1

Multiplexing Analysis of Millimeter-Wave Massive
MIMO Systems

arXiv:1801.02987v2 [] 21 Jan 2018

Dian-Wu Yue, Ha H. Nguyen and Shuai Xu

Abstract—This paper is concerned with spatial multiplexing
analysis for millimeter-wave (mmWave) massive MIMO systems.
For a single-user mmWave system employing distributed antenna
subarray architecture in which the transmitter and receiver
consist of Kt and Kr subarrays, respectively, an asymptotic
multiplexing gain formula is firstly derived when the numbers of
antennas at subarrays go to infinity. Specifically, assuming that
all subchannels have the same number of propagation paths L,
the formula states that by employing such a distributed antennasubarray architecture, an exact multiplexing gain of Ns can be
achieved, where Ns ≤ Kr Kt L is the number of data streams.
This result means that compared to the co-located antenna
architecture, using the distributed antenna-subarray architecture
can scale up the maximum multiplexing gain proportionally to
Kr Kt . In order to further reveal the relation between diversity
gain and multiplexing gain, a simple characterization of the
diversity-multiplexing tradeoff is also given. The multiplexing
gain analysis is then extended to the multiuser scenario. Moreover, simulation results obtained with the hybrid analog/digital
processing corroborate the analysis results.
Index Terms—Millimeter-wave communications, massive
MIMO, multiplexing gain, diversity gain, diversity-multiplexing
tradeoff, distributed antenna-subarrays, hybrid precoding.

I. I NTRODUCTION
Recently, millimeter-wave (mmWave) communication has
gained considerable attention as a candidate technology for
5G mobile communication systems and beyond [1]–[3]. The
main reason for this is the availability of vast spectrum in the
mmWave band (typically 30-300 GHz) that is very attractive
for high data rate communications. However, compared to
communication systems operating at lower microwave frequencies (such as those currently used for 4G mobile communications), propagation loss in mmWave frequencies is much
higher, in the orders-of-magnitude. Fortunately, given the
much smaller carrier wavelengths, mmWave communication
systems can make use of compact massive antenna arrays to
compensate for the increased propagation loss.
Nevertheless, the large-scale antenna arrays together with
high cost and large power consumption of the mixed analog/digital signal components makes it difficult to equip a
Dian-Wu Yue is with the College of Information Science and Technology, Dalian Maritime University, Dalian, Liaoning 116026, China (e-mail:
dwyue@dlmu.edu.cn), and also with the Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon,
SK, Canada S7N 5A9.
Ha H. Nguyen is with the Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK, Canada
S7N 5A9 (e-mail: ha.nguyen@usask.ca).
Shuai Xu is with the College of Information Science and Technology, Dalian Maritime University, Dalian, Liaoning 116026, China (e-mail:
xu shuai@dlmu.edu.cn).

separate radio-frequency (RF) chain for each antenna and
perform all the signal processing in the baseband. Therefore,
research on hybrid analog-digital processing of precoder and
combiner for mmWave communication systems has attracted
very strong interests from both academia and industry [4] −
[16]. In particular, a lot of work has been performed to
address challenges in using a limited number of RF chains. For
example, the authors in [4] considered single-user precoding
in mmWave massive MIMO systems and established the
optimality of beam steering for both single-stream and multistream transmission scenarios. In [10], the authors showed that
hybrid processing can realize any fully digital processing if the
number of RF chains is twice the number of data streams.
However, due to the fact that mmWave signal propagation
has an important feature of multipath sparsity in both the
temporal and spatial domains [17]–[20], it is expected that
the potentially available benefits of diversity and multiplexing
are indeed not large if the deployment of the antenna arrays is
co-located. In order to enlarge diversity/multiplexing gains in
mmWave massive MIMO communication systems, this paper
consider the use of a more general array architecture, called
distributed antenna subarray architecture, which includes lolocated array architecture as a special case. It is pointed out
that distributed antenna systems have received strong interest
as a promising technique to satisfy such growing demands for
future wireless communication networks due to the increased
spectral efficiency and expanded coverage [21] − [25].
It is well known that diversity-multiplexing tradeoff (DMT)
is a compact and convenient framework to compare different MIMO systems in terms of the two main and related
system indicators: data rate and error performance [26]–[31].
This tradeoff was originally characterized by Zheng and Tse
[26] for MIMO communication systems operating over i.i.d.
Rayleigh fading channels. The framework has then ignited a
lot of interests in analyzing various communication systems
and under different channel models. For a mmWave massive
MIMO system, how to quantify the diversity and multiplexing
performance and further characterize its DMT is a fundamental
and open research problem. In particular, to the best of
our knowledge, until now there is no unified multiplexing
gain analysis for mmWave massive MIMO systems that is
applicable to both co-located and distributed antenna array
architectures.
To fill this gap, this paper investigates the multiplexing
performance of mmWave massive MIMO systems with the
proposed distributed subarray architecture. The focus is on
the asymptotical multiplexing gain analysis in order to find
out the potential multiplexing advantage provided by multiple

