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A New Approach for Error Rate Analysis of Wide-Band DSSS-CDMA System with Imperfect Synchronization Under Jamming Attacks

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Abstract

Most of the researches on error rate analysis of direct sequence spread spectrum (DSSS-CDMA) systems assume that the synchronization is perfect. However, in practice, the synchronization is often imperfect due to various effects of channel parameters such as noise and fading. The degree of imperfection further increases due to jamming attacks. The present study, therefore, derives new expressions to compute the probability of error in DSSS-CDMA systems under imperfect synchronisation. It is assumed that the channel is wideband and is subjected to various jamming attacks. A new parameter, called as probability of successful synchronization, was introduced which includes the effects of both the probability of false alarm and detection under fast and slow jammers. Monte Carlo simulations were conducted in MATLAB to establish the validity of the derived mathematical expressions.

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Auckland, 38 Princes Street, Level 2, Room 240, Auckland, New Zealand

    Arash Tayebi, Stevan Berber & Akshya Swain

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  1. Arash Tayebi
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  2. Stevan Berber
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  3. Akshya Swain
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Correspondence to Arash Tayebi.

Appendices

Appendix A

$$\begin{aligned} {U_{syn}} &= { { {\sqrt{{E_c}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} \sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } +\,\,{ { {\sqrt{{E_c}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^0x_i^0} \right) } \sum \limits _{j = 1}^{h - 1} {\alpha _j^{}j} \nonumber \\& { { { + \sqrt{{E_c}} } } \big / { m }}{B_k}\sum \limits _{i = 1}^{2\beta } {x_i^kx_i^0} \sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } + { { {\sqrt{{E_c}} } } \big / { m }}{B_k}\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^kx_i^0} \right) } \sum \limits _{j = 1}^{h - 1} {\alpha _j^{}j} \nonumber \\& + { { {\sqrt{{E_N}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {m\left( {n_i^{}.\,x_i^0} \right) } + { { {\sqrt{{E_J}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {m\left( {J_i^{}.\,x_i^0} \right) } \end{aligned}$$
$$\begin{aligned} E\left[ {{U_{syn}}} \right] = E[A + B + C + D] = E[A] + E[B] + E[C] + E[D] + E[F] + E[G] \end{aligned}$$
$$\begin{aligned} E[A]=\,\,& {} E\left[ {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) \left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) } \right] \nonumber \\=\,\,& {} { { {\sqrt{{E_c}} } } \big / { m }}E\left[ {\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) } \right] E\left[ {\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) } \right] \nonumber \\=\,\,& {} \sqrt{{E_c}} 2\beta \sqrt{ { \pi } \big / { {h(2m + 1 - h)} } \big / { {2m} }}\\ E[B]=\,\,& {} E[C] = E[D] = E[F] = E[G] = 0\\ E[{U_{syn}}]= & {} \sqrt{{E_c}} 2\beta \sqrt{{ { \pi } \big / { 4 }}} { { {h(2m + 1 - h)} } \big / { {2m} }} \end{aligned}$$

and the variance is:

$$\begin{aligned} \sigma _{{U_{syn}}}^2=\,\,& {} E[{\left( {{U_{syn}}} \right) ^2}] - {\left( {E[{U_{syn}}]} \right) ^2}\nonumber \\=\,\,& {} E[{(A + B + C + D + F + G)^2}] - {\left( {E[{U_{syn}}]} \right) ^2}\nonumber \\=\,\,& {} E[{A^2}] + E[{B^2}] + E[{C^2}] + E[{D^2}] + E[{D^2}] + E[{D^2}] - {\left( {E[{U_{syn}}]} \right) ^2} \end{aligned}$$
$$\begin{aligned} E[A_{}^2]=\,\,& {} E\left[ {{{\left( {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) \left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) } \right) }^2}} \right] \nonumber \\=\,\,& {} { { {{E_c}} } \big / { {{m^2}} }}.E\left[ {{{\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) }^2}} \right] E\left[ {{{\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) }^2}} \right] \end{aligned}$$
$$\begin{aligned}&E\left[ {{{\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) }^2}} \right] = \,2\beta { { {{E_c}} } \big / { {{m^2}} }}\left( {E\left[ {{{\left( {x_i^0} \right) }^4}} \right] + (2\beta - 1)E\left[ {{{\left( {x_i^0} \right) }^2}} \right] E\left[ {{{\left( {x_i^0} \right) }^2}} \right] } \right) \nonumber \\&\qquad \qquad \qquad \quad = \,\left( {\beta + 4{\beta ^2}} \right) { { {{E_c}} } \big / { {{m^2}} }} \end{aligned}$$

