Performance of Cell-Free Systems with Channel Reciprocity Errors

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 237

 Performance of Cell-Free Systems with Channel
 Reciprocity Errors
 Rafael DUARTE 1 , Marcelo ALENCAR 1,3,4 , Waslon LOPES 2 , Fabrício CARVALHO 2 ,
 Wamberto QUEIROZ 1 , Danilo ALMEIDA 1
 1 Dept. of Electrical Engineering, Federal University of Campina Grande, Campina Grande, Brazil
 2 Dept. of Electrical Engineering, Federal University of Paraíba, João Pessoa, Brazil
 3 Dept. of Electrical Engineering, Federal University of Bahia, Salvador, Brazil
 4 Senai Cimatec, Salvador, Brazil

 {rafael.duarte, danilo.brito}@ee.ufcg.edu.br, {malencar, wamberto}@dee.ufcg.edu.br, {waslon, fabricio}@cear.ufpb.br

 Submitted May 2, 2020 / Accepted October 25, 2020

Abstract. Cell-free systems are characterized by the ab- number of access points (AP) that normally serve a lesser
sence of a cell-based spatial subdivision. In these systems amount of user equipment in comparison with that served in
a large number of access points may serve each user, which traditional systems [4].
contribute to improve signal transmission conditions. In this
 Two remarkable properties of massive MIMO systems,
context, it is important to obtain equations that describe the
 favorable propagation and channel hardening, were studied
behavior of the system, as a function of its main parameters.
 by Chen and Bjornson in the context of cell-free systems [5].
Such equations become more complete when more effects are
 The authors concluded that the value of the exponent of ge-
taken into account. One of these effects is the loss of channel
 ometric losses and the number of antennas per AP are deter-
reciprocity due to radiofrequency (RF) mismatch. This paper
 minant for the occurrence of channel hardening, even with
proposes the introduction of a multiplicative model for the
 a smaller number of AP in the system. In the study of cell-
reciprocity errors resulting from RF mismatch in all devices
 free systems, the use of carrier frequency around 1.9 GHz,
of a cell-free model. Additionally, it also proposes the use
 with a 20 MHz narrowband, is usual [3, 4, 6–8]. Despite
of different levels of mismatch for each device. The main
 this, Alonzo and Buzzi [9] opted to compare conventional
contribution of this work is an analytical expression for the
 cell-free systems with those using user-centric virtual cells
downlink achievable rates in the presence of multiplicative
 under millimeter waves, with a carrier frequency of 73 GHz
reciprocity errors due to RF mismatch. Based on it, one can
 and a 200 MHz bandwidth.
compute the approximate value of the achievable rates. The
analytical expression is used in scenarios with and without With the advancement of the study of cell-free commu-
line-of-sight. It is shown that the analytical expression is nication systems, an effort has also been made to obtain equa-
very close when there is line-of-sight, as it provides achiev- tions that allow the computation of parameters such as achiev-
able rate values closer to that obtained by using Monte Carlo able rates, spectral efficiency and energy efficiency [6, 10].
simulation. In addition, such equations can be used in the development
 of optimization methods, for example. Ngo et al. [11] de-
 rived an analytical expression for the uplink and downlink
 achievable rate on cell-free systems. This expression was
Keywords successfully used for the development of optimization meth-
 ods in the allocation of pilot sequences and power control,
 Achievable rates, cell-free systems, channel reciprocity, since they contributed to raise the rates, concentrating them
 line-of-sight, RF mismatch around a higher median. It proved to be an advantage even
 over small-cell systems.
 Therefore, the possibility of expanding the cell-free sys-
1. Introduction tem model was devised, considering hardware effects in the
 In recent years, research on co-located massive Multiple form of multiplicative reciprocity errors, in addition to reach-
Input Multiple Output (MIMO) systems has divided attention ing an expression that could be used to arrive at an approxi-
with those in which radiobase units (RAUs) are distributed mate performance of the system. The complete model relies
across the coverage area [1, 2]. A distributed system variant on obtaining the statistics of the channel estimators in the
is the cell-free system [3], in which there is no cell-based presence of such hardware effects. The entire procedure is
spatial division. Another feature of these systems is a large detailed in this work.