2

distributed antenna arrays. The obtained analysis can be used
conveniently to compare various mmWave massive MIMO
systems with different distributed antenna array structures.
The main contributions of this paper are summarized as
follows:
• For a single-user system with the proposed distributed
subarray architecture, a multiplexing gain expression is
obtained when the number of antennas at each subarray
increases without bound. This expression clearly indicates
that one can obtain a large multiplexing gain by employing the distributed subarray architecture.
• A simple DMT characterization is further given. It can
reveal the relation between diversity gain and multiplexing gain and let us obtain insights to understand the
overall resources provided by the distributed antenna
architecture.
• The multiplexing gain analysis is then extended to the
multiuser scenario with downlink and uplink transmission.
• Simulation results are provided to corroborate the analysis results and show that the distributed subarray architecture yields significantly better multiplexing performance
than the co-located single-array architecture.
The remainder of this paper is organized as follows. Section
II describes the massive MIMO system model and hybrid processing with the distributed subarray architecture in mmWave
fading channels. Section III and Section IV provides the
asymptotical achievable rate analysis and the multiplexing gain
analysis for the single-user mmWave system, respectively. In
Section V, the multiplexing gain analysis is extended to the
multiuser scenario. Section VI concludes the paper.
Throughout this paper, the following notations are used.
Boldface upper and lower case letters denote matrices and
column vectors, respectively. The superscripts (·)T and (·)H
stand for transpose and conjugate-transpose, respectively.
diag{a1 , a2 , . . . , aN } stands for a diagonal matrix with diagonal elements {a1 , a2 , . . . , aN }. The expectation operator
is denoted
N by E(·). [A]ij gives the (i, j)th entry of matrix
A. A B is the Kronecker product of A and B. We write a
function a(x) of x as o(x) if limx→0 a(x)/x = 0. We use (x)+
to denote max{0, x}. Finally, CN (0, 1) denotes a circularly
symmetric complex Gaussian random variable with zero mean
and unit variance.
II. S YSTEM M ODEL
Consider a single-user mmWave massive MIMO system as
shown in Fig. 1. The transmitter is equipped with a distributed
antenna array to send Ns data streams to a receiver, which
is also equipped with a distributed antenna array. Here, a
distributed antenna array means an array consisting of several
remote antenna units (RAUs) (i.e., antenna subarrays) that are
distributively located, as depicted in Fig. 2. Specifically, the
antenna array at the transmitter consists of Kt RAUs, each of
which has Nt antennas and is connected to a baseband processing unit (BPU) by fiber. Likewise, the distributed antenna
array at the receiver consists of Kr RAUs, each having Nr
antennas and also being connected to a BPU by fibers. Such

a MIMO system shall be referred to as a (Kt , Nt , Kr , Nr )
distributed MIMO (D-MIMO) system. When Kt = Kr = 1,
the system reduces to a conventional co-located MIMO (CMIMO) system.
The transmitter accepts as its input Ns data streams and is
(rf)
(rf)
equipped with Nt RF chains, where Ns ≤ Nt ≤ Nt Kt .
(rf)
Given Nt
transmit RF chains, the transmitter can apply a
(rf)
low-dimension Nt × Ns baseband precoder, Wt , followed
(rf)
by a high-dimension Kt Nt × Nt
RF precoder, Ft . Note
that amplitude and phase modifications are feasible for the
baseband precoder Wt , while only phase changes can be made
by the RF precoder Ft through the use of variable phase
shifters and combiners. The transmitted signal vector can be
written as
1/2
x = Ft Wt Pt s,
(1)
where Pt = [pij ] is a diagonal power allocation matrix with
PNs
l=1 pll = 1 and s is the Ns × 1 symbol vector such that
E[ssH ] = P INs . Thus P represents the average total input
power. Considering a narrowband block fading channel, the
Kr Nr × 1 received signal vector is
1/2

y = HFt Wt Pt s + n

(2)

where H is Kr Nr ×Kt Nt channel matrix and n is a Kr Nr ×1
vector consisting of i.i.d. CN (0, 1) noise samples. Throughout
this paper, H is assumed known to both the transmitter and
(rf)
(rf)
receiver. Given that Nr RF chains (where Ns ≤ Nr ≤
Nr Kr ) are used at the receiver to detect the Ns data streams,
the processed signal is given by
1/2

H H
z = WrH FH
r HFt Wt Pt s + Wr Fr n

(3)

(rf)

where Fr is the Kr Nr × Nr RF combining matrix, and Wr
(rf)
is the Nr × Ns baseband combining matrix. When Gaussian
symbols are transmitted over the mmWave channel, the the
system achievable rate is expressed as
H H
H H H
R = log2 |INs +P R−1
n Wr Fr HFt Wt Pt Wt Ft H Fr Wr |
(4)
where Rn = WrH FH
r Fr Wr .
Furthermore, according to the architecture of RAUs at the
transmitting and receiving ends, H can be written as

