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Energy Efficiency Analysis of Rank-1
Ricean Fading MIMO Channels
Jingya Li†, Michail Matthaiou?†, Shi Jin‡, and Tommy Svensson†
†Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden
?School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, U.K.
‡National Mobile Communications Research Laboratory, Southeast University, Nanjing, China
Email: jingya.li@chalmers.se, m.matthaiou@qub.ac.uk, jinshi@seu.edu.cn, tommy.svensson@chalmers.se
Abstract—This paper studies the energy efficiency (EE) of
a point-to-point rank-1 Ricean fading multiple-input-multiple-
output (MIMO) channel. In particular, a tight lower bound and
an asymptotic approximation for the EE of the considered MIMO
system are presented, under the assumption that the channel is
unknown at the transmitter and perfectly known at the receiver.
Moreover, the effects of different system parameters, namely,
transmit power, spectral efficiency (SE), and number of transmit
and receive antennas, on the EE are analytically investigated. An
important observation is that, in the high signal-to-noise ratio
regime and with the other system parameters fixed, the optimal
transmit power that maximizes the EE increases as the Ricean-K
factor increases. On the contrary, the optimal SE and the optimal
number of transmit antennas decrease as Kincreases.
I. INTRODUCTION
MIMO techniques can greatly improve the SE of wireless
communications systems. However, due to the extra signal
processing and circuit power consumption per transceiver RF
chain, the use of multiple antennas also leads to an increased
energy consumption. In future green communication systems,
the EE becomes an important performance metric. Therefore,
the trade-off between EE and SE has become a hot topic in
MIMO systems.
Most studies on the EE of MIMO systems adopted the
assumption of Rayleigh fading conditions [1]–[3]. However,
in many practical scenarios, there exists a deterministic or
strong line-of-sight (LoS) component. Some relevant examples
are: 1) short-range millimeter wave MIMO channels [4],
2) microwave backhauling between macro base stations and
outdoor small-cell base stations [5], 3) the channel between a
user and its connected small-cell base stations. In such cases,
Ricean fading conditions should be considered, which induces
several research challenges.
Motivated by the above discussion, in this paper, we study a
worst case scenario, where the Ricean fading MIMO channel
has a rank-1 mean. This scenario is relevant when the inter-
element distances at both ends are small and the impinging
LoS waveforms have identical phases [6]. A tight lower
bound for and an asymptotic approximation on the EE of the
considered MIMO system are presented, assuming that the
This work was supported in part by the Swedish Governmental Agency
for Innovation Systems (VINNOVA) within the VINN Excellence Center
Chase, and the Swedish Research Council VR under the project 621-2009-
4555 Dynamic Multipoint Wireless Transmission.
channel is unknown at the transmitter and perfectly known at
the receiver. With the help of the closed-form EE expressions,
the effects of different system parameters, i.e., transmit power,
SE, and number of transmit and receive antennas, on the EE
are analytically investigated. Interestingly, we observe that,
in the high signal-to-noise ratio (SNR) regime and with the
other system parameters fixed, the optimal transmit power that
maximizes the EE increases with the Ricean-Kfactor or with
the fixed transceiver power consumption. On the other hand,
the optimal SE and the optimal number of transmit antennas
decrease as the Ricean-Kfactor increases, i.e., when the LoS
component becomes more and more dominant.
II. SY STE M MOD EL
We consider a point-to-point MIMO system with nttransmit
antennas and nrreceive antennas. Here, we assume nt≤nr,
while all our results can be easily generalized to the case of
nt≥nr. Let s∈Cnt×1denote the transmitted signal vector
with Es†s≤P, where (·)†stands for conjugate transpose.
The received signal vector at the receiver, y∈Cnr×1, is
y=Hs +n(1)
where n∈Cnr×1denotes the circularly symmetric complex
Gaussian noise with n∼ CN 0, σ2Inr. The MIMO channel,
H∈Cnr×nt, is assumed to be Ricean flat-fading with rank-1
mean, such that
H=rK
K+ 1 ¯
H+r1
K+ 1 ˜
H(2)
where E{H}=qK
K+1 ¯
Hand Kis the Ricean K-factor.
