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A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems

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Millimeter wave (mmWave) communication in the 60 GHz band requires large antenna arrays at both the transmit and receive terminals to achieve beamforming gains, in order to counteract the high pathloss. Fully digital techniques are infeasible with large antenna arrays due to hardware constraints at such frequencies, while purely analog solutions suffer severe performance limitations. Hybrid analog/digital beamforming is a promising solution, especially when extended to a multi-user scenario. This paper conveys three main contributions: (i) a Kalman-based formulation for hybrid analog/digital precoding in multi-user environment is proposed, (ii) an analytical expression of the error between the transmitted and estimated data is formulated, so that the Kalman algorithm at the base station (BS) does not require information on the estimated data at the mobile stations (MSs), and instead, relies only on the precoding/combining matrix, (iii) an iterative solution is designed for the hybrid precoding scheme with affordable complexity. Simulation results confirm significant improvement of the proposed approach in terms of both BER and spectral efficiency- achieving almost 7 bps/Hz, at 20 dB with 10 channel paths with respect to the analogonly beamsteering, and almost 1 bps/Hz with respect to the hybrid minimum mean square error (MMSE) precoding under the same conditions.
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A Kalman based Hybrid Precoding for
Multi-User Millimeter Wave MIMO
Systems
ANNA VIZZIELLO1, (Member, IEEE), PIETRO SAVAZZI1, (Senior Member, IEEE), and
KAUSHIK R. CHOWDHURY2, (Senior Member, IEEE)
1Department of Electrical, Computer and Biomedical Engineering, University of Pavia, 27100, Pavia, Italy (e-mail: given name.surname@unipv.it)
2Department of Electrical and Computer Engineering, Northeastern University, Boston MA 02115, USA (e-mail: krc@ece.neu.edu)
Corresponding author: Anna Vizziello (e-mail: anna.vizziello@unipv.it).
This work was supported in part by the U.S. Office of Naval Research under grant number N00014-16-1-2651.
ABSTRACT Millimeter wave (mmWave) communication in the 60 GHz band requires large antenna arrays
at both the transmit and receive terminals to achieve beamforming gains, in order to counteract the severe
pathloss. Fully digital techniques are infeasible with large antenna arrays due to hardware constraints at
such frequencies, while pure analog solutions have severe performance limitations. Hybrid analog/digital
beamforming is a promising solution, especially when extended to a multi-user scenario. This paper makes
three main contributions: (i) a Kalman-based formulation for hybrid analog/digital precoding in multi-user
environment is proposed, (ii) a novel expression of the error between the transmitted and estimated data
is formulated, so that the Kalman algorithm at the base station (BS) does not require any detail on the
estimated data at the mobile stations (MSs), but only the precoding/combining matrix, (iii) an iterative
solution is designed for the hybrid precoding scheme with affordable complexity. Simulation results confirm
a significant improvement of the proposed approach with respect to the existing solutions, in terms of both
BER and spectral efficiency. As an example, almost 7 bps/Hz, at 20 dB with 10 channel paths, with respect
to the analog-only beamsteering, and almost one bps/Hz with respect to the hybrid minimum mean square
error (MMSE) precoding, in the same conditions.
INDEX TERMS Hybrid beamforming, Kalman filter, Millimiter Wave, massive MIMO
I. INTRODUCTION
Millimeter wave (mmWave) communication is a key en-
abling technology for solving the spectrum crunch in future
5G systems [1]–[7]. Due to limited available spectrum in
the sub-6 GHz band, conventional cellular and WiFi-based
solutions cannot be scaled up to meet the ever-growing data
demands of network densification, and emerging applications
associated with data centers and mobile devices. While in-
novative solutions such as utilizing licensed spectrum on an
opportunistic basis have been proposed [8], such approaches
are still subjected to frequent disruption and are limited
by the channel bandwidth available in the licensed bands,
such as the TV bands. Millimeter wave (mmWave) band
communication in the recently opened up contiguous block
of unlicensed spectrum in the 57-71GHz range is an opportu-
nity for achieving gigabit-per-second data rates [9]. Indeed,
existing standards like the IEEE 802.11ad operating in these
bands allow up to 2GHz wide channels for short-distance
communications.
MmWave challenges: Due to the high path loss that occurs
in the mmWave bands, directional beamforming exploiting
large antenna arrays at both base station (BS) and mobile
stations (MSs) is required [10]–[13]. This enables high qual-
ity and long-distance communication links, and increases
the signal power concentrated at the receiver end. The high
frequency of operation also supports massive multi-antenna
architectures from a design viewpoint, while reducing the
dimensions of each antenna and allowing many of them to
be packed in a small area.
While the hardware support from large antenna arrays
for beamforming functions is available today, the high fre-
quency of operation, expected sampling rates and channel
bandwidths make it difficult to deploy traditional fully digital
beamforming solutions [10], [14]. Thus, analog beamform-
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Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
FIGURE 1. MmWave multi-user system model: BS hybrid analog/digital
precoding and MS analog combining.
ing techniques is a possible alternative in mmWave systems,
such as [15]–[19], which coordinate the signal phases of the
antennas through analog phase shifters.
However, analog solutions cannot use adaptive gain con-
trol. Moreover, phase shifters can be digitally controlled with
only quantized phases [12], thus limiting the possibility of
advanced processing and resulting in poor performance. For
example, our experiments prove that analog beamforming
may achieve at most a spectral efficiency around 3bps/Hz at
20 dB, while our proposed hybrid analog/digital beamform-
ing gets 10 bps/Hz under the same conditions.
