CNN-based Precoder and Combiner Design in mmWave MIMO Systems

Abstract

Hybrid beamformer design is a crucial stage in millimeter wave (mmWave) MIMO systems. In this work, we propose a convolutional neural network (CNN) framework for the joint design of precoder and combiners. The proposed network accepts the input of channel matrix and gives the output of analog and baseband beamformers. Previous works are usually based on the knowledge of steering vectors of array responses which is not always accurately available in practice. The proposed CNN framework does not require such a knowledge and it provides higher performance in capacity as compared to the conventional greedy-and optimization-based algorithms.

Full text

1
CNN-based Precoder and Combiner Design in
mmWave MIMO Systems
Ahmet M. Elbir
Abstract—Hybrid beamformer design is a crucial stage in
millimeter wave (mmWave) MIMO systems. In this work, we
propose a convolutional neural network (CNN) framework for the
joint design of precoder and combiners. The proposed network
accepts the input of channel matrix and gives the output of analog
and baseband beamformers. Previous works are usually based
on the knowledge of steering vectors of array responses which is
not always accurately available in practice. The proposed CNN
framework does not require such a knowledge and it provides
higher performance in capacity as compared to the conventional
greedy- and optimization-based algorithms.
Index Terms—mmWave, MIMO, Hybrid beamforming, Deep
learning, Convolutional neural network.
I. INTRODUCTION
Hybrid beamforming is a promising architecture to be
used in next generation millimeter wave (mmWave) MIMO
(Multiple Input Multiple Output) systems where robust beam-
forming performance is provided with smaller cost and less
number of fully-digital beamformers [1]–[4]. Several methods
are proposed to design the hybrid beamformers [3]–[7]. In
[6], a greedy-based approach, orthogonal matching pursuit
(OMP), is proposed where the analog precoder and combiners
are selected from a dictionary of transmit and receive array
responses. This algorithm requires the knowledge of the user
direction-of-arrival/aperture (DOA/DOD) angles to construct
such a dictionary. Using the connection between the optimum
and the hybrid beamformers, [7] proposes an alternating
minimization approach to estimate the analog and baseband
beamformers based on phase extraction.
The above works provides optimization-based and greedy-
based solutions for hybrid beamforming problem. However
achieving the optimum solution and the computation time
are the main drawbacks of the above techniques. In order
to circumvent this issue, we consider deep learning (DL)-
based techniques for the hybrid beamforming problem [8]. DL
has several advantages such as low computational complex-
ity when solving optimization-based or combinatorial/greedy
search problems and the ability to extrapolate new features
from a limited set of features contained in a training set
[9]. A great deal of attention is received for DL-based tech-
niques in communications society for the problems such as
channel estimation [10]–[12] DOA estimation [12], antenna
selection [13], and analog beam selection [14]. An end-to-
end communication scenario is modeled in [15] and [16] by
using auto-encoders where single-input-single-output (SISO)
systems are considered. [17] also uses auto-encoders for the
A.M.E. is with the Department of Electrical and Electronics Engineering,
Duzce University, Duzce, Turkey. E-mail: ahmetmelbir@gmail.com.
channel state information (CSI) feedback problem. In [14], a
sub-optimum method is proposed based on the support vector
machines (SVMs) for analog beamforming vector selection.
Very recently, a DL based hybrid beamforming is considered
in [18] where only precoder design is considered whereas
joint precoder and combiner design is used in massive MIMO
system where the beamforming is required in both end of
the communication [6]. The proposed network architecture
in [18] is based on multi-layer perceptrons which do not
effectively extract the hidden features inherit in the input data
[9], [13]. In order to achieve feature extraction and obtain
better performance, we propose a convolutional neural network
(CNN) framework for mmWave massive MIMO systems.
In this study, we propose a CNN-based framework with
two CNNs, each of which is dedicated to estimate the analog
precoders and combiners respectively. The CNNs accept the
input of channel matrix and give the beamformer weights at
the output. In order to train the network we generated different
channel realizations with synthetic noise added to each input
data. A decoupled optimization problem is formulated and
solved to obtain the best beamformers providing the highest
spectral efficiency. Using the best analog beamformers, we
construct the input-output pairs of the network. Since the
best beamformers are the optimum solution of the problem,
the proposed CNN framwork enjoys better spectral efficiency
and less computation time. Furthermore, our CNN approach
does not require the knowledge of array responses of users’
DOA/DOD angles which are not always accurately available
in practical scenarios.