 √
√
g11 H11
···
g1Kt H1Kt


..
..
..
H=
 . (5)
.
.
.
√
√
g Kr 1 HKr 1 · · ·
g Kr Kt HKr Kt

In the above expression, gij represents the large scale fading
effect between the ith RAU at the receiver and the jth RAU
at the transmitter, which is assumed to be constant over many
coherence-time intervals. The normalized subchannel matrix
Hij represents the MIMO channel between the jth RAU at
the transmitter and the ith RAU at the receiver. We assume
that all of {Hij } are independent mutually each other.
A clustered channel model based on the extended SalehValenzuela model is often used in mmWave channel modeling
and standardization [4] and it is also adopted in this paper. For
simplicity of exposition, each scattering cluster is assumed to

3

N t(rf)

Ns

N r(rf)

Ns

Kr

Kt

Fig. 1. Block diagram of a mmWave massive MIMO system with distributed antenna arrays.

included in the argument of aULA since the response for an
ULA is independent of the elevation angle. In contrast, for a
uniform planar array (UPA), which is composed of Nh and
Nv antenna elements in the horizontal and vertical directions,
respectively, the array response vector is represented by
aUPA (φ, θ) = aULA
(φ) ⊗ aULA
(θ),
h
v

(8)

where
1 h j2π dh
λ
√
aULA
(φ)
=
1, e
h
Nh

sin(φ)

, . . . , ej2π(Nh −1)

dh
λ

sin(φ)

iT

(9)

and
1 h j2π dv
λ
1, e
aULA
(θ) = √
v
Nv
Fig. 2. Illustration of distributed antenna array deployment.

contribute a single propagation path.1 Using this model, the
subchannel matrix Hij is given by
s
Lij
Nt Nr X l
rl H
tl
tl
Hij =
αij ar (φrl
(6)
ij , θij )at (φij , θij ),
Lij
where Lij is the number of propagation
the
rl
complex gain of the lth ray, and φrl
(θ
)
are
ij
ij
its random azimuth (elevation) angles of arrival and departure,
respectively. Without loss of generality, the complex gains αlij
rl
are assumed to be CN (0, 1). 2 The vectors ar (φrl
ij , θij ) and
tl
tl
at (φij , θij ) are the normalized receive/transmit array response
vectors at the corresponding angles of arrival/departure. For an
N -element uniform linear array (ULA) , the array response
vector is
iT
du
1 h j2π du sin(φ)
λ
aULA (φ) = √
, . . . , ej2π(N −1) λ sin(φ)
1, e
N
(7)
where λ is the wavelength of the carrier and du is the interelement spacing. It is pointed out that the angle θ is not
1 This assumption can be relaxed to account for clusters with finite angular
spreads and the results obtained in this paper can be readily extended for such
a case.
2 The different variances of αl can easily accounted for by absorbing into
ij
the large scale fading coefficients gij .

, . . . , ej2π(N

v

−1) dλv sin(θ)

iT

.

(10)

III. A SYMPTOTIC ACHIEVABLE R ATE A NALYSIS
From the structure and definition of the P
channel
matrix
Kr P Kt
H in Section II, there is a total of Ls =
i=1
j=1 Lij
propagation paths. Naturally, H can be decomposed into a
sum of Ls rank-one matrices, each corresponding to one
propagation path. Specifically, H can be rewritten as

l=1

paths, αlij is
tl
and φtl
ij (θij )

sin(θ)

H=

Lij
Kt X
Kr X
X

tl
tl
rl H
α̃lij ãr (φrl
ij , θij )ãt (φij , θij ),

(11)

i=1 j=1 l=1

where
α̃lij

=

s

gij

Nt Nr l
α ,
Lij ij

(12)

rl
ãr (φrl
ij , θij ) is a Kr Nr × 1 vector whose bth entry is defined
as

r
rl
[ar (φrl
rl
ij , θij )]b−(i−1)Nr , b ∈ Qi
[ãr (φrl
,
θ
)]
=
(13)
ij ij b
0,
b∈
/ Qri
tl
where Qri = ((i − 1)Nr , iNr ]. And ãt (φtl
ij , θij ) is a Kt Nt × 1
vector whose bth entry is defined as

t
tl
[at (φtl
tl
tl
ij , θij )]b−(j−1)Nt , b ∈ Qj
(14)
[ãt (φij , θij )]b =
t
0,
b∈
/ Qj .
rl
where Qtj = ((j − 1)Nt , jNt ]. Regarding {ãr (φrl
ij , θij )} and
tl
tl
{ãt (φij , θij )}, we have the following lemma from [32].
Lemma 1: Suppose that the antenna configurations at
all RAUs are either ULA or UPA. Then all Ls vectors

