1

Full-Duplex Massive MIMO Multi-Pair
Two-Way AF Relaying: Energy Efficiency
Optimization
arXiv:1705.09043v2 [] 4 Oct 2017

Ekant Sharma, Rohit Budhiraja, K Vasudevan and Lajos Hanzo, Fellow IEEE
Abstract
We consider two-way amplify and forward relaying, where multiple full-duplex user pairs exchange
information via a shared full-duplex massive multiple-input multiple-output (MIMO) relay. Most of the
previous massive MIMO relaying works maximize the spectral efficiency (SE). By contrast, we maximize
the non-convex energy efficiency (EE) metric by approximating it as a pseudo-concave problem, which is
then solved using the classic Dinkelbach approach. We also maximize the EE of the least energy-efficient
user relying on the max-min approach. For solving these optimization problems, we derive closedform lower bounds for the ergodic achievable rate both for maximal-ratio combining and zero-forcing
processing at the relay, by using minimum mean squared error channel estimation. We numerically
characterize the accuracy of the lower bounds derived. We also compare the SE and EE of the proposed
design to those of the existing full-duplex systems and quantify the significant improvement achieved
by the proposed algorithm. We also compare the EE of the proposed full-duplex system to that of its
half-duplex counterparts, and characterize the self-loop and inter-user interference regimes, for which
the proposed full-duplex system succeeds in outperforming the half-duplex ones.
Index Terms
Energy efficiency, full-duplex, relay.

I. I NTRODUCTION
Relay-based communication has been extensively investigated to expand the coverage, improve
the diversity, increase the data rate, and reduce the power consumption of wireless communication
systems [1], [2]. The current generation of relays is predominantly half-duplex, as a benefit of
their implementational simplicity. A half-duplex relay requires two channel uses to send a data
packet from the transmitter to the receiver, since the relay cannot transmit and receive with in
Ekant Sharma, Rohit Budhiraja and K Vasudevan are with the Department of Electrical Engineering, Indian Institute of
Technology, Kanpur, 208016, India. email: {ekant, rohitbr, vasu@}iitk.ac.in.
Lajos Hanzo is with the School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ,
U.K. (e-mail: lh@ecs.soton.ac.uk).
Part of this work will be presented in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications
(PIMRC) 2017.

2

the same time slot. Full-duplex technology is becoming popular as a benefit of its increased
throughput [2]–[6]. A full-duplex one-way relay [4], [7] transmits and receives at the same time,
hence theoretically it doubles the spectral efficiency (SE) of a half-duplex one-way relay [8], [9].
Full-duplex two-way relaying [10]–[12], wherein two users exchange their data over a single
channel use via a relay, further improves the SE. Two-way full-duplex relaying has recently
been extended to multi-pair two-way full-duplex relaying [13]–[15] wherein multiple user pairs
exchange their data via a shared relay in a single channel use. A multi-pair two-way full-duplex
relay system has the following interference sources: i) co-channel (inter-pair) interference due to
multiple users simultaneously accessing the channel; ii) self-loop interference at the relay and at
the users; and iii) inter-user interference caused due to simultaneous transmission and reception
by full-duplex nodes.
Massive multiple-input multiple-output (MIMO) systems have become popular, since they
cancel co-channel interference by using simple linear transmit processing schemes, such as, zeroforcing transmission (ZFT) and maximal-ratio transmission (MRT) [16]–[19], which significantly
improve their SE. Massive MIMO technology is also being incorporated into multi-pair fullduplex relays for mitigating the self-loop interference at the relay, and the inter-pair co-channel
interference [13]–[15], [20], [21]. Ngo et al. [13] derived the achievable rate and a power
allocation scheme for maximizing the ergodic sum-rate for one-way decode and forward fullduplex massive MIMO-aided relaying. Zhang et al. [14] proposed four power scaling schemes
for two-way full-duplex massive MIMO relaying to improve both its SE and its energy efficiency
(EE). Zhang et al. [15] developed a power allocation scheme for maximizing the sum-rate of
multi-pair two-way full-duplex massive MIMO amplify-and-forward (AF) relaying by using
maximal-ratio combining (MRC)/MRT processing at the relay, and by using least squares (LS)
channel estimation. Dai et al. [20] considered a half-duplex multi-pair two-way massive MIMO
AF relay and derived closed-form achievable rate expressions and a power allocation scheme for
maximizing the sum-rate under realistic imperfect channel state information (CSI). Cui et al. [21]
developed power scaling schemes for half-duplex massive MIMO-aided one-way relay systems.
The energy cost of wireless network operation has increased to almost 50% of the total
operational cost [22]. The EE metric, which relies on the Pareto-optimality between throughput
and energy consumption, has recently drawn attention as a potent performance measure. The EE
has recently been optimized for both conventional MIMO [23], [24] and for single-hop massive

3

MIMO systems [25], [26]. Nguyen et al. [25] optimized the downlink EE of a cell-free single-hop
massive MIMO system using ZF precoding. Li et al. [26] optimized both the achievable rate and
the EE of a small-cell based massive MIMO single-hop full-duplex system. It is anticipated that
a paradigm shift towards multi-component Pareto-optimization is about to take place, leading to
an entire optimal Pareto-front of solutions [27], [28].
The existing literature of massive MIMO relays, on the other hand, has either optimized the
achievable rate [13], [15], [20] or analyzed the EE [14], [21]. The EE optimization for massive
MIMO relaying has not been investigated at the time of writing, except for a recent study in [29]
which optimized the asymptotic EE for a multi-pair one-way decode and forward massive MIMO
half-duplex relay. To the best of our knowledge, the EE of multi-pair two-way AF massive MIMO
full-duplex relay system has not been considered in the literature for a realistic finite-cardinality
antenna-set. Hence we fill this gap. Due to the self-loop interference and the coupled channels
encountered in two-way full-duplex AF relaying, the power allocation scheme of [29] cannot be
applied to our system. Against this backdrop, we list the main contributions of this paper.
1) We derive closed-form lower bounds for the achievable rate of the multi-pair two-way AF
full-duplex massive MIMO relay for an arbitrary number of relay antennas. We consider both
MRC/MRT and zero-forcing reception (ZFR)/ZFT processing at the relay, whilst relying on the
minimum mean square error (MMSE) relay channel estimation. In contrast to [15], which derives
a closed-form lower bound for MRC/MRT processing alone, which is based on the LS channel
estimation, we derive new bounds both for MRC/MRT and for ZFR/ZFT processing based on
the MMSE channel estimation. These closed-form achievable-rate expressions have not been
derived for an arbitrary number of relay antennas in the massive MIMO relaying literature.
2) We optimally allocate power to maximize the EE by using the closed-form achievable rate
expressions derived. The EE maximization, which has a non-convex objective, is solved by
proposing an algorithm wherein we first approximate the objective as a pseudo-convex function,
and later choose Dinkelbach’s approach. This contribution is significantly different from [29],
which considers asymptotic EE optimization for decode and forward one-way half-duplex relay.
The achievable rate expressions, and consequently the EE optimization, developed herein are
applicable to any antenna configuration. Furthermore, the expressions and the analysis developed
herein are significantly more complex due to the coupling of channels in AF relaying, and both the
self-loop as well as the inter-user interference imposed by the full-duplex nodes. We numerically

4

compare the performance of the proposed EE algorithm to the equal-power approach of [14].
Furthermore, we show that the EE optimization algorithm can also be used for optimizing the EE
under specific Quality-of-Service (QoS) constraints. We investigate the effect of QoS constraints
on the EE performance.
3) We also maximize the EE by using the max-min fairness criterion; the problem has a nondifferentiable objective. We solve this problem by first using the sequential convex programming
approach to approximate the objective function by a quasi-concave function, and later by using
the generalized Dinkelbach’s method of [30].
4) The proposed EE maximization framework can also maximize the SE for both MRC/MRT and
ZFR/ZFT processing. In contrast to [15], which derives the lower bound and maximizes the SE
(not EE) for the MRC/MRT processing alone relying on LS channel estimation, we conceive a
more general approach than that of [15]. We quantify the considerably improved SE of ZFR/ZFT
processing and MMSE channel estimation over the LS-based MRC/MRT processing of [15].
5) The proposed full-duplex EE optimization framework can also be used for evaluating the EE of
massive MIMO half-duplex AF systems which has not been investigated in the open literature.
We compare the EE of both the full-duplex and of the half-duplex systems and numerically
quantify the self-loop and inter-user interference values for which a full-duplex system has a
better EE than a half-duplex system. We also show the significantly improved EE of optimal
power allocation over the equal-power EE analyzed in [21].
The rest of the paper is organized as follows. We present our system model in Section II, and
discuss the MMSE channel estimation in Section III. The relay processing is discussed in Section IV, while the achievable rates are analyzed in Section V. Our energy-efficient optimization
problems are formulated in Section VI, while our performance improvements over [14], [15],
[21] are presented in Section VII. The paper is concluded in Section VIII.
Notations: The boldface capital and small letter represents matrix and vector, respectively,
Cr×s denotes a complex matrix of dimension r × s. The superscript (·)T , (·)H , (·)∗ denotes
the transpose, Hermitian and conjugate operations, respectively. The diag(x) denotes a (square)
diagonal matrix with elements x on its main diagonal, IQ denotes an Q × Q identity matrix, and
1k denotes a 2K ×1 vector consisting of value one at kth row and zero otherwise. The expectation
and trace operations are denoted by E[·] and tr{·}, respectively. The notation CN (0, K) represents
a circularly-symmetric complex Gaussian random vector with covariance matrix K.

5

II. S YSTEM M ODEL
We consider the multi-pair two-way AF full-duplex relaying shown in Fig. 1, where K fullduplex user pairs communicate via a single full-duplex relay within the same time-frequency
resource. We assume that the user S2m−1 for m = 1, · · · , K on one side of the relay, wants to send
as well as receive from the user S2m on the other side of the relay. We also assume that there are
no direct links between the user-pairs (S2m−1 , S2m ) due to a high path loss and heavy shadowing,
which is commonly assumed in the multi-user two-way massive MIMO relaying literature [13],
[14]. Furthermore, the relay has N transmit and N receive antennas, while each user has one
transmit and one receive antenna. The users on either side of the relay interfere with each other,
due to full-duplex architecture; the interference caused is termed as inter-user interference.

Fig. 1: Multi-pair two-way full-duplex AF massive MIMO relay system: All users and the relay simultaneously transmit and
receive which leads to self-loop interference at the relay GRR (shown by dotted ellipse), at the user Ωk,k (shown by dotted
circle) and inter-user interference (shown by solid red line). The users on either side of the relay (marked with bold dotted line
at the center) are isolated.

At time instant n, each user Sk , k = 1 to 2K,1 transmits the signal

√

pk xk (n) to the relay,

and simultaneously the relay broadcasts a vector xR (n) ∈ CN ×1 to all the users. Here the term
pk denotes the transmit power of the kth user. The signal received at the relay and at the user
Sk are given by
2K
X
√
pk gk xk (n) + GRR xR (n) + zR (n) = G̃x(n) + GRR xR (n) + zR (n),
yR (n) =
k=1
X
√
Ωk,i pi xi (n) + zk (n).
yk (n) = fkT xR (n) +
Here gk ∈ C

N ×1

i,k∈Uk
N ×1

and fk ∈ C

(1)
(2)

denote the channels spanning from the transmit antenna

of the kth user to the relay’s receive antenna array, and from the relay’s transmit antenna
1

To avoid repetition, we assume that k = 1 to 2K throughout this paper.

