We consider, as shown in Fig. 1, a FD CF mMIMO system where M
FD APs serve K = (K_{u} + K_{d})
single-antenna HD UEs on the same spectral resource, with K_{u}
and K_{d}
being the number of uplink and downlink UEs, respectively. Each AP has N_{t}
transmit and N_{r}
receive antennas, and is connected to the CPU using a limited-capacity fronthaul link which carries quantized uplink/downlink information to/from the CPU. We see from Fig. 1 that due to FD model
uplink receive signal of each AP is interfered by its own downlink transmit signal and that of other APs. These intra- and inter-AP interferences are shown using purple and brown dashed lines, respectively.
downlink UEs receive transmit signals from uplink UEs, causing uplink downlink interference (UDI) (shown as black dotted lines between uplink and downlink UEs). Additionally, the UEs experience multi-UE interference (MUI) as the APs serve them on the same spectral resource.
We next explain various channels, their estimation and data transmission. We assume a coherence interval of duration
T_{c}
(in s) with
\tau _{c}
samples, which is divided into: a) channel estimation phase of
\tau _{t}
samples, and b) downlink and uplink data transmission of (
\tau _{c}
-
\tau _{t}
) samples.
A. Channel Description
The channel of the k
th downlink UE to the transmit antennas of the m
th AP is \boldsymbol {g}^{d}_{mk} \in \mathbb {C}^{N_{t} \times 1}
, while the channel from the l
th uplink UE to the receive antennas of the m
th AP is \boldsymbol {g}^{u}_{ml} \in \mathbb {C}^{N_{r} \times 1}
. We model these channels as \boldsymbol {g}^{d}_{mk} = (\beta ^{d}_{mk})^{1/2} \tilde { \boldsymbol {g}}^{d}_{mk}
and \boldsymbol {g}^{u}_{ml} = (\beta ^{u}_{ml})^{1/2} \tilde { \boldsymbol {g}}^{u}_{ml}
. Here \beta ^{d}_{mk}
and \beta ^{u}_{ml} \in \mathbb {R}
are corresponding large scale fading coefficients, which are same for all antennas at the m
th AP [3], [12]. The vectors \tilde { \boldsymbol {g}}^{d}_{mk}
and \tilde { \boldsymbol {g}}^{u}_{ml}
denote small scale fading with independent and identically distributed (i.i.d.) \mathcal {CN}(0,1)
entries. The UDI channel between the k
th downlink UE and l
th uplink UE is modeled as h_{kl} = (\tilde {\beta }_{kl})^{1/2} \tilde {h}_{kl}
[12], [13], where \tilde {\beta }_{kl}
is the large scale fading coefficient and \tilde {h}_{kl} \sim \mathcal {CN}(0,1)
is the small scale fading. The inter- and intra-AP channels from the transmit antennas of the i
th AP to the receive antennas of the m
th AP are denoted as \boldsymbol {H}_{mi} \in \mathbb {C}^{N_{r} \times N_{t}}
for i=1\,\,\text {to}\,\,M
.
B. Uplink Channel Estimation
Recall that the channel estimation phase consists of \tau _{t}
samples. We divide them as \tau _{t} = \tau ^{d}_{t} + \tau ^{u}_{t}
, where \tau ^{d}_{t}
and \tau ^{u}_{t}
are samples used as pilots for the downlink and uplink UEs, respectively. All the downlink (resp. uplink) UEs simultaneously transmit \tau ^{d}_{t}
(resp. \tau ^{u}_{t}
)-length uplink pilots to the APs, which they use to estimate the respective channels. In this phase, both transmit and receive antenna arrays of each AP, similar to [12], operate in receive mode. The k
th downlink UE (resp. l
th uplink UE) transmits pilot signals \sqrt {\tau ^{d}_{t}} \boldsymbol {\varphi }^{d}_{k} \in \mathbb {C}^{\tau ^{d}_{t} \times 1}
(resp. \sqrt {\tau ^{u}_{t}} \boldsymbol {\varphi }^{u}_{l} \in \mathbb {C}^{\tau ^{u}_{t} \times 1}
). We assume, similar to [12], [18], that the pilots i) have unit norm, i.e., \left \lVert{ \boldsymbol {\varphi }^{u}_{l}}\right \rVert = \left \lVert{ \boldsymbol {\varphi }^{d}_{k}}\right \rVert = 1
; and ii) are intra-set orthonormal, i.e., (\boldsymbol {\varphi }^{u}_{l})^{H} \boldsymbol {\varphi }^{u}_{l'} = 0\, \forall l \neq l' {\,\,\text {and }} (\boldsymbol {\varphi }^{d}_{k})^{H} \boldsymbol {\varphi }^{d}_{k'} = 0\, \forall k \neq k'
. Therefore, we need \tau ^{d}_{t} \geq K_{d}
and \tau ^{u}_{t} \geq K_{u}
[12], [18].
The pilots received by transmit and receive antennas of the m
th AP are given respectively as \begin{align*} \boldsymbol {Y}^{tx}_{m}=&\sqrt {\tau ^{d}_{t} \rho _{t}} \sum _{k=1}^{K_{d}} \boldsymbol {g}^{d}_{mk} \left ({\boldsymbol {\varphi }^{d}_{k}}\right)^{H} + \boldsymbol {W}^{tx}_{m}, \\ \boldsymbol {Y}^{rx}_{m}=&\sqrt {\tau ^{u}_{t} \rho _{t}} \sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} \left ({\boldsymbol {\varphi }^{u}_{l}}\right)^{H} + \boldsymbol {W}^{rx}_{m}.\end{align*}
View Source
\begin{align*} \boldsymbol {Y}^{tx}_{m}=&\sqrt {\tau ^{d}_{t} \rho _{t}} \sum _{k=1}^{K_{d}} \boldsymbol {g}^{d}_{mk} \left ({\boldsymbol {\varphi }^{d}_{k}}\right)^{H} + \boldsymbol {W}^{tx}_{m}, \\ \boldsymbol {Y}^{rx}_{m}=&\sqrt {\tau ^{u}_{t} \rho _{t}} \sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} \left ({\boldsymbol {\varphi }^{u}_{l}}\right)^{H} + \boldsymbol {W}^{rx}_{m}.\end{align*}
Here \rho _{t}
is the normalized pilot transmit signal-to-noise-ratio (SNR). The matrices \boldsymbol {W}^{tx}_{m} \in \mathbb {C}^{N_{t} \times \tau ^{d}_{t}}
and \boldsymbol {W}^{rx}_{m} \in \mathbb {C}^{N_{r} \times \tau ^{u}_{t}}
denote additive noise with \mathcal {CN}(0,1)
entries. Each AP independently estimates its channels with the uplink and downlink UEs to avoid channel state information (CSI) exchange overhead [12], [21]. To estimate the channels \boldsymbol {g}^{d}_{mk}
and \boldsymbol {g}^{u}_{ml}
, the m
th AP projects the received signal onto the pilot signals \boldsymbol {\varphi } _{k}^{d}
and \boldsymbol {\varphi } _{l}^{u}
respectively, as \begin{align*} \hat { \boldsymbol {y}}^{tx}_{mk}=&\boldsymbol {Y}^{tx}_{m} \boldsymbol {\varphi }^{d}_{k} = \sqrt {\tau ^{d}_{t} \rho _{t}} \boldsymbol {g}^{d}_{mk} + \boldsymbol {W}^{tx}_{m} \boldsymbol {\varphi }^{d}_{k}\\ \hat { \boldsymbol {y}}^{rx}_{ml}=&\boldsymbol {Y}^{rx}_{m} \boldsymbol {\varphi }^{u}_{l} = \sqrt {\tau ^{u}_{t} \rho _{t}} \boldsymbol {g}^{u}_{ml} + \boldsymbol {W}^{rx}_{m} \boldsymbol {\varphi }^{u}_{l}.\end{align*}
View Source\begin{align*} \hat { \boldsymbol {y}}^{tx}_{mk}=&\boldsymbol {Y}^{tx}_{m} \boldsymbol {\varphi }^{d}_{k} = \sqrt {\tau ^{d}_{t} \rho _{t}} \boldsymbol {g}^{d}_{mk} + \boldsymbol {W}^{tx}_{m} \boldsymbol {\varphi }^{d}_{k}\\ \hat { \boldsymbol {y}}^{rx}_{ml}=&\boldsymbol {Y}^{rx}_{m} \boldsymbol {\varphi }^{u}_{l} = \sqrt {\tau ^{u}_{t} \rho _{t}} \boldsymbol {g}^{u}_{ml} + \boldsymbol {W}^{rx}_{m} \boldsymbol {\varphi }^{u}_{l}.\end{align*}
These projections are used to compute the corresponding linear minimum-mean-squared-error (MMSE) channel estimates [12] as \begin{align*} \hat { \boldsymbol {g}}^{d}_{mk}=&\mathbb {E}\left \{{ \boldsymbol {g}^{d}_{mk}\left ({\hat { \boldsymbol {y}}^{tx}_{mk}}\right)^{H}}\right \}\left ({\mathbb {E}\left \{{\hat { \boldsymbol {y}}^{tx}_{mk}\left ({\hat { \boldsymbol {y}}^{tx}_{mk}}\right)^{H}}\right \}}\right)^{-1} \hat { \boldsymbol {y}}^{tx}_{mk} = c^{d}_{mk} \hat { \boldsymbol {y}}^{tx}_{mk}, \\ \hat { \boldsymbol {g}}^{u}_{ml}=&\mathbb {E}\left \{{ \boldsymbol {g}^{u}_{ml}\left ({\hat { \boldsymbol {y}}^{rx}_{ml}}\right)^{H}}\right \}\left ({\mathbb {E}\left \{{\hat { \boldsymbol {y}}^{rx}_{ml}\left ({\hat { \boldsymbol {y}}^{rx}_{ml}}\right)^{H}}\right \}}\right)^{-1} \hat { \boldsymbol {y}}^{rx}_{ml} = c^{u}_{ml} \hat { \boldsymbol {y}}^{rx}_{ml},\end{align*}
View Source\begin{align*} \hat { \boldsymbol {g}}^{d}_{mk}=&\mathbb {E}\left \{{ \boldsymbol {g}^{d}_{mk}\left ({\hat { \boldsymbol {y}}^{tx}_{mk}}\right)^{H}}\right \}\left ({\mathbb {E}\left \{{\hat { \boldsymbol {y}}^{tx}_{mk}\left ({\hat { \boldsymbol {y}}^{tx}_{mk}}\right)^{H}}\right \}}\right)^{-1} \hat { \boldsymbol {y}}^{tx}_{mk} = c^{d}_{mk} \hat { \boldsymbol {y}}^{tx}_{mk}, \\ \hat { \boldsymbol {g}}^{u}_{ml}=&\mathbb {E}\left \{{ \boldsymbol {g}^{u}_{ml}\left ({\hat { \boldsymbol {y}}^{rx}_{ml}}\right)^{H}}\right \}\left ({\mathbb {E}\left \{{\hat { \boldsymbol {y}}^{rx}_{ml}\left ({\hat { \boldsymbol {y}}^{rx}_{ml}}\right)^{H}}\right \}}\right)^{-1} \hat { \boldsymbol {y}}^{rx}_{ml} = c^{u}_{ml} \hat { \boldsymbol {y}}^{rx}_{ml},\end{align*}
where c^{d}_{mk} = \frac {\sqrt {\tau ^{d}_{t} \rho _{t}}\beta ^{d}_{mk}}{\tau ^{d}_{t} \rho _{t}\beta ^{d}_{mk} + 1}
and c^{u}_{ml} = \frac {\sqrt {\tau ^{u}_{t} \rho _{t}}\beta ^{u}_{ml}}{\tau ^{u}_{t} \rho _{t}\beta ^{u}_{ml} + 1}
. The estimation error vectors are defined as \boldsymbol {e}^{u}_{ml} \triangleq \boldsymbol {g}^{u}_{ml} - \hat { \boldsymbol {g}}^{u}_{ml}
and \boldsymbol {e}^{d}_{mk} \triangleq \boldsymbol {g}^{d}_{mk} - \hat { \boldsymbol {g}}^{d}_{mk}
. With MMSE channel estimation, \hat { \boldsymbol {g}}^{d}_{mk}, \boldsymbol {e}^{d}_{mk}
and \hat { \boldsymbol {g}}^{u}_{ml}, \boldsymbol {e}^{u}_{ml}
are mutually independent and their individual terms are i.i.d. with pdf \mathcal {CN}(0,\gamma ^{d}_{mk}),\mathcal {CN}(0,\beta ^{d}_{mk}-\gamma ^{d}_{mk}), \mathcal {CN}(0,\gamma ^{u}_{ml}), \mathcal {CN}(0, \beta ^{u}_{ml} - \gamma ^{u}_{ml})
respectively, with \gamma ^{d}_{mk} = \frac {\tau ^{d}_{t} \rho _{t} (\beta ^{d}_{mk})^{2}}{\tau ^{d}_{t} \rho _{t} \beta ^{d}_{mk} + 1}
and \gamma ^{u}_{ml} = \frac {\tau ^{u}_{t} \rho _{t} (\beta ^{u}_{ml})^{2}}{\tau ^{u}_{t} \rho _{t} \beta ^{u}_{ml} + 1}
[12], [18].
After channel estimation, data transmission starts simultaneously on downlink and uplink.
C. Transmission Model
An objective of this work is to derive a SE lower bound for FD CF mMIMO systems, where the M
APs serve K_{u}
uplink UEs and K_{d}
downlink UEs simultaneously on the same spectral resource. We note that for the FD CF mMIMO systems, unlike the HD CF mMIMO systems [3], [15], [16], uplink and downlink transmissions interfere to cause UDI and inter-/intra-AP interferences. Further, unlike existing FD CF mMIMO literature [12], [13], [21], we consider a limited-capacity fronthaul. It is critical to model and analyze the UDI and inter-/intra-AP interferences and limited-capacity impairments while deriving the lower bound.
1) Downlink Data Transmission
The CPU chooses a message symbol s^{d}_{k}
for the k
th downlink UE, which is distributed as \mathcal {CN}(0,1)
. It intends to send this symbol to the m
th AP via the limited-capacity fronthaul link. Before doing that, it multiplies s^{d}_{k}
with a power-control coefficient \eta _{mk}
, and then quantizes the resulting signal. The m
th AP, due to its limited fronthaul capacity, is allowed to serve only a subset \kappa _{dm} \subset \{1, {\dots }, K_{d}\}
of downlink users, an aspect which is discussed later in Section II-D. The CPU consequently sends downlink symbols for UEs in the set \kappa _{dm}
to the m
th AP, which uses MMSE channel estimates to perform MRT precoding. The transmit signal of the m
th AP is therefore given as follows \begin{align*} \boldsymbol {x}^{d}_{m}=&\sqrt {\rho _{d}} \sum _{k \in \kappa _{dm}} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} \mathcal {Q}\left ({\sqrt {\eta _{mk}} s^{d}_{k}}\right) \\=&\sqrt {\rho _{d}} \sum _{k \in \kappa _{dm}} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{mk}} s^{d}_{k} + \varsigma ^{d}_{mk}}\right).\tag{1}\end{align*}
View Source\begin{align*} \boldsymbol {x}^{d}_{m}=&\sqrt {\rho _{d}} \sum _{k \in \kappa _{dm}} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} \mathcal {Q}\left ({\sqrt {\eta _{mk}} s^{d}_{k}}\right) \\=&\sqrt {\rho _{d}} \sum _{k \in \kappa _{dm}} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{mk}} s^{d}_{k} + \varsigma ^{d}_{mk}}\right).\tag{1}\end{align*}
Here \rho _{d}
is the normalized maximum transmit SNR at each AP. The function \mathcal {Q}(\cdot)
denotes the quantization operation, which is modeled as a multiplicative attenuation \tilde {a}
, and an additive distortion \varsigma ^{d}_{mk}
, for the k
th downlink UE in the fronthaul link between the CPU and the m
th AP [15], [19]. We have, from Appendix A, \mathbb {E}\{(\varsigma ^{d}_{mk})^{2}\} = (\tilde {b}-\tilde {a}^{2}) \mathbb {E}\{|\sqrt {\eta _{mk}} s^{d}_{k}|^{2}\} = (\tilde {b}-\tilde {a}^{2})\eta _{mk}
, where the scalar constants \tilde {a}
and \tilde {b}
depend on the number of fronthaul quantization bits.
The m
th AP must satisfy the average transmit SNR constraint, i.e., \mathbb {E}\{\| \boldsymbol {x}^{d}_{m}\|^{2}\} \leq \rho _{d}
. Using the expression of \boldsymbol {x}^{d}_{m}
from (1), and the above expression of quantization error variance, \mathbb {E}\{(\varsigma ^{d}_{mk})^{2}\}
, the constraint can be simplified as follows \begin{equation*} \rho _{d} \tilde {b} \sum _{k \in \kappa _{dm}} \eta _{mk} \mathbb {E}\left \{{\|\hat { \boldsymbol {g}}^{d}_{mk}\|^{2}}\right \} \leq \rho _{d} \Rightarrow \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} \eta _{mk} \leq \frac {1}{N_{t}}. \tag{2}\end{equation*}
View Source\begin{equation*} \rho _{d} \tilde {b} \sum _{k \in \kappa _{dm}} \eta _{mk} \mathbb {E}\left \{{\|\hat { \boldsymbol {g}}^{d}_{mk}\|^{2}}\right \} \leq \rho _{d} \Rightarrow \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} \eta _{mk} \leq \frac {1}{N_{t}}. \tag{2}\end{equation*}
The k
th downlink UE receives its desired message signal from a subset of all APs, denoted as \mathcal {M}^{d}_{k} \subset \{1, {\dots }, M\}
, along with various interference and distortion components, as in (5) (shown at the top of the next page). The m
th AP serves the k
th downlink UE iff k \in \kappa _{dm} \Leftrightarrow m \in \mathcal {M}^{d}_{k}
. Here x^{u}_{l}
is the transmit signal of the l
th uplink UE, which is modelled next.
