ISI Free Channel Equalization for Future 5G Mobile Terminal Using Bio-inspired Neural Networks

Abstract

This article presents the concept of future 5th generation (5G) wireless communications as an existing beyond 4G systems like long term evolution (LTE) which means the indicate of 5G scenarios can be introduced in near future. Therefore, this paper deals of a future 5G mobile terminal for applications to uplink transmission in a multiuser LTE scheme. Unfortunately, LTE-uplink inherently generates significant inter-symbol interference especially high bandwidth scenarios. The result is a rise to mutual interference among active users with an increased error rate. This incidence eventually causes non-orthogonal user spreading codes. Moreover, this drawback is known as the multiple access interference episodes which demonstrate high computational complexity and enhances symbol error rate at the receiving end and degrades the communication quality. Most of the related work has been claimed iterative linear minimum mean square error (LMMSE) detection requires a matrix inversion role which has a high computational complexity and contains a combinatorial optimization problem. Consequently, the LMMSE does not meet the requirement to implement real-time detection with low complexity and thus limiting its application. Therefore, we propose an acceptable bio-inspired neural network (NN) in the case of single and multilayer NNs with supervised learning particularly Levenberg–Marquardt backpropagation learning algorithm to improve the convergence speed. Simulation results performed with highest approaches highlights a better act for the proposed system.

Appendices

Appendix 1

A multilayer NNs model is a very useful for a neuromodeling application and based on linear or nonlinear models which are proposed by Mcculloch and Pitts, 1943 and Rosenblatt, 1958 as cited in Haykin, 1999. There are three main steps in multilayer perceptions—(1) outline; (2) weights update or learning rule; (3) generalized test. In this model, neurons are signal processing units composed by a linear combiner and an activation function that can be linear or nonlinear [51]. Usually a linear channel estimator that is a single computing element with estimated network weights $w_{i} = w_{0} ,w_{1} ,\text{w}_{2} , \ldots ,\text{w}_{q}$ and which for the input produces $w_{1} x_{1} + w_{2} x_{2} + \cdots + w_{q} x_{q}$. The neuron output signal is the value of the activation function f(·) in response to the neuron activation potential from j network net _j as follows

$$ net_{j} = \sum\limits_{i = 1}^{q} {W_{i} x_{i} } + b_{i} $$

(53)

The final output of the linear NNs is given by

$$ {\mathbf{o}} = f\left( {net_{j}} \right) $$

(54)

Similarly, x ₀ = ±1 is the polarization potential (or threshold) applied to the neurons. The non-liner NNs is given by

$$ net_{j} = \sum\limits_{i = 1}^{q} {W_{ji} x_{i} } + v_{i} $$

(55)

and the final output of non-linear NNs is given by

$$ {\mathbf{o}} = f\left( {net_{j}} \right) $$

(56)

We can represent the threshold detection using Eqs. (54–56) as follows

$$ {\text{o}} = \left\{ {\begin{array}{*{20}l} { + 1} \hfill & {if\,\,\sum\limits_{i = 1}^{q} { \ge 0} } \hfill \\ { - 1} \hfill & {if\,\,\sum\limits_{i = 1}^{q} { < 0} } \hfill \\ \end{array} } \right. $$

(57)

In multilayer NNs, a collection of neuron connected together in a network can be represented by a direction graph in Fig. 10.

Fig. 10

Signal flow of multilayer NNs

In Fig. 10, nodes represented neurons and arrows represent the links between them and each node has its number and a link connecting two nodes will have a pair (connecting nodes 1 and 4). The network without cycle (feedback loops) are called feed-forward network or perception. The input nodes of the network such as node 1, 2 and 3 are associated with the input variables $(x_{i} = x_{1} ,x_{2} , \ldots ,x_{q} )$. They do not compute anything but simply pass the value to the processing nodes. The output nodes (4 and 5) are associated with the output variables $(y_{j} = y_{1} ,y_{2} , \ldots ,y_{p} )$. NNs can have several hidden layers. Each j-th node in a network has a set of weights w _ij and the activation functions, f of all the nodes. For example of two layers feed-forward NN in Fig. 11.

