Challenges of Massive MIMO System

Non-asymptotic Analysis of Massive MIMO under
Different Wireless Scenario
Dr. Varun Kumar
Dr. Varun Kumar (NIT Rourkela)
Massive MIMO 1 / 89

Outlines
1 Introduction
2 Motivation
3 Problem Formulation
4 Work Done
5 Conclusion and Future Work
6 References
Massive MIMO 2 / 89

Introduction
Mobile Communication Background
Generation Bandwidth Data Rate
2G 200 KHz 144 Kbps
3G 5 MHz 384 Kbps- 2 Mbps
4G 10/20 MHz 100 Mbps- 1 GHz
5G > 100 MHz 1 Gbps-20 Gbps
Challenges
Mobile communication trends have shifted prioritizing data over voice
communication.
Exponentially growing in data demand
Available wireless resource are limited for mobile communication.
Working in interfering environment
Massive MIMO 3 / 89

Ongoing research trends in the mobile communication:
What will be the new in 5G ? [1]
1 Enhanced Mobile broadband (mmWave Communication)
2 Ultra reliable low latency (Network Densification)
3 Massive connectivity (IoT)
Massive MIMO 4 / 89

5G Research Direction: [2],[3]
Massive MIMO
1 It is mature wireless technology.
2 It incorporate all flavor of conventional MIMO with larger scale.
3 Current 4G standard incorporate only 8 antenna at Tx − Rx end, but
massive MIMO will have more flexibility.
Massive MIMO 5 / 89

Continued–
Why go to massive MIMO ?
Ans- Due to exploration of new band (mmWave beyond sub 6 GHz band)
allocation for mobile communication.
Note- Sub 6 GHz band also support with limited scale.
Features of massive MIMO:
It increases the degree of freedom.
Increases spectrum efficiency as well as energy efficiency.
It can support large number of user in same time-frequency slot.
Massive MIMO 6 / 89

Major obstacle in massive MIMO:
Challenges:
Hardware mismatch (HM) in TDD system.
Highly correlated spatial gain.
Channel estimation for large MIMO network.
Resource allocation for such a complex network.
Pilot contamination
Feasibility of this technology with existing one, like cooperative
network, spectrum sharing network (cognitive radio), heterogeneous
networks (Het-Net) and many more.
Massive MIMO 7 / 89

Dissertation Plan:
Introduction Conclusion
3. Feasibility of massive MIMO in
cooperative network
1. Effect of HM in TDD based
massive MIMO
2. Effect of antenna correlation in
massive MIMO
Thesis
Contributory
Massive MIMO 8 / 89

Why TDD not FDD for massive MIMO?
Let a BS have M number of antenna, which serves K users and each user has
single antenna, also M >> K.
FDD slot structure
TDD slot structure
Tc = Tch + TUL(Data) + TDL(Data) + Toth (1)
Tch = (2M + K)τs for FDD mode
Tch = Kτs for TDD mode
Massive MIMO 9 / 89

Impact of Hardware Mismatch in TDD Based Massive MIMO System
Uplink channel matrix of size M × K can be expressed as
HU = RB HTU (2)
RB → Hardware response (HR) matrix of size M × M during signal
reception at BS.
H → Rayleigh channel matrix of size M × K.
TU → HR matrix of size K × K during signal transmission across the users.
On the other side, downlink channel matrix
HD = RU HT
TB (3)
RU → HR matrix of size K × K during signal reception at user
TB → HR matrix of size M × M during signal transmission at BS
Massive MIMO 10 / 89

Continued–
Since propagation channel H is perfectly reciprocal [4].
HD = RU T−1
U
| {z }
Unknown
HT
U R−1
B TB
| {z }
Unknown
(4)
RU T−1
U → Hardware mismatch (HM) matrix across the user
R−1
B TB → HM matrix across the BS
Reciprocal and non-reciprocal channel:
Case 1: RU T−1
U = CuIK and R−1
B TB = CbIM
Hence from (4), the DL channel matrix is the constant multiple of the transpose
of UL channel matrix.
HD = CuCbHT
U (5)
This is the sufficient condition for the channel reciprocity.
Case 2: If RU T−1
U 6= CuIK and R−1
B TB 6= CbIM

Continued–
Channel Reciprocity Error Modeling
hD = (RU T−1
U ⊗ R−1
B TB )
| {z }
Unknown
hU (6)
hD = vec(HT
D ) of size MK × 1.
hU = vec(HU ) of size MK × 1
(RU T−1
U ⊗ R−1
B TB ) → Kronecker product between HM matrix across the
UE and BS side with size (MK × MK)
Let TB = diag(tb1
, tb2
, ....., tbM
). The coefficient of HR across the mth
antenna is
modeled as
tbm = <(tbm ) + j=(tbm ) (7)
where <(tbm
), =(tbm
)

∼ N(0, σ2
tbm
).
Assumption 1: σtb1
= σtb2
=, ....., = σtbM
= σ1.

Continued–
From (7), we assume that the amplitude and phase of the RV tbm follow the
Rayleigh distribution and uniform distribution respectively. So diagonal matrix TB
can also be expressed as
TB = Tb
a Tb
θ (8)
Tb
a = diag(|tb1
|, |tb2
|, ....., |tbM
|) → amplitude response matrix.
Tb
θ = diag(ejθb
t1 , ejθb
t2 , ......ejθb
tM ) → phase response matrix.
Similarly, RB = diag(rb1
, rb2
, ....., rbM
) and also expressed as
RB = Rb
a Rb
θ (9)
where (rbm
), =(rbm
)

∼ N(0, σ2
2).
Ro
b =(Rb
a Rb
θ )−1
(Tb
a Tb
θ ) = (Rb
a )−1
Tb
a

| {z }
A.M
(Rb
θ )−1
Tb
θ

| {z }
P.M
(10)
Here Ro
b = diag(ro
b1
ejθo
b1 , ro
b2
ejθo
b2 , ....., ro
bM
ejθo
bM ), where ro
bm
=
|tbm |
|rbm | and
θo
bm
= (θb
tm
− θb
rm
)

PDF of hardware mismatch coefficient
Theorem 1:
Consider x1 ∼ a1 + ib1, x2 ∼ a2 + ib2 are two complex random variables (RV),
where (a1, b1) ∼ N(0, σ2
1), and (a2, b2) ∼ N(0, σ2
2). |x1| and |x2| are considered
as Rayleigh distributed RV, with scale parameter σ1 and σ2. Let Z = |x1|
|x2| be
another RV, considering |x1| and |x2| are statistically independent, then the joint
PDF of Z is
fZ (z) =
2zb2
0
(z2 + b2
0)2
(11)
where b0 = σ1
σ2
. Similarly the mean and variance of Z are
E[Z] =
π
2
σ1
σ2