II. SI GNA L MOD EL A ND PROBLEM FORMULATION
In this work, we consider a single user mmWave MIMO
communication system with multiple antennas. Let NSbe
the number of data streams to be transmitted from the base
station (BS) with NTtransmit antennas to the user with NR
antennas. The BS is equipped with NRF
Tanalog phase shifters
with analog beamformer FRF ∈CNT×NRF
Tand baseband
beamformer FBB ∈CNRF
T×NS. Then the transmitted signal
becomes x=FRF FBB swhere s∈CNSis the symbol
vector desired to be transmitted and E{ssH}=INS/NS.
The analog beamformers are unitary matrices with equal-norm
elements, i.e., [[FRF ]:,i[FRF ]H
:,i]i,i = 1/NTand we have power
constraint on the transmitter as ||FRF FBB ||F=NS. We can
write the received signal at the NRantennas for a narrowband
block-fading channel as
y=√ρHFRF FBB s+n,(1)
where y∈CNRand ρis the average received power.
n∈CNRdenotes the additive white Gaussian noise (AWGN)

2
with n∼ CN(0, σ2
nINR)and H∈CNR×NTis the channel
matrix with E{||H||F}=NRNT. In mmWave transmission,
the channel can be represented by Saleh-Valenzuela (SV)
model [19] where the clustered channel model is used as the
contribution of Ncclusters of Nray paths as
H=γ
Nc
X
i=1
Nray
X
j=1
αij gR(Θ(ij)
R)gT(Θ(ij)
T)aR(Θ(ij)
R)aH
T(Θ(ij)
T),
where Θ(ij)
R= (φ(ij)
R, θ(ij)
R)and Θ(ij)
T= (φ(ij)
T, θ(ij)
T)
respectively denote the angle of arrivals and angle of
departures. We denote the angular parameters φand θ
as the azimuth and the elevation angles respectively.
γ=pNTNR/(NcNray)is the normalization factor and αij
is the complex channel gain associated with the ith scattering
cluster and jth path for i= 1, . . . , Ncand j= 1, . . . , Nray.
gR(Θ(ij)
R)and gT(Θ(ij)
T)are the antenna element gains for
receive and transmit antennas respectively. aR(Θ(ij)
R)and
aT(Θ(ij)
T)are NR×1and NT×1steering vectors representing
the array responses at the receiver and transmitters
respectively. The nth element of the steering vector
aR(Θ(ij)
R)is given as [aR(Θ(ij)
R)]n= exp{−2π
λpT
nr(Θ(ij)
R)},
where pn= [xn, yn, zn]Tis the position of the nth receive
antenna in Cartesian coordinate system and r(Θ(ij)
R) =
[sin(φ(ij)
R) cos(θ(ij)
R),sin(φ(ij)
R) sin(θ(ij)
R),cos(θ(ij)
R)]T. The
transmit side steering vector aT(Θ(ij)
T)can be defined in a
similar way as for aR(Θ(ij)
R). In order to generate the labels
(precoder and combiners) the proposed CNN frameworks
requires the perfect CSI. However, we use imperfect channel
matrices both in the training and testing stages.
The transmitted signal is received and processed by analog
and baseband combiners as ˜
y=WH
BB WH
RF y, i.e.,
˜
y=√ρWH
BB WH
RF HFRF FBB s+WH
BB WH
RF n,(2)
where WRF ∈CNR×NRF
Ris the analog combiner with
the constrained [[WRF ]:,i[WRF ]H
:,i]i,i = 1/NRand WBB ∈
CNRF
R×NSdenotes the baseband combiner matrix. By as-
suming that the Gaussian symbols are transmitted through
the mmWave channel, we can define the spectral efficiency
achieved by the hybrid beamforming [3]–[6] as
RHYB = log2
INS+ρ
NS
Λ−1
nWH
BB WH
RF H
×FRF FBB FH
BB FH
RF HHWRF WBB 
,(3)
where Λ=
nσ2
nWH
BB WH
RF WRF WBB ∈CNS×NSis the covari-
ance matrix of the noise term in (2) after combining.
