Beam-Domain Full-Duplex Massive MIMO Transmission in the Cellular System

Abstract

Co-time co-frequency uplink and downlink (CCUD) transmission was considered challenging in the cellular system due to the strong self-interference (SI) between the transmitter and receiver of base station (BS). In this chapter, by investigating the beam-domain representation of channels based on the basis expansion model, we propose a beam-domain full-duplex (BDFD) massive multiple-input multiple-output (MIMO) scheme to make the CCUD transmission possible. The key idea of the BDFD scheme lies in intelligently scheduling the uplink and downlink user equipments (UEs) based on the beam-domain distributions of their associated channels to mitigate SI and enhance transmission efficiency. We show that the BDFD scheme achieves significant saving in uplink/downlink training resource and achieves the uplink and downlink sum capacities simultaneously as the number of BS antennas approaches to infinity. The superiority of the BDFD scheme over the traditional time-division duplex/frequency-division duplex massive MIMO is evaluated through simulation for the macro-cell environment. The results show that the spectral efficiency gain can even exceed in the specific scenarios, since the BDFD scheme utilizes the time-frequency resource more efficiently in both training and data transmission phases.

Appendix

As in [6], we assume that the number of channel paths is very large within the DOA/DOD regions. As a result, we can assume that the uplink/downlink channel is Gaussian distributed from the law of large numbers, i.e., ${\mathbf{h}}_{g_{u,k}}\sim \mathcal{CN}\left(0,{\mathbf{C}}_{g_{u,k}}\right)$ and ${\mathbf{h}}_{g_{d,k}}\sim \mathcal{CN}\left(0,{\mathbf{C}}_{g_{d,k}}\right)$. According to (1) and (2), the correlation matrices can be expressed as

$$ \kern0em {\mathbf{C}}_{g_{u,k}}=\mathbb{E}\left[{\mathbf{h}}_{g_{u,k}}{\mathbf{h}}_{g_{u,k}}^H\right]=\sum \limits_{i=1}^{M_u}{\int}_{\theta_{g_{u,k},i}^{\mathrm{min}}}^{\theta_{g_{u,k},i}^{\mathrm{max}}}\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right){S}_{g_{u,k},i}\left(\theta \right) d\theta $$

$$ {\mathbf{C}}_{g_{d,k}}=\mathbb{E}\left[{\mathbf{h}}_{g_{d,k}}{\mathbf{h}}_{g_{d,k}}^H\right]=\sum \limits_{i=1}^{M_d}{\int}_{\theta_{g_{d,k},i}^{\mathrm{min}}}^{\theta_{g_{d,k},i}^{\mathrm{max}}}\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right){S}_{g_{d,k},i}\left(\theta \right) d\theta $$

(41)

Based on the above assumption, we have ${\tilde{\mathbf{h}}}_{g_{u,k}}^{\left\{{B}_{g_u^{\prime }},:\right\}}\sim \mathcal{CN}\left(0,{\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u^{\prime }}\right)$ and ${\tilde{\mathbf{h}}}_{g_{d,k}}^{\left\{{B}_{g_d^{\prime },:}\right\}}\sim \mathcal{CN}\left(0,{\tilde{\mathbf{C}}}_{g_{d,k}}^{g_d^{\prime }}\right)$, where ${\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u^{\prime }}={\left({\mathbf{F}}^{\left\{:,{B}_{g_u^{\prime }}\right\}}\right)}^H{\mathbf{C}}_{g_{u,k}}{\mathbf{F}}^{\left\{:,{B}_{g_u^{\prime }}\right\}}$ and ${\tilde{\mathbf{C}}}_{g_{d,k}}^{g_d^{\prime }}={\left({\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}\right)}^H{\mathbf{C}}_{g_{d,k}}{\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}$. With this and the standard result for LMMSE estimator [32, Ch. 12], the LMMSE estimates for uplink and downlink channels have distributions ${\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}\sim \mathcal{CN}\left(0,{\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}\right)$ and ${\tilde{\mathbf{h}}}_{g_{d,k},\mathrm{LM}}^{\left\{{B}_{g_d},:\right\}}\sim CN\left(0,{\tilde{\mathbf{C}}}_{g_{d,k},\mathrm{LM}}^{g_d}\right)$, where the correlation matrices can be expressed as

