Analysis of Full-Duplex Downlink Using Diversity Gain

Abstract

In this paper, we carry out performance analysis of a multiuser full-duplex (FD) communication system. Multiple FD user equipments (UEs) share the same spectrum resources, simultaneously, at both the uplink and downlink. This results in co-channel interference (CCI) at the downlink of a UE from uplink signals of other UEs. This work proposes the use of diversity gain at the receiver to mitigate the effects of the CCI. For this an architecture for the FD eNodeB (eNB) and FD UE is proposed and corresponding downlink operation is described. Finally, the performance of the system is studied in terms of downlink capacity of a UE. It is shown that through the deployment of sufficient number of transmit and receive antennas at the eNB and UEs, respectively, significant improvement in performance can be achieved in the presence of CCI.

Appendix

The noise and interference in the Eq. (6) is given by:

$$\begin{aligned} {\mathbf {I}}_{l,k} = {\mathbf {n}}_{l,k} + \left[\mathop{\mathop {\sum} \limits_{q=l}} \limits_{q \ne l} ^{K} \sum _{\acute{k}=1}^{N_r} {\mathbf {h}}^u_{q,\acute{k},l,k} \otimes{\mathbf {s}}^u_{q,\acute{k}} \right] \alpha ^{k-1}_{q} \bigtriangleup _k \end{aligned}$$

(16)

After removing the CP and converting it to the frequency domain, we have:

$$\begin{aligned} {\tilde{\mathbf {I}}}_{l,k}&= {\mathbf {F}}_N {\mathbf {I}}_{l,k} \\&= {\tilde{\mathbf {n}}}_{l,k} +\left[ \mathop{ \mathop {\sum} \limits_{q=l}} \limits_{q \ne l}^{K} \sum_{\acute{k}=1}^{N_r} {\mathbf {H}}^u_{q,\acute{k},l,k} {\mathbf{x}}^u_{q,\acute{k}} \right] \alpha ^{k-1}_{q} \bigtriangleup _k\end{aligned}$$

(17)

where ${\mathbf {H}}^u_{q,\acute{k},l,k} = diag({\mathbf {F}}_N {\mathbf {h}}^u_{q,\acute{k},l,k})$ is the NXN diagonal matrix whose diagonal elements are frequency domain coefficients between $\acute{k}{\text{th}}$ transmit antenna of the $q{\hbox {th}}$ UE and $k{\hbox {th}}$ receive antenna of the $l{\hbox {th}}$ UE. ${\mathbf {x}}^u_{q,\acute{k}}$ is the transmit signal from $\acute{k}$ antenna of the $q{\hbox {th}}$ UE. This signal is then subjected to the subcarrier deallocation and after simplifications, similar to Eq. (8), we get:

$$\begin{aligned} {\bar{\mathbf {I}}}_l (m) = {\bar{\mathbf {n}}}_l(m) + \sum\limits_{{\mathop {q = l}\limits_{{q \ne l}} }}^{K} {\mathbf {\alpha }}_q {\mathbf {\bigtriangleup }} {\mathbf {H}}^u_{q,l}(m) {\bar{\mathbf {x}}}^u_q (m) \end{aligned}$$

(18)

where ${\bar{\mathbf {I}}}_l (m) = [\bar{I}_{l,1}(m), \bar{I}_{l,2}(m),\ldots ,\bar{I}_{l,N_r}(m)]^T$, ${\mathbf {\alpha }}_q = diag(\alpha _q^{0}, \alpha _q^{1}, \ldots , \alpha _q^{N_r-1})$ and ${\mathbf {\bigtriangleup }} = diag(\bigtriangleup _1, \bigtriangleup _2, \ldots , \bigtriangleup _{N_r})$. ${\mathbf {H}}^u_{q,l} (m)$ is the $N_r \times N_r$ frequency domain channel coefficient vector between the $l{\hbox {th}}$ U and the $q{\hbox {th}}$ UE on the $m{\hbox {th}}$ subcarrier. This signal for $k{\hbox {th}}$ antenna of the $l{\hbox {th}}$ UE is given by:

$$\begin{aligned} \bar{I}_{l,k}(m) = \bar{n}_{l,k}(m) + \left[\mathop{\mathop {\sum}\limits_{q=l}} \limits {q \ne l }^{K} \sum_{\acute{k}=1}^{N_r} H^u_{q,\acute{k},l,k}(m)\bar{x}^u_{q,\acute{k}}(m) \right] \alpha ^{k-1}_{q} \bigtriangleup_k \end{aligned}$$

(19)

