Massive Multiple Input Massive Multiple Output for
5G Wireless Backhauling
D.-T. Phan-Huy1, P. Ratajczak1, R. D'Errico2, J. Järveläinen3*, D. Kong4, K. Haneda3, B. Bulut4, A. Karttunen3, M.
Beach4, E. Mellios4, M. Castañeda5, M. Hunukumbure6, T. Svensson7
1

Orange Labs, 2CEA-LETI, 3Aalto Univ.,*Premix Oy, 4Univ. of Bristol, 5Huawei, 6Samsung, 7Chalmers Univ. of Technology
dinhthuy.phanhuyorange.com

Abstract— In this paper, we propose a new technique for the
future fifth generation (5G) cellular network wireless
backhauling. We show that hundreds of bits per second per
Hertz (bits/s/Hz) of spectral efficiency can be attained at a high
carrier frequency (such as 26 GHz) between large antenna arrays
deployed along structures (such as lamp posts) that are close and
roughly parallel to each other. Hundreds of data streams are
spatially multiplexed through a short range and line of sight
“massive multiple input massive multiple output” (MMIMMO)
propagation channel thanks to a new low complexity spatial
multiplexing scheme, called “block discrete Fourier transform
based spatial multiplexing with maximum ratio transmission” (BDFT-SM-MRT). Its performance in real and existing
environments is assessed using accurate ray-tracing tools and
antenna models. In the best simulated scenario, 1.6 kbits/s/Hz of
spectral efficiency is attained, corresponding to 80% of Singular
Value Decomposition performance, with a transmitter and a
receiver that are 200 and 10,000 times less complex, respectively.

spectral efficiencies of hundreds of bits/s/Hz. In this paper, for
the first time, we evaluate MMIMMO deployments in real and
existing environments, using ray-tracing tools that accurately
model the scattering. Antenna radiation patterns are also
accurately modeled and practical deployment considerations
not exactly fulfilling (1) are also taken into account.
Legend:
L

ULA
D

Transmit ULA with N elements

Receive ULA with N elements

Fig. 1. Communication between two ULAs in LOS.
Transmit
ULA

Receive
ULA

Transmit
ULA

Receive
ULA

Keywords—5G, high carrier frequency, millimeter wave,
Massive MIMO, short range, Line-Of-Sight MIMO

I.

INTRODUCTION

Due to the availability of large spectrum at higher carrier
frequencies, the spectrum bands corresponding to millimeter
waves are good candidates for the self-backhauling of the
future 5th generation (5G) of mobile networks [1]. Their
coverage limitation could be overcome through a dense
deployment. In this paper, we propose to boost the spectral
efficiency of millimeter wave based backhaul links through a
new type of deployment. In theory [2-5], two uniform linear
arrays (ULAs) of 𝑁 antenna elements and of equal length and
parallel to each other, communicating through a line of sight
(LOS) multiple input multiple output (MIMO) propagation
channel (as illustrated in Fig. 1), can multiplex 𝑁 data streams
in the spatial domain, under the following conditions:
(1)
𝐿2
= 𝑁;
𝜆𝐷
(2)
𝐷 ≫ 𝐿,
where 𝜆 = 𝑐/𝑓 is the wavelength (𝑐 and 𝑓 being the speed of
light and the carrier frequency, respectively), 𝐷 is the distance
between the ULAs, 𝐿 is the ULAs’ length (as illustrated in Fig.
1.). According to [5], conditions (1) and (2) guarantee that the
MIMO channel matrix has 𝑁 equal eigenvalues. Recently, [6]
has shown that 𝑁 can reach values as high as several hundreds
of antenna elements, if 𝜆 corresponds to 5G (candidate) high
carrier frequencies and if 𝐷 and 𝐿 are chosen smartly. These
new types of deployments, that we call “massive multiple input
massive multiple output” (MMIMMO), could deliver gigantic

a) DFT over the entire
array

b) DFT per block (two
blocks example)

Fig. 2. 16×16 MIMO system mapping 16 streams into 16 angles.