In summary, hybrid schemes are promising candidate
solutions that overcome the limitations of pure digital or
analog beamforming, as they incorporate the advantages of
both methods [20], [21]. Hybrid schemes reduce the training
overhead compared to analog-only architectures by lever-
aging multiple simultaneous beam transmissions. In hybrid
solutions, the number of RF chains may be much lower than
the number of antennas [1], thus reducing the complexity
compared to fully digital solutions. It also allows more
freedom that classical analog beamforming by dividing the
multiple input multiple output (MIMO) optimization process
between analog and digital domains. An advantage of the
hybrid approach is that the digital precoder/combiner can
correct for lack of precision in the analog, for example to
cancel residual multi-stream interference [1]. In particular,
hybrid precoding/combining signals for a single-user has
been analyzed for mmWave systems [10], [22]–[25], showing
that hybrid designs are able to achieve similar performance
compared to fully digital solutions. As an example, the
hybrid solution [10] decomposes the optimal precoding and
combining matrix via an orthogonal matching pursuit, where
the transmit and receive response vectors represent the basis
vectors. However, to sustain many simultaneously active BS-
MS links, suitable for multi-user mmWave systems where
BS and MSs form multiple beams, new solutions need to be
developed for inter-user interference reduction.
Current hybrid multi-user mmWave solutions: Multi-
user solutions for sub-6GHz channels [26], [27], cannot be
directly applied since they do not consider hardware con-
straints and the specific mmWave channel features. Further-
more, given the difficulty in processing samples timely in
these bands, low-complexity solutions must be emphasized.
Some works specific for hybrid multi-user mmWave sys-
tems have been proposed, such as [12], [28]–[34]. The au-
thors in [12] propose a two-stage hybrid precoding scheme.
First, as in the single-user scheme, the BS and the MS jointly
select a ‘best’ combination of radio frequency (RF) beam-
former and RF combiner in order to maximize the channel
gain to that specific MS. Then, a zero-forcing (ZF) baseband
precoding algorithm is applied at the BS by inverting the
effective channel, in order to reduce the interference between
the users. Also [28] uses a digital ZF baseband precoder.
While [12] requires explicit channel state information (CSI)
feedback from users, [28] develops a non-feedback non-
iterative channel estimation. Specifically, the strongest angle
of arrivals (AoAs) at both BS and users are estimated, which
are exploited for analog beamforming at BS and MSs. Then,
the MSs send orthogonal pilot symbols to the BS along
the strongest AoA directions to ease the equivalent channel
estimation, used in the BS digital ZF precoder.
[29] proposes to first calculate the RF combiner for each
MS independently, and then designs the RF and baseband
precoder at the BS for all the MSs jointly. The analog/digital
precoder is developed by minimizing the mean-squared error
(MSE) of the data streams received at the MSs. Also other
solutions include minimum mean square error (MMSE) as
part of the approach. In particular, an iterative algorithm
for joint precoding and combining is proposed in [30]. At
the initial step, the analog precoder/combiner is selected
through an orthogonal matching pursuit (OMP) algorithm to
enhance the channel gain, while mitigating the multi user
interference, and the digital combiner is obtained via MMSE
criterion. Then, the iterative procedure is applied to improve
the performance. In [31], the authors first develop the digital
precoder/combiner leveraging minimum sum-mean-square-
error (min-SMSE) criterion to minimize the BER, and then
design an over-sampling codebook (OSC) based analog pre-
coder/combiner scheme to further reduce the SMSE. The
solution in [32] handles the inter-user interference at both
analog and digital beamforming: the analog beamforming
matrix is calculated through the low complexity Gram-
Schmidt algorithm and the digital one is obtained by MMSE
method with a low dimensional effective channel.
Block diagonalization solutions include methods such as
[33], [34]. Specifically, in [33] phase-only RF precoding
and combining is performed exploiting the large array gain,
and then the block diagonalization method is applied at the
equivalent baseband channel. The algorithm [34] designs an
hybrid beamforming in two steps: first it finds an equivalent
baseband channel by maximizing its capacity through the
analog RF processing, then the interference among the users
is deleted by the block diagonalization procedure.
However, prior works [12], [28]–[30], [32]–[34] have de-
veloped algorithms to maximize the spectral efficiency rather
than to minimize the bit error rate (BER) performance. Few
2VOLUME 4, 2016
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
BER optimization solutions include works such as [31], [35].
To the best of our knowledge, this is the first work that
designs a hybrid precoder for mmWave multiuser massive
MIMO based on Kalman criterion to reduce inter-user inter-
ference, evaluating both spectral and BER performance.
Proposed Approach: We propose an iterative Kalman-
based multi-user hybrid solution that minimizes the error be-
tween the preamble transmitted by the BS and the estimated
received data at the MS.
We mathematically define the error formulation as a func-
tion of only the precoding, combining and channel matrices.
In this way the algorithm does not require any explicit data
estimation. Then, a two step procedure is carried out: first,
the RF analog precoding/combining step is performed as in
single-user systems based on energy maximum principle;
then an iterative Kalman-based approach is applied to es-
timate the digital baseband precoder at the BS in order to
reduce inter-user interference.
Contributions: The main contributions of this work are:
1) we devise a Kalman-based hybrid precoding/combining
scheme for the discovery phase in multi-user mmWave
massive MIMO systems, where the precoding base-
band matrix is considered as the state matrix in the
Kalman formulation;
2) we define the error between transmitted and estimated
data as a function only of the precoding, combining and
channel matrices, so that data estimation is not needed;
3) we show comparative simulation results both in terms
of spectral efficiency and BER performance, which
confirm that the proposed approach performs better
than other existing hybrid solutions.