Now, considering the fact that the $\bar{n}_{l,k}(m)$ and $H^u_{q,\acute{k},l,k}(m)$ are i.i.d, the power in the above signal is given by:

$$\begin{aligned} E[|\bar{I}_{l,k}(m)|^2]&= \sigma^2_{\bar{n}_{l}(m)} + \mathop{\mathop{\sum} \limits_{q=l}} \limits_{q \ne l}^{K} \sum _{\acute{k}=1}^{N_r} \sigma _q^2 (\beta ^q_mN_r^{-1}) \nonumber \\&= \sigma ^2_{\bar{n}_{l}(m)} + \mathop{\mathop{\sum}\limits_{q=l}} \limits_{q \ne l}^{K} \sigma _q^2 \beta^q_m, \forall l, k \end{aligned}$$

(20)

The signal is now subjected to post-processing from equation (10), .i.e, multiplying with $({\mathbf {U}}^d_{m,l})^H$:

$$\begin{aligned} {\tilde{\mathbf {I}}}_l(m) = ({\mathbf {U}}^d_{m,l})^H {\bar{\mathbf {I}}}_l(m) \end{aligned}$$

(21)

where ${\tilde{\mathbf {I}}}_l(m)=[{\tilde{I}}_{l,1}(m), {\tilde{I}}_{l,2}(m), \ldots ,{\tilde{I}}_{l,Q}(m)]^T$. As $Q=N_r$ and ${\mathbf {U}}^d_{m,l}$ is an unitary matrix, we have $|{\tilde{\mathbf {I}}}_l(m)|^2 = |{\bar{\mathbf {I}}}_l(m)|^2$. The power of the signal at the $j{\hbox {th}}$ data stream of the $l{\hbox {th}}$ UE is given by:

$$\begin{aligned} E[|\tilde{I}_{l,j}(m)|^2]&=E[|\bar{I}_{l,k}(m)|^2] \\&= \sigma^2_{\bar{n}_{l}(m)} +\mathop{\mathop {\sum} \limits_{q=l}} \limits_{q \ne l}^{K} \sigma _q^2 \beta ^q_m, \forall j \end{aligned}$$

(22)

The received signal for the $l{\hbox {th}}$ user at the $m{\hbox {th}}$ subcarrier obtained by MRC is given in Eq. (12) as:

$$\begin{aligned} \hat{\bar{x}}^d_l(m)&=(({\bar{\mathbf {E}}}^d_l(m))^H ({\bar{\mathbf {E}}}^d_l(m))^{-1} ({\bar{\mathbf {E}}}^d_l(m))^H {\tilde{\mathbf {y}}}^d_l(m) \nonumber \\&= \bar{x}_l^d(m) + (({\bar{\mathbf {E}}}^d_l(m))^H ({\bar{\mathbf {E}}}^d_l(m))^{-1} ({\bar{\mathbf {E}}}^d_l(m))^H {\tilde{\mathbf {I}}}_l(m) \end{aligned}$$

(23)

From this signal, the interference and noise term is represented by:

$$\begin{aligned} I_n = (({\bar{\mathbf {E}}}^d_l(m))^H ({\bar{\mathbf {E}}}^d_l(m))^{-1} ({\bar{\mathbf {E}}}^d_l(m))^H {\tilde{\mathbf {I}}}_l(m) \end{aligned}$$

(24)

As the elements of ${\tilde{\mathbf {I}}}_l(m)$ are i.i.d, the power in the above signal is given by:

$$\begin{aligned} E[|I_n|^2]&= [||{\mathbf{E}}_{m,l}^d||^2]^{-1} |\tilde{I}_{l,k}(m)|^2 \\&= [||{\mathbf{E}}_{m,l}^d||^2]^{-1} (\sigma ^2_{\bar{n}_{l}(m)} + \mathop{\mathop {\sum}\limits_{q=l}} \limits_ {q \ne l}^{K} \sigma _q^2 \beta^q_m) \end{aligned}$$

(25)

Hence, the effective SINR $(\gamma _{eff^m_l})$ for the $l{\hbox {th}}$ UE on the $m{\hbox {th}}$ subcarrier can now be given by:

$$\begin{aligned} \gamma _{eff^m_l}&= [\sigma_{\bar{n}_l}^2 + \mathop{\mathop{\sum} \limits_{q=l}} \limits_ {q \ne l}^{K} \sigma _q^2 \beta ^q_m ]^{-1}||{\bar{\mathbf{E}}}^d_{m,l}||^2 |\bar{x}_l^d (m)|^2 \\&= [\sigma_{\bar{n}_l}^2 +\mathop{\mathop{\sum} \limits_{q=l}} \limits_{ q \ne l}^{K} \sigma _q^2 \beta ^q_m ]^{-1}||{\bar{\mathbf{E}}}^d_{m,l}||^2 (\beta ^l_m Q^{-1}) \end{aligned}$$

(26)