution kernel over the 3D data and multiplying the convolution kernel and the local region of the input, and finally adding all the results to get the output feature sequence.

Figure 6.

Structure of 3D-CsiNet network model

The true value is the result of placing the SRS in all time slots with perfect channel estimation; the observed value is the result obtained by interpolating all time slots after the actual channel estimation of the SRS under the actual transmission period of the SRS. The predicted values are the results predicted by inputting the observations into the network as a training set. In this experiment the uplink SNR is set to 25 dB and the SRS period is 6 slots. The comparison of the predicted and real values of the channel in the walking scenario is shown in Fig. 7. From the figure it can be seen that both the observed and predicted values are very close to the true values, which indicates the accuracy of the channel estimation and the fact that the proposed MIMO-based 3D-CsiNet model is very favorable for channel prediction.

Figure 7.

The channel prediction is compared to the real value in the walking scenario

The results of the comparison between the predicted and true values of the channel in the cycling scenario are shown in Fig. 8. As can be seen from the figure, as the Doppler shift increases (f_d=250Hz), the channel changes faster over time and the error in channel prediction gradually increases. Part of the reason is that the error of the observed channel estimation increases with the increase of the Doppler shift, i.e., there is an error in the dataset itself, and part of the reason is that it is more difficult for the deep learning network to capture the change rule of the channel as it changes at a faster rate, which leads to an increase in the prediction error.

Figure 8.

Channel prediction and true value of cycling scenarios

The results of the comparison between the prediction performance of the 3D-CsiNet-based model prediction and the traditional second-order AR model prediction are shown in Fig. 9. The simulation sets the SRS period to 4 time slots, the Doppler shift to 250 Hz, and the prediction time length to 28. It can be seen that the estimation error of the 3D-CSINET-based prediction algorithm is lower than that of the second-order AR algorithm, and the MSE value of the 3D-CSINET algorithm is reduced by 59.70% compared to the second-order AR algorithm at the signal-to-noise ratio of 35 dB.

Figure 9.

The 3D-CsiNet model and the AR model predictive can compare the results

This paper also analyzes the effect of different SRS transmission periods on the channel prediction results, which are simulated with SRS periods of 2, 4, 8, 16, and 32 time slots respectively. The channel prediction errors for different SRS transmission periods are shown in Fig. 10. The results show the MSE comparison between the predicted and unpredicted schemes for five SRS cycles at a fixed Doppler shift of 250 Hz and an SNR of 30 dB, with a selected prediction time length of 24. It can be seen from the figure that the predicted scheme outperforms the unpredicted scheme for each SRS cycle, and the advantage increases with the increase of the SRS cycle from SRS cycle of 16 time slots. The proposed prediction method alone improves the MSE performance by 54.29% compared to the non-prediction scheme.

Figure 10.

The channel prediction error of different SRS transmission cycles3

Two typical bitstream generation methods, namely quantization and binarization, are first evaluated and compared. The simulation scenario is set up as follows: the base station is equipped with 256 ULA antennas, and the user is a single antenna. In this paper, the number of bits per dimension (BPD) is used to characterize the feedback prediction effect. The NMSE performance of CSI amplitude feedback under different bitstream generation methods is shown in Fig. 11. The results show that the best feedback performance is achieved when the number of quantization bits is set to 4, while the total number of feedback bits is fixed. Consequently, the number of quantization bits in the quantization layer in this section is set to 4. To some extent, binarization can be regarded as a special case of quantization, where the quantization bits are set to 1. This special 1-bit quantization (i.e., binarization) is far superior to the ordinary 1-bit quantization. At the same time, binarization operates better than 4-bit quantization when the feedback bits are extremely limited, and far worse than 4-bit quantization in other cases.

Figure 11.

The NMSE performance of the CSI amplitude feedback

This paper is dedicated to evaluating the performance of two different CSI phase feedback mechanisms proposed, namely MDPF-1 and MDPF-2, where the most important difference between the two is whether statistical or instantaneous CSI amplitude information is introduced into the phase feedback NN. Unlike the compression and feedback of CSI amplitude, the compression of CSI phase is dependent on CSI amplitude. Therefore, when evaluating the feedback accuracy of the CSI phase, the NMSE or MSE between the original CSI phase and the reconstructed CSI phase is not directly computed, but rather the NMSE between the original complex CSI and the reconstructed complex CSI is computed, and the CSI amplitude is assumed to be perfect because of the dedication to the evaluation of the feedback accuracy of the CSI phase at this point.

Assuming that the base station side is equipped with 272 antennas and the user side is equipped with a single antenna, and the number of channel paths is set to 5. The performance of NMSE of different CSI phase feedback mechanisms is shown in Fig. 12. Where the raw feedback indicates that the CSI phase is fed back directly with the self-encoder and the loss function is the most primitive MSE function. When the BPD is very low, the NMSE of the primitive phase feedback method is > 0 dB, which means that very little useful information is fed back, comparable to noise. This is mainly because it is not known which information is important, so we just try to provide feedback on all the phase information, but this requires a large number of feedback bits. When the BPD is 0.5-0.6, the performance gains of MDPF-1 and MDPF-2 are 10.41-10.53 dB and 10.03-10.36 dB, respectively, compared to the original phase feedback method.In addition, utilizing the instantaneous CSI amplitude information, MDPF -2 outperforms MDPF-1 that utilizes statistical CSI magnitude information, and the performance gain is especially evident when the BPD is low. As the BPD increases, the performance gap between the two slowly becomes smaller. This is due to the fact that the feedback bits are sufficient for all the phase information required by both MDPF-1 and MDPF-2. In order to exploit the correlation in CSI amplitude between neighboring users, CSI phase and amplitude are fed back separately, which leads to the problem of bit allocation for CSI phase and amplitude feedbacks, and an unoptimized allocation strategy leads to a significant degradation of CSI feedback accuracy. Therefore, the optimal bit allocation strategy can be found by extensive simulation, similar to exhaustive search.

Figure 12.

NMSE performance of different CSI phase feedback mechanisms