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DOI: 10.1109/ACCESS.2019.2956828
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BER Evaluation of Linear Detectors in Massive
MIMO Systems Under Imperfect
Channel Estimation Effects
CARLOS DANIEL ALTAMIRANO 1,4, (Student Member, IEEE), JUAN MINANGO 2, HENRY CARVAJAL MORA 3, (Member, IEEE), AND CELSO DE ALMEIDA 4
1Electrical, Electronics and Telecommunications Department (DEEL), Universidad de las Fuerzas Armadas-ESPE, Sangolquí 171103, Ecuador 2Institute for the Promotion of the Human Talent (IFTH), Secretaria Nacional de Educación Superior, Ciencia, Tecnología e Innovación (SENESCYT), Quito 1701518, Ecuador
3Faculty of Engineering and Applied Sciences (FICA), School of Telecommunications Engineering, Universidad de Las Américas (UDLA), Quito 170125, Ecuador
4School of Electrical and Computer Engineering, University of Campinas (UNICAMP), Campinas-SP 13083-970, Brazil
Corresponding author: Carlos Daniel Altamirano (cdaltamirano@espe.edu.ec)
ABSTRACT New perspectives for wireless communications have brought new techniques, such as a very large number of antennas at a base station (BS) serving multiple user terminals (UTs) with a single antenna each, known as massive MIMO (M-MIMO). M-MIMO linear detectors, such as maximal-ratio combining (MRC), zero-forcing (ZF) or minimum-mean-square error (MMSE) can achieve excellent performance with low complexity due to the channel hardening property. However, imperfect channel estimation produces a penalty in the performance. An average bit error rate (BER) performance analysis over time-invariant channel is presented for M-MIMO systems under imperfect channel estimation in contrast with most of M-MIMO literature that uses the ergodic capacity approach. Closed-form expressions and bounds to evaluate the average BER are derived for MRC, ZF and MMSE detectors in a unicellular environment considering
M-QAM modulation. Furthermore, an expression to evaluate the normalized signal-to-noise ratio (Eb/N0) penalty due to the imperfect channel estimation is presented. Montecarlo numerical simulations are used to verify the tightness of the derived equations which are a function of the number of BS antennas, number of users, coherence time interval, number of pilot symbols and the Eb/N0of pilot and data symbols used for channel estimation and data detection.
INDEX TERMS BER, imperfect channel estimation, massive MIMO, maximal-ratio combining, minimum-mean-square error, zero-forcing.
I. INTRODUCTION
The technological transition to fifth generatio (5G) systems is expected to increase a thousand-fold higher through-put [1]–[3]. Massive MIMO (M-MIMO) has emerged as one of the most promising technologies towards this direction, because M-MIMO includes a very large number of antennas at the base station (BS) serving a reduced number of user ter-minals (UTs) which are tipically equipped with one antenna. Thus, offers more degrees of freedom at BS multiplying the cellular throughput [4], [5].
Several works show that M-MIMO had brought bene-fits over ordinary MIMO, allowing reliable communication, higher throughput, and power efficiency by using simple The associate editor coordinating the review of this manuscript and approving it for publication was Chenhao Qi .
linear processing. In a unicellular scenario, linear detec-tors, such as maximal-ratio combining (MRC), zero-forcing (ZF) and minimum-mean-square error (MMSE) behave nearly optimal1, as the number of BS antennas goes to infinity [4]–[7]. M-MIMO take advantage of two concepts: the channel favorable propagation regime [8], defined by the mutual orthogonality among channels and the channel hardening [9], which means that the channel randomness impact is negligible on the communication performance, i.e. the channel behaves as if there is no fading.
By taking advantage of the M-MIMO properties, some algorithms to reduce the complexity of linear detectors are proposed in [10] and [11]. In particular, in [10] is proposed a low-complexity MMSE detector algorithm based
1The performance is similar to the Maximum-Likelihood detector
on the Damped Jacobi method to determine the optimum and quasi-optimum damped parameter by exploiting the mas-sive MIMO channel property of asymptotic orthogonality. Moreover, in [11], the authors propose a novel computation-ally efficient data detection algorithm based on the modified Richardson method that outperforms the existing methods and achieves near-MMSE performance with a significantly reduced computational complexity.