In this paper, we also propose a new practical signal
processing scheme for these new MMIMMO deployments. As
applying singular value decomposition (SVD) to a MMIMMO
system is too complex, we propose to re-use a practical low
complexity spatial multiplexing (SM) scheme that combines
discrete Fourier transform (DFT) and maximum ratio
transmission (MRT) precoding, called DFT-SM-MRT [7]. Fig.
2-a) illustrates the use of DFT-SM alone (without MRT) with
two ULAs parallel to each other and in LOS. In this case, data
streams are mapped into angles. In DFT-SM-MRT, the role of
MRT is to mitigate the effect of scattering and to deal with
cases where the ULAs might not be perfectly parallel. However,
DFT-SM-MRT [7] still suffers from residual interference,
especially when condition (2) is not met. In this paper, we
present a new low complexity scheme called block DFT-SMMRT (B-DFT-SM-MRT), with a similar complexity as the one
of DFT-SM-MRT. As illustrated in Fig. 2, it applies the DFT
per block. The main idea of this scheme is to approximately
fulfill condition (2) on a per-block basis.
The outline of the paper is as follows. Section II defines a
set of practical MMIMMO links that could be deployed in
existing environments, and their corresponding antenna and
propagation models. Section III presents the novel B-DFT-SM-

MRT scheme and recalls the definitions of the DFT-SM-MRT
and SVD schemes. Section IV compares these schemes in
terms of performance and complexity, for all links defined in
Section II. Section V concludes this paper.

A. MMIMMO links in the City Center of Bristol
Legend:
ULA

The following notations are used. 𝐈 (𝑁) is the identity matrix
of size 𝑁. 𝐌 DFT(𝑁) , 𝐌 IDFT(𝑁) ∈ ℂ𝑁×𝑁 are the Butler matrices
of size 𝑁, for the DFT and the IDFT operations, respectively. If
𝐀 ∈ ℂ𝑁×𝑀 , 𝐀∗ is the conjugate of 𝐀 , 𝐀† is the transpose
conjugate of 𝐀, rank(𝐀) is the rank of 𝐀, 𝐀𝑛,𝑚 is the element in
the 𝑛-th row and 𝑚-th column, with 1 ≤ 𝑛 ≤ 𝑁 and 1 ≤ 𝑚 ≤
𝑀. If 𝑥 ∈ ℂ, then |𝑥| is the module 𝑥.
II.

a) College Green Area
(Google maps)

MMIMMO LINKS IN EXISTING ENVIRONMENTS

To assess the performance of MMIMMO links in real and
existing environments, we have built environment-specific
channel models. We consider
𝑓 = 26 ∙ 109 Hz, as it
corresponds to a candidate carrier frequency for 5G in Europe.
We consider a narrowband signal. Such signal can either be
obtained with a narrowband single carrier waveform or a
narrow sub-band of a wideband multi-carrier waveform. With
this assumption, the propagation between the transmit array
and the receive array can be considered as frequency flat and
can be modeled with a complex channel matrix. Let 𝐇 ∈ ℂ𝑁×𝑁
be the MMIMMO propagation channel matrix. We model the
propagation between the 𝑛-th transmit antenna element and the
rays
𝑚-th receive antenna with a finite number of rays 𝑁𝑛,𝑚 that
depends on 𝑛 and 𝑚 . Indeed, different pairs of receive and
transmit antenna, which are very far apart in the arrays, may
see different numbers of scatterers. The 𝑟-th ray (with 1 ≤ 𝑟 ≤
rays
𝑁𝑛,𝑚 ) has a path gain 𝛼𝑛,𝑚,𝑟 ∈ ℂ, a direction of arrival vector
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐃𝐨𝐀 𝑛,𝑚,𝑟 ∈ ℝ𝟑 and a direction of departure vector ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐃𝐨𝐃𝑛,𝑚,𝑟 ∈
ℝ𝟑 . We assume that all transmit and receive antenna elements
𝟑
have the same antenna gain function 𝛤 ∈ ℝℝ , 𝛤 being a
function of the direction of arrival (or departure). With these
notations, the channel coefficient 𝐇𝑛,𝑚 between the receive
antenna n and the transmit antenna m is given by:
rays

𝑁𝑛,𝑚

(3)

⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐇𝑛,𝑚 = ∑ 𝛼𝑛,𝑚,𝑟 𝛤(𝐃𝐨𝐀
𝑛,𝑚,𝑟 )𝛤(𝐃𝐨𝐃𝑛,𝑚,𝑟 ).
𝑟=1