6

array to the receive antenna of the kth user, respectively. We now introduce the matrix G =
[g1 , g2 , g3 , · · · , g2K ] ∈ CN ×2K and the matrix F = [f1 , f2 , f3 , · · · , f2K ] ∈ CN ×2K (to be used
√ √
√
later in the sequel). Furthermore we have, G̃ = GP where P = diag
p1 , p2 , · · · p2K

with 0 ≤ pk ≤ P max . The signal received at the relay and at the users are interfered by their
own transmit signal, which is termed as the self-loop interference. In (1) and (2), GRR and Ωk,k
denote the self-loop interference at the relay and at the user Sk , respectively. The entries of the
2
matrix GRR are independent and identically distributed (i.i.d.) with distribution CN (0, σLIR
)

2
). The terms Ωk,i (k, i ∈ Uk , i 6= k), which
[14]. Furthermore Ωk,k is distributed as CN (0, σk,k
2
denotes the inter-user interference channel, are independent and are distributed as CN (0, σk,i
)

[14], where the set obeys Uk = [1, 3, 5, · · · , 2K − 1] for odd k and Uk = [2, 4, 6, · · · , 2K]

for even k. We also define the vector x(n) = [x1 (n), x2 (n), x3 (n), · · · , x2K (n)]T ∈ CN ×1


with E x(n)xH (n) = I2K . The vector zR (n) ∈ CN ×1 and the scalar zk (n) are additive white
Gaussian noise (AWGN) processes at the relay and at the user Sk . The elements of zR and the
2
scalar zk (n) are modeled as i.i.d. CN (0, σnr
) and CN (0, σn2 ), respectively.

Remark 1. The channel matrices account for both small-scale and large-scale fading; we therefore
1/2

have G = Hu Du

1/2

and F = Hd Dd . Here the small-scale fading matrices Hu and Hd have

i.i.d. CN (0, 1) elements, while the kth element of the large-scale diagonal fading matrices Du

2
2
and Dd are denoted as σg,k
and σf,k
, respectively.
In the first time slot (n = 1), the relay only receives the signal from the users, and does not

transmit. The signals received at the relay and at the user Sk are given respectively as
X
√
yR (1) = G̃x(1) + zR (1), yk (1) =
Ωk,i pi xi (1) + zk (1).

(3)

W, such that

(4)

i,k∈Uk

At the nth time slot, the relay linearly precodes its received signal yR (n − 1) using a matrix
xR (n) = αWyR (n − 1),

where α is the scaling factor chosen to satisfy the relay’s power constraint. The relay’s transmit
signal xR (n), before applying any self-loop interference cancellation technique, can be reformulated by iteratively substituting (1) into (4) as
xR (n) = s [x(n − ν) + x(n − 2ν) + · · · + zR (n − ν) + zR (n − 2ν) + · · · ] .

(5)

Here s[·], as discussed in [14], is a function involving both vector and matrix operations, while
ν is the relay’s processing delay (ν = 1 in this paper). The relay transmit signal, in the above
form which assumes no interference cancellation, is difficult to analyze [14]. The recent full-

7

duplex studies e.g. in [4], [6], [14] have shown that the self-loop interference can be significantly
suppressed, and the residual self-loop interference at the relay can be modeled as an additional
Gaussian noise source. We also apply these cancellation techniques, and replace xR (n) in the
self-loop interference term GRR xR (n) in (1) with a Gaussian noise x̃R (n), which represents
 

the residual self-loop interference with the power constraint E x̃R (n)x̃H
(n)
= PNR IN [14].
R
We therefore have

ỹR (n) = G̃x(n) + GRR x̃R (n) + zR (n).

(6)

After loop interference suppression, the relay’s transmit signal in (4) is re-expressed using (6) as
xR (n) = αWỹR (n − 1).

(7)

We observe from (7) that xR (n) now does not have infinite memory of x(n), which was the
case earlier in (5). It is a function of x(n − 1). The time labels from the model in (7) can now
be dropped for the sake of brevity. We re-write (7), using (6), after dropping the labels as
xR = αWỹR = αWG̃x + αWGRR x̃R + αWzR .
The relay’s transmit signal should satisfy its transmit power constraint, so we have
h
i
 





2
PR = Tr E xR xH
=
E
kαW
G̃xk
+ E kαWGRRx̃R k2 + E kαWzR k2 ,
R

which leads to

v
u
PR
u
i
.
α=t h
E kWG̃xk2 + E [kWGRR x̃R k2 ] + E [kWzR k2 ]

(8)

(9)

(10)

We next re-formulate the signal received at the user Sk in (2), by using (8), as
X
√
yk = αfkT WG̃x + αfkT WGRR x̃R + αfkT WzR +
Ωk,i pi xi + zk
i,k∈U k
2K
X

√
√
√
pi gi xi + αfkT WGRR x̃R
= αfkT W pk′ gk′ xk′ + αfkT W pk gk xk + αfkT W
|
{z
}
{z
}
|
{z
} |
′
i6=k,k
amplified self-loop interference
self-interference
desired signal
{z
}
|
inter-pair interference
X
√
+
αfkT WzR
+
Ωk,i pi xi +
.
(11)
zk
|{z}
| {z }
i,k∈U k
amplified noise from relay
{z
} AWGN at Sk
|
self-loop interference
and inter-user interference

′

Here we have (k, k ) = (2m − 1, 2m) or (2m, 2m − 1), where m = 1, · · · , K denotes the user
pair, which exchange information with one another.
In this work we assume that the relay estimates the channels G and F and uses them to design
the precoder W. The relay then transmits the self-interference cancellation (SIC) coefficient
f̂kT Wĝk for each user, where f̂k and ĝk are the estimated channel coefficients. The signal received

8

at user Sk after SIC is
2K
X
√
√
√
pi gi xi + αfkT WGRR x̃R
ỹk = αfkT W pk′ gk′ xk′ + α pk λk xk + αfkT W
{z
}
|
| {z }
{z
}
|
i6=k,k ′
amplified self-loop interference
residual interference
desired signal
{z
}
|
inter-pair
interference
X
√
+
αfkT WzR
Ωk,i pi xi +
.
(12)
zk
+
|{z}
| {z }
i,k∈U k
amplified noise from relay
|
{z
} AWGN at Sk
self-loop interference
and inter-user interference

Here λk = fkT Wgk − f̂kT Wgˆk is the residual self-interference. Before designing the relay’s
precoder W, we briefly digress to discuss the MMSE channel estimation process.

III. C HANNEL E STIMATION
During the channel estimation phase, all 2K users simultaneously transmit pilot sequences of
p
length τ ≤ T symbols to the relay, where T is the channel’s coherence interval. Let τ Pρ ϕ ∈

C2K×τ denote the pilot symbols transmitted from 2K users with Pρ being the transmit power of
each pilot symbol. The pilots are assumed to be orthogonal so that ϕϕH = I2K , which requires
that τ ≥ 2K [31]. The MMSE channel estimates Ĝ and F̂ are given as [13], [32]
G = Ĝ + Eg , and F = F̂ + Ef .

(13)

Here Eg and Ef represents the estimation error matrices of G and F, respectively. The estimated
channel matrices Ĝ and F̂ are independent of the error matrices Eg and Ef , respectively
[32]. The rows of the matrices Ĝ and F̂ are distributed as CN (0, D̂u ) and CN (0, D̂d ) re 2
 2
2
2
2
2
spectively, where D̂u = diag σ̂g,1
, σ̂g,2
, · · · , σ̂g,2K
and D̂d = diag σ̂f,1
, σ̂f,2
, · · · , σ̂f,2K
,
2
with σ̂g,k
=

4
τ Pρ σg,k
2
τ Pρ σg,k +1

2
and σ̂f,k
=

4
τ Pρ σf,k
2
τ Pρ σf,k +1

[32]. Hence Eg ∼ CN (0, Du − D̂u ) and Ef ∼
 2
2
2
2
CN (0, Dd − D̂d ), with Du − D̂u = diag σξ,g,1 , σξ,g,2
, σξ,g,3
, · · · , σξ,g,2K
and Dd − D̂d =
2
2
 2
σf,k
σ
2
2
2
2
2
diag σξ,f,1 , σξ,f,2
, σξ,f,3
, · · · , σξ,f,2K
with σξ,g,k
= τ Pρ σg,k
2 +1 and σξ,f,k = τ P σ 2 +1 .
ρ
g,k

f,k

IV. R ELAY PRECODER DESIGN
We design our relay precoder based on: i) MRC/MRT; and ii) ZFR/ZFT processing.

A. MRC/MRT processing
The MRC/MRT precoder using the estimated CSI is formulated as
W = F̂∗ TĜH ,

(14)

where T = blkdiag {T1 , T2 , T3 , · · · , TK }. The permutation matrix Tm = [0 1; 1 0] permutes
the transmit data of each user pair to ensure that the transmit data reaches the intended receiver.
We next state the following proposition, whose proof is relegated to Appendix A, to simplify
the scaling factor α in (10).

9

Proposition 1. For the MRC/MRT precoder, we have
s
PR
α =
,
2
2
2
N (Ψ + σnr + PR σLIR
) Φ̂ + N 3 Υ̂
K
K


P
P
2
2
2
2
2
2
where Ψ =
p2i−1 σg,2i−1
+ p2i σg,2i
, Φ̂ =
σ̂g,2i−1
σ̂f,2i
+ σ̂g,2i
σ̂f,2i−1
and
Υ̂ =

K
P

i=1

i=1

(15)

i=1



4
2
4
2
p2i−1 σ̂g,2i−1
σ̂f,2i
+ p2i σ̂g,2i
σ̂f,2i−1
.

B. ZFR/ZFT processing
The ZFR/ZFT precoder using the estimated CSI is formulated as

−1
ˆ = F̂ F̂H F̂
where F̄

ˆ ∗ TḠ
ˆ H,
W = F̄
(16)

−1
ˆ = Ĝ ĜH Ĝ
and Ḡ
. In the next proposition, which is proved in

Appendix B, we simplify the scaling factor α in (10).

Proposition 2. For the ZFR/ZFT precoder, we have
v
u
PR
 2K
,
α = u
u
P
t
2
2
2
λ̂ + η̂
pi σξ,g,i + σnr + PR σLIR

(17)

i=1

where λ̂ =

2K
P

i=1

p

′
i

2
(N −2K−1)σ̂f,i

, η̂ =

P2K

1
j=1 (N −2K−1)2 σ̂2 σ̂2
f,j

g,j

.

′

V. ACHIEVABLE SUM - RATE OF MRC/MRT AND ZFR/ZFT PRECODERS
In this section, we calculate lower bounds on the instantaneous sum-rate for both MRC/MRT
and ZFR/ZFT precoders. The instantaneous SNRk at the user Sk can be expressed using (12) as in
SNRk =
|2

pk |λk +

2K
P

pk′ |fkT Wgk′ |
|fkT Wgi |2

i6=k,k ′

+

2 PR
kfkT WG
( 2KRR k N
X

+

kfkT Wk)2 σn2 R

log2 (1 + SNRk ) .

The sum-rate of the system is R = E

k=1

. (18)
+

P

1
2
pi+ α12 σn2
σk,i
α2
i,k∈U k

(19)
′

Next we derive a lower bound on the achievable rate using the method of [33]. For the (k, k )
pair, the signal received by the kth user can be written as (see (12))


√
,
ñk
ỹk = α pk′ E fkT Wgk′ xk′ +
|{z}
|
{z
}
effective noise
2K
signal
X

desired
√
√
√
T
T
T
′
′
′
′
where ñk = α pk fk Wgk − E fk Wgk xk + α pk λk xk + αfk W
pi gi xi
′
X
p
i6=k,k
+αfkT WGRR x̃R + αfkT WzR +
Ωk,i p(k)xi + zk .

(20)

(21)

i,k∈U k



The value of E fkT Wgk′ can be calculated by exploiting the knowledge of channel distribution.

