Enhancing 5G Massive MIMO Systems Using a Compressive Sensing-Based Approach

The suggested pilot decontamination method is covered in this part. It uses a huge MIMO system and compressive sensing and deep learning algorithms. Figure 1 shows the general design for pilot decontamination.

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Figure 1. Proposed architecture for compressive sensing with deep learning technique

In DL multicell massive MIMO systems, where each cell contains M antennas of BS to service K, (M＞K), and runs at the same frequency, BS transmits signals to UEs [22]. When the IID (Independent and Identically Distributed) channel model for correlated Rayleigh fading is used, together with the characteristics of MMSE for channel estimation $h_{l j k} \in C^{M \times 1}$ assign various antenna correlations to each channel between users in BS $l$ and $M$ in BS $j$. In $\theta_{l j k} \in C^M, M \times 1$ is mini scale fading channel and $\psi_{l j k} \in C^{M \times M}$ accounts for the corresponding channel correlation matrix for massive scale fading [23]. In $\left[D_{l j}\right]_{k, k}=\sqrt{\Psi_{l j k}}, D_{l j k}$ is a diagonal matrix whose diagonal elements are $\sqrt{\Psi_{l j}}=\left[\sqrt{\Psi_{l j 1}}, \sqrt{\Psi_{l j 2}}, \cdots, \sqrt{\Psi_{l j K}}\right]$. Channel between BS $l$ and $k$-th user in cell $j$ is represented as Eq. (1):

$\underline{h}_{l j k}=\sqrt{\Psi_{l j k}} \theta_{l j k}$    (1)

Let us consider channel as a superposition of a large number of paths in Eq. (2) and Eq. (3). The $h_{i j k}$ refers to the channel estimation, and the $\beta_{i j k}$ refers to the transmission length, and B refers to the maximum bandwidth of the nodes, which is counted from 1 to the maximum value, and $a,d,p$  refers to the different channels for the elements $\theta_{i j k}^{(b)}$.

$h_{l j k}=\sqrt{\frac{\beta_{l j k}}{B}} \sum_{b=1}^B a\left(\theta_{l j k}^{(b)}\right) \alpha_{l j k}^{(b)}$    (2)

$\begin{gathered}R_{l j k}=E\left[h_{l j k}\left(h_{l j k}\right)^H\right]= \frac{\beta_{l j k}}{B} \int_0^{2 \pi} p\left(\theta_{i j k}\right) a\left(\theta_{i j k}\right) a^H\left(\theta_{i j k}\right) d \theta_{i j k}\end{gathered}$    (3)

M×1 vector of the received signal in DL by $k$-th user in cell $i$, assuming linear filters is represented by Eq. (4). The coherence interval length is $\rho$ , and the $\rho_D$ represent the signal to noise error. The $H_{j, l_k}, V_{j_k}, s_{j_k}$ represent the channel lengths, and represent the $z_{l_k}$ represent the overall elements.

$y_{i_k}^D=\sqrt{\rho_D} \sum_{j=1}^L \sum_{i=1}^K H_{j, i_k} V_{j_i} S_{j_i}+z_{i_k}$    (4)

The desired signal is decoded at the user side by utilizing a combiner matrix. $\underline{U}_{l_k} \subset C^{M \times d}$  as  $\hat{y}_{i_k}^D=U_{i_k}^H y_{i_k}^D \underline{\Phi}^H$.

2.1 Uplink training

Users broadcast pilot symbols in the uplink, and each BS then evaluates its users' channels. Users from different cells transmit the same set of pilots due to the pilot reuse factor being one. $K$ users' pilot signals were given by a $K \times \tau$ matrix $\underline{\Phi}^H$ with the orthogonality characteristic $\underline{\Phi}^H \Phi=I_K$, $(K \leq \tau)$. $G_{i l}$ refers to the initial channel which is then added with the calculation of channel $g_{i j k}$. An $M \times \tau$ matrices $Y_i$ represents received pilot symbols at BS $i$ as in Eq. (5):

$Y_i=\sum_{i=1}^L \sqrt{q} G_{i l} \Phi^H+N_i g_{i i k}$    (5)

For calculation of channel $g_{i j k}$ at BS $i$, a sufficient statistic is given by Eq. (6) and Eq. (7):

$Z_{i k}=\frac{1}{\sqrt{q} \tau} Y_i \phi_k=\sum_{i=1}^L g_{i i k}+C N\left(0, \frac{1}{\tau q} I_M\right)$    (6)

$\zeta_{i k} \triangleq \sum_{i=1}^L \beta_{i l k}+\frac{1}{\tau q}$    (7)

It is well known that $\tau$ is the coherence interval, and $q$ is the threshold value for the same interval, which is added and calculated for the channel $\beta$ which is directly proportional to the $\zeta_{i k}$ channel [24].