4

rl
{ãr (φrl
ij , θij )} are orthogonal to each other when Nr → ∞.
tl
Likewise, all Ls vectors {ãt (φtl
ij , θij )} are orthogonal to each
other when Nt → ∞.
Mathematically, the distributed massive MIMO system can
be considered as a co-located massive MIMO system with
Ls paths that have complex gains {α̃lij }, receive array rerl
sponse vectors {ãr (φrl
ij , θij )} and transmit response vectors
tl
tl
{ãt (φij , θij )}. Furthermore, order all paths in a decreasing
order of the absolute values of the complex gains {α̃lij }. Then
the channel matrix can be written as

H=

Ls
X

SNRl = P pll |α̃l |2 = γ̃l |β̃l |2 , l = 1, 2, . . . , Ns .

l

rl

rl

tl

tl H

α̃ ãr (φ , θ )ãt (φ , θ ) ,

(15)

where α̃1 ≥ α̃2 ≥ · · · ≥ α̃Ls .
One can rewrite H in a matrix form as
H = Ar DAH
t

(16)

where D is a Ls × Ls diagonal matrix with [D]ll = α̃l , and
Ar and At are defined as follows:
Ar = [ãr (φr1 , θr1 ), . . . , ãr (φrLs , θrLs )]

(17)

At = [ãt (φt1 , θt1 ), . . . , ãt (φtLs , θtLs )].

(18)

and
rl

tl

tl

Since both {ãr (φ , θ )} and {ãt (φ , θ )} are orthogonal
vector sets when Nr → ∞ and Nt → ∞, Ar and At are
asymptotically unitary matrices. Then one can form a singular
value decomposition (SVD) of matrix H as
H = UΣV

H

=

⊥ H
[Ar |A⊥
r ]Σ[Ãt |Ãt ]

(19)

where Σ is a diagonal matrix containing all singular values
on its diagonal, i.e.,
 l
|α̃ |, for 1 ≤ l ≤ Ls
(20)
[Σ]ll =
0,
for l > Ls
and the submatrix Ãt is defined as
Ãt = [e−jψ1 ãt (φt1 , θt1 ), . . . , e−jψLs ãt (φtLs , θtLs )]

(21)

where ψl is the phase of complex gain α̃l corresponding to the
lth path. Based on (19), the optimal precoder and combiner
are chosen, respectively, as
[Ft Wt ]opt = [e−jψ1 ãt (φt1 , θt1 ), . . . , e−jψLs ãt (φtNs , θtNs )]
(22)
and
[Fr Wr ]opt = [ãr (φr1 , θr1 ), . . . , ãr (φrNs , θrNs )].

(23)

To summarize, when Nt and Nr are large enough, the
massive MIMO system can employ the optimal precoder and
combiner given in (22) and (23), q
respectively.
′
N r l′
αij for a given l.
Now suppose that α̃l = α̃lij = gij NLt ij
We introduce two notations:
Nt Nr
γ̃l = P pll gij
(24)
Lij

(26)

So we obtain another lemma.
rl
Lemma 2: Suppose that both sets {ãr (φrl
ij , θij )} and
tl
tl
{ãt (φij , θij )} are orthogonal vector sets when Nr → ∞ and
Nt → ∞. Let Ns ≤ Ls . In the limit of large Nt and Nr , then
the system achievable rate is given by
R=

l=1

rl

Then it follows from the above SVD analysis that the instantaneous SNR of the lth data stream is given by

Ns
X
l=1

log2 (1 + γ̃l |β̃l |2 ).

(27)

Remark 1: (22) and (23) indicate that when Nt and Nr
is large enough, the optimal precoder and combiner can be
implemented fully in RF using phase shifters [4]. Furthermore,
(13) and (14) imply that for each data stream only a couple
of RAUs needs the operation of phase shifters at each channel
realization.
Remark 2: By using the optimal power allocation (i.e., the
well-known waterfilling power allocation [33]), the system can
achieve a maximum achievable rate, which is denoted as Ro .
We use Re (P/Ns ) to denote the achievable rate obtained by
using the equal power allocation, namely, pll = NPs , l =
1, 2, . . . , Ns . Then
Re (P/Ns ) ≤ Ro ≤ Re ((P Ns )/Ns ) = Re (P ).

(28)

By doing expectation operation on (28), (28) becomes,
R̄e (P/Ns ) ≤ R̄o ≤ R̄e (P ).

(29)

In what follows, we derive an asymptotic expression of
the ergodic achievable rate with the equal power allocation,
R̄e (P/Ns ) (or R̄e for simplicity). For this reason, we need to
define an integral function
Z +∞
1
∆(x) =
log2 (1 + t)e−t/x dt
x
0
= log2 (e)e1/x E1 (1/x)
(30)
where E1 (·) is the exponential integral of a first-order function
defined as [34], [35]
Z +∞ −yt
e
E1 (y) =
dt
t
1
∞
X
(−y)k
= −E + ln(y) −
(31)
k · k!
k=1

with E being the Euler constant.
rl
Theorem 1: Suppose that both sets {ãr (φrl
ij , θij )} and
tl
tl
{ãt (φij , θij )} are orthogonnal vector sets when Nr → ∞ and
Nt → ∞. Let Ns ≤ Ls and γ̃1 = γ̃2 = . . . = γ̃Ls = γ̃. Then
in the limit of large Nt and Nr , the ergodic achievable rate
with homogeneous coefficient set {γ̃l }, denoted R̄eh , is given
by


γ̃
Ns L
s −l
X
X
(−1)Ls −l−k Ls ! Ls − l ∆( Ls −k )
. (32)
R̄eh =
(Ls − l)!(l − 1)!
Ls − k
k
l=1 k=0

When Ns = Ls , R̄eh can be simplified to

and
β̃l =

′
αlij .

