Multiuser Detection in SDMA–OFDM Wireless Communication System Using Complex Multilayer Perceptron Neural Network

Abstract

The space division multiple access–orthogonal frequency division multiplexing (SDMA–OFDM) wireless system has become very popular owing high spectral efficiency and high load capability. The optimal maximum likelihood multiuser detection (MUD) technique suffers from high computational complexity. On the other hand the linear minimum mean square error (MMSE) MUD techniques yields poor performance and also fails to detect users in overload scenario, where the number of users are more than that of number of receiving antennas. By contrast, the differential evolution algorithm (DEA) aided minimum symbol error rate (MSER) MUD can sustain in overload scenario as it can directly minimizes probability of error rather than mean square error. However, all these classical techniques are still complex as these do channel estimation and multiuser detection sequentially. In this paper, complex multi layer perceptron (CMLP) neural network model is suggested for MUD in SDMA–OFDM system as it do both channel approximation and MUD simultaneously. Simulation results prove that the CMLP aided MUD performs better than the MMSE and MSER techniques in terms of enhanced bit error rate performance with low computational complexity.

Appendix: Development of the Complex BP Algorithm for CMLP Network Training

During the proposed CMLP network training, the conventional BP algorithm cannot be applied directly as all connection weights and biases are complex in form. For BP algorithm to be applied for CMLP network training, it is required to compute the gradient of the instantaneous error with respect to real and imaginary components of all network weights and biases. Thus, the modifications incorporated in this algorithm are derived as follows. Figure 7 represents a typical training model of CMLP model with $Q$ number of layers each consisting $n$ neurons. Let, $g_n^q$ is the activation output of the $n$th neuron in the $q$th layer, then the net activation value of the $n$th neuron in the $q$th layer is given by:

$$\begin{aligned} s_n^q =\sum \limits _{m=0}^{N_{q-1} } {W_{nm}^q } g_m^{q-1} , n=1,2,\ldots ,N_q , q=1,2,\ldots ,Q \end{aligned}$$

where $g_n^q =SGM\left( {s_n^q } \right)$, and $n$ = 0 refers to the bias input: i.e. $g_0^q =1$. Also, when $q$ = 0, then $g_n^0 , n=1,2,\ldots ,N_0$ refers the input signal. Assuming $g_n^q =u_n^q +jv_n^q$,

$$\begin{aligned} s_n^q&= \sum \limits _{m=0}^{N_{q-1} } {\left( {W_{nm}^{qR} +jW_{nm}^{qI} } \right) \left( {u_m^{q-1} +jv_m^{q-1} } \right) }\\&= \sum \limits _{m=0}^{N_{q-1} } {\left( {W_{nm}^{qR} u_m^{q-1} -W_{nm}^{qI} v_m^{q-1} } \right) } +j\left( {W_{nm}^{qR} v_m^{q-1} +W_{nm}^{qI} u_m^{q-1} } \right) \end{aligned}$$

The superscripts $R$ and $I$ corresponds to real and imaginary values, respectively. In the complex BP algorithm, the network weights are adjusted such that the error in the output layer is minimized. Hence, the sum squared global instantaneous error at the output layer ‘$Q$’ is expressed as:

$$\begin{aligned} E=\frac{1}{2}\sum \limits _{n=1}^{N_Q } {\left\| {e_n } \right\| ^{2}} \end{aligned}$$

where $e_n =d_n -g_n^Q , n=1,2,\ldots ,N_Q$ and $d_n$ is the desired response. The gradient of $E$ with respect to $g_n^q$ is:

$$\begin{aligned} \nabla _{g_n^q } E=\frac{\partial E}{\partial g_n^q }=\frac{\partial E}{\partial g_n^{qR} }+j\frac{\partial E}{\partial g_n^{qI} } \end{aligned}$$

(28)

Similarly, the gradient of $E$ with respect to $s_n^q$ is:

$$\begin{aligned} \nabla _{s_n^q } E=\frac{\partial E}{\partial s_n^q }=\frac{\partial E}{\partial s_n^{qR} }+j\frac{\partial E}{\partial s_n^{qI} } \end{aligned}$$

(29)