DOI: 10.13164/re.2021.0237 SYSTEMS

238 R. DUARTE, M. ALENCAR, W. LOPES, ET AL., PERFORMANCE OF CELL-FREE SYSTEMS . . .

1.1 Related Works age of a modified MMSE precoder has increased downlink
 sum rates. In that work, the authors also present an analytical
 A study on the impact of different levels of hardware expression for sum rates. The authors considered that the RF
 A study on theon
impairments impact of different
 cell-free systemslevels of hardware
 is presented impair-
 in [12] and mismatch varies jointly with the large-scale fading.
 mentsIn
[13]. onthe
 cell-free systems
 first paper, is presented
 closed in [12]
 expressions forand
 the[13].
 uplinkIn
 the first paper, closedand
 expressions for the uplink Spectral efficiency of cell-free massive MIMO systems
spectral efficiency energy efficiency werespectral
 obtainedef-
 ficiency and energy
considering efficiency
 the presence of were obtained
 hardware consideringAd-
 impairments. the in which some links present line-of-sight (LoS) was studied
 presence ofthe
ditionally, hardware impairments.
 variation Additionally,
 of these metrics with the varia-
 num- by Ozdogan et al [7]. They tested the system performance
 tion of
ber of these
 accessmetrics
 points,with the numberdifferent
 considering of accessscaling
 points, factors,
 consid- under two channel estimation methods: MMSE and Least-
 eringalso
was different scalingHardware
 analyzed. factors, was also analyzed.
 impairments wereHardware
 charac- Squares (LS). They also derived a spectral efficiency analyt-
 impairments
terized werequality
 by using characterized by using
 coefficients quality
 that rangecoefficients
 from 0 to ical expression for that scenario, when operating in uplink
 thatand
1, range from distortion
 additive 0 to 1, andnoise.
 additive
 Thedistortion noise. The
 authors adapted the mode. In that paper, the equalization method considered was
 authors Minimum
Linear adapted the Linear
 Mean Minimum
 Square Mean Square
 Error (LMMSE) Error
 estimator the maximum-ratio combining (MRC). They observed, as in
 (LMMSE)
to take intoestimator
 accounttosuch
 take into account suchInimperfections.
 imperfections. [13], down- systems with no line-of-sight, that MMSE estimators out-
 In [13],
link downlink
 spectral spectral
 efficiency efficiency were
 expressions expressions
 obtainedwere ob-
 under perform LS. Shortly thereafter, the same authors expanded
 tainedconditions.
these under theseThese
 conditions. These expressions
 expressions were used towere used
 develop the work, including downlink mode and the LMMSE esti-
ato max-min
 develop a power
 max-min poweroptimization
 control control optimization algorithm,
 algorithm, which mator [20]. Downlink mode was studied in coherent and
 which proved
proved to be tosuperior
 be superior to heuristicmethods.
 to heuristic methods.The The results
 results non-coherent precoding schemes. An important result of
 obtained in that work show
obtained show thethe importance
 importance of of considering
 considering this work is that when the phase of transmission is unknown,
 hardware effects.
hardware effects. when MMSE estimation is used, the error is narrowly asso-
 ciated to the pilot contamination due to sequences length.
 One of the aspects studied in recent years is the ab-
sence of channel reciprocity. It affects the performance of The development of an SINR equation for a cell-free
the system, since, in the Time Division Duplexing (TDD) system under no line-of-sight (NLoS) conditions and with
mode, the estimated channel tends to be used for various fast varying RF mismatch only in the access points is pre-
purposes [14]. A possible cause of non-reciprocity is the oc- sented in [21], while the performance analysis of the scenario
currence of radiofrequency (RF) mismatch [15]. Assuming with no NLoS and slow varying mismatch only in the access
that the transmitting and receiving RF gains affect the signals points is presented in [22]. The importance of improving the
in different ways, the estimated uplink channel will not match model is justified, because optimization methods are based
the downlink channel. on analytical expressions. In particular, when each device
 has an error level, neglecting the existence of mismatch can
 The influence of the radiofrequency mismatch, that is, harm even more the optimization procedures. The objective
gain mismatches of the transceiver radiofrequency circuits of this work is to extend the cell-free model with reciprocity
(mixers, amplifiers and analog to digital converters, for ex- errors, performing analyzes and obtaining expressions differ-
ample), on simplified communication systems, was analyzed ent from those presented in the cited works. The proposal, in
by Wei et al [16]. They also proposed the use of antenna the present work, is to use multiplicative models to represent
calibration to overcome channel non-reciprocity and derived reciprocity errors, modeled by Truncated Gaussian r.v., in
ergodic sum rates for evaluating the impact of calibration er- both UEs and APs; also, are obtained analytical expressions
ror on system performance. A similar study was conducted for the downlink SINR and for channel estimators variance
by De Mi et al [17] to obtain approximate expressions for in such a scenario, considering different levels for reciprocity
the downlink signal-to-interference-plus-noise ratio (SINR) errors.
when considering Zero-Forcing (ZF) and Maximum-Ratio
Transmission (MRT) precoding schemes. In that work, the
analysis was performed considering only small-scale fading, 1.2 Key Contributions
white additive Gaussian noise and an estimation error repre-
sented by an coefficient that ranges from 0 to 1 by a complex • In this work, the cell-free model is extended to take into
Gaussian random variable (r.v.) with zero mean and unitary account the reciprocity errors resulting from RF mis-
variance. In that paper, it is assumed that amplitude and match in both ends of a link, i.e. user equipment and
phase of the device response coefficients vary jointly with access points. Using similar approach of [17] and [23],
the small-scale fading. Their study did not consider neither errors due hardware effects are include as multiplica-
the occurrence of large-scale fading nor the influence of mis- tive complex coefficients, instead the approach used to
match on channel estimation. The impact of RF mismatch model hardware impairments in [12] and [13], based
depends on environmental conditions, such as humidity and on additive terms and hardware quality coefficients. As
temperature [18]. done in [17] and [24], the amplitude and phase of reci-
 procity errors are modeled as truncated Gaussian ran-
 Another example of how the introduction of RF mis- dom variables instead of uniform r.v., because the latter
match into models can be advantageous is the robust esti- has been pointed as unrealistic.
mation error precoder proposed by Chen et al [19]. It is
demonstrated that, in the presence of RF mismatch, the us- • Analytical expressions are obtained for the downlink

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 239

 achievable rate of this system, with line-of-sight and
 with NLoS, for scenarios in which the amplitude and
 phase of mismatch coefficients vary jointly with the 
  
 
 small-scale (fast varying) and large-scale (slow vary-
  
 
 ing) fading.

  
 The remaining of this paper is organized as follows. In 
 

Sec. 2, the features of the model used to simulate the cell- 
 
free system are presented, including the fading and downlink 

 

signal model as well as the RF mismatch. In Sec. 3, the  
characteristics of the estimators are studied. Sections 4 and   

5 are dedicated to the analytical expressions of the achiev- 
able downlink rates, considering the mismatches of fast and
slow variation, respectively. In Sec. 6, the results of the sim-
ulations are presented and discussed. Finally, Section 7 is Fig. 1. Cell-free system.
devoted to the conclusions of this work. - factor
 -factor is given
 is given by 10by 101.3−0.003 
 1.3−0.003d , in which
 mk, inmkwhich mk
 dmk is theisdis-
 the
 distance between
 tance between k-th -th
 UE UE m-th -th
 andand AP AP[28].[28]. In scenario,
 In this this sce-
 nario,
 the the large-scale
 large-scale fadingfading coefficient
 coefficient andmagnitude
 and the the magnitude of
 of the
2. System Model the specular
 specular component
 component are given,
 are given, respectively,
 respectively, by [20,
 by [20, 28]:28] :
 In order to verify the validity of the obtained expres- mk
sions, a cell-free system needs to be modeled. The model uses mk = , (1)
 1 + mk
Monte Carlo simulations to compute the rates provided by
 3
analytical expressions. A cell-free system can have more than mk 4
one antenna per access point and per user equipment [25,26]. ℎmk = mk . (2)
 1 + mk
In this research, there are single-antenna access points
 For the no line-of-sight condition, mk = 0. The path-loss
(AP) and single-antenna user equipment (UE) randomly
 COST231 Walfish-Ikegami model, presented in the 3rd Gen-
distributed in a square region with area × . All access
 eration Partnership Project (3GPP) Report [28] consists of
points are connected to the same central processing unity
 two equations, which can be used to model the large-scale
(CPU) via the backhaul. This system is illustrated in Fig. 1.
 fading. The first corresponds to the condition in which there
Although the system is not divided into cells, in order to
 is line-of-sight (LoS) and the second when there is no line-of-
simulate the system, only access points and user equipment
 sight (NLoS). In the first condition, the shadowing standard
located within the square region are considered. No spatial
 deviation ( ) is 4 dB and, considering 1.9 GHz as carrier
correlation model was considered for shadowing, because,
 frequency, the path-loss is given by:
from the results obtained by Hien et al [11], it can be seen
that it only worsens the performance of the system, reducing = 10 (−34.53−38 log10 ( mk )+ s mk )/10
 (−34.53−38 log10 ( mk )+ s mk )/10 , (3)
 mk = 10
 mk , (3)
its the achievable rates.
 In this model, the system is narrowband and operates in in which mk is the distance between the -th user equipment
TDD mode. Assuming channel reciprocity, uplink and down- and the -th access point. The shadowing coefficient, mk , is
link channel gains are equal in a coherence interval. It is as- characterized by a random variable with Log-normal distri-
sumed that large-scale fading gain varies every 40 small-scale bution [27,29]. When there is no line-of-sight, the shadowing
fading coherence intervals [4], and that the fading coefficient standard deviation is 10 dB, and the path-loss is given by:
remains constant in a coherence interval. Each coherence = 10 (−30.18−26 log10 ( mk )+ s mk )/10
 (−30.18−26 log10 ( mk )+ s mk )/10 . (4)
 mk = 10
 mk . (4)
interval can be dedicated to four steps: uplink and downlink
channel estimation and uplink and downlink data transmis- In order to obtain an expression for its SINR, the down-
sion [14]. Here, channel estimation in the downlink was not link signal received by the -th UE is defined by:
considered. The gain between the -th user equipment and
 4 
 