(25)

R̄eh = Ls ∆(γ̃).

(33)

5

F (γ) = 1 − e

−γ
γ̃

1 γ
, f (γ) = e− γ̃ .
γ̃

(34)

For the lth best data stream, based on the theory of order
statistics [36], the PDF of the instantaneous receive SNR at
the receiver, denoted γl , is given by
Ls !
fl:Ls (γl ) =
[F (γl )]Ls −l [1 − F (γl )]l−1 f (γl ).
(Ls − l)!(l − 1)!
(35)
Inserting (34) into (35), we have that


L
s −l
X
Ls !(−1)Ls −l−k Ls − l e−γl (Ls −k)/γ̃
fl:Ls (γl ) =
.
(Ls − l)!(l − 1)!
γ̃
k
k=0
(36)
By the definition of the function ∆(·), the ergodic available
rate for the lth data stream can therefore be written as
Z +∞
(l)
Reh =
log2 (1 + γl )fl:Ls (γl )dγl
0
L
s −l 
X


Ls − l Ls !(−1)Ls −l−k
gk (γ̃)
(Ls − l)!(l − 1)!
k
k=0

γ̃
L
s −l 
X
Ls − l Ls !(−1)Ls −l−k ∆( Ls −k )
(37)
=
(Ls − l)!(l − 1)! Ls − k
k

=

k=0

where

gk (γ̃) =

Z

0

+∞

log2 (1 + γl )

e−γl (Ls −k)/γ̃
dγl .
γ̃

(38)

So we can obtain the desired result (32).
Finally, when Ns = Ls , we can readily prove (33) with the
help of the knowledge of unordered statistics.

Remark 3: Now let Ns = Ls and assume that Lij = L for
any i and j (i.e., all subchannels Hij have the same number of
propagation paths). When Nr → ∞ and Nt → ∞, the ergodic
achievable rate of the distributed MIMO system, R̄eh , can be
rewritten as
R̄eh (Kt , Kr ) = Kt Kr L∆(γ̃(Kt , Kr )).

(39)

Furthermore, consider a co-located MIMO system in which
the numbers of transmit and receiver antennas are equal to
Kt Nt and Kr Nr , respectively. Then its asymptotic ergodic
achievable rate can be expressed as
R̄eh (1, 1) = L∆(Kt Kr γ̃(Kt , Kr )).

(40)

Remark 4: Generally, the coefficient set {γ̃l } is inhomogeneous. Let γ̃max = max{γ̃l } and γ̃min = min{γ̃l }. Then
the ergodic achievable rate with inhomogeneous coefficient set
{γ̃l }, R̄e , has the following upper and lower bounds:
R̄eh (γ̃min ) ≤ R̄e ({γ̃l }) ≤ R̄eh (γ̃max ).

(41)

Nt=Nr=5

70

Nt=Nr=10
Nt=Nr=50

60

Limit Results
50
Rate

Proof: Due to the assumptions that each complex gain αlij is
CN (0, 1) and the coefficient set {γ̃l } is homogeneous, thus
the instantaneous SNRs in the Ls available data streams are
i.i.d.. Let F (γ) and f (γ) denote the cumulative distribution
function (CDF) and the probability density function (PDF) of
the unordered instantaneous SNRs, respectively. Then γ̃ is just
the average receive SNR of each data stream. Furthermore,
F (γ) and f (γ) can be written as

40

30

20

10
5

10

15

20

25

30

SNRa

Fig. 3. Rate versus SNRa for different numbers of antennas.

(rf)

(rf)

= 2Ns [10] and Nr =
Assuming that Nt
= Nr
Nt = N , Fig.3 plots the ergodic achievable rate curves versus
SNRa = γ̃ for different numbers of antennas, N = 5, 10, 50.
In Fig.3, we set Ns = 6, Kr = Kt = 2, and L = 3. As
expected, it can be seen that the rate performance is improved
as N increases. Obviously, the rate curve with N = 10 is
very close to the curve with limit results obtained based on
(30) while the rate curve with N = 50 is almost the same as
the curve with limit results. This verifies Theorem 1.
IV. M ULTIPLEXING G AIN A NALYSIS AND
D IVERSITY-M ULTIPLEXING T RADEOFF
A. Multiplexing Gain Analysis
P
Definition 1: Let γ̄ = L1s Ls
l=1 γ̃l . The distributed MIMO
system is said to achieve spatial multiplexing gain Gm if its
ergodic date rate with optimal power allocation satisfies
Gm (R̄o ) = lim

γ̄→∞

R̄o (γ̄)
.
log2 γ̄

(42)

rl
Theorem 2: Assume that both sets {ãr (φrl
ij , θij )} and
tl
tl
{ãt (φij , θij )} are orthogonal vector sets when Nr and Nt are
very large. Assume that Nr and Nt are always very large but
fixed and finite when γ̄ → ∞. Let Ns ≤ Ls . Then the spatial
multiplexing gain is given by

Gm (R̄o ) = Ns .