Here, E{·} stands for expectation. The rank-1 matrix ¯
H
corresponds to the coherent line-of-sight component, and it
is normalized such that
¯
H
2
F=ntnr, where k·kFde-
notes the Frobenius norm. The term ˜
Hrepresents the non-
coherent scattered contributions. In this paper, we focus on
the case without spatial correlation between receive antennas
and transmit antennas. Thus, ˜
Hcan be modeled as a random
fading matrix, whose entries are circularly symmetric complex
Gaussian random variables with unit variance. Therefore,
Hhas a matrix-variate complex Gaussian distribution with
H∼ CN M,Inr⊗ε2Int, where ε,1/√K+ 1 and
M,ε√K¯
H. Here, ⊗is defined as the Kronecker product
between matrices.
Under the assumption of no channel state information (CSI)
at the transmitter and perfect CSI at the receiver, the SE, which
is defined as the ergodic capacity per unit bandwidth, is [7]
SE =EHlog2det Int+P
σ2nt
H†H (3)
where the expectation is taken with respect to Hand det (·)
denotes the matrix determinant. The EE is defined as the ratio
between the SE and total power consumption, that is,
EE =SE
Ptot
.(4)
In this paper, we adopt the linear approximated power con-
sumption model proposed in [8], where
Ptot =αP +ntPct +nrPcr.(5)
The static terms, Pct and Pcr, denote the fixed power con-
sumption for each transmit and receiver chain, respectively.
The scaling factor, α, models the impact of the output power
Pon the efficiency of the power amplifiers used at the
transmitter. The values of Pct,Pcr and αfor different base
station (BS) types can be found in [8, Table 2].
III. SPECTRAL RES ULTS
The exact expressions for SE and EE are difficult to obtain.
To make the EE analysis tractable, we consider the following
two approximations of the SE.
Lemma 1 [9, Theorem 9]: The SE of the uncorrelated Ricean
fading MIMO channel with rank-1 mean matrix Mand nt≤
nr, is tightly lower bounded for arbitrary SNR =P
σ2by SE ≥
SE1, where
SE1,
nt
X
i=1
log2(1 + µiP)(6)
with positive values of µigiven as
µi=
exp(gnr(4))
σ2nt(K+1) , i = 1
exp(Pnr−i
p=1
1
p−γ)
σ2nt(K+1) , i = 2, . . . , nt
where γ= 0.5772... is Euler’s constant and 4,Kntnr. The
function gnr(4)is defined as
gnr(4),ln (4)−Ei (−4) +
nr−1
X
i=1 −1
4i
×e−4 (i−1)! −(nr−1)!
i(nr−1−i)!(7)
where Ei (−x) = −´∞
x
e−t
tdt,x > 0, is the exponential
integral function.
Note that for the special case of K→ ∞, (6) reduces to
SEK→∞
1= log21 + nr
P
σ2(8)
which is the exact capacity of deterministic MIMO systems,
with multiplexing gain equal to 1 [7]. In the high SNR regime,
we have the following approximation for the SE.
Lemma 2 [9, Eq. (24)]: The SE of the uncorrelated Ricean
fading MIMO channel in the high SNR regime, with rank-1
mean matrix Mand nt≤nr, can be approximated as SE ≈
SE2, where
SE2,ntlog2P
σ2nt+c(9)
where c,log2nr!
(nr−nt)! + log21+ntK
(K+1)nt.
IV. ENERGY EFFIC IEN CY MA XIM IZ ATION
In this section, we utilize the closed-form expressions in
Section III to analyze the impact of different system parame-
ters on the EE of rank-1 Ricean fading MIMO channels.