Hence the aim in this work is to estimate the hybrid
beamformers FRF ,WRF ,FBB and WRF that maximize the
spectral efficiency as in (3) given the channel matrix H.
III. HYBRID BEA MF OR ME R DESIGN
The optimization problem for joint estimation of hybrid
beamformers ˆ
FRF ,ˆ
FBB ,ˆ
WRF ,ˆ
WBB can be stated as follows
argmax
FRF ,FBB ,WRF ,WBB
RHYB
s.t.: FRF ∈ FRF ,||FRF FBB ||2
F=NS,WRF ∈ WRF ,(4)
where FRF and WRF denote the feasible sets of analog beam-
formers which obey the constraints defined for FRF and WRF .
Obtaining the real-time solution to the problem in (4) is im-
practical due to the complexity of several matrix variables. To
cast the problem in (4) more effectively, we first define the sets
FRF and WRF . Note that the analog beamformers FRF ,WRF
are related with the array responses aT(Θ(ij)
T),aR(Θ(ij)
R)
through linear transformation [6]. Hence the feasible RF
beamformer sets can be formed as FRF ={F(1)
RF ,...,F(QF)
RF }
where F(qF)
RF =aT(Θ(ij)
T), i = 1, . . . , Nc, j = 1, . . . , Nray for
qF= 1, . . . , QF.QF=Npath
NRF
Tis the number RF precoder
candidates and Npath =NcNray. The feasible set for RF
combiner is similarly defined as WRF ={W(1)
RF ,...,W(QW)
RF }
where W(qW)
RF =aR(Θ(ij)
R), i = 1, . . . , Nc, j = 1, . . . , Nray
and QW=Npath
NRF
R. Now we can present the joint precoder
and combiner design problem as follows
¯qF,¯qW= argmax
qF,qW
log2
INS+ρ
NS
Λ−1
nWH
BB WH
RF
×HFRF FBB FH
BB FH
RF HHWRF WBB 
,s.t.:
FRF =F(qF)
RF ,WRF =W(qW)
RF ,FBB = (FH
RF FRF )−1FH
RF Fopt,
WBB = (WH
RF ΛWRF )−1(WH
RF ΛWopt),(5)
where ¯qF,¯qWrepresent the selected elements in the feasible
sets. Λis the covariance of the array output in (1) which is
given by Λ=ρ
NSHFRF FBB FH
BB FH
RF HH+σ2
nINRS .Fopt and
Wopt are the optimum baseband beamformers which can be
obtained from the singular value decomposition (SVD) of the
channel matrix. Let U∈CNR×rank(H)and V∈CNT×rank(H)be
the left and the right singular value matrices of Hrespectively,
where the SVD of H∈CNR×NTis H=UΣVHwhere Σ
is rank(H)×rank(H)matrix composed of the singular values
of Hin descending order. By decomposing Σand Vas Σ=
diag{Σ(1),Σ(2) },V= [V(1),V(2) ]where V(1) ∈CNT×NS
and V(2) ∈CNT×NR−NSrespectively, one can readily select
the unconstrained precoder as Fopt =V(1) [6]. Using the
unconstrained beamformer Fopt,Wopt can be computed as [20]
Wopt =1
ρFoptHHHHFopt +NSσ2
n
ρINS−1FoptHHHH.
The solution of (5) requires to visit QFQWnodes which is
computationally prohibitive. In order to reduce the complexity,
(5) is decomposed into two different problems where precoders
(FRF and FBB ) and combiners (WRF and WB B ) are sepa-
rately estimated. By doing so, the complexity is reduced from
QFQWto QF+QW. In order to find the precoders we solve
the following problems, i.e.,
¯qF= argmax
qF
log2|INS+ρ
NSσ2
n
(WoptHWopt)−1WoptH
×HFRF FBB FH
BB FH
RF HHWopt|,s.t.:
FRF =F(qF)
RF ,FBB = (FH
RF FRF )−1FH
RF Fopt,(6)

3
Algorithm 1 Training data generation.
Input: L,N,NT,NR,NRF
T,NRF
R, SNRTRAIN.
Output: Training data DFand DW.
1: Generate {H(n)}N
n=1 with {F(n)
RF }N
n=1 and {W(n)
RF }N
n=1.