In most of the literature, the performance of M-MIMO is evaluated in terms of the ergodic capacity, where the bit error rate (BER) and modulation do not take part [4]–[7], [12]–[15]. However, other more practical performance evaluations could be used, such as the BER, pair-wise error probability (PEP) or outage probability [16]–[19]. Further-more, those performance measures are a function of the signal-to-interference plus noise ratio (SNIR) [20], [21]. For M-MIMO, the SNIR is derived in [8] for MRC, ZF and MMSE detectors and it is used to assess the ergodic capacity. In contrast, in this work, the SNIR is used to evaluate the average BER.
The BER performance of MIMO with perfect channel estimation (or perfect channel state information - PCSI) for MRC and ZF detectors is analyzed in [22] and [23], respec-tively. For the MMSE detector, with PCSI, a closed-form BER expression is derived in [24] and a symbol error rate (SER) in [25].
For M-MIMO, the BER is evaluated in [26] for the MRC detector considering PCSI. The authors derive a SNIR expression and present an approximate expression to evaluate the BER for multilevel quadrature amplitude modu-lations (M -QAM). For the ZF detector with PCSI, the SER is obtained in [18] and [27], considering multilevel quadrature amplitude modulations (M -QAM) and phase-shift keying (M -PSK), respectively.
In a real scenario, it is not possible to have PCSI and channel estimation is performed to obtain an imperfect chan-nel state information (ICSI). The chanchan-nel estimation can be performed by using the maximum-likelihood (ML) estima-tor or the minimum-mean-square (MMSE) estimaestima-tor over a set of pilot symbols known at the receiver [28]. For channel estimation, one technique consist in transmitting pilot sym-bols multiplexed with data symsym-bols. The multiplexed pilot estimation technique is studied in [29], it shows how many pilot symbols are required for a reliable communication in MIMO and the effects on the ergodic capacity.
In [30], an approximated expression is derived to evaluate the average BER of the MRC detector with ICSI, with this aim the MMSE channel estimator is employed considering the minimum number of pilot symbols in a relay network with M-MIMO. On the other hand, in [31], the ZF detector is studied, and a simple average BER expression is pre-sented for scenarios with ICSI and M -QAM modulation. The derived expression is valid for MIMO and M-MIMO. However, in [31], the ICSI depends on an arbitrary parameter introduced to model the channel estimation error. In addition, MMSE channel estimation is not considered in the analysis.
Finally, in [32], the PEP is evaluated for ZF and MMSE detectors considering ICSI using the MMSE detector with a fixed number of pilot symbols. In this work, M-MIMO is considered in a multi-cell scenario where M -PSK modulation is used.
By the above, in the literature, there is a lack of studies of linear detectors for M-MIMO systems at bit-level, evaluating studying the BER for M-QAM modulation, under the effects of ICSI using the MMSE channel estimator for different number of pilot symbols. Besides, in the literature we have not found expressions that shows the penalty in Eb/N0when ICSI systems are compared with PCSI systems. Finally, for systems with ICSI it is necessary to the energy expended during channel estimation and data detection, and the trade-off between the BER and the spectral efficiency.
In this paper, the average BER for the uplink of unicellular M-MIMO systems using MRC, ZF and MMSE detectors under ICSI for flat fading time-invariant channel is presented. The MMSE estimator is employed for channel estimation using orthogonal pilot symbol sequences of different lengths. Simple and exact expressions are derived for each detector in order to evaluate the average BER. For this purpose, M-QAM modulation, N antennas at the BS and K UTs are considered. Besides, BER lower bounds for M-MIMO systems, i.e. N K , by considering the channel hardening properties are introduced, as well as BER upper bounds in the asymptotic region. Furthermore, expressions that evaluate the system performance loss due to the ICSI, in terms of
Eb/N0, are derived for the ZF detector. Finally, an study of channel estimation with pilot symbol sequences of different lengths, in order to maximize the system spectral efficiency and minimize average BER is presented.
The average BER expressions show similar performance between ZF and MMSE detector for N K and poor performance for the MRC detector which presents BER floor. Furthermore, the lower bounds are tighter as the ratio
N/K → ∞. Besides, as the channel estimation quality
wors-ens, the BER performance can be improved by increasing the energy at no cost of spectral efficiency or increasing the number of pilot symbols at the cost of spectral efficiency.
The remainder of this paper is organized as follows. SectionIIpresents the system and channel models. SectionIII
shows the channel estimation strategy and the system model considering imperfect channel estimation. SectionIV
presents the SNIR of MRC, ZF and MMSE detectors. The exact performance evaluation in terms of the average bit error rate is derived in SectionV. Section VIpresents the BER bound for M-MIMO systems. Numerical results and simulations are presented in SectionVII. Finally, the conclu-sions are drawn in SectionVIII.