Also

$$\begin{aligned}&E\left[ {{{\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) }^2}} \right] = E\left[ {{{\left( {\alpha _j^{}} \right) }^2}} \right] E\left[ {{{\left( {{{\sum \limits _{j = 1}^h {\left( {m + 1 - j} \right) } }^2}} \right) }^{}}} \right] \nonumber \\&\qquad + E\left[ {\alpha _j^{}} \right] E\left[ {\alpha _j^{}} \right] 2E\left[ {\sum \limits _{l = 1}^{h - 1} {\sum \limits _{j = l + 1}^h {\left( {m + 1 - l} \right) \left( {m + 1 - j} \right) } } } \right] \nonumber \\ \quad&={ { {h(2{h^2} - 3h(2m + 1) + 6{m^2} + 6m + 1)} } \big / { 6 }}\nonumber \\&\quad + { { \pi } \big / { 4 }}\left( {{ { {\left( {{h^2} - h} \right) \left( {3{h^2} - 12hm - 7h + 12{m^2} + 12m + 2} \right) } } \big / { {12} }}} \right) \end{aligned}$$

Thus

$$\begin{aligned} E[A^2]=\,\,& {} \left( {\beta + 4{\beta ^2}} \right) { { {{E_c}} } \big / { {{m^2}} }}\\&\quad \left( \begin{array}{l} { { {h(2{h^2} - 3h(2m + 1) + 6{m^2} + 6m + 1)} } \big / { 6 }}\\ + { { \pi } \big / { {\left( {{h^2} - h} \right) \left( \begin{array}{l} 3{h^2} - 12hm - 7h\\ + 12{m^2} + 12m + 2 \end{array} \right) } } \big / { {12} }} \end{array}\right) \end{aligned}$$
$$\begin{aligned} E[B_{}^2]=\,\,& {} E\left[ {{{\left( {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^0x_i^0} \right) } } \right) \left( {\sum \limits _{j = 1}^{h - 1} {\alpha _j^{}j} } \right) } \right) }^2}} \right] \nonumber \\= \,& {} { { {2\beta {E_c}} } \big / { {{m^2}} }}.\left( {E\left[ {{{\left( {\alpha _j^{}} \right) }^2}} \right] E\left[ {\left( {\sum \limits _{j = 1}^{h - 1} {{{\left( j \right) }^2}} } \right) } \right] + E\left[ {\alpha _j^{}} \right] E\left[ {\alpha _j^{}} \right] 2E\left[ {\sum \limits _{l = 1}^{h - 2} {\sum \limits _{j = l + 1}^{h - 1} {jl} } } \right] } \right) \nonumber \\= \,& {} { { {2\beta {E_c}} } \big / { {{m^2}} }}.\left( {\frac{{h(h - 1)(2h - 1)}}{6} + \frac{{\pi (h - 2)(h - 1)h(3h - 1)}}{{48}}} \right) \end{aligned}$$
$$\begin{aligned} E[C_{}^2]=\,\,& {} E\left[ {{{\left( {{{{\sqrt{{E_c}} }} \big / {m}}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^0x_i^0} \right) } } \right) \left( {\sum \limits _{j = 1}^{h - 1} {\alpha _j^{}j} } \right) } \right) }^2}} \right] \\= &\, {} {{{2\beta {E_c}}}\big /{{{m^2}}}}.\left( \begin{array}{l} E\left[ {{{\left( {\alpha _j^{}} \right) }^2}} \right] E\left[ {{{\left( {{{\sum \limits _{j = 1}^h {\left( {m + 1 - j} \right) } }^2}} \right) }^{}}} \right] \\ + 2E\left[ {\alpha _j^{}} \right] E\left[ {\alpha _j^{}} \right] 2E\left[ {\sum \limits _{l = 1}^{h - 2} {\sum \limits _{j = l + 1}^{h - 1} {\left( {m + 1 - l} \right) \left( {m + 1 - j} \right) } } } \right] \end{array} \right) \\=\,\,& {} {{{2\beta {E_c}}} \big /{{{m^2}}}}.\left( \begin{array}{l} {{{h(2{h^2} - 3h(2m + 1) + 6{m^2} + 6m + 1)}} \big /{6}}\\ + {{\pi } \big /{4}}\left( {{{{\left( {{h^2} - h} \right) \left( {3{h^2} - 12hm - 7h + 12{m^2} + 12m + 2} \right) }} \big / {{12}}}} \right) \end{array} \right) \end{aligned}$$
$$\begin{aligned} E[D_{}^2]= 2\beta { { {{E_c}} } \big / { {{m^2}} }}.\left( {\frac{{h(h - 1)(2h - 1)}}{6} + \frac{{\pi (h - 2)(h - 1)h(3h - 1)}}{{48}}} \right) \end{aligned}$$
$$\begin{aligned} E(F) = E\left[ {{{\left( {{ { {\sqrt{{E_N}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {m\left( {n_i^{}.\,x_i^{}} \right) } } \right) }^2}} \right] = { { {{E_N}} } { {{m^2}} }}{m^2}E\left[ {{{\left( {\sum \limits _{i = 1}^{2\beta } {\left( {n_i^{}.\,x_i^{}} \right) } } \right) }^2}} \right] = 2\beta { { {{N_0}} } \big / { 2 }} \end{aligned}$$
$$\begin{aligned} E(G) = E\left[ {{{\left( {\sum \limits _{i = 1}^{2\beta } {m\left( {J_i^{}.\,x_i^{}} \right) } } \right) }^2}} \right] = 2\beta \varphi { { {{J_0}} } \big / { 2 }} \end{aligned}$$