2) Uplink Data Transmission
The K_{u}
uplink UEs also simultaneously transmit to all M
APs on the same spectral resource as that of the K_{d}
downlink UEs. The l
th uplink UE transmits its signal x^{u}_{l} = \sqrt {\rho _{u} \theta _{l}} s^{u}_{l}
with s^{u}_{l}
being its message symbol with pdf \mathcal {CN}(0,1)
, \rho _{u}
being the maximum uplink transmit SNR and \theta _{l}
being the power control coefficient. To satisfy the average SNR constraint, \mathbb {E}\{|x^{u}_{l}|^{2}\} \leq \rho _{u}
, the l
th uplink UE satisfies the constraint \begin{equation*} 0 \leq \theta _{l} \leq 1. \tag{3}\end{equation*}
View Source\begin{equation*} 0 \leq \theta _{l} \leq 1. \tag{3}\end{equation*}
The FD APs not only receive the uplink UE signals but also their own downlink transmit signals and that of the other APs, referred to as intra-AP and inter-AP interference, respectively. Using (1), the received uplink signal at the m
th AP is\begin{align*} \boldsymbol {y}^{u}_{m}=&\sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} x^{u}_{l} + \sum _{i=1}^{M} \boldsymbol {H}_{mi} \boldsymbol {x}^{d}_{i} + \boldsymbol {w}^{u}_{m} = \sqrt {\rho _{u}} \sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} \sqrt {\theta _{l}} s^{u}_{l} \\&{}+\,\,\sqrt {\rho _{d}} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k}+ \varsigma ^{d}_{ik}}\right) + \boldsymbol {w}^{u}_{m}. \tag{4}\end{align*}
View Source\begin{align*} \boldsymbol {y}^{u}_{m}=&\sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} x^{u}_{l} + \sum _{i=1}^{M} \boldsymbol {H}_{mi} \boldsymbol {x}^{d}_{i} + \boldsymbol {w}^{u}_{m} = \sqrt {\rho _{u}} \sum _{l=1}^{K_{u}} \boldsymbol {g}^{u}_{ml} \sqrt {\theta _{l}} s^{u}_{l} \\&{}+\,\,\sqrt {\rho _{d}} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k}+ \varsigma ^{d}_{ik}}\right) + \boldsymbol {w}^{u}_{m}. \tag{4}\end{align*}
Here
\boldsymbol {w}^{u}_{m} \in \mathbb {C}^{N_{r} \times 1}
is the additive receiver noise at the
m
th AP with i.i.d. entries
\sim \mathcal {CN}(0,1)
.
The intra and inter-AP interference channels vary extremely slowly and thus can be estimated with very low pilot overhead [13]. The receive antenna array of each AP, with estimated channel, can only partially mitigate the intra- and inter-AP interference [12], [13]. The residual intra-/inter-AP interference (RI) channel \boldsymbol {H}_{mi} \in \mathbb {C}^{N_{r} \times N_{t}}
is modeled as Rayleigh-faded with i.i.d. entries and pdf \mathcal {CN}(0, \gamma _{\text {RI},mi})
[6], [12], [13], [24]. Here \gamma _{\text {RI},mi} \triangleq \beta _{\text {RI},mi} \gamma _{\text {RI}}
, with \beta _{\text {RI},mi}
being the large scale fading coefficient from the i
th AP to the m
th AP, and \gamma _{\text {RI}}
being the RI power after its suppression.
The m
th AP receives the signals from all the uplink UEs, and performs MRC for the l
th uplink UE with (\hat { \boldsymbol {g}}^{u}_{ml})^{H}
. Due to its limited fronthaul: i) AP quantizes the combined signal before sending it to CPU; ii) as discussed in detail later in Section II-D, the CPU receives contributions for the l
th uplink UE only from the subset of APs serving it, denoted as \mathcal {M}^{u}_{l} \subset \{1, {\dots }, M\}
. Using (4), the signal received by the CPU for the l
th uplink UE is expressed as in (6) (shown at the top of the next page).
We denote the subset of uplink UEs served by the m
th AP as \kappa _{ {\mu {\mathrm{ m}}}} \subset \{1, {\dots }, K_{u}\}
. The m
th AP serves the l
th uplink UE iff l \in \kappa _{ {\mu {\mathrm{ m}}}} \Leftrightarrow m \in \mathcal {M}^{u}_{l}
. The quantization operation \mathcal {Q}(\cdot)
is mathematically modeled using constant attenuation \tilde {a}
, and additive distortion \varsigma ^{u}_{ml}
which, as shown in Appendix A, has power \mathbb {E}\{(\varsigma ^{u}_{ml})^{2}\} = (\tilde {b} - \tilde {a}^{2}) \mathbb {E}\{|(\boldsymbol {g}^{u}_{ml})^{H} \boldsymbol {y}_{m}|^{2}\}
.
D. User-Centric Behavior Through Limited Fronthaul
Initial CF mMIMO literature considered system models where all APs can serve all UEs [3]–[5]. However, for geographically large areas, each UE can only have practically feasible channels with a subset of APs in its vicinity. Therefore, recent CF mMIMO literature has increasingly focused on user-centric CF mMIMO system design [2, and the references therein]. In the subsequent discussion, we show that a user-centric CF deployment, as desired by us, is a natural outcome of the design choice to impose fronthaul capacity constraints on the CF mMIMO system model, as shown in Fig. 1.
The fronthaul between the m
th AP and the CPU uses \nu _{m}
bits to quantize the real and imaginary parts of transmit signal of the m
th downlink UE and the uplink receive signal after MRC, i.e., \sqrt {\eta _{mk}} s^{d}_{k}
, and (\hat { \boldsymbol {g}}^{u}_{ml})^{H} \boldsymbol {y}^{u}_{m}
, respectively. Due to the limited-capacity fronthaul, the m
th AP serves only K_{ {\mu {\mathrm{ m}}}} (\triangleq |\kappa _{ {\mu {\mathrm{ m}}}}|)
and K_{dm} (\triangleq |\kappa _{dm}|)
UEs on the uplink and downlink, respectively [15], [19]. For each UE, we recall that there are (\tau _{c} - \tau _{t})
data samples in each coherence interval of duration T_{c}
. The fronthaul data rate between the m
th AP and the CPU is \begin{equation*} R_{\text {fh},m} = \frac {2\nu _{m}(K_{dm} + K_{ {\mu {\mathrm{ m}}}}) (\tau _{c} - \tau _{t})}{T_{c}}.\tag{7}\end{equation*}
View Source\begin{equation*} R_{\text {fh},m} = \frac {2\nu _{m}(K_{dm} + K_{ {\mu {\mathrm{ m}}}}) (\tau _{c} - \tau _{t})}{T_{c}}.\tag{7}\end{equation*}
The fronthaul link between the m
th AP and the CPU has capacity C_{\text {fh},m}
which implies that \begin{equation*} R_{\text {fh},m} \leq C_{\text {fh},m} \Rightarrow \nu _{m} \cdot (K_{ {\mu {\mathrm{ m}}}} + K_{dm}) \leq \frac {C_{\text {fh},m} T_{c}}{2(\tau _{c} - \tau _{t})}.\tag{8}\end{equation*}
View Source\begin{equation*} R_{\text {fh},m} \leq C_{\text {fh},m} \Rightarrow \nu _{m} \cdot (K_{ {\mu {\mathrm{ m}}}} + K_{dm}) \leq \frac {C_{\text {fh},m} T_{c}}{2(\tau _{c} - \tau _{t})}.\tag{8}\end{equation*}
We propose the following lemma where we consider a proportionally fair approach to calculate K_{dm}
and K_{ {\mu {\mathrm{ m}}}}
in proportion to the total number of downlink and uplink UEs, respectively. We use \varepsilon \triangleq \{d,u\}
to denote downlink and uplink, respectively, and define the total number of UEs, K \triangleq K_{u} + K_{d}
.
Lemma 1:
The maximum number of uplink and downlink UEs served by the m
th AP when connected via a limited optical fronthaul to the CPU with capacity C_{\text {fh},m}
are given as \begin{equation*} \bar {K}_{\epsilon m} = \left \lfloor{ {\frac {K_{\epsilon }}{K} \frac {C_{\text {fh},m} T_{c}}{4 (\tau _{c} - \tau _{t}) \nu _{m}}}}\right \rfloor. \tag{9}\end{equation*}
View Source\begin{equation*} \bar {K}_{\epsilon m} = \left \lfloor{ {\frac {K_{\epsilon }}{K} \frac {C_{\text {fh},m} T_{c}}{4 (\tau _{c} - \tau _{t}) \nu _{m}}}}\right \rfloor. \tag{9}\end{equation*}
Proof:
Let \bar {K}_{ {\mu {\mathrm{ m}}}}
and \bar {K}_{dm}
denote the maximum number of uplink and downlink UEs served by the m
th AP. We consider \bar {K}_{ {\mu {\mathrm{ m}}}} \propto K_{u}
and \bar {K}_{dm} \propto K_{d}
for proportional fairness on the uplink and downlink. Using (8), we get, \begin{equation*} \bar {K}_{\epsilon m} \leq \frac {K_{\epsilon }}{K} \frac {C_{\text {fh},m} T_{c}}{2(\tau _{c} - \tau _{t}) \nu _{m}}.\end{equation*}
View Source\begin{equation*} \bar {K}_{\epsilon m} \leq \frac {K_{\epsilon }}{K} \frac {C_{\text {fh},m} T_{c}}{2(\tau _{c} - \tau _{t}) \nu _{m}}.\end{equation*}
The lemma follows from definition of floor function \lfloor { \cdot }\rfloor
.
Using the maximum limits obtained in (9), we assign K_{ {\mu {\mathrm{ m}}}} = \min \{K_{u}, \bar {K}_{ {\mu {\mathrm{ m}}}}\}
and K_{dm} = \min \{K_{d}, \bar {K}_{dm}\}
. We see that the constraint imposed in (8) is similar to a UE-centric (UC) CF mMIMO system, wherein each UE is served by a subset of the APs [2]. We now define the procedure for AP selection to obtain the best subset of APs to serve each uplink and downlink UE, while satisfying (8). For this, we extend the procedure in [15] for a FD system as follows:
The m
th AP sorts the uplink and downlink UEs connected to it in descending order based on their channel gains (\beta ^{u}_{ml}
and \beta ^{d}_{mk}
, respectively) and chooses K_{ {\mu {\mathrm{ m}}}}
uplink UEs and K_{dm}
downlink UEs, with the largest channel gains, to populate the sets \kappa _{ {\mu {\mathrm{ m}}}}
and \kappa _{dm}
, respectively.
For the l
th uplink UE and the k
th downlink UE, we populate the sets \mathcal {M}^{u}_{l}
and \mathcal {M}^{d}_{k}
, respectively, using the axioms l \in \kappa _{ {\mu {\mathrm{ m}}}} \Leftrightarrow m \in \mathcal {M}^{u}_{l}
and k \in \kappa _{dm} \Leftrightarrow m \in \mathcal {M}^{d}_{k}
.
If an uplink or downlink UE is found with no serving AP, we use the procedure in Algorithm 1 to assign it the AP with the best channel conditions, while satisfying (8).
Clearly, Lemma II-D ensures that each AP can only serve a limited number of UEs which do not violate the fronthaul capacity constraints. This makes the system effectively a user-centric system. Algorithm 1 ensures that, under limited fronthaul constraints, the strongest AP-UE connections are retained and the UE-centric cell-free system delivers good performance.
E. Self-Interference Mitigation Methods
To ensure that our proposed FD CF mMIMO system has substantial performance improvement over an equivalent HD CF mMIMO system, we need effective techniques to cancel the self-interference (SI) caused due to inter-AP transmissions. We show in Eq. (5)–(6), shown at the bottom of the page that this SI cancellation results in a residual interference (RI) due to the multiplication of a suppression factor, \gamma _{\text {RI}}
. We now discuss SI cancellation techniques from the existing literature, which makes the SI suppression easier, by not requiring its instantaneous channel knowledge.
Passive cancellation: Reference [7], [30] suggests that a careful utilization of the passive self-interference suppression mechanisms (directional isolation, absorptive shielding, and cross polarization) can significantly suppress the SI. Reference [7] also showed that by additionally assuming statistical SI channel knowledge and by using antennas arrays of sources/destinations, the passive cancellation techniques can further suppress the SI.
Large antenna array: Reference [31] argued that with large N, channel vectors of the desired signal and the SI become nearly orthogonal. The beamforming techniques, e.g., MRC/MRT inherently project the desired signal to the orthogonal complement space of the SI, which significantly reduces the SI.
Lower transmit power: Reference [31] also demonstrated that an alternative way to reduce interference could be to reduce transmit power, since the SI depends strongly on the transmit power. A cell free massive MIMO system, due to large number of transmit antennas, uses radically less transmit power/antenna than conventional MIMO systems, which significantly reduces the SI.
We therefore, similar to existing massive MIMO FD literature [7], [31], [32], assume that the SI can be significantly mitigated by utilizing the above mentioned SI cancellation techniques, and without requiring the knowledge of SI channel. However, if required, the residual SI can be further reduced by employing active (time-domain and spatial suppression) techniques developed in [33], which require SI channel knowledge.
Active cancellation: The authors in [33] present an algorithm for SI channel estimation at the relay, which is equipped with large number of antennas. It also noted that the APs, which are infrastructure devices, are in a stationary environment. The SI channel changes much more slowly than the channel from users to the APs. It is therefore reasonable to assume that i) the SI channel remains constant for multiple consecutive blocks; and ii) inter-AP pilot overhead is affordable because of the sufficiently longer coherence time of the residual SI channels. Similar to [33], one can estimate the SI channel by utilizing its slowly-varying nature using a cost-efficient expectation-maximization algorithm with reduced complexity.
SECTION III.
Achievable Spectral Efficiency
We now derive the ergodic SE for the k
th downlink UE and the l
th uplink UE, denoted respectively as \bar {S}^{d}_{k}
and \bar {S}^{u}_{l}
. The AP employs MRC/MRT in the uplink/downlink and optimal uniform fronthaul quantization. We use \varepsilon \triangleq \{d,u\}
to denote downlink and uplink, respectively; \phi \triangleq \{k,l\}
to denote k
th downlink UE and l
th uplink UE, respectively; and \upsilon ^{\varepsilon }_{m\phi } \triangleq \{\eta _{mk}\,\,\text {for}\,\,\phi =k, \theta _{l}\,\,\text {for}\,\,\phi =l\}
. The ergodic SE expressions are calculated using (5) and (6), as\begin{align*}&\bar {S}^{\varepsilon }_{\phi } = \left ({\frac {\tau _{c} - \tau _{t}}{\tau _{c}}}\right) \mathbb {E}\left \{{\log _{2}\left ({1 + \frac {P^{\varepsilon }_{\phi }}{I^{\varepsilon }_{\phi } + \left ({\sigma ^{\varepsilon }_{\phi,0}}\right)^{2}}}\right)}\right \},~\text {where} \\&\,P^{\varepsilon }_{\phi } = \Big |\tilde {a}\sum _{m \in \mathcal {M}^{\varepsilon }_{\phi }} \sqrt {\rho _{\varepsilon }} \sqrt {\upsilon ^{\varepsilon }_{m\phi }} \left ({\hat { \boldsymbol {g}}^{\varepsilon }_{m\phi }}\right)^{H} \boldsymbol {g}^{\varepsilon }_{m\phi } s_{\phi }^{\varepsilon }\Big |^{2}, \\&\,\left ({\sigma ^{d}_{k,0}}\right)^{2} = \left |{w_{k}^{d}}\right |^{2}, \left ({\sigma ^{u}_{l,0}}\right)^{2} = \Big |\tilde {a} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}\Big |^{2}, \\&\,I^{d}_{k} =\left |{\sum _{l=1}^{K_{u}} h_{kl} \sqrt {\rho _{u} \theta _{l}} s^{u}_{l}}\right |^{2} \\&\quad +\,\,\Big |\tilde {a}\sqrt {\rho _{d}} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k}~\sqrt {\eta _{mq}}\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} s^{d}_{q}\Big |^{2} \\&\quad +\,\,\Big |\sqrt {\rho _{d}} \sum _{m = 1}^{M} \sum _{q \in \kappa _{dm}} \left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} \varsigma ^{d}_{mq}\Big |^{2}, \\&\,I^{u}_{l} = \Big |\tilde {a}~\sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q = 1, q \neq l}^{K_{u}}~\sqrt {\rho _{u}} \sqrt {\theta _{q}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} s_{q}^{u}\Big |^{2} + \Big |\sum _{m \in \mathcal {M}^{u}_{l}}~\varsigma ^{u}_{ml}\Big |^{2} \\&\quad +\,\,\Big |\tilde {a}~\sum _{m \in \mathcal {M}^{u}_{l}}~\sum _{i=1}^{M}~\sqrt {\rho _{d}}~\sum _{k \in \kappa _{di}}~\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k} + \varsigma ^{d}_{ik}}\right)\Big |^{2}, \\ {}\tag{10}\end{align*}
View Source\begin{align*}&\bar {S}^{\varepsilon }_{\phi } = \left ({\frac {\tau _{c} - \tau _{t}}{\tau _{c}}}\right) \mathbb {E}\left \{{\log _{2}\left ({1 + \frac {P^{\varepsilon }_{\phi }}{I^{\varepsilon }_{\phi } + \left ({\sigma ^{\varepsilon }_{\phi,0}}\right)^{2}}}\right)}\right \},~\text {where} \\&\,P^{\varepsilon }_{\phi } = \Big |\tilde {a}\sum _{m \in \mathcal {M}^{\varepsilon }_{\phi }} \sqrt {\rho _{\varepsilon }} \sqrt {\upsilon ^{\varepsilon }_{m\phi }} \left ({\hat { \boldsymbol {g}}^{\varepsilon }_{m\phi }}\right)^{H} \boldsymbol {g}^{\varepsilon }_{m\phi } s_{\phi }^{\varepsilon }\Big |^{2}, \\&\,\left ({\sigma ^{d}_{k,0}}\right)^{2} = \left |{w_{k}^{d}}\right |^{2}, \left ({\sigma ^{u}_{l,0}}\right)^{2} = \Big |\tilde {a} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}\Big |^{2}, \\&\,I^{d}_{k} =\left |{\sum _{l=1}^{K_{u}} h_{kl} \sqrt {\rho _{u} \theta _{l}} s^{u}_{l}}\right |^{2} \\&\quad +\,\,\Big |\tilde {a}\sqrt {\rho _{d}} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k}~\sqrt {\eta _{mq}}\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} s^{d}_{q}\Big |^{2} \\&\quad +\,\,\Big |\sqrt {\rho _{d}} \sum _{m = 1}^{M} \sum _{q \in \kappa _{dm}} \left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} \varsigma ^{d}_{mq}\Big |^{2}, \\&\,I^{u}_{l} = \Big |\tilde {a}~\sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q = 1, q \neq l}^{K_{u}}~\sqrt {\rho _{u}} \sqrt {\theta _{q}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} s_{q}^{u}\Big |^{2} + \Big |\sum _{m \in \mathcal {M}^{u}_{l}}~\varsigma ^{u}_{ml}\Big |^{2} \\&\quad +\,\,\Big |\tilde {a}~\sum _{m \in \mathcal {M}^{u}_{l}}~\sum _{i=1}^{M}~\sqrt {\rho _{d}}~\sum _{k \in \kappa _{di}}~\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k} + \varsigma ^{d}_{ik}}\right)\Big |^{2}, \\ {}\tag{10}\end{align*}
are signal, noise and interference powers respectively, for the k
th downlink and l
th uplink UEs. The expectation outside logarithm in the SE expressions in (10) is mathematically intractable, and it is difficult to simplify them further [3], [12], [15]. We, similar to [3], employ use-and-then-forget (UatF) technique to derive SE lower bounds. To use UatF, we rewrite the received signal at the k
th downlink UE in (5), and at the CPU for the l
th uplink UE in (6) as \begin{equation*} r^{\varepsilon }_{\phi } = \underbrace {\tilde {a} \sum _{m \in \mathcal {M}^{\varepsilon }_{\phi }} \sqrt {\rho _{\varepsilon }} \sqrt {\upsilon ^{\varepsilon }_{m\phi }} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{\varepsilon }_{m\phi }}\right)^{H} \boldsymbol {g}^{\varepsilon }_{m\phi }}\right \}s^{\varepsilon }_{\phi }}_{\text {desired signal, DS}^{\varepsilon }_{\phi }} + n^{\varepsilon }_{\phi }, \tag{11}\end{equation*}
View Source\begin{equation*} r^{\varepsilon }_{\phi } = \underbrace {\tilde {a} \sum _{m \in \mathcal {M}^{\varepsilon }_{\phi }} \sqrt {\rho _{\varepsilon }} \sqrt {\upsilon ^{\varepsilon }_{m\phi }} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{\varepsilon }_{m\phi }}\right)^{H} \boldsymbol {g}^{\varepsilon }_{m\phi }}\right \}s^{\varepsilon }_{\phi }}_{\text {desired signal, DS}^{\varepsilon }_{\phi }} + n^{\varepsilon }_{\phi }, \tag{11}\end{equation*}
where the effective additive noise terms n^{\varepsilon }_{\phi }
are expressed in (12)–(13) (shown at the bottom of the page).