The paper is organized as follows: Sec. II describes the sys-
tem model, Sec. III details the proposed Kalman formulation.
Sec. IV presents the proposed Kalman hybrid analog/digital
precoding algorithm, and Sec. V gives the simulations re-
sults. Finally, some concluding remarks are given in Sec. VI.
II. SYSTEM MODEL
For ease of explanation, we list the following notations that
are used throughout the paper: Ais a matrix, ais a vector,
ais a scalar, and Ais a set. kAkFis the Frobenius norm of
A, whereas AT,AH,A1are its transpose, Hermitian, and
inverse respectively. Iis the identity matrix, and N(m,R)
is a complex Gaussian random vector with mean mand
covariance R.E[·]is used to denote expectation.
Differently from fully digital schemes, the number of RF
chains may be much lower than the number of antennas
in hybrid solutions [1], thus reducing the signal processing
complexity and the energy consumption of RF chains. The
network architecture is a mmWave-based massive MIMO
cellular system where the BS is sending Nbstreams through
NBS antennas and NtRF chains for serving Mmobile sta-
tions (MS), each with NMS antennas and one RF chain, with
Nb< Nt< NBS [12] [29]. Without loss of generality, we
assume a simpler configuration of only one RF chain at the
MS, similar to [12], [29], [35]. This assumption is justified
FIGURE 2. MmWave multi-user system with hybrid analog/digital precoding
and analog combining.
since the implementation of user devices is influenced by
the need for low complexity, cost, and power consumption.
On the other hand, the BS may have more sophisticated
digital signal processing (DSP) capability to support multiple
concurrent data streams.
The BS communicates with each MS via one stream, so
that the total number of streams is Nb=Mand the maxi-
mum number of user Mthat can be served simultaneously by
the BS is equal to the number of RF chains at the BS, that is
MNt.Note that this architectural assumption is possible
through the hybrid scheme, that allows spatial multiplexing
and multi-user MIMO. In this way the BS may communicate
simultaneously with multiple MSs through multiple beams
[1].
At the downlink, the BS sends a synchronization message
applying both the baseband precoder FBB , with size Nt×Nb,
and the analog precoder FRF , with size NBS ×Nt, so that
the sampled transmitted signal is:
x=FRF FBB s(1)
where sis the Nb×1transmitted symbol vector, such that
E[ssH] = P
MIM, being Pthe transmitted power and Nb=
M. We assume that Pis equally allocated among different
users’ streams.
As highlighted in [20], following the same assumptions of
[10], [36]–[39], for simplicity we adopt a narrowband block-
fading channel.
Thus, the received signal at MS-mis:
rm=HmFRF FBB s+nm(2)
where Hmis the NMS ×NBS matrix of the mmWave
channel between the BS and the MS-m, and nm
N(0, σ2I)is the Gaussian noise vector.
The received signal rmin (2) can be rewritten showing the
desired contribution and the interference as follows:
rm=HmFRF fBBmsm+Hm
M
X
j6=m
FRF fBBjsj+nm(3)
where FRF fBBmis the BS precoding vector for MS-m,
fBBmis the column mof the matrix FBB . and sjis the jth
element of s.
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Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
Since the MS employ only RF analog combining wRFm=
wm, after the combining process, the estimated symbol of the
MS-mcan be expressed as:
ˆsm=wH
mHmFRF FBB s+wH
mnm(4)
where (·)Hrepresents the conjugate transpose.
The estimated signal ˆsmin (4) can be written showing the
desired contribution and the interference ones:
ˆsm=wH
mHmFRF fBBmsm+wH
mHm
M
X
j6=m
FRF fBBjsj+wH
mnm
(5)
On the uplink, the signal model is similar to (4) but the role
of the precoders and combiners are exchanged, and we can
replace Hmwith HT
m, where (·)Trepresents the transpose,
using the principle of channel reciprocity [12].
Since mmWave channels show limited scattering [10], we
use a geometric channel model with Lmscatterers for the m-
th mobile station. Assuming that each scatterer contributes
to a single propagation path between the BS and the MS-m
[10], the channel Hmin (4) is expressed as
Hm=rNBS NM S
Lm
Lm
X
l=1
αm,laM S (θm,l )aH
BS (φm,l )(6)
where αm,l is the complex gain of the l-th path includ-
ing the path loss between the BS and the MS-m, with
E[|αm,l|2] = ¯α. The variables θm,l and φm,l [0,2π]are
the l-th path’s angles of arrival and departure (AoAs/AoDs)
respectively. aBS (φm,l)and aM S (θm,l )are the antenna array
response vectors at the BS and MS-m, respectively.
If uniform linear arrays (ULA) are assumed at BS and
MSs, aBS (φm,l )can be expressed as:
aBS (φ) = 1
NBS h1,expj2π
λdsin(φ),··· ,expj(NBS 1) 2π
λdsin(φ)i
(7)
where λis the signal wavelength and dis the distance
between antenna elements, and the array response vectors at
the MS-maMS (θm,l )can be written in a similar way.
III. KALMAN FORMULATION FOR
BEAMFORMING/COMBINING
Kalman filter [40], [41] is a powerful tool that has been
exploited for several physical layer applications, such as car-
rier frequency synchronization [42], [43] and phase recovery
[44], [45].