We observe that the desired signal and the effective noise are uncorrelated. Similar to [34], [35],

10



we only exploit the knowledge of E fkT Wgk′ in the detection. We use the central limit theorem

to treat the uncorrelated additive noise ñ(k) as the worst-case Gaussian noise, when computing
the sum-rate. For massive MIMO systems, the central limit theorem provides a tight statistical
lower bound on the achievable rate. This fact is extensively exploited for deriving the SE and
EE expressions in massive MIMO systems [13], [20], [35]. The lower bound on the achievable
sum-rate, consequently, becomes
R lower =

" 2K
X
k=1

where
SNRk,lower =

#


log2 1 + SNRk,lower ,

(22)

2

α2 pk′ E fkT Wgk′
. (23)
α2 pk′ var [fkT Wgk′ ] + α2 pk SIk + α2 IPk + α2 NRk + α2 LIRk + UIk + NUk
In (23), the residual self-interference after SIC (SI), the inter-pair interference (IP), the amplified
noise from the relay (NR), the amplified self-loop interference at relay (LIR), self-loop interference and inter-user interference (UI), and the noise at the user (NU), are given as follows:
2K
i
h
X


T
T
2
pi E |fkT Wgi |2 ,
SIk = E |fk Wgk − f̂k Wĝk | , IPk =
′

NRk
UIk

i6=k,k




= E |fkT WzR |2 , LIRk = E |fkT WGRR x̃|2 ,
X




pi E |Ωk,i xi |2 , NUk = E |zk |2 .
=

(24)

i,k∈U k

We further simplify the SNRk,lower expressions for both the MRC/MRT and ZFR/ZFT precoders
in the following theorems.
Theorem 1. The achievable rate of user Sk for a finite number of receive antennas at the relay
relying on MMSE channel estimate based MRC/MRT processing is lower bounded as follows:

(pk , PR ) , where we have:
log2 1 + SNRmrc
k
SNRmrc
(pk , PR ) =
k
2K
X
i=1

X
(3)
(1)
(2)
pi PR−1 bk,i
bk,i + bk,i PR−1 +
i,k∈Uk

ak pk′
, (25)

 X
(1)
(2)
(3)
pi fk,i (PR )
pi+ck pk+ dk + dk PR + dk PR−1 +

!

i,k∈U

k


2
4
2
2
2
2
4
2
2 4
4
= ηk,i = Φ̂σf,k σg,i + N σf,k σ̂g,i σ̂f,i′ + σg,i σ̂f,k σ̂g,k′ ,
where ak = N σ̂f,k σ̂g,k′ ,




(3)
(2)
2
2
4
2
2
4
2
Φ̂σg,i
+ N σ̂g,i
σ̂f,i
bk,i = σn2 Φ̂σg,i
+ N σ̂g,i
σ̂f,i
′ ,
′ , bk,i = σk,i



2
4
2
2
2
4
2
2
,
+
σ̂
σ̂
σ̂
ck = − Φ̂σ̂f,k
σ̂g,k
+ N σ̂f,k
σ̂g,k
σ̂g,k
′
f,k g,k f,k ′



(3)
(1)
(2)
2
2
2
2
2
2
2
4
2
2
4
2
σn2 Φ̂,
dk = σLIR σn + σnr σf,k Φ̂+Nσnr σ̂f,k σ̂g,k′ , dk = σLIR σf,k Φ̂ + N σ̂f,k σ̂g,k′ , dk = σnr


(1)
(2)
(1)
(2)
2
2
2
2
σk,i
Φ̂, ek,i = σLIR
σk,i
Φ̂.
fk,i (PR ) = PR−1ek,i + ek,i , ek,i = σnr

(1)
bk,i

Proof: Refer to Appendix C.

11

Theorem 2. The achievable rate of user Sk for a finite number of receive antennas at the relay
with MMSE channel estimate based ZFR/ZFT processing is lower bounded as
n
o
(p
,
P
)
, where
log2 1 + SNRzf
k
R
k
SNRzf
k (pk , PR ) =
2K
X
i=1

X
(3)
(1)
(2)
pi PR−1 dk,i
dk,i + dk,i PR−1 +
i,k∈Uk

(1)



!

2
σξ,f,k
σ̂2 ′
f,i

u k pk ′
, (26)
 X

(1)
(2)
(3)
pi uk,i(PR )
pi + vk + vk PR + vk PR−1 +
2
σξ,g,i
σ̂2 ′
g,k



i,k∈Uk

1
2
2
where uk = 1, dk,i = (N −2K−1)
+ σξ,f,k
σξ,g,i
η̂,
+




(3)
(2)
1
1
2
2
2
2
dk,i = σn (N −2K−1)σ̂2 + η̂σξ,g,i , dk,i = σk,i (N −2K−1)σ̂2 + η̂σξ,g,i ,
′
′
f,i
f,i 



(1)
(2)
1
1
2
2
2
2
2
2
vk = σnr (N −2K−1)σ̂2 + σξ,f,k η̂ + η̂σLIR σn , vk = σLIR (N −2K−1)σ̂2 + σξ,f,k η̂ ,
′
′
g,k
g,k


(1)
(2)
(3)
(1)
−1 (2)
2
2
2
2
2
2
vk = η̂σnr σn , uk,i (PR ) = wk,i + PR wk,i , wk,i = η̂σk,i σLIR , wk,i = η̂σk,i σnr

Proof: Refer to Appendix D.
VI. E NERGY- EFFICIENT OPTIMIZATION
We now optimally allocate the power for maximizing the EE subject to the rate required by

each user. The EE (in bits/Joule/Hz) is defined as [14], [24]
R̃(pk , PR )
.
(27)
EE =
PT (pk , PR )
The numerator in the EE is the SE, which also includes the channel estimation overhead, and
is given by



2K



τ X
R̃(pk , PR ) = 1 −
log2 1 + SNRζk (pk , PR ) .
T k=1

(28)

The expression of SNRζk , where ζ ∈ (mrc, zf) are given in (25) and (26), respectively. The
denominator in the EE denotes the overall power consumed by the system [26]
2K
X
pk + P R + P c .
PT (pk , PR ) =

(29)

k=1

The terms pk and PR denote the transmit power of user Sk and of the relay, respectively. The
term Pc represents the fixed circuit power used by the system.
Remark 2. We assume the power amplifiers efficiency to be unity both at the user and at the
relay as in [26], [36], [37] for mathematical simplicity. Also, Pc takes into account the power
required by the different components, namely by the transceiver’s radio-frequency chain, the
oscillator and the power consumption of channel estimation [26], [36], [37]. This assumption
is commonly exploited in the massive MIMO literature for mathematical simplicity [26], and it
does not affect the overall behavior of the system considered.

12

Before formulating the related optimization problem, we briefly discuss the terminologies used
in geometric and fractional programming from [38] and [39] respectively, which will be used in
the sequel. Geometric and fractional programming have earlier been used for power allocation
in [39]–[42].
A real valued function f (x) of the form f (x) = c xa11 xa22 · · · xann , where c > 0, ai ∈ R and

x ∈ Rn++ , is referred to as a monomial function. The sum of one or more monomials, i.e.
P
a1m a2m
· · · xannm , where cm > 0 is termed as a posynomial. Monomials are
f (x) = K
m=1 cm x1 x2

closed both under multiplication and division, whereas posynomials are closed under addition
and multiplication, but not under division. The ratio of a posynomial and monomial is a posynomial. A geometric program (GP) has a posynomial objective and upper bounded posynomial
inequality constraints.
A fractional program is of the form u(x)/v(x), so that the optimization variable obeys x ∈ X ,

where u : C ⊂ Rn → R, v : C ⊂ Rn → Rx+ and X ⊂ C ⊂ Rn . Since the objective is a fraction,

the problem is not guaranteed to be always convex, even if both u and v are affine functions.
For maximizing fractions, mostly two classes of generalized concave functions, namely quasiconcave (QC) functions and pseudo-concave (PC) functions are used. If C ⊂ Rn is a convex set,
then r : C → R is QC if r(λx1 + (1 − λ)x2 ) ≥ min{r(x1 , x2 )} for all x1 , x2 ∈ C and λ ∈ [0; 1].

Similarly, r : C → R is PC if it is differentiable and, r(x2 ) < r(x1 ) ⇒ ▽(r(x2 ))T (x1 − x2 ) > 0

for all x1 , x2 ∈ C. A local maximum of optimization associated with the PC objective constitutes
a global maximum, whereas under a QC objective it is not necessarily a global maximum. For
example, when the objective has a concave numerator and a convex denominator, the fractional
program is a PC and its stationary point is its global maximizer. Such a problem belongs to the
class of concave-convex fractional programs (CCFP) [39].
With this information in mind, we optimize the EE in the next section.
A. EE maximization
The EE maximization problem is formulated as


2K
P
ζ
τ
1− T
log2 1 + SNRk (pk , PR )
k=1
P1 :Max
2K
pk ,PR
P
pk + P R + P c

(30a)

k=1

s.t. 0 ≤ pk ≤ P max, 0 ≤ PR ≤ PRmax
2K
X
pk + PR ≤ Ptmax .
k=1

(30b)
(30c)

13

The first two constraint specify the peak transmit power of the user and the relay i.e. P max and
PRmax respectively. The last constraint impose a constraint of Ptmax on the total system transmit
power. The problem P1 can be re-cast as
P2 : Max

1−

τ
T

2K
P

pk ,PR ,Γk



log2

2K
Q

(1 + Γk )

k=1

(31a)

pk + P R + P c

k=1

s.t.

0 ≤ pk ≤ P max , 0 ≤ PR ≤ PRmax
2K
X
pk + PR ≤ Ptmax

(31b)
(31c)

k=1

ISNRζk (pk , PR ) ≤ Γ−1
k .

(31d)

The symbol Γk = SNRζk denotes an auxiliary variable and the term ISNRζk denotes the inverse
of SNRζk , i.e. ISNRζk = 1/SNRζk . We observe from (25) and (26) that the SNRζk is a ratio
of a monomial and of a posynomial, the numerator of the objective in P2 becomes a ratio
of two posynomials, which is not a posynomial and hence non-convex. We also note that the
constraints in P2 are upper-bounded posynomials and are therefore convex. We now approximate
the numerator as a monomial such that P2 becomes a CCFP. To this end, we use the following
lemma from [43].
Lemma 1. Consider a monomial function q(νk ) = δk νkαk (νk > 0), which is used for approximating s(νk ) = 1 + νk near an arbitrary point ν̃k > 0. For the above approximation, the following
two conditions hold.
1) For the best monomial local approximation, the parameters αk and δk are given by
αk = ν̃k (1 + ν̃k )−1 , δk = ν̃k−αk (1 + ν̃k ).

(32)

2) For all νk > 0, s(νk ) ≥ q(νk ).
Using Lemma 1, the numerator of the objective function can be approximated as
i

−1
2K
Qh
αk (pk ,PR )
, where αk (pk , PR ) = Γ̃k 1 + Γ̃k
δk (pk , PR )Γk
and
log2
k=1
 −αk (pk ,PR )

δk (pk , PR )= Γ̃k
1 + Γ̃k , where Γ̃k is an initial value approximation for Γk . Given
the approximated objective, the optimization P2 can be formulated as follows.
i
2K
Qh
α (p ,P )
δk (pk , PR )Γk k k R
log2
k=1
P3 : Max
2K
pk ,PR ,Γk
P
pk + P R + P c
k=1

s.t.

(31b), (31c), (31d).

(33)

14

Here we have dropped the constant 1 −

τ
T



from the objective. The optimization P3 is now a

CCFP. We use the following result, proved in [39], [44], to solve it.
Proposition 3. Consider the CCFP g(x) = u(x)/v(x), with u being non-negative, differentiable
and concave, while v being positive, differentiable and convex. Then the function g(x) is a PC
and a stationary point x∗ of g(x) is its global maximizer. The problem of maximizing g(x) is
equivalent to finding the positive zero of D(λ), which is defined as
D(λ) , max {u(x) − λv(x)} .
x

(34)

The function D(λ) is convex, continuous and strictly monotonically decreasing and its zero is
found using Dinkelbach’s algorithm [44].
Let us now exploit the monomial approximation and Dinkelbach’s algorithm to solve the EE
problem P3, as illustrated in Algorithm 12 .
Remark 3. The first inequality constraint in Algorithm 1, also termed as the trust region constraint
[38], is added to confine the domain of variable Γ to a region around the current guess Γ̃[m] .
In most practical cases, α = 1.1 provides a good accuracy/speed trade-off for the monomial
approximations [38], [43].
B. Energy efficient optimization relying on the max-min approach
We next maximize the EE of the least energy-efficient user using the classic fractional programming approach. The max-min optimization can be cast as
i
h
α (p ,P )
log2 δk (pk , PR )Γk k k R
P4 : Max
min
PR + Pc
pk ,PR ,Γk 1≤k≤2K
pk +
2K
s.t.
(31b), (31c), (31d).