2.2 Uplink pilot transmission

This proposed work is based on a pilot multiplexing assignment since each cell uses the identical set of pilot sequences that are provided to each user $\Phi=\left(\varphi_1, \varphi_2, \cdots, \varphi_k, \cdots, \varphi_K\right)^T$. T is $k$-dimensional pilot sequence matrix were assigned to entire K users in cell i. τ is pilot sequence length that fulfils $\Phi \underline{\Phi}^H=I_K$, orthogonality between pilots. A $K \times K_1$ where K dimension unit matrix is $I_K$. Finally, BS in a cell I received the pilot $Y_i^p$, which can be represented as in Eq. (8):

$Y_i^p=\sqrt{P_p} \sum_{j=1}^L \sum_{k=1}^K h_{j k i} \varphi_k+N_i^p$    (8)

The received signal is represented in Eq. (9):

$r_n^{\wedge}=\sqrt{p_u} a_n^{\wedge H} g_n x_n$ + $\sqrt{p_u} \sum_{i=1, i \neq n}^N \quad a_n^{\wedge H} g_i x_i + n a_n^{\wedge H}$   (9)

The signal is obtained by adding the interval length of the channels $\rho_u$, within the channels $a, g, x$ up to $n$ number of elements, along with the addition of highest threshold value $a_n^H$.

2.3 Compressive Sensing-Based Location Scheduling System (CSLSS)

This part shows a position scheduling method based on compressed sensing to increase the massive MIMO network's uplink sum rate. We presume that the BS has access to location information for all network users. Now, look at UL rate expression in (10). The derivation continues for the channels with maximum threshold $a_n^H$ to $g_i$, with LOS representing the loss of signal in that channel, and the NLOS to the net signal loss in the channel. When using ZF detection, BS represents the interference term in the denominator of (11) as follows:

$a_n^{\wedge H} g_i=\left[\frac{g_n^{\wedge }}{g_n^{\wedge H} g_n^{\wedge}}\right]^H, g_i=\frac{g_n^{\wedge H} g_i}{g_n^{\wedge H} g_n^{\wedge}}$    (10)

$\begin{gathered}g_n^{\wedge H} g_i=\left(g_n^{L O S}\right)^H g_i^{L O S}+\left(g_n^{L O S}\right)^H g_i^{N L O S}+\sum_{j \in m, j \neq n}\left(g_j^{N L O S}\right)^H\left(g_i^{L O S}+g_i^{N L O S}\right)+ \left(\frac{1}{\sqrt{p_p} N_n}\right)^H g_i^{L O S}+\frac{1}{\sqrt{p_p}\left(N_n\right)^H g_i^{N L O S}}\end{gathered}$ (11)

LOS interference $I_{n i} \triangleq\left(g_n^{L O S}\right)^H g_i^{L O S} / g_n^{\wedge H} g_n^{\wedge}$ is defined as Eq. (12):

$\left(g_n^{L O S}\,\right)^H g_i^{L O S}=\Omega_{n i}\left[1+\cdots+e^{j(M-1) d \theta_{n i}}\,\,\,\,\right]$    (12)

It is depicted using the property of the sum of exponentials in Eq. (13) and Eq. (14):

$\begin{aligned} & \left(g_n^{L O S}\right)^H g_i^{L O S}=\Omega_{n i} \frac{e^{M j d \theta_{n i-1}}}{e^{j d \theta_{n i-1}}}= \Omega_{n i} \frac{-e^{j \frac{p}{2} d \theta_{n i}}\left(-e^{-j \frac{\mu}{2} d \theta_{n i}}-e^{j \frac{\mu}{2} d \theta_{n i}}\right)}{-e^{j \frac{j}{2} d \theta_{n i}}\left(-e^{-j \frac{1}{2} d \theta_{n i}}-e^{j \frac{j}{2} d \theta_{n i}}\right)}= \Omega_{n i}\left[\frac{\sin \sin \left(\frac{M d \theta_{n i}}{2}\right)}{\sin \sin \left(\frac{d \theta_{n i}}{2}\right)}\right] e^{j d \theta_{n i}\left(\frac{M-1}{2}\right)} \\ & \end{aligned}$ (13)