We next formulate the beamforming/combining problem
using a Kalman filter-based approach, and then we propose a
Kalman-based hybrid precoding solution. The system archi-
tecture consists of a hybrid analog/digital precoder at the BS,
and simple MS devices with RF analog combining only.
The BS sends the preamble message s, and the estimated
signal ˆ
s= [ˆs1, ..., ˆsM]Tat the MS represents the observation
vector, whose element ˆsmat MS-mcan be defined at the
iteration nas
ˆsm(n) = (wH
mHmFRF FBB )s(n) + nm(n)(8)
where s(n)is the training vector transmitted from the BS
as expressed in (1), and wH
mHmFRF .
=hH
mrepresents the
effective downlink channel from the BS to MS-m, where
He= [h1, ..., hM]His the effective channel matrix and
represents the observer matrix.
The Kalman filter (KF) algorithm minimizes the sum-MSE
E{ksˆ
sk2}of the training vector, defined as the squared
difference between the signal stransmitted by the BS on
different beams as expressed in (1), and the estimated signal
ˆ
sat the MSs, that is the collection of all the estimates ˆsm
represented in (8). The error e(n)at the n-th Kalman iteration
is thus formulated as
e(n) = s(n)ˆ
s(n)
ks(n)ˆ
s(n)k2
F
(9)
We consider the baseband precoding matrix FBB as the
Kalman filter state, while the analog precoder FRF is com-
puted in the previous step of the algorithm 1 as detailed in
Sec. IV-B.
The proposed Kalman state equation is given as follows:
FBB (n|n) = FBB (n|n1) + K(n)E{diag[e(n)]}(10)
where K(n)represents the Kalman gains, and
diag {s(n)ˆ
s(n)}is the matrix representation of the error
defined in (9), whose mean is computed using (8), and the
effective channel representation He, so that
E{diag[e(n)]}Iˆ
HeFBB (n|n1)
kIˆ
HeFBB (n|n1)k2
F
(11)
where ˆ
Heis estimated as specified in Algorithm 1. Substi-
tuting (11) in (10) we obtain:
FBB (n|n) = FBB (n|n1)+K(n)Iˆ
HeFBB (n|n1)
kIˆ
HeFBB (n|n1)k2
F
(12)
In this way, the Kalman algorithm can be divided in the
steps detailed in Sec. IV-C. The error in eq. (11) is normalized
with respect its Frobenius norm.
Note that the observation vector ˆ
s(n)is needed for the
formulation of the procedure, while, as detailed in the final
formula (14) and (15)-(17) in Sec. IV, the algorithm at the
BS does not require any details on the estimated data at the
MSs, but only the precoding/combining matrices.
IV. KALMAN-BASED HYBRID PRECODING
In the hybrid multi-user system, we must compute the analog
combining wmmatrix for each mobile station and the hybrid
analog and digital precoding FRF and FBB matrices at the
BS.
4VOLUME 4, 2016
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
A. HYBRID PRECODING OPTIMIZATION FORMULATION
We now aim to design the hybrid mmWave precoding matrix
through the Kalman-based approach by minimizing the error
defined in (9):
minimize
FRF ,FBB
Eksˆ
sk2
subject to kFRF FBB k2
FP
FRF ∈ {f1, ..., fL}
(13)
where s(n)is the training vector stransmitted from the BS
as expressed in (1), at the n-th Kalman iteration, and ˆ
s(n)is
the collection ˆ
sof all the estimates ˆsmat MS-mexpressed in
(5) at iteration n.
The first condition kFRF FBB k2
FPin (13) refers to
power constraint, while the second one is to limit the search
for the columns of the RF precoder within a set of Lbasis
vectors {f1, ..., fL}. These basis vectors can be chosen from
the transmit array response vectors at the angle of departure
(AoD) of the mmWave channel, under the hypothesis of per-
fect AoD knowledge at the transmitter, or from a codebook
Fof quantized RF precoding vectors [10].
Given the error calculation in (11), the minimization prob-
lem (13) becomes
minimize
FRF ,FBB kIHeFBB (n|n1)k2
F
subject to kFRF FBB k2
FP
FRF ∈ {f1, ..., fL}
(14)
The optimization formulation (14) does not involve any
data transmission/estimation s(n)and ˆ
s(n)but only the
precoding/combining matrices, i. e., FBB ,FRF , and the
collection of wmcontained in He, that is the equivalent
channel matrix defined as He= [h1, ..., hM]Hin which
hH
m=wH
mHmFRF represents the effective downlink chan-
nel to MS-min (8).
The problem (14) is nonconvex due to the multiplication of
the variables FRF ,FBB , and wm. However, if we fix FRF
and wm, we can solve the optimization problem and calculate
FBB . Specifically, we first design the RF beamforming and
combining matrices (FRF ,wmm) in Sec. IV-B, and then
we compute the digital precoding FBB through the iterative
Kalman procedure in Sec. IV-C.
B. RF PRECODING AND COMBINING MATRIX
We determine first the RF beamforming/combining matrices
for each BS-MS link independently, similarly to [12], and
then continue with the baseband precoding to reduce the
multi-user interference.
In the first step, the BS and each MS-mcalculate the
RF beamforming and combining vectors, fRFmand wm, by
maximizing the signal power for the MS-m(line 3-6in Al-
gorithm 1). Existing single-user RF beamforming solutions
can be used on this purpose, such as [46], [47], in order to de-
sign the RF beamforming/combining vectors without explicit
channel estimation and maintain a low training overhead.
Once the combining vectors wmare determined for all
MSs, as well as the the analog precoder FRF at the BS, the
digital baseband precoder FBB is computed as follows.