(36)

Here δk (pk , PR ) and αk (pk , PR ) are calculated similar to (32). The problem P4 now belongs to
a class of a max-min fractional program (MMFP). We next use the following proposition from
[30], [39] to solve our MMFP.
2

The Algorithm 1 uses the monomial, obtained using Lemma 1, to cast the problem P3 as a GP. Note that this algorithm

is not optimal as it approximates the posynomial with a monomial. It is a heuristic that often computes the globally optimal
power allocation [45]. The monotonic optimization, which obtains global optimal solution in finite time [45], can also be used
to solve problem P2 without any approximation. But the worst-case complexity of monotonic optimization increases at least
exponentially in 2K [45]. The proposed sequential fractional solution can solve the problem in polynomial time with affordable
complexity. Further the proposed algorithm – as concluded by exhaustive numerical studies in [45] – is an excellent heuristic
that achieves a global optimum solution.

15

Algorithm 1: EE maximization algorithm
Input: Given a tolerance ǫ > 0 and the maximum number of iterations L, set m = 1 and
[1]

λ[1] = 0. Calculate the initial values pk , PR and Γ̃k by allocating equal power to all
users i.e. pk = PR /2K, and PR = P/2.
Output: p∗k and PR∗ as the solutions.
1

for m ← 1 to L do

[m]

[m]

2

Given a feasible pk , PR , compute αk (pk , PR ) and δk (pk , PR ).

3

Solve the GP to calculate pk , PR and Γk .
!)
(
2K
2K
X
Y
[m]
αk (pk ,PR )
[m]
pk + P R + P c
− λ[m]
δk (pk , PR )Γk
Max
log2
pk ,PR ,Γk

s.t.

4

5

k=1
−1 [m]
β Γ̃k ≤

k=1

Γk ≤

[m]
β Γ̃k

(35)

(31b), (31c), (31d).
Based on the solutions PR and pk of GP, compute Γ̃k , and
!
2K
2K
X
Y
[m]
α (p ,P )
[m]
pk + P R + P c .
δk (pk , PR )Γ̃k k k R − λ[m]
D(λ[m] ) = log2
[m]

k=1

k=1

Do until convergence

[m+1]

if D(λk ) > ǫ then Γ̃k

6

else break.

7

return p∗k and PR∗ .

= Γ̃k , and
2K
[m]
Q [m]
α (p ,P )
δk (pk , PR )Γ̃k k k R
log2
k=1
λ[m+1] =
.
2K
P
pk + P R + P c
k=1

Proposition 4. The MMFP optimization problem is stated as
uk (x)
Max min
x
1≤k≤2K vk (x)
s.t. ci (x), i = 1, 2, 3, · · · , I.

(37)

For a non-negative and concave uk , positive and convex vk , ∀k = 1, 2, 3, · · · , 2K, as well as
convex ci , ∀i = 1, 2, 3, · · · , I, each ratio of the objective is a QC function. Furthermore, if
z(x) =

min yk (x), so that yk (x) is a QC, ∀k = 1, 2, 3, · · · , 2K, then z(x) is also a QC

1≤k≤2K

function. The auxiliary function of our real variable is defined as


D(λ) , Max
min {u(x) − λv(x)} .
x

1≤k≤2K

(38)

The generalized Dinkelbach algorithm [30] solves a sequence of convex problems to obtain

the global solution of MMFP, as illustrated in Algorithm 2.

16

Algorithm 2: Max-min EE optimization algorithm
Input: Given a tolerance ǫ > 0 and the maximum number of iterations L, set m = 1 and
[1]

λ[1] = 0, calculate the initial values pk , PR and Γ̃k by allocating equal power to all
users i.e. pk = PR /2K, and PR = P/2.
Output: p∗k and PR∗ as the solutions.
1

for m ← 1 to L do

[m]

[m]

2

Given a feasible pk , PR , compute αk (pk , PR ) and δk (pk , PR ).

3

Solve the GP to calculate pk , PR and Γk .





[m]
PR + Pc
αk (pk ,PR )
[m]
[m]
pk +
−λ
Max min log2 δk (pk , PR )Γk
pk ,PR ,Γk
2K
s.t.
(31b), (31c), (31d)
[m]

[m]

β −1 Γ̃k ≤ Γk ≤ β Γ̃k .
4

5

6

Based on the solutions PR and pk of GP, compute Γ̃k , and





[m]
PR + Pc
αk (pk ,PR )
[m]
[m]
[m]
pk +
−λ
.
log2 δk (pk , PR )Γ̃k
D(λ ) = min
1≤k≤2K
2K


[m]
[m+1]
αk (pk ,PR )
[m]
Update Γ̃k
= Γ̃k , and
log2 δk (pk , PR )Γ̃k
[m+1]
λ
= min
.
+Pc
1≤k≤2K
pk + PR2K
return p∗k and PR∗ .
VII. S IMULATION R ESULTS
Let us now investigate the EE and the max-min algorithms using Monte-Carlo simulations to

highlight their advantage. For this study, we set the length of the coherence interval to T = 200
symbols and the pilot length to τ = 2K. We also set the large scale fading matrices Du and
Dd , similar to [17], [20] as
Du = diag [0.749, 0.045, 0.246, 0.121, 0.125, 0.142, 0.635, 0.256, 0.021, 0.123] , and
Dd = diag [0.257, 0.856, 1.000, 0.899, 0.014, 0.759, 0.315, 0.432, 0.195, 0.562] .

(39)

Furthermore, the maximum transmit power of each user is P max=10 dBm, the maximum relay
transmit power is PRmax=23 dBm, and the circuit power is Pc=30 dBm [20]. For this analysis, we
2
consider K=5 user pairs and N = 500 relay antennas. The noise variances are set to σn2 =σnr
=σ 2

and we define η=Ptmax /σ 2 , where Ptmax is the maximum total transmit power of the system.
A. EE maximization
We investigate the performance of the EE maximization algorithm for both MRC/MRT and
2
ZFR/ZFT processing. The values of the self-loop interference σLIR
and inter-user interference

17
2
σU2 I , σk,j
for k, j = 1, · · · , 2K are assumed to be 0 dB with respect to σ 2 . Before investigating

the performance, we have to decide the pilot transmission power Pρ , which affects the channel
estimation and consequently the overall performance. To determine Pρ , we maximize the EE by
varying Pρ from −10 dBm to 40 dBm with two different values of η, i.e. 0 dB and 20 dB. We see
from Fig. 2a that Pρ = 20 dBm is capable of achieving the maximum EE for both MRC/MRT
and ZFR/ZFT designs. We therefore set Pρ = 20 dBm for the rest of the EE analysis. We also see
that a lower pilot power degrades the ZFR/ZFT EE performance more than that of the MRC/MRT

0.7

0.8

0.6

0.7

MRC/MRT, proposed
MRC/MRT, [14]
ZFR/ZFT, proposed
ZFR/ZFT, [14]

0.6

0.5 MRC/MRT

EE (bits/Joule/Hz)

EE (bits/Joule/Hz)

design. This is because the ZFR/ZFT design depends on the composite channels of the users.

0.4
0.3

0.5
0.4
0.3

0.2
0.1
0
−10

ZFR/ZFT
0

10
20
30
Pilot power (P [dBm])
ρ

(a)

0.2

η = 0 dB
η = 20 dB

0.1
40

0
−10

0

10

η [dB]

20

30

40

(b)

Fig. 2: EE comparison of MRC/MRT and ZFR/ZFT processing versus a) Pρ ; b) η=Ptmax/σ 2 for Pρ = 20 dBm

pilot transmit power. For both these figures we have K = 5 user pairs and N = 500 relay antennas.

We plot in Fig. 2b the EE versus η for both MRC/MRT and ZFR/ZFT processing. We compare
the performance of the proposed EE algorithm to [14], where Zhang et al. have derived the
asymptotic SE and EE for two-way amplify-and-forward full-duplex relaying systems assuming
that all the 2K users are allocated equal power. We see that the proposed algorithm yields a higher
EE for both MRC/MRT and ZFR/ZFT than that in [14]. We also see that the achieved EE remains
constant for η ≥ 7 dB for the MRC/MRT scheme, since with η = 7 dB, the system achieves the
maximum EE. After attaining the maximum EE, the system does not use any additional power,
since any additional power usage would reduce the EE. We observed this undesired behavior
in [14], where the system keeps using the available power beyond η = 7 dB and hence the EE
reduces. We also see that for η ≤ 0 dB the MRC/MRT has better EE than ZFR/ZFT.
The proposed EE optimization algorithm (Algorithm 1) can also optimize the EE under QoS

18

constraints that are specified as the rate rk required by each user


log2 1 + SNRζk (pk , PR ) ≥ rk .

(40)

It is readily seen that these constraints are posynomial and the GP in Step-3 of the algorithm
is still a GP. We now maximize in Fig. 3a the EE for ZFR/ZFT processing under per-user QoS
requirements of 0.2 bps/Hz, 0.5 bps/Hz and 0.7 bps/Hz. We see that the QoS constraints of
0.2 bps/Hz, 0.5 bps/Hz and 0.7 bps/Hz cannot be met for η ≤ −5 dB, η ≤ 0 dB and η ≤ 5 dB,
respectively. We observe that for η ≥ 5 dB, the EE is the same, both with and without QoS. This
0.7

0.65

0.6

0.6
0.55

0.5

EE (bits/Joule/Hz)

EE (bits/Joule/Hz)

implies that the system is capable of satisfying the QoS constraints, while maximizing the EE.

0.4
0.3
0.2
Proposed, without Qos
Proposed, Qos = 0.2 bps/Hz
Proposed, Qos = 0.5 bps/Hz
Proposed, Qos = 0.7 bps/Hz

0.1
0
−10

0

10

η [dB]

(a)

20

30

40

0.5
0.45
0.4
MRC/MRT, η = 10 dB
ZFR/ZFT, η = 10 dB
MRC/MRT, η = 0 dB
ZFR/ZFT, η = 0 dB

0.35
0.3
0.25
0

20
40
60
Number of iterations (L)

80

(b)

Fig. 3: a) EE versus η=Ptmax /σ 2 for the proposed optimal power allocation subject to the QoS constraint, considering

ZFR/ZFT processing associated with QoS requirements of 0.2 bps/Hz, 0.5 bps/Hz and 0.7 bps/Hz; b) EE versus
the number of iterations. For both these figures we have K = 5, N = 500, Pρ = 20 dBm

B. Convergence of the proposed algorithm
We now investigate the number of iterations required for the algorithm to converge for two
different η values, namely for η = 0 dB, and η = 10 dB. We observe from Fig. 3b that the
algorithm converges within 20 iterations for MRC/MRT and within 40 iterations for ZFR/ZFT.
The algorithm has a similar convergence behavior for other η values as well.
C. EE optimization under the max-min approach
We now evaluate the performance of the max-min algorithm for ZFR/ZFT processing. For this
2
study, we set both the self-loop interference σLIR
and the inter-user interference σU2 I to be 5 dB

below σ 2 . Again, we have to decide the pilot transmission power Pρ , which affects the channel
estimation. To decide its value, we first solve our max-min optimization problem by varying Pρ
from −10 dBm to 40 dBm for a fixed η = 10 dB. We observe from Fig. 4a that Pρ = 20 dBm
attains the maximum EE for the worst user. We therefore set Pρ = 20 dBm for the max-min

19

analysis. The max EE provided by the proposed optimization is higher than [14] for all values
of pilot power. We see from Fig. 4b that the proposed algorithm significantly improves the EE
of the worst user, when compared to equal power allocation. We also see that the system is
capable of satisfying the QoS of 0.5 bps/Hz above η = 10 dB without compromising the EE of
0.8

0.6

0.7

0.5
0.4
0.3
0.2
ZFR/ZFT, proposed
ZFR/ZFT, [14]

0.1
0
−10

0

10
20
30
Pilot power (P [dBm])

40

[14], no QoS
Proposed, no QoS
Proposed, QoS = 0.5 bps/Hz

60

Spectral efficiency (bps/Hz)

0.7

Minimum EE (bits/Joule/Hz)

Minimum EE (bits/Joule/Hz)

the worst user.