A. The Impact of Transmit Power on EE
1) Arbitrary SNR: We start with finding the optimal trans-
mit power Pby using the lower bound EE expression, i.e.,
EE1,SE1
Ptot
=Pnt
i=1 log2(1 + µiP)
αP +ntPct +nrPcr
(10)
which is tight for arbitrary SNR. From (10), we see that the
sublevel sets Sν={−EE1≤ν}={Pnt
i=1 log2(1 + µiP)
+ν(αP +ntPct +nrPcr)≤0}are convex for ν∈R. Thus,
EE1is a quasi-concave function of P. The first derivative of
EE1with respect to Pis
∂EE1
∂P =f1(P)
(αP +ntPct +nrPcr)2ln 2 (11)
where
f1(P),(αP +ntPct +nrPcr)
nt
X
i=1
µi
1 + µiP
−α
nt
X
i=1
ln (1 + µiP)(12)
which is a decreasing function of P. Note that f1(0) =
(ntPct +nrPcr)Pnt
i=1 µi>0. Thus, there exists a unique
P∗such that f1(P∗) = 0, i.e., ∂EE1
∂P |P=P∗= 0. Recall that
EE1in (10) is a quasi-concave function of P. Thus, the unique
P∗is the global optimal power which maximizes EE1. Unfor-
tunately, the optimal value of P∗= arg (f1(P) = 0) cannot
be expressed in closed-form. However, with the expression in
(12) and noting that f1(P)is a decreasing function of P, we
can easily obtain the optimal power value numerically, e.g.,
via a bisection search shown in Algorithm 1.
Algorithm 1 Bisection method for finding the optimal transmit
power that maximizes EE1
given Pl≤P∗and Pu≥P∗, with f1(Pl)≥0and f1(Pu)≤0,
given tolerance > 0.
repeat
1: set P=Pl+Pu
2, and calculate f1(P)by using (12).
2: if f1(P)>0then
3: Pl:= P
4: else
5: Pu:= P
until |f1(P)|< .
An initial lower bound can be easily found as Pl= 0.
To find an initial upper bound Pusuch that g(Pu)≤0, we
define µmax ,max (µi)and µmin ,min (µi). Then, we have
f1(P)≤nt˜
f1(P), where
˜
f1(P),(αP +ntPct +nrPcr)µmax
1 + µminP
−αln (1 + µminP).(13)
Therefore, an upper bound Pucan be found by setting
˜
f1(Pu) = 0, from which we get
Pu=
exp x+µmax
µmin −1
µmin
(14)
where
x,W
µmax (αP +ntPct +nrPcr)
αexp µmax
µmin
(15)
and Wis the Lambert Wfunction satisfying W(x)eW(x)=x.
With the closed-form expression of Puin (14), Algorithm 1
is guaranteed to converge to the optimal P∗.
2) High SNR Approximation: In order to gain more physi-
cal insights, we now elaborate on the high SNR regime. Plug-
ging (9) into (4), the EE can be approximated as EE ≈EE2,
where
EE2,SE2
Ptot
=
ntlog2P
ntσ2+c
αP +ntPct +nrPcr
.(16)
Proposition 1: For the uncorrelated Ricean fading MIMO
channel with rank-1 mean matrix Mand nr≤nt, in the high
SNR regime, the optimal transmit power that maximizes EE2
is
P∗=ntσ2exp WntPct +nrPcr
ntσ2αed+d (17)
where d,1−c
ntln 2, and cis defined in Lemma 2.
Proof: From (16), we see that EE2is a quasi-concave
function of P. By calculating the first order derivative of EE2
with respect to P, and setting it to zero, we get
f2(P) = ntPct +nrPcr
αP −ln P
ntσ2−d= 0.(18)
Let z= ln P
ntσ2−d, then (18) can be rewritten as zez=
ntPct+nrPcr
ntσ2αed, from which, we get z=WntPct+nrPcr
ntσ2αed.
Therefore, the optimal transmit power can be obtained by
substituting the value of zinto P∗=ntσ2exp (z+d). The
optimal solution (17) can also be found by using a different
methodology presented in [10].
Note that W(x)is an increasing function of xfor x > 0.
Thus, from Proposition 1, we observe that in the high SNR
regime, the optimal transmit power increases with increasing
the fixed power consumption Pct and/or Pcr. Moreover, with
the aid of [1, Lemma 3], we can show that the optimal transmit
power is approximately equal to ntσ2edfor small ntPct +
nrPcr. For large ntPct +nrPcr, the optimal transmit power
increases linearly with ntPct+nrPcr . This is intuitive, because
when ntPct +nrPcr is small, the effect of the fixed power
consumption on the EE is negligible. On the contrary, when
the fixed power consumption is large and dominates the total
power consumption, then more transmit power can be used to
increase the SE, thereby, increasing the EE.