$\begin{gathered}\frac{1}{M} g_n^{\wedge H} g_n=\frac{1}{M}\left(g_n^{L O S}+g_n^{N L O S}+\sum_{\frac{j \in \psi_\,m,}{j \neq n}} g_j^{N L O S}+\right.\left.\frac{1}{\sqrt{p_p}} N_n\right)^H \times\left(g_n^{L O S}+g_n^{N L O S}+\sum_{\frac{j \in N_\,m}{j \neq n},} g_j^{N L O S}+\right.\left.\frac{1}{\sqrt{p_p}} N_n\right)\end{gathered}$ (14)

The channels grow progressively orthogonal in the M-MIMO regime, as represented by Eq. (15).

$\frac{1}{M} h_i^H h_j=\{1, \forall i=j 0$, otherwise.    (15)

Using this property, Eq. (16) and Eq. (17) shows:

$\frac{1}{M} g_n^{\wedge H} g_n^{\wedge} \rightarrow^{a . s} \beta_n+\sum_{j \in m, j \neq n} \,\,\frac{\beta_j}{1+K_j}+\frac{1}{p_p}$  (16)

$\left|I_{n i}\right|^2=\frac{\Omega_{n i}^2}{M^2\left(\beta_n+\sum_{\frac{j \in i}{j \neq n}} \frac{\beta_j}{1+K_j}+\frac{1}{p_p}\right)^2}\left[\frac{\sin \left(\frac{M d \theta_{n i}}{2}\right)}{\sin \left(\frac{d \theta_{m i}}{2}\right)}\right]^2$    (17)

The proposed channel estimator is represented by Eq. (18):

$g_{i i k}^{\wedge p r o p}=M \beta_{i i k} \frac{z_{i k}}{\left\|z_{i k}\right\|^2}$    (18)

About M, it approaches the ideal MMSE estimator asymptotically. It is represented by $\eta_{i k}^{\text {prop }} \triangleq \frac{1}{M} E\left\{\left\|g_{i i k}^{\wedge \text { prop }}-g_{i i k}\right\|^2\right\}$ which is given in Eq. (19):

$\eta_{i k}^{\text {prop }}=\frac{M}{M-1} \frac{\beta_{i i k}^2}{\zeta_{i k}}+\beta_{i i k}-2 \beta_{i i k} \theta_{i k}$    (19)

where, the reward matrix is $k$, and the shape of the matrix is $ixk$, $t$ is the time taken for the transmission of the signal. 

$\theta_{i k}=\int_0^1 \int_{-1}^1 \frac{\kappa_{i k}^2(1-t)+\kappa_{i k}\, w \sqrt{t(1-t)}}{\kappa_{i k}^2(1-t)+2 \kappa_{i k}\, w \sqrt{t(1-t)}+t} f_T(t) f_W(w) d w d t$ with $\kappa_{i k} \triangleq \sqrt{\frac{\beta_{i i k}}{\zeta_{i k}-\beta_{i i k}}}$, and $f_T(t)$ and $f_W(w)$ are given by Eq. (20):

$\begin{gathered}f_T(t)=\frac{\Gamma(2 M)}{(\Gamma(M))^2}(t(1-t))^{M-1}, 0<t<1 \,\,f_W(w)= \frac{M}{\pi} B\left(\frac{1}{2}, M\right)\left(1-w^2\right)^{M-\frac{1}{2}},|w|<1\end{gathered}$    (20)

Algorithm for Pilot scheduling:

Algorithm for Location scheduling system:

Algorithm for compressive sensing-based location scheduling system:

2.4 Feed-Forward Convolutional Neural Network (FFCNN) based interference alignment

The deep learning technology is used to locate the pilots in an effective manner and, thus, reduce the traffic between them. The main objective of this paper is to optimize the pilot sequence for the user. The Feed Forward Convolution Neural Network model does interference alignment in MIMO. The internal weight parameters of NN compute input-output correspondence and underlying loss function are described as follows (21).

$L=E\left[\left\|h_{l l k}-f\left(h_{l l k}^{\wedge LS}\right)\right\|_2^2\right]$    (21)

NN-based IA: There are 4 million nodes in the first layer, 64 million in the second layer, and 2 million in the final layer of the NN-based IA model. ReLU is the activation function, and r(x)=max (x, 0). The weight parameters for the i-th layer of the NN-based estimate FFCNN model are given as Wi and bi (22).