C. ITERATIVE KALMAN BASEBAND PRECODING
At this step, the BS utilizes the effective channels hH
m
.
=
wH
mHmFRF m. Each effective channel vector hH
mhas di-
mension M×1, which is much lower than the original
channel matrix Hmwith size NMS ×NBS [12]. Each MS-m
uses a codebook Hto quantize its effective channel response,
and sends the index of the quantized channel vector to the BS
(line 8-10 in Algorithm 1).
As the last step, the BS designs its Kalman-based digital
precoder FBB based on the quantized channels (line 11-18 in
Algorithm 1). The sparse mmWave channels and the narrow
beamforming ensure that the effective MIMO channel is
well-conditioned [48]. This allows the Kalman-based digital
beamforming approach to achieve near-optimal performance,
as shown in Sec. V.
In particular, we consider FRF with wmcalculated in Sec.
IV-B. Thus, the Kalman algorithm can be incorporated in the
following measurements update equations to compute FBB .
We calculate the conditional mean FBB (n|n)and the
variance R(n|n) = E[FBB (n)F
BB (n)] of the state matrix
FBB (n), the Kalman gains K(n), at time instant n, accord-
ing to (15), (16), and (17) respectively:
FBB (n|n) = FBB (n|n1)+K(n)IHDFB B (n|n1)
kIHDFBB (n|n1)k2
F
(15)
K(n) = R(n|n1)HH
D[HDR(n|n1)HH
D+Qn]1(16)
R(n|n)=[IK(n)HD]R(n|n1) (17)
where HDis set equal to the equivalent channel matrix
estimate ˆ
He, and Qnis the covariance matrix of the noise
n(n). We set Qn= (1/SN R)I, where SNR is the signal
to noise ratio. Although the proposed solution requires some
iterations compared to the ZF [12] and MMSE [29] closed
form equations, it gives better performance in adjusting the
precoding matrix in a hybrid architecture. Moreover, as will
be detailed in Sec. V, the number of needed iterations is
limited to only few trials. Finally, we note that all the matrices
involved in the Kalman formulation, i. e., HD,K,Qn, and
R, have small size M×Mwhere Mis the number of users,
and are independent from the large number of antennas NBS
and NMS of the massive MIMO system.
The pseudo-code of the proposed Kalman-based hybrid
precoding is summarized in the following Algorithm 1.
V. PERFORMANCE EVALUATION
In this section, we evaluate the performance of the proposed
Kalman hybrid analog/digital precoding algorithm.
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Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
Algorithm 1 Kalman based hybrid beamforming
1: Input: BS RF codebook F, MS RF codebook W
2: Output: FBB ,FRF , and wmm= 1, ..., M
3: Step 1 - RF Analog design: Single-user FRF and wm
m
4: BS and MS-mselect ˜
vm,˜
gmmso that
5: ˜
gm,˜
vm= arg max
gm∈W,vm∈F kgH
mHmvmk
6: BS sets FRF = [˜
v1, ..., ˜
vM]and MS-msets wm=
˜
gmm
7: Step 2 - BB Digital design: Multi-user FBB
8: MS-mestimates ˜
hH
m
.
=wH
mHmFRF and quantizes
˜
hmusing a codebook H∀m
9: MS-mcalculate and sends to BS ˆ
hmmwhere
10: ˆ
hm= arg max
ˆ
hm∈H k˜
hH
mˆ
hmk
11: BS sets HD=ˆ
He= [ˆ
h1, ..., ˆ
hM]H
12: At BS: for nNdo
13: (n) = IHDFBB (n|n1)
kIHDFBB (n|n1)k2
F
14: FBB (n|n) = FBB (n|n1) + K(n)(n)
15: K(n) = R(n|n1)HH
D[HDR(n|n1)HH
D+
Qn]1
16: R(n|n)=[IK(n)HD]R(n|n1)
17: Normalize FBB =PFBB
kFRF FBB kF
A. SIMULATION ENVIRONMENT
We consider the system described in Sec. II with a BS
employing an 8×8UPA and associated with 4MSs, each
having a 4×4UPA, unless stated otherwise. Simulations
are performed assuming both single-path channels (Lm= 1
in (6)), and multi-path channels (Lm= 10). The azimuth
AoAs/AoDs are supposed to be uniformly distributed in
[0,2π], the elevation AoAs/AoDs are uniformly distributed
in [π/2, π/2], and perfect channel knowledge is assumed.
The performance of the proposed solution is shown
in terms of the average achievable rates per user,
E1
MPM
m=1 Am, with Am:
A(m) = log2
1 +
P
M
wH
mHmFRF fBBm
2
P
MPn6=m|wH
mHmFRF fBBn|2+σ2
(18)
B. IMPACT OF VARYING SN R
Fig. 3 compares the rate achieved by the proposed hybrid
Kalman precoding algorithm with the one got by the sim-
pler analog beamforming, as well as with the single-user
rate (i. e., when there is no interference due to multi-user
environment), and a fully-digital MSE beamforming. The
figure illustrates the averaged achievable rates versus the
SNR (signal to noise ratio) in multi-path scenario (Lm= 10).
FIGURE 3. Achievable Rate varying SNR in multi-path scenario (Lm= 10) in
the range SN R = [20,20]dB.
FIGURE 4. Comparison among hybrid precoding solutions: Achievable Rate
varying SNR in the range SN R = [20,20]dB with number of antennas
NBS = 256, number of MS antennas NMS = 64, number of users M= 8.