From Eqs. (28) and (29), the error gradient of $n$th neuron in the $q$th layer is derived as:

$$\begin{aligned} \delta _n^q&= \frac{\partial E}{\partial g_n^{qR} }\frac{\partial g_n^{qR} }{\partial s_n^{qR} }+j\frac{\partial E}{\partial g_n^{qI} }\frac{\partial g_n^{qI} }{\partial s_n^{qI} } \\ \delta _n^{qR} +j\delta _n^{qI}&= e_n^{qR} \varphi {\prime }\left( {s_n^{qR} } \right) +je_n^{qI} \varphi {\prime }\left( {s_n^{qI} } \right) \end{aligned}$$

Since the weight vector in complex valued, the gradient of instantaneous error can be obtained by calculating the gradient of the error with respect to both the real and imaginary components of the $W_{nm}^q$, which is according to:

$$\begin{aligned} \nabla _{W_{nm}^q } E=\frac{\partial E}{\partial W_{nm}^q }=\frac{\partial E}{\partial W_{nm}^{qR} }+j\frac{\partial E}{\partial W_{nm}^{qI} } \end{aligned}$$

Using the chain rule,

$$\begin{aligned} \frac{\partial E}{\partial W_{nm}^{qR} }&= \frac{\partial E}{\partial s_n^{qR} }\frac{\partial s_n^{qR} }{\partial W_{nm}^{qR} }+\frac{\partial E}{\partial s_n^{qI} }\frac{\partial s_n^{qI} }{\partial W_{nm}^{qR} } \\ \frac{\partial E}{\partial W_{nm}^{qI} }&= \frac{\partial E}{\partial s_n^{qR} }\frac{\partial s_n^{qR} }{\partial W_{nm}^{qI} }+\frac{\partial E}{\partial s_n^{qI} }\frac{\partial s_n^{qI} }{\partial W_{nm}^{qI} } \end{aligned}$$

In the above equations

$$\begin{aligned} \frac{\partial s_n^{qR} }{\partial W_{nm}^{qR} }&= u_m^{q-1} ;\frac{\partial s_n^{qI} }{\partial W_{nm}^{qR} }=-v_m^{q-1} ;\frac{\partial s_n^{qR} }{\partial W_{nm}^{qI} }=v_m^{q-1} ;\frac{\partial s_n^{qI} }{\partial W_{nm}^{qI} }=u_m^{q-1} \\ \frac{\partial E}{\partial W_{nm}^{qR} }&= -\delta _n^{qR} u_m^{q-1} -\delta _n^{qI} v_m^{q-1} \\ \frac{\partial E}{\partial W_{nm}^{qI} }&= -\delta _n^{qR} (-v_m^{q-1} )-\delta _n^{qI} u_m^{q-1} \end{aligned}$$

From above equations $\nabla _{W_{nm}^q } E$ can be obtained as:

$$\begin{aligned} \nabla _{W_{nm}^q } E&= \left( {-\delta _n^{qR} u_m^{q-1} -\delta _n^{qI} v_m^{q-1} } \right) +j\left( {\delta _n^{qR} v_m^{q-1} -\delta _n^{qI} u_m^{q-1} } \right) \\&= -\left( {\delta _n^{qR} +j\delta _n^{qI} } \right) \left( {g_m^{q-1} } \right) ^{{*}} \\&= -\left( {g_m^{q-1} } \right) ^{{*}}\delta _n^q \end{aligned}$$

where (.)$^{*}$ denote the complex conjugate. The correction $\Delta W_{nm}^q$ applied to $W_{nm}^q$ is defined by delta rule as:

$$\begin{aligned} \Delta W_{nm}^q =-\mu \nabla _{W_{nm}^q } E \end{aligned}$$

(30)

where $\mu$ is the learning parameter.

Fig. 7

Training model of CMLP network

The generalized error gradient for $q=Q,Q-1,\ldots ,1$ and $n=1,2,\ldots ,N_q$ as:

$$\begin{aligned} e_n^q&= \left\{ \begin{array}{ll} e_n,&{} {\hbox {for }}q=Q \\ \sum _{r=1}^{N_{q+1}} W_{rn}^{*(q+1)} \delta _r^{q+1},&{} {\hbox {for }}q=Q-1,\ldots ,1 \end{array} \right. \end{aligned}$$

(31)

$$\begin{aligned} \delta _n^q&= e_n^{qR} \varphi ^{\prime }\left( {s_n^{qR} } \right) +je_n^{qI} \varphi {\prime }\left( {s_n^{qI} } \right) \end{aligned}$$

(32)