(43)

Proof: We first consider the simple homogeneous case with
γ̃1 = γ̃2 = . . . = γ̃Ls = γ̃ and derive the spatial multiplexing
gain with respect to R̄eh . In this case, γ̄ = γ̃. Obviously,
N

s
R̄eh (γ̄) X
R̄l (γ̄)
=
.
lim eh
γ̄→∞ log2 γ̄
γ̄→∞ log2 γ̄

Gm (R̄eh ) = lim

(44)

l=1

For the lth data stream, under the condition of very large Nt
and Nr , the individual ergodic rate can be written as
(l)

Reh (γ̄) = E log2 (1 + γ̄|β̃l |2 ).

(45)

6

B. Diversity-Multiplexing Tradeoff
18
16
14

Multiplexing Gain

The previous subsection shows how much the maximal
spatial multiplexing gain we can extract for a distributed
mmWave-massive MIMO system while our previous work in
[32] indicates how much the maximal spatial diversity gain
we can extract. However, maximizing one type of gain will
possibly result in minimizing the other. We need to bridge
between these two extremes in order to simultaneously obtain
both types of gains. We firstly give the precise definition of
diversity gain before we carry
Pon the analysis.
Definition 2: Let γ̄ = L1s Ls
l=1 γ̃l . The distributed MIMO
system is said to achieve spatial diversity gain Gd if its average
error probability satisfies

K=1: Ns=1
K=1: Ns=2
K=1: Ns=3
K=2: Ns=3
K=2: Ns=6
K=2: Ns=12

12
10
8
6
4
2
0
10

20

30

40

50
60
SNRa

70

80

90

100

Gd (P̄e ) = − lim

γ̄→∞

Fig. 4. Multiplexing gain versus SNRa for different numbers of data streams.

Gd (P̄out ) = − lim

γ̄→∞

(l)

(l)

(46)

and Reh (1) is a finite value, we can have that
(l)

Reh (γ̄)
γ̄→∞ log2 γ̄
log2 γ̄ + E log2 (|β̃l |2 )
= lim
γ̄→∞
log2 γ̄
= 1.
(47)
PNs (l)
Therefore, Gm (R̄eh ) = l=1 Gm (R̄eh ) = Ns .
Now we consider the general inhomogeneous case with
equal power allocation. Because cmin = γ̃min /γ̄ and cmax =
γ̃max /γ̄ be finite when γ̄ → ∞. Consequently, it readily
follows that both of the two systems with the achievable rates
R̄eh (γ̃min ) and R̄eh (γ̃max ) can achieve a multiplexing gain of
Gm (R̄eh ) = Ns . So we conclude from (41) that the distributed
MIMO system with the achievable rate R̄e (γ̄) can achieve a
multiplexing gain of Gm (R̄e ) = Ns .
Finally, it can be readily shown that the system with the
optimal achievable rate R̄o (γ̄) can only achieve multiplexing
gain Gm (R̄o ) = Ns since both of the equal power allocation
systems with the achievable rates R̄e (P/Ns ) and R̄e (P ) have
the same spatial multiplexing gain Ns .

Remark 5: Assume that Lij = L for any i and j. Theorem 2
implies that the distributed massive MIMO system can obtain
a maximum spatial multiplexing gain of Kr Kt L while the colocated massive MIMO system can only obtain a maximum
spatial multiplexing gain of L.
(rf)
(rf)
Now we let Nt = Nr = 2Ns [10] and Kr = Kt = K,
and set L = 3 and Nr = Nt = 50. We consider the
eh (γ̄)
. In order to
homogeneous case and define Ψ(γ̄) = R̄log
2 γ̄
verify Theorem 2, Fig.4 plots the curves of Ψ(γ̄) versus
SNRa = γ̄ for different numbers of data streams, namely,
Ns = 1, 2, 3 when K = 2 and Ns = 3, 6, 12 when K = 2. It
can be seen that for any given Ns , the function Ψ(γ̄) converges
to the limit value Ns as γ̄ grows large. This observation is
expected and agrees with Theorem 2.
G(l)
m (R̄eh ) =

lim

(48)

P̄out (γ̄)
.
log2 γ̄

(49)

or its outage probability satisfies

Noting that
E log2 (|β̃l |2 ) ≤ E log2 (1 + |β̃l |2 ) = Reh (1)