 
the -th access point is given by mk = ℎmk ej mk + mk , in = d mk ℎbt,m ℎur,k ( ˆ 1/2 ∗
 mi ) mi + , (5)
which mk is a random variable uniformly√ distributed be- =1 =1
tween and − , and mk = ℎmk mk represents the no
line-of-sight components [27]. The symbols ℎmk and mk in which is the additive white Gaussian noise in the -th
are, respectively, the small-scale and large-scale fading coef- user equipment. In (5), d is the normalized downlink trans-
 
ficients. The small-scale fading coefficient is characterized mission power, given by d / n , in which n is the noise
by a complex Gaussian r.v. with zero mean and unit variance. power given by B 0 . In this expression, is the noise
Therefore, mk can be described by a Gaussian r.v. with zero factor, 0 is the absolute temperature, is the bandwidth and
mean and variance mk . Thus, the term mk represents the B is the Boltzmann constant. Since is the noise figure,
specular component
 component of ofthe
 thesignal.
 signal.For
 Forthethe
 LoS condition,
 LoS condition, the noise factor is given by = 10 /10 . In this research,

240 R. DUARTE, M. ALENCAR, W. LOPES, ET AL., PERFORMANCE OF CELL-FREE SYSTEMS . . .

as in [6], the bandwidth is 20 MHz, the noise figure is 9 dB, coefficient, given by:
and 290 K is the environment temperature.
 ! bf − f f % ! a f − f f %
 Considering the occurrence of RF mismatch at the ac- erf "" - − j √ && − erf "" - − j √ &&
cess point of transmitters and user equipment receivers, the 2 f2 2 2 f2 2
downlink received signal of the -th UE can be rewritten as: = # ' # '. (12)
  ! bf − f % ! %
 4 
 
 erf "" - & − erf " a-f − f &
 = d ℎur,k vT Hbt 1/2 Hbt 1/2 & " &
 w + w + 2 f2 2 f2
 ≠ # ' # '
 (6) This coefficient is the mean of exp(j ), in which is the
 phase of the reciprocity error.
in which Hbt is the diagonal matrix [24] of the response co-
 After deriving these equations, the next step is to in-
efficients (ℎbt,m ) of all access point transmitters, ℎur,k is the
 troduce the impact of reciprocity errors resulting from the
receiver response of the -th user equipment, v is the channel
 mismatch on the channel estimation.
vector between all access points and the -th user equipment,
 is the signal destined to the -th user equipment, w is the
 -th user equipment precoding vector and is the diagonal
matrix of power control coefficients corresponding to the -th 3. Channel Estimation
user equipment, whose elements depend only on m. In this In order to generate the estimated channel coefficients
work, only the MRT precoding method is used. Therefore, that are used in the precoding vector and in achievable rate
the power control coefficient between the -th AP and the calculation, some statistical parameters associated with its
 -th UE is given by [21]: estimators are necessary. For channel training based on the
 1 transmission of pilot sequences, its symbols must also be
 mk = 1 (7) known by the receiver [31]. In channel estimation phase, the
 
 =1 mi pilot signal received by the -th AP in estimation coher-
and mi = ( ˆ ∗ ence intervals is projected onto the pilot sequence H [32].
 mi ) . Since ˆ mi = mi + ˜ mi , and these r.v.
are zero mean and independent, (6) can be rewritten as: This projection results in:

 4 
 
 4 4 4 
 
 ˜ P,mk = p ml ℎut,l ℎbr,m H l + H nP (13)
 = d k,s + d k,e + d k,i + , (8)
 =1
 ≠ 
 in which nP is the noise vector of the -th AP during the sam-
in which: ples, p is the normalized pilot symbol transmission power
 and is the pilot sequence used by the -th user in the
 ∗
 k,s = ℎur,k vT Hbt 1/2
 (v ) , (9) estimation phase. The normalized pilot symbol transmission
 
 power is given by p / n , in which p is the pilot symbol
 transmission power. Finally, ℎbr,m and ℎut,l are the response
 ∗ coefficients of the -th AP receiver and the -th user equip-
 k,e = ℎur,k vT Hbt 1/2
 ( ṽ ) , (10)
 ment, respectively. When the strength of the faded signal is
 as low or lower than that of the additive noise, the values of
 the fading coefficients cannot be determined precisely. The
 k,i = ℎur,k vT Hbt 1/2
 w . (11)
 same occurs with the reduction of the number of samples of
 the pilot sequences, as this reduces the precision of the cal-
In the computation of SINR, the channel estimation error, culated averages. In (13), 
 ml = ml ℎbr,m ℎut,l is the effective
 k,e , will be considered as part of the noise, along with k,i uplink channel coefficient.
and .
 In this research, the pilot sequences were distributed
 The parameters of the Gaussian function (used to gen- randomly among users, but avoiding unnecessary repeti-
erate the truncated Gaussian r.v. that represent magnitude tions. No optimal sequence allocation method was used.
and phase of the device response coefficients) are expressed, However, pilot contamination [33] has been considered in
respectively, by ( g ; g2 ; [ g , g ]) and ( f ; f2 ; [ f , f ]), in some scenarios. This effect occurs when two or more equal
which is the Gaussian mean, 2 is its variance, and are or non-orthogonal strings are assigned to different devices.
the truncation limits. In [30], it is mentioned that, after an- A simpler way to avoid such an effect would be to use more
tenna calibration, residual reciprocity errors (amplitude and sequences, which would reduce the spectral efficiency of the
phase) remain constant for all subcarriers. The reciprocity system, since it would reduce the number of samples in the
error levels adopted in this research are the same as in [17], as coherence interval dedicated to data transmission [34]. The
well as the equation used to calculate the phase error related channel estimation was made before uplink data transmission,

