Suppression method of inter-symbol interference in communication system based on mathematical chaos theory

2.1.1 Principle analysis of inter-symbol interference in communication system

In the optical communication system, the optical channel is divided into direct channel and diffuse channel, that is, the optical pulse signal will arrive at the detector through different paths, and the signals of different paths will produce time delay, which will cause the overlap and crosstalk between different optical pulse signals in time (Wu et al., 2015), as shown in Fig. 1.

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Fig. 1

Inter-symbol interference due to multi-path transmission delay.

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In Fig. 1, the signal transmitted by path 2 has a certain time delay compared with the signal transmitted by path 1. When the time delay is larger than the transmission interval of the pulse signal, the adjacent two pulse signals will be aliased when they arrive at the detector (Deng and Chen, 2017; Gao and Wang, 2017a,2017b; Jin and Mi, 2019). When accepting the sampling decision, it may lead to inter-symbol interference which not only causes the decision error, but also weakens the signal energy and reduces the signal-to-noise ratio (Akhtar et al., 2015; Li et al., 2018; Zhang et al., 2015a,2015b). Relevant studies show that the average signal-to-noise ratio (SNR) of the channel decreases due to the inter-symbol interference caused by diffuse light.

In addition to multi-path delay, channel distortion can also cause ISI. Because of the unsatisfactory channel, symbol widening and delay occur in the transmission process. In the waveform, symbol pulse trailing occurs, and the trailing of adjacent pulses overlaps with each other, which makes the receiving symbol error, as shown in Fig. 2.

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Fig. 2

Inter-symbol interference due to symbol broadening due to channel distortion.

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In fact, for communication systems, inter-symbol interference can be neglected at low transmission rates. This is because the signal delay is far less than the symbol period, and the adjacent symbols will not produce crosstalk. With the increase of transmission rate (Li and Zheng, 2015), the signal delay can be compared with the symbol period, or even greater than the symbol period. At this time, the aliasing between adjacent symbols will occur (Dar et al., 2015; Gao et al., 2017; Mi et al., 2015; Harraga and Yebdri, 2018). This shows that the effect of ISI on the performance of communication system increases with the increase of the rate. Relevant research results show that when the transmission rate is below 100 Mb/s, ISI has little effect on the performance of communication system; when the transmission rate of the system is above 100 Mb/s, the performance of communication system is seriously deteriorated by ISI, which can no longer meet the requirements of communication, and the ISI of communication system needs to be suppressed (Li and Li, 2015; Zhang et al., 2015a,2015b).

2.1.2 Analysis of channel impulse response in communication system

In order to achieve the design of ISI suppression for communication system, the transmission channel model of communication system is constructed. Assuming that the maximum time interval of multi-path components is s, the communication terminal receives signal samples and generates sample sets:

Under the constraint of multi-path propagation characteristics, it is assumed that the energy attenuation of communication channel is as follows:

In the formula, A is the output signal amplitude of the optical fiber communication channel, f is the initial frequency of channel attenuation, $k=\frac{B}{T}$ is the slope of frequency modulation, and B is the phase offset of each path. An adaptive noise canceller is used to suppress multi-path interference. The multi-path channel model of the communication system is constructed as shown in Fig. 3.

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Fig. 3

Multi-path channel model of communication system.

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In the multi-path channel model of the communication system shown in Fig. 3, the channel impulse response is expressed as follows:

In the formula, ${\theta }_i\left(t\right)$ is the frequency doubling offset of each path, ${\theta }_i\left(t\right)$ is positively correlated with symbol rate $f_s$ . The angle between each path and the direct path is an integral multiple of symbol width $T_s$ . By adding phase offset of multi-path component, the channel impulse response of communication system is described as follows:

2.1.3 Calculation of multi-path component delay of the channel

Because the delay of channel’s multi-path component causes ISI in communication system, the delay of channel’s multi-path component is calculated based on the communication channel model constructed above, and the ISI is suppressed by setting reasonable delay (Chen and Wei, 2016; Rui et al., 2016). The signal $S\left(t\right)$ received by the communication channel is composed of many pulses with different time delays. According to the phase offset of each path, the multi-path components of multi-channel are processed with time weights. According to the tapped delay line model constructed above, time-frequency characteristic decomposition is performed on different transmission paths of communication channel (Bono, 2016; Gao and Wang, 2017a,2017b; Konwar and Debnath, 2017; Pyskunov et al., 2016; Schulze, 2016; Turgut, 2016), Assuming that the arrival time delay of each multi-path is an integer multiple of $T_s$ , the cluster component is taken as the initial eigenvector of impulse response, and the signal’s sparsity representation model of communication system is constructed.

In the formula, ${\tau }_{\mathit{mk}}$ is the impulse coefficient.

According to the sparse representation model, the ith path delay of communication channel for data transmission in communication network is obtained as follows:

In the formula, $k$ is the sparsity of vector $w_i$ , $\lambda$ is the impulse convergence factor in stable state and $G$ is the tap of equalizer in communication channel, where the tap of equalizer is an unknown variable. In order to obtain a reasonable multi-path component delay of channel, the tap of equalizer is optimized. Therefore, the mathematical chaos optimization theory is introduced to optimize the tap parameters of equalizer in communication channel.