$$ {\displaystyle \begin{array}{c}{\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}={\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u}\left(\frac{\sigma }{\tau_u{p}_u}{\mathbf{I}}_{b_u}+\sum \limits_{g_u^{\prime}\in {G}_u}{\tilde{\mathbf{C}}}_{g_{u,k}^{\prime}}^{g_u}\right.+\sum \limits_{g_d^{\prime },{g}_d^{{\prime\prime}}\in {G}_d}\sum \limits_{i=1}^{M_{SI}}\underset{\theta_{R,i}^{\mathrm{min}}}{\overset{\theta_{R,i}^{\mathrm{max}}}{\int }}\underset{\theta_{T,i}^{\mathrm{min}}}{\overset{\theta_{T,i}^{\mathrm{max}}}{\int }}{\mathbf{G}}_{\theta_R,{\theta}_T}^{\left\{{B}_{u,g},{B}_{d,{g}^{\prime }}\right\}}\\ {}{\left.\times {\Psi}_k{\left({\mathbf{G}}_{\theta_R,{\theta}_T}^{\left\{{B}_{u,g},{B}_{d,{g}^{{\prime\prime} }}\right\}}\right)}^H{S}_{SI,i}\left({\theta}_R,{\theta}_T\right)d{\theta}_Rd{\theta}_T\right)}^{-1}{\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u}\end{array}} $$

$$ {\displaystyle \begin{array}{c}{\tilde{\mathbf{C}}}_{g_{d,k},\mathrm{LM}}^{g_d}=\sum \limits_{g_d^{\prime}\in {G}_d}\sum \limits_{i=1}^{M_d}\underset{\theta_{g_{d,k},i}^{\mathrm{min}}}{\overset{\theta_{g_{d,k},i}^{\mathrm{max}}}{\int }}{\left({\mathbf{F}}^{\left\{:,{B}_{g_d}\right\}}\right)}^H\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right){\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}{S}_{g_{d,k},i}\left(\theta \right) d\theta \\ {}\times \left(\sum \limits_{g^{\prime },{g}^{{\prime\prime}}\in {G}_d}\sum \limits_{i=1}^{M_d}\underset{\theta_{g_{d,k},i}^{\mathrm{min}}}{\overset{\theta_{g_{d,k},i}^{\mathrm{max}}}{\int }}{\left({\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}\right)}^H\right.{\left.\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right){\mathbf{F}}^{\left\{:,{B}_{g_d^{{\prime\prime} }}\right\}}{S}_{g_{d,k},i}\left(\theta \right) d\theta {+}\frac{\sigma }{\tau_d{p}_d}{\mathbf{I}}_{b_d}\right)}^{-1}\\ {}\times \sum \limits_{g_d^{\prime}\in {G}_d}\sum \limits_{i=1}^{M_d}\underset{\theta_{g_{d,k},i}^{\mathrm{min}}}{\overset{\theta_{g_{d,k},i}^{\mathrm{max}}}{\int }}{\left({\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}\right)}^H\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right){\mathbf{F}}^{\left\{:,{B}_{g_d}\right\}}{S}_{g_{d,k},i}\left(\theta \right) d\theta \end{array}} $$

(42)

Moreover, as N → ∞, we have [3]

$$ \frac{1}{N}{\left({\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}\right)}^H{\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}=\frac{1}{N}\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}\right) $$

$$ \frac{1}{N}{\left({\tilde{\mathbf{h}}}_{g_{d,k},\mathrm{LM}}^{\left\{{B}_{g_d},:\right\}}\right)}^H{\tilde{\mathbf{h}}}_{g_{d,k},\mathrm{LM}}^{\left\{{B}_{g_d},:\right\}}=\frac{1}{N}\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{d,k},\mathrm{LM}}^{g_d}\right) $$

(43)

With (41), (42), and (43), the scaling behaviors for average powers of useful signal, channel estimation, IUI, and IGI can be readily obtained by using the technique in [38, Proof of Theorem 4] as shown in (44) at the bottom of the page. Then we focus on the scaling behavior of SI power. According to Table 3, the asymptotic result in (43) and the property tr(AB) = tr(BA), we can rewrite the average SI power as

$$ {\begin{aligned}&\mathbb{E}\left[{\mathrm{SI}}_{g_{u,k}}\right]={\left(\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}\right)\right)}^{-2}\sum \limits_{g_d^{\prime}\in {G}_d}\sum \limits_{k^{\prime }=1}^{K_{g_d}}{p}_{g_{d,{k}^{\prime}}^{\prime }}{\left(\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{d,{k}^{\prime}}^{\prime },\mathrm{LM}}^{g_d^{\prime }}\right)\right)}^{-1}\\ &{\times}\underset{X_{g_{u,k},{g}_{d,k}^{\prime }}}{\underbrace{\mathrm{tr}\!\left(\!{\mathbb{E}}\!\left[{\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}{\left({\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}\right)}^H\!{\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}{\tilde{\mathbf{h}}}_{g_{d,{k}^{\prime}}^{\prime },\mathrm{LM}}^{\left\{{B}_{g_d^{\prime }},:\right\}}{\left({\tilde{\mathbf{h}}}_{g_{d,{k}^{\prime}}^{\prime },\mathrm{LM}}^{\left\{{B}_{g_d^{\prime }},:\right\}}\right)}^H{\left({\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}\right)}^H\right]\right)}}\end{aligned}} $$