(12)
and
var(Z) = ∞ (13)

PDF of ro
bm
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Z
0
0.2
0.4
0.6
0.8
1
1.2
1.4
f
Z
(z)
Amplitude Mismatch PDF
1
=1, 2
=2
1
=2, 2
=3
1
=2, 2
=4
1
=3, 2
=2

Truncated PDF:
gZ (z) =
fZ (z)
F(b) − F(a)
(14)
where F(z) =

1 −
b2
0
z2+b2
0

and b0 = σ1
σ2
. The mean and variance for the
truncated PDF can be expressed as
E(z) =
(a2
+ b2
0)(b2
+ b2
0)
b2
0(b2 − a2)
×

b0

tan−1
b
b0

− tan−1
a
b0

− b2
0
b
b2 + b2
0
−
a
a2 + b2
0
(15)
and
var(z) = E(z2
) − E(z)2
(16)
where E(z2
) = 1
F(b)−F(a)

b4
0

1
b2+b2
0
− 1
a2+b2
0

+ log

b2
+b2
0
a2+b2
0

.

Continued–
Assumption 2:
Hardware component of Tx for all UE have same statistical RF response, where
(tu
k ), =(tu
k )

∼ N(0, σ2
3). Similarly hardware component of Rx for all UE have
same statistical RF response, where (ru
k ), =(ru
k )

∼ N(0, σ2
4).
Overall HM due to mth
transmit antenna from the BS to kth
user
lk,m = lo
k,mejφk,m
(17)
where lo
k,m = ro
uk
ro
bm
and φk,m = (θo
bm
+ θo
uk
). The PDF of overall A.M lo
k,m has
been reported in Theorem 2.

Continued–
Theorem 2
Given z1 and z2 are ratio of Rayleigh distributed RVs, whose statistical
parameters are (σ1, σ2) and (σ3, σ4), respectively. The PDF of these two
RV are fZ1 (z1, σ1, σ2) =
2z1b2
0
(z2
1 +b2
0)2 and fZ2 (z2, σ3, σ4) =
2z2u2
0
(z2
2 +u2
0)2 , where
b0 = σ1
σ2
and u0 = σ3
σ4
. Let w = z1z2 be the product of these two RV.
Considering z1 and z2 are statistically independent then the joint PDF of
w can be expressed as
fW (w) =2Dcw

u2
0 +
w2
b2
0
log w2
b2
0u2
0

(w2
b2
0
− u2
0)3
−
2
(w2
b2
0
− u2
0)2

= 2t
(1 + t2) log(t2)
(t2 − 1)3
−
2
(t2 − 1)2

(18)
where Dc =
u2
0
b2
0
, t = w
u0b0
.

PDF of coefficient of amplitude mismatch, when DL channel is
affected by both side HM
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w
0
1
2
3
4
5
6
7
f
W
(w)
(σ
1
, σ
2
, σ
3
, σ
4
) ∼ (1, 2, 1, 5)
(σ
1
, σ
2
, σ
3
, σ
4
) ∼ (2, 2, 1, 5)
(σ
1
, σ
2
, σ
3
, σ
4
) ∼ (1, 1, 2, 5)

PDF of the phase Mismatch of a toy example
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Phase mismatch across
either side i.e BS or UE
per terminal
Phase response in
UL/DL
Phase mismatch,
when UE and BS per
terminal side consideration

Trapezoidal rule for numerical integration
E(w) =
Z b
a
wfW (w)dw =
b − a
2N
h
w(1)fW (w(1)) + w(N)
fW (w(N)) + 2{w(2)fW (w(2))....., w(N − 1)fW (N − 1)}
i (19)
CDF of random variable w:
FW (w) = 1 +

(u0b0)5
+ (u0b0)3
w2

log

w2
(u0b0)2

− 1

w2 − u2
0b2
0
2 (20)
Mean based calibration:
Qk = diag(Lo
k ) =





ro
uk ro
b1 0 ... 0
0 ro
uk ro
b2 ... 0
.
.
.
.
.
.
...
.
.
.
0 0 ... ro
uk ro
bM





M×M
(21)
Let Ao
= vec([Lo
1, Lo
2, .....Lo
K ]) is the overall mismatch coefficient vector with size
MK × 1. From (6) the unknown HM matrix can be expressed as
Lo
= Ro
u ⊗ Ro
b = diag(Ao
) (22)

SINR calculation using different precoder
The received signal for kth
user in DL scenario can be expressed as
yk =
√
pk (hd )k (gk )I xk +
K
X
j=1,j6=k
√
pj (hd )k (gj )I xj + nk (23)
pk → Signal power of kth
user.
(hd )k → DL channel vector of size 1 × M
(gk )I → Precoding vector for kth
user of size M × 1 ∀ I = {MF, RZF, ZF}
nk → Additive white noise with zero mean and variance σ2
xk → kth
user transmit symbol, where E(|xk |2
) = 1 ∀ k = 1, 2, ...K,
Let GI = λI Gi = λI [(g1)I , ....(gK )I ] is the precoding matrix of size M × K.

The normalization factor for Ith
precoding scheme can be expressed as
λI =
s
PT
E

tr(PGH
i Gi )
=
r
PT
ΨI
(24)
PT → Upper bound for total transmit power from the BS.
P = diag(p1, p2, ...., pK ) is a diagonal matrix of size K × K.
The observed SINR for kth
user under Ith
precoding scheme can be expressed as
(γk )I =
pk |(hd )k (gk )I |2
PK
j=1,j6=k pj |(hd )k (gj )I |2 + ζ
(25)
where
ζ =
σ2
λ2
I
=
σ2
ΨI
PT
=
ΨI
ρ
(26)

Continued-
Let UL channel matrix is
HU = [(hu)1, (hu)2, ...., (hu)K ] (27)
where (hu)k is the perfectly estimated UL channel vector of size M × 1. The
imperfectly estimated DL channel vector can be expressed as
(hd )k = ruk
t−1
uk
(hu)T
k R−1
b Tb (28)
Imperfectly estimated DL channel vector can also be expressed as
(hd )k = (ĥd )k + (h̄d )k (29)
From (21), the DL channel vector for kth
user can be expressed as
(hk )d = (hu)T
k Qk = (hu)T
k (Q̂k + Q̄k ) (30)
where (ĥd )k = (hu)T
k Q̂k , (h̄d )k = (hu)T
k Q̄k and Qk is the overall HM experienced
by the kth
user.