We have used two different ray-tracing tools to obtain the
rays
parameters 𝑁𝑛,𝑚 , 𝛼𝑛,𝑚,𝑟 , ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐀𝐨𝐀 𝑛,𝑚,𝑟 and ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐀𝐨𝐃𝑛,𝑚,𝑟 in two
different environments: an outdoor and an indoor environment,
described in the Sections II-A and II-B, respectively. Each tool
is modelling a real and existing environment in which antenna
elements can be positioned. The aforementioned parameters are
then generated based on the chosen positions of the antenna
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
elements. 𝛤(𝐃𝐨𝐀
𝑛,𝑚,𝑟 ) and 𝛤(𝐃𝐨𝐃𝑛,𝑚,𝑟 ) are determined
based on an accurate antenna model presented in the Section IIC. Finally, the method for the setting of the MMIMMO
parameters (such as the number of antenna elements N) is
given in Section II-D.

c) Link N 1

b) Link N 0

d) Link N 2

Fig. 3. Modeled links for outdoor environment (in Bristol’s City Center).

Fig. 3 a) illustrates the considered outdoor environment.
The chosen outdoor environment is an existing road of the
“College Green Area”, in the city center of Bristol, in the
United Kingdom. The ULAs are assumed to be deployed on
existing lamp-posts. Three different links between lamp-posts,
named “Link N° 0”, “Link N° 1” and “Link N° 2” are
considered and illustrated in Fig. 3 b), c) and d), respectively.
The employed ray tracer identifies the radio wave scatterers
using an accurate geometrical database of the physical
environment [8], [9]. A similar scenario has been adopted in
[10]. Point-source three dimensional (3D) ray-tracing is
performed from each antenna element of the transmit array to
each antenna element of the receive array assuming isotropic
elements. The tool provides the necessary information to
compute the parameters of equation (3) for each ray.
B. MMIMMO links in the Helsinki Airport
As an example of indoor environment, we chose to model
the existing Helsinki airport check-in hall. Again, the used raytracing tool uses an accurate geometrical database of the
physical environment, a so called point cloud model [11]. The
point cloud model includes small objects (e.g. self-check-in
machines) which scatter energy at high carrier frequencies. Our
simulator is calibrated with experimental measurements made
in the Helsinki airport check-in hall [12].

Legend:

scatterer (larger circle means stronger)
distance between ULAs

C. Model of antennas
The radiation patterns at 26 GHz of two different antennas
are generated by simulation: a ‘basic antenna’ (a classical
printed dipole on a ground plane) illustrated in Fig. 6-a) and a
“directional antenna” illustrated in Fig. 6-b). The directional
antenna consists of five units of the basic antenna, separated by
1.5 wavelengths. As a finite number of discrete spatial samples
of these radiation patterns are generated by simulations, an
interpolation between samples is necessary to obtain the exact
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
values of 𝛤(𝐃𝐨𝐀
𝑛,𝑚,𝑟 ) and 𝛤(𝐃𝐨𝐃𝑛,𝑚,𝑟 ) used in Equation (3).

Fig. 4. Point cloud model of the Helsinki Airport Check-In Hall, illustrating
the propagation between one point on the giant screen and one point on
the canopy: the main LOS direction is indicated by the light green
straight line, the scatterers identified by the tool are indicated by the
yellow circles. Larger yellow circle means that the scatterer has a
stronger impact.

L = 25 cm

a) Link N 3 between laptops in Helsinki Airport Check-In
Hall Airport (view of the hall from above)

Fig. 6. Antenna radiation pattern of an antenna element and ULAs.

D. MMIMMO parameters
For each MMIMMO link listed in Section II-A and Section
II-B, we compute the ULAs parameters depending on the
physical structures (lamp posts, signboards etc.) on which the
ULAs are deployed.
Link N 6
b) Links N 4, 5, 6 and 7 in Helsinki Airport Check-In Hall

Fig. 5. Modeled links for indoor environment in Helsinki Airport Check-In
Hall.

As illustrated in Fig. 4, our simulator accurately identifies
the locations and the reflection (or scattering) coefficients of
the scatterers. The information on the scatterers allows us to
derive the parameters of equation (3). Different MMIMMO
links illustrated in Fig. 5 are considered: between nearby
devices (“Link N° 3”); between signboards (“Link N° 4” and
“Link N° 5”); between self-check-in-machines (“Link N° 6”)
and finally between a signboard and a canopy (“Link N° 7”).

