Code-Time Diversity for Direct Sequence Spread Spectrum Systems

Abstract

Time diversity is achieved in direct sequence spread spectrum by receiving different faded delayed copies of the transmitted symbols from different uncorrelated channel paths when the transmission signal bandwidth is greater than the coherence bandwidth of the channel. In this paper, a new time diversity scheme is proposed for spread spectrum systems. It is called code-time diversity. In this new scheme, N spreading codes are used to transmit one data symbol over N successive symbols interval. The diversity order in the proposed scheme equals to the number of the used spreading codes N multiplied by the number of the uncorrelated paths of the channel L. The paper represents the transmitted signal model. Two demodulators structures will be proposed based on the received signal models from Rayleigh flat and frequency selective fading channels. Probability of error in the proposed diversity scheme is also calculated for the same two fading channels. Finally, simulation results are represented and compared with that of maximal ration combiner and multiple-input and multiple-output systems.

Appendices

Appendix 1

The generalized hypergeometric function _pF_q has a series expansion as shown in the following equation.

$$ _{\text{p}} {\text{F}}_{\text{q}} \left( {\left\{ {a_{1 ,} a_{2 ,} \ldots , a_{p } } \right\};\left\{ {b_{1 ,} b_{2 ,} \ldots , b_{q} } \right\};z} \right) = \mathop \sum \limits_{k = 0}^{\infty } \frac{{\left( {a_{1} } \right)_{k} \cdots \left( {a_{p} } \right)_{k} }}{{\left( {b_{1} } \right)_{k} \cdots \left( {b_{q} } \right)_{k} }}.\frac{{z^{k} }}{k!} $$

This mathematical function is suitable for both symbolic and numerical manipulation. (a) _k is the Pochhammer symbol defined as:

$$ \left( a \right)_{k} = \frac{{\Gamma \left( {a + k} \right)}}{\Gamma \left( a \right)} $$

Appendix 2

Starting from Eq. (38)

$$ \begin{aligned} \bar{P}_{QAM} & = \mathop \int \limits_{0}^{\infty } P_{QAM} .p\left( {SINR\left( {kT_{s} } \right)} \right).dSINR \\ & =&\,2\left( {1 - \frac{1}{\sqrt M }} \right)\frac{2}{\Gamma \left( m \right)}.\left( {\frac{m}{\Omega }} \right)^{m} \mathop \int \limits_{0}^{\infty } erfc\left( {\sqrt {\frac{3}{{2\left( {M - 1} \right)}}SINR} } \right)SINR^{2m - 1} e^{{ - \left( {\frac{m}{\Omega }} \right)SINR^{2} }} .dSINR \\ \end{aligned} $$

(61)

From [26] the erfc() can be approximated to:

$$ erfc\left( x \right) \approx \frac{1}{6}e^{{ - x^{2} }} + \frac{1}{2}e^{{ - \frac{4}{3}x^{2} }} $$

(62)

$$ \bar{P}_{QAM} = 2\left( {1 - \frac{1}{\sqrt M }} \right)\frac{2}{\Gamma \left( m \right)}.\left( {\frac{m}{\Omega }} \right)^{m} \left[ {\mathop \int \limits_{0}^{\infty } \frac{1}{6}SINR^{2m - 1} e^{{ - \left( {\frac{m}{\Omega }} \right)SINR^{2} }} e^{{ - \left( \frac{a}{2} \right)SINR}} .dSINR + \mathop \int \limits_{0}^{\infty } \frac{1}{2}SINR^{2m - 1} e^{{ - \left( {\frac{m}{\Omega }} \right)SINR^{2} }} e^{{ - \left( \frac{2a}{3} \right)SINR}} .dSINR} \right] $$

(63)

$$ where\quad a = \frac{3}{M - 1} $$

From 3.462 in [25]

$$ \mathop \int \limits_{0}^{\infty } x^{v - 1} .e^{{ - Bx^{2} }} .e^{ - \gamma x} .dx = \left( {2B} \right)^{{ - \frac{v}{2}}} \Gamma \left( v \right)e^{{\left( {\frac{{\gamma^{2} }}{8B}} \right)}} D_{ - v} \left( {\frac{\gamma }{{\sqrt {2B} }}} \right) $$

(64)

Referring to equation (63)

$$ v = 2m , \quad B = \frac{2}{\Omega } , \quad \gamma = \frac{a}{2}\,for\, the\, first\, integral\quad and\quad \gamma = \frac{2a}{3}\,for\, the\, second\, one. $$

By substituting equation (64) in (63), the integration can be solved and the final value of the average probability of error can be represented as:

$$ \bar{P}_{QAM} = \left( {1 - \frac{1}{\sqrt M }} \right)\frac{{2^{1 - m} \Gamma \left( {2m} \right)}}{\Gamma \left( m \right)}.\left( {\frac{1}{3}.e^{{\left( {\frac{9}{{32\left( {M - 1} \right)^{2} }}.\frac{\Omega }{m}} \right)}} .D_{ - 2m} \left( {\sqrt {\frac{9}{{8\left( {M - 1} \right)^{2} }}.\frac{\Omega }{m}} } \right) + e^{{\left( {\frac{1}{{2\left( {M - 1} \right)^{2} }}.\frac{\Omega }{m}} \right)}} .D_{ - 2m} \left( {\sqrt {\frac{2}{{\left( {M - 1} \right)^{2} }}.\frac{\Omega }{m}} } \right)} \right) $$

(65)