$f_{F F C N N}(x)=W_3 \cdot r\left(W_2 \cdot r\left(W_1 x+b_1\right)+b_2\right)+b_3$    (22)

Here, $x \in R^{2 M}$ is input for FFCNN, and is converted from the complex into real, $\underline{x}=\left[\left(h_{l l k}^{\wedge LS}\right)^T, \operatorname{Im}\left(h_{l l k}^{\wedge LS}\right)^T\right]^T$. Thus, the desired channel is attained from $h^{\wedge}{   }_{l l k}^{{ }^{p r o p p}}={ }^{u n} f^{\prime}\left(h_{l l k}^{\wedge LS}\right)= f_{F F C N N}(x)$. Numerous batches of datasets are handled at the same time in practice.

Consider a fully-connected ReLU FFCNN with a DL estimator D. D's input and outputs are indicated by the letters xRd and hRd, respectively. We'll use the term "denote" in the discussion that follows by Eq. (23).

$Z=\left\{\left(x_m, h_m\right)\left|x_m, h_m \in R^d, m=1, \ldots,\right| Z \mid\right\}$    (23)

The ReLU activation function, the hidden layers, and the neuron assignment with $d=\left(d_0, d_1, \ldots, d_l, d_{l+1}\right) \in N^{l+2}$ with $d_0=d_{l+1}=d$. The depth of D is equal to the number of hidden layers l. $\max \left(d_1, \ldots, d_l\right)$ and $\sum_{i=1}^l d_l$ denotes the width and size of D, respectively by Eq. (24).

$\Theta=\left\{\theta=\left(\operatorname{vec}\left(W_0\right), b_0, \ldots, \operatorname{vec}\left(W_l\right), b_l\right) \in R^{d_u}\right\}$    (24)

$b$ represent the bias of the weights. Set of all D parameters, where $d_u=\sum_{i=0}^l d_{i+1} \times\left(d_i+1\right), W_i \in R^{d_{i+1} \times d_i}$ and $b_i \in R^{d_{i+1}}$ are weight matrix and bias vector of $i$-th layer for $i \in\{0,1 \ldots, l\}$. The underlying function that D represents for a constant network structure d can be written as Eq. (25):

$f_\theta(x)=A_l \circ \varphi_{d_l} \circ A_{l-1} \circ \varphi_{d_{l-1}} \circ \cdots \circ \varphi_{d_1} \circ A_0(x)$    (25)

where, $A_i: R^{d_i} \rightarrow R^{d_{i+1}}$.

The affine transformation depends on the weight. $W_i$ and bias $b_i, \varphi_{d_i}: R^{d_i} \rightarrow R^{d_i}$. The purpose of DL estimator is to enhance to approximate MMSE estimator. The neurons in D have only two possible states since the ReLU function is piecewise linear: replicating input or zero output. Whenever the θ is fixed, a set $\underline{K} \subseteq\{0,1\}^d$ presents all potential activation patterns of neurons in D, where $d^{-}=\sum_{i=1}^l d_i$ is the whole neurons in D. The fact that $|K|$ is upper bounded by $2^d$ is self-evident by Eq. (26).

$X_k \subseteq X, k=1, \ldots, K=|K|$    (26)

Let $x_i^{\sim}=\left[x_{i, 1}, \ldots, x_{i, d_i}\right]^T$ be $i$-th layer output with ${\tilde{x}}_0=x$. Next, for all input $x \in X_k, A_i\left({\tilde{x_i}}\right)$ computed by using Eq. (27):

$\begin{gathered}A_i\left(\tilde{x}_i\right)=\left\{W_0 x+b_0, i=0 \,\,W_i A_{i-1}\left(\tilde{x_{i-1}}\right)+b_i, i\right. \geq 1\end{gathered}$    (27)

Further express $A_i\left(\tilde{x}_i\right)$ as by recursively increasing $A_i\left(\tilde{x}_i\right)$ layer by Eq. (28):

$\begin{aligned} & A_i\left(\tilde{x}_i^{\sim}\right)=\prod_{j=0}^i W_j^{\sim} x+\sum_{j=0}^{i-1}\left(\prod_{p=0}^j W_{j+1-p}^{\sim}\right) b_{i-1-j} +b_i=W_i^{\wedge} x+b_i^{\wedge}, b_i \quad=\sum_{j=0}^{i-1}\left(\prod_{p=0}^j W_{j+1-p}^{\sim}\right) b_{i-1-j}+b_i, x \in X_k\end{aligned}$(28)

As a result, $f_\theta(x)$ becomes an affine function for $x \in X_k$ is expressed as in Eq. (29):

$\mathrm{f}_\theta(\mathrm{x})=\mathrm{f}_{\mathrm{X}_{\mathrm{k}}}(\mathrm{x})=\mathrm{W}_{\mathrm{X}_{\mathrm{k}}} \mathrm{x}+\mathrm{b}_{\mathrm{X}_{\mathrm{k}}}$    (29)

where, $W_{X_k}=W_l^{\wedge}$ and $b_{X_k}=b_l^{\wedge}$ by using Eq. (30).

