Multi-source, multi-object and multi-domain (M-SOD) electromagnetic interference system optimised by intelligent optimisation approaches

Abstract

With the wide use of electromagnetic information equipment, a large number of wireless radiation systems coexisting in the same region produce intentional or unintentional interference on electronic receivers. For the purpose of intentional electromagnetic interference, it is necessary to realise the efficient suppression of other receivers at little cost. When multiple transmitting sources are used to interfere with multiple receivers, the parameters of multiple transmitting sources are required to be comprehensively optimised and set so as to achieve a desired high-efficiency interference. Therefore, we propose a novel method to optimise the setting of parameters of a multi-source, multi-object and multi-domain (M-SOD) interference system based on intelligent optimisation approaches. Furthermore, this study also builds an intelligent optimisation model, which contains multiple transmitters and receivers which involved many parameters include position, direction of space domain, frequency, bandwidth, and power. Then the model is abstracted to the problem of single-objective optimisation with constraints and optimised through a traditional GA and an improved FWA method. The extensive experiments and comparisons show that the proposed algorithm is an effective approach for setting the parameters of an M-SOD electromagnetic interference system and superior to the conventional method.

Appendices

Appendices

Appendix 1: Derivation of instantaneous SINR $\gamma$

In the research, the communication mode used in the interfered communication system is TD-SCDMA and the uplink data frame structure is shown in Fig. 10 (Liu et al. 2011b). The structure comprises two data blocks in the length of 352 chips, a midamble code (training sequence) in the length of 144 chips and a guard interval in the length of 16 chips.

Fig. 10

Data frame structure of TD-SCDMA system (GP represents the guard interval and $T_c$ indicates the chip period)

Assuming that the spread spectrum code sequence $c\left( t \right)$ is a PN code, the autocorrelation function of this PN code can be expressed as:

$$\begin{aligned} R_c \left( t \right) =\left\{ \begin{array}{ll} 1-{\left| \tau \right| } / {T_c } &{} \left| \tau \right| \le T_c \\ 0&{} \left| \tau \right| >T_c \\ \end{array} \right. \end{aligned}$$

(15)

where $T_c$ represents the chip period. The power spectral density function of PN code is obtained via the Fourier transform of the autocorrelation function:

$$\begin{aligned} S_c \left( t \right) =\mathrm{\mathbf{F}}\left( {R_c \left( t \right) } \right) =T_c Sa^2\left( {fT_c } \right) \end{aligned}$$

(16)

where $\mathrm{\mathbf{F}}$ indicates the Fourier transform, and

$$\begin{aligned} Sa\left( x \right) =\frac{\sin \left( {\pi x} \right) }{\pi x} \end{aligned}$$

(17)

Supposing that the interference signals are independent, the correlation function and the power spectral density of $J_{m,t} \left( t \right)$ are given by:

$$\begin{aligned} R_J \left( \tau \right)&= \left( {U_m^2 +R_{U_n } \left( \tau \right) } \right) e^{j2\pi f_m \tau } \end{aligned}$$

(18)

$$\begin{aligned} S_J \left( f \right)&= U_m^2 \delta \left( {f-f_m } \right) +S_{U_n } \left( {f-f_m } \right) \nonumber \\&= \frac{F\left( {\varDelta \beta _m } \right) G_r \lambda ^2L_b }{\left( {4\pi R_m } \right) ^2}\left( {U_m^2 \delta \left( {f-f_m } \right) +S_{U_n } \left( {f-f_m } \right) } \right) \end{aligned}$$

(19)

The power of $J_{m,t} \left( t \right)$ is $P_m (P_m =U_m ^2+R_{U_n } \left( 0 \right) )$, where $R_{U_n } \left( \tau \right)$ indicates the autocorrelation function.

