Performance analysis of cross layer design with imperfect channel information in distributed antenna systems

Abstract

In this paper, based on the imperfect channel state information (CSI), a cross layer design (CLD) scheme is developed for distributed antenna system (DAS) by combining adaptive modulation (AM) at the physical layer and automatic repeat request (ARQ) at the data link layer. The performance of DAS with CLD is investigated over composite fading channel which considers large-scale path loss and small-scale Rayleigh fading. With the performance analysis, the probability density function of the estimated signal-to-noise ratio (SNR) is derived, and then, the switching thresholds under a target packet error rate constraint are further derived. According to these results, and using numerical calculation, the closed form analytical expressions of average packet error rate and spectrum efficiency of DAS with CLD are, respectively, achieved, which will provide better evaluation way for the DAS performance. To decrease the performance loss caused by the conventional single estimation in the presence of imperfect CSI, the multi-estimation method is proposed to increase the system performance by exploiting previous channel estimation information. Numerical results corroborate our theoretical analysis, and the simulation is in consistence with the theoretical result. Moreover, the system performance can be increased by decreasing the estimation error and/or path loss. Especially, the multi-estimation method can enhance the performance effectively and enable the system to tolerate large estimation errors.

Appendix

In this appendix, we give the derivation of (17). According to the total probability theorem [35], ${\overline{PER}}_n$ can be calculated as

$$ {\overline{PER}}_n={\int}_{\gamma_n}^{\gamma_{n+1}}{\int}_0^{+\infty }{\sum}_{i=1}^{N_t}{PER}_{M_n}\left(\gamma \right){f}_{\gamma_i\mid {\hat{\gamma}}_i}\left(\gamma |\hat{\gamma}\right){\mathrm{P}}_{\mathrm{r}}\left({A}_i\right){f}_{{\hat{\gamma}}_i}\left({\hat{\gamma}}_i=\hat{\gamma}|{A}_i\right) d\gamma d\hat{\gamma} $$

(A1)

where A_i is defined as the event that the i-th RA is the selected transmit antenna, and P_r(A_i) = 1/N_t according to fair selection properties.${f}_{{\hat{\gamma}}_i}\left({\hat{\gamma}}_i=\hat{\gamma}|{A}_i\right)$ can be expressed as

$$ {f}_{{\hat{\gamma}}_i}\left({\hat{\gamma}}_i=\hat{\gamma}|{A}_i\right)={dF}_{{\hat{\gamma}}_i}\left(\hat{\gamma}|{A}_i\right)/d\hat{\gamma} $$

(A2)

where ${F}_{{\hat{\gamma}}_i}\left(\hat{\gamma}|{A}_i\right)$ can be given by

$$ {F}_{{\hat{\gamma}}_i}\left(\hat{\gamma}|{A}_i\right)={\mathrm{P}}_{\mathrm{r}}\left({\hat{\gamma}}_i<\hat{\gamma}|{A}_i\right)=\frac{{\mathrm{P}}_{\mathrm{r}}\left({\hat{\gamma}}_i<\hat{\gamma},{A}_i\right)}{{\mathrm{P}}_{\mathrm{r}}\left({A}_i\right)}={N}_t\ {\mathrm{P}}_{\mathrm{r}}\left({\hat{\gamma}}_i<\hat{\gamma},\kern0.5em {\hat{\gamma}}_i\max \right) $$

(A3)

where ${\mathrm{P}}_{\mathrm{r}}\left({\hat{\gamma}}_i<\hat{\gamma},\kern0.5em {\hat{\gamma}}_i\max \right)$ is written as [36]

$$ {\mathrm{P}}_{\mathrm{r}}\left({\hat{\gamma}}_i<\hat{\gamma},\kern0.5em {\hat{\gamma}}_i\max \right)={\int}_0^{\hat{\gamma}}{f}_{{\hat{\gamma}}_i}(y)\prod \limits_{p=1,p\ne i}^{N_t}{F}_{{\hat{\gamma}}_p}(y) dy. $$

(A4)

Substituting (A4) and (A3) into (A2) yields

$$ {f}_{{\hat{\gamma}}_i}\left({\hat{\gamma}}_i=\hat{\gamma}|{A}_i\right)={N}_t{f}_{{\hat{\gamma}}_i}\left(\hat{\gamma}\right)\prod \limits_{p=1,p\ne i}^{N_t}{F}_{{\hat{\gamma}}_p}\left(\hat{\gamma}\right). $$

(A5)

Then, substituting (A5) into (A1) yields

$$ {\overline{PER}}_n={\sum}_{i=1}^{N_t}{\int}_{\gamma_n}^{\gamma_{n+1}}{\int}_0^{\infty }{PER}_{M_n}\left(\gamma \right){f}_{\gamma_i\mid {\hat{\gamma}}_i}\left(\gamma |\hat{\gamma}\right){f}_{{\hat{\gamma}}_i}\left(\hat{\gamma}\right)\prod \limits_{p=1,p\ne i}^{N_t}{F}_{{\hat{\gamma}}_p}\left(\hat{\gamma}\right) d\gamma d\hat{\gamma}. $$

(A6)