Notation: Column vector and matrices are represented
by lower, x, and upper case boldface X. In addition, (·)T, (·)H,(·)−1,(·)+
, Tr [·] and k·k denote transpose, conjugate transpose, matrix inversion, pseudo-inverse matrix, matrix trace and vector norm, respectively. INrepresents the N × N identity matrix and xij is the i-th row, j-th column element
of matrix X. Finally, E {·} or x denotes expectation, Var {·} variance and Cov {·} covariance operators of its argument.
II. SYSTEM AND CHANNEL MODELS
Consider the uplink of a unicellular M-MIMO system employing N antennas at the BS to simultaneously serve K single-antenna UTs, where N K . Thus, at each symbol time interval, the received signal vector, y of dimension N ×1, at the BS is given by2: y = Hx 2 +w, = K X j=1 hj xj 2 +w, (1)
where x = [x1x2 · · · xK]T is the K × 1 transmitted symbol vector from K users selected from a M -QAM constellation with power |xj|2=23(M − 1)A2j, where Ajis the signal ampli-tude and M is the modulation order. H = [h1h2 · · · hK] is the N × K slow flat Rayleigh fading channel matrix, whose entries hi,j = αi,jexpjφi,j are independent and identically distributed (i.i.d.) complex Gaussian random variables, i.e.
hi,j ∼CN
0, α2, whereαi,jis the Rayleigh fading ampli-tude with second momentα2,φi,jis the uniformly distributed phase over the interval [0, 2π) and j =
√
−1. Finally, w rep-resents the complex additive white Gaussian noise (AWGN) vector, whose entries are i.i.d random variables with distri-bution CN 0, σ2, whereσ2 = N0
2Ts is the noise variance
3,
N0is the one-sided noise power spectral density and Tsis the symbol time interval.
A. TIME-INVARIANT FADING CHANNEL MODEL
The time-invariant channel model considers that the fading remains invariant during a time-interval TB = LTs, which is known as coherence-time interval, where L = Lp+ Ld is the block of transmitted symbols from each UT to the BS with Lppilot symbols and Lddata symbols. The pilot symbols are employed to perform channel estimation and they do not convey information, reducing the effective average symbol energy used in data transmission and the spectral efficiency.
B. AVERAGE SYMBOL ENERGY
For comparing systems with perfect channel estimation and imperfect channel estimation, it is necessary define the aver-age symbol energy of each one of them.
1) PERFECT CHANNEL ESTIMATION
For ideal PCSI, the total block energy of the jth UT transmit-ted symbols is given by4:
EsT = LD |x|2
2 Ts, (2)
2The1
2factor is introduced by carriers of bandpass transmission. 3The complex variance isσ2=σ2
I +σQ2whereσI2=σQ2 = N0
4Ts is the noise variance per dimension.
4For easiest notation the susbscript j it is not written down. This is
reasonable for perfect power control systems, where the signals of all UT are received with same power, which is considered in this work.
FIGURE 1. Block for Multiplexed Pilot Estimation.
where LD is the block length, |x|2 is mean power of data symbol and Tsis the symbol time period. The average symbol energy of the jth UT is given by:
Es =
EsT
LD = Es (3)
where Es =|x|
2
2 Tsis the symbol energy. The average energy per bit is given by Eb= Es/ log2M
The block symbols time-interval duration is given by:
TB= LDTs (4)
2) MULTIPLEXED PILOT ESTIMATION
For MP channel estimation, data symbols are time multi-plexed with pilot symbols as shown in Fig.1. The total block energy of the jth UT transmitted symbols is given by:
EsT = LpEsp+ LdEsd, (5)
where Esp = |xp|
2
2 T 0
s is the pilot symbol energy5, Esd = |xd|2
2 T 0
s is the data symbol energy and T 0
s is the symbol time period with imperfect estimation.
For a fair comparison between systems with PCSI and ICSI, note that both systems must transmit the same number of data symbols, i.e., LD= Ld. Furthermore, by considering the same TBfor PCSI and ICSI, the time symbol period is:
Ts=(1 + η) Ts0 (6) whereη = LpL
d. Observe that the time symbol period T 0 s for MP is reduced which represents a bandwidth expansion.
For a fixed EsT the relation between energies of PCSI and ICSI is given by6:
LDEs = LpEsp+ LdEsd (7)
Es =ηEsp+ Esd (8)
Finally, by introducing the ratioµ = EEsp
sd, the jth UT pilot symbol energy in terms of the average symbol energy is given by:
Esp = µ
(1 +ηµ)Es (9)
and the multiplexed data symbol energy is:
Esd = 1
(1 +ηµ)Es (10) 5The pilot symbol is deterministic because it is known at the receiver. 6In terms of the symbol power |x|2= 1+ηη |xp|2+1+η1 |xd|2.