Let 𝑁 U and 𝑑 be the number of data streams multiplexed in
the spatial domain and the inter-antenna spacing, respectively.
The number of antenna elements 𝑁 is set equal to 𝑁 U in the
case of the DFT-SM-MRT scheme and the SVD scheme. As it
will later be explained in Section III, 𝑁 potentially very slightly
exceeds 𝑁 U in the case of B-DFT-SM-MRT. Although, from
the ray-tracing tools, we know the exact value of 𝐷 , we
compute the MMIMMO system parameters based on an
̂ (with an arbitrarily small chosen error in the
approximation 𝐷
order of a decimeter), since we assume that in a real
deployment situation, one can only obtain an imperfect
measurement of 𝐷.
We choose 𝑑 and 𝑁 U , where 𝑁 U = 𝑁 (for all cases except
some configurations of B-DFT-SM-MRT), so that condition
(1) is met as much as possible and with the following

additional constraint: 𝑁 U must be a power of 2. Note that this
constraint only applies to 𝑁 U and does not apply to 𝑁 when
𝑁 > 𝑁 U . This latter requirement ensures a low complexity
implementation of the DFT. Two different methods to chose
𝑁 U are tested in this paper.
In the first method (applied to the outdoor links), we
arbitrarily set 𝑁 U = 64 = 26 . In the second method (applied to
the indoor links), we determine the length 𝐿 (in meters) of the
physical structure on which we deploy the ULA. We then
compute the largest 𝑁 U that is deployable within 𝐿 and that is
close to fulfilling condition (1), as follows:
𝑁 U = 2𝐾 and

(4)

2

𝐿
)}.
𝑘∈ℕ
𝜆𝐷
In practice, one cannot position antennas with an infinite
precision. We thus define 𝛿 as the spatial step for the
positioning of antennas. 𝑑 is then determined as follows, for
both methods:
𝐾 = arg {max (2𝑘 ≤

𝛿 1
⌊ √𝜆𝐷𝑁 U ⌋.
𝑁U 𝛿
Table I lists the parameters of the considered MMIMMO links.
̂ /(dND)
Note that for some links, condition (2) is not met (i.e. 𝐷
is not much higher than 1). Compared to DFT-SM-MRT, BDFT-SM-MRT is therefore expected to improve these links.
For the Link N°4, we allow the deployment of antennas 30 cm
above the check-in machine height.
𝑑=

TABLE I. MMIMMO PARAMETERS WITH 𝛿 = 0.1 𝑚𝑚

Link
N°
3
4
5
0, 1
and 2
6
7

̂
𝐷
(m)
0.5
17.4
8.9

𝐿
(m)
0.30
1.55+0.30
3.7

25

NA

0.9
18.7

1.3
8.85

III.

𝑁U
8
16
64
128
256

𝑑
(mm)
26.8
112
45.8

𝑁 U𝑑
(m)
0.2144
1.792
2.9312

67.1

4.2944

9
28.7

1.152
7.3472

̂
𝐷

Fig. 7. Common and scheme specific parameters.

To assess the maximum achievable performance of the
MMIMMO system, we assume that the signal to noise ratio is
very large, and that the system is only limited by the signal to
interference ratio (SIR). The SIR of one data stream can be
derived based on the MIMO equivalent channel matrix 𝐆. The
SIR 𝐬𝐢𝐫𝑛 of each data stream 𝑛 is given by:
2

𝐬𝐢𝐫𝑛 =

|𝐆𝑛,𝑛 |

2
U
∑𝑁
𝑝=1,𝑝≠𝑛|𝐆𝑛,𝑝 |

The theoretical attainable spectral efficiency 𝐜𝑛 for the data
stream number 𝑛 is given by: 𝐜𝑛 = log 2 (1 + 𝐬𝐢𝐫𝑛 ), 1 ≤ 𝑛 ≤
𝑁 U . Practical modulations (such as 256 QAM or QPSK) and
coding schemes have a bounded spectral efficiency. We
therefore define the minimum and maximum spectral
efficiencies, 𝑠 MIN and 𝑠 MAX , accordingly. We define the
practical
spectral
efficiency
𝐜𝑛P
as
follows:
P
MAX )
MAX )
𝐜𝑛 = min(𝐜𝑛 , 𝑠
if min(𝐜𝑛 , 𝑠
> 𝑠 MIN and 𝐜𝑛P = 0
otherwise. The resulting total spectral efficiency 𝑠 is therefore:

D

𝑑𝑁
~2
~10
~3
~6
~13
~3

STUDIED SCHEMES

This section describes the three following spatial
multiplexing schemes: A) DFT-SM-MRT [7] (as the baseline
method); B) B-DFT-SM-MRT (as the new proposed method);
C) SVD spatial multiplexing (as an upper bound). To make a
fair comparison, we impose the following common constraint:
the number of streams 𝑁U and the inter-antenna spacing 𝑑
defined in Section II are common to all schemes. As illustrated
in Fig. 7, only the following parameters can be schemespecific: the number of antenna elements 𝑁 , the spatial
precoder and the spatial decoder. As a consequence, the
propagation channel matrix 𝐇 ∈ ℂ𝑁×𝑁 and the equivalent
channel matrix 𝐆 ∈ ℂ𝑁U×𝑁U (that includes precoding,
propagation, and decoding) are also scheme-specific.

, 1 ≤ 𝑛 ≤ 𝑁 U.

𝑁U

(5)

𝑠 = ∑ 𝐜𝑛P .
𝑝=1

For each spatial multiplexing scheme, the spectral efficiency is
determined using equation (5), this equation being fed with a
scheme-specific expression of 𝐆.
Next sub-sections provide the expressions of the schemespecific parameters 𝐇 , 𝐆 and 𝑁 . The same transmit power
constraint is assumed for all transmitters.
A. DFT-SM-MRT
For DFT SM-MRT [7], the number of antenna elements is
U
𝑁 = 𝑁 U . As illustrated in Fig. 8, 𝜌𝐇 † 𝐌 IDFT(𝑁 ) is the
U
precoder and 𝐌 DFT(𝑁 ) is the decoder, with 𝜌 being a schemespecific normalising factor to satisfy the power constraint. The
equivalent MIMO channel 𝐆 is thus:
U

U

𝐆 = 𝜌𝐌 DFT(𝑁 ) 𝐇𝐇 † 𝐌 IDFT(𝑁 ) .

(6)

𝐆 = 𝜌𝐑𝐇𝐇 † 𝐓,

Fig. 8. DFT-SM-MRT spatial multiplexing scheme [7].

B. B-DFT-SM-MRT
As illustrated in Fig. 9, compared to DFT-SM-MRT, BDFT-SM-MRT applies the DFT to 𝑁 S blocks of 𝑁 D data
symbols separately, with 𝑁 D = 𝑁 U /𝑁 S . 𝑁 S is selected so that
condition (2) is better fulfilled, at least on a per-block basis, i.e.
̂ /(𝑑𝑁 D ) > 1. We optionally append a cyclic prefix
such that: 𝐷
(CP) [13] of 𝑁 CP symbols in the spatial domain (with 0 ≤
𝑁 CP ≤ 𝑁 D), after each ‘per block’ DFT operation. As symbols
are mapped onto antennas, this has a direct impact on the role
of each antenna. The 𝑁 S blocks of 𝑁 D data symbols are
mapped onto 𝑁 S blocks of 𝑁 D ‘data antennas’. This constitutes
a set of 𝑁 U ‘useful antennas’. 𝑁 S blocks of 𝑁 CP symbols are
mapped onto 𝑁 S blocks of 𝑁 CP ‘CP antennas’. Each block of
𝑁 CP ‘CP antennas’ is inserted between two successive blocks
of 𝑁D ‘data antennas’. Each block of 𝑁 D data streams goes
through an inverse DFT, which is equivalent to a multiplication
D
by 𝐌 IDFT(𝑁 ) . We set 𝑁 E = 𝑁 D + 𝑁 CP . This time, 𝑁 ≥ 𝑁 U
antenna elements (instead of 𝑁 U ) are used at both the
transmitter and receiver sides, with:
𝑁 = 𝑁 S (𝑁 D + 𝑁 CP ) = 𝑁 S 𝑁 E = 𝑁 U + 𝑁 S 𝑁 CP .

(7)

(8)

where 𝜌 is a scheme-specific normalising factor to satisfy the
power constraint. Note that when 𝑁 S = 1 and 𝑁 CP = 0 , BDFT-SM-MRT is identical to DFT-SM-MRT.
C. SVD

Fig. 10. SVD spatial multiplexing scheme.