(44)

The expression of ${X}_{g_{u,k},{g}_{d,k}^{\prime }}$ can be rewritten as

$$ {\displaystyle \begin{array}{c}{X}_{g_{u,k},{g}_{d,k}^{\prime }}=\mathrm{tr}\left(\mathbb{E}\left[{\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}{\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}{\tilde{\mathbf{C}}}_{g_{d,{k}^{\prime }},\mathrm{LM}}^{g_d}{\left({\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}\right)}^H\right]\right)\\ {}\le \mathrm{tr}\left(\mathbb{E}\left[{\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u}{\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}{\tilde{\mathbf{C}}}_{g_{d,{k}^{\prime}}}^{g_d}{\left({\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}\right)}^H\right]\right)\\ {}=\mathrm{tr}\left(\kern0em {\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u}\kern0em \sum \limits_{i=1}^{M_{SI}}\kern0em \underset{\theta_{R,i}^{\mathrm{min}}}{\overset{\theta_{R,i}^{\mathrm{max}}}{\int }}\kern0em \underset{\theta_{T,i}^{\mathrm{min}}}{\overset{\theta_{T,i}^{\mathrm{max}}}{\int }}\kern0em {S}_{SI,i}\left({\theta}_R,{\theta}_T\right){\left({\mathbf{F}}^{\left\{:,{B}_{g_u}\right\}}\right)}^H\kern0em \mathbf{a}\left({\theta}_R\right){\mathbf{a}}^H\left({\theta}_T\right)\right.\\ {}\kern0.75em \left.\times {\mathbf{F}}^{\left\{:,{B}_{d,g}\right\}}{\tilde{\mathbf{C}}}_{g_{d,{k}^{\prime}}^{\prime}}^{g_d^{\prime }}{\left({\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}\right)}^H\mathbf{a}\left({\theta}_T\right){\mathbf{a}}^H\left({\theta}_R\right){\mathbf{F}}^{\left\{:,{B}_{g_u}\right\}}d{\theta}_Rd{\theta}_T\right)\\ {}\le {I}_R\times {I}_T\underset{\theta_R\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right],{\theta}_T\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{T,i}^{\mathrm{min}},{\theta}_{T,i}^{\mathrm{max}}\right]}{\max }{S}_{SI,i}\left({\theta}_R,{\theta}_T\right)\end{array}} $$

(45)

where

$$ {I}_R=\underset{\theta_R\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right]}{\int}\kern0em {\mathbf{a}}^H\left({\theta}_R\right){\mathbf{F}}^{\left\{:,{B}_{g_u}\right\}}{\tilde{\mathbf{C}}}_{g_{u,k}}^{g_u}{\left({\mathbf{F}}^{\left\{:,{B}_{g_u}\right\}}\right)}^H\mathbf{a}\left({\theta}_R\right)d{\theta}_R $$

$$ {I}_T=\underset{\theta_T\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{T,i}^{\mathrm{min}},{\theta}_{T,i}^{\mathrm{max}}\right]}{\int}\kern0em {\mathbf{a}}^H\left({\theta}_T\right){\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}{\tilde{\mathbf{C}}}_{g_{d^{\prime },{k}^{\prime}}}^{g_d^{\prime }}{\left({\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}\right)}^H\mathbf{a}\left({\theta}_T\right)d{\theta}_T $$

(46)

In (45), the first step is based on the independence between ${\tilde{\mathbf{h}}}_{g_{u,k},\mathrm{LM}}^{\left\{{B}_{g_u},:\right\}}$, ${\tilde{\mathbf{h}}}_{g_{d,k}^{\prime },\mathrm{LM}}^{\left\{{B}_{g_d^{\prime }},:\right\}}$, and ${\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}$. The second step is based on the relation ${\tilde{\mathbf{C}}}_{g_{m,k},\mathrm{LM}}^{g_m}={\tilde{\mathbf{C}}}_{g_{m,k}}^{g_m}-\left({\tilde{\mathbf{C}}}_{g_{m,k}}^{g_m}-{\tilde{\mathbf{C}}}_{g_{m,k},\mathrm{LM}}^{g_m}\right)$(m ∈ {u, d}) and the positive definiteness of ${\tilde{\mathbf{C}}}_{g_{m,k}}^{g_m}$ and ${\tilde{\mathbf{C}}}_{g_{m,k}}^{g_m}-{\tilde{\mathbf{C}}}_{g_{m,k},\mathrm{LM}}^{g_m}$. The third step is obtained by using the equation ${\tilde{\mathbf{H}}}_{SI}^{\left\{{B}_{g_u},{B}_{g_d^{\prime }}\right\}}={\left({\mathbf{F}}^{\left\{:,{B}_{g_u}\right\}}\right)}^H{\mathbf{H}}_{SI}{\mathbf{F}}^{\left\{:,{B}_{g_d^{\prime }}\right\}}$ and the SI channel model (3).