0.6
0.5
0.4
0.3
0.2

50
40 N = 256
30
N = 64

20

Monte−Carlo (MRC/MRT)
Lower bound (MRC/MFT)
Monte−Carlo (ZFR/ZFT)
Lower bound (ZFR/ZFT)

10

0.1
0
−10

0

10

ρ

(a)

η [dB]

20

30

40

0
−10

0

(b)

10
20
P [dBm]

30

40

R

(c)

Fig. 4: Minimum EE for both equal and optimal power allocation for ZFR/ZFT processing a) versus Pρ , where

η = 10 dB; and b) versus η, where Pρ = 20 dBm. For both these figures we have K = 5 and N = 500; c) SE
efficiency of MRC/MRT and ZFR/ZFT versus PR , where we have K = 10, Pρ = 10 dBm.

In this paper, we also derived the analytical lower bounds of the achievable rate for both
MRC/MRT and ZFR/ZFT processing with MMSE channel estimation which were formulated
in Theorem 1 and Theorem 2. In this section, we compare the SE3 lower bounds with their
exact expressions in (19) for N = 64 and N = 256 relay antennas and for K = 10 user pairs.
We set Pρ = 10 dBm and allocate equal power to all users i.e. we have pi = PR /2K, ∀i =

2
2
1, 2, 3, · · · , 2K. The large scale fading coefficients σg,i
= σf,i
, ∀i, j = 1, 2, 3, · · · , 2K, and the

2
self-loop interference σLIR
as well as the inter-user interference σU2 I = σ 2 are set to 0 dB

with respect to σ 2 . We see from Fig. 4c that the derived lower bound and the exact expression
overlap for ZFR/ZFT processing for N = 256 relay antennas. For MRC/MRT, the lower bound
marginally differs from the exact expression. We also observe that the SE, of high PR values
saturates for both MRC/MRT and ZFR/ZFT. This is because the relay self-loop interference also
increases proportionally upon increasing PR .
D. SE comparison with existing full-duplex designs
We note that the EE maximization framework considered in this work can also maximize
the SE by maximizing the numerator of the Problem P3 – the modified problem is a GP. We
3

Recall from (28) that SE is obtained by scaling the achievable rate with pilot overhead.

20

can consequently also maximize the SE both for MRC/MRT and for ZFR/ZFT processing. In
contrast to [15], which derives the lower bound and maximizes the SE (and not the EE and
max-min EE), for the MRC/MRT processing alone, whilst relying on LS channel estimation.
The current work is therefore more general than [15]. The objective of this study is to compare
the SE of the proposed system to that of the existing full-duplex relaying system in [15], and
to that of [14] where the latter has developed equal-power MRC/MRT and ZFR/ZFT for fullduplex relaying designs.
40

30
σ2 =1
nr

30
25
20

σ2nr=10

15
10

ZFR/ZFT, proposed
MRC/MRT, proposed
MRC/MRT, [15]

5
0

−5

0

5
10
P (dBm)

15

20

ρ

(a)

Spectral efficiency (bps/Hz)

Spectral efficiency (bps/Hz)

35

25
20
15
10
MRC/MRT, proposed
MRC/MRT, [14]
MRC/MRT, [15]
ZFR/ZFT, proposed
ZFR/ZFT, [14]

5
0
−20

−10

0

10
η [dB]

20

30

40

(b)

Fig. 5: SE of MRC/MRT and ZFR/ZFT versus: a) Pρ , comparing the performance of MMSE and LS estimators,

where K = 5, N = 256 and PR = 10 dBm; b) η, where K = 5, Pρ = 20 dBm, N = 500.

We commence by comparing the SE to that of the design in [15]. We plot in Fig. 5a the SE
versus Pρ , where we observe that for MRC/MRT processing the MMSE estimator outperforms the
2
LS estimator [15] both for a lower pilot power Pρ , and a higher noise power σnr
. We also see that

the ZFR/ZFT associated with Pρ > 0 dB performs much better than the MRC/MRT processing.
We now evaluate the SE by varying η for the proposed algorithm considering both MRC/MRT
and ZFR/ZFT processing and compare it to the designs in references [14], [15]. We see from
Fig. 5b that the algorithm proposed for both the MRC/MRT and ZFR/ZFT designs considerably
improves the SE, when compared to their counterparts from [14], which considers an equal power
allocation. We also see that the MRC/MRT associated with the proposed algorithm outperforms
even ZFR/ZFT for η < 5 dB. Furthermore, the performance of the MRC/MRT processing
associated with the proposed algorithm overlaps with that of [15]. This fact can also be justified
by observing Fig. 5a, where for Pρ = 20 dBm, the performance of the proposed system and
[15] is the same for the MRC/MRT processing.

21

E. Full-duplex versus half-duplex comparison
In this section, we compare both the SE and the EE of the proposed full-duplex system to
those of the existing half-duplex systems [20], [21] for different values of self-loop interference
2
2
σLIR
and inter-user interference σU2 I . The objective is to characterize the values of σLIR
and σU2 I

for which the proposed full-duplex relaying system using MMSE channel estimation as well as
MRC/MRT and ZFR/ZFT processing has better SE and EE than their half-duplex counterparts.
Reference [21] derives the asymptotic spectral and energy efficiencies of the half-duplex system
by allocating equal power to all users, whereas reference [20] allocates power to maximize
the SE of the half-duplex system. We observe from Fig. 6a that for the full-duplex system
2
using MRC/MRT processing, the σLIR
dB and σU2 I dB should be around 0 dB for it to achieve

a better SE than the half-duplex system with optimal power allocation [20]. As expected for
ZFR/ZFT processing, the full-duplex system requires much higher self-loop interference and
2
inter-user interference suppression. We see from Fig. 6b that both σLIR
and σU2 I should have

values around −10 dB.
25

60

20

50

10
σ2

5

LIR
σ2
LIR
2
σ
LIR

0
−20

= σ2 = 0 dB, proposed
=
=

UI
σ2
UI
2
σ
UI

= 5 dB, proposed
= 10 dB, proposed

Half−duplex [20]
Half−duplex [21]

−10

0

10
η [dB]

(a)

20

30

40

2
LIR

σ

2
UI

=σ

= 0 dB,

FD proposed

40
30
20

2
LIR
2
LIR
2
σ
LIR

10
0
−20

2
UI
2
UI
2
σ
UI

σ

=σ

= −10 dB, proposed

σ

=σ

= −5 dB, proposed

=

= −10 dB, proposed

EE (bits/Joule/Hz)

15

Spectral efficiency (bps/Hz)

Spectral efficiency (bps/Hz)

0.5

0.4

HD, proposed

0.3

HD [21]
0.2
2
LIR

σ

2
UI

=σ

= 10 dB,

FD proposed

0.1

Half−duplex [20]
Half−duplex [21]

−10

0

10
η [dB]

(b)

20

30

40

0
−10

0

10

η [dB]

20

30

40

(c)

Fig. 6: Performance comparison of full- and half-duplex systems: a) SE versus η = Ptmax /σ 2 for MRC/MRT

processing b) SE versus η = Ptmax /σ 2 for ZFR/ZFT processing; c) EE versus η for MRC/MRT processing. For all
of these figures we have K = 5, N = 500 and Pρ = 20 dB.
2
We now investigate the EE of full- and half-duplex relaying for different values of σLIR
and

σU2 I in conjunction with MRC/MRT processing. We note that the EE of an AF half-duplex massive MIMO relaying has not been investigated in the literature. In the proposed framework, the
EE of half-duplex massive MIMO relaying can be evaluated by setting the self-loop interference
2
=σU2 I =0 dB the full
and inter-user interference to zero. We observe from Fig. 6c that for σLIR

duplex system has higher EE than a half-duplex system. However, the half-duplex performs better
2
2
for σLIR
= σLIR
= 10 dB. The proposed EE power allocation algorithm, which works for a

22

half-duplex system, has better EE than the equal-power half-duplex MRC/MRT scheme of [21].
VIII. C ONCLUSION
We considered a multi-pair AF full-duplex massive MIMO two-way relay associated with fullduplex users. We derived closed-form lower bounds for the achievable rate of both MRC/MRT
and ZFR/ZFT relay processing in conjunction with MMSE channel estimation and an arbitrary
number of relay antennas. We numerically showed that these lower bounds match with their exact
expressions. Furthermore, we used these bounds to design algorithms for jointly allocating power
for the users and the relay based on the EE maximization and max-min fairness criterion. We
investigated the performance of both algorithms, with and without QoS constraints, and demonstrated their improved performance over existing full-duplex systems. We exhaustively compared
the spectral and energy efficiencies of the proposed relaying system to the existing state-of-the-art
half-duplex relaying systems, and characterized the values of self-loop and inter-user interference,
for which the proposed system becomes capable of outperforming the existing ones.
A PPENDIX A
i
h
2
The term E kWG̃xk in the denominator of (10) can be expressed using (14) as


n h
io
2
∗
H
= Tr E (F̂∗ TĜH GPPH GH ĜTH F̂T )
E F̂ TĜ G̃x
n h
io
∗
H
H
H
H
H T
= Tr E F̂ TĜ (Ĝ + Eg )PP (Ĝ + Eg )ĜT F̂
io
n h
(a)
H H
H T ∗
H
= Tr E ĜPP Ĝ ĜT F̂ F̂ TĜ
io
n h
H
H
H T ∗
H
.
+ Tr E Eg PP Eg ĜT F̂ F̂ TĜ

(41)
(42)

The equality in (a) holds, because Ĝ and Eg are independent and Tr (AB) = Tr (BA). We now

want to simplify (41) and (42). We first expand (41) as in
n h
io
H H
H T ∗
H
Tr E ĜPP Ĝ ĜT F̂ F̂ TĜ
"
#)
( 2K
K 


X
X
T
∗
H
H
pk E ĝk ĝkH
ĝ2i−1 f̂2iT + ĝ2i f̂2i−1
f̂2i−1
ĝ2i
+ f̂2i∗ ĝ2i−1
.
= Tr
k=1

(43)

i=1

To simplify (43), we decompose the summation therein for k 6= 2i or 2i − 1, k = 2i − 1 and

k = 2i. For these k values, Eq. (43) can respectively be simplified as
n h


io

T
2
2
2
2
2
∗
H
H
Tr E ĝk ĝkH ĝ2i−1 f̂2iT + ĝ2i f̂2i−1
= N 2 σ̂g,k
σ̂g,2i
σ̂f,2i−1
+σ̂g,2i−1
σ̂f,2i
, (43)
f̂2i−1
ĝ2i
+ f̂2i∗ ĝ2i−1
n h
io
H
T
∗
H
H
Tr E ĝ2i−1 ĝ2i−1
(ĝ2i−1 f̂2iT + ĝ2i f̂2i−1
)(f̂2i−1
ĝ2i
+ f̂2i∗ ĝ2i−1
)

2
2
2
2
2
= N 2 σ̂g,2i−1
σ̂g,2i
σ̂f,2i−1
+ (N + 1)σ̂g,2i−1
σ̂f,2i
, (44)
n h
io
H
T
∗
H
H
Tr E ĝ2i ĝ2i
(ĝ2i−1 f̂2iT + ĝ2i f̂2i−1
)(f̂2i−1
ĝ2i
+ f̂2i∗ ĝ2i−1
)

2
2
2
2
2
= N 2 σ̂g,2i
σ̂g,2i−1
σ̂f,2i
+ (N + 1)σ̂g,2i
σ̂f,2i−1
.
(45)

23

In deriving the equalities in (43), (44), and (45), we exploit the following facts: i) for any arbitrary
i
h




2
2
vectors x and y, x∗ yT = yxH ; and ii) E ĝiH ĝi = N σ̂g,i
, E ĝiH ĝj = 0, E ĝiH ĝj
=


2
2
4
N σ̂g,i
σ̂g,j
, E ĝi ĝiH ĝi ĝiH = (N + 1)σ̂g,i
IN [46]. Furthermore, we can simplify (43) as
n h
io
Tr E ĜPPH ĜH ĜTH F̂T F̂∗ TĜH
"
#)
( 2K
K 