If it is assumed that the channels are classical Clarke spectra, the autocorrelation function of $h_i \left( t \right)$ is:

$$\begin{aligned} R_h \left( \tau \right) =J_0 \left( {2\pi f_{d,i} \tau } \right) \end{aligned}$$

(20)

where $J_0 \left( t \right)$ denotes the first-class zero-order Bessel function. The power spectral density of $h\left( t \right)$ is:

$$\begin{aligned} S_h \left( f \right) =\left\{ \begin{array}{ll} \frac{1}{\pi f_{d,i} }\frac{1}{\sqrt{1-\left( {f / {f_{d,i} }} \right) ^2} }&{}\left| f \right| <f_{d,i} \\ 0&{} \text {else} \\ \end{array} \right. \end{aligned}$$

(21)

Assuming that $h_i \left( t \right)$, $J_i \left( t \right)$, and $c\left( t \right)$ are independent, the autocorrelation function of $\sum \limits _{i=1}^{Q_n } {h_i \left( t \right) J_i \left( t \right) } c^*\left( t \right)$ is:

$$\begin{aligned} R\left( \tau \right) =\sum \limits _{i=1}^{Q_n } {R_h \left( \tau \right) R_J \left( \tau \right) R_c \left( \tau \right) } \end{aligned}$$

(22)

According to the property of convolution, the power spectral density function of

$\sum \limits _{i=1}^{Q_n } {h_i \left( t \right) J_i \left( t \right) } c^*\left( t \right)$ is:

$$\begin{aligned} \begin{aligned}&S\left( f \right) =\sum \limits _{m=1}^M {S_h \left( f \right) *S_J \left( f \right) *S_c \left( f \right) } \\&\quad =\sum \limits _{m=1}^M {\frac{1}{\pi f_{d,m} }S_h \left( f \right) *\frac{F\left( {\varDelta \beta _m } \right) G_r \lambda ^2L_b }{\left( {4\pi R_m } \right) ^2}\left( {U_m^2 \delta \left( {f-f_m } \right) +S_{U_n } \left( {f-f_m } \right) } \right) *T_c Sa^2\left( {fT_c } \right) } \\&\quad =\sum \limits _{m=1}^M {\frac{T_c }{\pi f_{d,m} }\frac{F\left( {\varDelta \beta _m } \right) G_r \lambda ^2L_b }{\left( {4\pi R_m } \right) ^2}\left\{ {U_m^2 S_h \left( f \right) *Sa^2\left[ {\left( {f-f_m } \right) T_c } \right] +S_h \left( f \right) *S_{U_n } \left( {f-f_m } \right) *Sa^2\left( {fT_c } \right) } \right\} } \\&\quad =\sum \limits _{m=1}^M {\frac{T_c }{\pi f_{d,m} }\frac{F\left( {\varDelta \beta _m } \right) G_r \lambda ^2L_b }{\left( {4\pi R_m } \right) ^2}\left\{ {\frac{U_m^2 }{\sqrt{1-\left( {f / {f_{d,m} }} \right) ^2} }*Sa^2\left[ {\left( {f-f_m } \right) T_c } \right] +S_h \left( f \right) *S_{U_n } \left( {f-f_m } \right) *Sa^2\left( {fT_c } \right) } \right\} } \\&\quad =\sum \limits _{m=1}^M {\frac{T_c }{\pi f_{d,m} }\frac{F\left( {\varDelta \beta _m } \right) G_r \lambda ^2L_b }{\left( {4\pi R_m } \right) ^2}\left\{ {\int _{-f_{d,m} }^{f_{d,m} } {\frac{U_m^2 Sa^2\left[ {\left( {f-t-f_m } \right) T_c } \right] }{\sqrt{1-\left( {t / {f_{d,m} }} \right) ^2} }dt} +S_h \left( f \right) *S_{U_n } \left( {f-f_m } \right) *Sa^2\left( {fT_c } \right) } \right\} } \\ \end{aligned} \end{aligned}$$

(23)

Supposing that $f_0$ is the carrier frequency, the power spectral density of $\sum \nolimits _{m=1}^M {h_m \left( t \right) J_m } c^*\left( t \right)$ at $f_0$ is:

$$\begin{aligned} N_I =S\left( {f_0 } \right) \end{aligned}$$

(24)