The number of antennas 𝑁 for this scheme is: 𝑁 = 𝑁 U . As
U
U
illustrated in Fig. 10, 𝐔, 𝐕, 𝚫 ∈ ℂ𝑁 ×𝑁 are matrices obtained
from the singular value decomposition of 𝐇𝐇 † , i.e., such that
U
𝐇𝐇 † = 𝐔𝚫𝐕, with 𝚫 being diagonal, and 𝐕𝐕 † =𝐔𝐔 † = 𝐈 (𝑁 ) .
Let 𝜌 be a scheme-specific normalising factor to satisfy the
power constraint. 𝜌𝐇𝐕 † is the precoder and 𝐔 † is the decoder.
With these notations, the equivalent MIMO channel 𝐆 is:
𝐆 = 𝜌𝚫.
IV.

(9)

PERFORMANCE AND COMPLEXITY EVALUATION

The performance analysis and the complexity analysis are
performed for the MMIMMO links defined in Section II and
the schemes described in Section III. Section IV. A lists the
simulated scenarios. Section IV-B and Section IV-C describe
the spectral efficiency and complexity evaluation methods,
respectively. Finally, Section IV-D provides the results.
A. Simulation scenarios
Table II lists the simulated scenarios and their
corresponding parameters. In this table, and throughout this
paper, the notations * and ** indicate that B-DFT-SM-MRT
without CP and B-DFT-SM-MRT with CP are used,
respectively. The absence of these notations indicates that
DFT-SM-MRT is used.

Fig. 9. B-DFT-SM-MRT spatial ultiplexing scheme.

,

For all scenarios, the channel and antenna models described
in Section II are used to generate H.

 𝐀𝑘,𝑙 = 1 if 𝑙 = 𝑁 + 𝑘 and 1 ≤ 𝑘 ≤ 𝑁 or if 𝑙 = 𝑘 and
𝑁 CP + 1 ≤ 𝑘 ≤ 𝑁 CP + 𝑁 D ; 𝐀(𝑘, 𝑙) = 0 otherwise;

The performance is also evaluated in a free space (FS)
propagation scenario (i.e. a pure LOS scenario). Let 𝛿𝑛,𝑞 be
the distance between the receive antenna element 𝑛 and the
transmit antenna element 𝑞. For FS, 𝐇𝑛,𝑞 is given by: 𝐇𝑛,𝑞 =

We define the matrices 𝐀, 𝐀′ ∈ ℂ𝑁
U
U
𝐓 ∈ ℂ𝑁×𝑁 and 𝐑 ∈ ℂ𝑁 ×𝑁 as follows:

E ×𝑁D

D

, 𝐁, 𝐁′ ∈ ℂ𝑁

D×

CP

 𝐁𝑘,𝑙 = 1 if 𝑙 = 𝑁 CP + 𝑘 and 1 ≤ 𝑘 ≤ 𝑁 D ; 𝐁(𝑘, 𝑙) = 0
otherwise;
 𝐀′ = 𝐀𝐌 IDFT(𝑁

D)

(

𝜆

4π𝛿𝑛,𝑞

)𝑒 −𝑗2𝜋𝛿𝑛,𝑞/𝜆 .

D

and 𝐁′ = 𝐌 DFT(𝑁 ) 𝐁;

 𝐓𝑘+(𝑛−1)𝑁E ,𝑙+(𝑛−1)𝑁D = 𝐀′𝑘,𝑙 , for 1 ≤ 𝑛 ≤ 𝑁 S , 1 ≤ 𝑘 ≤ 𝑁 E
and 1 ≤ 𝑙 ≤ 𝑁 D;
′
 𝐑 𝑘+(𝑛−1)𝑁D ,𝑙+(𝑛−1)𝑁E = 𝐁𝑘,𝑙
, for 1 ≤ 𝑛 ≤ 𝑁 S , 1 ≤ 𝑘 ≤
D
E
𝑁 and 1 ≤ 𝑙 ≤ 𝑁 .