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 241

using the Discrete Fourier Transform (DFT) coefficients, as the sequence length tends to infinity, the estimator behavior
it can assume any length, unlike Hadamard sequences [35]. resembles to a system with perfect CSI. When this occurs,
Both sequences would allow the same quality as the estima- the uplink channel is perfectly known. In the downlink, us-
tion. However, due to the fact that the length of the sequences ing this channel estimate in place of mk results in precoding
is also their number, and because DFT is more flexible with errors. The relationship of the mean square error with the
regard to its length, it was used. Since is a matrix of orthog- transmission power, in estimators based on pilot sequences,
onal pilot sequences, in which each column ( ) corresponds was studied by Biguesh et al [31].
to a sequence, then H l = 0.
 Usually, when LMMSE estimator is used to estimate
 Among the estimation methods that could be used, the channel coefficient , estimated channel is given by
 4 1 2
Least-Squares (LS) and LMMSE were chosen. In the absence mk ˜ mk , in which mk = p mk /( p =1 ml | H l | +1)
of pilot interference, the Least-Squares estimator can be con- [3, 4]. In this work, the pilot signals capture not only the in-
sidered without loss of optimality. However, the LMMSE formation related to , but that related to 
 mk , i.e., the
estimator provides better performance when there is pilot effective uplink channel. In this case, minimization of the
interference [11]. mean square error occurs with ˆ 
 mk = mk ˜ P,mk , in which:
 Using the least-squares method
 √ [31], the estimated up- { ˜ ∗P,mk mk ℎut,k ℎbr,m }
link channel is given by: ˜ mk / d [32]. That is, mk = . (20)
 1 4 {| ˜ mk | 2 }
 =1 p ml ℎut,l ℎbr,m l H nP 
 H
 ˆ mk = √ + √ . (14) In { ˜ ∗P,mk mk ℎut,k ℎbr,m }, due to the fact that the channel
 P P 
 coefficients m,l are zero mean, the summation in P,mk re-
Since the estimated channel variance is given by mk = duces to | mk | 2 |ℎ . | 2 |ℎ . | 2 , even with the occurrence of
 {| ˆ mk | 2 }, when the RF mismatch is fast and slow varying, pilot contamination. Then, considering fast varying radiofre-
respectively, it is: quency mismatch, mk becomes:
 
 4
 1 p mk br,m ut,k
 mk,a = ml t,l r,m H l + , (15) mk = 1 . (21)
 =1
 P 2
 p =1 ml ut,l br,m | H l | + 1
 Without this adaptation of mk , the LMMSE estimator would
 
 
 1
 provide worst results. On the other hand, when radiofre-
 
 mk,a = ml |ℎut,l | 2 |ℎbr,m | 2 H l + . (16) quency mismatch varies slowly:
 =1
 P 4
 p mk |ℎut.k | 2 |ℎ . | 2
. mk = . (22)
 1 2
 2
In (15), ml = ml + ℎml , br,m = br,m
 2 + br,m
 2 and ut,l = p =1 ml |ℎbr,m | 2 |ℎ . | 2 | H l | + 1
 ut,l + ut,l . The terms and are, respectively, the stan-
 2 2 The variance of the estimated uplink channel is given
dard deviation and the mean of the truncated Gaussian r.v. by {| mk ˜ P,mk | 2 }, i.e., {| mk | 2 | ˜ P,mk | 2 }. Calculating the
that models the magnitude and phase of the device response average, it is obtained, for the cases in which the mismatch
coefficients. is fast and slow, respectively:
 Uplink channel estimation error is given by mk = 
 p mk
 2 2 2
 ut,k br,m
 2 mk,b = 1 , (23)
 {| ˜ mk | } [20]. Considering again Least-Squares estima- p
 2
 =1 ml ut,l br,m | l | +1
 H
tion, it can be shown that:
 / 5
 ∗ ∗
 | 
 ˜ | 2 
 ˜ ( ) 
 ˜
 mk,a = | 2 mk mk mk mk mk
 p mk
 2 |ℎ
 ut,k | |ℎbr,m |
 4 4
 mk | + − √ − √ . 
 mk,b = .
 P P P 1 2
 (24)
 p =1 ml |ℎut,l |
 2
 |ℎbr,m | 2 | H l | + 1
 (17)
 Finally, as was done in [20], the mean-square error,
Therefore, when the RF mismatch is fast varying, estimation and, therefore, the variance of the estimation error is given
error variance is given by: by {| 
 mk |} − mk . Therefore, when the mismatch is fast or
 
 slow, respectively:
 1
 mk,a = ml br,m ut,l H l − mk br,m ut,k + . (18) 
 P mk,b = r,m , mk − mk,b , (25)
 =1

On the other hand, when mismatch is slow varying: 
 mk,b = mk |ℎbr,m | 2 |ℎut,k | 2 − mk,b . (26)
 
 mk,a = mk,a − mk |ℎbr,m | |ℎut,k | .
 2 2
 (19)
 By using these parameters it is possible to generate ˜ mk
 
Based on mk,a and mk,a , when no pilot contamination oc- and ˆ mk . In addition, the error variance will be necessary for
curs, only the term referring to SNR remains. When p or the calculation of theoretical achievable rates.

242 R. DUARTE, M. ALENCAR, W. LOPES, ET AL., PERFORMANCE OF CELL-FREE SYSTEMS . . .

4. SINR for Fast RF Mismatch In (33), the product can be can be splitted into a sum:
 / 5 / 5
 In order to assess the performance of a communica-  
tion system one of the possible evaluation metrics is the { k,s } = 1 + 2 , (34)
achievable rate. When instantaneous channel coefficients are =1 ≠ 
known, it is given by [4]:
 in which:
 d,k =
 1 2 1 = mk | mk | 4 |ℎbt,m | 2 |ℎur,k | 2 |ℎbr,m | 2 |ℎut,k | 2 (35)
 ⎧
 ⎪  1/2 ∗  ⎫
 ⎪
 ⎪
 ⎨
 ⎪ ! d  =1 mk ℎbt,m ℎur,k mk ˆ 
 mk  %⎪⎬
 ⎪
 log2 ""1 +   & .
 &
 ⎪
 ⎪ 1 1 1/2 ∗ 
 2 ⎪
 ⎪ 2 = mk nk | mk | 2 | nk | 2 ℎbt,m ℎ∗bt,n
 1/2 1/2
 ⎪
 ⎩ # d ≠  =1 mi mk ℎbt,m ℎur,k ˆ 
 mi  + 1 ' ⎭
 ⎪
 (27) ×|ℎur,k | 2 ℎ∗br,m ℎbr,n |ℎut,k | 2 . (36)

 In (27), the mean is calculated over each block of 40 Computing the means in (34), one obtains:
small-scale coherence intervals. Assuming that the large
scale fading coefficients are known, the theoretical achiev- 
 
able rate of the -th user can be obtained from [11]: { k,s } = mk {| mk | 4 }ℎbr,n ur,k br,m ut,k +
 =1
 d,k = log2 (1 + ), (28) 
 