C. SPECTRAL EFFICIENCY
The spectral efficiency defined as the ratio between the total throughput and the total system bandwidth and evaluated in bits/s/Hz, is given by:
ε = PK
j=1Rb,j
B (11)
where K is the number of user terminals in the cell, Rb,jis the
kth user bit rate and B is the system bandwidth.
1) PREFECT ESTIMATION
The cellular spectral efficiency for a system with PCSI is given by:
ε = K log2M (12)
where the K users transmit at the same bit rate Rb=log2 M, the bandwidth B = 1/Ts = 1 is normalized and M is the modulation order.
2) MULTIPLEXED PILOT
For MP the bandwidth is B = 1/T0
s. There is a bandwidth expansion as shown in (6), and the bandwidth increases to
B =1+η. By considering the increased bandwidth, the spec-tral efficiency is given by:
ε = Klog2M
1 +η (13)
whereη = Lp Ld.
III. CHANNEL ESTIMATION
For performing coherent detection the channel state infor-mation is required by the receiver at the BS. In practice, this is obtained by estimating αi,j and φi,j from the chan-nel matrix H. From estimation theory, chanchan-nel estimation can be performed by using the maximum-likelihood (ML) estimator or the minimum-mean-square (MMSE) estimator. At the cost of some a-priory knowledge and complexity, the MMSE estimator presents better performance [28] and is used in this work.
For the MP estimation technique, the multiplexed pilot symbols are orthogonal sequences of length Lp ≥ K, which guarantee a reliable MMSE channel estimation free of inter-ference [29]. Thus, the received pilot matrix Ypof dimension
N × Lpis given by:
Yp=H Xp
2 +W, (14)
where Xpis a K ×Lpmatrix whose elements are the transmit-ted pilot symbols from K users, which are known by the BS, and W is an N × Lpcomplex additive white Gaussian noise matrix.
Applying the MMSE estimator [28], the conjugate trans-pose of the estimated channel matrix is given by:
b HH =RHHYH pR −1 YH pYHp YHp, (15) where RHHYH p = EH HYp and R YH pYHp = E n YH pYp o
are the covariance matrices. Using (14) and the covariance matrices in (15), the conjugate transpose of the estimated channel matrix is:
b HH =α 2 2 Xp α2 4 X H pXp+σ2ILp !−1 YHp, (16) where we have considered that EHHH = α2N I
K and EWHW = σ2 N I
Lp. Notice that a-priory knowledge of the noise varianceσ2and the fading second momentα2are required to evaluate (16).
Considering perfect power control, the transmitted pilot symbols matrix Xpthat minimize the channel estimation error and eliminates the interference during the channel estimation is an orthogonal matrix (e.g. a Hadamard matrix) satisfy-ing the condition XpXHp = |xp|2LpIK, where K must be a power of 2 that is K = 2n for n ≥ 0 and Lp = mK for m ≥ 1.
Applying the matrix inversion theorem7, considering orthogonal pilot symbols matrix and performing the conju-gate transpose operation in (16), the estimated channel matrix is given by: b H = α2 2 α2 4|xp|2Lp+σ2 YpX H p. (17)
Accordingly to the MMSE estimator principles [28], the estimated channel matrix bH and the estimation error matrix eH = H − bH are independent random matrices with the same distribution of H. This aspect is shown in AppendixA. Therefore, the entries of bH and eH have distributions:
ˆ hi,j ∼CN 0, α2% , (18) ˜ hi,j ∼CN h 0, α2(1 − %)i , (19) whereα2% is the variance of the estimated channel and % is the imperfection introduced by the channel estimation, defined in AppendixAas: % = 1 + (1 +µη) α2Lpµ log 2(M )NEb0 −1 . (20)
The channel estimation imperfection is bounded by 0< % ≤ 1, where the upper bound stands for perfect channel estimation and the lower bound for completely imperfect channel estimation.
A. SYSTEM WITH CHANNEL ESTIMATION ERROR
The received signal vector, given by (1), during the data detection, can be rewritten in terms of the estimated channel
7AAHA + I N −1 = AAH+IM −1 A, where A is a matrix of dimension M × N
and the channel estimation error as: yd =Hb xd 2 +eH xd 2 +w, (21) = K X j=1 ˆ hj xd,j 2 + K X j=1 ˜ hj xd,j 2 +w. (22) Observe that the channel estimation error ˜h can be seen as an additional interference.