Fig. 11

Example of signal flow diagram of multilayer NNs

Let the input variables x ₁ = 1 and x ₂ = 0. Node 3, 4 and 5 has weights w ₁₃ = 2, w ₂₃ = −3, w ₁₄ = 1, w ₂₄ = 4, w ₃₅ = 2 and w ₄₅ = −1, respectively. The output of final node 5 is obtain

$$ y_{5} = f\left( {v_{5} } \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {if\,\,v_{5} \ge 0} \hfill \\ 0 \hfill & {otherwise} \hfill \\ \end{array} } \right. $$

(58)

Now we calculate weighted sum in the first hidden layer

$$ v_{3} = w_{13} x_{1} + w_{23} x_{2} = 2.1 - 3.0 = 2 $$

(59)

$$ v_{4} = w_{14} x_{1} + w_{24} x_{2} = 1.1 - 4.0 = 2 $$

(60)

After activation function in Eqs. (59) and (60) and given by

$$ y_{3} = f\left( {v_{3} } \right) = f\left( 2 \right) = 1 $$

(61)

and

$$ y_{4} = f\left( {v_{4} } \right) = f\left( 1 \right) = 1 $$

(62)

The weighted sum of node 5 is

$$ v_{5} = w_{35} y_{3} + w_{45} y_{4} = 2.1 - 1.1 = 1 $$

(63)

The output of final node 5 is given by

$$ y_{5} = f\left( {v_{5} } \right) = f(1) = 1. $$

(64)

Appendix 2

The SVD decomposition [52] Theorem 1 as follows

Theorem 1

Let ${\mathbf{H}} \in {\mathbf{R}}^{{M_{R}\times M_{T}}}$ or ${\mathbf{H}} \in {\mathbf{C}}^{{M_{R}\times M_{T}}}$ , then there exist orthogonal or unitary matrices such as $U \in {\mathbf{R}}^{{M_{R}\times M_{R}}}$ or $U \in {\mathbf{C}}^{{M_{R}\times M_{R}}}$ and $V \in {\mathbf{R}}^{{M_{T}\times M_{T}}}$ or $V \in {\mathbf{C}}^{{M_{T}\times M_{T}}}$ and $\Sigma \in {\mathbf{R}}^{{M_{R}\times M_{T}}}$ or $\Sigma\in {\mathbf{C}}^{{M_{R}\times M_{T}}}$ is a rectangular matrix whose diagonal elements are non-negative real numbers and whose off-diagonal elements are zero whereas the matrix H has the SVD as follows:

$$ {\mathbf{H}} = U\Sigma V^{T} $$

(65)

or,

$$ {\mathbf{H}} = U\Sigma V^{H} $$

(66)

where $\Sigma = \left[{\begin{array}{*{20}l} S & 0\\ 0 & 0\\ \end{array}}\right]$ and S = diag{σ ₁ , σ ₂ ,…,σ _k } since σ ₁  ≥ σ ₂  ≥ ···>σ _k .

Proof

Let ${\mathbf{H}}^{T} {\mathbf{H}} \ge 0$ we have $\sigma \left( {{\mathbf{H}}^{T} {\mathbf{H}}} \right) \le \left[ {0,\infty}\right]$ where operator {·}^T represent the transpose. Denoting $\sigma \left( {{\mathbf{H}}^{T} {\mathbf{H}}} \right)$ by $\left\{ {\sigma_{i}^{2} ,i = 1,2, \ldots ,n} \right\}$, we can arrange that,

$$ \sigma_{1} \ge \sigma_{2} \ge \cdots \ge \sigma_{v} > 0 = \sigma_{k + 1} = \cdots = \sigma_{n} . $$

Suppose $V_{1} ,\text{V}_{2}, \ldots ,\text{V}_{n}$ be a corresponding set of orthogonal eigenvectors and let,

$$ V_{1} = \left( {U_{1} , \ldots ,U_{k} } \right)\;\;{\text{and}}\;\;V_{2} = \left( {U_{k + 1} , \ldots ,U_{m} } \right), $$

then if S = diag{σ ₁, σ ₂,…,σ _k } we have:

$$ {\mathbf{H}}^{T} {\mathbf{H}}V_{1} = V_{1} S^{2} , $$

where $S^{-1} V_{1}^{T} {\mathbf{H}}^{T} {\mathbf{H}}V_{1} S^{-1} = I$.

Also ${\mathbf{H}}^{T} {\mathbf{H}}V_{2} = V_{2} 0$, So that, $V_{2}^{T} {\mathbf{H}}^{T} {\mathbf{H}}V_{2} = 0$, thus H V ₂ = 0.