MF precoding
Here Gmf = HH
D and (gk )MF = (hd )H
k . From (25), we expand the numerator and
denominator terms individually. Hence the numerator can be expressed as
(ps)MF = pk |(ĥd )k (gk )MF |2
= pk |(hu)T
k Q̂k {(hu)T
k (Q̂k + Q̄k )}H
|2
(31)
Let Ak = (hu)T
k Q̂k Q̂H
k (hu)∗
k and Bk = (hu)T
k Q̂k Q̄H
k (hu)∗
k are mutually independent
and uncorrelated then
(ps)MF = pk |Ak |2
| {z }
Signal power
+ pk |Bk |2
| {z }
HMpower
(32)
Now inter-user interference power can be modeled as
(PI )MF =
K
X
j=1,j6=k
pj (hu)T
k Q̂k QH
j (hu)∗
j (hu)T
j Qj Q̂H
k (hu)∗
k (33)

From (25) and (26),
Nmf = (ζ)mf =
Ψmf σ2
PT
(34)
where PT and σ2
are considered to be fixed and Ψmf − Ψo
mf → 0, when M is
considered to be very large.
Ψo
mf =MK

(E(r−1
bm
tbm
))2
+ var(r−1
bm
tbm
)

(E(t−1
uk
ruk
))2
+ var(t−1
uk
ruk
)
(35)
RZF precoding:
Grzf = (HH
D HD + αIM )−1
HH
D , where α is the regularization parameter. Precoding
vector (gk )rzf = (HH
D HD + αIM )−1
(hd )k = Wrzf (hd )k . From (25) signal power
can be modeled as
(ps)rzf = |(ĥd )k Wrzf (hd )H
k |2
(36)
Since HH
D HD = HD
H
[k]HD[k] + (hd )H
k (hd )k , where
HD[k] = [(hd )1; ....; (hd )k−1; (hd )k+1; ....; (hd )K ] is a matrix of size (K − 1) × M

Continued–
Γ = HH
D HD + αIM , Γ[k] = HD
H
[k]HD[k] + αIM then Γ = Γ[k] + (hd )H
k (hd )k and
Wrzf = Γ−1
. We also use the matrix inversion lemma for finding the signal power.
(ĥd )k Wrzf (hd )H
k =
(ĥd )k Γ−1
[k]
(hd )H
k
(1+(hd )k Γ−1
[k]
(hd )H
k )
From (29), the above relation can also be modified as
(ĥd )k Wrzf (hd )H
k =
h (ĥd )k Γ−1
[k] (ĥd )H
k
(1 + (hd )k Γ−1
[k] (hd )H
k )
+
(ĥd )k Γ−1
[k] (h̄d )H
k
(1 + (hd )k Γ−1
[k] (hd )H
k )
i
≈
mΓ,Q2
k
(−α)
1 + mΓ,Q2
k
(−α)
(37)
where dependent parameter can be expressed as [5]
mΓ,Q2
k
(−α) =
1
M
trQ2
k
1
M
K
X
j=1
Q2
k
1 + eM,j (−α)
+ αIM
−1
(38)
eM,i (−α) =
1
M
trQ2
i
1
M
K
X
k=1
Q2
k
1 + eM,k (−α)
+ αIM
−1
(39)

Inter-user interference power
From (25), inter-user interference power can also be expressed as
K
X
j=1,j6=k
pj |(ĥd )k (gj )rzf |2
= (ĥd )k Wrzf HD
H
[k]P[k]HD[k]Wrzf (ĥd )H
k (40)
where P[k] = diag(p1, ....pk−1, pk+1, ...., pK ). From [5], the deterministic
equivalent yields can be expressed as
(ĥd )k Wrzf HD
H
[k]P[k]HD[k]Wrzf (ĥd )H
k −
h
u0
−
uu0
(1 + u)2
i
→ 0 (41)
where dependent parameter u = trQ2
k Γ−1
[k] and u0
= tr(P[k]HD[k]Γ−1
[k] Q2
k Γ−1
[k] HD
H
[k]).
For large value of M,
h
u0
− Υo
k
i
→ 0 and u − mΓ,Q2
k
(−α) → 0, where
Υo
k =
1
M
K
X
j=1,j6=k
pj e0
j,k
(1 + ej )2
(42)

Noise power
From (25) and (26), noise component mainly depends on the parameter
Ψrzf . Mathematically it can be expressed as
Ψrzf = E{tr(HD(HH
D HD + αIM)−2
HH
D )}
K
X
j=1
(hd )k(HH
D HD + αIM)−2
(hd )H
k
(43)
Using Lemma 2
Ψrzf =
(hd )kΓ−2
[k] (hd )k
(1 + (hd )kΓ−2
[k] (hd )k)2
=
m0
Γ,Q2
k
(−α)
1 + mΓ,Q2
k
(−α)
2
(44)

where
h
m0
Γ,Q2
k
(−α) − 1
M tr(Q2
k T0
)
i
→ 0 and
T0
= eM,i (−α)
h 1
M
K
X
j=1
Q2
j e0
j
(1 + ej )2
+ IM
i
eM,i (−α) (45)
For large value of M, Ψrzf − Ψo
rzf → 0 where
Ψo
rzf =
1
M
K
X
k=1
pk e0
k
(1 + ek )2
(46)
In case of very large number of BS antenna, (γk )rzf − (γo
k )rzf → 0. The
closed-form expression for kth
user SNR can be
(γo
k )rzf =
pk (mo
k )2
Υo
k +
Ψo
rzf
ρ (1 + mo
k )2
(47)

ZF precoding
Gzf = HH
D (HDHH
D )−1
and (gk )ZF = HH
D (HDHH
D )−2
HD(hd )H
k = Wzf (hd )H
k , where
Wzf = HH
D (HDHH
D )−2
HD. From [5], the derived SINR using ZF precoding can be
modeled as
(γk )zf − (γk )o
zf
M → ∞
−
−
−
−
−
→ 0
So closed-form expression using ZF precoding scheme for SINR can be expressed
as
(γk )o
zf =
pk
Υo
k +
Ψo
zf
ρ
(48)
where
Ψo
zf = 1
M
PK
j=1 and Υo
k = 1
M
PK
j=1,j6=k
phe0
j,k
e2
j
Sum-rate Capacity:
Sum-rate capacity in different precoding schemes can be expressed as
Rsum =
K
X
k=1
log2(1 + (γk )I ) ∀ I = {zf , rzf , mf } (49)