2: for 1≤n≤Nand 1≤l≤Ldo
3: [H(l,n)]i,j ∼ CN ([H(l)]i,j, σ2
TRAIN).
4: Find ¯qFby solving (6) for F(qF,l,n)
RF ,1≤qF≤QF.
5: Construct ˆ
F(l,n)
RF and F(l,n)
BB from F( ¯qF,l,n)
RF .
6: Find ¯qWby solving (7) for W(qW,l,n)
RF ,1≤qW≤QW.
7: Construct ˆ
W(l,n)
RF and W(l,n)
BB from W( ¯qW,l,n)
RF .
8: [[X(l,n)]:,:,1]i,j =|[H(l,n)]i,j |.
9: [[X(l,n)]:,:,2]i,j = Re{[H(l,n)]i,j }.
10: [[X(l,n)]:,:,3]i,j = Im{[H(l,n)]i,j } ∀ij.
11: z(l,n)
F=∠vec{ˆ
F(l,n)
RF },z(l,n)
W=∠vec{ˆ
W(l,n)
RF }.
12: end for n,l
13: Training data for CNNFand CNNWis obtained as
DF= ((X(1,1),z(1,1)
F),...,(X(L,N),z(L,N )
F)),
DW= ((X(1,1),z(1,1)
W),...,(X(L,N),z(L,N )
W)).
Fig. 1. The proposed CNN framework for precoder (CNNFat the top) and
combiner design (CNNWat the bottom).
¯qW= argmax
qW
log2|INS+ρ
NSσ2
n
(WH
BB WH
RF
×WRF WBB WH
RF WH
BB )−1HFopt FoptHHHWRF WBB |,s.t.:
WRF =W(qW)
RF ,WBB = (WH
RF ΛWRF )−1(WH
RF ΛWopt),
Λ=ρ
NS
HFoptFoptHHH+σ2
nINR.(7)
Once (6) and (7) are solved, the analog beamformers are
constructed as ˆ
FRF =F(¯qF)
RF and ˆ
WRF =W(¯qW)
RF . The
baseband beamformers can also be obtained accordingly.
IV. CNN-BASED APPROACH
In this section, we present our CNN framework for joint
precoder and combiner design which is shown in Fig. 1. The
proposed network is composed of two CNNs with 8 layers
which have identical structures except the last layer. The first
layer is the input layer of size NR×NT×3with c= 3
channels. The first channel of the input is the element-wise
absolute value of the channel matrix as [[X]:,:,1]i,j =|[H]i,j|.
The second and the third channels are defined as the real and
the imaginary parts of the channel matrix as [[X]:,:,2]i,j =
Re{[H]i,j }and [[X]:,:,3]i,j = Im{[H]i,j }. The second and third
layer are the convolutional layers with 32 filters of size 2×2.
The fourth and sixth layers are fully connected layers with
1024 units. There are dropout layers after each fully connected
layers (the fifth and seventh layers) with %50 probability. The
output layer of CNNFis of size NTNRF
T×1which is the
vectorized version of the phases of FRF . Similarly, the size of
the output layer of CNNWis NRNRF
R×1. The complexity of
a CNN is directly proportional with the number of parameters
which, in our case, calculated as C2(2Ncv(wh+1) + 2(Nf c +
1) ·50
100 )[21]. Here C= 3 is the number of channels, w=
h= 2 is the filter size, Ncv = 32 is the number of filters,
Nfc = 1024 is the number of units in the fully connected
layer for %50 dropout probability. Hence the CNN structure
in Fig. 1 has 12105 parameters.
In data generation, Ndifferent realizations of channel
matrices H(n)for different user locations are generated to-
gether with the corresponding sets F(n)
RF and W(n)
RF . Then
for each realization, Lnoisy channel matrices are obtained
where the added element-wise synthetic noise is defined by
SNRTRAIN = 20 log10(|[H]i,j |2
σ2
TRAIN
). To account for the changes
in the wireless environment, we use three different SNRTRAIN
levels. Hence the total size of the training input data becomes
NR×NT×3×3NL. In order to obtain the output data the
problems in (6) and (7) are solved ∀n, l. Then the output data
of each network is obtained. We summarize the algorithmic
steps of the training data generation in Algorithm 1.