Corollary 1: The optimal transmit power, P∗in (17),
increases as the Ricean K-factor, K, increases.
Proof: Note that c, which is defined in Lemma 2, de-
creases with increasing K. Thus, d,1−c
ntln 2 increases
as Kincreases. Then, from (18), we see that f2(P)is an
increasing function of K. Note that f2(P)is a decreasing
function of P. Therefore, the optimal transmit power, P∗,
which satisfies f2(P∗)=0, increases as Kincreases.
Corollary 1 implies that, in the high SNR regime, in order
to maximize the EE, the transmit power should increase in
order to compensate for the SE loss due to the dominating
LoS component.
B. The Impact of SE on EE
For arbitrary SNR, the optimal SE that maximizes EE1can
be obtained by substituting P∗obtained from Algorithm 1
into (6). For the high SNR regime, we present the following
proposition.
Proposition 2: For the uncorrelated Ricean fading MIMO
channel with rank-1 mean matrix Mand nr≤nt, in the high
SNR regime, the optimal value of SE2that maximizes EE2is
SE∗
2=nt
ln 2 W(ntPct +nrPcr)e−d
ntασ2+ 1.(19)
Proposition 2 implies that increasing the SE does not always
increase the EE. Recall that in the high SNR regime, the
optimal transmit power increases as ntPct +nrPcr increases.
As expected, the optimal SE also increases with increasing
the fixed power consumption. For small ntPct +nrPcr, the
optimal SE is approximately equal to nt
ln 2 , which increases
linearly with nt. However, when the fixed power consumption
is large, the optimal SE increases only logarithmically with
ntPct +nrPcr. Recall that W(x)is an increasing function of
xfor x > 0, and dis an increasing function of K. From (19),
we have the following corollary.
Corollary 2: The optimal SE, SE∗
2in (19), decreases as the
Ricean K-factor, K, increases.
Corollary 2 claims that, in the high SNR regime, the SE
decreases as the LoS component becomes more and more
dominant, which is expected due to the reduced multiplexing
gain [9].
C. The Impact of Number of Antennas on EE
Since the optimal ntand nrthat maximize EE1for arbitrary
SNR are difficult, if not impossible, to obtain, we now focus
on the high SNR regime to gain insights. The values of nt
and nrare assumed to be continuous with 0< nt≤nr.
1) The Impact of nton EE: We start with investigating
the impact of the number of transmission antennas, nt, on the
EE. The number of receive antennas, nr, is assumed to be
fixed. Utilizing Stirling’s approximation, ln n!≈nln n−n,
(9) becomes
SE2≈1
ln 2 ntln P
σ2nt(1 + K)−1+ ln (1 + ntK)
+1
ln 2 nrln nr−(nr−nt) ln (nr−nt).(20)
By doing so, it can be shown that EE2in (16) is a quasi-
concave function of nt, with
∂EE2
∂nt≈(αP +nrPcr )
(αP +ntPct +nrPcr)2ln 2 J1(nt)(21)
where
J1(nt),(1 + nrβ) ln (nr−nt)−ln nt−βln (1 + ntK)
+(1 + ntβ)K
1 + ntK+ ln P
σ2(1 + K)−1−βnrln nr
(22)
with β,Pct
αP +nrPcr . Note that J1(nt)is a decreasing function
of nt. Recall that 0< nt≤nrand note that J1(nt→0) >0
and J1(nt→nr)<0. Thus, there exists a unique optimal ˆnt,
where
ˆnt= arg
0<nt≤nr
(J1(nt) = 0) .(23)
The optimal value of ˆntcannot be expressed in closed-form.
However, similar to Algorithm 1, we can easily obtain ˆnt
numerically via a bisection search. If the value of ˆntis non-
integer, the optimal n∗
tis attained at one of the two closest pos-
itive integers. Note that J1(nt)is also a decreasing function
of βand K. Therefore, ˆntdecreases as βor/and Kincreases.
Intuitively, this implies that, in the high SNR regime, the
optimal number of transmit antennas decreases when the fixed
power consumption at the transmitter increases. Moreover, it
indicates that less transmit antennas should be used when the
LoS component becomes more and more dominant. This is
expected since when the Ricean K-factor is large, the SE gain
achieved by adding more transmit antennas is negligible. Note
that the total power consumption increases with increasing nt.