Simulation results:
0 5 10 15 20 25 30 35 40 45 50
Number of Users
10
20
30
40
50
60
70
Total
Sum-rate
capacity
in
bps/HZ
Perfect reciprocity ZF
α=0.01
α=0.1
α=2
α=5
α=∞
Perfect reciprocity MF
Figure: Impact of regularization parameter α in case of perfect reciprocity with
RZF precoder along with ZF and MF precoder

5 10 15 20 25 30 35 40 45 50
Number of Users
10
20
30
40
50
60
70
Total
Sum-rate
capacity
in
bps/HZ
BS side HM consideration
Perfect reciprocity RZF
BS side HM consideration RZF
BS side HM consideration MF
Figure: HM across the BS when UE experience no HM and σ1 = 2, σ2 = 3. Here
α = 0.1 for RZF precoder

0 5 10 15 20 25 30 35 40 45 50
Number of Users
10
20
30
40
50
60
70
Total
Sum-rate
capacity
in
bps/HZ
UE side HM consideration
UE side HM consideration RZF
UE side HM consideration MF
Figure: HM across the UE with σ3 = 2, σ4 = 3 and α = 0.1 for RZF precoder

5 10 15 20 25 30 35 40 45 50
Number of Users
20
30
40
50
60
70
Total
Sum-rate
capacity
in
bps/HZ
Both side HM consideration
Both side HM consideration RZF
Both side HM consideration MF
Figure: HM across the UE and BS with σ1 = 2, σ2 = 3, σ3 = 2, σ4 = 3

5 10 15 20 25 30 35 40 45 50
Number of Users
1
2
3
4
5
6
7
8
9
10
%
Deviation
in
sum-rate
capacity
ZF
RZF α=0.01
RZF α=0.1
RZF α=10
MF
Figure: Percentage deviation of sum rate capacity for different α with 5% error
variance of the mean of HM coefficients and HM is considered only across the BS
side

Dissertation plan:
cooperative network
massive MIMO
massive MIMO
Thesis
Contributory

Antenna Correlation Effect:
Brief Introduction
Massive MU-MIMO consist of a single BS with a large number of
antennas.
Increasing the number of antenna across BS increases the degree of
freedom in terms of receive/transmit diversity.
If physical spacing between adjacent antenna is closer gradually then
observed wireless channel does not remain statistically uncorrelated.
Advantage of statistically uncorrelated wireless channel:
Favorable propagation
Linear data processing can be possible through linear detector.
Spatial multiplexing gain increases leading to linearly increase in the
capacity.

Channel model for point-to-point MIMO and MU-MIMO
Considering the complex baseband channel, the received signal vector for the
point-to-point MIMO network can be modeled as
Y = GpX + n (50)
Gp → Channel matrix of size Nr × Nt.
X → Transmitted symbol vector of size Nt × 1.
n → AWGN noise vector of size Nr × 1.
Note: It is also considered that (Nr , Nt) both are very large and
trace(E{XXH
}) = PT , where PT is the total transmitted power.
The observed channel matrix in correlated scenario can be expressed as
Gp = ζ
1
2
r
˜
Gpζ
1
2
t (51)

Kronecker Model [6]
In such scenario, (51) can be further modified as
gp = (ζ
1
2
r ⊗ ζ
1
2
t ) ˜
gp (52)
gp = vec(Gp) is a vector of size (NtNr × 1)
˜
gp = vec( ˜
Gp) is a vector of size (NtNr × 1)
(ζ
1
2
r ⊗ ζ
1
2
t ) → Kronecker product between receive and transmit
correlation matrix.
Let Ac is correction matrix of size (Nr Nt × Nr Nt) then the estimated
channel vector can be further modeled as
ˆ
gp = Ac(ζ
1
2
r ⊗ ζ
1
2
t ) ˜
gp (53)

Multi-user MIMO (MU-MIMO):
We consider a cellular network consisting of one BS with M number of antennas
and K users. The received signal vector in UL scenario can be expressed as
Y = GmuX + n (54)
Gmu = [g1, ....., gK ] → UL channel matrix of size M × K
X → Transmitted symbol vector of size K × 1 such that E[XXH
] =
diag(p1, ....., pK ).
n is the AWGN noise vector of size M × 1.
Considering antenna correlation, the kth
user channel vector can be expressed as
gk = Θ
1
2
k hk β
1
2
k (55)
Θk → Receive correlation matrix for kth
user of size M × M.
hk → Rayleigh faded channel vector of size M × 1.
βk is the large scale fading coefficient for kth
user [7].

Continued–
Note: In case of MU-MIMO, the inter-user separation is greater than O(10λ) in
general practice.
Correlation Matrix Generation:
From [6], if two RV are the received voltage across the ith
and jth
antenna
terminal assuming that the mean voltage µVi
and µVj
are zero can be modified as
ρ =
E{Vi V ∗
j }
q
E(Vi V ∗
i )E(Vj V ∗
j )
(56)
(Vi , Vj ) ∼ f (φ, ψ), where φ → Polar angle and ψ → Azimuth angle
Let P(φ, ψ) be the joint PDF for two RV φ and ψ. The expected correlation
between ith and jth antenna terminal can be expressed as [8]
ρi,j =
Z
φ
Z
ψ
vi (φ, ψ)vj (φ, ψ)∗
P(φ, ψ)sin(φ)dφdψ (57)

Correlation function:
1 2 3 4 5 6 7 8 9
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Antenna Relative Spacing across Base Station
Correlation
d0
=.3828λ d
10
=d
0
+5λ
d1
=d0
+λ/2 d2
=d0
+λ
Figure: Correlation Function vs Relative Antenna Spacing

A correlation scenario based on antenna placement:
64
d
(a)
32
d
2
(b)
8
d
8
(c)
d d
dt
2
r
r
r
r
r
r
r
r
r
Figure: Antenna placement in (ULA), (URA) (32 × 2), (URA) (8 × 8)