 1/2 1/2 ∗ ∗
 mk nk mk nk bt,m bt,n ur,k br,m br,n ut,k , (37)
 ≠ 
in which is the downlink SINR of the -th UE. In order
to obtain an analytical expression for the achievable rate, it is in which br,m = br,m br,m , bt,m = bt,m bt,m ,
necessary to obtain an expression for the SINR. Based on (8), br,n = br,n r,n , bt,n = t,n bt,n , and:
it is given by:
 4 2
 / 5 {| mk | 4 } = ℎmk + 2 mk ( mk + ℎmk ). (38)
 d k,s
 = 1 , (29)
 d k,e + ≠ d k,i + N As mentioned earlier, the power of the estimation error
 is considered as part of the noise power. Since this signal is
in which N = | | , k,s is the power of k,s , k,e is the power
 2
 defined in (10), its power is given by:
of k,e and k,i is the power of the interuser interference.
 / 5
 In order to obtain an analytical expression for the SINR, ∗ 2
 { k,e } = |ℎur,k vT Hbt 1/2
 ( ṽ ) | , (39)
one can approximate (29) as a ratio of the means of its numer-
ator (power of the signal) and denominator (power of noise
plus interference), which can lead to loss of accuracy [17] which, after some manipulation, leads to:
given by: / 5
 
 2 2
 d { k,s } { k,e } = mk | mk | |ℎbt,m | |ℎur,k | | ˜ mk | +
 2 2
 ≈ 1 . (30)
 =1
 d { k,e } + d ≠ { k,i } + 1 / 5
 
 
 ∗
 First, the strength of the signal of interest is calculated. mk nk mk ∗nk ℎbt,m ℎ∗bt,n |ℎur,k | 2 ˜ 
 1/2 1/2
 mk ( ˜ nk ) . (40)
It is given by: ≠ 

 ∗
 { k,s } = {ℎur,k vT Hbt 1/2
 (v ) |} .
 2
 (31) The second sum of (40) has independent r.v. with zero
 mean. For this reason, only the first term remains. So, its
Transforming this equation from vector form to summation mean is:
form, it is given by: 
 
 / 2 5 { k,e } = mk mk ℎbr,n ur,k mk . (41)
  
  1/2 ∗ ∗  =1
 { k,s } =  ℎur,k | mk | ℎbt,m mk ℎbr,m ℎut,k  , (32)
 2
  
 =1
 Finally, based on (11), the power of the interuser inter-
which can be rewritten as: ference is given by:
 / / 
  
 
 { k,s } = 1/2
 mk | mk | 2 ℎbt,m ℎur,k ℎ∗br,m ℎ∗ut,k × { k,i } =  1/2
 mi mk ∗mi ℎbt,m ℎur,k ℎ∗br,m ℎ∗ut,i
 
 =1 =1
  5 2 5
 
 
 
 ∗
 1/2
 nk | nk | 2 ℎ∗bt,n ℎ∗ur,k ℎbr,n ℎut,k . (33) + 1/2
 mi mk ℎbt,m ℎur,k ( ˜ mi )  . (42)
 
 =1 =1

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 243

Due to the independence of some random variables in these one is used to represent the estimation error level. In this
two sums, the modulus can be splitted into a sum of two other research the variances of the estimation errors are used in the
modulus, that is: analytical expression. Considering the absence of line-of-
 sight and large-scale fading ( mk = 0, mk = 1 and mk = 
 { k,i } = { 3 } + { 4 } , (43) ∀ , ), no RF mismatch in user equipment and the same
 reciprocity error level at all access points, the equations pre-
in which:
 sented here become similar to those presented in that work.
  2
   So, taking as an example (35), one can obtain:
  1/2 ∗ ∗ ∗ 
 3 =  mi mk mi ℎbt,m ℎur,k ℎbr,m ℎut,i  , (44) √
 
 =1
  { k,s } = [2 t + ( − 1) ∗ t t∗ ]. (52)

 On the other hand, assuming that RF mismatch occurs only
  2
   in the transceivers of access points in a cell-free system, the
  1/2 ∗
 4 =  mi mk ℎbt,m ℎur,k ( ˜ mi )  . (45) development of (34) provides:
  
 =1
 
 
The term 3 can be rearranged as: { k,s } = mk {| mk | 4 }ℎbr,n br,m +
  =1
  
 
 3 = 1/2
 mi mk ∗mi ℎbt,m ℎur,k ℎ∗br,m ℎ∗ut,i × 1/2 1/2
 mk ∗
 nk mk nk bt,m bt,n ∗
 br,m br,n . (53)
 =1 ≠ 
 
 
 
 1/2 ∗
 ni nk ni ℎ∗bt,n ℎ∗ur,k ℎbr,n ℎut,i . (46) It is worth to mention that the SINR expression obtained
 =1 in this section can also be manipulated in order to test other
The development of 3 leads to: scenarios, with diverse parameter arrangements.
 / 5 / 5
  
 { 3 } = 5 + 6 , (47)
 =1 ≠ 
 5. SINR for Slow RF Mismatch
in which: In a scenario in which there is slow RF mismatch, the
 SINR of the -th user can be obtained in a similar way used
 5 = mi | mk | 2 | mi | 2 |ℎbt,m | 2 |ℎur,k | 2 |ℎbr,m | 2 |ℎut,i | 2 , (48) for the rate with fast variation. The difference is that the
 terms referring to mismatch can be taken as constant [19].
 Therefore, considering the RF mismatch in both ends of the
 6 = mi ni mk ∗nk ∗mi ni ℎbt,m ℎ∗bt,n
 1/2 1/2
 link, one can obtain:
 ×|ℎur,k | 2 ℎ∗br,m ℎbr,n |ℎut,i | 2 . (49) 
 
 { k,s }= mk {| mk | 4 }|ℎbt,m | 2 |ℎbr,m | 2 |ℎut,k ||ℎur,k | 2
Calculating the mean of 3 , the term 6 disappears, because =1
it has independent r.v. with zero mean ( ni and ∗mi , for 
 
example). Therefore, it leads to: + mk nk mk nk ℎbt,m ℎ∗bt,n ℎ∗br,m ℎbr,n |ℎut,k ||ℎur,k | 2 ,
 1/2 1/2

 ≠ 
 
 
 (54)
 { 3 } = mi mk mi ℎbr,n ur,k br,m ut,i . (50)
 =1

 
 
 Applying the same procedure to the second term of 4 , { k,e } = 
 mk mk |ℎbt,m | 2 |ℎur,k | 2 mk , (55)
its mean is given by: =1

 
 
 { 4 } = mi mk ℎbr,n ur,k mi . (51)
 =1
 
 
 { 3 } = mi mk mi |ℎbt,m | 2 |ℎbr,m | 2 |ℎut,i | 2 |ℎur,k | 2 , (56)
 =1
 Replacing (50), (51) in (43), it is obtained the expres-
sion for the interuser interference power.
 