Supposing that the power spectral density function of received noise $n\left( t \right)$ is $S_n \left( f \right)$, the power spectral density of $n\left( t \right) c^*\left( t \right)$ at $f_0$ is:

$$\begin{aligned} S_{nc} \left( {f_0 } \right) =S_n \left( {f_0 } \right) *S_c \left( {f_0 } \right) =\int _{-\infty }^\infty {N_0 S_c \left( {f-f_0 } \right) df} =N_0 \end{aligned}$$

(25)

The received instantaneous SINR $\gamma$ is:

$$\begin{aligned} \gamma =\frac{P_s T_b }{\left| \alpha \right| ^2N_I +N_0 }=\frac{E_b }{\left| \alpha \right| ^2N_I +N_0 }=\frac{1}{\left| \alpha \right| ^2{N_I } / {N_0 }+1}\frac{E_b }{N_0 } \end{aligned}$$

(26)

where $E_b =P_s T_b$ denotes the bit energy and $T_b$ represents the symbol period.

Appendix 2: Derivation of BER

Based on the literature (Goldsmith 2005), the average BER of the nth receiver is:

$$\begin{aligned} p_{b,n} =\int _0^\infty {Q\left( {\sqrt{2\gamma } } \right) p\left( \gamma \right) d\gamma } \end{aligned}$$

(27)

where $Q\left( x \right) =\frac{1}{2\pi }\int _x^\infty {\exp \left( {-\frac{u^2}{2}} \right) du}$ and $p\left( \gamma \right)$ indicates the probability density function of $\gamma$. According to the probability density function of $\left| \alpha \right| ^{[57]}$:

$$\begin{aligned} p\left( {\left| \alpha \right| } \right) =\frac{\left| \alpha \right| }{\sigma ^2}\exp \left( {-\frac{\left| \alpha \right| ^2}{2\sigma ^2}} \right) \end{aligned}$$

(28)

Thus:

$$\begin{aligned} p\left( \gamma \right) =\frac{{E_b } / {N_0 }}{2\sigma ^2\gamma ^2{N_I } / {N_0 }}\exp \left( {-\frac{{E_b } / {\gamma N_0 }-1}{2\sigma ^2{N_I } / {N_0 }}} \right) \end{aligned}$$

(29)

$Q\left( x \right)$ can be expressed as (Goldsmith 2005):

$$\begin{aligned} Q\left( x \right) =\frac{1}{\pi }\int _0^{\pi / 2} {\exp \left( {\frac{-x^2}{2\sin ^2\varphi }} \right) d\varphi } \end{aligned}$$

(30)

In accordance with the literature (Goldsmith 2005), the average BER of QPSK modulation is:

$$\begin{aligned} p_{b,n} =\frac{1}{\pi }\int _0^{\pi / 2} {\int _0^\infty {\exp \left( {\frac{-\gamma ^2}{\sin ^2\varphi }} \right) p\left( \gamma \right) d\gamma d\varphi } } \end{aligned}$$

(31)

Substituting Formula (29) into Formula (31) gives:

$$\begin{aligned} p_{b,n} =\frac{1}{\pi }\int _0^{\pi / 2} {\int _0^{{E_b } / {N_0 }} {\frac{{E_b } / {N_0 }}{2\sigma ^2\gamma ^2{N_I } / {N_0 }}\exp \left( {\frac{-\gamma ^2}{\sin ^2\varphi }} \right) \times } } \exp \left( {-\frac{{E_b } / {\gamma N_0 }-1}{2\sigma ^2{N_I } / {N_0 }}} \right) d\gamma d\varphi \end{aligned}$$

(32)

Appendix 3: Parameter design table

1. Solutions in Case 1

(1) Parameter optimisation result

See Table 6.

Table 6 Parameter optimisation results of different algorithms in Case 1

(2) Sketch maps

See Fig. 11.

Fig. 11

Parameter optimisation result of evolutionary algorithm in Case 1 (the red and blue rectangles represent the interference transmitter and communication receiver and the short line indicates the antenna direction)

2. Solutions in Case 2

(1) Parameter optimisation result

See Table 7.

Table 7 Parameter optimisation results of different algorithms in Case 2

(2) Sketch maps

See Fig. 12.

Fig. 12

Parameter optimisation results of evolutionary algorithms in Case 2