The term DS
^{\varepsilon }_{\phi }
in
(11) denotes the desired signal received over the channel mean, and the term BU
^{\varepsilon }_{\phi }
in
(12)–(13) denotes beamforming uncertainty, i.e., the signal received over deviation of channel from mean. It is easy to see that
n^{\varepsilon }_{\phi }
are uncorrelated with their respective
\text {DS}^{\varepsilon }_{\phi }
terms. We, similar to
[12], treat them as worst-case additive Gaussian noise, an approximation which is tight for mMIMO systems
[12]. Using
(11)–
(12), we next derive an achievable SE lower bound.
Theorem 1:
An achievable lower bound to the SE for the k
th downlink UE with MRT and the l
th uplink UE with MRC can be expressed respectively as\begin{align*} S^{d}_{k}=&\tau _{f} \log _{2} \left ({1 + \frac {\left ({\sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \sqrt {\eta _{mk}}}\right)^{2}}{\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \eta _{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1}}\right), \tag{14}\\ S^{u}_{l}=&\tau _{f} \log _{2} \left ({1 + \frac {A^{u}_{l} \theta _{l}}{\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \eta _{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l}}}\right), \\ {}\tag{15}\end{align*}
View Source\begin{align*} S^{d}_{k}=&\tau _{f} \log _{2} \left ({1 + \frac {\left ({\sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \sqrt {\eta _{mk}}}\right)^{2}}{\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \eta _{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1}}\right), \tag{14}\\ S^{u}_{l}=&\tau _{f} \log _{2} \left ({1 + \frac {A^{u}_{l} \theta _{l}}{\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \eta _{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l}}}\right), \\ {}\tag{15}\end{align*}
where \tau _{f} = \left({\frac {\tau _{c} - \tau _{t}}{\tau _{c}}}\right)
, A^{d}_{mk} = \tilde {a} N_{t} \sqrt {\rho _{d}} \gamma ^{d}_{mk}
, B^{d}_{kmq} = \tilde {b} N_{t} \rho _{d} \beta ^{d}_{mk} \gamma ^{d}_{mq}
, D^{d}_{kl} = \rho _{u} \tilde {\beta }_{kl}
, A^{u}_{l} = \tilde {a}^{2} N^{2}_{r} \rho _{u} \left({\sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}}\right)^{2}
, B^{u}_{lq} = \tilde {b} N_{r} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}\beta ^{u}_{mq}
, D^{u}_{lik} = \tilde {b}^{2} N_{r} N_{t} \rho _{d} \gamma ^{d}_{ik} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml} \beta _{\text {RI},mi} \gamma _{\text {RI}}
, E^{u}_{l}=(\tilde {b}-\tilde {a}^{2})N^{2}_{r} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} (\gamma ^{u}_{ml})^{2}
, and F^{u}_{l} = \tilde {b} N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}
. Here \boldsymbol {\eta } \triangleq \{\eta _{mk}\} \in \mathbb {C}^{M \times K_{d}}
, \boldsymbol {\Theta } \triangleq \{ \theta _{l}\} \in \mathbb {C}^{K_{u} \times 1}
and \boldsymbol {\nu } \triangleq \{\nu _{m}\} \in \mathbb {C}^{M \times 1}
are the variables on which the SE is dependent. We recall from Section II that \tilde {a}
and \tilde {b}
in (14)–(15) depend on the number of quantization bits, \boldsymbol {\nu }
.
Proof:
Refer to Appendix B. The SE expressions are functions of large scale fading coefficients, \gamma ^{d}_{mk}
and \gamma ^{u}_{ml}
, which we will use to optimize WSEE. This is unlike [21] which requires instantaneous channel while optimizing SE-GEE metric.
MRC/MRT has tractable SE expression that depend solely on large-scale channel statistics, which remain constant over hundreds of coherence intervals [28]. This is in contrast to zero-forcing designs which yield better SE but not tractable SE expressions [2]. Further, MRC/MRT can be implemented in a distributed fashion with low complexity.
SECTION IV.
Two-Layer Decentralized WSEE Optimization for FD CF mMIMO
We now devise a decentralized algorithm which maximizes WSEE by calculating the optimal downlink and uplink power control coefficients \boldsymbol {\eta }^{*}
and \boldsymbol {\Theta }^{*}
, respectively. We use “two-layered” approach to decompose WSEE maximization into a sequential process with two distinct individual steps, each of which is called a “layer”. The first layer simplifies the non-convex WSEE maximization into a successive convex approximation (SCA) setting. Its output is a generalized convex program (GCP) which needs to be solved iteratively for the optimal solution. The second layer optimally solves above GCP, either centrally through standard interior-point approaches or decentrally using ADMM method. The proposed procedure is outlined in Algorithm 2.
SECTION Algorithm 2:
Two-Layer Decentralized WSEE Maximization Algorithm
1AP selection: Select APs that serve each UE while satisfying limited fronthaul constraints.
2:SCA framework (first layer): Apply a series of transformations and approximations to recast the non-convex WSEE maximization using successive convex approximation (SCA) framework. The output of first layer is a GCP.
3Decentralized ADMM approach (second layer): Introduce global and local variables to decouple the problem into multiple sub-problems. Each sub-problem is solved at a distributed (or “D”) server, whose solutions are coordinated to obtain the global solution at the central (or “C”) server. This procedure is implemented using ADMM.
We use \varepsilon \triangleq \{d,u\}
for the downlink and uplink, respectively; \phi \triangleq \{k,l\}
for the k
th downlink UE and l
th uplink UE, respectively; and first define the individual EE for each UE as \text {EE}^{\varepsilon }_{\phi } = \frac {B \cdot S^{\varepsilon }_{\phi }}{p^{\varepsilon }_{\phi }}
[23], where B
is the system bandwidth, and p^{\varepsilon }_{\phi }
denotes the power consumed by each UE. The fronthaul links consume power for both downlink and uplink transmission. The APs consume power while transmitting data to the downlink UEs, and the uplink UEs consume power while transmitting their data. The power consumed by the system to transmit data to the k
th downlink UE and the power consumed by the l
th uplink UE are given respectively as [19], [21]\begin{align*} p^{d}_{k}=&P_{\text {fix}} + N_{t} \rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {1}{\alpha _{m}} \gamma ^{d}_{mk} \eta _{mk} + P^{d}_{\text {tc},k}, \tag{16}\\ p^{u}_{l}=&P_{\text {fix}} + \rho _{u} N_{0} \frac {1}{\alpha '_{l}} \theta _{l} + P^{u}_{\text {tc},l}. \tag{17}\end{align*}
View Source\begin{align*} p^{d}_{k}=&P_{\text {fix}} + N_{t} \rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {1}{\alpha _{m}} \gamma ^{d}_{mk} \eta _{mk} + P^{d}_{\text {tc},k}, \tag{16}\\ p^{u}_{l}=&P_{\text {fix}} + \rho _{u} N_{0} \frac {1}{\alpha '_{l}} \theta _{l} + P^{u}_{\text {tc},l}. \tag{17}\end{align*}
Here \alpha _{m}, \alpha '_{l}
are power amplifier efficiencies at the m
th AP and the l
th uplink UE respectively [12], N_{0}
is the noise power and P^{d}_{\text {tc},k}, P^{u}_{\text {tc},l}
are the powers required to run the transceiver chains at each antenna of the k
th downlink UE and the l
th uplink UE, respectively. The power consumed by the AP transceiver chains and the fronthaul between APs and CPU: \begin{equation*} P_{\text {fix}} = \frac {1}{K} \sum _{m=1}^{M} \left ({P_{0,m} + (N_{t} + N_{r}) P_{\text {tc},m} + P_{\text {ft}}\frac {R_{\text {fh},m}}{C_{\text {fh},m}} }\right). \tag{18}\end{equation*}
View Source\begin{equation*} P_{\text {fix}} = \frac {1}{K} \sum _{m=1}^{M} \left ({P_{0,m} + (N_{t} + N_{r}) P_{\text {tc},m} + P_{\text {ft}}\frac {R_{\text {fh},m}}{C_{\text {fh},m}} }\right). \tag{18}\end{equation*}
Here P_{\text {tc},m}
is the power required to run the transceiver chains at each antenna of the m
th AP. The fronthaul power consumption for the m
th AP has a fixed component, P_{0,m}
, and a traffic-dependent component, which attains a maximum value of P_{\text {ft}}
at full capacity C_{\text {fh},m}
. The term R_{\text {fh},m}
, given in (7), is the fronthaul data rate of the m
th AP. The WSEE is now defined as the weighted sum of EEs of individual UEs [22].\begin{equation*} \text {WSEE} = \sum _{k=1}^{K_{d}} w^{d}_{k} \text {EE}^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} \text {EE}^{u}_{l} \stackrel {\bigtriangleup }{=}B \left ({\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}}{p^{d}_{k}} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}}{p^{u}_{l}}}\right),\end{equation*}
View Source\begin{equation*} \text {WSEE} = \sum _{k=1}^{K_{d}} w^{d}_{k} \text {EE}^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} \text {EE}^{u}_{l} \stackrel {\bigtriangleup }{=}B \left ({\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}}{p^{d}_{k}} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}}{p^{u}_{l}}}\right),\end{equation*}
where w^{\varepsilon }_{\phi }
are weights assigned to the UEs to account for their heterogeneous EE requirements. The WSEE metric can prioritize the EE requirements of individual UEs by assigning them different weights [23], [24]. For example, it could assign a higher weight to a UE that is more energy-scarce. After omitting the constant B
from the objective, the WSEE maximization problem can now be formulated as follows \begin{align*} \mathbf {P1}~:~\underset {\substack { \boldsymbol {\eta, \Theta, \nu }}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}\left ({\boldsymbol {\eta, \Theta,\nu }}\right)}{p^{d}_{k}(\boldsymbol {\eta,\nu })} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta, \nu })}{p^{u}_{l}(\boldsymbol {\Theta,\nu })} \\ \text {s.t.}\,\, &S^{d}_{k} (\boldsymbol {\eta, \Theta,\nu }) \geq S^{d}_{ok}, S^{u}_{l} (\boldsymbol {\eta, \Theta,\nu }) \geq S^{u}_{ol}, \tag{19a}\\&R_{\text {fh},m} \leq C_{\text {fh},m},~(2), {(3) }. \tag{19b}\end{align*}
View Source\begin{align*} \mathbf {P1}~:~\underset {\substack { \boldsymbol {\eta, \Theta, \nu }}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}\left ({\boldsymbol {\eta, \Theta,\nu }}\right)}{p^{d}_{k}(\boldsymbol {\eta,\nu })} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta, \nu })}{p^{u}_{l}(\boldsymbol {\Theta,\nu })} \\ \text {s.t.}\,\, &S^{d}_{k} (\boldsymbol {\eta, \Theta,\nu }) \geq S^{d}_{ok}, S^{u}_{l} (\boldsymbol {\eta, \Theta,\nu }) \geq S^{u}_{ol}, \tag{19a}\\&R_{\text {fh},m} \leq C_{\text {fh},m},~(2), {(3) }. \tag{19b}\end{align*}
quality-of-service (QoS) constraints in (19a) guarantee a minimum SE, denoted by the constants S^{d}_{ok}
and S^{u}_{ol}
, for each downlink and uplink UE respectively. The first constraint in (19b) ensures that the fronthaul transmission rate for all APs is within the capacity limit. We observe that the number of quantization bits \boldsymbol {\nu }
, if included in problem P1, will make it a difficult-to-solve integer optimization problem [15], [19], [34]. We therefore solve it to optimize the power control coefficients \{ \boldsymbol {\eta }, \boldsymbol {\Theta }\}
, by fixing \boldsymbol {\nu }
such that it satisfies the first constraint in (19b) [15], [19], and numerically investigate \boldsymbol {\nu }
in Section V. We reformulate P1 as follows \begin{align*} \mathbf {P2}~:~\underset {\substack { \boldsymbol {\eta, \Theta }}}{\text {max}}\,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}(\boldsymbol {\eta, \Theta })}{p^{d}_{k}(\boldsymbol {\eta })} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta })}{p^{u}_{l}(\boldsymbol {\Theta })} \\ \text {s.t.}\,\,&S^{d}_{k}(\boldsymbol {\eta, \Theta }) \geq S^{d}_{ok}, S^{u}_{l}(\boldsymbol {\eta, \Theta }) \geq S^{u}_{ol}, \\&(2),~(3).\tag{20}\end{align*}
View Source\begin{align*} \mathbf {P2}~:~\underset {\substack { \boldsymbol {\eta, \Theta }}}{\text {max}}\,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} \frac {S^{d}_{k}(\boldsymbol {\eta, \Theta })}{p^{d}_{k}(\boldsymbol {\eta })} + \sum _{l=1}^{K_{u}} w^{u}_{l} \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta })}{p^{u}_{l}(\boldsymbol {\Theta })} \\ \text {s.t.}\,\,&S^{d}_{k}(\boldsymbol {\eta, \Theta }) \geq S^{d}_{ok}, S^{u}_{l}(\boldsymbol {\eta, \Theta }) \geq S^{u}_{ol}, \\&(2),~(3).\tag{20}\end{align*}
The objective in P2 is a sum of ratios, each of which is a PC function (concave-over-linear) of power control coefficients \{ \boldsymbol {\eta }, \boldsymbol {\Theta }\}
. It is, therefore, not guaranteed to be a PC function and Dinkelbach’s algorithm cannot be applied to maximize it [22]. This makes it a much harder objective to optimize as opposed to the more commonly studied GEE metric, which is a PC function [22] and has been investigated for CF mMIMO systems [18]–[21].
We now maimize WSEE centrally and decentrally using a two-layered approach. The first layer comprises an SCA framework, which formulates a GCP by approximating the non-convex objective and constraints in P2 as convex. In the second layer, the approximate GCP formed in the n
th SCA iteration is either solved centrally or decentrally using ADMM.
Since the approximate GCP obtained in the first layer, due to coupled optimization variables, is not in the standard ADMM form, we introduce their local and global versions. The sub-problems to update local variables are solved independently, and the local variables are coordinated to calculate the global solution [27], [35]. The updation of variables and coordination continues till ADMM converges. The obtained solution is then used to formulate GCP for the (n+1)
th SCA iteration.