Antenna Misalignment:
If jth
antenna is not properly aligned then the current location of the misaligned
jth
antenna can be expressed as
d0
j = ˆ
dj + ˜
dj ∀ 0 ≤ ˜
dj λ/2 (58)
d0
j → Current position of the misaligned antenna.
˜
dj → Deviation of misaligned antenna terminal from the reference location.
ˆ
dj → True position of the jth antenna from the reference location.
Mathematically the percentage displacement can be expressed as
Percentage displacement =
( ˆ
dj − d0
j ) × 100
ˆ
dj − ˆ
dj+1
(59)

Achievable Rate and Power Efficiency
The detected signal vector can be expressed as
r = AmuY (60)
Here, the three linear detector matrices i.e MRC, ZF and MMSE can be expressed
as
Amu =





GH
mu for MRC
(GH
muGmu)−1
GH
mu for ZF
(GH
muGmu + IK )−1
GH
mu for MMSE
(61)
r = AmuGmuX + Amun (62)
Let rk and xk be the kth
elements of the vector r and X, respectively. Hence,
rk = ak gk xk
| {z }
Desired Signal
+
K
X
i=1,i6=k
ak gi xi
| {z }
Inter−user Intf ..
+ ak n
|{z}
Noise
(63)

Continued–
Assumption 2 :
Let p1β1 = p2β2 = .... = pK βK = pu
(ak)MRC = gH
k for MRC (64)
(ak)ZF = gH
k Gmu(GH
muGmu)−1
GH
mu for ZF (65)
Since
(GH
muGmu + IK )−1
GH
mu = GH
mu(GmuGH
mu + IM)−1
(66)
Using matrix inversion identity, from (66), the decoding vector in MMSE
detection scheme can be expressed as
(ak)MMSE = gH
k (GmuGH
mu + IM)−1
for MMSE (67)

PS,k = |akgkxk|2
= pu|akΘ
1
2
k hk|2
(68)
PI,k =
K
X
i=1,i6=k
|akgi xi |2
= pu
K
X
i=1,i6=k
|akΘ
1
2
i hi |2
(69)
PN,k = |akn|2
= |ak|2
(70)
Capacity Formulation
The achievable rate for kth user, using three different decoding vector
(64), (65), (67)

is given by (Rk)×, where × = MRC, ZF, MMSE. In UL
scenario, the total system throughput in correlated environment can be
expressed as
R =
K
X
k=1
(Rk)× =
K
X
k=1
E
n
log2

1 +
PS,k
PI,k + PN,k
o
(71)

Continued–
Power Efficiency Formulation
Here EE is evaluated either in TDMA or FDMA based multiple access
schemes. Using FDMA based multiple access technique, the EE can be
expressed as
ηEE =
PK
k=1(Rk)×Bk

tr(E(xxH)
(72)
where (Rk)×, Bk are the capacity and bandwidth allocation for the kth
user. On the other side, EE in TDMA based multiple access can be
expressed as
ηEE =
K
X
k=1
τk
TC
(Rk)×B
tr(E(xxH))
(73)

Simulation Parameter
S/n System Parameters Remarks
1 Cell radius 1 KM
2 UE to BS separation r1 = ... = rK = 500 m
3 Carrier frequency 3 GHz
4 Small scale fading distribution Rayleigh fading with unit variance
5 Reference distance, r0 35 m [9]
6 Linear detector MRC, ZF, MMSE
7 Channel path loss model 3GPP-Urban Micro
8 Path loss exponent 3.2
9 Lognormal shadowing σshadow = 8 dB
10 Number of channel realization 10000
11 Antenna gain (BS) 0 dBm
12 Multiple access technique in UL SC-FDMA

Simulation Results:
5 10 15 20 25 30
Number of User
10
15
20
25
30
35
40
45
50
System
throughput
in
bps/HZ MRC detector performance in IID and correlated scenario
IID channel condition
ULA 64X1, From Figure 3.4(a)
URA 32X2, From Figure 3.4(b)
URA 8X8, From Figure 3.4(c)
Figure: System throughput vs total number of users in IID and different
correlated environment

5 10 15 20 25 30
Number of User
20
40
60
80
100
120
System
throughput
in
bps/HZ
ZF detector performance in IID and correlated scenario

5 10 15 20 25 30
Number of User
20
40
60
80
100
120
System
throughput
in
bps/HZ
MMSE detector performance in IID and correlated scenario

0 5 10 15 20 25
Total transmit power in dB
0
5
10
15
20
25
η
EE
/Hz,
bps/Hz/Joule
EE using MRC detector
Correlated channel with 64X1 antenna configuration
Figure: EE Vs Total Transmit Power under Different Correlation Scenario, when
K=10

0 5 10 15 20 25
0
5
10
15
20
25
30
35
40
η
EE
/Hz,
bps/Joule
EE using ZF detector
Figure: EE Vs Number of Users for Different Correlation Scenario

0 5 10 15 20 25
0
5
10
15
20
25
30
35
40
η
EE
/Hz,
bps/Joule
EE using MMSE detector
Figure: EE Vs Number of Users for Different Correlation Scenario

0 5 10 15 20 25
0
5
10
15
20
η
EE
bps/Joule
EE using ZF detector
Properly aligned within 8× 8 grid
10% misaligned
20% misaligned
30% misaligned
Figure: EE vs SNR in Antenna Alignment Mismatch for Properly Aligned, 10%,
20%, 30% Displacement from its Original Position

Dissertation Plan:
cooperative network
massive MIMO
massive MIMO
Thesis
Contributory

Massive MIMO in Cooperative Networks
Motivation:
Q- Can massive MIMO be implemented for cooperative communication.
Q- What type of existing relaying mechanism?
Q- What are the different cooperation protocols?
Types of relay:
1 Coordinated multi-point (CoMP): Data and CSI are shared among
neighboring cellular BS to coordinate their transmission in the DL and
jointly processed the received signal in the UL.
2 Fixed relays : In this scenario, a fixed transceiver unit (relay) with low cost
and without wired backhaul connections induct into a wireless network,
where relay store data received from the mobile station (MS) and forwarded
to BS in UL and vice versa.
3 Mobile relay: Here relays are mobile and are not deployed as the
infrastructure of a network. It is more flexible in the variable traffic pattern
and adapting to different propagation environment.