P̄e (γ̄)
.
log2 γ̄

With the help of a result of diversity analysis from [32], we
can derive the following DMT result.
rl
Theorem 3: Assume that both sets {ãr (φrl
ij , θij )} and
tl
tl
{ãt (φij , θij )} are orthogonal vector sets when Nr and Nt are
very large. Assume that Nr and Nt are always very large but
fixed and finite when γ̄ → ∞. Let Ns ≤ Ls . For a given
d ∈ [0, Ls ], by using optimal power allocation, the distributed
MIMO system with Ns data streams can reach the following
maximum spatial multiplexing gain at diversity gain Gd = d
Gm =

Ls
X
l=1

(1 −

d
)+ .
Ls − l + 1

(50)

Proof: We first consider the simple case where the distributed system is the one with equal power allocation and
the channel is the one with homogeneous large scale fading
coefficients. The distributed system has Ls available link paths
in all. For the lth best path, its individual maximum diversity
(l)
gain is equal to Gd = Ls − l + 1 [32]. Due to the fact that
(l)
each path can not obtain a multiplexing gain of Gm > 1
(l)
[31], we therefore design its target data rate R = rl log2 γ̄
with 0 ≤ rl ≤ 1. Then the individual outage probability is
expressed as
(l)

Pout

= P(log2 (1 + γ̄|β̃l |2 ) < rl log2 γ̄)
γ̄ rl − 1
).
= P(|β̃l |2 <
γ̄

(51)

From [37], [38], the PDF of the parameter µ = |β̃l |2 can be
written as
fµ = aµLs −l + o(µLs −l )
(52)
(l)

where a is a positive constant. So Pout can be rewritten as
(l)

Pout = (cγ̄)−(Ls −l+1)(1−rl ) + o((γ̄)−(Ls −l+1)(1−rl ) ) (53)
where c is a positive constant. This means that this path now
can obtain diversity gain
(l)

Gd = (Ls − l + 1)(1 − rl ).

(54)

7

Diversity−multiplexing Tradeoff

Diversity−multiplexing Tradeoff
14

(0, Ls)

Multiplexing Gain: Fractions
Multiplexing Gain: Integers

12
(0, 12)
10

Diversity Gain

Diversity Gain

(1/Ls, Ls−1)

(Gm(Ls−Ns+1), Ls−Ns+1)

8
6
4

(1, 7)
(2, 6)
(3, 4)
(5, 3)

2

(Ls, 0)

(6, 2)
0

(8, 1)
(12, 0)

−2
−2

Multiplexing Gain

Fig. 5. Diversity-multiplexing tradeoff Gm (d) for a general integer d.

(l)

or say
rl ≤ (1 −

1
)+ .
Ls − l + 1

Ls
X
l=1

rl =

Ls
X
l=1

(1 −

d
)+ .
Ls − l + 1

(56)

(57)

This proves that (50) holds under the special case. We
readily show that for a general case, the lth best path can also
reach a maximum diversity gain of Ls − l + 1. So applying
(41) and (29) leads to the desired result.

Remark 6: When d is an integer, Gm (d) can be expressed
simply. In particular, Gm (0) = Ls if d = 0; Gm (1) =
P
Ls −1 Ls −l
1
l=1 Ls −l+1 if d = 1; Gm (Ls − 1) = Ls if d = Ls − 1;
Gm (Ls ) = 0 if d = Ls . In general, if d = Ls − Ns + 1 for a
given integer Ns ≤ Ls , then
Gm (Ls − Ns + 1) =

NX
s −1
l=1

Ns − l
.
Ls − l + 1

4
6
8
Multiplexing Gain

10

12

14

V. M ULTIPLEXING G AIN A NALYSIS
S CENARIO

FOR THE

M ULTIUSER

(55)

To this end, under the condition that the diversity gain satisfies
Gd = d, the maximum spatial multiplexing gain of the
distributed system must be equal to
Gm (R̄eh ) =

2

Fig. 6. Diversity-multiplexing tradeoff when L = 3 and Kt = Kr = 2.

Since the distributed system requires the system diversity gain
Gd ≥ d, this implies that
Gd = (Ls − l + 1)(1 − rl ) ≥ d

0

(58)

The function Gm (d) is plottedPin Fig.5. Note that Gm (Ls −
Ns
1
Ns ) − Gm (Ls − Ns + 1) = l=1
Ls −l+1 . Generally, when
d ∈ [Ls − Ns , Ls − Ns + 1) , the multiplexing gain is given
by
Ns
X
d
.
(59)
Gm (d) = Ns −
Ls − l + 1
l=1

Example 1: We set that L = 3 and Kt = Kr = 2. So
Ls = 12. The DMT curve with fractional multiplexing gains
is shown in Fig.6. If the multiplexing gains be limited to
integers, the corresponding DMT curve is also plotted in Fig.6
for comparison.