Let $U_{1} = {\mathbf{H}}V_{1} S^{-1}$ we have $U_{1}^{T} U_{1} = I$ choose any U ₂ such that U = (U ₁, U ₂) is orthogonal then:

$$ U^{T} {\mathbf{H}}V = \left[ {\begin{array}{*{20}l} {U_{1}^{T} {\mathbf{H}}V_{1}} \hfill & {U_{1}^{T} {\mathbf{H}}V_{2}} \\ {U_{2}^{T} {\mathbf{H}}V_{1}} & {U_{2}^{T} {\mathbf{H}}V_{2}}\\ \end{array}} \right] = \left[ {\begin{array}{*{20}l} S & 0 \\ 0 & 0 \\ \end{array}} \right] = \Sigma $$

(67)

So H = UΣV ^T as desired.

For an example in [53], If the matrix product H = UΛV ^H and M _min = min(M _T , M _R ) can be expressed when M _min = M _T as follows:

$$ {\mathbf{H}} = \left[ {\underbrace {{\begin{array}{*{20}c} {U_{{M_{\min}}}} & {U_{{M_{R} - M_{\min}}}}\\ \end{array}}}_{U}} \right]\underbrace {{\left[ {\begin{array}{*{20}c} {\Sigma_{{M_{\min}}}}\\ {0_{{M_{R} - M_{\min}}}}\\ \end{array}} \right]}}_{\Sigma}V^{H} = U_{{M_{\min}}} \Sigma_{{M_{\min}}} V^{H} $$

(68)

where $U_{{M_{\min}}}\in {\mathbf{C}}^{{M_{R}\times M_{\min}}}$ is composed of M _min left singular vectors corresponding to the maximum possible singular values and $\Sigma_{{M_{\min}}}\in {\mathbf{C}}^{{M_{\min}\times M_{\min}}}$ square matrix.

Similarly if M _min = M _R then we can be written as:

$$ {\mathbf{H}} = U\left[ {\underbrace {{\begin{array}{*{20}c} {\Sigma_{{M_{\min}}}} & {0_{{M_{T}- M_{\min}}}}\\ \end{array}}}_{\Sigma}} \right]\underbrace {{\left[ {\begin{array}{*{20}c}{V_{{M_{\min}}}^{H}}\\{V_{{M_{T}- M_{\min}}}^{H}}\\ \end{array}} \right]}}_{{V^{H}}} = U\Sigma_{{M_{\min}}} V_{{M_{\min}}}^{H} $$

(69)

where $V_{{M_{\min}}}\in {\mathbf{C}}^{{M_{T}\times M_{\min}}}$ is composed of M _min right singular vectors.

Therefore, the eigenvalue decomposition as below:

$$ {\mathbf{HH}}^{H} = U\Sigma \Sigma^{H} U^{H} = Q\Lambda Q^{H} $$

(70)

where $\Lambda \in {\mathbf{C}}^{{M_{R} \times M_{R} }}$ a diagonal matrix is whose diagonal elements is given by:

$$ \lambda_{i} = \left\{{\begin{array}{*{20}l}{\sigma_{i}^{2} ,} \hfill & {if\;i = 1,2, \ldots ,M_{\min}} \hfill\\ {0,} \hfill & {if\;i = M_{\min}+ 1, \ldots ,M_{R}} \hfill\\ \end{array}} \right. $$

(71)

where $\{ \lambda_{i} \}_{i = 1}^{{M_{R} }}$ is the eigenvalues of Λ and the unitary matrix U = Q and $QQ^{H} = I_{{M_{R} }}$.

The squared Frobenius norm of the MIMO channel becomes:

$$ \left\| {\mathbf{H}} \right\|_{F}^{2} = tr\left( {{\mathbf{HH}}^{H}} \right) = \sum\limits_{i = 1}^{{M_{R}}} {\sum\limits_{j = 1}^{{M_{T}}} {\left| {h^{(i,j)}} \right|^{2}}} $$

(72)

Using Eq. (70) in Eq. (71) and we have:

$$ \left\| {\mathbf{H}} \right\|_{F}^{2} = \left\| {Q^{H} {\mathbf{H}}} \right\|_{F}^{2} = tr\left( {Q^{H} {\mathbf{HH}}Q} \right) = tr(\Lambda ) = \sum\limits_{i = 1}^{{M_{\min}}} {\sigma_{i}^{2}} $$

(73)