2.2.1 Mathematical chaos optimization theory

Chaotic optimization algorithm takes advantage of the spatial ergodicity of chaos and expresses the optimization variables needed by the system with the chaotic variables generated by the chaotic map. At the same time, the ergodic range of chaos is transformed to the definition domain of optimization variables, and chaotic variables are used to search the global optimal solution, which is regarded as the global optimal solution of optimization variables (Liu et al., 2018; Sen Gupta, 2017; Yassaee et al., 2015; Yang et al., 2018). However, there is a disadvantage of chaotic motion, that is, it does not traverse the space in a uniformly distributed way, and chaotic operators need a lot of computation to traverse all the space. The sensitivity of initial values of chaotic motion makes the domain of independent variables sensitive and affects the trajectory of chaotic motion. Therefore, the performance of chaotic optimization algorithm is unstable and takes a long time. Thus, when chaotic variables are used to disturb the current point of the weight vector, time-varying parameters are introduced, and the disturbance amplitude decreases gradually with the updating of the time-varying parameters, so as to gradually achieve the goal of local fine search near the optimal solution, and achieve the global optimal value of the weight vector.

2.2.2 Weight vector optimization of equalizer in communication channel based on chaos optimization algorithm

Chaotic variables are introduced by using carrier-like method, and chaotic variables are used to search. Time-varying parameters are introduced in the process of secondary carrier. A time-varying parameter $\mathrm{\Psi }\left(t\right)$ is defined and introduced into the optimization variables by formula (7) and (8):

In the formulas, $J^{\ast }$ and $J_{max}$ are the corresponding minimum objective function values and the maximum objective function values of all feasible solutions obtained in the first stage of chaotic search, so that the parameter $P$ is selected in the range of 0 < P < 1.

The chaotic algorithm is used to solve the minimum value $J^{\ast }$ of the cost function to optimize the tap. When chaotic variables are generated, one-dimensional Logistic mapping is used to generate chaotic signals. The mapping formula is as follows:

In the formula, when $\rho \in \left[3.57,4\right]$ , the mapping is chaotic.

Let the objective function of the continuous object to be optimized be $J^{\ast }=J\left(f^{\ast }\right)=minJ\left(f\right)$ , $J$ represent the objective function, $f_{Re}\in \left[d_{Re},e_{Re}\right]$ , $f_{\mathit{lm}}\in \left[d_{\mathit{lm}},e_{\mathit{lm}}\right]$ , $f$ represents the optimization variable, $f_{Re}$ and $f_{\mathit{lm}}$ are the real and imaginary parts, $e_{Re}$ and $d_{Re}$ are the upper and lower limits of $f_{Re}$ , $e_{\mathit{lm}}$ and $d_{\mathit{lm}}$ are the upper and lower limits of $f_{\mathit{lm}}$ . When the number of iterations in the first stage of chaotic search is $N_1$ and the number of iterations in the second stage of chaotic search is $N_2$ , the steps of optimizing the weight vector of equalizer in communication channel by chaotic optimization algorithm are as follows:

• (1)

  The difference value of [0,1] interval is given to form vector $c\left(0\right)$ (not 0, 0.25, 0.5, 0.75, 1), $n$ is the dimension of the weight vector, the optimal solution of the channel’s equalizer weight vector is set to f*, and the corresponding cost function is the initial value of f* and $f_{max}$ , so that $K=0$ , $K^{\prime }=0$ , $t=0$ .

• (2)

  n chaotic variables are obtained by the iteration of formula (9), and their trajectories are different from each other. They are enlarged to the range of optimal variables by formula (11), (12):

• (3)

  After iteration of chaotic variables, $J\left(f\left(K\right)\right)$ is calculated and $J_{max}$ and J* are retained. If $J\left(f\left(K\right)\right)⩽J^{\ast }$ , then $J^{\ast }=J\left(f\left(K\right)\right)$ , $f_{Re}^{\ast }=f_{Re}\left(K\right)$ , $f_{\mathit{lm}}^{\ast }=f_{\mathit{lm}}\left(K\right)$ ; if $J\left(f\left(K\right)\right)>J_{max}$ , then $J_{max}=J\left(f\left(K\right)\right)$ , $K=K+1$ , if $K⩽N_1$ , then turn to step (2).

• (4)

  After the above $N_1$ -step search to J*, the parameter $P$ is selected, to calculate the initial value of the time-varying parameter $\mathrm{\Psi }\left(t\right)$ :, and carry $\mathrm{\Psi }\left(0\right)=\left(J^{\ast }-J_{max}\right)/lnP$ out the second carrier according to the following formula:

In the formula, $f_{Re}^{\ast }$ and $f_{\mathit{lm}}^{\ast }$ are the real and imaginary parts of the current optimal solution $f^{\ast }$ .

• (5)

  The second carrier of chaotic variable is searched carefully and $J\left(f\left(K^{\prime }\right)\right)$ is calculated. If $J\left(f\left(K^{\prime }\right)\right)⩽J^{\ast }$ , then $J^{\ast }=J\left(f\left(K^{\prime }\right)\right)$ , $f_{Re}^{\ast }=f_{Re}\left(K^{\prime }\right)$ , $f_{\mathit{lm}}^{\ast }=f_{\mathit{lm}}\left(K^{\prime }\right)$ ; otherwise, give up $f_{Re}\left(K^{\prime }\right)$ and $f_{\mathit{lm}}\left(K^{\prime }\right)$ . $K^{\prime }=K^{\prime }+1$ , $\mathrm{\Psi }\left(t+1\right)=\lambda \mathrm{\Psi }\left(t\right)$ , $t=t+1$ .

• (6)

  If the termination condition $K^{\prime }>N_2$ is satisfied, then the search is stopped and J* and the corresponding global optimal solution f* are output; otherwise, step 5 is returned.

Therefore, the above steps (1)-(6) are the process of optimizing the weight vector of equalizer in communication channel by chaotic optimization algorithm, and the output result f* is the tap of equalizer. Based on the tap of equalizer, reasonable multi-path component delay of channel is set to effectively suppress inter-symbol interference.