This section considers the downlink communication in a
multiuser massive MIMO system as illustrated in Fig. 7. Here
the base station (BS) employs Kb RAUs with each having Nb
(rf)
antennas and Nb RF chains to transmit data streams to Ku
mobile stations. Each mobile station (MS) is equipped with
(rf)
Nu antennas and Nu RF chains to support the reception of
its own Ns data streams. This means that there is a total of
Ku Ns data streams transmitted by the BS. The numbers of
(rf)
data streams are constrained as Ku Ns ≤ Nb ≤ Kb Nb for
(rf)
the BS, and Ns ≤ Nu ≤ Nu for each MS.
(rf)
At the BS, denote by Fb the Kb Nb × Nb RF precoder
(rf)
and by Wb the Nb × Ns Ku baseband precoder. Then under
the narrowband flat fading channel model, the received signal
vector at the ith MS is given by
yi = Hi Fb Wb s + ni , i = 1, 2, . . . , Ku

(60)

where s is the signal vector for all Ku mobile stations,
which satisfies E[ssH ] = KuPNs IKu Ns and P is the average
transmit power. The Nu ×1 vector ni represents additive white
Gaussian noise, whereas the Nu × Kb Nb matrix Hi is the
channel matrix corresponding to the ith MS, whose entries
Hij are described as in Section II. Furthermore, the signal
vector after combining can be expressed as
H H
H H
zi = Wui
Fui Hi Fb Wb s + Wui
Fui ni , i = 1, 2, . . . , Ku
(61)
(rf)
where Fui is the Nu × Nu RF combining matrix and Wui
(rf)
is the Nu × Ns baseband combining matrix for the ith MS.
Theorem 4: Assume that all antenna array configurations
for the downlink
transmission are ULA. For the ith user, let
P b
(i)
(i)
Ls = K
L
and 0 ≤ d(i) ≤ Ls . In the limit of large
ij
j=1
Nb and Nu , the ith user can achieve the following maximum
spatial multiplexing gain when its individual diversity gain

8

Ku

N b(rf)

Ku N s

Kb

Fig. 7. Block diagram of a multiuser mmWave system with distributed antenna arrays.

(i)

satisfies Gd = d(i)

VI. C ONCLUSION
i

(i)
Gm

=

Ls
X
l=1

d(i)

(1 −

(i)

Ls − l + 1

+

) .

(62)

Proof: For the downlink transmission in a massive MIMO
multiuser system, the overall equivalent multiuser basedband
channel can be written as

 H

H1
Fu1 0
···
0
 0
  H2 
FH
0
u2 · · ·



Heq =  .
 Fb . (63)
  ..
.
.
.
..
..
..

 ..
 .
H
0
0
· · · FuKu
HKu

On the other hand, when both Nb and Nu are very large, both
rl
receive and transmit array response vector sets, {ãr (φrl
ij , θij )}
tl
tl
and {ãt (φij , θij )}, are orthogonal sets. Therefore the multiplexing gain for the ith user can depend only on the subchannel
matrix Hi and the choices of Fui and Fb . The subchannel
(i)
matrix Hi has a total of Ls propagation paths. Similar to the
proof of Theorem 2, by employing the optimal RF precoder
and combiner for the ith user, when its diversity gain satisfies
Gid = d(i) , the user can achieve a maximum multiplexing gain
of
Ls(i)
X
d(i)
(i)
(1 − (i)
Gm =
)+ .
(64)
L
−
l
+
1
s
l=1
So we obtain the desired result.

Remark 7: Consider the case that all antenna array configurations for the downlink transmission are ULA and Lij = L
for any i and j . Let 0 ≤ d ≤ Kb L. In the limit of large
Nb and Nu , the downlink transmission in the massive MIMO
multiuser system can achieve the following maximum spatial
multiplexing gain at diversity gain Gd = d
Gm =

Ku
X

G(i)
m = Ku

i=1

K
bL
X
l=1

(1 −

d
)+ .
Kb L − l + 1

(65)

Remark 8: In a similar fashion, it is easy to prove that the
uplink transmission in the massive MIMO multiuser system
can also achieve simultaneously a diversity gain of Gd = d
(0 ≤ d ≤ Kb L) and a spatial multiplexing gain of
Gm = Ku

Ls
X
l=1

(1 −

d
)+ .
Kb L − l + 1

(66)

This paper has investigate the distributed antenna subarray architecture for mmWave massive MIMO systems and
provided the asymptotical multiplexing analysis when the
number of antennas at each subarray goes to infinity. In
particular, this paper has derived the closed-form formulas
of the asymptotical available rate and spatial maximum multiplexing gain under the assumption which the subchannel
matrices between different antenna subarray pairs behave
independently. The spatial multiplexing gain formula shows
that mmWave systems with the distributed antenna architecture
can achieve potentially rather larger multiplexing gain than
the ones with the conventional co-located antenna architecture.
On the other hand, using the distributed antenna architecture
can also achieve potentially rather higher diversity gain. For a
given mmWave massive MIMO channel, both types of gains
can be simultaneously obtained. This paper has finally given
a simple DMT tradeoff solution, which provides insights for
designing a mmWave massive MIMO system.
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