With these definitions, the equivalent channel 𝐆 is given by:

TABLE II. SIMULATED SCENARIOS

N°
3
3*
4
4*

Link

N

𝑁𝑑 (m)

3

8

0.2144

4

16

1.792

𝑁S

𝑁D

1
2
1
2

8
4
16
8

𝑁 CP
0
0

̂
𝐷
D

𝑑𝑁
~2
~5
~10
~19

0

4.2944
64

1

4.2944

2

4.2944

6

7

128

1.152

144

1.296

256

7.3472

264

7.5768

1
4
1
2
1
2
1
2
1
16
16
1
8

64
16
64
32
64
32
64
32
128
8
8
256
32
32

0
0
0
0
0
0
1
0
1

~3
~12
~6
~12
~6
~12
~6
~12
~1
~13
~13
~3
~20
~20

B. Spectral efficiency evaluation methodology
The spectral efficiency 𝑠 is computed using Equation (5)
and the method given in Section III. We set 𝑠 =8 bits/s/Hz
(corresponding to 256-QAM and a coding rate of 1) and
𝑠 MIN = 1 bit/s/Hz (corresponding to QPSK and a coding rate
of 1/2). Note that, for SVD, the spectral efficiency is simply
given by 𝑠 = 𝑁 U 𝑠 MAX . For each scheme, the two following
metrics are computed:
 the ratio 𝜙 SVD between the spectral efficiency of the
considered scheme and the spectral efficiency of SVD;

in Table IV. The larger these metrics, the better they are.
Indeed, the transmitter (respectively the receiver) of the
considered scheme, is 𝜇 TX (respectively 𝜇 RX ) less times
complex than the one of SVD.
TABLE III. SVD SCHEME COMPLEXITY SCALING LAWS (TX= TRANSMITTER,
RX= RECEIVER)

Computations
Tx
SVD

2.9312

Rx
Tx

DFT-SMMRT

5

𝐕,

𝐕 † 𝐱,

Complexity scaling law
3

𝐇† 𝐕† 𝐱

3

𝐌 IDFT(𝑁U) 𝐱

2

𝑂 (𝑁 U log 2 (𝑁 U )).

𝐌 DFT(𝑁U) 𝐳 DFT

Tx

𝐇 † 𝐓. 𝐱

𝑂 ((𝑁 U + 𝑁 S 𝑁 CP ) + 𝑁 U log 2 (

Rx

𝐑𝐳 BDFT

𝑂 (𝑁 U log 2 (

2

B-DFT-SM-MRT

We finally define 𝜇 𝑇𝑋 (and 𝜇 𝑅𝑋 , respectively) as the ratio of
the complexity scaling law of the transmitter (the receiver
respectively) of SVD, over the complexity scaling law of the
transmitter (the receiver respectively) of the considered scheme.
Using the expressions in Table III, we obtain the expressions of
𝜇 𝑇𝑋 and 𝜇 𝑅𝑋 , for the DFT-SM-MRT and the B-DFT-SM-MRT,

𝑁S

𝑁S

))

))

𝜇TX

𝜇RX

log2 (𝑁U )+𝑁U

log2 (𝑁U )

𝑁U +2𝑁U

𝑁U +2𝑁U

2

2

(𝑁U +𝑁S 𝑁CP ) +𝑁U log2 (
𝑁

U

𝑁U

𝑁U

TABLE IV. EXPRESSIONS OF 𝜇TX AND 𝜇RX

The closer to these metrics, the better the schemes are.

We define 𝐱 ∈ ℂ𝑁 ×1 as the vector of transmitted symbols.
U
U
U
S CP
SVD
𝐳
∈ ℂ𝑁 ×1 , 𝐳 DFT ∈ ℂ𝑁 ×1 and 𝐳 BDFT ∈ ℂ𝑁 +𝑁 𝑁 are the
vectors of symbols received at the receive antenna array for the
SVD, the DFT-SM-MRT and the B-DFT-SM-MRT,
respectively. Using these notations, we derive the complexities
scaling laws for the transmitter (taking into account the spatial
precoding only) and the receiver (taking into account the
spatial decoding) and report them in Table III. Our analysis
excludes the MRT block as it appears in all the compared
schemes (SVD included).

2

𝑂 (𝑁 U log 2 (𝑁 U ) + 𝑁 U ) .

𝐇 † 𝐌 IDFT(𝑁U) 𝐱

Scheme
DFT-SM-MRT

We recall that the complexities of the DFT of size 𝑁, of the
SVD of a matrix of size 𝑁 × 𝑁 and of the multiplication of two
matrices of sizes 𝑁 × 𝑀 and 𝑀 × 𝑃, scale with 𝑂(𝑁log 2 (𝑁)),
𝑂(𝑁 3 ) and 𝑂(𝑁𝑀𝑃) , respectively. As a consequence, 𝑁 S
DFTs of complexity that scales with 𝑂((𝑁 U /𝑁 S )log 2 (𝑁 U /
𝑁 S )) each, result in a total complexity that scales with
𝑂(𝑁 U log 2 (𝑁 U /𝑁 S )).

