Performance Analysis of Adaptive Chaos Based CDMA System with Antenna Diversity in Frequency Selective Channel

Abstract

This paper presents the theoretical analysis of the adaptive multiuser DS-CDMA system with antenna diversity in frequency selective channel environment. Pilot symbols are added to users’ data to estimate the complex fading coefficients. Least mean square algorithm is used for estimating the channel coefficients. Performance of receiver antenna diversity using maximal ratio combining is investigated. Effect of channel estimation and intersymbol interference is studied analytically and probability of error in closed form is derived. Under the perfect synchronization assumption, simulation results show that the performance of system deteriorates with increase in number of multipaths components and can be improved using antenna diversity.

Appendices

Appendix 1: LMS Algorithm

In this research work, extracted pilot symbols are normalized by $\sum \nolimits _{k = 2(i - 1)\beta +1 }^{k = 2i\beta } {\left( {x_k^{p} } \right) ^2 }$before estimation process. Therefore LMS algorithm for this particular system can be represented as

$$\begin{aligned}&e_{i,m} = z_{i}^{mp}\bigg /\sum \limits _{k = 2(i - 1)\beta +1 }^{k = 2i\beta } {\left( {x_k^{p} } \right) ^2 } - \hat{a}_{i,m} e^{j\hat{\phi }_{i,m} } \end{aligned}$$

(21)

$$\begin{aligned}&\hat{a}_{i + 1,m} e^{j\hat{\phi }_{i + 1,m} } = \hat{a}_{i,m} e^{j\hat{\phi }_i } + \mu e_{i,m} \end{aligned}$$

(22)

where $z_{i}^{mp}$ is the extracted pilot symbol. $e_{i,m}$ and $\mu$ are known as error and step size in estimation theory [13].

Appendix 2: Derivation of Eq. (12)

Since the chaotic spreading sequences and AWGN noise are zero mean uncorrelated processes. Hence

$$\begin{aligned}&{E\left[ {Z_i^{g}\;\left| {\;\gamma _i^{g} = 1} \right. } \right] = 2\beta E\left[ {{{\left( {x_k^{g}} \right) }^2}} \right] \times }\nonumber \\&\quad {\sum \limits _{m = 1}^M {{a_{i,m,0}}{{\hat{a}}_{i,m,0}}\cos ({\phi _{i,m,0}} - {{\hat{\phi }}_{i,m,0}})} }\nonumber \\&\quad { = 2\beta {P_c}\sum \limits _{m = 1}^M {{a_{i,m,0}}{{\hat{a}}_{i,m,0}}\cos ({\phi _{i,m,0}} - {{\hat{\phi }}_{i,m,0}})} } \end{aligned}$$

(23)

All the four terms in (5) are uncorrelated to each other, hence ${\mathop {\mathrm{var}}} \left[ {Z_i^{g} \;\left| {\;\gamma _i^{g} = 1} \right. } \right]$ is given by

$$\begin{aligned} {\mathrm{var[}}Z_i^{g}\;\left| {\;\gamma _i^{g}=1} \right. ] = {\mathrm{var[}}Z_i^{ag}] + {\mathrm{var[}}Z_i^{bg}] + {\mathrm{var[}}Z_i^{cg}]+{\mathrm{var[}}Z_i^{dg}] \end{aligned}$$

(24)

Variance of the first term $Z_i^{ag}$ can be written as

$${\mathop {\mathrm{var}}} Z_i^{ag} = 2\beta \left[ {{\mathrm{var}}{{\left( {x_k^g} \right) }^2}} \right] {\left\{ {\sum \limits _{m = 1}^M {{a_{i,m,0}}{{\hat{a}}_{i,m,0}}\cos ({\phi _{i,m,0}} - {{\hat{\phi }}_{i,m,0}})} } \right\} ^2} $$

(25)

Channel is assume to slowly varying therefore we can assume that ${h_{i - {\tau _l},m,l}} \approx {h_{i,m,l}}$. Under this assumption, since chaotic sequences are uncorrelated with it’s shifted version therefore

$${\mathop {\mathrm{var}}} Z_i^{bg} = 2\beta P_c^2{\left\{ {\sum \limits _{m = 1}^M {\sum \limits _{l = 1}^{L - 1} {{a_{i,m,l}}{{\hat{a}}_{i,m,0}}\cos ({\phi _{i,m,l}} - {{\hat{\phi }}_{i,m,0}})} } } \right\} ^2} $$

(26)

Since all the chaotic sequences are uncorrelated hence variance of $Z_i^{cg}$ will be

$${\mathop {\mathrm{var}}} Z_i^{cg} = 2\beta NP_c^2{\left\{ {\sum \limits _{m = 1}^M {\sum \limits _{l = 1}^{L - 1} {{a_{i,m,l}}{{\hat{a}}_{i,m,0}}\cos ({\phi _{i,m,l}} - {{\hat{\phi }}_{i,m,0}})} } } \right\} ^2} $$

(27)

Real and Imaginary part of complex AWGN have power spectral density equal to $N_0 /2$ and independent to chaotic sequences, therefore variance of $Z_i^{dg}$ can be defined as

$$\begin{aligned} {\mathop {\mathrm{var}}} Z_i^{dg}&= {\mathrm{var}}\left[ {\sum \limits _{m = 1}^M {{{\hat{a}}_{i,m,0}}\cos ({{\hat{\phi }}_{i,m,0}})} } \right. \sum \limits _{k = 2(i - 1)\beta + 1}^{2i\beta } {{\mathrm{Re}}({\xi _{k,m}})x_k^g} \nonumber \\ & \quad+ \sum \limits _{m = 1}^M {{{\hat{a}}_{i,m,0}}\sin ({{\hat{\phi }}_{i,m,0}})} \left. {\sum \limits _{k = 2(i - 1)\beta + 1}^{2i\beta } {{\mathrm{Im}}({\xi _{k,m}})x_k^g} } \right] \nonumber \\&= \beta {N_0}{P_c}\sum \limits _{m = 1}^M {\hat{a}_{i,m,0}^2} \end{aligned}$$

(28)

Putting Eqs. (23), (25), (26), (27) and (28) in Eq. (11) and after rearranging the equation we have Eq. (12).