 
 The analytical SINR obtained in this work can be seen { 4 } = mi mk |ℎbt,m | 2 |ℎur,k | 2 mi . (57)
as an expansion of that presented in [21] and that obtained by =1
De Mi et al and presented in [17]. An important difference
for the latter is how the estimation error is introduced in the In the case in which the mismatch is slow, it has been
model. In that work, a variable coefficient between zero and observed that the analytical expression becomes a function of

244 R. DUARTE, M. ALENCAR, W. LOPES, ET AL., PERFORMANCE OF CELL-FREE SYSTEMS . . .

the devices response coefficients, instead of the distribution
parameters that characterize the reciprocity errors. (a)NLoS
 1

 0.8

6. Results 0.6

 ECDF
 K=25, MC in (28), LS
 This section presents simulations results in order to K=25, Theoretical, LS
 0.4 K=25, MC in (28), LMMSE
assess the impact of RF mismatch on the performance of K=25, Theoretical, LMMSE
cell-free systems considering the model with multiplicative 0.2
 K=15, MC in (28), LS
 K=15, Theoretical, LS
errors. All simulations are based on Matlab ®codes. The K=15, MC in (28), LMMSE
 K=15, Theoretical, LMMSE
analyzes are based on the Empirical Cumulative Density 0
 0 1 2 3 4 5 6 7 8 9 10
Function (ECDF) curves for the downlink achievable rates Achievable Rates (bits/s/Hz)
of a cell-free system affected by reciprocity error due RF
mismatch. The objective is not only to evaluate the influence (b)LoS
 1
of multiplicative reciprocity errors due the RF mismatch on
achievable rates in cell-free systems, but also the quality of 0.8

the estimate obtained with the usage of the extended analyt-
 0.6
ical expressions proposed in this paper. To plot each ECDF
 ECDF
 K=25, MC in (28), LS
 K=25, Theoretical, LS
curve, 10,000 large-scale fading realizations were used. This 0.4 K=25, MC in (28), LMMSE
includes varying the AP and UE coordinates, as well as K=25, Theoretical, LMMSE
 K=15, MC in (28), LS
shadowing. At each realization, achievable rate samples 0.2 K=15, Theoretical, LS
 K=15, MC in (28), LMMSE
were produced. Hien et al [11] used 200 channel realiza- K=15, Theoretical, LMMSE
tions, despite considering larger values of . With = 40, it 0
 0 1 2 3 4 5 6 7 8 9 10
would be 8,000 achievable rate samples. Therefore, the num- Achievable Rates (bits/s/Hz)
ber of achievements used here is sufficient to obtain almost
the same number of samples that would be obtained with the Fig. 2. ECDF curves of the achievable rates obtained with Monte
values adopted in that work. Two homogeneous scenarios Carlo simulation in (27) and by using the analytical ex-
were considered for visibility conditions: NLoS for all links pression in a cell-free system affected by high level and
and LoS for all links. fast reciprocity errors in the AP and UE.

 As in [4], the downlink symbol transmission power is
 
 d = 200 mW and the pilot symbol transmission power (a)NLoS
 1
 
is p = 100 mW. Besides that, there are = 100 ac-
cess points and = 500 m. In order to observe the impact 0.9

of multiplicative errors and the validity of the equation, the 0.8

values used to model phase and amplitude errors were the 0.7
double of those used by De mi et al [17]. Such values are
 0.6
in the range of those presented in [24]. The high level reci-
 ECDF

procity errors was simulated using (0; 2; [−8, +8]) dB and 0.5 MC in (28), RFx2, K=25
 Theoretical, RFx2, K=25
(0; 1; [−50, +50])° as parameters of the Gaussian r.v. associ- 0.4
 MC in (29), RFx2, K=25
 MC in (28), RFx1, K=25
ated with the truncated Gaussian r.v. that describe magnitude Theoretical, RFx1, K=25
 0.3 MC in (29), RFx1, K=25
and phase of a device response, respectively. For the normal MC in (28), RFx2, K=15
level of reciprocity, these parameters are (0; 1; [−2, +2]) dB 0.2 Theoretical, RFx2, K=15
 MC in (29), RFx2, K=15
and (0; 0.5; [−20, +20])°. 0.1
 MC in (28), RFx1, K=15
 Theoretical, RFx1, K=15
 MC in (29), RFx1, K=15
 The first evaluation was performed for a homogeneous 0
scenario with and without line-of-sight, considering = 15 0 1 2 3 4 5 6
 Achievable Rates (bits/s/Hz)
 7 8 9 10

and considering the occurrence of high level reciprocity er-
rors in the access point and user equipment transceivers.
Were considered two different quantities of user equipment: Fig. 3. ECDF curves of the achievable rates obtained with
 Monte Carlo simulation in (27), analytical expressions
15 and 25. When = 15, there is no reuse of pilot se- and an Monte Carlo simulation in (30), considering only
quences, while in the second case, ten pilot sequences are high level reciprocity errors in all transceivers and only
reused, which results in the occurrence of pilot contamina- in access points, for NLoS condition.
tion. According to Buzzi and D’andrea [32], the first quantity In this first test, fast varying RF mismatch and two chan-
would correspond to a sparsely populated scenario, while the nel gain estimation methods (LMMSE and LS) were consid-
second would correspond to an intermediate scenario. The ered. In Fig. 2, for = 15, it is shown that both curves
impact of the number of users on the values obtained with overlap, indicating that the performance of the two estima-
the use of analytical expression will be evaluated later. tion methods is similar in the absence of pilot interference,

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 245

and this is an indication that they were correctly derived. From them, it appears that, in the absence of pilot contam-
Besides that, there is a gap between the curves obtained by ination, the variation in the number of users does not affect
using Monte Carlo method in (27) and by using the analytical the quality of the data obtained by using the analytical ex-
expression. When = 25, this gap also exist. However, the pression, because the gap relatively to the curve obtained
gaps observed in the LoS scenario were smaller. with data generated by using Monte Carlo method does not
 increase considerably. Without and with line-of-sight, it is
 When the terms of the numerator (power of the signal of
 ≈ 0.4 and 0.2 bits/s/Hz, for the lowest rate observed among
interest) and the denominator (interference and noise) of (27)
 60% of users.
are obtained by using the Monte Carlo method in (30), both
curves coincide, as observed in Fig. 3 (for NLoS condition) The next test considered different levels of reciprocity
and Fig. 4 (for LoS). This reveals that the gap stems from the errors. In Fig. 6, the ECDF curves of achievable rates ob-
approximations made in (29) and (30), which were necessary tained with three conditions of the reciprocity error are illus-
to obtain the SINR expression. This error occurs because the trated, considering fast varying mismatch: only normal level,
coefficients representing the reciprocity error increase the de- only high level and a mixed scenario, where each transceiver
pendency among the signal of interest and the interference. can have its own reciprocity errors level. It is noticed that
Also in Figs. 3 and 4, the curves obtained with the occur- the increase in the level of mismatch impair the adherence of
rence of RF mismatch are shown only in the access point the data provided by the analytical expression. The relative
transceivers (RFx1). Comparing these with those obtained gap among the curves obtained with the analytical expres-
with the mismatch at both ends (RFx2), it is observed that the sion is less than that observed in the curves obtained with the
system performance worsens even more. On the other hand, application of the Monte Carlo method in (27). This is also
the values obtained with the analytical expression almost did explained by the dependence between the terms of the SINR.
not vary with the use of mismatch at one or two ends of the
 Considering the slow mismatch and high level reci-
link. This is confirmed by the usage of the Monte Carlo
 procity errors in a scenario with no line-of-sight, the impact
method in (30), for RFx1 and RFx2.
 on transmission rates is different than that observed in a sce-
 Even though there is a dependency between terms in the nario in which the mismatch is fast and reciprocity errors
numerator, it is important to observe whether the increase in level is high, as can be observed in Fig. 7.
the number of users affects the gap between the data obtained
with the Monte Carlo simulation in (27) and the theoretical (a)NLoS
values. For that, four values were considered, without the 1

