A new flower pollination algorithm for equalization in synchronous DS/CDMA multiuser communication systems

Abstract

This work proposes a modified version of an emerging nature-inspired technique, named flower pollination algorithm, for equalizing digital multiuser channels. This equalization involves two different tasks: (1) estimation of the channel impulse response, and (2) estimation of the users’ transmitted symbols. The new algorithm is developed and applied in a direct sequence/code-division multiple-access multiuser communications system. Important issues such as robustness, convergence speed and population diversity control have been in deep investigated. A method based on the entropy of the flowers’ fitness is proposed for in-service monitoring and adjusting population diversity. Numerical simulations analyze the performance, showing comparisons with well-known conventional multiuser detectors such as matched filter, minimum mean square error estimator or several Bayesian schemes, as well as with other nature-inspired strategies. Numerical analysis shows that the proposed algorithm enables transmission at higher symbol rates under stronger fading and interference conditions, constituting an attractive alternative to previous algorithms, both conventional and nature-inspired, whose performance is frequently sensible to near–far effects and multiple-access interference problems. These results have been validated by running hypothesis tests to confirm statistical significance.

A Particularization to a chip-rate algorithm

In this case, taking into account that the energy of the chips is $\mathcal {E}_\gamma = \int _0^{T_\mathrm{c}} |\gamma (t)|^2 \hbox {d}t$, a sequence of N samples is obtained for each symbol, whose components are calculated as

$$\begin{aligned} r_{n,j}= & {} \int _{nT+jT_\mathrm{c}}^{nT+(j+1)T_\mathrm{c}} r(t) \gamma (t-nT-jT_\mathrm{c}) \hbox {d}t \nonumber \\= & {} \sum _{u=1}^U a_u(n) d_u(n) s_u(j) \mathcal {E}_\gamma + \int _0^{T_\mathrm{c}} g(t+nT-jT_\mathrm{c}) \gamma (t) \hbox {d}t, \nonumber \\&\qquad \qquad j=0,1,\dots ,N-1 \end{aligned}$$

(19)

This set of N samples is normalized by the chip energy $\mathcal {E}_\gamma$ and grouped into a vector $\mathbf{r}_n^\mathrm{chip}$. Thus, we can write

$$\begin{aligned} \mathbf{r}_n^\mathrm{chip} = \sum _{i=1}^U \mathbf{s}_i a_i(n) d_i(n) + \mathbf{g}(n) = \mathbf{SA}{} \mathbf{d}(n) + \mathbf{g}(n) \end{aligned}$$

(20)

where $\mathbf{S}=[\mathbf{s}_1,\mathbf{s}_2,\dots ,\mathbf{s}_U]$ is the $N\times U$ matrix whose columns contain the users’ signatures $\mathbf{s}_i$, and $\mathbf{g}(n) = [g_{n,0}, g_{n,1},\dots , g_{n,L-1}]^\mathrm{T}$ stands for the normalized noise vector with components

$$\begin{aligned} g_{n,j} = \frac{1}{\mathcal {E}_\gamma } \int _0^{T_\mathrm{c}} g(t+nT+j T_\mathrm{c}) \gamma (t) \hbox {d}t, \qquad j=0,1,\dots ,N-1 \end{aligned}$$

(21)

Since g represents a zero-mean, white and Gaussian noise process, its covariance matrix is $E\{ \mathbf{g}(n) \mathbf{g}(n)^H \} = \sigma ^2 \mathbf{I}_N$, with $\mathbf{I}_N$ being the $N \times N$ identity matrix.