Cooperation protocols:
1 Non-cooperative Decoding
2 Cooperative Decoding
3 Adaptive Decoding
Figure: Relay assisted cooperative network having large number of antenna
across RS and BS

Key features, analysis and observation:
Random matrix theory
Perfect CSI
Imperfect CSI
Multiple relay scenario

Achievable Rate and Power Efficiency of Cooperative
Massive MIMO Network Under Perfect CSI
Signal reception at BS end in 1st
TS
Signal transmission from K number of MS to the BS. The received signal BS can
be expressed as
Yb1
=
√
puH1D
1/2
1 X + nub (74)
H1 → Circularly symmetric complex Gaussian random channel matrix from
MS to the BS of size Ms × K.
D1 = diag(β11, ..., β1K ) → Large scale fading (LSF) matrix of size
K × K,where β1k is the LSF coefficient between kth
user to BS for 1st
link.
X → Transmitted symbol from the MS, where E(XXH
) = IK
nub → AWGN noise vector of size Ms × 1.

Continued–
ZF Signal Detection
Let G1 = H1D
1/2
1 and using ZF detector, Aub = (G†
1 G1)−1
G†
1 , the detected signal
vector can be expressed as
Ybd1
= AubYb1
=
√
puX + D1d
nubd (75)
Applying the property of random matrix theory:
Tr E[{(G†
1 G1)−1
}]

=
K
X
k=1
1
(Ms − K)β1k
(76)
D1d
= diag 1
√
(Ms −K)β11
, 1
√
(Ms −K)β12
,........, 1
√
(Ms −K)β1K

→ K × K diagonal
matrix.
nubd → K × 1 noise vector, where E[nubd n∗
ubd ] = IK .

Signal reception and detection at relay end in 1st
TS:
The received signal across RS can also be expressed as
Yur =
√
puH2D
1/2
2 X + nur (77)
Yurd
= Aur Yur =
√
puX + D2d
nurd (78)
D2d
= diag 1
√
(Mr −K)β21
, 1
√
(Mr −K)β22
,........, 1
√
(Mr −K)β2K

→ K × K
diagonal matrix.
nurd → K × 1 noise vector, where E[nurd n∗
urd ] = IK .

Signal reception and detection in 2nd
TS:
In 2nd
time slot the received signal for P2P MIMO network across the BS can be
expressed as
Yb2
=
1
√
Mr
χrbH3WYurd
+ nrb
⇒
1
√
Mr
χrbH3W (
√
puX + D2d
nurd ) + nrb
(79)
H3 → Channel matrix between relay to BS of size Ms × Mr
W → Isometric precoder matrix of size Mr × K
nrb → AWGN noise vector of size Ms × 1
χrb → Amplification factor
Let GD=H3W an equivalent matrix of size Ms × K

Theorem 2:
If U ∼ CN(0, 1)M×N
and V ∼ CN(0, 1)N×P
are two random matrices and Z is
the another random matrix such that Z = UV , where M N P. In such a
scenario trace of the inverse of covariance matrix ie (Z†
Z)−1
converges as follows
Tr E

(Z†
Z)
−1
≈
P
(M − P)(N − P)
(80)
Referring the above relation
Tr E[(GH
d Gd )−1
]

=
K
(Mr − K)(Ms − K) βr
(81)
where βr = β31 = β32..... = β3Mr is the LSF coefficient.
Detected signal in 2nd
TS:
Ybd2 = ArsYurd
Ybd2
=
1
√
Mr
χrb
√
puX
| {z }
Signal Component
+ χrb

G†
d Gd
−1
G†
d D2d
nurd
| {z }
Interference
+

G†
d Gd
−1
Gd nrb
| {z }
Noise
(82)

Applying cooperative diversity scheme:
Yd = [Ybd1 Ybd2 ] (83)
Observed power after linearly and coherently combining:
Pobs = Tr E{Y †
d Yd }

(84)
Pd1 = puIK E(X†
X)
| {z }
Signal Power matrix
+ D†
1d
D1d
E(n†
ubnub)
| {z }
Noise Power Matrix
(85)
Observed covariance matrix of SNR can be expresses as
γ = puIK E(X†
X){D†
1d
D1d
E(n†
ubnub)}−1
(86)
where γ is a K × K matrix. Simplified form of (86) can be expressed as
γ =









pu(Ms − K)β11 12 . . . 1K
21 pu(Ms − K)β12 . . . 2K
.
.
.
.
.
.
...
.
.
.
K1 . . . . . . pu(Ms − K)β1K









(87)

From (82), observed signal power, interference power and noise power in 2nd
time
slot have been given below.
Signal Power (S) = pu
χ2
rb
Mr
IK
Similarly interference power after mathematical simplification can be expressed as
Interference (I) =
χ2
rb
Mr (Mr − K)
D†
2d
D2d
E(nurd n∗
urd )
Whereas expected noise power is formulated using Theorem 2 that can be
expressed as
Noise (N) =
1
(Ms − K)(Mr − K)βr
E(nrbn∗
rb)
So observed SINR in 2nd
time slot across the BS can be expressed as
γ2 =
S
I + N
(88)
Hence simplified covariance matrix of SNR can be expressed as
γ2 =pu
χ2
rb
Mr
IK
n χ2
rb
Mr (Mr − K)
D†
2d
D2d
E(nurd n∗
urd ) +
1
(Ms − k)(Mr − k)βr
E(nrbn∗
rb)
o−1
(89)

Achievable rate analysis:
CDirect =
K
X
i=1
log2(1 + ps(Ms − K)β1K ) (90)
Similarly lower bound for non-cooperative decoding can be expressed as
Cnon−coop =
K
X
k=1
1
2
log2(1 + γ2k ) (91)
Here the degree of freedom in such scheme is 0.5 because E2E signal transmission
occurs in the two-time slot. The observed achievable rate under cooperative
decoding scheme with ZF detector is as follow
Ccoop =
K
X
k=1
1
2
log2(1 + γ1k + γ2k ) (92)
Power Efficiency Analysis
In case of uniform power assignment to each user, the power efficiency for such a
system can be expressed as
ηee =
CB
Kpu + pr
(93)

Simulation Results:
10 15 20 25
2
2.5
3
3.5
4
4.5
5
5.5
6
Number of RS antenna
Achievable
Rate
in
bps/Hz
Exact, Monte Carlo Simulation
Lower bound Analytical
Non−Cooperative Decoding
Direct Decoding (No relay)
Cooperative Decoding
Direct Decoding (relay assisted)
Figure: Achievable rate vs total number of RS antenna where K = 3, Ms = 50,
pr = 15 dB and
PK
i=1 pui
= 15 dB