occurrence of pilot interference. In Fig. 5, the ECDF curves
of the downlink achievable rates obtained with = 30, 15, 10 0.8

and 8 are illustrated. This simulation considered a set of 30
 0.6
 ECDF

pilot sequences with length 30 samples.
 0.4 K=15, MC in (28)
 K=15, Theoretical
 K=10, MC in (28)
 0.2 K=10, Theoretical
 (b)LoS K=8, MC in (28)
 1 K=8, Theoretical
 0
 0.9 0 1 2 3 4 5 6 7 8 9 10
 Achievable Rates (bits/s/Hz)
 0.8
 (b)LoS
 0.7 1

 0.6
 0.8
 ECDF

 MC in (28), RFx2, K=25
 0.5 Theoretical, RFx2, K=25
 MC in (29), RFx2, K=25 0.6
 ECDF

 0.4 MC in (28), RFx1, K=25
 Theoretical, RFx1, K=25
 MC in (29), RFx1, K=25 K=15, MC in (28)
 0.3 0.4
 MC in (28), RFx2, K=15 K=15, Theoretical
 Theoretical, RFx2, K=15 K=10, MC in (28)
 0.2 MC in (29), RFx2, K=15 K=10, Theoretical
 0.2 K=8, MC in (28)
 MC in (28), RFx1, K=15
 0.1 Theoretical, RFx1, K=15 K=8, Theoretical
 MC in (29), RFx1, K=15
 0
 0 0 1 2 3 4 5 6 7 8 9 10
 0 1 2 3 4 5 6 7 8 9 10
 Achievable Rates (bits/s/Hz)
 Achievable Rates (bits/s/Hz)

 Fig. 4. ECDF curves of the achievable rates obtained with
 Monte Carlo simulation in (27), analytical expressions Fig. 5. ECDF curves of the achievable rates obtained with Monte
 and an Monte Carlo simulation in (30), considering only Carlo simulation in (27) and by using the analytical ex-
 high level reciprocity errors in all transceivers and only pression in a cell-free system affected by high level reci-
 in access points, for LoS condition. procity errors in the AP and UE, with = 8, 10 and 15.

246 R. DUARTE, M. ALENCAR, W. LOPES, ET AL., PERFORMANCE OF CELL-FREE SYSTEMS . . .

 This can be seen in the ECDF curves in which slow mis-
 (a)K=15, NLoS match impacted the variance of the achievable rates more
 1
 than the fast mismatch. In terms of the 10%-outage rate, that
 0.8 is, the smallest rate among 90% of the users, systems with
 slow mismatch, fast mismatch and without mismatch provide
 0.6
 ≈ 1.5, 2.0 and 2.5 bits/s/Hz. It makes sense that slow and fast
ECDF

 0.4 High, MC in (28) mismatches provide different results. In the former, the prod-
 High, Theoretical
 Mixed, MC in (28)
 uct of the reciprocity error coefficients through the complex
 0.2 Mixed, Theoretical Gaussian channel mk , results in Gaussian r.v. with different
 Normal, MC in (28)
 Normal, Theoretical statistics. In the second, the product of the error coefficients
 0
 0 1 2 3 4 5 6 7 8 9 10
 by the Gaussian channel results in r.v. with another distribu-
 Achievable Rates (bits/s/Hz) tion and another variance. For this scenario, assuming that
 the mismatch coefficients are also known, because it varies
 (b)K=15, LoS
 1 slowly, (30) can be used to estimate the achievable rates and
 obtain the ECDF curves. In this case, the SINR expression
 0.8 terms are those obtained in Sec. 5.
 0.6 In Fig. 8, it is shown the ECDF curves of the achievable
ECDF

 rates obtained considering = 15 and 25 for slow mismatch
 High, MC in (28)
 0.4
 High, Theoretical with homogeneous error level scenarios, with high level RF
 Mixed, MC in (28)
 Mixed, Theoretical mismatch, when applying the LS and LMMSE channel es-
 0.2
 Normal, MC in (28)
 Normal, Theoretical
 timators. In this case, the values provided by the analytical
 0 expression, when compared to that obtained with fast mis-
 0 1 2 3 4 5 6 7 8 9 10
 Achievable Rates (bits/s/Hz)
 match, provided better adherence to the data provided by
 Monte Carlo method applied in (27). In the case where the
 mismatch is slow, the terms related to the signal of inter-
 Fig. 6. ECDF curves of the achievable rates of cell-free systems est and interference are reduced to Gaussian products, as in
 with = 15 and different levels of RF mismatch.
 systems without mismatch.

 (a)NLoS
 (a)NLoS, K=15 1
 1

 0.8
 0.8

 0.6
 ECDF

 0.6
ECDF

 0.4 K=25, MC in (28), LS
 0.4 K=25, Theoretical, LS
 K=25, MC in (28), LMMSE
 MC in (28), fast mismatch 0.2 K=25, Theoretical, LMMSE
 0.2 MC in (28), slow mismacth K=15, MC in (28)
 MC in (28), without mismatch K=15, Theoretical
 0
 0 0 1 2 3 4 5 6 7 8 9 10
 0 1 2 3 4 5 6 7 8 9 10 Achievable Rates (bits/s/Hz)
 Achievable Rates (bits/s/Hz)
 (b)LoS
 (b)LoS, K=15 1
 1

 0.8
 0.8

 0.6
 ECDF

 0.6
ECDF

 K=25, MC in (28), LS
 0.4
 K=25, Theoretical, LS
 0.4
 K=25, MC in (28), LMMSE
 K=25, Theoretical, LMMSE
 MC in (28), fast mismatch 0.2
 K=15, MC in (28)
 0.2 MC in (28), slow mismatch
 K=15, Theoretical
 MC in (28), without mismatch
 0
 0 0 1 2 3 4 5 6 7 8 9 10
 0 1 2 3 4 5 6 7 8 9 10 Achievable Rates (bits/s/Hz)
 Achievable Rates (bits/s/Hz)

 Fig. 7. ECDF curves of the achievable rates of cell-free systems
 Fig. 8. ECDF curves of the achievable rates obtained with Monte
 with = 15, without mismatch and with high level reci-
 Carlo simulation in (27) and by using the analytical ex-
 procity errors due slow and fast varying RF mismatch.
 pression in a cell-free system affected by high level slow
 reciprocity error in the AP and UE.