Results show that a classical analog-only solution is not
sufficient, while the achievable rate obtained by the proposed
scheme approaches close to the more complex fully-digital
one. Moreover, the proposed hybrid algorithm is able to
achieve performance closer to the single-user scenario, which
means it is able to reduce the interference from the multi-
user environment. Although sparse mmWave channels and
large number of antennas at BS and MSs help in reducing
multi-user interference, there is still a non-negligible amount
of interference present [12].
Fig. 4 compares several hybrid solutions with the pro-
posed Kalman based precoding scheme. Besides ZF [12] and
MMSE [29], a hybrid block diagonalization (BD) algorithm
[33] and a Sparse Precoding&Combining method [10] are
included. In particular, the work [10] has been initially pro-
posed for SU-MIMO and then extended to MU-MIMO in
[33]. All the algorithms refer to the system architecture in
the multipath scenario described in Sec. II, with multiple
RF chains at the BS and one RF chain per user under the
simulation setting [33]. The proposed Kalman based precod-
ing shows the best performance along with the MMSE and
Hybrid BD methods, while ZF spectral efficiency is lower
due to its failure in multipath environment, as detailed in
the following Fig. 5. Sparse Precoding&Combining method
results in the lowest performance.
Fig. 5 compares the performance of the proposed hybrid
6VOLUME 4, 2016
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
(a) Achievable rate in 1 path scenario
(b) Achievable rate in 5 paths scenario
(c) Achievable rate in 10 paths scenario
FIGURE 5. Achievable Rate for Hybrid Precoding solutions varying SNR and
the number of channel paths.
Kalman solution with other two hybrid schemes (ZF [12]
and MMSE [29]) that consider the same system model, and a
fully-digital MSE solution shown as a benchmark. Note that
all the algorithms consider the same channel estimate.
In particular Fig. 5 shows different simulation scenarios,
from single-path (Fig. 5(a)) to multi-path environment with
a number of specific paths identified for closer scrutiny
(Fig. 5(b)-5(c)). These particular cases are of interest, since
mmWave channels are sparse, which means that only few
paths exist [49]. While for a single-path scenario the three hy-
brid algorithms show similar performance (Fig. 5(a)), almost
the same of the fully-digital one, when increasing the number
of paths, their spectral efficiencies move away from the fully
FIGURE 6. Achievable Rate varying the number of users with SN R = 20dB .
digital curve. However, the proposed hybrid Kalman solution
outperforms both ZF and MMSE schemes (Fig. 5(b)-5(c)). In
particular, ZF is the algorithm whose performance becomes
worse in multipath environment, due to its lack in exploiting
multipath channel gains. These latter scenarios are more
realistic study cases, for recent measurement campaigns in
mmWave bands reveal that the channels in a city environment
can be well approximated with a pre-calculated number of
paths clusters [50].
C. IMPACT OF NUMBER OF USERS
Fig. 6 presents the averaged achievable rates by varying the
number of users for a fixed SN R = 20dB in a multi-path
scenario (Lm= 10). While the ZF procedure decrease its
performance for higher number of users, both the proposed
Kalman and MMSE algorithms show good outcomes with
a similar trend, close to the fully digital bound. Anyway,
the proposed Kalman solution is the algorithm that gives the
best results. This is due to the iterative Kalman procedure
that allows to better adjust the precoding baseband matrix, as
well as the Kalman paramenters, in a hybrid scheme. Note
that only a small number of iterations is required, which we
set as 10. We note that fewer iterations, i.e. 4-5, give similar
results. The distance between the Kalman and fully-digital
curve remains constant when increasing the number of users.
D. IMPACT OF THE NUMBER OF ANTENNAS
Fig. 7 shows the rates with different number of BS and MS
antennas, assuming that NBS =NM S . When the number
of antennas increases, the three hybrid solution show similar
performance.
Given the implementation challenges and energy con-
sumption limits, a practical considerations require higher
number of antennas at the BS than at the MS. For this
purpose, we simulate the scenario with a fixed number of MS
antennas NMS = 16 in Fig. 8, and a fixed number of BS
antennas NBS = 64 in Fig. 9, while varying the number of
antennas at the BS and MS respectively.
Fig. 8 confirms the improvement of the proposed solutions
for higher number of BS antennas (over the simple case of
assuming NBS =NM S ) as shown in Fig. 7.
VOLUME 4, 2016 7
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
FIGURE 7. Achievable Rate varying the number of BS and MS antennas
(N=NBS =NM S ). SN R = 20 dB and 10 paths
FIGURE 8. Achievable Rate varying the number of BS antennas NBS.
Number of MS antennas NMS = 16,S N R = 20 dB ,Lm= 10 paths.
FIGURE 9. Achievable Rate varying the number of MS antennas. Number of
BS antennas NBS = 64,SN R = 20 dB,Lm= 10 paths.
The trend from Fig. 8 also appears in Fig. 9 when varying
the number of MS antennas. However, note that the case of
the number of MS antennas being greater than BS antennas
(NMS = 64,NM S = 256 in Fig. 9) may not be applied in
practical cases.
Finally, Fig. 10, 11 illustrate the effect of the number of
antennas on both the system architecture and the precoding
schemes. We have included all the possible configuration
scenarios, going from the practical setting NBS NM S
to the theoretical case NBS < NM S , while varying the
number of antennas to cover both MIMO and massive MIMO
implementation.