We define \(\xi\), \(\psi\) and \(\eta\) as:

$$\begin{aligned} \xi=\,\,& {} { { {h(2{h^2} - 3h(2m + 1) + 6{m^2} + 6m + 1)} } \big / { {6{m^2}} }}\\&+ { { \pi } \big / { 4 }}\left( {{ { {\left( {{h^2} - h} \right) \left( {3{h^2} - 12hm - 7h + 12{m^2} + 12m + 2} \right) } } \big / { {12{m^2}} }}} \right) \\ \psi= & {} \left( {\frac{{h(h - 1)(2h - 1)}}{{6{m^2}}} + \frac{{\pi (h - 2)(h - 1)h(3h - 1)}}{{48{m^2}}}} \right) \\ \eta= & {} { { {h(2m + 1 - h)} } \big / { {2m} }}\sqrt{{ { \pi } \big / { 4 }}} \end{aligned}$$
$$\begin{aligned} \sigma _{{U_{syn}}}^2 &= \left( {3\beta + 4{\beta ^2}} \right) {E_c}\xi + 4\beta {E_c}\psi + 2\beta { { {{N_0}} } \big / { {{J_0}} } \big / { 2 }}\\&-\, {\left( {\sqrt{{E_c}} 2\beta \eta } \right) ^2}\\ \sigma _{{U_{syn}}}^2 &= {E_b}\left( {\left( {1.5 + 2\beta } \right) \xi + 2\psi + \beta {{\left( {SNR} \right) }^{ - 1}} + \beta \varphi {{\left( {SJR} \right) }^{ - 1}} - 2\beta {\eta ^2}} \right) \end{aligned}$$

where

$$\begin{aligned} {E_b} = 2\beta {E_c} \end{aligned}$$

Also, we define

$$\begin{aligned} SNR = {{{{N_0}}}/{{{E_b}}}} \end{aligned}$$
$$\begin{aligned} SJR = {{{{J_0}}}/{{{E_b}}}} \end{aligned}$$
$$\begin{aligned} {U_{syn}} \approx \left( {\sqrt{2\beta {E_b}} \eta ,{E_b}\left( {\left( {1.5 + 2\beta } \right) \xi + 2\psi + \beta {{\left( {SNR} \right) }^{ - 1}} + \beta \varphi {{\left( {SJR} \right) }^{ - 1}} - 2\beta {\eta ^2}} \right) } \right) \end{aligned}$$