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 247

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[20] OZDOGAN, O., BJÖRNSON, E., ZHANG, J. Performance of cell- actions on Communications, 2013, vol. 61, no. 4, p. 5205–5219.
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[21] DUARTE, R. M., ALENCAR, M. S., LOPES, W. T.
 A., et al. Cell-free systems performance under RF mis-
 match. In Proceedings of the IEEE Latin-American Confer-
 ence on Communications (LatinCom). Salvador (Brazil), 2019.
 About the Authors . . .
 DOI: 10.1109/LATINCOM48065.2019.8938026
 Rafael DUARTE was born in Recife, Pernambuco, Brazil.
[22] DUARTE, R. M., ALENCAR, M. S., LOPES, W. T. A., et al. Perfor- He received the B.Sc. and M.Sc. degrees in Electrical Engi-
 mance of a cell-free MIMO under RF mismatch. In Proceedings of
 the 22nd ACM International Conference on Modeling, Analysis and neering from Federal University of Paraíiba, Brazil, in 2014
 Simulation of Wireless and Mobile Systems (MSWiM). Miami Beach and 2016, respectively. He currently belongs to the post grad-
 (USA), 2019, p. 207–210. DOI: 10.1145/3345768.3355943 uation in Electrical Engineering of the Federal University of
[23] ROGALIN, R., BURRSALIOGLU, O. Y., PAPADOPOULOS, Campina Grande (UFCG), in order to obtain a D.Sc. degree
 H., et al. Scalable synchronization and reciprocity calibration in electronics and telecommunications.
 for distributed multiuser MIMO. IEEE Transactions on Wire-
 less Communications, 2014, vol. 13, no. 4, p. 1815–1831. Marcelo ALENCAR received his Bachelor Degree in Elec-
 DOI: 10.1109/TWC.2014.030314.130474 trical Engineering, from Federal University of Pernambuco
 (UFPE), Brazil, his Master Degree in Electrical Engineering,
[24] ALCATEL-LUCENT. document TSG RAN WG159, R1-
 100426: Channel Reciprocity Modeling and Performance from Federal University of Paraíıba (UFPB), Brazil, and his
 Evaluation. [Online] Cited 2020-08-05. Available at: Ph.D. from the University of Waterloo, Canada. He worked
 https://www.3gpp.org/DynaReport/TDocExMtg–R1-59–27294.html for the Federal University of Paraiba (UFPB), for the Fed-
[25] IBRAHIM, A. A. I., ASHKHMIN, A., MARZETTA, T. L., et al. Cell-
 eral University of Campina Grande (UFCG), for the State
 free massive MIMO systems utilizing multi-antenna access points. In University of Santa Catarina (UDESC), and for the Federal
 2017 51st Asilomar Conference on Signals, Systems, and Computers. University of Bahia, Brazil. Since 2019, he is with the Senai
 Pacific Grove (USA), 2017. DOI: 10.1109/ACSSC.2017.8335610 Cimatec, Salvador, Brazil. He is founder and President of the
[26] MAI, T. C., NGO, H. Q., DUONG, T. Q. Cell-free massive MIMO Institute for Advanced Studies in Communications (Iecom).
 with multi-antenna users. In 2018 IEEE Global Conference on Sig- He published 26 books and over 450 papers, and received
 nal and Information Processing (GlobalSIP). Anaheim (USA), 2018. achievement awards from the Brazilian Telecommunications
 DOI: 10.1109/GlobalSIP.2018.8646330
 Society (SBrT), from the Medicine College of the Federal
[27] ZHENG, K., OU, S., YIN, X. Massive MIMO channel models: University of Campina Grande (UFCG), and from the Col-
 a survey. International Journal of Antennas and Propagation, 2014, lege of Engineering of the Federal University of Pernambuco
 vol. 2014. DOI: 10.1155/2014/848071
 (UFPE).
[28] 3RD GENERATION PARTNERSHIP PROJECT (3GPP).
 Waslon LOPES was born in Petrolina, Pernambuco, Brazil.
 Technical Specification Group Radio Access Network: Spa-
 tial Channel Model for Multiple Input Multiple Output He received the B.Sc. and M.Sc. degrees in Electrical Engi-
 (MIMO) Simulations (Release 15) - TR 25.996 V15.0.0 neering from Federal University of Paraíba, Brazil, in 1998

RADIOENGINEERING, VOL. 30, NO. 1, APRIL 2021 249

and 1999, respectively. He received his D.Sc. degree in York, US. He has published more than 100 papers in journals
Electrical Engineering from Federal University of Campina and conferences.
Grande, Brazil in June, 2003. Prof. Waslon Terllizzie was
 Wamberto QUEIROZ received the B.Sc. degree in 1997
with AREA1 College of Science and Technology, Salvador,
 and the M.Sc. degree in 2002 from Federal University of
Brazil from August 2003 to December 2009. From Novem-
 Paraíba and the D.Sc. degree from Federal University of
ber 2018 to October 2019 he was visiting professor in the
 Campina Grande, Campina Grande, PB, in 2004, all in Elec-
University of Toronto, Canada. Currently, he is with de-
 trical Engineering. From July 2004 to January 2005 he
partment of Electrical Engineering of Federal University of
 was with University Tiradentes, Aracaju, SE. From February
Paraíba, Brazil. His research interests include robust vector
 2005 to January 2007, he was with University of Fortaleza,
quantization, wireless communication systems, communica-
 CE, and from February 2007 to May 2010 he was with Fed-
tion theory, and digital signal processing.
 eral University of Ceará. Since June 2010 he has been with
Fabrício CARVALHO received his D.Sc. degree in 2015, the Department of Electrical Engineering of Federal Univer-
his M.Sc. degree in 2006 and his B.Sc. degree in 2003 all in sity of Campina Grande. His research interests include signal
Electrical Engineering from Federal University of Campina spatial processing, diversity systems, channel modeling and
Grande (UFCG), in Brazil. He is Assistant Professor at the estimation of parameters for wireless communication.
Federal University of ParaĂba (UFPB), in Brazil. He is the
 Danilo ALMEIDA was born in Itabaiana, Paraııba, Brazil in
founder and the leader of the Research Group in Communi-
 1989. He received his B.Sc. and M.Sc degrees in Electrical
cations and Signal Processing (GCOMPS). He is an IEEE
 Engineering from the University Federal of Campina Grande,
Senior Member. He received the Young Professional Award
 Brazil, in 2016 and 2019, respectively. He is currently work-
2018 from IEEE ComSoc Latin America. He is one of the
 ing towards the D.Sc. degree at the same university. His
authors of the book "Spectrum Sensing Techniques and Ap-
 research interests include channel modeling and estimation
plications", published in 2017 by Momentum Press, in New
 of parameters for wireless communication.