IV. LINEAR DETECTION
Linear detectors are presented in this section. In particular, MRC, ZF and MMSE detection can be performed after the channel estimation, bH is known by the receiver at the BS.
A. MAXIMAL RATIO COMBINING DETECTOR
The detected data symbol vector at MRC output is given by: ˆ
xd =bHHyd. (23)
The detection of the kth user data symbol can be rewritten as:
ˆ
xd,k = ˆhHkyd, (24)
which is equivalent to:
ˆ xd,k = ˆhHkhˆk xd,k 2 | {z } Signal + K X j=1 j6=k ˆ hHkhˆj xd,j 2 | {z } MAI + K X j=1 ˆ hHk h˜j xd,j 2 | {z } CEEI + ˆhHk w | {z } Noise , (25)
where (22) was substituted in (24). In (25) the first term is the
kth user signal, MAI is the the multiple access interference, CEEI is the channel estimation error interference and the last term is the noise.
Appendix B shows that the instantaneous SNIR condi-tioned on the kth user vector channel is given by:
γs| ˆh k = |xd,k| 2k ˆh kk2 α2|x d|2% (K − 1)+α2(1 − %) |xd|2K +4σ2 , (26) where k ˆhkk2 ∼ χ2(2N ) follows a chi-square distribution with 2N degrees of freedom.
It is clear that the SNIR also follows a chi-square dis-tribution γs ∼ χ2(2N ). Previously, we have indicated that σ2 = N0
2Ts and since perfect power control is considered, |xd|2 =
2Esd
Ts , Es =(1 +µη)Esd and Eb = Es/ log2M, for all UT. Employing these results in (26), the mean SNIR can be written as: γs= N% (K − %) + (1 +µη) α2log 2(M ) Eb N0 −1 , (27) and the mean SNIR per channel isγc=γs/N.
The SNIR can be reduced to the PCSI scenario by using % = 1 µ = 1 and η = 0 which is equivalent to the SNIR expression derived in [26].
B. ZERO FORCING DETECTOR
The detected data symbol vector at the ZF output is given by: ˆ xd =Hb+yd, (28) where bH+ = b HH b H−1 b
HH is the Moore-Penrose pseudo-inverse of bH. The detection of the kth user symbol can be rewritten as: ˆ xd,k = ˆh+kyd, (29) or equivalently, ˆ xd,k = ˆh+khˆkxd2,k +PKj=1 j6=k ˆ h+khˆj xd,j 2 +PKj=1hˆ+kh˜jxd2,j + ˆh+kw, ˆ xd,k = xd,k 2 |{z} Signal + K X j=1 ˆ h+kh˜j xd,j 2 | {z } CEEI + ˆhHkw | {z } Noise , (30)
where (22) was substituted in (29) and ˆh+k is the kth row of bH+. Notice that the MAI is eliminated due to the channel inversion. Specifically, it has been employed that:
ˆ h+khˆj=
1 if k = j
0 if k 6= j (31) Appendix C shows that the instantaneous SNIR condi-tioned on the kth user vector channel is given by:
γs| ˆh k = |xd,k|2 k ˆh+kk2α2|x d|2K(1 − %) + 4σ2 , (32) where 1 kh+kk2 ∼ χ 2[2(N − K + 1)] follows a chi-square distribution with 2(N − K + 1) degrees of freedom [33].
The SNIR also follows a chi-square distribution γs ∼ χ2[2(N − K + 1)], where the mean SNIR is given by:
γs=(N − K +1)% K(1 −%) + (1 +µη) α2log 2(M )EbN0 −1 , (33) and the mean SNIR per channel isγc=γs/(N − K + 1).
Just as for the MRC detector, the SNIR of ZF detector can be reduced to the PCSI scenario by using% = 1, µ = 1 and η = 0. In this case, our derived SNIR expression is equivalent to the SNIR given in [18] and [27] for unicellular systems.