A. SCA Framework
We now first linearize the non-convex objective in P2 using epigraph transformation as [34] \begin{align*} \mathbf {P3}~:~\underset {\substack { \boldsymbol {\eta, \Theta, f^{d}, f^{u}}}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\ \text {s.t.} \,\, &f^{d}_{k} \leq \frac {S^{d}_{k}(\boldsymbol {\eta, \Theta })}{p^{d}_{k}(\boldsymbol {\eta })}, f^{u}_{l} \leq \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta })}{p^{u}_{l}(\boldsymbol {\Theta })}, \\&(2),~(3),~(20).\tag{21}\end{align*}
View Source\begin{align*} \mathbf {P3}~:~\underset {\substack { \boldsymbol {\eta, \Theta, f^{d}, f^{u}}}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\ \text {s.t.} \,\, &f^{d}_{k} \leq \frac {S^{d}_{k}(\boldsymbol {\eta, \Theta })}{p^{d}_{k}(\boldsymbol {\eta })}, f^{u}_{l} \leq \frac {S^{u}_{l}(\boldsymbol {\eta, \Theta })}{p^{u}_{l}(\boldsymbol {\Theta })}, \\&(2),~(3),~(20).\tag{21}\end{align*}
Here \boldsymbol {f}^{\varepsilon } \triangleq [f^{\varepsilon }_{1} {\dots } f^{\varepsilon }_{K_{\varepsilon }}] \in \mathbb {C}^{K_{\varepsilon } \times 1}
are slack variables [34]. To approximate the non-convex constraints in (20) and (21) as convex, we substitute S^{d}_{k}
and S^{u}_{l}
from (14)–(15) and cross-multiply the terms p^{d}_{k}, p^{u}_{l}
and f^{d}_{k}, f^{u}_{l}
in (21). We also introduce slack variables \boldsymbol {\Psi }^{\varepsilon } \triangleq [\Psi ^{\varepsilon }_{1}, {\dots }, \Psi ^{\varepsilon }_{K_{\varepsilon }}] \in \mathbb {C}^{K_{\varepsilon } \times 1}
, \boldsymbol {\zeta }^{\varepsilon } \triangleq [\zeta ^{\varepsilon }_{1}, {\dots }, \zeta ^{\varepsilon }_{K_{\varepsilon }}] \in \mathbb {C}^{K_{\varepsilon } \times 1}
and equivalently cast P3 as [22]\begin{align*}&2\mathbf {P4}~:~\underset {\substack { \boldsymbol { \eta,\Theta,f^{d},f^{u},}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},\zeta ^{u}}}}{\text {max}} \sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.} ~p^{d}_{k} \leq \frac {(\Psi ^{d}_{k})^{2}}{f^{d}_{k}}, p^{u}_{l} \leq \frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}, \tag{22a}\\&\, \left ({\Psi ^{d}_{k}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{d}_{k}}\right),\left ({\Psi ^{u}_{l}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{u}_{l}}\right), \qquad \tag{22b}\\&\,\zeta ^{d}_{k} \leq \frac {\left ({\sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \sqrt {\eta _{mk}}}\right)^{2}}{\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \eta _{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1}, \tag{22c}\\&\, \zeta ^{u}_{l} \leq \frac {A^{u}_{l} \theta _{l}}{\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \eta _{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l}}, \qquad \tag{22d}\\&\, \log _{2}\left ({1 + \zeta ^{d}_{k}}\right) \geq S^{d}_{ok}/\tau _{f}, \log _{2}\left ({1 + \zeta ^{u}_{l}}\right) \geq S^{u}_{ol}/\tau _{f}, \\&(2),~(3).\tag{22e}\end{align*}
View Source\begin{align*}&2\mathbf {P4}~:~\underset {\substack { \boldsymbol { \eta,\Theta,f^{d},f^{u},}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},\zeta ^{u}}}}{\text {max}} \sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.} ~p^{d}_{k} \leq \frac {(\Psi ^{d}_{k})^{2}}{f^{d}_{k}}, p^{u}_{l} \leq \frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}, \tag{22a}\\&\, \left ({\Psi ^{d}_{k}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{d}_{k}}\right),\left ({\Psi ^{u}_{l}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{u}_{l}}\right), \qquad \tag{22b}\\&\,\zeta ^{d}_{k} \leq \frac {\left ({\sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \sqrt {\eta _{mk}}}\right)^{2}}{\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \eta _{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1}, \tag{22c}\\&\, \zeta ^{u}_{l} \leq \frac {A^{u}_{l} \theta _{l}}{\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \eta _{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l}}, \qquad \tag{22d}\\&\, \log _{2}\left ({1 + \zeta ^{d}_{k}}\right) \geq S^{d}_{ok}/\tau _{f}, \log _{2}\left ({1 + \zeta ^{u}_{l}}\right) \geq S^{u}_{ol}/\tau _{f}, \\&(2),~(3).\tag{22e}\end{align*}
We introduce the variable c_{mk} \triangleq \sqrt {\eta _{mk}}
and denote \boldsymbol {C} \triangleq \{c_{mk}\} \in \mathbb {C}^{M \times K_{d}}
to remove concave terms in (22c) arising due to \sqrt {\eta _{mk}}
and facilitate its conversion into a convex constraint. We introduce additional slack variables \boldsymbol {\lambda }^{\varepsilon } \triangleq [\lambda ^{\varepsilon }_{1}, {\dots }, \lambda ^{\varepsilon }_{K_{\varepsilon }}] \in \mathbb {C}^{K_{\varepsilon } \times 1}
to further simplify the non-convex constraints (22c)–(22d). We now cast P4 equivalently as \begin{align*}&\mathbf {P5}~:~\underset {\substack { \boldsymbol {C,\Theta,f^{d},f^{u}}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},}\\ \boldsymbol {\zeta ^{u},\lambda ^{d},\lambda ^{u}}}}{\text {max}} \sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.}~\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} c^{2}_{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1 \leq \frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}, \tag{23a}\\&\,\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} c^{2}_{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l} \leq \frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}, \qquad \tag{23b}\\&\,\lambda ^{d}_{k} \leq \sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} c_{mk}, \left ({\lambda ^{u}_{l}}\right)^{2} \leq A^{u}_{l} \theta _{l}, \tag{23c}\\&\,\lambda ^{d}_{k} \geq 0, \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} c^{2}_{mk} \leq \frac {1}{N_{t}}, c_{mk} \geq 0, \\& \text{(22a), (22b), (22e), (3)}.\tag{23d}\end{align*}
View Source\begin{align*}&\mathbf {P5}~:~\underset {\substack { \boldsymbol {C,\Theta,f^{d},f^{u}}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},}\\ \boldsymbol {\zeta ^{u},\lambda ^{d},\lambda ^{u}}}}{\text {max}} \sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.}~\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} c^{2}_{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1 \leq \frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}, \tag{23a}\\&\,\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} c^{2}_{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l} \leq \frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}, \qquad \tag{23b}\\&\,\lambda ^{d}_{k} \leq \sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} c_{mk}, \left ({\lambda ^{u}_{l}}\right)^{2} \leq A^{u}_{l} \theta _{l}, \tag{23c}\\&\,\lambda ^{d}_{k} \geq 0, \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} c^{2}_{mk} \leq \frac {1}{N_{t}}, c_{mk} \geq 0, \\& \text{(22a), (22b), (22e), (3)}.\tag{23d}\end{align*}
We note that P5 has all convex constraints except (22a) and (23a)–(23b). Since a first-order Taylor approximation is a global under-estimator of a convex function [34], we now linearize the right-hand side of these constraints. At the n
th iteration, we substitute first-order Taylor approximation \frac {f^{2}_{1}}{f_{2}} \geq 2 \frac {f^{(n)}_{1}}{f^{(n)}_{2}} f_{1} - \frac {(f^{(n)}_{1})^{2}}{(f^{(n)}_{2})^{2}}f_{2} \triangleq \Lambda ^{(n)} \left({\frac {f^{2}_{1}}{f_{2}}}\right)
and use (16)–(17) to recast P5 into a GCP: \begin{align*}&\mathbf {P6}~:~\underset {\substack { \boldsymbol {C,\Theta,f^{d},f^{u}}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},}\\ \boldsymbol {\zeta ^{u},\lambda ^{d},\lambda ^{u}}}}{\text {max}}~\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.}~\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} c^{2}_{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1 \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}}\right), \\ \tag{24a}\\&\,\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} c^{2}_{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}}\right), \\ \tag{24b}\\&\,P_{\text {fix}} + N_{t}\rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {\gamma ^{d}_{mk} c^{2}_{mk}}{\alpha _{m}}+ P^{d}_{\text {tc},k} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{d}_{k}}\right)^{2}}{f^{d}_{k}}}\right), \\ \tag{24c}\\&\,P_{\text {fix}} + \rho _{u} N_{0} \frac {\theta _{l}}{\alpha '_{l}} + P^{u}_{\text {tc},l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}}\right), \\& \text{(3), (22b), (22e), (23c), (23d)}.\tag{24d}\end{align*}
View Source\begin{align*}&\mathbf {P6}~:~\underset {\substack { \boldsymbol {C,\Theta,f^{d},f^{u}}\\ \boldsymbol {\Psi ^{d},\Psi ^{u},\zeta ^{d},}\\ \boldsymbol {\zeta ^{u},\lambda ^{d},\lambda ^{u}}}}{\text {max}}~\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\&\text {s.t.}~\sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} c^{2}_{mq} + \sum _{l=1}^{K_{u}} D^{d}_{kl} \theta _{l} + 1 \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}}\right), \\ \tag{24a}\\&\,\sum _{q=1}^{K_{u}} B^{u}_{lq} \theta _{q} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} c^{2}_{ik} + E^{u}_{l} \theta _{l} + F^{u}_{l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}}\right), \\ \tag{24b}\\&\,P_{\text {fix}} + N_{t}\rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {\gamma ^{d}_{mk} c^{2}_{mk}}{\alpha _{m}}+ P^{d}_{\text {tc},k} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{d}_{k}}\right)^{2}}{f^{d}_{k}}}\right), \\ \tag{24c}\\&\,P_{\text {fix}} + \rho _{u} N_{0} \frac {\theta _{l}}{\alpha '_{l}} + P^{u}_{\text {tc},l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}}\right), \\& \text{(3), (22b), (22e), (23c), (23d)}.\tag{24d}\end{align*}
We next provide a centralized SCA to solve P6 in the second layer in Algorithm 3.
The SCA procedure converges when \| \boldsymbol {r}_{\text {SCA}}^{(n)}\| = \sqrt {\| \boldsymbol {C}^{(n+1)} - \boldsymbol {C}^{(n)}\|^{2}_{F} + \| \boldsymbol {\Theta }^{(n+1)} - \boldsymbol {\Theta }^{(n)}\|^{2}}
has magnitude \| \boldsymbol {r}_{\text {SCA}}^{(n)}\| \leq \epsilon _{\text {SCA}}
, where \epsilon _{\text {SCA}}
is the convergence threshold.
At the n
th SCA iteration, P6 is obtained from P5 by applying first-order Taylor approximations to the constraints (22a) and (23a)–(23b). These approximations are of the form \Lambda (\boldsymbol {x}) \triangleq \frac {x_{1}^{2}}{x_{2}} \geq 2 \frac {x_{1}^{(n)}}{x_{2}^{(n)}} x_{1} - \left({\frac {x_{1}^{(n)}}{x_{2}^{(n)}}}\right)^{2} x_{2} \triangleq \bar {\Lambda }(\boldsymbol {x}, \boldsymbol {x}^{(n)})
. It is easy to show that P6 is the inner-approximation problem for P5, where we replace each of the constraints (22a) and (23a)–(23b), denoted here as g_{i}(\boldsymbol {x}) \leq 0, i = 1, 2, 3
, with a convex approximation of the form \bar {g}_{i}(\boldsymbol {x}, \boldsymbol {x}^{(n)}) \leq 0, i = 1, 2, 3
. For each of the approximations, it can be easily shown that the following properties hold [36]: i) g_{i}(\boldsymbol {x}) \leq \bar {g}_{i}(\boldsymbol {x}, \boldsymbol {x}^{(n)})
for all feasible \boldsymbol {x}
; ii) g_{i}(\boldsymbol {x}^{(n)}) = \bar {g}_{i}(\boldsymbol {x}^{n}, \boldsymbol {x}^{(n)})
; and \frac {\partial g_{i}(\boldsymbol {x}^{(n)})}{\partial x_{j}} = \frac {\partial \bar {g}_{i}(\boldsymbol {x}^{n}, \boldsymbol {x}^{(n)})}{\partial x_{j}}
, j = 1, 2
. The constraints in P6 also satisfy Slater’s conditions [34].
This implies that Algorithm 3, by solving the inner-approximation problem, always converges to a KKT point of P2 due to [36]. It must be noted here that even though Algorithm 3 solves the approximate problem P6 in each SCA iteration, it is provably optimal after sufficient number of iterations. This is due to the fact that it provably converges to a KKT point of P2 which is an optimal solution [34].
B. Decentralized ADMM Approach
We now use ADMM to decentrally solve P6 in the second layer, an approach well-suited for CPUs with multiple distributed D-servers, connected via a central C-server [25], [26]. The ADMM method decomposes a central problem into multiple sub-problems, each of which is solved by a D-server locally and independently. The C-server combines the local solutions to obtain a global solution. We observe that the constraints in (24a)–(24b) couple the power control coefficients of different uplink and downlink UEs. We next introduce global variables for the power control coefficients at the C-server, with local copies at the D-servers to decouple P6 into sub-problems for each UE. We observe that the constraints in P6 for the downlink and uplink UEs can be divided between downlink and uplink D-servers, respectively. The D-servers solve sub-problems defined for each downlink and uplink UE. We first define local feasible sets at the n
th SCA iteration for them, which are denoted as \mathcal {S}_{k}^{d,(n)}
and \mathcal {S}_{l}^{u,(n)}
, respectively. These sets are given as follows \begin{align*}&\mathcal {S}_{k}^{d,(n)} = \Biggl \{{f^{d}_{k}, \Psi ^{d}_{k}, \zeta ^{d}_{k}, \lambda ^{d}_{k}, \widetilde{\boldsymbol {C}}^{d}_{k}, \widetilde{\boldsymbol {\Theta }}^{d}_{k} } {\vert} \tilde {b} \sum _{q \in \kappa _{dm}} \gamma ^{d}_{mk} \left ({\tilde {c}^{d}_{mq,k}}\right)^{2} \leq \frac {1}{N_{t}}, \\ \tag{25a}\\&\, \lambda ^{d}_{k} \leq \sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \tilde {c}^{d}_{mk,k}, \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \left ({\tilde {c}^{d}_{mq,k}}\right)^{2} \\&\quad +\,\,\sum _{l=1}^{K_{u}} D^{d}_{kl} \tilde {\theta }^{d}_{l,k} + 1 \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}}\right), \tag{25b}\\&\, \left ({\Psi _{k}^{d}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{d}_{k}}\right), P_{\text {fix}} + N_{t} \rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {1}{\alpha _{m}} \gamma ^{d}_{mk} \left ({\tilde {c}^{d}_{mk,k}}\right)^{2} \\&\quad +\,\,P^{d}_{\text {tc},k} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{d}_{k}}\right)^{2}}{f^{d}_{k}}}\right), \tag{25c}\\&\,\tilde {c}^{d}_{mq,k} \geq 0 \, \forall q = 1~\text {to}~K_{d}, 0 \leq \tilde {\theta }^{d}_{l,k} \leq 1, \lambda ^{d}_{k} \geq 0, \\&\,{\log _{2}\left ({1 + \zeta ^{d}_{k}}\right) \geq S^{d}_{ok}/\tau _{f}}\Biggr \}, \tag{25d}\\&\, \mathcal {S}_{l}^{u,(n)} = \Biggl \{{f^{u}_{l}, \Psi ^{u}_{l}, \zeta ^{u}_{l}, \lambda ^{u}_{l}, \widetilde{\boldsymbol {C}^{u}_{l},\widetilde {\Theta }^{u}_{l}} } { \vert} \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} \left ({\tilde {c}^{u}_{mk,l}}\right)^{2} \leq \frac {1}{N_{t}}, \\ \tag{25e}\\&\,\left ({\lambda ^{u}_{l}}\right)^{2} \leq A^{u}_{l} \tilde {\theta }^{u}_{l,l}, \sum _{q=1}^{K_{u}} B^{u}_{lq} \tilde {\theta }^{u}_{q,l} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \left ({\tilde {c}^{u}_{ik,l}}\right)^{2} \\&\quad +\,\,E^{u}_{l} \tilde {\theta }^{u}_{l,l} + F^{u}_{l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}}\right), \tag{25f}\\&\,\left ({\Psi _{l}^{u}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{u}_{l}}\right), \\&\,P_{\text {fix}} + \rho _{u} N_{0} \frac {1}{\alpha '_{l}} \tilde {\theta }^{u}_{l,l} + P^{u}_{\text {tc},l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}}\right), \tag{25g}\\&\, \tilde {c}^{u}_{mk,l} \geq 0, 0 \leq \tilde {\theta }^{u}_{q,l} \leq 1 \, \forall q = 1~\text {to}~K_{u}, \\&\,{\log _{2}\left ({1 + \zeta ^{u}_{l}}\right) \geq S^{u}_{ol}/\tau _{f} \vphantom {f^{u}_{l}, \Psi ^{u}_{l}, \zeta ^{u}_{l}, \lambda ^{u}_{l}, \widetilde{\boldsymbol {C}^{u}_{l},\widetilde {\Theta }^{u}_{l}}}}\Biggr \}.\tag{25h}\end{align*}
View Source\begin{align*}&\mathcal {S}_{k}^{d,(n)} = \Biggl \{{f^{d}_{k}, \Psi ^{d}_{k}, \zeta ^{d}_{k}, \lambda ^{d}_{k}, \widetilde{\boldsymbol {C}}^{d}_{k}, \widetilde{\boldsymbol {\Theta }}^{d}_{k} } {\vert} \tilde {b} \sum _{q \in \kappa _{dm}} \gamma ^{d}_{mk} \left ({\tilde {c}^{d}_{mq,k}}\right)^{2} \leq \frac {1}{N_{t}}, \\ \tag{25a}\\&\, \lambda ^{d}_{k} \leq \sum _{m \in \mathcal {M}^{d}_{k}} A^{d}_{mk} \tilde {c}^{d}_{mk,k}, \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} B^{d}_{kmq} \left ({\tilde {c}^{d}_{mq,k}}\right)^{2} \\&\quad +\,\,\sum _{l=1}^{K_{u}} D^{d}_{kl} \tilde {\theta }^{d}_{l,k} + 1 \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{d}_{k}}\right)^{2}}{\zeta ^{d}_{k}}}\right), \tag{25b}\\&\, \left ({\Psi _{k}^{d}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{d}_{k}}\right), P_{\text {fix}} + N_{t} \rho _{d} N_{0} \sum _{m \in \mathcal {M}^{d}_{k}} \frac {1}{\alpha _{m}} \gamma ^{d}_{mk} \left ({\tilde {c}^{d}_{mk,k}}\right)^{2} \\&\quad +\,\,P^{d}_{\text {tc},k} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{d}_{k}}\right)^{2}}{f^{d}_{k}}}\right), \tag{25c}\\&\,\tilde {c}^{d}_{mq,k} \geq 0 \, \forall q = 1~\text {to}~K_{d}, 0 \leq \tilde {\theta }^{d}_{l,k} \leq 1, \lambda ^{d}_{k} \geq 0, \\&\,{\log _{2}\left ({1 + \zeta ^{d}_{k}}\right) \geq S^{d}_{ok}/\tau _{f}}\Biggr \}, \tag{25d}\\&\, \mathcal {S}_{l}^{u,(n)} = \Biggl \{{f^{u}_{l}, \Psi ^{u}_{l}, \zeta ^{u}_{l}, \lambda ^{u}_{l}, \widetilde{\boldsymbol {C}^{u}_{l},\widetilde {\Theta }^{u}_{l}} } { \vert} \tilde {b} \sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk} \left ({\tilde {c}^{u}_{mk,l}}\right)^{2} \leq \frac {1}{N_{t}}, \\ \tag{25e}\\&\,\left ({\lambda ^{u}_{l}}\right)^{2} \leq A^{u}_{l} \tilde {\theta }^{u}_{l,l}, \sum _{q=1}^{K_{u}} B^{u}_{lq} \tilde {\theta }^{u}_{q,l} + \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} D^{u}_{lik} \left ({\tilde {c}^{u}_{ik,l}}\right)^{2} \\&\quad +\,\,E^{u}_{l} \tilde {\theta }^{u}_{l,l} + F^{u}_{l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\lambda ^{u}_{l}}\right)^{2}}{\zeta ^{u}_{l}}}\right), \tag{25f}\\&\,\left ({\Psi _{l}^{u}}\right)^{2} \leq \tau _{f} \log _{2} \left ({1 + \zeta ^{u}_{l}}\right), \\&\,P_{\text {fix}} + \rho _{u} N_{0} \frac {1}{\alpha '_{l}} \tilde {\theta }^{u}_{l,l} + P^{u}_{\text {tc},l} \leq \Lambda ^{(n)} \left ({\frac {\left ({\Psi ^{u}_{l}}\right)^{2}}{f^{u}_{l}}}\right), \tag{25g}\\&\, \tilde {c}^{u}_{mk,l} \geq 0, 0 \leq \tilde {\theta }^{u}_{q,l} \leq 1 \, \forall q = 1~\text {to}~K_{u}, \\&\,{\log _{2}\left ({1 + \zeta ^{u}_{l}}\right) \geq S^{u}_{ol}/\tau _{f} \vphantom {f^{u}_{l}, \Psi ^{u}_{l}, \zeta ^{u}_{l}, \lambda ^{u}_{l}, \widetilde{\boldsymbol {C}^{u}_{l},\widetilde {\Theta }^{u}_{l}}}}\Biggr \}.\tag{25h}\end{align*}
Here \widetilde { \boldsymbol {C}}^{d}_{k}, \widetilde { \boldsymbol {C}}^{u}_{l} \in \mathbb {C}^{M \times K_{d}}
and \widetilde { \boldsymbol {\Theta }}^{d}_{k}, \widetilde { \boldsymbol {\Theta }}^{u}_{l} \in \mathbb {C}^{K_{u} \times 1}
are local copies at the D-server of the corresponding global variables at the C-server, which are denoted as \widetilde { \boldsymbol {C}} \in \mathbb {C}^{M \times K_{d}}
and \widetilde { \boldsymbol {\Theta }} \in \mathbb {C}^{K_{u} \times 1}
respectively, and represent the downlink and uplink power control coefficients, \boldsymbol {C}
and \boldsymbol {\Theta }
, in P6. We note that each D-server has its local power control variables and hence the constraints in (25), which are all convex, are independent for each D-server. This ensures that the sets \mathcal {S}_{k}^{d,(n)}
and \mathcal {S}_{l}^{u,(n)}
are convex. We define the sets of local variables for the D-servers corresponding to the downlink and uplink UEs as \boldsymbol {\Omega }^{d}_{k} \triangleq [\widetilde { \boldsymbol {C}}^{d}_{k}, \widetilde { \boldsymbol {\Theta }}^{d}_{k}, f^{d}_{k}, \Psi ^{d}_{k}, \lambda ^{d}_{k}, \zeta ^{d}_{k}]
and \boldsymbol {\Omega }^{u}_{l} \triangleq [\widetilde { \boldsymbol {C}}^{u}_{l},\widetilde { \boldsymbol {\Theta }}^{u}_{l}, f^{u}_{l}, \Psi ^{u}_{l}, \lambda ^{u}_{l}, \zeta ^{u}_{l}]
respectively.