X
X
(a)
T
∗
H
H
pk E ĝk ĝkH
ĝ2i−1 f̂2iT + ĝ2i f̂2i−1
f̂2i−1
ĝ2i
+ f̂2i∗ ĝ2i−1
= Tr
k=1
2K
X

(b)

= N2

i=1

pk

i=1

k6=2i,2i−1

+(N +
= N2

K
X


2
2
2
2
2
2
2
2
σ̂g,k
σ̂g,2i
σ̂f,2i−1
+ σ̂g,2i−1
σ̂f,2i
+ N2
p2i σ̂g,2i
σ̂g,2i−1
σ̂f,2i

2
2
1)σ̂g,2i
σ̂f,2i−1

2K X
K
X

(c)

K
X

k6=2i,2i−1i=1
K
X
3

+N



+N

2
pk σ̂g,k
Φ̂i + N 2

2

K
X
i=1

K
X

i=1

2
p2i−1 σ̂g,2i−1


2
2
p2i−1 σ̂g,2i−1
+ p2i σ̂g,2i
Φ̂i

i=1

(d)

=N

(e)

2

=N 2

2
p2j−1 σ̂g,2j−1

+

2
p2j σ̂g,2j

j=1,j6=i i=1
K X
K
X
j=1 i=1

2
2
+ (N + 1)σ̂g,2i−1
σ̂f,2i

i=1

4
2
4
2
p2i−1 σ̂g,2i−1
σ̂f,2i
+ p2i σ̂g,2i
σ̂f,2i−1

K
K
X
X

2
2
σ̂g,2i
σ̂f,2i−1





Φ̂i + N

2


2
2
p2j−1 σ̂g,2j−1
+p2j σ̂g,2j
Φ̂i+N 3 Υ̂ = N 2

K
X




2
2
p2i−1 σ̂g,2i−1
+ p2i σ̂g,2i
Φ̂i + N 3 Υ̂

i=1
K X
K
X

Ψ̂j Φ̂i+N 3 Υ̂= N 2 Ψ̂Φ̂ + N 3 Υ̂. (46)

j=1 i=1

We now discuss the steps used for simplifying (43). Equality in (b) is obtained by using the results

2
2
2
2
σ̂f,2i
+ σ̂g,2i
σ̂f,2i−1
,
in (43), (44) and (45). Equality in (c) is obtained by defining Φ̂i = σ̂g,2i−1

and by rearranging the terms. Equality in (d) is obtained by exploiting the fact that

PK
PK
P2K
4
2
4
2
p
σ̂
σ̂
+
p
σ̂
σ̂
.
(x
+
x
),
and
by
defining
Υ̂
=
x
=
2i−1
2i
2j−1
2j
k
g,2i−1
g,2i
f,2i
f,2i−1
i=1
j=1
k=1
PK
PK
P2K 2 2
Equality in (e) is obtained by defining Φ̂ = i=1 Φ̂i = i=1 σ̂g,i σ̂f,i′ and Ψ̂ =
i=1 Ψ̂i =

PK
P
2K
2
2
2
i=1 p2i−1 σ̂g,2i−1+p2i σ̂g,2i =
i=1 pi σ̂g,i . By using an approach similar to (41), we simplify (42) as
n h
io
n 
io
 h
H T ∗
H
Tr E Eg PPH Eg H ĜTH F̂T F̂∗ TĜH
= Tr E Eg PPH EH
E
ĜT
F̂
F̂
T
Ĝ
g
( 2K
" K
#)




X
X
2
T
T
∗
H
∗ H
= Tr
pk σ̂ξg,k E
ĝ2i−1 f̂2i + ĝ2i f̂2i−1 f̂2i−1 ĝ2i + f̂2i ĝ2i−1
=

2K
X
k=1

k=1

2
pk σ̂ξg,k

K
X

i=1

N

2

2
2
σ̂g,2i−1
σ̂f,2i

i=1

+

2
2
σ̂g,2i
σ̂f,2i−1



2

= N Φ̂

2K
X

2
pk σ̂ξg,k
.

We now simplify the second term in the denominator of (10).
io

 PR n h ∗
H T
Tr E F̂ TĜH GRR GH
ĜT
F̂
E kWGRR x̂R k2 =
RR
N
K
X

2
2
2
2
2
2
N 2 σ̂g,2i−1
σ̂f,2i
+ σ̂g,2i
σ̂f,2i−1
= N 2 PR σLIR
Φ̂.
= PR σLIR
i=1

(47)

k=1

(48)

24

The last term E [kWzR k2 ] in the denominator of (10) can be simplified as
n
h
io


2
E F̂∗ TĜH ĜTH F̂T
E kWzR k2 = Tr σnr
K
X


2
2
2
2
2
N 2 σ̂g,2i−1
σ̂f,2i
+ σ̂g,2i
σ̂f,2i−1
= N 2 σnr
Φ̂.
(49)
i=1
i
h
By substituting the expression of E kWG̃xk2 from (46) as well as (47), and the expressions
=

2
σnr

of E [kWGRR x̂R k2 ] and E [kWzR k2 ] from (48) and (49) respectively in (10), we get (15).
A PPENDIX B
i
h
2
To derive this result, we will first simplify E kWG̃xk in the denominator of (10) using (16).
i
n h
io
i
h
h
ˆ ∗ TḠ
ˆ H GPPH GH Ḡ
ˆ TH F̄
ˆT)
ˆ ∗ TḠ
ˆ H G̃xk2 = Tr E (F̄
E kWG̃xk2 = E kF̄
n h
io
ˆ ∗ TḠ
ˆ H (Ĝ + E )PPH (ĜH + EH )Ḡ
ˆ TH F̄
ˆT )
= Tr E (F̄
g
g
n h
io
io
n h
H ˆ H ˆT ˆ∗
H
∗
H
H H ˆ H ˆT
H
ˆ
ˆ
ˆ
= Tr E F̄ TḠ ĜPP Ĝ ḠT F̄
(50)
+ Tr E Eg PP Eg ḠT F̄ F̄ TḠ
2K
io X
n h
n h
io
2
ˆ ∗ TPPH TF̄
ˆT
ˆ TH F̄
ˆ T F̄
ˆ ∗ TḠ
ˆH
= Tr E F̄
+
pi σξ,g,i
Tr E Ḡ

(a)

i=1

(b)

=

2K
X
i=1

2K
2K
h
i X
n h
io (c)
X
Hˆ
2
∗
2
ˆ
pi′ E f̄i f̄i +
pi σξ,g,i Tr E Λ̂F TΛ̂G T = λ̂ +
pi σξ,g,i
η̂.
i=1

(51)

i=1

ˆ H Ĝ = ĜH Ḡ
ˆ = I , and
The equality in (a) is obtained by exploiting the fact that Ḡ
2K


P
ˆ H F̄
ˆ ) = F̄H F̄−1 as
2
E Eg PPH Eg H = 2K
pi σξ,g,i
IN . In the equality (b), we define Λ̂F , (F̄
i=1


ˆ H Ḡ
ˆ = ḠH Ḡ−1 and exploits the fact that TPPH T = diag{p , p , · · · , p , p
well as Λ̂G , Ḡ
2 1
2K
2K−1 }.
To derive the equality in (c), we first note that the random matrices Λ̂F and Λ̂G have inverse

−1
Wishart distribution, i.e. we have Λ̂F ∼ W −1 (D̂−1
(D̂−1
u , 2K), where D̂d and
d , 2K), Λ̂G ∼ W

D̂u are covariance matrices of the rows of the estimated channels Ĝ and F̂, which are given
 
 
h i
, ŵg,i,j = Λ̂G
, ∀i, j = 1, 2, 3, ..., 2K and E Λ̂F =
after (13). We denote ŵf,i,j = Λ̂F
i,j
h i
h i,j i
D̂−1
D̂−1
1
u
d
, E Λ̂G = N −2K−1 [47]. We also have E ˆf̄iHˆf̄j = E [ŵf,i,j ] = (N −2K−1)σ̂
2 , ∀i =
N −2K−1
f,i
 H 
1
ˆ i ḡ
ˆ j = E [ŵg,i,j ] =
j, and 0 otherwise. Similarly, E ḡ
2 , ∀i = j, and 0 otherwise.
(N −2K−1)σ̂g,i
P
p′
i
Given the above equalities, we obtain the equality in (c), where λ̂ = 2K
2 . The
i=1 (N −2K−1)σ̂f,i
n h
io
expression Tr E Λ̂∗F TΛ̂G T is simplified as
2K  h
i 
n h
io
X

 ∗  

∗
∗
E wg,j,j ′ + E wf,j,j
E wg,j ′ ,j ′
E wf,j,j
Tr E Λ̂F TΛ̂G T
=
′
j=1

=

2K
X
j=1

1
, η̂.
2
2
σ̂g,j
(N − 2K − 1)2 σ̂f,j
′

(52)

25

Following similar lines, the second term in the denominator of (10) can be simplified as
i
h
n h
io


ˆ ∗ TḠ
ˆ H G x̃ k2 = PR Tr E F̄
ˆ ∗ TḠ
ˆ H G GH Ḡ
ˆ H ˆT
E kWGRR x̃R k2 = E kF̄
RR R
RR RR T F̄
N
n h
io
n h
io
2
2
2
ˆ T F̄
ˆ ∗ TḠ
ˆ H Ḡ
ˆ TH
= PR σLIR
Tr E F̄
= PR σLIR
Tr E Λ̂∗F TΛ̂G T = PR σLIR
η̂.
(53)

The last term in the denominator of (10) can be simplified to
h
i
n h
io
n h
io
2
∗
2
ˆ ∗ TḠ
ˆ H z k2 = σ 2 Tr E F̄
ˆ ∗ TḠ
ˆ H Ḡ
ˆ TH F̄
ˆT
E kF̄
=
σ
Tr
E
Λ̂
T
Λ̂
T
= σnr
η̂.
R
G
nr
nr
F

(54)

By using (51), (53) and (54), we get the simplified expression of α in (17).

A PPENDIX C
We will start by simplifying the numerator of (23), with fk = f̂k + ef,k , gk = ĝk + eg,k , and
using the fact that f̂k , ĝk , ef,k , eg,k are independent, as follows


T
i
h

 T

T ∗
H
′
′
′
′
E |fk Wgk | = E f̂k + ef,k W ĝk + eg,k
= E f̂k F̂k TĜ ĝk
"
#
K 
i
h

X
T
∗
H
∗ H
T ∗ H
T ∗ H
′
′
′
= E f̂k
f̂2i−1 ĝ2i + f̂2i ĝ2i−1 ĝk = E f̂k f̂k ĝk′ ĝk + f̂k f̂k′ ĝk ĝk
h i=1
h
i 
i 

 (a)
2
2
= E f̂kH f̂k E ĝkH′ ĝk′ + E f̂kH′ f̂k E ĝkH ĝk′ = N 2 σ̂f,k
σ̂g,k
′.
 H 


2
The equality in (a) is because E ĝi ĝi = N σ̂g,i
, E ĝiH ĝj = 0 [46].

(55)

We now simplify the first term in the denominator of (23), the variance of fkT Wgk′ =
2





var fkT Wgk′ = E |fkT Wgk′ |2 − E |fkT Wgk′ | , where we have:




E |fkT Wgk′ |2 = E fkT Wgk′ gkH′ WH fk∗
h
i
h
i
H ∗
= E f̂kT Wĝk′ ĝkH′ WH f̂k∗ + E f̂kT Weg,k′ eH
′ W f̂k
g,k
{z
} |
{z
}
|
(I)
(II)
i
h
 T

H ∗
H
H ∗
H
T
′
′
+ E ef,k Wĝk ĝk′ W ef,k + E ef,k Weg,k eg,k′ W ef,k .
(56)
|
{z
} |
{z
}
(III)

(IV)

The term-(I) in (56) is simplified as following.
"
#
K 
h
i (a)