30 35 40 45 50
1.5
2
2.5
3
3.5
4
4.5
5
Number of BS antenna
Achievable
Rate
in
bps/Hz
Exact Monte Carlo Simulation
Closed form ZF
Direct Decoding (relay assisted)
Direct Decoding (No relay) Non−cooperative Decoding
Cooperative Decoding
Figure: Achievable rate vs total number of BS antenna where K = 3, Mr = 15,
pr = 15 dB and
PK
i=1 pui
= 15 dB

10 15 20 25 30
0
1
2
3
4
5
6
7
Total Power in dB
η
EE
in
bps/Hz/Joule×
100 M
r
=10
M
r
=15
Mr
=20
Mr
=25
M
s
=50 K=3
Figure: EE vs total transmit power PT = pr +
PK
i=1

in dB where K = 3,
Ms = 50

10 15 20 25 30
0
1
2
3
4
5
6
Total Power in dB
η
EE
in
bps/Hz/Joule×
100
Ms
=40
M
s
=50
M
s
=60
Ms
=70
M
r
=15 K=3
Figure: EE vs total transmit power PT = pr +
PK
i=1

in dB where K = 3,
Mr = 15

Large-Scale Antenna System Performance with Imperfect
CSI in Cooperative Networks
The received signal across RS in 1st
time slot can be expressed as
Yr =
√
puĤur X + nur (94)
pu → Transmitted power by each user
Ĥur → Channel matrix of size Mr × K .
X → K × 1 transmitted symbol vector such that E(XH
X) = IK .
nur → AWGN noise vector of size Mr × 1 where E(nH
ur nur ) = IMr
.

Channel Modeling and Signal Detection:
Let imperfectly estimated channel matrix is Hur .
Hur =
q
1 − τ2
1 Ĥur + τ1H̄ur (95)
τ1 → Degree of channel accuracy
H̄ur is the Mr × K independent uncorrelated channel matrix.
Applying ZF detection technique at RS:
Ydr = (HH
ur Hur )−1
HH
ur (
√
puĤur X + nur ) (96)
The detected signal vector under imperfect channel condition can be expressed as
Ydr =
q
1 − τ2
1
√
pu(HH
ur Hur )−1
ĤH
ur Ĥur X
| {z }
S1
+ τ1
√
pu×
(HH
ur Hur )−1
H̄H
ur Ĥur X
| {z }
I1
+ (HH
ur Hur )−1
HH
ur nur
| {z }
N1
(97)

Hence, in the 2nd
TS, the received signal vector at the BS end can be expressed as
Yb =
1
√
Mr
ζrbĤrbWrbYdr + nrb (98)
where Ĥrb and Wrb are the perfect channel matrix between RS to BS of size
Ms × Mr and precoding matrix of the size Mr × K such that 1
Mr
W H
rb Wrb = IK

.
nrb is the AWGN noise vector of size Ms × 1, where E(nH
rbnrb) = IMs
. From (98),
the amplification factor ζrb can be represented as
ζrb =
r
pr
PS1
+ PI1
+ PN1
(99)
where pr is the power transmitted by the RS. PS1 , PI1 and PN1 are the expected
total power of signal, interference due to channel error and noise, respectively.
PS1
= Tr
h
pu(1 − τ2
1 )E

HH
ur Hur
−2
E

ĤH
ur Ĥur
2 i
(100)

20 22 24 26 28 30 32 34 36 38 40
Number of Relay Antenna
8
9
10
11
12
13
14
15
Uplink
rate
in
bps/Hz
σ2
e1
=0, σ2
e2
=0
σ2
e1
=0.01, σ2
e2
=0
σ2
e1
=0.04, σ2
e2
=0
σ2
e1
=0.1, σ2
e2
=0
Figure: Uplink rate vs. Mr , when MU → RS link have perfect channel and RS →
BS link have imperfect CSI

50 52 54 56 58 60 62 64 66 68 70
Number of BS Antenna
8.12
8.13
8.14
8.15
8.16
8.17
8.18
8.19
8.2
8.21
Uplink
rate
in
bps/Hz
σ2
e1
=0, σ2
e2
=0
σ
2
e1
=0, σ
2
e2
=0.01
σ
2
e1
=0, σ
2
e2
=0.04
σ
2
e1
=0, σ
2
e2
=0.1
Figure: Capacity per user vs Ms, when MU→ RS link have perfect channel and
RS → BS link have imperfect CSI

20 22 24 26 28 30 32 34 36 38 40
9
10
11
12
13
14
15
Uplink
rate
in
bps/Hz
σ2
e1
=0, σ2
e2
=0
σ2
e1
=0, σ2
e2
=0.01
σ2
e1
=0, σ2
e2
=0.04
σ2
e1
=0, σ2
e2
=0.1
Figure: Uplink rate vs Mr , when RS → BS link have perfect channel and MU →
RS link have imperfect CSI

50 52 54 56 58 60 62 64 66 68 70
Number of BS Antenna
8.197
8.1975
8.198
8.1985
8.199
8.1995
8.2
Uplink
rate
in
bps/Hz
σ2
e1
=0, σ2
e2
=0
σ
2
e1
=0.01, σ
2
e2
=0
σ
2
e1
=0.04, σ
2
e2
=0
σ
2
e1
=0.1, σ
2
e2
=0
Figure: Capacity per user vs Ms, when RS → BS link have perfect channel and
MU → RS link have imperfect CSI

20 22 24 26 28 30 32 34 36 38 40
8
9
10
11
12
13
14
Uplink
rate
in
bps/Hz
σ2
e1
=0.01, σ2
e2
=0.01
σ2
e1
=0.01, σ2
e2
=0.04
σ2
e1
=0.04, σ2
e2
=0.01
σ2
e1
=0.04, σ2
e2
=0.04
Figure: Uplink rate vs Mr , when MU → RS link and RS → BS link both have
imperfect CSI

20 22 24 26 28 30 32 34 36 38 40
12
13
14
15
16
17
18
19
Uplink
rate
in
bps/Hz
σ2
e1
=0.01, σ2
e2
=0.01
σ2
e1
=0.01, σ2
e2
=0.04
σ2
e1
=0.04, σ2
e2
=0.01
σ2
e1
=0.04, σ2
e2
=0.04
Figure: Capacity per user vs Ms, when MU → relay link and relay → BS link
both have imperfect CSI

Thesis conclusion and future scope:
Thesis conclusion-1
The statistical modeling for HM coefficient can be a good solution for
hardware calibration, although the proper feedback from UE side is
unknown.
Statistical parameters like mean and variance of analytically derived
PDF of HM coefficient gives the estimated value of calibration point
and the residual reciprocity errors after calibration.
System performance can also be improved by the proper selection of
precoders in the HM condition.
RZF precoder with optimal regularization parameter gives the superior
performance compare to MF and ZF precoder.
MF precoder is more robust and less sensitive towards HM, but we
observe great capacity loss compare to RZF and ZF.