We now reformulate P6 as follows \begin{align*}{2} \mathbf {P7}~:~\underset {\substack { \widetilde{\boldsymbol {C},\widetilde {\Theta }, \boldsymbol {\Omega }^{d}_{k}, \boldsymbol {\Omega }^{u}_{l}}}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\ \text {s.t.} \,\, &\boldsymbol {\Omega }^{d}_{k} \in \mathcal {S}_{k}^{d,(n)}, \boldsymbol {\Omega }^{u}_{l} \in \mathcal {S}_{l}^{u,(n)}, \tag{26a}\\&\widetilde { \boldsymbol {C}}^{d}_{k} = \widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {C}}^{u}_{l} = \widetilde { \boldsymbol {C}}, \tag{26b}\\&\widetilde { \boldsymbol {\Theta }}^{d}_{k} = \widetilde { \boldsymbol {\Theta }}, \widetilde { \boldsymbol {\Theta }}^{u}_{l} = \widetilde { \boldsymbol {\Theta }}. \tag{26c}\end{align*}
View Source\begin{align*}{2} \mathbf {P7}~:~\underset {\substack { \widetilde{\boldsymbol {C},\widetilde {\Theta }, \boldsymbol {\Omega }^{d}_{k}, \boldsymbol {\Omega }^{u}_{l}}}}{\text {max}} \,\, &\sum _{k=1}^{K_{d}} w^{d}_{k} f^{d}_{k} + \sum _{l=1}^{K_{u}} w^{u}_{l} f^{u}_{l} \\ \text {s.t.} \,\, &\boldsymbol {\Omega }^{d}_{k} \in \mathcal {S}_{k}^{d,(n)}, \boldsymbol {\Omega }^{u}_{l} \in \mathcal {S}_{l}^{u,(n)}, \tag{26a}\\&\widetilde { \boldsymbol {C}}^{d}_{k} = \widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {C}}^{u}_{l} = \widetilde { \boldsymbol {C}}, \tag{26b}\\&\widetilde { \boldsymbol {\Theta }}^{d}_{k} = \widetilde { \boldsymbol {\Theta }}, \widetilde { \boldsymbol {\Theta }}^{u}_{l} = \widetilde { \boldsymbol {\Theta }}. \tag{26c}\end{align*}
To ensure that the global variables at the C-server have identical local copies maintained at the D-servers, we introduce the consensus constraints (26b)–(26c). The ADMM algorithm can now be readily applied to P7 as it is in the global consensus form [35].
We use \varepsilon \triangleq \{d,u\}
to denote the downlink and uplink respectively, and \phi \triangleq \{k,l\}
to denote k
th the downlink UE and l
th uplink UE, respectively. The sub-problems of individual D-servers can now be written as follows \begin{align*} \mathbf {P7b}~:~\underset {\substack { \widetilde{\boldsymbol {C}}, \widetilde{\boldsymbol {\Theta }}, \boldsymbol {\Omega }^{\varepsilon }_{\phi }}}{\text {max}} \,\, &w^{\varepsilon }_{\phi } f^{\varepsilon }_{\phi } \\ \text {s.t.} \,\, &\boldsymbol {\Omega }^{\varepsilon }_{\phi } \in \mathcal {S}_{\phi }^{\varepsilon,(n)}, \widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } = \widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } = \widetilde { \boldsymbol {\Theta }}.\end{align*}
View Source\begin{align*} \mathbf {P7b}~:~\underset {\substack { \widetilde{\boldsymbol {C}}, \widetilde{\boldsymbol {\Theta }}, \boldsymbol {\Omega }^{\varepsilon }_{\phi }}}{\text {max}} \,\, &w^{\varepsilon }_{\phi } f^{\varepsilon }_{\phi } \\ \text {s.t.} \,\, &\boldsymbol {\Omega }^{\varepsilon }_{\phi } \in \mathcal {S}_{\phi }^{\varepsilon,(n)}, \widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } = \widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } = \widetilde { \boldsymbol {\Theta }}.\end{align*}
We now define auxiliary functions for the objective in P7b as follows \begin{align*} q^{\varepsilon }_{\phi }\left ({\boldsymbol {\Omega }^{\varepsilon }_{\phi }}\right)\triangleq\begin{cases} w^{\varepsilon }_{\phi } f^{\varepsilon }_{\phi }, & \boldsymbol {\Omega }^{\varepsilon }_{\phi } \in S_{\phi }^{\varepsilon,(n)}, \\ -\infty, & \text {otherwise}. \end{cases} \tag{27}\end{align*}
View Source\begin{align*} q^{\varepsilon }_{\phi }\left ({\boldsymbol {\Omega }^{\varepsilon }_{\phi }}\right)\triangleq\begin{cases} w^{\varepsilon }_{\phi } f^{\varepsilon }_{\phi }, & \boldsymbol {\Omega }^{\varepsilon }_{\phi } \in S_{\phi }^{\varepsilon,(n)}, \\ -\infty, & \text {otherwise}. \end{cases} \tag{27}\end{align*}
We write, using (27), the augmented Lagrangian function for P7 as \begin{align*}&\mathcal {L}^{(n)} \left ({\widetilde{\boldsymbol {C}, \widetilde {\Theta }}, \left \{{ \boldsymbol {\Omega ^{d}_{k}, \chi ^{d}_{k}, \xi ^{d}_{k}}}\right \}, \left \{{ \boldsymbol {\Omega ^{u}_{l}, \chi ^{u}_{l}, \xi ^{u}_{l}}}\right \} }\right) \\&\,= \sum _{k=1}^{K_{d}}\Biggl ({q^{d}_{k}\left ({\boldsymbol {\Omega }^{d}_{k}}\right) - \langle \boldsymbol {\chi }^{d}_{k}, \widetilde { \boldsymbol {C}}^{d}_{k} - \widetilde { \boldsymbol {C}}\rangle - \frac {\rho _{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{d}_{k} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} } \\&\quad { - \left \langle{ \boldsymbol {\xi }^{d}_{k}, \widetilde { \boldsymbol {\Theta }}^{d}_{k} - \widetilde { \boldsymbol {\Theta }}}\right \rangle - \frac {\rho _{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{d}_{k} - \widetilde { \boldsymbol {\Theta }}}\right \|^{2} }\Biggr ) \\&\quad +\,\,\sum _{l=1}^{K_{u}}\Biggl ({q^{u}_{l}\left ({\boldsymbol {\Omega }^{u}_{l}}\right) - \left \langle{ \boldsymbol {\chi }^{u}_{l}, \widetilde { \boldsymbol {C}}^{u}_{l} - \widetilde { \boldsymbol {C}} }\right \rangle - \frac {\rho _{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{u}_{l} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} } \\&\quad { - \left \langle{ \boldsymbol {\xi }^{u}_{l}, \widetilde { \boldsymbol {\Theta }}^{u}_{l} - \widetilde { \boldsymbol {\Theta }} }\right \rangle - \frac {\rho _{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{u}_{l} - \widetilde { \boldsymbol {\Theta }}}\right \|^{2} }\Biggr ), \tag{28}\end{align*}
View Source\begin{align*}&\mathcal {L}^{(n)} \left ({\widetilde{\boldsymbol {C}, \widetilde {\Theta }}, \left \{{ \boldsymbol {\Omega ^{d}_{k}, \chi ^{d}_{k}, \xi ^{d}_{k}}}\right \}, \left \{{ \boldsymbol {\Omega ^{u}_{l}, \chi ^{u}_{l}, \xi ^{u}_{l}}}\right \} }\right) \\&\,= \sum _{k=1}^{K_{d}}\Biggl ({q^{d}_{k}\left ({\boldsymbol {\Omega }^{d}_{k}}\right) - \langle \boldsymbol {\chi }^{d}_{k}, \widetilde { \boldsymbol {C}}^{d}_{k} - \widetilde { \boldsymbol {C}}\rangle - \frac {\rho _{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{d}_{k} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} } \\&\quad { - \left \langle{ \boldsymbol {\xi }^{d}_{k}, \widetilde { \boldsymbol {\Theta }}^{d}_{k} - \widetilde { \boldsymbol {\Theta }}}\right \rangle - \frac {\rho _{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{d}_{k} - \widetilde { \boldsymbol {\Theta }}}\right \|^{2} }\Biggr ) \\&\quad +\,\,\sum _{l=1}^{K_{u}}\Biggl ({q^{u}_{l}\left ({\boldsymbol {\Omega }^{u}_{l}}\right) - \left \langle{ \boldsymbol {\chi }^{u}_{l}, \widetilde { \boldsymbol {C}}^{u}_{l} - \widetilde { \boldsymbol {C}} }\right \rangle - \frac {\rho _{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{u}_{l} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} } \\&\quad { - \left \langle{ \boldsymbol {\xi }^{u}_{l}, \widetilde { \boldsymbol {\Theta }}^{u}_{l} - \widetilde { \boldsymbol {\Theta }} }\right \rangle - \frac {\rho _{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{u}_{l} - \widetilde { \boldsymbol {\Theta }}}\right \|^{2} }\Biggr ), \tag{28}\end{align*}
where \rho _{C}, \rho _{\theta } > 0
are the penalty parameters corresponding to the global variables \widetilde { \boldsymbol {C}}
and \widetilde { \boldsymbol {\Theta }}
respectively, and \boldsymbol {\chi }^{\varepsilon }_{\phi } \in \mathbb {C}^{M \times K_{d}}, \boldsymbol {\xi }^{\varepsilon }_{\phi } \in \mathbb {C}^{K_{u}\times 1}
are the Lagrangian variables associated with the equality constraints (26b) and (26c), respectively. The quadratic penalty terms are added to the objective to penalise equality constraints violations, and to enable the ADMM to converge by relaxing constraints of finiteness and strict convexity [35].
We note that the augmented Lagrangian in (28) is not decomposable in general for the problem formulation in P7b [34]. The auxiliary functions defined in (27) enable us to decompose it and formulate sub-problems for the D-servers. In ADMM method, the D-servers independently solve the sub-problems and update the local variables, which are collected by the C-server to update the global variables [35]. In the (p+1)
th iteration, following steps are executed in succession.
1) Local Computation:
The D-servers for each UE solve P8 to update the local variables as\begin{align*} \mathbf {P8}~:~ \boldsymbol {\Omega }_{\phi }^{\varepsilon,(p+1)}=&\underset {\substack { \boldsymbol {\Omega }^{\varepsilon }_{\phi }}}{ \mathop {\mathrm {arg\,max}} } \quad q^{\varepsilon }_{\phi }\left ({\boldsymbol {\Omega }^{\varepsilon }_{\phi }}\right) - \left \langle{ \boldsymbol {\chi }^{\varepsilon,(p)}_{\phi }, \widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {C}}^{(p)} }\right \rangle \\&{}-\,\,\left \langle{ \boldsymbol {\xi }^{\varepsilon,(p)}_{\phi }, \widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)} }\right \rangle \\&{}-\,\,\frac {\rho ^{(p)}_{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {C}}^{(p)}}\right \|^{2}_{F} - \frac {\rho ^{(p)}_{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right \|^{2}. \tag{29}\end{align*}
View Source\begin{align*} \mathbf {P8}~:~ \boldsymbol {\Omega }_{\phi }^{\varepsilon,(p+1)}=&\underset {\substack { \boldsymbol {\Omega }^{\varepsilon }_{\phi }}}{ \mathop {\mathrm {arg\,max}} } \quad q^{\varepsilon }_{\phi }\left ({\boldsymbol {\Omega }^{\varepsilon }_{\phi }}\right) - \left \langle{ \boldsymbol {\chi }^{\varepsilon,(p)}_{\phi }, \widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {C}}^{(p)} }\right \rangle \\&{}-\,\,\left \langle{ \boldsymbol {\xi }^{\varepsilon,(p)}_{\phi }, \widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)} }\right \rangle \\&{}-\,\,\frac {\rho ^{(p)}_{C}}{2} \left \|{\widetilde { \boldsymbol {C}}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {C}}^{(p)}}\right \|^{2}_{F} - \frac {\rho ^{(p)}_{\theta }}{2} \left \|{\widetilde { \boldsymbol {\Theta }}^{\varepsilon }_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right \|^{2}. \tag{29}\end{align*}
2) Lagrangian Multipliers Update:
The D-servers now update the Lagrangian multipliers as \begin{align*} \boldsymbol {\chi }_{\phi }^{\varepsilon,(p+1)}=&\boldsymbol {\chi }_{\phi }^{\varepsilon,(p)} + \rho ^{(p)}_{C} \left ({\widetilde { \boldsymbol {C}}^{\varepsilon,(p+1)}_{\phi } - \widetilde { \boldsymbol {C}}^{(p)}}\right) \tag{30}\\ \boldsymbol {\xi }_{\phi }^{\varepsilon,(p+1)}=&\boldsymbol {\xi }_{\phi }^{\varepsilon,(p)} + \rho ^{(p)}_{\theta } \left ({\widetilde { \boldsymbol {\Theta }}^{\varepsilon,(p+1)}_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right). \tag{31}\end{align*}
View Source\begin{align*} \boldsymbol {\chi }_{\phi }^{\varepsilon,(p+1)}=&\boldsymbol {\chi }_{\phi }^{\varepsilon,(p)} + \rho ^{(p)}_{C} \left ({\widetilde { \boldsymbol {C}}^{\varepsilon,(p+1)}_{\phi } - \widetilde { \boldsymbol {C}}^{(p)}}\right) \tag{30}\\ \boldsymbol {\xi }_{\phi }^{\varepsilon,(p+1)}=&\boldsymbol {\xi }_{\phi }^{\varepsilon,(p)} + \rho ^{(p)}_{\theta } \left ({\widetilde { \boldsymbol {\Theta }}^{\varepsilon,(p+1)}_{\phi } - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right). \tag{31}\end{align*}
3) Global Aggregation and Computation:
The C-server now collects the updated local variables and Lagrangian multipliers from the D-servers and updates the global variables \{\widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {\Theta }}\}
.\begin{align*}&\mathbf {P9}~:~\left \{{\widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {\Theta }}}\right \}^{(p+1)} = \underset {\substack { \widetilde{\boldsymbol {C}, \widetilde {\Theta }}}}{ \mathop {\mathrm {arg\,max}} } \quad \mathcal {L}^{(n)} \Biggl ({\widetilde{\boldsymbol {C}, \widetilde {\Theta }}, } \\&\qquad \qquad \qquad {\left \{{ \boldsymbol {\Omega ^{d}_{k}, \chi ^{d}_{k}, \xi ^{d}_{k}}}\right \}^{(p+1)}, \left \{{ \boldsymbol {\Omega ^{u}_{l}, \chi ^{u}_{l}, \xi ^{u}_{l}}}\right \}^{(p+1)} }\Biggr ).\end{align*}
View Source\begin{align*}&\mathbf {P9}~:~\left \{{\widetilde { \boldsymbol {C}}, \widetilde { \boldsymbol {\Theta }}}\right \}^{(p+1)} = \underset {\substack { \widetilde{\boldsymbol {C}, \widetilde {\Theta }}}}{ \mathop {\mathrm {arg\,max}} } \quad \mathcal {L}^{(n)} \Biggl ({\widetilde{\boldsymbol {C}, \widetilde {\Theta }}, } \\&\qquad \qquad \qquad {\left \{{ \boldsymbol {\Omega ^{d}_{k}, \chi ^{d}_{k}, \xi ^{d}_{k}}}\right \}^{(p+1)}, \left \{{ \boldsymbol {\Omega ^{u}_{l}, \chi ^{u}_{l}, \xi ^{u}_{l}}}\right \}^{(p+1)} }\Biggr ).\end{align*}
Using (28) and maximizing w.r.t. each global variable, we obtain a closed form solution\begin{align*} \widetilde { \boldsymbol {C}}^{(p+1)}=&\frac {1}{K} \Biggl ({\sum _{k=1}^{K_{d}} \left [{\widetilde { \boldsymbol {C}}^{d,(p+1)}_{k} + \frac {1}{\rho ^{(p)}_{C}} \boldsymbol {\chi }^{d,(p+1)}_{k}}\right] } \\& { +\,\,\sum _{l=1}^{K_{u}} \left [{\widetilde { \boldsymbol {C}}^{u,(p+1)}_{l} + \frac {1}{\rho ^{(p)}_{C}} \boldsymbol {\chi }^{u,(p+1)}_{l} }\right] }\Biggr ), \tag{32}\\ \widetilde { \boldsymbol {\Theta }}^{(p+1)}=&\frac {1}{K} \Biggl ({\sum _{k=1}^{K_{d}} \left [{\widetilde { \boldsymbol {\Theta }}^{d,(p+1)}_{k} + \frac {1}{\rho ^{(p)}_{\theta }} \boldsymbol {\xi }^{d,(p+1)}_{k} }\right] } \\& { +\,\,\sum _{l=1}^{K_{u}} \left [{\widetilde { \boldsymbol {\Theta }}^{u,(p+1)}_{l} + \frac {1}{\rho ^{(p)}_{\theta }} \boldsymbol {\xi }^{u,(p+1)}_{l} }\right] }\Biggr ). \tag{33}\end{align*}
View Source\begin{align*} \widetilde { \boldsymbol {C}}^{(p+1)}=&\frac {1}{K} \Biggl ({\sum _{k=1}^{K_{d}} \left [{\widetilde { \boldsymbol {C}}^{d,(p+1)}_{k} + \frac {1}{\rho ^{(p)}_{C}} \boldsymbol {\chi }^{d,(p+1)}_{k}}\right] } \\& { +\,\,\sum _{l=1}^{K_{u}} \left [{\widetilde { \boldsymbol {C}}^{u,(p+1)}_{l} + \frac {1}{\rho ^{(p)}_{C}} \boldsymbol {\chi }^{u,(p+1)}_{l} }\right] }\Biggr ), \tag{32}\\ \widetilde { \boldsymbol {\Theta }}^{(p+1)}=&\frac {1}{K} \Biggl ({\sum _{k=1}^{K_{d}} \left [{\widetilde { \boldsymbol {\Theta }}^{d,(p+1)}_{k} + \frac {1}{\rho ^{(p)}_{\theta }} \boldsymbol {\xi }^{d,(p+1)}_{k} }\right] } \\& { +\,\,\sum _{l=1}^{K_{u}} \left [{\widetilde { \boldsymbol {\Theta }}^{u,(p+1)}_{l} + \frac {1}{\rho ^{(p)}_{\theta }} \boldsymbol {\xi }^{u,(p+1)}_{l} }\right] }\Biggr ). \tag{33}\end{align*}
The updated global variables in (32)–(33) are broadcasted by the C-server to all the D-servers.