X
T
∗
H
H
E f̂kT Wĝk′ ĝkH′ WH f̂k∗ = E f̂kT
f̂2i−1
+ ĝ2i−1 f̂2iT f̂k∗
ĝ2i
+ f̂2i∗ ĝ2i−1
ĝk′ ĝkH′ ĝ2i f̂2i−1
i=1

(b)

= N2

K
X

i=1,i6=⌈ k2 ⌉
(c)

=

2
2
N 2 σ̂f,k
σ̂g,k
′

(d)

2
2
N 2 σ̂f,k
σ̂g,k
′


4
4
2
2
2
2
2 2
2
2
2
σ̂g,k
σ̂ 2 σ̂ 2 + N 2 (N + 1)2 σ̂f,k
σ̂f,k
σ̂g,k
′
′ σ̂f,2i−1 σ̂g,2i + σ̂f,2i σ̂g,2i−1 + N σ̂f,k σ̂
f,k ′ g,k g,k ′

K
X
i=1

=




2
2
2
2
4
4
σ̂f,2i−1
σ̂g,2i
+ σ̂f,2i
σ̂g,2i−1
+ N 3 (N + 2)σ̂f,k
σ̂g,k
′


2
2
.
N(N + 2)σ̂f,k
σ̂g,k
′ + Φ̂

(57)
′

The equality in (b) is obtained by expanding (a) for two different cases: i) {k, k } = {2i, 2i−1};

26

h
i


′
2
2
2
ii) {k, k } =
6 {2i, 2i−1}, and by using the fact that E ĝiH ĝj
= N σ̂g,i
σ̂g,j
and E ĝi ĝiH ĝi ĝiH =

4
(N +1)σ̂g,i
IN [46]. The equality in (c) is obtained by simple arithmetic manipulations and by re
P
2
2
2
2
arranging terms. The equality in (d) is obtained by defining Φ̂ = K
i=1 σ̂g,2i−1 σ̂f,2i + σ̂g,2i σ̂f,2i−1 .

The term-(II) in (56) is simplified using similar ideas as follows:
h
i
H ∗
H
T
′
E f̂k Weg,k eg,k′ W f̂k


K
X

2
2
2
2
2
2
2
4
2
2
2
2
2
σ̂ξg,k
= N 2 σ̂f,k
σ̂f,k
′ σ̂f,k σ̂f,2i−1 σ̂g,2i + σ̂f,2i σ̂g,2i−1 
′ +
′ σ̂
′ + (N + 1)σ̂f,k σ̂
′ σ̂g,k σ̂
g,k ξg,k
ξg,k
2
2
= N 2 σ̂f,k
σ̂ξg,k
′

K
X



i6=⌈ k2 ⌉

2
2
2
2
2
4
2
σ̂f,2i−1
σ̂g,2i
+ σ̂f,2i
σ̂g,2i−1
+ N 3 σ̂f,k
σ̂ξg,k
′ σ̂
g,k ′

i=1
2
2
2
2
.
N σ̂f,k
σ̂g,k
= N 2 σ̂f,k
σ̂ξg,k
′ + Φ̂
′

The term-(III) in (56) is simplified as following.


E eTf,k
Wĝk′ ĝkH′ WH e∗f,k

2
2
4
2
2
2
2
+ (N + 1)σ̂g,k
= N 2 σ̂f,k
′ σ̂f,k σ̂ξf,k +
′ σ̂ξf,k σ̂g,k σ̂
g,k ′
2
2
= N 2 σ̂ξf,k
σ̂g,k
′

K
X



K
X

2
2
σ̂ξf,k
σ̂g,k
′

i6=⌈ k2 ⌉

2
2
2
2
2
2
4
σ̂f,2i−1
σ̂g,2i
+ σ̂f,2i
σ̂g,2i−1
+ N 3 σ̂ξf,k
σ̂g,k
′ σ̂f,k

i=1
2
2
2 2
2
= N σ̂ξf,k σ̂g,k′ N σ̂g,k′ σ̂f,k + Φ̂ .

(58)



2
2
2
2

σ̂f,2i−1
σ̂g,2i
+ σ̂f,2i
σ̂g,2i−1
(59)

Finally the term-(IV) in (56) is simplified as follows:
K
i
h
X

H ∗
2
2
2
2
2
2 2
2
2
2
W
e
=N
Φ̂. (60)
E eTf,k Weg,k′ eH
σ̂
σ̂
′
′ σ̂f,2i−1 σ̂g,2i + σ̂f,2i σ̂g,2i−1 =N σ̂ξf,k σ̂
f,k
ξf,k
g,k
ξg,k ′
ξg,k
i=1

Further using (57)-(60), (56) can be re-written as




2
2
2
2
2
2
2
2
σ̂g,k
E |fkT Wgk′ |2 = N 2 Φ̂ σ̂f,k
′
′ + σ̂ξf,k σ̂
′ + σ̂ξf,k σ̂
′ + σ̂f,k σ̂
ξg,k
g,k
ξg,k


2
2
2
2
2
2
2
2
.
(N + 2)σ̂f,k
σ̂g,k
+ N 3 σ̂f,k
σ̂g,k
′
′ + σ̂ξf,k σ̂
′ + σ̂f,k σ̂
′
g,k
ξg,k

(61)

2
2
2
2
2
2
By using (55), (61), and considering the fact that σg,k
= σ̂ξg,k
+ σ̂g,k
, σf,k
= σ̂ξf,k
+ σ̂f,k
, we have





2
2
2
2
2
2
2
2
.
(62)
σf,k
σ̂g,k
σg,k
var fkT Wgk′ = N 2 Φ̂σf,k
′ + σ̂f,k σ
′ + N σ̂f,k σ̂
g,k ′
g,k ′

Similar to the above calculations, we can simplify other terms in the denominator of (23) as




2 2
2
2
2
2
2
2
2
2
2
+
Φ̂
, (63)
+
Φ̂
+
N
σ
σ̂
N
σ̂
σ̂
SIk = N 2 σξf,k
σξg,k
Φ̂ + N 2 σ̂f,k
σξg,k
N σ̂f,k
σ̂g,k
′
ξf,k g,k
g,k f,k ′
2K
X

i
h

2
4
2
2
2
2
4
2
pj σf,k σg,j Φ̂ + N σf,k σ̂g,j σ̂f,j ′ + σg,j σ̂f,k σ̂g,k′ ,
IPk = N
′
j6=k,k 



2
2
2
4
2
2
4
2
,
, LIRk = N 2 PR , σLIR
N σ̂f,k
σ̂g,k
NRk = N 2 σnr
N σ̂f,k
σ̂g,k
′ + σf,k Φ̂
′ + σf,k Φ̂
X
X




2
UIk =
pi E |Ωk,i xi |2 =
pi σk,i
, NUk = E |zk |2 = σn2 .
2

i,k∈U k

i,k∈U k

(64)
(65)
(66)

27

We can further simplify the denominator of (23) using (62)-(66), as


1
1
pk′ var fkT Wgk′ + pk SIk + IPk + NRk + LIRk + 2 UIk + 2 NUk
!#
( 2K "
α
α


X
X
2
−1
2
2
4
2
2
pi σk,i
+ ck p k
σn +
pi ηk,i + Φ̂σg,i + N σ̂g,i σ̂f,i′ PR
=N
i=1
i,k∈Uk
!


X
X
2
2
2
2
2
4
2
2
pi σk,i
σLIR
Φ̂
+
pi σk,i
+ PR−1σnr
Φ̂ σn2 +
σf,k
Φ̂ + N σ̂f,k
σ̂g,k
+PR σLIR
′
i,k∈Uk
i,k∈Uk
o

2
2
2
2
2
4
2
(67)
+ σLIR σn + σnr σf,k Φ̂ + Nσnr σ̂f,k σ̂g,k′ ,


2
2
2
2
4
4
2
2
and,
where ηk,i = Φ̂σf,k
σg,i
+ N σf,k
σ̂g,i
σ̂f,i
′ + σg,i σ̂f,k σ̂
′
g,k



2
4
2
4
2
2
2
2
. The proof of Theorem (1) is complete.
σ̂f,k
σ̂g,j
σ̂g,k
ck = − Φ̂σ̂f,k
σ̂g,j
′
′ + σ̂f,k σ̂g,j σ̂
′ + N
f,k

A PPENDIX D
Starting with the numerator of (23) considering ZFR/ZFT processing, we have


i
h
T


 T
∗
H
ˆ
ˆ
ˆ ∗ TḠ
ˆ H ĝ ′
E fk Wgk′ = E f̂k + ef,k F̄ TḠ ĝk′ + eg,k′ = E f̂kT F̄
k
 (b)
 T
(a)
(68)
= E 1f,k T1g,k′ = E[1] = 1.
ˆ H ĝ ′ = 1 ′ ,
ˆ = 1T ′ , Ḡ
The equality in (a) is obtained by using the following results: ĝkH′ Ḡ
g,k
k
g,k
ˆ
T ˆ∗
T
T ∗
T
f̂k F̄ = 1f,k , F̄ f̂k = 1f,k . Equality in (b) is attained because 1f,k T1g,k′ = 1.


The expression of var fk T Wgk′ in the denominator of (23) is given by
i
h
2



2
− E fk T Wgk′
var fk T Wgk′ = E fk T Wgk′
h
i
h
i
 T

(a)
H ∗
H
H ∗
H
T
H ∗
H
T
′
′
′
= E f̂k Wĝk ĝk′ W f̂k − 1 + E ef,k Wĝk ĝk′ W ef,k + E f̂k Weg,k eg,k′ W f̂k
i
h
H ∗
W
e
+ E eTf,k Weg,k′ eH
′
f,k
g,k
h
i

 


(b) 2
2
H
T
H ∗
2
2
+ σξ,f,k
σξ,g,k
= σξ,f,k E ĝkH′ WH Wĝk′ + σξ,g,k
′ E f̂k WW f̂k
′ Tr E WW
h
i
i
h
(c) 2
ˆ ∗ TḠ
ˆ H Ḡ
ˆ TF̄
ˆ T f̂ ∗
ˆ TF̄
ˆ T F̄
ˆ H ĝ ′ + σ 2 ′ E f̂ T F̄
ˆ ∗ TḠ
= σξ,f,k E ĝkH′ Ḡ
k
k
k
ξ,g,k
n h
io
2
2
ˆ ∗ TḠ
ˆ H Ḡ
ˆ TF̄
ˆT
E F̄
+ σξ,f,k
σξ,g,k
′ Tr
n h
h
i
i
io
h
(d) 2
T
2
2
2
E Λ̂∗F TΛ̂G T
+ σξ,f,k
σξ,g,k
= σξ,f,k E 1Tg,k′ TΛ̂∗F T1g,k′ + σξ,g,k
′ Tr
′ E 1f,k TΛ̂G T1f,k
!
2K  h

i 
X







 ∗ 
(e) 2
∗
∗
2
2
E wf,j,j
= σξ,f,k E ŵf,k,k + σξ,g,k
′ E wg,j,j ′ +E wf,j,j E wg,j ′ ,j ′
′ E ŵg,k ′ ,k ′ + σξ,f,k
(f )

=

2
σξ,f,k

2
(N − 2K − 1)σ̂f,k

+

2
σξ,g,k
′

2
(N − 2K − 1)σ̂g,k
′

j=1

2
2
+ σξ,f,k
σξ,g,k
′ η̂.