C. MINIMUM-MEAN-SQUARE ERROR
The detected data symbol vector at the MMSE output is given by:
bxd=Ayd, (34) where A = RxdydR
−1
ydyd is the compensation matrix, which is the MMSE estimator, defined by product of the covariances
matrices Rxdyd and Rydyd. By using (21), the covariance matrices are given respectively by:
Rxdyd = |xd|2 2 Hb H, (35) Rydyd = |xd|2 4 HbbH H+" α2 4 (1 −%)|xd| 2K +σ2 # IN. (36) The detection of the kth user data symbol, can be rewritten as:
ˆ
xd,k=aHkyd, (37)
which is equivalent to: ˆ xd,k=aHk hˆk xd,k 2 | {z } Signal + K X j=1 j6=k aHkhˆj xd,j 2 | {z } MAI + K X j=1 aHk h˜j xd,j 2 | {z } CEEI +aHkw | {z } Noise , (38) where (22) was substituted in (37) and akis the kth row of A. Appendix D shows that the instantaneous SNIR condi-tioned on the kth user vector channel is given by:
γs| ˆh k = K −1 X j=1 |xd,k|2| ˆhj,k|2 |xd|2λj+α2(1 −%)|xd|2K +4σ2 + N X j=K |xd,k|2| ˆhj,k|2 α2(1 −%)|xd|2K +4σ2, (39) whereλjis the jth eigenvalue of bHkbH
H
k and bHkis defined as b
H without its kth column.
The SNIR follows a generalized chi-square distribu-tion [25]. The SNIR is also condidistribu-tioned on the eigenval-uesλjand since the eigenvalues distribution is an elaborated expression [34], obtaining the mean SNIR is a quite complex task as shown in [25]. However, from the results of [24], for our scenario, it is possible to show that the mean SNIR per channel isγc=γs/(N −K +1) where γsis the same obtained for the ZF detector in (33). For the PCSI scenario, it must be used% = 1, µ = 1 and η = 0 on the SNIR expressions.
In M-MIMO systems, the SNIR distribution of the MMSE detector given by (39) can be approximated by a chi-square distributionγs∼χ2[2(N −K +1)]. Notice that (39) is divided in two summations depending on j, where the denominator of the first sum depends on the eigenvaluesλj 6= 0 and the denominator of the second sum on λj = 0, as indicated in the Appendix D. Thus, the elements of the second summation are greater than the elements on the first summation. Besides, the first summation has K − 1 terms and the second summa-tion N − K + 1 terms. Finally, by the above and considering
N Kthe second summation has more elements, and hence,
this sum is dominant, and the first summation can be consid-ered negligible compared with the second summation. As a consequence, the SNIR distribution of the MMSE detector in M-MIMO systems is equivalent to the chi-square distribution of the ZF detector given by (32)
V. EXACT AVERAGE BIT ERROR RATE
Closed-form expressions to evaluate the exact average BER are derived in this section. Linear detectors are presented in this section. Using the SNIR, it is possible to derive the average BER by [21]:
Pb= Z ∞
0
P(b|γs) f0s(γs) dγs (40) where P(b|γs) is the bit error probability conditioned on the instantaneous SNIR and f0s(γs) is the probability density function (PDF) of the SNIR.
For a M -QAM modulation using Gray mapping, the exact BER, conditioned on the instantaneous SNIR is given by [35]:
P(b|γs) = log2√M X κ=1 1 √ Mlog2 √ M (1−2−κ)√M −1 X i=0 −1b i·√2κ−1 M c × 2κ−1− i · 2 κ−1 √ M +1 2 ×erfc " (2i + 1) s 3 2(M − 1) γs #) , (41) where erfc(x) = √2 π R∞ x e−t 2
dtis the complementary error function and bxc is floor operation, that is, the greatest integer less than or equal to x.
Using (41) in (40), the average BER is given by:
Pb= log2√M X κ=1 2 √ Mlog2 √ M (1−2−κ)√M −1 X i=0 −1b i·√2κ−1 M c × 2κ−1− i · 2 κ−1 √ M +1 2 I(νi) , (42) where I(νi) is an integral defined as:
I(νi) = 1 2 Z ∞ 0 erfc rγs νi f0s(γs) dγs (43)
which depends on the SNIR PDF and onνi= 2(M−1) 3(2i+1)2. A. BER FOR MRC DETECTOR
For the MRC detector, the solution of (43) is obtained employing the SNIR PDF given by (26). Hence, from [33] the solution of (43) is:
I(νi) = pN N −1 X j=0 N − 1 + j j (1 − p)j (44) where p = 121 −q γs Nνi+γs ,γsis given by (27) and(nx) = n!
x!(n−x)! is the binomial expansion. Consequently, the average BER for the MRC detector is obtained by substituting (44) in (42). Performing asymptotic expansion of (44), it is easy to show that the diversity order of the MRC detector is N .