4) Residue Calculation and Penalty Parameter Updates:
The C-server calculates the squared magnitude of the primal and dual residuals, denoted as \boldsymbol {r}_{\text {ADMM}}
and \boldsymbol {s}_{\text {ADMM}}
respectively, as [35]\begin{align*} \left \|{ \boldsymbol {r}_{\text {ADMM}}^{(p+1)}}\right \|_{2}^{2}=&\sum _{k=1}^{K_{d}} \left ({\left \|{\widetilde { \boldsymbol {C}}_{k}^{d} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} + \left \|{\widetilde { \boldsymbol {\Theta }}_{k}^{d} - \widetilde { \boldsymbol {\Theta }}}\right \|_{2}^{2}}\right)^{(p+1)} \\&{}+\,\,\sum _{l=1}^{K_{u}} \left ({\left \|{\widetilde { \boldsymbol {C}}_{l}^{u} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} + \left \|{\widetilde { \boldsymbol {\Theta }}_{l}^{u} - \widetilde { \boldsymbol {\Theta }}}\right \|_{2}^{2}}\right)^{(p+1)}, \qquad \tag{34}\\ \left \|{ \boldsymbol {s}_{\text {ADMM}}^{(p+1)}}\right \|_{2}^{2}=&K \left ({\left \|{\widetilde { \boldsymbol {C}}^{(p+1)} - \widetilde { \boldsymbol {C}}^{(p)}}\right \|_{F}^{2} + \left \|{\widetilde { \boldsymbol {\Theta }}^{(p+1)} - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right \|_{2}^{2}}\right). \tag{35}\end{align*}
View Source\begin{align*} \left \|{ \boldsymbol {r}_{\text {ADMM}}^{(p+1)}}\right \|_{2}^{2}=&\sum _{k=1}^{K_{d}} \left ({\left \|{\widetilde { \boldsymbol {C}}_{k}^{d} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} + \left \|{\widetilde { \boldsymbol {\Theta }}_{k}^{d} - \widetilde { \boldsymbol {\Theta }}}\right \|_{2}^{2}}\right)^{(p+1)} \\&{}+\,\,\sum _{l=1}^{K_{u}} \left ({\left \|{\widetilde { \boldsymbol {C}}_{l}^{u} - \widetilde { \boldsymbol {C}}}\right \|^{2}_{F} + \left \|{\widetilde { \boldsymbol {\Theta }}_{l}^{u} - \widetilde { \boldsymbol {\Theta }}}\right \|_{2}^{2}}\right)^{(p+1)}, \qquad \tag{34}\\ \left \|{ \boldsymbol {s}_{\text {ADMM}}^{(p+1)}}\right \|_{2}^{2}=&K \left ({\left \|{\widetilde { \boldsymbol {C}}^{(p+1)} - \widetilde { \boldsymbol {C}}^{(p)}}\right \|_{F}^{2} + \left \|{\widetilde { \boldsymbol {\Theta }}^{(p+1)} - \widetilde { \boldsymbol {\Theta }}^{(p)}}\right \|_{2}^{2}}\right). \tag{35}\end{align*}
The C-server now compares the primal and dual residual norms obtained in (34)–(35). To accelerate convergence, it updates the penalty parameters for the (p+1)
th ADMM iteration, \rho ^{(p+1)}_{\{C\}}
and \rho ^{(p+1)}_{\{\theta \}}
, appropriately as follows [37]:\begin{align*} \rho ^{(p+1)}_{\{C,\theta \}}=\begin{cases} \rho ^{(p)}_{\{C,\theta \}}\vartheta ^{\text {incr}}, & \|r^{(p+1)}\|_{2} > \mu \|s^{(p+1)}\|_{2}, \\ \rho ^{(p)}_{\{C,\theta \}}/\vartheta ^{\text {decr}}, & \|s^{(p+1)}\|_{2} > \mu \|r^{(p+1)}\|_{2}, \\ \rho ^{(p)}_{\{C,\theta \}}, & \text {otherwise}. \end{cases} \tag{36}\end{align*}
View Source\begin{align*} \rho ^{(p+1)}_{\{C,\theta \}}=\begin{cases} \rho ^{(p)}_{\{C,\theta \}}\vartheta ^{\text {incr}}, & \|r^{(p+1)}\|_{2} > \mu \|s^{(p+1)}\|_{2}, \\ \rho ^{(p)}_{\{C,\theta \}}/\vartheta ^{\text {decr}}, & \|s^{(p+1)}\|_{2} > \mu \|r^{(p+1)}\|_{2}, \\ \rho ^{(p)}_{\{C,\theta \}}, & \text {otherwise}. \end{cases} \tag{36}\end{align*}
The parameters \mu > 1, \vartheta ^{\text {incr}} > 1, \vartheta ^{\text {decr}} > 1
are tuned to obtain good convergence [37].
Initialization for ADMM: At the (n+1)
th SCA iteration, we initialize the global variables at the C-server and their local copies at the D-servers with the SCA iteration variables as \begin{align*} \tilde {c}^{(1) }_{mk}=&c^{(n+1)}_{mk}, \tilde {\theta }^{(1) }_{l} = \theta ^{(n+1)}_{l}, \widetilde { \boldsymbol {C}}^{d,(1) }_{k} = \widetilde { \boldsymbol {C}}^{u,(1) }_{l} = \widetilde { \boldsymbol {C}}^{(1) }, \\ \widetilde { \boldsymbol {\Theta }}^{d,(1) }_{k}=&\widetilde { \boldsymbol {\Theta }}^{u,(1) }_{l} = \widetilde { \boldsymbol {\Theta }}^{(1) }. \tag{37}\end{align*}
View Source\begin{align*} \tilde {c}^{(1) }_{mk}=&c^{(n+1)}_{mk}, \tilde {\theta }^{(1) }_{l} = \theta ^{(n+1)}_{l}, \widetilde { \boldsymbol {C}}^{d,(1) }_{k} = \widetilde { \boldsymbol {C}}^{u,(1) }_{l} = \widetilde { \boldsymbol {C}}^{(1) }, \\ \widetilde { \boldsymbol {\Theta }}^{d,(1) }_{k}=&\widetilde { \boldsymbol {\Theta }}^{u,(1) }_{l} = \widetilde { \boldsymbol {\Theta }}^{(1) }. \tag{37}\end{align*}
ADMM Convergence Criterion: The ADMM can be said to have converged at iteration P
if the primal residue is within a pre-determined tolerance limit \epsilon _{\text {ADMM}}
, i.e., \|r^{(P)}\|_{2} \leq \epsilon _{\text {ADMM}}
. The steps (29), (30)–(31), (32)–(33) and (36) are iterated until convergence, after which we obtain the locally optimal power control coefficients \{\widetilde { \boldsymbol {C}}^{*},\widetilde { \boldsymbol {\Theta }}^{*}\}
. We assign them to the iterates for the (n+1)
th SCA iteration, i.e., \boldsymbol {C}^{(n+1)} = \widetilde { \boldsymbol {C}}^{*}, \boldsymbol {\Theta }^{(n+1)} = \widetilde { \boldsymbol {\Theta }}^{*}
. This concludes the n
th SCA iteration. The SCA is iterated till convergence. The steps for the decentralized WSEE maximization using SCA and ADMM are summarized in Algorithm 4.
The maximal ratio combiner/beamformer considered herein is the simplest receiver/transmitter for a distributed cell-free mMIMO system [2]. Further, the power optimization algorithms require only long-term fading channel coefficients, which remain constant for hundreds of coherence intervals [28]. This is in contrast to the existing work in SE-GEE maximization of FD cell-free massive MIMO systems in [21], which requires instantaneous channel. The current optimization problem whose reduced complexity is discussed below, therefore, needs to be solved over a relaxed time frame, which makes it easily implementable.
C. Computational Complexity of Centralized and Decentralized Algorithms
Before beginning this study, it is worth noting that both centralized Algorithm 3 and decentralized Algorithm 4 comprise of multiple steps that involve solving simple closed form expressions. These steps consume much lesser time than the ones which solve a GCP, typically using interior points methods [34]. We therefore compare the per-iteration complexity of centralized and decentralized algorithms by calculating the complexity of solving the respective GCPs.
Algorithm 3 solves P6 in step-1 of each SCA iteration, which has 4(K_{u}+K_{d}) + K_{u} + MK_{d}
real variables and 6(K_{u} + K_{d}) + M + MK_{d}
linear constraints. It has a worst-case computational complexity \mathcal {O}((10(K_{u}\!+\!K_{d})\! +\! K_{u} \!+\! M \!+\! 2MK_{d})^{3/2}(4(K_{u}\!+\!K_{d})\!+\! K_{u} \!+\! MK_{d})^{2} \!)
[38].
Algorithm 4, in step-2 of each ADMM iteration, solves P8 at the D-servers in parallel to update the local variables. We, therefore, need to analyse the computational complexity at any one of the D-servers. Since the downlink has an additional constraint (second one in (25d)), we consider a downlink D-server for worst-case complexity analysis, which in P8 has MK_{d} + K_{u} + 4
real variables and MK_{d} + M + K_{u} + 6
linear constraints. It will have a worst-case computational complexity [38]: \mathcal {O}((2MK_{d} + M + 2K_{u} + 10)^{3/2} (MK_{d} + K_{u} + 4)^{2})
.
We consider
K_{d} = K_{u} = K/2
uplink and downlink UEs for this analysis. We observe that for a large
K
,
Algorithm 4 has a much lower computational complexity than
Algorithm 3.
SECTION V.
Simulation Results
We now numerically investigate the SE and WSEE of a FD CF mMIMO system with limited-capacity fronthaul links. We assume a realistic system model wherein the M
APs, K_{d}
downlink UEs and K_{u}
uplink UEs are all scattered randomly in a square of size D
km \times D
km. To avoid the boundary effects [3], we wrap the APs and UEs around the edges [12]. We use \varepsilon \triangleq \{d,u\}
to denote downlink and uplink respectively, and \phi \triangleq \{k,l\}
to denote k
th downlink UE and l
th uplink UE, respectively. The large-scale fading coefficients, \beta ^{\varepsilon }_{m\phi }
, are modeled as [18] \begin{equation*} \beta ^{\varepsilon }_{m\phi } = 10^{\frac {\text {PL}^{\varepsilon }_{m\phi }}{10}} 10^{\frac {\sigma _{\text {sd}} z^{\varepsilon }_{m\phi }}{10}}.\tag{38}\end{equation*}
View Source\begin{equation*} \beta ^{\varepsilon }_{m\phi } = 10^{\frac {\text {PL}^{\varepsilon }_{m\phi }}{10}} 10^{\frac {\sigma _{\text {sd}} z^{\varepsilon }_{m\phi }}{10}}.\tag{38}\end{equation*}
Here 10^{\frac {\sigma _{\text {sd}} z^{\varepsilon }_{m\phi }}{10}}
is the log-normal shadowing factor with a standard deviation \sigma _{\text {sd}}
(in dB) and z^{\varepsilon }_{m\phi }
follows a two-components correlated model [3]. The path loss \text {PL}^{\varepsilon }_{m\phi }
(in dB) follows a three-slope model [3], [12].
We, similar to [12], model the large-scale fading coefficients for the inter-AP RI channels, i.e., \beta _{\text {RI}, mi}, \, \forall i \neq m
, as in (38), and assume that the large-scale fading for the intra-AP RI channels, which do not experience shadowing, are modeled as \beta _{\text {RI},mm} = 10^{\frac {\text {PL}_{\text {RI}} \text {(dB)}}{10}}
. The inter-UE large scale fading coefficients, \tilde {\beta }_{kl}
, are also modeled similar to (38). We consider, for brevity, the same number of quantization bits \nu
, and the same fronthaul capacity C_{\text {fh}}
for all links. We, henceforth, denote the transmit powers on the downlink and uplink as p_{d}\,\,(= \rho _{d} N_{0})
and p_{u}\,\,(= \rho _{u} N_{0})
, respectively, and the pilot transmit power as p_{t} (= \rho _{t} N_{0})
. We fix the system model values and power consumption model parameters, unless mentioned otherwise, as given in Table 1. These values are commonly used in the literature, e.g., [3], [12], [15].
Validation of SE expressions: We consider an FD CF mMIMO system with i) M = \{16, 32\}
APs, each having N_{t} = N_{r} = 8
transmit and receive antennas, K_{d} = 12
downlink UEs and K_{u} = 8
uplink UEs; and ii) unequal uplink and downlink transmit power, i.e., p_{d} = 2p_{u} = p
. We verify in Fig. 2 the tightness of the SE lower bound derived in (14)–(15), labeled as LB, by comparing it with the numerically-obtained ergodic SE in (10), labeled as upper-bound (UB) as it requires instantaneous CSI. The large-scale fading coefficients are set according to a practical FD CF channel model with parameters specified in Table 1. We, similar to [3], [18], allocate equal power to all downlink UEs and full power to all uplink UEs, i.e., \eta _{mk} = \left({{b}N_{t}\left({\sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk}}\right)}\right)^{-1}, \forall k \in \kappa _{dm}
and \theta _{l} = 1
. We see that the derived lower bound is tight for both values of M
.
Sum SE - FD and HD comparison: We consider an FD CF mMIMO system with M = 32
APs, K_{d} = 12
downlink UEs, K_{u} = 8
uplink UEs and with transmit powers p_{d} = 30
dBm, p_{u} = 27
dBm on the downlink and uplink. We compare in Fig. 3(a) the FD CF mMIMO system with varying levels of RI suppression factor \gamma _{\text {RI}}
and an equivalent HD system which serves uplink and downlink UEs in time-division duplex mode. For the HD system, we i) set \gamma _{\text {RI}} = 0
and inter-UE channel gains \tilde {\beta }_{kl} = 0
; ii) use all AP antennas, i.e., N = (N_{t} + N_{r})
, during uplink and downlink transmission; and iii) multiply sum SE with a factor of 1/2. We see that the FD system has a significantly higher sum SE than an equivalent HD system, provided the RI suppression is good, i.e., \gamma _{\text {RI}} \leq -10
dB. It is important to reemphasize here that the gains in sum SE achieved by the FD transmissions completely vanish with poor RI suppression, i.e., \gamma _{\text {RI}} > -10
dB. Moreover we note that, contrary to intuitive expectations, the sum SE does not double, even with significant RI suppression \gamma _{\text {RI}} \leq -40
dB. This is due to the UDI experienced by the downlink UEs in a FD CF mMIMO system as shown in Fig. 1, which cannot be mitigated by RI suppression at APs.
Sum SE - variation with quantization bits: We plot in Fig. 3(b) the sum SE by varying the number of fronthaul quantization bits \nu
. We consider M = 32
APs, K_{d} = 12
downlink UEs, K_{u} = 8
uplink UEs, and p_{d} = 2p_{u} = 30
dBm power for downlink and uplink, N_{t} = N_{r} = \{8,16\}
transmit and receive antennas on each AP, and fronthaul capacities C_{\text {fh}} = \{10, 100\}
Mbps. We observe that for both antenna configurations, sum SE increases with increase in \nu
initially and then saturates. Increasing \nu
reduces the quantization distortion and attenuation, which improves the sum SE. This effect, however, saturates as after a limit most of the information is retrieved. We observe that reducing the fronthaul capacity from C_{\text {fh}} = 100
Mbps to C_{\text {fh}} = 10
Mbps reduces the sum SE slightly, as the procedure outlined in Section II-D fairly retains the AP-UE links with the highest channel gains and helps maintain the sum SE.
Sum SE - impact of channel estimation error: We know that the channel estimation error is a function of pilot transmit power p_{t}
. We now vary p_{t}
and evaluate its impact on the sum SE for a full-duplex cell-free massive MIMO system in Fig. 3(c). For this study, we considered M = 32
APs, K_{d} = 12
downlink UEs, K_{u} = 8
uplink UEs and transmit power p_{d} = 2p_{u} = 30
dBm. We see that the sum SE increases for p_{t} \le -10
dB but saturates beyond that. This is because the channel estimation error reduces with increase in pilot power till p_{t} = -10
dB. Any further increase in p_{t}
, only marginally reduces the channel estimation error, which does not affect the sum SE. Our choice of p_{t} = 0.2
W in the numerical studies is, therefore, practical.
WSEE metric - influence of weights: We now demonstrate that the WSEE metric can accommodate the heterogeneous EE requirements of both uplink and downlink UEs. For this study, we consider a particular realization of a FD CF mMIMO system with a transmit power p_{d} = 2p_{u} = 30
dBm, M = 32
APs, K_{d} = K_{u} = K/2 = 2
uplink and downlink UEs and N_{t} = N_{r} = N = 2
transmit and receive antennas on each AP, with QoS constraints S_{ok} = S_{ol} = 0.1
bits/s/Hz. We plot the individual EEs of the uplink (UL) and downlink (DL) UEs versus the SCA iteration index for centralized WSEE maximization, using Algorithm 3, for two different combinations of UE weights. Weights w_{1}
and w_{2}
are associated with DL UE 1 and DL UE 2, while weights w_{3}
and w_{4}
are associated with UL UE 1 and UL UE 2, respectively.