Thus, as Kincreases, less transmit antennas should be used in
order to achieve a better trade-off between the SE and power
consumption.
2) The Impact of nron EE: Now, we try to find the optimal
number of receive antennas, nr, by keeping ntfixed. Utilizing
(20), it can been shown that EE2in (16) is also a quasi-concave
function of nr, with
∂EE2
∂nr≈αP +ntPct
(αP +ntPct +nrPcr)2ln 2 J2(nr)(24)
where
J2(nr),ln nr−a1ln (nr−nt)−a2(25)
with a1,1 + ntζand a2,
ζntln P
ntσ2−1+ ln 1+ntK
(1+K)nt >0in the high
SNR regime. Here, ζ,Pcr
αP +ntPct . It is easy to show that
J2(nr)is a decreasing function of nr. Recall that nr≥nt
and noting that J2(nr→+∞)<0and J2(nr→nt)>0.
Thus, there exists a unique optimal ˆnr, where
ˆnr= arg
nr≥nt
(J2(nr) = 0) .(26)
The optimal value of ˆnrcannot be expressed in closed-
form. However, similar to Algorithm 1, we can easily obtain
ˆnrnumerically via a bisection search. Note that J2(nr)≤
ln nr
nr−nt−a2. Thus, by setting ln nr
nr−nt−a2= 0, an upper
bound of ˆnrcan be found as ˆnr≤nt
1−ea2. Finally, the optimal
n∗
ris attained at one of the two closest integers.
From (25), we see that J2(nr)decreases as ζincreases.
Therefore, in order to satisfy J2(ˆnr)=0,ˆnrshould decrease
as ζincreases. This implies that the optimal number of
receive antennas decreases with increasing the fixed power
consumption at the receiver. Note that a2is proportional to
ζ. Thus, when the fixed power consumption at the receiver
side dominates the total power consumption, the upper bound,
nt
1−ea2will approach to nt. In this case, we have n∗
r=nt.
3) Special Case of nt=nr:For the special case of nt=
nr=n, we present the following corollary.
Proposition 3: When nt=nr=n, the optimal n∗that
maximizes EE2is given by n∗= arg max
{bˆnc,dˆne}
EE2, where
ˆn=
exp W αP K
Pct+Pcr −11
eb+b−1
K(27)
with b,1 + αP
Pct+Pcr ln P
σ2(1+K)−1.
Proof: Plugging nt=nr=ninto (9),
and utilizing Stirling’s approximation, we get SE2≈
1
ln 2 nln ρ
1+K−n+ ln (1 + nK), which is a concave func-
tion of n. Therefore, EE2in (16) is a quasi-concave function
of n, with ∂EE2
∂n ≈J3(n)
(αP +n(Pct+Pcr ))2ln 2 , where
J3(n),αP ln P
σ2(1 + K)−1 + K
1 + nK
−(Pct +Pcr)ln (1 + nK)−nK
1 + nK .(28)
It can be shown that J3(n)is a decreasing function of n, with
J3(n= +∞)<0and J3(n= 1) >0. Thus, there exists a
unique optimal ˆnsuch that J3(ˆn) = 0, and the optimal n∗is
attained at one of the two closest integers from ˆn.
Similarly, utilizing (28), we can show that the optimal
number of antennas, ˆn, decreases as the Ricean K-factor
and/or the fixed power consumption Pct +Pcr increase.
V. NUMERICAL RE SU LTS
Numerical results are presented to verify our analytical re-
sults and to illustrate the impact of different system parameters
(P,SE,ntand nr) on the EE of rank-1 Ricean fading MIMO
systems. The noise power is σ2= 10mW.