In particular, Fig. 10(a) shows the rates when NMS = 4
(solid lines), and when NBS = 4 (dashed lines), with varying
NBS and NM S respectively. In this way, we derive the insight
that apart from the implementation ease and energy cost, it is
better to have higher NBS than NMS also from the viewpoint
of spectral efficiency. Moreover, in the practical case of NBS
higher than NMS (solid lines) the proposed Kalman solution
outperforms the other two hybrid schemes. In the opposite
scenario of NBS being lower than NM S (dashed lines) the
algorithms show similar, lower performance. This is due to
the fact that increasing the number of antennas at the MS
influences the analog combining, which is less efficient than
the BS hybrid precoding. Moreover, in the simulation study,
the MS combining vectors are assumed the same for the three
algorithms. Thus, the performance of the hybrid algorithms
improves more when increasing the number of BS antennas,
rather then of MS ones, and also the difference among the
algorithms’ performance are more evident.
Similar to the studies in Fig. 10(a), in Fig. 10(b)-11(b) we
increase the fixed number of antennas to 16,64 and 256,
respectively. In particular, Fig. 10(b) shows two simulation
settings, the first one with NMS = 16 (solid lines) and the
second one with NBS = 16 (dashed lines). Similarly Fig.
11(a) has NMS = 64 (solid lines) and NBS = 64 (dashed
lines), and Fig. 11(b) has NMS = 256 (solid lines) and
NBS = 256 (dashed lines).
Comparing Fig. 10(a) and Fig. 10(b), when increasing the
number of fixed antennas from 4(Fig. 10(a)) to 16 (Fig.
10(b)), the gap between solid and dashed lines decreases
and hence this lowers the spectral efficiency gain. This is
simply due to the lower difference between the number of
antennas in this new setting. As an example, let us consider
in Fig. 10(a) the point of the green solid line for NMS = 4
and NBS = 64 that has spectral efficiency equal to 7.5
bps/Hz, which becomes 4.4bps/Hz for the dashed green
line. In this way there is a gain of 3.1bps/Hz for a
difference NBSsolid NBSdashed = 60. In Fig. 10(b) the
gap becomes 97.8=2.2bps/Hz due to the difference
NBSsolid NBSdashed = 48 lower than 60 as in Fig. 10(a).
Anyhow, the better behavior of the proposed Kalman solution
is confirmed also in such scenario.
In Fig. 11(a) both the spectral efficiency gain of having
NBS higher than NM S and the gain of the Kalman algorithm
over the other two hybrid solutions decrease, since the abso-
8VOLUME 4, 2016
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
(a) Achievable rate when MS (or BS) antennas are fixed to 4
(b) Achievable rate when MS (or BS) antennas are fixed to 16
FIGURE 10. Achievable Rate for Hybrid Precoding when the number of BS (or
MS) antennas are fixed to a certain value. SN R = 20dB and 10 paths.
lute value of the difference NBSsolid NBSdashed is lower
than in the previous figures.
Finally, in Fig. 11(b), when NBS (or NMS ) is high, equal
to 256, the position of dashed and solid lines is inverted
compared to the previous Fig. 10(a)-11(a), confirming the
advantage of having an higher number of antennas at the BS
than at the MSs. The trend of the hybrid precoding scheme is
always the same, although the difference performance of the
Kalman solution compared to the other two algorithms is less
evident.
As a practical configuration implies higher number of
antennas at the BS than at MS, as in Fig. 10(a) and 10(b),
we see the performance of the proposed Kalman solution is
much better compared to the other schemes.
E. BER PERFORMANCE EVALUATION
Fig. 12 compares BER performance of different schemes for
QPSK modulation in the simulation setting [35]. Hybrid tree
search method [35] calculates the optimal beam subset via
a tree search, while maximum magnitude (MM) algorithm
[51] selects the beams that allows the maximum received
power for each MS. The proposed Kalman solution shows
the best performance, which however is closed to the MMSE
(a) Achievable rate when MS (or BS) antennas are fixed to 64
(b) Achievable rate when MS (or BS) antennas are fixed to 256
FIGURE 11. Achievable Rate for Hybrid Precoding when the number of BS (or
MS) antennas are fixed to a certain value. SN R = 20dB and 10 paths.
FIGURE 12. BER evaluation with number of BS antennas NBS = 64,
number of MS antennas NMS = 4, number of users M= 4.
and hybrid tree search ones. Poor ZF trend is due to its failure
in multipath scenario as detailed in Fig. 5, while the low MM
performance depends on its inability to accurately choose the
beam subset that has produced performance loss.
VI. CONCLUSION
We proposed a multi-user beamforming solution for
mmWave massive MIMO systems, based on a Kalman for-
mulation and a hybrid precoding designed by minimizing
VOLUME 4, 2016 9
Vizziello et al.: A Kalman based Hybrid Precoding for Multi-User Millimeter Wave MIMO Systems
the error between the transmitted and estimated data. The
method uses a specially designed formulation of the error,
and following this, a two step procedure is carried out to
first calculate the RF precoding/combining matrix, and then
design the digital baseband precoder at the BS.
Simulation results show that the proposed algorithm out-
performs existing solutions, in terms of both spectral ef-
ficiency and BER, due to its ability to better adjust the
precoding matrix in hybrid architectures. In future work,
we will incorporate a channel estimation scheme inn the
Kalman hybrid precoding algorithm, and extend the proposed
solution to mobile scenarios. Moreover, we will focus on
extending the proposed solution to joint precoders/combiners
iterative optimization [52] when user devices employ multi-
ple streams.
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ANNA VIZZIELLO received the Laurea degree
in Electronic Engineering and the Ph.D. degree
in Electronics and Computer Science from the
University of Pavia, Italy, in 2007 and in 2011,
respectively.