B. BER FOR ZF DETECTOR
For the ZF detector, the solution of (43) using the SNIR PDF given by (32) is: I(νi) = pN −K +1 N −K X j=0 N − K + j j (1 − p)j (45) where p = 121 −q γs (N −K +1)νi+γs andγsis given by (33). The average BER for the ZF detector is obtained by sub-stituting (45) in (42). Note that the solution of (43), given by (45), is similar to (44), except for the diversity order of N − K + 1.
C. BER FOR MMSE DETECTOR
For the MMSE detector, the solution of the integral expres-sion given in (43) is quite complex. Nevertheless, it is solved in [24] using the channel reliability approach8, which is given by: I(νi) = 1 2 1 − s 1 2νi N −K +1 X j=1 1 0(j) N − K + 1 γs j−1 ×J j − 1 2, 1 νi + N − K +1 γs + N X j=N −K +2 1 0(j) N − K + 1 γs j−1 N −j X l=0 K − 1 l ×L j + l −1 2, K − 1, 1 νi + N − K +1 γs , (46) where J (x, y) = 0(x)y−x, (47) L(x, y, z) = 0(x) 0(y − x) 0(y) 1F1(x; x − y + 1; z) + zy−x0(x − y) 0(x) 1F1(y; y − x + 1; z) , (48) 0(·) is the gamma function,1F1(·; ·; ·) is the confluent hyper-geometric function [36] and γs is given by (33) for our scenario.
VI. MASSIVE MIMO BER
In the SectionV, the derived average BER expressions are exact and they are function of N and K for MRC, ZF and MMSE detectors. In this section, a lower bound of the aver-age BER is derived for M-MIMO systems by considering that N K .
Exploiting the channel hardening properties of M-MIMO [8], it is possible to apply Jensen’s inequality9 in (40). Therefore, the lower bound of the average BER is given by:
PbLB≤ P(b|γs). (49) 8The integral can also be solved by using the SNIR PDF approach shown
in [25]. However, the resulting expression is more elaborated.
9f(x) ≤ f (x)
Hence, it has been employed that P(b|E {γs}) ≤ E {P(b|γs)}.
By using (49) in (41), the average BER lower bound for M-MIMO systems is given by:
PbLB ≤ log2 √ M X κ=1 1 √ Mlog2 √ M (1−2−κ) √ M −1 X i=0 −1b i·√2κ−1 M c × 2κ−1− i · 2 κ−1 √ M +1 2 ×erfc (2i + 1) s 3 2(M − 1)γs !) , (50)
The average BER lower bound for the MRC detector is obtained by replacing (27) in (50) and for the ZF detector by replacing (33) in (50). For the MMSE detector, the average BER lower bound is similar to the ZF detector, because both mean SNIR are equivalent.
The average BER expressions derived in SectionsVandVI
are also valid to evaluate the performance with perfect CSI by using% = 1, µ = 1 and η = 0 on the SNIR expressions.
A. PENALTY DUE TO IMPERFECT ESTIMATION
In this section, the penalty in SNR for comparing M-MIMO systems with PCSI and ICSI at the same average BER is pre-sented. The penalty analyzes for the ZF detector is performed in the asymptotic region, that is, the high SNR region10
Appendix E shows that for high Eb/N0, the asymptotic average BER for the ZF detector with ICSI is approximated by: Pb,I≈ 2( √ M −1) √ Mlog2(M ) 2(N − K + 1) − 1 N − K +1 × (M − 1) 6 (N − K + 1) γs,I N −K +1 , (51) whereγs,Iis given by (33).
For PCSI the asymptotic average BER is approximated by:
Pb,P≈ 2( √ M −1) √ Mlog2(M ) 2(N − K + 1) − 1 N − K +1 × (M − 1) 6 (N − K + 1) γs,P N −K +1 , (52) whereγs,Pis the ZF detector mean SNIR for PCSI, obtained by using% = 1 and η = 0 in (33).
On the asymptotic region, for the same average BER,
Pb,I = Pb,P, the AppendixEshows that the penalty due to the ICSI in dB is given by:
P = Eb N0 I,dB − Eb N0 P,dB '10 log10 (1 +µη) 1 + K µLp . (53) Note in (53) that penalty depends onη, µ, Lpand K and is independent on the number of antennas N at BS and on the modulation order M .
FIGURE 2. Average BER as a function of Eb/N0for MRC, ZF and MMSE
detectors for M-MIMO systems with ICSI for N = 128 antennas at BS, K = 16 UTs, L = 250 symbols, Lp=16 pilot symbols,µ = 1 and M-QAM.