$\|f(x)\|_2=\left[\sum_{i=1}^d E\left\{\left\|f_i(x)\right\|_2^2\right\}\right]^{1 / 2}<+\infty$    (30)

Define in Eq. (31):

$J(f)=E\left\{\|f(x)-h\|_2^2\right\}$    (31)

MSE is estimation of f(x). From the orthogonal principle given in Eq. (32):

$\begin{gathered}J(f)=E\left\{\left\|f(x)-h_{M M S E}+h_{M M S E}-h\right\|_2^2\right\}= \\E\left\{\left\|f(x)-h_{M M S E}\right\|_2^2\right\}+E\left\{\left\|h_{M M S E}-h\right\|_2^2\right\}+2 E\left\{\left(f(x)-h_{M M S E}\right)^T\left(h_{M M S E}-h\right)\right\}=\\E\{\| f(x)-\left.f_o(x) \|_2^2\right\}+J_{M M S E}\end{gathered}$     (32)

The DL estimator's input-output relationship is expressed as a function, $f_\theta(x)$, given a ReLU DNN with parameter.

Then, $\left(J\left(f_\theta\right)=E\left\{\left\|f_\theta(x)-h\right\|_2^2\right\}\right.$ by Eq. (33).

Denote,

$\theta_o=\arg \min _{\theta \in \Theta_R} J\left(f_\theta\right), J\left(f_{\theta_o}\right)=\min _{\theta \in \Theta_R} J\left(f_\theta\right)$    (33)

Similarly, denoted by Eq. (34):

$\theta_Z=\arg \min _{\theta \in \Theta_R} J_Z\left(f_\theta\right), J_Z\left(f_{\theta_Z}\right)=\min _{\theta \in \Theta_R} J_Z\left(f_\theta\right)$    (34)

where, Eq. (35) as follows:

$J_Z\left(f_\theta\right)=\frac{1}{|Z|} \sum_{\left(x_m\,, h_m\right) \in Z}\left\|f_\theta\left(x_m\right)-h_m\right\|_2^2$    (35)

Therefore, Eq. (36) and Eq. (37) are:

$h_{D L}=f_{\theta_Z}(x)$    (36)

$x=f_u(\tau h+n)$    (37)

The first term $J\left(f_{\theta_o}\right)$ in (33) is again decomposed into, shown by Eq (38):

$J\left(f_{\theta_o}\right)=E\left\{\left\|f_{\theta_o}(x)-f_o(x)\right\|_2^2\right\}+J\left(f_o\right)$    (38)

Since, $J\left(f_o\right)$ has lowest MSE, $J\left(f_{\theta_0}\right)$ identified by $E\left\{\left\|f_{\theta_o}(x)-f_o(x)\right\|_2^2\right\}$, also known as approximation error.

2.5 Architecture of the FFCNN model

The following is the architecture of the given FFCNN algorithm consisting of 3 fully connected layers, that point towards 17 parameters for optimizing the network. The final value obtained refers to the position of the placement of the pilot in the network.

In a large MIMO method, this research provided a novel technique for pilot decontamination and interference alignment-based channel estimation. Here, the goal is to lessen PC attacks by adopting a Compressive Sensing-Based Location Scheduling System (CSLSS) for a user-friendly pilot sequence. The pilot set will be optimised with an additional sequence thanks to our location-based decontamination technique. A feed-forward convolutional neural network is used to do interference alignment in MIMO (FFCNN). The results of the experiment are reviewed, and the parameters used for analysis include BER, sum rate, SINR, and spectral efficiency in terms of interference alignment. Compressive sensing parameters include MSE, BER, throughput versus the number of antennas in BS, and throughput versus SNR. The suggested technique has eliminated pilot contamination based on compressive sensing and interference alignment has been achieved by deep learning technique in channel modelling, according to experimental analysis based on both compressive sensing and deep learning techniques.