By substituting (41) into (46), the integral I _R can be rewritten as

$$ {\displaystyle \begin{array}{c}\frac{1}{N}{I}_R=\frac{1}{N}\sum \limits_{i=1}^{M_u}\underset{\theta_{g_{u,k},i}^{\mathrm{min}}}{\overset{\theta_{g_{u,k},i}^{\mathrm{max}}}{\int }}{S}_{g_{u,k},i}\left(\theta \right)\underset{\theta_R\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right]}{\int }{\mathbf{a}}^H\left({\theta}_R\right)\mathbf{a}\left(\theta \right){\mathbf{a}}^H\left(\theta \right)\mathbf{a}\left({\theta}_R\right)d{\theta}_R d\theta \\ {}=N\sum \limits_{i=1}^{M_u}\underset{\theta_{g_{u,k},i}^{\mathrm{min}}}{\overset{\theta_{g_{u,k},i}^{\mathrm{max}}}{\int }}{S}_{g_{u,k},i}\left(\theta \right)\underset{\theta_R\in \underset{i=1}{\overset{M_{SI}}{\cup }}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right]}{\int }{\mathrm{asinc}}_N^2\left(\frac{d}{\lambda}\sin {\theta}_R-\frac{d}{\lambda}\sin \theta \right)d{\theta}_R d\theta \end{array}} $$

(47)

Note that according to Lemma 1 and Lemma 2, we have ${\cap}_{i=1}^{M_u}\left[{\theta}_{g_{u,k},i}^{\mathrm{min}},{\theta}_{g_{u,k},i}^{\mathrm{max}}\right]\cap {\cap}_{i=1}^{M_{SI}}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right]=\varnothing$ or ${\cap}_{i=1}^{M_d}\left[{\theta}_{g_{d,k},i}^{\mathrm{min}},{\theta}_{g_{d,k},i}^{\mathrm{max}}\right]\cap {\cap}_{i=1}^{M_{SI}}\left[{\theta}_{T,i}^{\mathrm{min}},{\theta}_{T,i}^{\mathrm{max}}\right]=\varnothing$, otherwise, Criterion 2 will be violated. If ${\cap}_{i=1}^{M_u}\left[{\theta}_{g_{u,k},i}^{\mathrm{min}},{\theta}_{g_{u,k},i}^{\mathrm{max}}\right]\cap {\cap}_{i=1}^{M_{SI}}\left[{\theta}_{R,i}^{\mathrm{min}},{\theta}_{R,i}^{\mathrm{max}}\right]=\varnothing$, with a same procedure as that in the proof of Lemma 1, we can obtain $\frac{1}{N}{I}_R<\mathcal{O}(1)$ or ${I}_R<\mathcal{O}(N)$ as N → ∞; otherwise, ${I}_R=\mathcal{O}(N)$. In the same way, we can prove that ${I}_T<\mathcal{O}(N)$ if ${\cap}_{i=1}^{M_d}\left[{\theta}_{g_{d,k},i}^{\mathrm{min}},{\theta}_{g_{d,k},i}^{\mathrm{max}}\right]\cap {\cap}_{i=1}^{M_{SI}}\left[{\theta}_{T,i}^{\mathrm{min}},{\theta}_{T,i}^{\mathrm{max}}\right]=\varnothing$, and ${I}_T=\mathcal{O}(N)$ otherwise. Combining the results in the above, we have ${I}_R\times {I}_T<\mathcal{O}\left({N}^2\right)$. Moreover, it has been shown in [38] that $\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{u,k},\mathrm{LM}}^{g_u}\right)$ and $\mathrm{tr}\left({\tilde{\mathbf{C}}}_{g_{d^{\prime },{k}^{\prime }},\mathrm{LM}}^{g_d^{\prime }}\right)$ scale with $\mathcal{O}(N)$ as N → ∞. Using these results on (44), we have $\mathbb{E}\left[{\mathrm{SI}}_{g_{u,k}}\right]<\mathcal{O}\left({N}^{-1}\right)$, which is exactly the result in Table 3.