Performance of adaptive modulation with optimal switching thresholds for distributed antenna system in composite channels

Abstract

In this paper, the performance of distributed antenna system (DAS) with adaptive modulation (AM) over a composite fading channel which takes path loss, Rayleigh fading, and log-normal shadowing into account is studied. Based on target bit error rate (BER), the AM scheme for DAS with average BER constraints is presented. The optimum switching thresholds (STs) for attaining maximum spectrum efficiency (SE) are derived by using Lagrange optimization method. An effective iterative algorithm based on Newton method for finding the optimal STs is proposed. With these thresholds, the closed-form expression of SE and average BER are derived for performance evaluation. Simulation results for SE and BER are in good agreement with the theoretical analysis. The results show that DAS-AM with optimal STs has higher SE than that with conventional fixed thresholds. Moreover, the proposed AM can fulfill the target BER for different signal-to-noise ratios (SNRs).

Appendix

In this Appendix, we give the specific expression of Jacobi matrix DG(y ^(l)) in (29) by means of theoretical analysis and derivation. With (28), using the partial derivative operation, (29) can be rewritten as follows:

$$ DG\left({y}^{(l)}\right)=\left[\begin{array}{c}\hfill {\left[\nabla {G}_1\left({y}^{(l)}\right)\right]}^T\hfill \\ {}\hfill {\left[\nabla {G}_2\left({y}^{(l)}\right)\right]}^T\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\left[\nabla {G}_U\left({y}^{(l)}\right)\right]}^T\hfill \\ {}\hfill {\left[\nabla {G}_{U+1}\left({y}^{(l)}\right)\right]}^T\hfill \end{array}\right]=\left[\begin{array}{ccccc}\hfill \partial {G}_1/\partial {\tilde{\gamma}}_1^{(l)}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill \partial {G}_1/\partial {\eta}^{(l)}\hfill \\ {}\hfill 0\hfill & \hfill \partial {G}_2/\partial {\tilde{\gamma}}_2^{(l)}\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill \partial {G}_2/\partial {\eta}^{(l)}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \partial {G}_U/\partial {\tilde{\gamma}}_U^{(l)}\hfill & \hfill \partial {G}_U/\partial {\eta}^{(l)}\hfill \\ {}\hfill \partial {G}_{U+1}/\partial {\tilde{\gamma}}_1^{(l)}\hfill & \hfill \partial {G}_{U+1}/\partial {\tilde{\gamma}}_2^{(l)}\hfill & \hfill \cdots \hfill & \hfill \partial {G}_{U+1}/\partial {\tilde{\gamma}}_U^{(l)}\hfill & \hfill 0\hfill \end{array}\right] $$

(30)

Using (12) and (28), the nonzero elements of DG(y ^(l)) in (30) can be expressed as:

$$ \partial {G}_1/\partial {\tilde{\gamma}}_1^{(l)}=-{\eta}^{(l)}{b}_1{\alpha}_1\sqrt{\tau_1/\left(\pi {\tilde{\gamma}}_1^{(l)}\right) \exp \left(-{\tau}_1{\tilde{\gamma}}_1^{(l)}\right)} $$

(31)

$$ \partial {G}_1/\partial {\eta}^{(l)}={b}_1{\alpha}_1 erfc\left\{\sqrt{\tau_1{\tilde{\gamma}}_1^{(l)}}\right\}-{b}_1 BE{R}_0 $$

(32)

$$ \partial {G}_i/\partial {\tilde{\gamma}}_i^{(l)}={\eta}^{(l)}{\left(\pi {\tilde{\gamma}}_i^{(l)}\right)}^{-0.5}\left[{b}_{i-1}{\alpha}_{i-1}\sqrt{\tau_{i-1} \exp \left(-{\tau}_{i-1}{\tilde{\gamma}}_i^{(l)}\right)}-{b}_i{\alpha}_i\sqrt{\tau_i \exp \left(-{\tau}_i{\tilde{\gamma}}_i^{(l)}\right)}\right],\kern1em i=2,3,\dots, U. $$

(33)

$$ \partial {G}_i/\partial {\eta}^{(l)}=\left({b}_{i-1}-{b}_i\right) BE{R}_0+{b}_i{\alpha}_i erfc\left\{\sqrt{\tau_i{\gamma}_i^{(l)}}\right\}-{b}_{i-1}{\alpha}_{i-1} erfc\left\{\sqrt{\tau_{i-1}{\tilde{\gamma}}_i^{(l)}}\right\},\kern1em i=2,3,\dots, U. $$

(34)

$$ \partial {G}_{U+1}/\partial {\tilde{\gamma}}_1^{(l)}={b}_1\left[ BE{R}_0-{\alpha}_1 erfc\left\{\sqrt{\tau_1{\tilde{\gamma}}_1^{(l)}}\right\}\right]{f}_{\gamma}\left({\tilde{\gamma}}_1^{(l)}\right) $$

(35)

$$ \partial {G}_{U+1}/\partial {\tilde{\gamma}}_i^{(l)}=\left[\left({b}_i-{b}_{i-1}\right) BE{R}_0+{b}_{i-1}{\alpha}_{i-1} erfc\left\{\sqrt{\tau_{i-1}{\tilde{\gamma}}_i^{(l)}}\right\}-{b}_i{\alpha}_i erfc\left\{\sqrt{\tau_i{\tilde{\gamma}}_i^{(l)}}\right\}\right]{f}_{\gamma}\left({\tilde{\gamma}}_i^{(l)}\right),\kern1em i=2,3,\kern0.5em \dots, U. $$

(36)

The other elements are zero (as shown in (30)), where f _γ (r) is the PDF of γ (see (11)). According to the derived expressions above, the corresponding elements will be nonzero. Based on this, using the elementary row transformation of matrix or calculating the determinant of matrix, we can conclude that (30) is a full-rank matrix. Thus, the matrix DF(x ^(l)) will be invertible.