She is currently a research fellow in the
Telecommunication & Remote Sensing Labora-
tory at the University of Pavia, Italy. From 2007 to
2009 she also collaborated with European Centre
for Training and Research in Earthquake Engi-
neering (EUCENTRE) working in the Telecommunications and Remote
Sensing group. From 2009 to 2010 she has been a visiting researcher at
Broadband Wireless Networking Lab at Georgia Institute of Technology,
Atlanta, GA, in summer 2009 and 2010 at Universitat Politècnica de
Catalunya, Barcelona, Spain, and in winter 2011 and in summer 2016 at
Northeastern University, Boston MA. Her research interests are Cognitive
Radio Networks, 5G Systems, Intra-body Networks.
PIETRO SAVAZZI received the Laurea degree
in Electronics Engineering and the Ph.D. degree
in Electronics and Computer Science from the
University of Pavia, Italy, in 1995 and in 1999,
respectively.
In 1999, he joined Ericsson Lab Italy, in Milan,
as a system designer, working on broadband mi-
crowave systems. In 2001 he moved to Marconi
Mobile, Genoa, Italy, as a system designer in the
filed of 3G wireless systems. Since 2003 he has
been working at the University of Pavia where he is currently teaching, as
an assistant professor, two courses on signal processing and wireless sensor
networks. His main research interests are in wireless communication and
sensor systems.
PROF. KAUSHIK R. CHOWDHURY received
the PhD degree from the Georgia Institute of Tech-
nology, Atlanta, in 2009.
He is currently Associate Professor and Fac-
ulty Fellow in the Electrical and Computer En-
gineering Department at Northeastern University,
Boston, MA. He was awarded the Presidential
Early Career Award for Scientists and Engineers
(PECASE) in Jan. 2017, the DARPA Young Fac-
ulty Award in 2017, the Office of Naval Research
Director of Research Early Career Award in 2016, and the NSF CAREER
award in 2015. He received multiple best paper awards, including the IEEE
INFOCOM 2018, the IEEE ICC conference, in 2009, ’12 and ’13, and
ICNC conference in 2013. He is presently a co-director of the Platforms
for Advanced Wireless Research project office, a joint $ 100 million public-
private investment partnership between the US NSF and wireless industry
consortium to create city-scale testing platforms.
VOLUME 4, 2016 11
... During the training period, the UE uplink transmit power is set at 30 dB m. From Figure 4, it is visualized that the DLIRL beamforming has achieved better spectral efficiency as compared to the existing conventional beamformers in [19]. As seen from the curves, the spectral efficiency with analogue beamforming is found to be around 2 bps/Hz, and close convergence is observed between ZF hybrid precoding, MMSE hybrid precoding, and Kalman hybrid precoding techniques. ...
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In this paper, a deep learning integrated reinforcement learning (DLIRL) algorithm is proposed for comprehending intelligent beamsteering in Beyond Fifth Generation (B5G) networks. The smart base station in B5G networks aims to steer the beam towards appropriate user equipment based on the acquaintance of isotropic transmissions. The foremost methodology is to optimize beam direction through reinforcement learning that delivers significant improvement in signal to noise ratio (SNR). This includes alternate path finding during path obstruction and steering the beam appropriately between the smart base station and user equipment. The DLIRL is realized through supervised learning with deep neural networks and deep Q‐learning schemes. The proposed algorithm comprises of an online learning phase for training the weights and a working phase for carrying out the prediction. Results confirm that the performance of the B5G system is improved considerably as compared to its counterparts with a spectral efficiency of 11 bps/Hz at SNR = 10 dB for a bit error rate performance of 10⁻⁵. As compared to reinforced learning and deep neural network with a deviation of ±3o and ±5°, respectively, the DLIRL beamforming displays a deviation of ±2o. Moreover, the DLIRL can track the user equipment and steer the beam in its direction with an accuracy of 92%.
... Thus, the desired precoding matrix cab be obtained by multiplying vector t with β CRZFH H .Thus, it is clear that the computationally complex matrix inversion can be achieved through iterative methods. 8 [Nr, Nt] = size(P) 9 a = [1 + sqrt(Nr/Nt)] 2 − 1; ω = 2/(1 + sqrt((1 − a 2 ))); relaxation parameter 10 Symbol = eye(Nr, Nr) 11 s0 = zeros(Nr, length(Symbol)) 12 i_iter = 1 13 s2 = s0; 14 In this paper, as a part of hybrid precoder design, analog RF precoder (F r f ) is designed first and low dimensional digital precoder (SSOR-ZF) is implemented based on the effective channel matrix (H). Step 3 is for generating the CRZF filtering matrix and steps 5 to 19 are for the SSOR process to obtain the desired SSOR-CRZF precoder (F BB ). ...
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In this paper, the performance of the millimeter-wave (mmWave) massive multiple-input multiple-output (mMIMO) non-orthogonal multiple access (NOMA) systems is investigated under multiple user scenarios. The performance of the system has been analyzed in terms of spectral efficiency (SE), energy efficiency (EE), and computational complexity. In the case of the mMIMO system, the linear precoder with matrix inversion becomes less efficient due to its high computational complexity. Therefore, the design of a low-complex hybrid precoder (HP) is the main aim of this paper. Here, the authors have proposed a symmetric successive over-relaxation (SSOR) complex regularized zero-forcing (CRZF) linear precoder. Through simulation, this paper demonstrates that the proposed SSOR-CRZF-HP performs better than the conventional linear precoder with reduced complexity.
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