V. NUMERICAL SIMULATIONS
In this section, we evaluate the performance of our CNN
framework (referred to as HBDL, Hybrid Beamforming via
Deep Learning) and compare it with the state-of-the-art tech-
niques such as SOMP [6] and PE-Alt-Min [7]. Uniform
square arrays are considered with half wavelength spacing with
NR=NT= 36 antennas. The number of analog beamformers
are NRF
R=NRF
T= 4. The feasible sets FRF ,WRF are used
for training only, and the output from CNN can be directly
used for analog beamforming since the analog beamformer
does not have to lie in the set of array response vectors.
The CNNs are fed with the training data generated for N=
L= 100. For each channel matrix realization, the propagation
environment is modeled with Nc= 4 and Nray = 5 for each
clusters with σ2
Θ= 5◦for all transmit and receive azimuth
and elevation angles which are uniform randomly selected
from the interval [−60◦,60◦]and [−20◦,20◦]respectively.
The proposed network is realized in MATLAB on a PC
with 768-core GPU. Stochastic gradient decent algorithm is
used to update the network parameters with the learning rate
0.005 and mini-batch size 500 for 200 epochs. As a loss
function, we use the negative log-likelihood or cross-entropy
loss [9]. In the training process, 70% and 30% of all data
generated are selected as the training and validation datasets,
respectively. Validation aids in hyperparameter tuning during
the training phase to avoid the network simply memorizing the
training data rather than learning general features for accurate
prediction with new data. The validation data is used to test the
performance of the network in the simulations for JT= 100
Monte Carlo trials. In order to prevent the similarity between
the test data and the training data we also add synthetic
noise to the test data where the SNR in testing is defined

4
-20 -15 -10 -5 0 5 10 15 20
SNR, [dB]
2
4
6
8
10
12
14
16
18
Spectral Efficiency [bits/s/Hz]
OPT
Best
HBDL
PE-Alt-Min
SOMP
-0.5 0 0.5
9.6
9.7
9.8
9.9
(a)
-20 -15 -10 -5 0 5 10 15 20
SNR, [dB]
0
5
10
15
20
25
30
35
Spectral Efficiency [bits/s/Hz]
OPT
Best
HBDL
PE-Alt-Min
SOMP
-0.1 0 0.1
18
18.2
18.4
18.6
(b)
-20 -15 -10 -5 0 5 10 15 20
SNR, [dB]
0
5
10
15
20
25
30
35
40
45
Spectral Efficiency [bits/s/Hz]
OPT
Best
HBDL
PE-Alt-Min
SOMP
(c)
Fig. 2. Spectral efficiency versus SNR for (a) NR=NT= 25,NS= 1; (b) NR=NT= 36,NS= 2; (c) NR=NT= 36,NS= 3.
similar to SNRTRAIN as SNRTEST = 20 log10(|[H]i,j |2
σ2
TEST
)and
SNRTRAIN ∈ {10,15,20}dB is selected.
In Fig. 2, the spectral efficiency for different algorithms is
presented for NS={1,2,3}and SNRTEST = 10dB. As it
is seen, HBDL provides better performance as compared to
the optimization-based method PE-Alt-Min and greedy-based
algorithm SOMP. The performance plot ”Best” denotes the
performance of the test data without prediction. We observe
that HBDL is very close to the best performance as well as the
fully-digital beamformer. HBDL effectively selects the analog
beamformers from the feasible sets which maximizes the spec-
tral efficiency. The effectiveness of HBDL is attributed to the
best selection of analog beamformers which are the optimum
solution of (4) through the SVD of the channel matrix [6].
SOMP has poor performance due the the fact that it cannot
select the ”best” set of array responses from the dictionary.
While PE-Alt-Min has sufficiently good performance, HBDL
performs better even when the output of PE-Alt-Min is inserted
to the feasible sets used for HBDL.
To compare the computation time of the algorithms we
consider the same settings and observe that HBDL spends
about 0.020s to compute both precoder and combiners whereas
SOMP and PE-Alt-Min take about 0.450s and 1.200s respec-
tively.
VI. CONCLUSIONS
In this work, a CNN framework is proposed for the joint
estimation of precoder and combiners in hybrid beamform-
ing problem. We show that the proposed network archi-
tecture provides better spectral efficiency as compared to
the optimization-based and greedy-based algorithm. In future
work, we reserve the case when the training data is small
where transfer learning-like approaches can be developed.
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