2

𝑂 (𝑁 U + 𝑁 U )

𝐔, 𝐔† 𝐳 SVD

 the ratio 𝜙 FS between the spectral efficiency of the
considered scheme and the spectral efficiency of the same
scheme in a FS environment.

C. Complexity evaluation
We assume a fully digital architecture and we base our
complexity evaluation on [14].

2

𝑂 (𝑁 U + 𝑁 U + 𝑁 U )

Rx

B-DFT-SMMRT

5
5*
0
0*
1
1*
2
2*
6
6*
6**
7
7*
7**

U3

+2𝑁

U2

2

𝑁U
)
𝑁S

𝑁U
)
𝑁S

log2 (
𝑁

U2

+𝑁U

D. Simulation results
Table V provides the simulation results for all scenarios
listed in Section IV-A). DFT-SM-MRT and B-DFT-SM-MRT
both attain spectral efficiencies of several hundreds of bits/s/Hz
(that are close to the ones of SVD) with much less complex
transmitters and receivers. B-DFT-SM-MRT outperforms
DFT-SM-MRT with an even simpler receiver and a slightly
more complex transmitter. For all MMIMMO links, except for
Link N° 7, the performance is close to the FS performance.
This confirms that in the chosen existing environments, the
propagation is dominated by LOS and that simple spatial
multiplexing schemes (such as DFT-SM-MRT or B-DFT-SMMRT) can be used. However, for Link N° 7, the performance is
much lower than the FS one. The scatterers of scenario 7 are
visible on Fig. 4. One can observe that a strong dominating
scatterer is located on the metallic ceiling of the canopy. For
this particular scenario, we replace the basic antennas by
directional antennas (defined in section III-C) oriented along
the main LOS direction. We obtain an improved performance
that is reported in Table VI. In particular, for scenario 7*,
around 1.6 kbits/s/Hz of spectral efficiency is attained,
corresponding to 80% of SVD performance with a transmitter
and a receiver that are 200 and 10000 less complex,
respectively. The CP insertion slightly improves the
performance in scenarios 6** and in scenario 7** (with
directional antennas). An extensive study of the CP insertion is
for further study.

TABLE V. RESULTS († INDICATES DIRECTIONAL ANTENNAS ARE USED)
N°

SE (bits/s/Hz)

𝜙 FS (%)

𝜙 SVD (%)

𝜇 TX

𝜇 RX

0
0*
1
1B*
2B
2B*
3B
3B*
4C
4C*
5D
5D*
6D
6D*
6D**
7
7*
7**

246
258
296
342
194
277
47
52
81
83
349
431
290
318
429
113/1174†
281/1651†
266/1681†

65
62
75
81
64
75
100
100
63
65
91
97
97
99
88
9/94†
16/93†
15/90†

48
50
58
67
38
54
74
81
63
65
68
84
57
31
42
6/57†
14/81†
13/82†

60
61
60
61
60
61

693
832
693
832
693
832

NA

NA

V.

NA

NA

60
62
52
123
127
101
253
238

693
1040
2359
5504
5504
8224
13158
13158

CONCLUSION

In this paper, we showed that there is an opportunity for
future 5G networks operators to exploit the existing urban
architecture to transport, on the wireless media, huge data rates
with gigantic spectral efficiencies. A new precoding/decoding
scheme is proposed, called “block discrete Fourier transform
based spatial multiplexing with maximum ratio transmission”,
B-DFT-SM-MRT, which has a low complexity compared to
singular value decomposition. The performance of this scheme
at 26 GHz is assessed in existing environments that are
accurately modeled with ray-tracing tools. Antennas as well,
are accurately modeled. In the best scenario, 1.6 kbits/s/Hz is
attained, corresponding to 80% of SVD performance, with a
transmitter and a receiver that are 200 and 10000 times less
complex, respectively. Further studies will be conducted with
measured MMIMMO channel data.
ACKNOWLEDGMENTS
This work has been partially funded by the 5G PPP project
mmMAGIC [15] under grant ICT-671650. We warmly thank
Mr Antonio Clemente and Mrs Marie-Hélène Hamon for their
support on this activity.
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