(69)
′

The equality in (b) therein is because F̂T WĜ = T, i.e., f̂kT Wĝj = 1, ∀j = k and 0, otherwise.
The equality in (c) is obtained by substituting the value of W from (16). Equalities in (d) and (e)
are obtained by simple manipulations, and equality in (f ) is because E [ŵf,k,k ] =

1
2 ,
(N −2K−1)σ̂f,k

28



E ŵg,k′ ,k′ =

1
(N −2K−1)σ̂2

g,k

′

, and η̂ ,

P2K

1
j=1 (N −2K−1)2 σ̂2 σ̂2
f,j

g,j

.
′

Remember that there is no need to perform SIC in the case ZFR/ZFT processing, and hence

the self-interference term in the denominator of (23) can be expressed as
!
2
2
σξ,g,k
σξ,f,k
1
2
2
SIk =
+ σξ,f,k
σξ,g,k
η̂.
+ 2
2
(N − 2K − 1) σ̂f,k
σ̂g,k′
′
Similarly, the other terms in the denominator of (23) can be written as follows
"
!
#
2K
2
2
X
σξ,g,i
σξ,f,k
1
2
2
pi
+ σξ,f,k
σξ,g,i
η̂ ,
IPk =
+ 2
2
(N
−
2K
−
1)
σ̂
σ̂
′
′
′
f,i
g,k
i6=k,k
!
!
1
1
2
2
2
2
+σξ,f,k η̂ , LIRk =PR σLIR
+ σξ,f,k η̂ ,
NRk =σnr
2
2
(N − 2K − 1)σ̂g,k
(N − 2K − 1)σ̂g,k
′
′
X
X




2
pi E |Ωk,i xi |2 =
pi σk,i
, NUk = E |zk |2 = σn2 .
UIk =
i,k∈U k

(70)

(71)

(72)
(73)

i,k∈U k

Substituting the values obtained from (69-73) in the denominator of (23), we obtain


1
1
pk′ var fkT Wgk′ + pk SIk + IPk + NRk + LIRk + 2 UIk + 2 NUk
!α
!
α
2K
X
1
1
2
2
2
2
=
θk,i pi + σnr
+ σξ,f,k
η̂ + PR σLIR
+ σξ,f,k
η̂
2
2
(N
−
2K
−
1)σ̂
(N
−
2K
−
1)σ̂
′
′
g,k
g,k
i=1 "
!
!#
2K
X
X
1
1
2
2
2
pi PR−1 σk,i
+
+ η̂σξ,g,i
+ η̂σξ,g,i
+ pi PR−1σn2
2
2
(N
−
2K
−
1)
σ̂
(N
−
2K
−
1)
σ̂
f,i′
f,i′
i=1
i,k∈Uk
!
X
X
2
2
2
2
2
2
pi σk,i
η̂σLIR
+ η̂σLIR
σn2 ,
(74)
pi PR−1 σk,i
η̂σnr
+
+ PR−1 η̂σnr
σn2 +
i,k∈U
i,k∈U
k
k


2
2
σξ,f,k
σξ,g,i
2
2
where θk,i = (N −2K−1)σ̂2 + (N −2K−1)σ̂2 + σξ,f,k σξ,g,i η̂ .
′
f,i

g,k

′

R EFERENCES
[1] K. Lee and L. Hanzo, “Resource-efficient wireless relaying protocols,” IEEE Wireless Commun., vol. 17, no. 2, pp. 66–72,
2010.
[2] Z. Zhang, X. Chai, K. Long, A. V. Vasilakos, and L. Hanzo, “Full duplex techniques for 5G networks: self-interference
cancellation, protocol design, and relay selection,” IEEE Commun. Mag., vol. 53, no. 5, pp. 128–137, 2015.
[3] Z. Zhang, K. Long, A. V. Vasilakos, and L. Hanzo, “Full-duplex wireless communications: Challenges, solutions, and
future research directions,” Proc. IEEE, vol. 104, no. 7, pp. 1369–1409, 2016.
[4] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback self-interference in full-duplex MIMO relays,” IEEE
Trans. Signal Process., vol. 59, no. 12, pp. 5983–5993, Dec. 2011.
[5] A. Sabharwal, P. Schniter, D. Guo, D. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges
and opportunities,” IEEE J. Sel. Areas Commun., vol. 32, no. 9, pp. 1637–1652, Sep. 2014.
[6] A. Nadh, J. Samuel, A. Sharma, S. Aniruddhan, and R. K. Ganti, “A taylor series approximation of self-interference
channel in full-duplex radios,” IEEE Trans. Wireless Commun., vol. 16, no. 7, pp. 4304–4316, 2017.
[7] Y. Y. Kang, B.-J. Kwak, and J. H. Cho, “An optimal full-duplex AF relay for joint analog and digital domain self-interference
cancellation,” IEEE Trans. Commun., vol. 62, no. 8, pp. 2758–2772, Aug. 2014.

29

[8] W. Zhang, U. Mitra, and M. Chiang, “Optimization of amplify-and-forward multicarrier two-hop transmission,” IEEE
Trans. Commun., vol. 59, no. 5, pp. 1434–1445, 2011.
[9] I. Hammerström and A. Wittneben, “Power allocation schemes for amplify-and-forward MIMO-OFDM relay links,” IEEE
Trans. Wireless Commun., vol. 6, no. 8, pp. 2798–2802, 2007.
[10] G. Zheng, “Joint beamforming optimization and power control for full-duplex MIMO two-way relay channel,” IEEE Trans.
Signal Process., vol. 63, no. 3, pp. 555–566, 2015.
[11] D. Choi and J. H. Lee, “Outage probability of two-way full-duplex relaying with imperfect channel state information,”
IEEE Commun. Lett., vol. 18, no. 6, pp. 933–936, 2014.
[12] Z. Zhang, Z. Ma, Z. Ding, M. Xiao, and G. K. Karagiannidis, “Full-duplex two-way and one-way relaying: Average rate,
outage probability, and tradeoffs,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 3920–3933, 2016.
[13] H. Q. Ngo, H. A. Suraweera, M. Matthaiou, and E. G. Larsson, “Multipair full-duplex relaying with massive arrays and
linear processing,” IEEE J. Sel. Areas Commun., vol. 32, no. 9, pp. 1721–1737, 2014.
[14] Z. Zhang, Z. Chen, M. Shen, and B. Xia, “Spectral and energy efficiency of multipair two-way full-duplex relay systems
with massive MIMO,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 848–863, 2016.
[15] Z. Zhang, Z. Chen, M. Shen, B. Xia, W. Xie, and Y. Zhao, “Performance analysis for training-based multipair two-way
full-duplex relaying with massive antennas,” IEEE Trans. Veh. Technol., vol. 66, no. 7, pp. 6130–6145, 2017.
[16] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless
Commun., vol. 9, no. 11, pp. 3590–3600, 2010.
[17] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO systems,”
IEEE Trans. Commun., vol. 61, no. 4, pp. 1436–1449, 2013.
[18] L. Lu, G. Y. Li, A. L. Swindlehurst, A. E. Ashikhmin, and R. Zhang, “An overview of massive MIMO: benefits and
challenges,” IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 742–758, 2014.
[19] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:
opportunities and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, 2013.
[20] Y. Dai and X. Dong, “Power allocation for multi-pair massive MIMO two-way AF relaying with linear processing,” IEEE
Trans. Wireless Commun., vol. 15, no. 9, pp. 5932–5946, 2016.
[21] H. Cui, L. Song, and B. Jiao, “Multi-pair two-way amplify-and-forward relaying with very large number of relay antennas,”
IEEE Trans. Wireless Commun., vol. 13, no. 5, pp. 2636–2645, May 2014.
[22] L. M. Correia, D. Zeller, O. Blume, D. Ferling, Y. Jading, I. Gódor, G. Auer, and L. V. der Perre, “Challenges and enabling
technologies for energy aware mobile radio networks,” IEEE Commun. Mag., vol. 48, no. 11, pp. 66–72, 2010.
[23] Z. Hasan, H. Boostanimehr, and V. Bhargava, “Green cellular networks: A survey, some research issues and challenges,”
IEEE Commun. Surveys Tuts., vol. 13, no. 4, pp. 524–540, Fourth Quarter 2011.
[24] A. Zappone, P. Cao, and E. Jorswieck, “Energy efficiency optimization in relay-assisted MIMO systems with perfect and
statistical CSI,” IEEE Trans. Signal Process., vol. 62, no. 2, pp. 443–457, Jan. 2014.
[25] L. D. Nguyen, T. Q. Duong, H. Q. Ngo, and K. Tourki, “Energy efficiency in cell-free massive MIMO with zero-forcing
precoding design,” IEEE Commun. Lett., vol. 21, no. 8, pp. 1871–1874, 2017.
[26] Y. Li, P. Fan, A. Leukhin, and L. Liu, “On the spectral and energy efficiency of full-duplex small-cell wireless systems
with massive MIMO,” IEEE Trans. Veh. Technol., vol. 66, no. 3, pp. 2339–2353, 2017.
[27] Z. Fei, B. Li, S. Yang, C. Xing, H. Chen, and L. Hanzo, “A survey of multi-objective optimization in wireless sensor
networks: Metrics, algorithms, and open problems,” IEEE Commun. Surveys Tuts., vol. 19, no. 1, pp. 550–586, 2017.

30

[28] D. Alanis, P. Botsinis, Z. Babar, S. X. Ng, and L. Hanzo, “Non-dominated quantum iterative routing optimization for
wireless multihop networks,” IEEE Access, vol. 3, pp. 1704–1728, 2015.
[29] F. Tan, T. Lv, and S. Yang, “Power allocation optimization for energy-efficient massive MIMO aided multi-pair decodeand-forward relay systems,” IEEE Trans. Commun., vol. 65, no. 6, pp. 2368–2381, June 2017.
[30] J.-P. Crouzeix and J. A. Ferland, “Algorithms for generalized fractional programming,” Mathematical Programming, vol. 52,
no. 1, pp. 191–207, 1991.
[31] M. Biguesh and A. B. Gershman, “Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal
training signals,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 884–893, March 2006.
[32] S. M. Kay, “Fundamentals of statistical signal processing, volume I: estimation theory,” 1993.
[33] J. Jose, A. E. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD
systems,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2640–2651, 2011.
[34] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inf.
Theory, vol. 49, no. 4, pp. 951–963, 2003.
[35] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we
need?” IEEE Journal on Selected Areas in Communications, vol. 31, no. 2, pp. 160–171, 2013.
[36] A. Zappone, Z. Chong, E. A. Jorswieck, and S. Buzzi, “Energy-aware competitive power control in relay-assisted
interference wireless networks,” IEEE Trans. Wireless Commun., vol. 12, no. 4, pp. 1860–1871, 2013.
[37] A. Zappone, E. A. Jorswieck, and S. Buzzi, “Energy efficiency and interference neutralization in two-hop MIMO
interference channels,” IEEE Trans. Signal Process., vol. 62, no. 24, pp. 6481–6495, 2014.
[38] S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric programming,” Optimization and
engineering, vol. 8, no. 1, p. 67, 2007.
[39] A. Zappone and E. A. Jorswieck, “Energy efficiency in wireless networks via fractional programming theory,” Foundations
and Trends in Communications and Information Theory, vol. 11, no. 3-4, pp. 185–396, 2015.
[40] R. Budhiraja, Karthik KS, and B. Ramamurthi, “Linear precoders for non-regenerative asymmetric two-way relaying in
cellular systems,” IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 5002–5014, Sep. 2014.
[41] R. Budhiraja and B. Ramamurthi, “Joint precoder and receiver design for AF non-simultaneous two-way MIMO relaying,”
IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 2942–2955, Jun. 2015.
[42] ——, “Joint transceiver design for QoS-constrained MIMO two-way non-regenerative relaying using geometric programming,” IEEE Trans. Wireless Commun., vol. 15, no. 5, pp. 3453–3465, May 2016.
[43] P. C. Weeraddana, M. Codreanu, M. Latva-aho, and A. Ephremides, “Resource allocation for cross-layer utility
maximization in wireless networks,” IEEE Trans. Veh. Technol., vol. 60, no. 6, pp. 2790–2809, 2011.
[44] W. Dinkelbach, “On nonlinear fractional programming,” Management science, vol. 13, no. 7, pp. 492–498, 1967.
[45] A. Zappone, E. Björnson, L. Sanguinetti, and E. A. Jorswieck, “Globally optimal energy-efficient power control and
receiver design in wireless networks,” IEEE Trans. Signal Process., vol. 65, no. 11, pp. 2844–2859, 2017.
[46] A. M. Tulino and S. Verdú, “Random matrix theory and wireless communications,” Foundations and Trends R in
Communications and Information Theory, vol. 1, no. 1, pp. 1–182, 2004.
[47] P. Graczyk, G. Letac, and H. Massam, “The complex wishart distribution and the symmetric group,” Annals of Statistics,
pp. 287–309, 2003.