Continued–
Thesis Conclusion-2
λ
2 spacing is not the only way for making wireless channel
uncorrelated.
Increasing the degree of compactness, still by keeping the antenna
spacing λ
2 causes significant loss in SE and EE.
Linear decoder like ZF and MMSE still gives better performance in
antenna correlation environment.
Thesis Conclusion-3
In AF-relaying, larger the number of relay antenna have the greater
impact on end-to-end SNR and capacity compare to BS antenna.
Small channel error variance causes significant loss for the end
terminals.

Future scope
Since HM causes the amplitude and phase impairment. This
dissertation describes only the impact of amplitude impairment.
Phase mismatch is the another challenge of TDD system, which can
also affect the sum-rate capacity in DL scenario.
Improper calibration causes additional interference, this reduces the
end user SNR. Hence for better BER and SER performance, there is a
need of robust calibration and precoding strategy.
Similarly, improper hardware calibration may also be the cause of
outage, because sum-rate capacity decreases due to reduction of per
user SNR.

Continued–
Future scope
Impact of antenna correlation in UL scenario has already been
analyzed in terms of achievable rate and power efficiency. Antenna
correlation and mutual coupling also influences the proper beam
formation and beam steering in DL scenario.
This dissertation has already incorporated the massive MIMO
technology in the cooperative network. Similar intuition can also be
drawn by extending this approach for the cognitive radio.
Channel estimation is very tedious for massive MIMO. Mobility is also
a serious concern in massive MIMO.
Non-coherent detector and random precoder is also an emerging area
for solving the problem of massive MIMO.

Publication–
Journals
1 Varun Kumar, Sarat Kumar Patra, Poonam Singh,“Mean Based Reciprocity Calibration in
TDD Massive MIMO system, IEEE System Journal 2018, Under review from 29 Nov 2018
2 Varun Kumar, Sarat Kumar Patra, Poonam Singh,“Massive MIMO in Cooperative
Network with Multiple Relay Under Imperfect CSI, IET Communication 2018,
Communicated
3 Varun Kumar, Sarat Kumar Patra, Poonam Singh,“Outage Probability in Large Antenna
System for Cooperative Network under Cooperative Decoding, IET Communication 2018,
Communicated
Conference
1 Varun Kumar, Sarat Kumar Patra, Poonam Singh, “Large Antenna Performance with
Imperfect CSI in Cooperative Networks, IEEE Conference Region 10 Connect 2018
2 Varun Kumar, Sarat Kumar Patra, Poonam Singh, “Achievable Rate and Power Efficiency
of Massive MIMO in Cooperative Network with ZF Receivers, IEEE Conference Region 10
Tencon 2017
3 Varun Kumar, Sudhansu Arya, Sarat K Patra, “Achievable Rate and Power Efficeincy in
Uplink Massive MIMO System Under Antenna Correlation IEEE Conference ANTS 2017

References I
,,,,
M. Shafi, A. F. Molisch, P. J. Smith, T. Haustein, P. Zhu, P. De Silva, F. Tufvesson, A. Benjebbour, and G. Wunder,
“5G: A tutorial overview of standards, trials, challenges, deployment, and practice,” IEEE J. Sel. Areas Commun.,
vol. 35, no. 6, pp. 1201–1221, 2017.
F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski, “Five disruptive technology directions for 5G,”
IEEE Commun. Mag, vol. 52, no. 2, pp. 74–80, 2014.
A. Osseiran, F. Boccardi, V. Braun, K. Kusume, P. Marsch, M. Maternia, O. Queseth, M. Schellmann, H. Schotten,
H. Taoka et al., “Scenarios for 5G mobile and wireless communications: the vision of the METIS project,” IEEE
Commun. Mag., vol. 52, no. 5, pp. 26–35, 2014.
C. A. Balanis, Antenna Theory: Analysis and Design; Hoboken, NJ, USA. Wiley-2005.
S. Wagner, R. Couillet, M. Debbah, and D. T. Slock, “Large system analysis of linear precoding in correlated MISO
broadcast channels under limited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509–4537, 2012.
C. Oestges, B. Clerckx, D. Vanhoenacker-Janvier, and A. Paulraj, “Impact of diagonal correlations on MIMO capacity:
application to geometrical scattering models,” in Vehicular Technology Conference, 2003. VTC 2003-Fall. 2003 IEEE
58th, vol. 1. IEEE, 2003, pp. 394–398.
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.

References II
J.-H. Lee and C.-C. Cheng, “Spatial correlation of multiple antenna arrays in wireless communication systems,” Progress
In Electromagnetics Research, vol. 132, pp. 347–368, 2012.
D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in ofdma systems with large numbers of
base station antennas,” IEEE Trans. Wireless Commun, vol. 11, no. 9, pp. 3292–3304, 2012.
E. Björnson, J. Hoydis, M. Kountouris, and M. Debbah, “Massive MIMO systems with non-ideal hardware: Energy
efficiency, estimation, and capacity limits,” IEEE Trans. Inf. Theory, vol. 60, no. 11, pp. 7112–7139, 2014.
E. Björnson, E. G. Larsson, and M. Debbah, “Massive MIMO for maximal spectral efficiency: How many users and pilots
should be allocated?” IEEE Trans. Wireless Commun, vol. 15, no. 2, pp. 1293–1308, 2016.
S. Biswas, C. Masouros, and T. Ratnarajah, “Performance analysis of large multiuser MIMO systems with
space-constrained 2-D antenna arrays,” IEEE Trans. Wireless Commun, vol. 15, no. 5, pp. 3492–3505, 2016.
Y. Chen, X. Gao, X.-G. Xia, and L. You, “Robust MMSE precoding for massive MIMO transmission with hardware
mismatch,” Science China Information Sciences, vol. 61, no. 4, p. 042303, 2018.
J. Choi, “Downlink multiuser beamforming with compensation of channel reciprocity from RF impairments,” IEEE Trans.
Commun, vol. 63, no. 6, pp. 2158–2169, 2015.

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Challenges of Massive MIMO System

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