Figure 1 shows the EE as a function of the transmit power,
P, for different Ricean-Kfactors. The numbers of transmit
and receive antennas are nt= 2 and nr= 4, respectively. The
power model parameters are Pct = 56W, Pcr = 130W and
α= 2.6, which correspond to a backhaul channel between a
micro BS to a macro BS [8, Table 2]. The lower bound EE1and
the high SNR approximated EE2are compared with Monte-
Carlo simulations. We see that both analytical EE expressions
agree perfectly with the numerical results. As expected, the
EE decreases as the Ricean-Kfactor increases. Utilizing
Proposition 1, for K= 0,10,100, the optimal power values
are 32.45, 35.49 and 41.14W, respectively, which agree with
the simulated optimal values as shown in Fig. 1. Moreover,
in agreement with Corollary 1, we observe that the optimal
transmit power increases as the Ricean-Kfactor increases.
This indicates that, in order to maximize the EE, the transmit
power should increase so as to compensate for the SE loss
due to the increased LoS component.
Figure 2 demonstrates the EE versus the SE considering the
same system model parameters used in Fig. 1. We see that even
though the optimal transmit power increases as Kincreases,
the optimal value of SE decreases when increasing K. This
observation is in agreement with Corollary 2. Moreover, we
see that when the SE is less than 12 bits/s/Hz, the EE increases
linearly with the SE. This is because, for small values of
SE, the corresponding transmit power Pis small. Thus, by
0 10 20 30 40 50 60
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
X: 31.98
Y: 0.03416
Transmit power, P[W]
EE [bits/Hz/Joule]
X: 35.88
Y: 0.0309
X: 41.14
Y: 0.02667
Simulation
EE1
EE2
K=0
K=10
K=10 0
Fig. 1. Energy Efficiency vs. transmit power.
0 4 8 12 16 20 24 28
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
SE [bits/s/Hz]
EE [bits/Hz/Joule]
Simulation
EE1
EE2
K=10 0
K=0
K=10
Fig. 2. Energy efficiency vs. spectral efficiency.
increasing P, the increase of the total power consumption Ptot
is negligible. Therefore, the EE, which is defined as the SE-
to-Ptot ratio, increases linearly as the SE increases.
Finally, Fig. 3 investigates the impact of the number of
transmit antennas on the EE of rank-1 Ricean fading MIMO
channels. The number of receive antennas nris set to 10.
The EE is plotted as a function of ntfor different values of
Kand Pcr respectively. We consider a micro BS transmitter
(Pct = 56W and α= 2.6), and investigate three different
backhaul MIMO channels with Pcr = 6.8,56 and 130W,
which correspond to three different types of receivers, i.e.,
pico BS, micro BS and macro BS, respectively [8, Table 2].
As expected, for a fixed value of Pcr, the optimal ntdecreases
as Kincreases, since with increasing the Ricean K-factor,
the contribution to the SE by adding more transmit antennas
becomes negligible, while the total power consumption will
increase linearly. On the other hand, for a fixed value of K,
the optimal ntincreases as Pcr increases. This observation
is also intuitive, because with increasing Pcr, the total power
consumption becomes dominated by the power consumption at
the receiver. Therefore, more transmit antennas can be added
to increase the SE, thereby, increasing the EE.
1 2 3 4 5 6 7 8 9 10
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Number of transmit antennas, nt
EE [bits/Hz/Joule]
Simulation
EE1
EE2
12345678910
0
0.05
0.1
0.15
Number of transmit antennas, nt
EE [bits/Hz/Joule]
Simulation
EE1
EE2
K=0, Pcr=56W
K=10 , Pcr=56W
K=10 0, Pcr=56 W
K=10 , Pcr=6.8W
K=10 , Pcr=56W
K=10 , Pcr=130 W
Fig. 3. Energy efficiency vs. the number of transmit antennas.
VI. CONCLUSIONS
In this paper, the EE of rank-1 Ricean fading MIMO chan-
nels was analyzed. More specifically, the effects of different
system parameters, i.e., transmit power, SE, and number of
transmit and receive antennas, on the EE were investigated.
When keeping the other parameters fixed, closed-from expres-
sions have been derived for the optimal transmit power and
the optimal SE, respectively, with the objective of maximizing
the EE in the high SNR regime. We analytically show that
the optimal transmit power and the optimal SE increase as
the fixed power consumption per transceiver chain increases.
Moreover, both our theoretical analysis and numerical results
indicated that, as the Ricean-Kfactor increases, the optimal
transmit power will increase; on the contrary, the optimal SE
and the optimal number of transmit antennas will decrease
with increasing K.
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