We plot in Fig. 4(a) and Fig. 4(b) the individual EEs of UL and DL UEs, with: i) equal weights (w_{1} = w_{2} = w_{3} = w_{4} = 0.25
), and ii) w_{1} = 0.08
, w_{2} = 0.02
, w_{3} = 0.5
, w_{4} = 0.4
, respectively. In Fig. 4(a), with equal weights, UEs attain an EE depending on their relative channel conditions, which clearly indicates that in terms of channel conditions, DL UE 2~\gg
DL UE 1 > UL UE 2 > UL UE 1. In Fig. 4(b), the weights are chosen in an order which is opposite to the channel conditions. The EEs of the UL UEs now dominate the EE of DL UE 1, while reversing their relative order. The DL UE 2, with excellent channel, still attains a high EE, although lower than in Fig. 4(a).
Convergence of decentralized ADMM algorithm: plot in Fig. 4(c) the WSEE obtained using decentralized Algorithm 4 with SCA iteration index. We consider M = 10
APs, K_{u} = K_{d} = K/2 = 2
uplink and downlink UEs and N_{t} = N_{r} = \{1, 2\}
transmit and receive antennas on each AP at transmit power p_{d} = 2p_{u} = p = 30
dBm. We assume the following: i) penalty parameters \rho _{C} = \rho _{\theta } = 0.1
; ii) penalty parameter update threshold factor \mu = 10
; iii) ADMM convergence threshold \epsilon _{\text {ADMM}} = 0.01
; and iv) SCA convergence threshold \epsilon _{\text {SCA}} = 0.001
. We consider two values of the penalty update parameter: \vartheta = \{1.2, 1.8\}
. We note that the algorithm in both cases converges marginally quicker with \vartheta = 1.2
. A smaller penalty update parameter is therefore beneficial as then changes in the penalty parameters are not too abrupt, and a bad ADMM iteration which causes the primal and dual residues to diverge is, consequently, not overly responded to [37]. We therefore fix \vartheta = 1.2
for the rest of the simulations.
Comparison with existing schemes: We now compare our proposed FD CF mMIMO WSEE optimization strategy with some existing approaches. In particular, we compare the
proposed fair AP selection algorithm, Algorithm 1, with the optimal AP selection scheme proposed in [29].
maximum-ratio combining (MRC)/maximal ratio transmission (MRT) considered herein with zero-forcing reception (ZFR)/zero-forcing transmission (ZFT) [39].
We observe from Fig. 5(a) that the proposed fair AP selection approach has almost as well as the optimal one in [29]. The proposed procedure efficiently eliminates the AP-UE links that do not have sufficient channel gain and thus contribute little to the system throughput while consuming a significant amount of power. Turning off APs according to the optimal AP selection procedure in [29], thus only provides marginally better WSEE.
MRC/MRT and ZFR/ZFT comparison: For this study, we considered a FD CF mMIMO system with M = 32
multi-antenna APs having N_{t} = N_{r} = 8
transmit and receive antennas each, K_{d} = 12
downlink UEs and K_{u} = 8
uplink UEs. We consider two fronthaul cases: i) perfect high-capacity with \tilde {a} = \tilde {b} = 1
, and ii) limited C_{\text {fh}} = 10
Mbps capacity with \nu = 2
quantization bits. We see from Fig. 5(b) that for both fronthaul capacities, the MRC/MRT transceiver for the scenario considered herein, although slightly inferior at high transmit power, performs reasonably well when compared with computationally-intensive ZFR/ZFT transceiver.
WSEE variation with parameters: We now vary WSEE with important system parameters and obtain crucial insights into energy-efficient FD CF mMIMO system designing. We consider M = 32
APs, N_{t} = N_{r} = N = 8
AP transmit and receive antennas, K_{d} = 12
downlink UEs, K_{u} = 8
uplink UEs and QoS constraints S_{ok} = S_{ol} = 0.1
bits/s/Hz, unless mentioned otherwise.
We plot in Fig. 6(a) the WSEE by simultaneously varying downlink and uplink transmit power as p_{d} = 2p_{u} = p
. We consider centralized and decentralized optimal power allocation (OPA) approaches from Algorithm 3 and Algorithm 4, respectively. We compare them with three sub-optimal power allocation schemes: i) equal power allocation of type 1, labeled as “EPA 1”, where \eta _{mk} = \left({{b}N_{t}\left({\sum _{k \in \kappa _{dm}} \gamma ^{d}_{mk}}\right)}\right)^{-1}, \forall k \in \kappa _{dm}
and \theta _{l} = 1
[18], [19], ii) equal power allocation of type 2, labeled as “EPA 2”, where \eta _{mk} = ({b} N_{t} K_{dm} \gamma ^{d}_{mk})^{-1}, \forall k \in \kappa _{dm}
and \theta _{l} = 1
[18], and iii) random power allocation, labeled as “RPA”, where power control coefficients are chosen randomly from a uniform distribution between 0 and the “EPA 1” value. We note that the existing literature has not yet optimized the WSEE metric for CF mMIMO systems, and hence we can only compare with above sub-optimal schemes. Further, the decentralized ADMM approach, with lower computational complexity, has the same WSEE as that of the centralized one. Also, both decentralized and centralized approaches far outperform the baseline schemes.
We next characterize in Fig. 6(b) the joint variation of WSEE and sum SE with the number of quantization bits \nu
in the fronthaul links. The WSEE is obtained using decentralized Algorithm 4. We consider transmit power p_{d} = 2p_{u} = p = 30
dBm and take two different cases: i) high fronthaul capacity, C_{\text {fh}} = 100
Mbps, which is sufficiently high to support all the UEs, and ii) limited fronthaul capacity, C_{\text {fh}} = 10
Mbps, which limits the number of UEs a single AP can serve. We observe that for C_{fh} = 100
Mbps, the WSEE falls with increase in \nu
, even though the corresponding sum SE increases. For C_{fh} = 10
Mbps, both sum SE and WSEE simultaneously increase with increase in \nu
. To explain this behavior, we note from Fig. 3(b) that increasing \nu
improves the sum SE for C_{\text {fh}} = 100
Mbps and C_{\text {fh}} = 10
Mbps. For C_{fh} = 100
Mbps, the APs serve all the UEs, i.e., K_{dm} = K_{d}
and K_{ {\mu {\mathrm{ m}}}} = K_{u}
, so increasing \nu
linearly increases the fronthaul data rate, R_{fh}
(see (7)). This, as seen from (18), increases the traffic-dependent fronthaul power consumption. Using lower number (1-2) of quantization bits is therefore more energy-efficient, as it provides sufficiently good SE with a low energy consumption. However, for C_{fh} = 10
Mbps, K_{ {\mu {\mathrm{ m}}}}
and K_{dm}
have an upper limit, given by (9), which is inversely related to \nu
. The product, \nu (K_{ {\mu {\mathrm{ m}}}} + K_{dm})
, remains nearly constant for all values of \nu
. Thus, R_{fh}
(see (7)) doesn’t increase with increase in \nu
and remains close to the capacity, C_{fh}
. The traffic-dependent fronthaul power consumption, given in (18), hence, remains close to P_{\text {ft}}
. A higher number of quantization bits (3 – 4) therefore provides a higher sum SE and hence, also maximizes the WSEE.
Latency: The per-iteration complexity of the decentralized Algorithm 4, as observed earlier in Section IV-C, is lower than the centralized Algorithm 3. We now demonstrate the same by comparing their per-iteration runtime. For this simulation, as shown in Fig. 6(c), we consider an FD CF mMIMO system with M = 32
APs, each having N_{t} = N_{r} = 8
transmit and receive antennas, and plot the average runtime of each iteration by varying the total number of UEs, K
, with K_{d} = K_{u} = K/2
. We note that the decentralized algorithm has significantly lower per-iteration runtime, particularly for large K
. Both these algorithms require only large-scale channel coefficients and hence need to be executed only once in hundreds of coherence intervals.
Appendix B
We now derive the achievable SE expression for the k
th downlink UE in (14). From Section II-B, we know that \boldsymbol {g}^{d}_{mk} = \hat { \boldsymbol {g}}^{d}_{mk} + \boldsymbol {e}^{d}_{mk}
, where \hat { \boldsymbol {g}}^{d}_{mk}
and \boldsymbol {e}^{d}_{mk}
are independent and \mathbb {E}\{\|\hat { \boldsymbol {g}}^{d}_{mk}\|^{2}\} = N_{t}\gamma ^{d}_{mk}
. We can express the desired signal for the k
th downlink UE as \begin{align*}&\mathbb {E}\left \{{|\text {DS}^{d}_{k}|^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{d} \mathbb {E}\left \{{|\sum _{m \in \mathcal {M}^{d}_{k}} \sqrt {\eta _{mk}} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*}}\right \} s^{d}_{k}|^{2}}\right \} \\&\,= \tilde {a}^{2} N^{2}_{t} \rho _{d} \left ({\sum _{m \in \mathcal {M}^{d}_{k}} \sqrt {\eta _{mk}} \gamma ^{d}_{mk}}\right)^{2}. \tag{39}\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{|\text {DS}^{d}_{k}|^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{d} \mathbb {E}\left \{{|\sum _{m \in \mathcal {M}^{d}_{k}} \sqrt {\eta _{mk}} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*}}\right \} s^{d}_{k}|^{2}}\right \} \\&\,= \tilde {a}^{2} N^{2}_{t} \rho _{d} \left ({\sum _{m \in \mathcal {M}^{d}_{k}} \sqrt {\eta _{mk}} \gamma ^{d}_{mk}}\right)^{2}. \tag{39}\end{align*}
We now calculate the beamforming uncertainty for the k
th downlink UE as follows\begin{align*}&\mathbb {E}\left \{{\left |{\text {BU}^{d}_{k}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \mathbb {E}\left \{{\left |{ \left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} - \mathbb {E}\left \{{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T}\left.{ \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*}}\right)}\right \}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \tilde {a}^{2} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \Biggl ({N_{t}(N_{t} +1) \left ({\gamma ^{d}_{mk}}\right)^{2}} \\&\quad {+\,\,N_{t} \gamma ^{d}_{mk}\left ({\beta ^{d}_{mk} - \gamma ^{d}_{mk}}\right) - N^{2}_{t} \left ({\gamma ^{d}_{mk}}\right)^{2}}\Biggr ) \\&\,= \tilde {a}^{2} N_{t} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \beta ^{d}_{mk} \gamma ^{d}_{mk}. \tag{40}\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\text {BU}^{d}_{k}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \mathbb {E}\left \{{\left |{ \left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*} - \mathbb {E}\left \{{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T}\left.{ \left ({\hat { \boldsymbol {g}}^{d}_{mk}}\right)^{*}}\right)}\right \}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \tilde {a}^{2} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \Biggl ({N_{t}(N_{t} +1) \left ({\gamma ^{d}_{mk}}\right)^{2}} \\&\quad {+\,\,N_{t} \gamma ^{d}_{mk}\left ({\beta ^{d}_{mk} - \gamma ^{d}_{mk}}\right) - N^{2}_{t} \left ({\gamma ^{d}_{mk}}\right)^{2}}\Biggr ) \\&\,= \tilde {a}^{2} N_{t} \rho _{d} \sum _{m \in \mathcal {M}^{d}_{k}} \eta _{mk} \beta ^{d}_{mk} \gamma ^{d}_{mk}. \tag{40}\end{align*}
Equality (a)
is because i) \hat { \boldsymbol {g}}^{d}_{mk}
are zero-mean and uncorrelated; and ii) \mathbb {E}\{\|\hat { \boldsymbol {g}}^{d}_{mk}\|^{4}\} = N_{t}(N_{t}+1) (\gamma ^{d}_{mk})^{2}
[12] and \mathbb {E}\{\| \boldsymbol {e}^{d}_{mk}\|^{2}\} = (\beta ^{d}_{mk} - \gamma ^{d}_{mk})
.
We now simplify MUI for the k
th downlink UE: \begin{align*}&\mathbb {E}\left \{{\left |{\text {MUI}^{d}_{k}}\right |^{2}}\right \} \\&\,=\tilde {a}^{2} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k} \eta _{mq} \mathbb {E}\left \{{\left |{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \tilde {a}^{2} N_{t} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k} \beta ^{d}_{mk} \eta _{mq} \gamma ^{d}_{mq}. \tag{41}\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\text {MUI}^{d}_{k}}\right |^{2}}\right \} \\&\,=\tilde {a}^{2} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k} \eta _{mq} \mathbb {E}\left \{{\left |{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \tilde {a}^{2} N_{t} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm} \setminus k} \beta ^{d}_{mk} \eta _{mq} \gamma ^{d}_{mq}. \tag{41}\end{align*}
Equality (a) is because: i) \hat { \boldsymbol {g}}^{d}_{mq}
and \boldsymbol {g}^{d}_{mk}
are mutually independent; and \text {ii) }\mathbb {E}\{|(\boldsymbol {g}^{d}_{mk})^{T} (\hat { \boldsymbol {g}}^{d}_{mq})^{*}|^{2}\}\!\! =\!\! \mathbb {E}\{ (\hat { \boldsymbol {g}}^{d}_{mq})^{T} \mathbb {E}\{(\boldsymbol {g}^{d}_{mk})^{*} (\boldsymbol {g}^{d}_{mk})^{T}\} (\hat { \boldsymbol {g}}^{d}_{mq})^{*}\} = N_{t} \beta ^{d}_{mk} \gamma ^{d}_{mq}
.
We next calculate UDI for the k
th downlink UE: \begin{equation*} \mathbb {E}\left \{{\left |{\text {UDI}^{d}_{k}}\right |^{2}}\right \} = \rho _{u} \sum _{l=1}^{K_{u}} \mathbb {E}\left \{{|h_{kl}|^{2}}\right \} \theta _{l} = \rho _{u} \sum _{l=1}^{K_{u}} \tilde {\beta }_{kl} \theta _{l}. \tag{42}\end{equation*}
View Source\begin{equation*} \mathbb {E}\left \{{\left |{\text {UDI}^{d}_{k}}\right |^{2}}\right \} = \rho _{u} \sum _{l=1}^{K_{u}} \mathbb {E}\left \{{|h_{kl}|^{2}}\right \} \theta _{l} = \rho _{u} \sum _{l=1}^{K_{u}} \tilde {\beta }_{kl} \theta _{l}. \tag{42}\end{equation*}
We express the total quantization distortion (TQD) for the k
th downlink UE as follows\begin{align*} \mathbb {E}\left \{{\left |{\text {TQD}^{d}_{k}}\right |^{2}}\right \}\approx&\rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} \mathbb {E}\left \{{\left |{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} \varsigma ^{d}_{mq}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{t} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} \beta ^{d}_{mk} \eta _{mq} \gamma ^{d}_{mq}. \tag{43}\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\text {TQD}^{d}_{k}}\right |^{2}}\right \}\approx&\rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} \mathbb {E}\left \{{\left |{\left ({\boldsymbol {g}^{d}_{mk}}\right)^{T} \left ({\hat { \boldsymbol {g}}^{d}_{mq}}\right)^{*} \varsigma ^{d}_{mq}}\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{t} \rho _{d} \sum _{m=1}^{M} \sum _{q \in \kappa _{dm}} \beta ^{d}_{mk} \eta _{mq} \gamma ^{d}_{mq}. \tag{43}\end{align*}
Equality (a)
is because: i) \mathbb {E}\{|\varsigma ^{d}_{mk}|^{2}\} = (\tilde {b} - \tilde {a}^{2}) \eta _{mk}
; ii) distortion \varsigma ^{d}_{mq}
is independent of channels \boldsymbol {g}^{d}_{mk}
and \hat { \boldsymbol {g}}^{d}_{mq}
; and iii) \mathbb {E}\{|(\boldsymbol {g}^{d}_{mk})^{T} (\hat { \boldsymbol {g}}^{d}_{mq})^{*} \varsigma ^{d}_{mq}|^{2}\} = (\tilde {b} - \tilde {a}^{2}) \eta _{mq} \beta ^{d}_{mk} \mathbb {E}\{(\hat { \boldsymbol {g}}^{d}_{mq})^{T}(\hat { \boldsymbol {g}}^{d}_{mq})^{*}\} = (\tilde {b} - \tilde {a}^{2}) N_{t} \beta ^{d}_{mk} \eta _{mq} \gamma ^{d}_{mq}
. The result in (14) follows from the expression for the achievable SE lower bound\begin{align*} S^{d}_{k} = \tau _{f} \log _{2} \left (1 + \frac {\mathbb {E}\left \{{\left |{\text {DS}^{d}_{k}}\right |^{2}}\right \}}{\left \{\begin{array}{l}{\mathbb {E}\left \{{\left |{\text {BU}^{d}_{k}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {MUI}^{d}_{k}}\right |^{2}+ \mathbb {E}\left \{{|\text {UDI}^{d}_{k}|^{2}}\right \} }\right \}} \\ \qquad \quad \qquad \qquad + \mathbb {E}\left \{{\left |{\text {TQD}^{d}_{k}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{w^{d}_{k}}\right |^{2}}\right \} \end{array}\right \}}\right).\end{align*}
View Source\begin{align*} S^{d}_{k} = \tau _{f} \log _{2} \left (1 + \frac {\mathbb {E}\left \{{\left |{\text {DS}^{d}_{k}}\right |^{2}}\right \}}{\left \{\begin{array}{l}{\mathbb {E}\left \{{\left |{\text {BU}^{d}_{k}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {MUI}^{d}_{k}}\right |^{2}+ \mathbb {E}\left \{{|\text {UDI}^{d}_{k}|^{2}}\right \} }\right \}} \\ \qquad \quad \qquad \qquad + \mathbb {E}\left \{{\left |{\text {TQD}^{d}_{k}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{w^{d}_{k}}\right |^{2}}\right \} \end{array}\right \}}\right).\end{align*}
We now derive the achievable SE expression for the l
th uplink UE in (15). We know from Section II-B that \boldsymbol {g}^{u}_{ml} = \hat { \boldsymbol {g}}^{u}_{ml} + \boldsymbol {e}^{u}_{ml}
, where \hat { \boldsymbol {g}}^{u}_{ml}
and \boldsymbol {e}^{u}_{ml}
are independent and \mathbb {E}\{\|\hat { \boldsymbol {g}}^{u}_{ml}\|^{2}\} = N_{r}\gamma ^{u}_{ml}
. We can express the desired signal for the l
th uplink UE as given next\begin{align*} \mathbb {E}\left \{{\left |{\text {DS}^{u}_{l}}\right |^{2}}\right \}=&\mathbb {E}\left \{{\left |{\tilde {a} \sum _{m \in \mathcal {M}^{u}_{l}} \sqrt {\rho _{u}} \mathbb {E}\left \{{\sqrt {\theta _{l}}\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \left ({\hat { \boldsymbol {g}}^{u}_{ml} + \boldsymbol {e}^{u}_{ml}}\right) s^{u}_{l}}\right \}}\right |^{2}}\right \} \\=&\tilde {a}^{2} N^{2}_{r} \rho _{u} \theta _{l} \left ({\sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}}\right)^{2}. \tag{44}\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\text {DS}^{u}_{l}}\right |^{2}}\right \}=&\mathbb {E}\left \{{\left |{\tilde {a} \sum _{m \in \mathcal {M}^{u}_{l}} \sqrt {\rho _{u}} \mathbb {E}\left \{{\sqrt {\theta _{l}}\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \left ({\hat { \boldsymbol {g}}^{u}_{ml} + \boldsymbol {e}^{u}_{ml}}\right) s^{u}_{l}}\right \}}\right |^{2}}\right \} \\=&\tilde {a}^{2} N^{2}_{r} \rho _{u} \theta _{l} \left ({\sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}}\right)^{2}. \tag{44}\end{align*}
The beamforming uncertainty for the l
th uplink UE is\begin{align*}&\mathbb {E}\left \{{\left |{\text {BU}^{u}_{l}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{u} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left \|{ \left ({\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} - \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} }\right \}}\right)}\right \|^{2}}\right \} \\&\left.{\stackrel {(a)}{=} \tilde {a}^{2}\rho _{u} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\mathbb {E}\left \{{\left \|{\hat { \boldsymbol {g}}^{u}_{ml}}\right \|^{4}}\right \} + \mathbb {E}\left \{{\left |{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {e}^{u}_{ml}}\right |^{2}}\right \} - N^{2}_{r}\left ({\gamma ^{u}_{ml}}\right)^{2}}\right \}}\right) \\&\,\stackrel {(b)}{=} \tilde {a}^{2}\rho _{u} N_{r} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}\beta ^{u}_{ml}. \tag{45}\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\text {BU}^{u}_{l}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \rho _{u} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left \|{ \left ({\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} - \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} }\right \}}\right)}\right \|^{2}}\right \} \\&\left.{\stackrel {(a)}{=} \tilde {a}^{2}\rho _{u} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\mathbb {E}\left \{{\left \|{\hat { \boldsymbol {g}}^{u}_{ml}}\right \|^{4}}\right \} + \mathbb {E}\left \{{\left |{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {e}^{u}_{ml}}\right |^{2}}\right \} - N^{2}_{r}\left ({\gamma ^{u}_{ml}}\right)^{2}}\right \}}\right) \\&\,\stackrel {(b)}{=} \tilde {a}^{2}\rho _{u} N_{r} \theta _{l} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}\beta ^{u}_{ml}. \tag{45}\end{align*}
Equality (a)
is because: i) \boldsymbol {e}^{u}_{ml}
and \hat { \boldsymbol {g}}^{u}_{ml}
are zero-mean and uncorrelated; ii) \mathbb {E}\{ |\hat { \boldsymbol {g}}^{u}_{ml}|^{2}\} = N_{r} \gamma ^{u}_{ml}
. Equality (b)
is because \mathbb {E}\{\|\hat { \boldsymbol {g}}^{u}_{ml}\|^{4}\} = N_{r}(N_{r}+1) (\gamma ^{u}_{ml})^{2}
[12] and \mathbb {E}\{\| \boldsymbol {e}^{u}_{ml}\|^{2}\} = \,\,(\beta ^{u}_{ml} - \gamma ^{u}_{ml})
.