Appendix B

$$\begin{aligned} {U_{syn\_int}} &= {{{\sqrt{{E_c}}}} /{m}}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {{{\left( {x_i^0} \right) }^2}\sum \limits _{j = 1}^h {\alpha _{ji}^{}\left( {m + 1 - j} \right) } } \right) } } \right) + {{{\sqrt{{E_c}}}}/ { m }}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^0x_i^0} \right) \sum \limits _{j = 1}^{h - 1} {\alpha _{ij}^{}j} } } \right) \\&{ { { + \sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {x_i^kx_i^0\sum \limits _{j = 1}^h {\alpha _{ji}^{}\left( {m + 1 - j} \right) } } } \right) + { { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^kx_i^0} \right) \sum \limits _{j = 1}^{h - 1} {\alpha _{ij}^{}j} } } \right) \\&+\, { { {\sqrt{{E_N}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {m\left( {n_i^{}.\,x_i^0} \right) } + { { {\sqrt{{E_J}} } } \big / { m }}\sum \limits _{i = 1}^{2\beta } {m\left( {J_i^{}.\,x_i^0} \right) } \\=\,\,& {} A' + B' + C' + D' + F' + G' \end{aligned}$$
$$\begin{aligned} E[A'^2]=\,\,& {} E\left[ {{{\left( {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {{{\left( {x_i^0} \right) }^2}} } \right) \left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) } \right) }^2}} \right] \\=\,\,& {} { { {{E_c}} } \big / { {{m^2}} }}\left( \begin{array}{l} 2\beta E\left[ {{{\left( {x_i^0} \right) }^4}} \right] E\left[ {{{\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) }^2}} \right] \\ + 2\beta \left( {2\beta - 1} \right) E\left[ {{{\left( {x_i^0} \right) }^2}} \right] E\left[ {{{\left( {x_i^0} \right) }^2}} \right] \\ E\left[ {\alpha _j^{}} \right] E\left[ {\alpha _p^{}} \right] \sum \limits _{j = 1}^h {\left( {m + 1 - j} \right) } \sum \limits _{p = 1}^h {\left( {m + 1 - p} \right) } \end{array} \right) \\ \\&\quad 2\beta E\left[ {{{\left( {x_i^0} \right) }^4}} \right] E\left[ {{{\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) }^2}} \right] = 2\beta E\left[ {{{\left( {x_i^0} \right) }^4}} \right] \left( \xi \right) \\ \\&\quad 2\beta \left( {2\beta - 1} \right) E\left[ {{{\left( {x_i^0} \right) }^2}} \right] E\left[ {{{\left( {x_i^0} \right) }^2}} \right] \sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } \\&\quad \sum \limits _{p = 1}^h {\alpha _p^{}\left( {m + 1 - p} \right) } = 2\beta \left( {2\beta - 1} \right) { { \pi } \big / { 4 }}{ { 1 } \big / { 4 }}{h^2}{\left( {h - 2m - 1} \right) ^2}\\ \\&\quad E[A'^2] = {E_c}3\beta \xi + {E_c}2\beta \left( {2\beta - 1} \right) {\eta ^2} \end{aligned}$$
$$\begin{aligned} E[B'^2]=\,\,& {} E\left[ {{{\left( {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {\left( {x_{i - 1}^0x_i^0} \right) \sum \limits _{j = 1}^{h - 1} {\alpha _{ij}^{}j} } } \right) } \right) }^2}} \right] \nonumber \\=\,\,& {} { { {{E_c}} } \big / { {{m^2}} }}\left( {2\beta E\left[ {{{\left( {x_i^0} \right) }^2}} \right] E\left[ {{{\left( {x_{i - 1}^0} \right) }^2}} \right] E\left[ {{{\left( {\sum \limits _{j = 1}^{h - 1} {\alpha _{ij}^{}j} } \right) }^2}} \right] } \right) = 2\beta {E_c}\psi \end{aligned}$$
$$\begin{aligned} E[C'^2]=\,\,& {} E\left[ {{{\left( {{ { {\sqrt{{E_c}} } } \big / { m }}\left( {\sum \limits _{i = 1}^{2\beta } {x_i^kx_i^0\sum \limits _{j = 1}^h {\alpha _{ji}^{}\left( {m + 1 - j} \right) } } } \right) } \right) }^2}} \right] \nonumber \\=\,\,& {} { { {{E_c}} } \big / { {{m^2}} }}\left( {2\beta E\left[ {x_i^kx_i^0} \right] E\left[ {{{\left( {\sum \limits _{j = 1}^h {\alpha _j^{}\left( {m + 1 - j} \right) } } \right) }^2}} \right] } \right) = 2\beta {E_c}\xi \end{aligned}$$
$$\begin{aligned} E[D_{}^2] = 2\beta {E_c}\psi \end{aligned}$$
$$\begin{aligned} {U_{syn\_int}} \approx \left( {\sqrt{2\beta {E_b}} ,{E_b}\left( {{ { 5 } \big / { 2 }}\xi + - {\eta ^2} + 2\psi + \beta {{\left( {SNR} \right) }^{ - 1}} + \beta \varphi {{\left( {SJR} \right) }^{ - 1}}} \right) } \right) \end{aligned}$$

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Tayebi, A., Berber, S. & Swain, A. A New Approach for Error Rate Analysis of Wide-Band DSSS-CDMA System with Imperfect Synchronization Under Jamming Attacks. Wireless Pers Commun 98, 3583–3610 (2018). https://doi.org/10.1007/s11277-017-5030-5

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