We simplify the MUI for the l
th uplink UE as \begin{align*} \mathbb {E}\left \{{\left |{\text {MUI}^{u}_{l}}\right |^{2}}\right \}=&\tilde {a}^{2} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \theta _{q} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq}}\right |^{2}}\right \} \\\stackrel {(a)}{=}&\tilde {a}^{2}\rho _{u} N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \gamma ^{u}_{ml}\beta ^{u}_{mq}\theta _{q}. \tag{46}\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\text {MUI}^{u}_{l}}\right |^{2}}\right \}=&\tilde {a}^{2} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \theta _{q} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq}}\right |^{2}}\right \} \\\stackrel {(a)}{=}&\tilde {a}^{2}\rho _{u} N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \gamma ^{u}_{ml}\beta ^{u}_{mq}\theta _{q}. \tag{46}\end{align*}
Equality (a)
is obtained by using these facts: i) \hat { \boldsymbol {g}}^{u}_{ml}
, \boldsymbol {g}^{u}_{mq}
are mutually independent; and ii)\begin{align*} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq}}\right |^{2} }\right \}=&\mathbb {E}\left \{{\left ({\boldsymbol {g}^{u}_{mq}}\right)^{H} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {g}^{u}_{mq}}\right \} \\&\,= \gamma ^{u}_{ml} \mathbb {E}\left \{{|| \boldsymbol {g}^{u}_{mq}||^{2}}\right \} = N_{r} \gamma ^{u}_{ml} \beta ^{u}_{mq}. \tag{47}\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq}}\right |^{2} }\right \}=&\mathbb {E}\left \{{\left ({\boldsymbol {g}^{u}_{mq}}\right)^{H} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {g}^{u}_{mq}}\right \} \\&\,= \gamma ^{u}_{ml} \mathbb {E}\left \{{|| \boldsymbol {g}^{u}_{mq}||^{2}}\right \} = N_{r} \gamma ^{u}_{ml} \beta ^{u}_{mq}. \tag{47}\end{align*}
We next obtain the noise power for the l
th uplink UE as\begin{align*} \mathbb {E}\left \{{\left |{\text {N}^{u}_{l}}\right |^{2}}\right \}=&\tilde {a}^{2} \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}\right |^{2}}\right \} = \tilde {a}^{2} N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml},~\text {where} \\ \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}\right |^{2}}\right \}=&\mathbb {E}\left \{{\left ({\boldsymbol {w}^{u}_{m}}\right)^{H} \mathbb {E}\left \{{ \hat { \boldsymbol {g}}^{u}_{ml} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {w}^{u}_{m}}\right \} = N_{r} \gamma ^{u}_{ml}. \tag{48}\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\text {N}^{u}_{l}}\right |^{2}}\right \}=&\tilde {a}^{2} \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}\right |^{2}}\right \} = \tilde {a}^{2} N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml},~\text {where} \\ \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}\right |^{2}}\right \}=&\mathbb {E}\left \{{\left ({\boldsymbol {w}^{u}_{m}}\right)^{H} \mathbb {E}\left \{{ \hat { \boldsymbol {g}}^{u}_{ml} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {w}^{u}_{m}}\right \} = N_{r} \gamma ^{u}_{ml}. \tag{48}\end{align*}
The undistorted MR-combined uplink signal at the m
th AP is expressed as \begin{align*}&\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {y}^{u}_{m} \\&\,= \sum _{q=1}^{K_{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} x^{u}_{q} + \sum _{i=1}^{M} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \boldsymbol {x}^{d}_{i} + \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m} \\&\,= \underbrace {\sqrt {\rho _{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} \sqrt {\theta _{l}} s^{u}_{l}}_{\text {message signal}} + \underbrace {\sqrt {\rho _{u}} \sum _{q=1, q \neq l}^{K_{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} \sqrt {\theta _{q}} s^{u}_{q}}_{\text {multi-user interference, MUI}^{u}_{l}} \\&\quad +\,\,\underbrace {\sqrt {\rho _{d}} \sum _{i=1}^{M}\sum _{k \in \kappa _{di}}\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k} + \varsigma ^{d}_{ik}}\right)}_{\text {intra-/inter-AP residual interference, RI}^{u}_{l}} \\&\quad +\,\,\underbrace {\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}_{\text {additive noise at APs, N}^{u}_{l}}.\tag{49}\end{align*}
View Source\begin{align*}&\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {y}^{u}_{m} \\&\,= \sum _{q=1}^{K_{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} x^{u}_{q} + \sum _{i=1}^{M} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \boldsymbol {x}^{d}_{i} + \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m} \\&\,= \underbrace {\sqrt {\rho _{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{ml} \sqrt {\theta _{l}} s^{u}_{l}}_{\text {message signal}} + \underbrace {\sqrt {\rho _{u}} \sum _{q=1, q \neq l}^{K_{u}} \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {g}^{u}_{mq} \sqrt {\theta _{q}} s^{u}_{q}}_{\text {multi-user interference, MUI}^{u}_{l}} \\&\quad +\,\,\underbrace {\sqrt {\rho _{d}} \sum _{i=1}^{M}\sum _{k \in \kappa _{di}}\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a}\sqrt {\eta _{ik}}s^{d}_{k} + \varsigma ^{d}_{ik}}\right)}_{\text {intra-/inter-AP residual interference, RI}^{u}_{l}} \\&\quad +\,\,\underbrace {\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {w}^{u}_{m}}_{\text {additive noise at APs, N}^{u}_{l}}.\tag{49}\end{align*}
We assume, similar to [19], that the quantization distortion is uncorrelated across the fronthaul links. The TQD power for the l
th uplink UE is accordingly expressed as \begin{align*} \mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}\approx&\sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{|\zeta ^{u}_{ml}|^{2}}\right \}\\\approx&\left ({\tilde {b} - \tilde {a}^{2}}\right) \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left |{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {y}_{m}}\right |^{2}}\right \}.\end{align*}
View Source\begin{align*} \mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}\approx&\sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{|\zeta ^{u}_{ml}|^{2}}\right \}\\\approx&\left ({\tilde {b} - \tilde {a}^{2}}\right) \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\left \{{\left |{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {y}_{m}}\right |^{2}}\right \}.\end{align*}
Using arguments similar to (44)–(48), the contributions of the message signal (DS + BU), MUI and noise (N) to the TQD for the l
th uplink UE are \begin{align*}&\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {DS+BU}} \\&\,\approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \rho _{u} \theta _{l} \left ({N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\gamma ^{u}_{ml}}\right)^{2} + \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml} \beta ^{u}_{ml}}\right). \\&\,\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {MUI}} \approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \gamma ^{u}_{ml} \beta ^{u}_{mq} \theta _{q}, \\&\,\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {N}} \approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}.\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {DS+BU}} \\&\,\approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \rho _{u} \theta _{l} \left ({N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \left ({\gamma ^{u}_{ml}}\right)^{2} + \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml} \beta ^{u}_{ml}}\right). \\&\,\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {MUI}} \approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \rho _{u} \sum _{m \in \mathcal {M}^{u}_{l}} \sum _{q=1, q \neq l}^{K_{u}} \gamma ^{u}_{ml} \beta ^{u}_{mq} \theta _{q}, \\&\,\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {N}} \approx \left ({\tilde {b} - \tilde {a}^{2}}\right) N_{r} \sum _{m \in \mathcal {M}^{u}_{l}} \gamma ^{u}_{ml}.\end{align*}
To accurately model the RI with limited fronthaul capacity and compute the corresponding power, as well as its contribution to the quantization distortion, we propose a lemma.
Lemma 2:
The intra-/inter-AP RI power and the RI contribution to the TQD power for the l
th uplink UE in a FD CF mMIMO system with MRT/MRC transceiver are expressed as \begin{align*}&\mathbb {E}\left \{{\left |{\text {RI}^{u}_{l}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \tilde {b} N_{r} N_{t} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \gamma ^{d}_{ik} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} N_{r} \gamma ^{u}_{ml}. \tag{50}\\&\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {RI}} \\&\,\approx \left ({\tilde {b} - \tilde {a}^{2}}\right) \tilde {b} N_{r} N_{t} \rho _{d} \!\sum _{m \in \mathcal {M}^{u}_{l}} \sum _{i=1}^{M} \!\sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} \gamma ^{d}_{ik}.\tag{51}\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\text {RI}^{u}_{l}}\right |^{2}}\right \} \\&\,= \tilde {a}^{2} \tilde {b} N_{r} N_{t} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \gamma ^{d}_{ik} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} N_{r} \gamma ^{u}_{ml}. \tag{50}\\&\mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \}_{\text {RI}} \\&\,\approx \left ({\tilde {b} - \tilde {a}^{2}}\right) \tilde {b} N_{r} N_{t} \rho _{d} \!\sum _{m \in \mathcal {M}^{u}_{l}} \sum _{i=1}^{M} \!\sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} \gamma ^{d}_{ik}.\tag{51}\end{align*}
Proof:
We express the RI power of the undistorted, MR combined received signal for the l
th uplink UE as \begin{align*}&\mathbb {E}\left \{{\left |{\widetilde {\text {RI}}^{u}_{l}}\right |^{2}}\right \} \\&\,= \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a} \sqrt {\eta _{ik}} s^{d}_{k} + \zeta ^{d}_{ik}}\right) }\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right |^{2} \tilde {b} \eta _{ik}}\right \} \\&\,\stackrel {(b)}{=} \tilde {b} N_{r} N_{t} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \gamma ^{d}_{ik} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} N_{r} \gamma ^{u}_{ml}.\end{align*}
View Source\begin{align*}&\mathbb {E}\left \{{\left |{\widetilde {\text {RI}}^{u}_{l}}\right |^{2}}\right \} \\&\,= \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*} \left ({\tilde {a} \sqrt {\eta _{ik}} s^{d}_{k} + \zeta ^{d}_{ik}}\right) }\right |^{2}}\right \} \\&\,\stackrel {(a)}{=} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right |^{2} \tilde {b} \eta _{ik}}\right \} \\&\,\stackrel {(b)}{=} \tilde {b} N_{r} N_{t} \rho _{d} \sum _{i=1}^{M} \sum _{k \in \kappa _{di}} \gamma ^{u}_{ml} \gamma ^{d}_{ik} \beta _{\text {RI},mi} \gamma _{\text {RI}} \eta _{ik} N_{r} \gamma ^{u}_{ml}.\end{align*}
Equality (a)
is because signal \tilde {a} \sqrt {\eta _{ik}} s^{d}_{k}
and quantization noise \varsigma ^{d}_{ik}
, are uncorrelated, and \mathbb {E}\{|\varsigma ^{d}_{ik}|^{2}\} = (\tilde {b} - \tilde {a}^{2}) \eta _{ik}
. Equality (b)
is because: i) \hat { \boldsymbol {g}}^{u}_{ml}
, \boldsymbol {H}_{mi}
and \hat { \boldsymbol {g}}^{d}_{mk}
are mutually independent, \begin{align*}&\text {ii)}~\mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right |^{2}}\right \} \\&\,= \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T} \mathbb {E}\left \{{ \boldsymbol {H}^{H}_{mi} \mathbb {E}\left \{{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right) \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {H}_{mi}}\right \} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,= \gamma ^{u}_{ml} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T} \mathbb {E}\left \{{ \boldsymbol {H}^{H}_{mi} \boldsymbol {H}_{mi}}\right \} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,=N_{r} \gamma ^{u}_{ml}\beta _{\text {RI},mi} \gamma _{\text {RI}} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T}\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,= N_{r} N_{t} \gamma ^{u}_{ml} \gamma ^{d}_{k} \beta _{\text {RI},mi} \gamma _{\text {RI}}. \tag{52}\end{align*}
View Source\begin{align*}&\text {ii)}~\mathbb {E}\left \{{\left |{\left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H} \boldsymbol {H}_{mi} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right |^{2}}\right \} \\&\,= \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T} \mathbb {E}\left \{{ \boldsymbol {H}^{H}_{mi} \mathbb {E}\left \{{ \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right) \left ({\hat { \boldsymbol {g}}^{u}_{ml}}\right)^{H}}\right \} \boldsymbol {H}_{mi}}\right \} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,= \gamma ^{u}_{ml} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T} \mathbb {E}\left \{{ \boldsymbol {H}^{H}_{mi} \boldsymbol {H}_{mi}}\right \} \left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,=N_{r} \gamma ^{u}_{ml}\beta _{\text {RI},mi} \gamma _{\text {RI}} \mathbb {E}\left \{{\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{T}\left ({\hat { \boldsymbol {g}}^{d}_{ik}}\right)^{*}}\right \} \\&\,= N_{r} N_{t} \gamma ^{u}_{ml} \gamma ^{d}_{k} \beta _{\text {RI},mi} \gamma _{\text {RI}}. \tag{52}\end{align*}
We obtain the i) attenuated intra-/inter-AP RI power as \mathbb {E}\{|\text {RI}^{u}_{l}|^{2}\} = \tilde {a}^{2}\mathbb {E}\{|\widetilde {\text {RI}}^{u}_{l}|^{2}\}
; and intra-/inter-AP RI contribution to the TQD power as \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\}_{\text {RI}} \approx (\tilde {b} - \tilde {a}^{2}) \sum _{m \in \mathcal {M}^{u}_{l}} \mathbb {E}\{|\widetilde {\text {RI}}^{u}_{l}|^{2}\}
.
The total quantization distortion for the l
th uplink UE is given as \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\} \!=\! \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\}_{\text {DS+BU}} + \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\}_{\text {MUI}} + \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\}_{\text {RI}} + \mathbb {E}\{|\text {TQD}^{u}_{l}|^{2}\}_{\text {N}}
.
The result in (15) follows from the expression\begin{align*} S^{u}_{l} = \tau _{f} \log _{2} \left (1 + \frac {\mathbb {E}\left \{{\left |{\text {DS}^{u}_{l}}\right |^{2}}\right \}}{\left \{\begin{array}{l} \mathbb {E}\left \{{\left |{\text {BU}^{u}_{l}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {MUI}^{u}_{l}}\right |^{2}}\right \} \\ \qquad \qquad \qquad~~ + \mathbb {E}\left \{{\left |{\text {RI}^{u}_{l}}\right |^{2}}\right \}+ \mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {N}^{u}_{l}}\right |^{2}}\right \} \end{array}\right \}}\right).\end{align*}
View Source\begin{align*} S^{u}_{l} = \tau _{f} \log _{2} \left (1 + \frac {\mathbb {E}\left \{{\left |{\text {DS}^{u}_{l}}\right |^{2}}\right \}}{\left \{\begin{array}{l} \mathbb {E}\left \{{\left |{\text {BU}^{u}_{l}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {MUI}^{u}_{l}}\right |^{2}}\right \} \\ \qquad \qquad \qquad~~ + \mathbb {E}\left \{{\left |{\text {RI}^{u}_{l}}\right |^{2}}\right \}+ \mathbb {E}\left \{{\left |{\text {TQD}^{u}_{l}}\right |^{2}}\right \} + \mathbb {E}\left \{{\left |{\text {N}^{u}_{l}}\right |^{2}}\right \} \end{array}\right \}}\right).\end{align*}
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