From (53) observe that a penalty of 0 dB is present for η → 0 (since η = Lp/Ld, for Ld → ∞thenη → 0) by considering Lp K andµ = 1 or Lp = K andµ 1. However, by limiting Lp = K andµ = 1 there is a penalty of 3 dB. Besides, for a bounded Ldthere is always a penalty.
VII. NUMERICAL RESULTS
In this section, simulation and numerical results are presented for M-MIMO in a unicellular scenario. For this, the fad-ing second moment is normalized, that isα2=1.
Fig.2presents the average BER as a function of Eb/N0for the MRC, ZF and MMSE detectors in M-MIMO systems with ICSI considering N = 128 antennas at the BS, K = 16 UTs,
L = 250 symbols, Lp = 16 pilot symbols, µ = 1 and
M-QAM modulation with M = 4, 16, 64. In M-MIMO systems, MMSE and ZF detectors have similar perform,11 because for N K , the SNIR of the MMSE detector given by (39) is similar to the SNIR of the ZF detector given by (32). Since the average BER depends on the SNIR, both perfor-mances are similar. This result for M-MIMO contrast with the well-known result of MIMO systems where the MMSE detector has better performance than the ZF detector [37]. On the other hand, both MMSE and ZF detectors, present superior performance over MRC. In particular, there is a BER floors caused by the MAI when MRC is used. Besides, the MAI is eliminated and there is no BER floors when ZF and MMSE is employed. Moreover, as expected, as the modulation order increases, more Eb/N0is necessary in order to maintain the same BER. The M-MIMO lower bounds are also plotted. Notice that for the ZF detector, the lower bound is tighter in the low Eb/N0 region. The same lower bound is valid for ZF and MMSE detector. Finally, the results are validated through Monte Carlo simulations.
11In Fig.2the curves for ZF and MMSE detectors are superposed
FIGURE 3. Average BER as a function of Eb/N0for MRC detector in
M-MIMO system with N = 128, 512, 1024 antennas, K = 16 UTs, L = 250 symbols, Lp=16 pilot symbols,µ = 1 and 16-QAM.
Once the theoretical expressions are validated by simula-tion in Fig.2, in the reminder of this paper, for simplicity, only theoretical curves will be plotted to analyze the performance. Fig.3shows the average BER as a function of Eb/N0for MRC detector in a unicellular M-MIMO system with PCSI and ICSI for N = 128, 512, 1024 antennas at the BS,
K = 16 UTs, L = 250 symbols, Lp = 16 pilot symbols, µ = 1 and 16-QAM modulation. Observe that the BER improves as N grows and achieves lower floor at the cost of a huge number of antennas. Furthermore, the BER penalty due to ICSI vanishes as Eb/N0increases. It occurs because the BER is limited by the MAI and not by the imperfect estimation (%). This can be verified in (27). As a consequence of that, the BER floor appears even for low Eb/N0dB. This result is also presented in [26].
In the next figures we focus only on the ZF performance. Fig.4compares the exact closed-form BER expression and the M-MIMO BER lower bound obtained in (42) and (50), respectively, with the average BER expressions derived in [18, Eq. (53)] and [27, Eq. (25)] for the PCSI case. Notice that the cited works present approximations. Specifically, [18] presents an upper bound, and [27] presents a BER lower bound. In [27], the general expression is derived for M -PSK modulation and multi-cell scenario. Hence, our comparison is limited to 4-QAM modulation and unicellular scenario.
Fig.5shows the average BER as a function of Eb/N0for M-MIMO systems with PCSI and ICSI for N = 64 and 128 antennas at the BS, K = 16 UTs, L = 250 symbols, Lp=16 pilot symbols, µ = 1 and 16-QAM. Observe that in the asymptotic region the penalty is independent of the number of BS antennas and it is approximately 3 dB as shown in (53). Furthermore, due to the high diversity, note the upper bound is tighter when the BER is lesser than 10−N. For example, for
N =128 antennas the upper bound is tight for Pb< 10−128. On the other hand, in the low Eb/N0 region, the penalty is slightly greater than that in the higher Eb/N0 region as
FIGURE 4. Average BER as a function of Eb/N0for a ZF detector in
M-MIMO systems with PCSI for N = 60, K = 10 and 4-QAM modulation.
FIGURE 5. Average BER as a function of Eb/N0for ZF detector in M-MIMO system with N = 64, 128 antennas, K = 16 UTs, L = 250 symbols, Lp=16 pilot symbols,µ = 1 and 16-QAM.
shown in the inside box of Fig.5. Notice that in the low SNR region, a BER in the order of 10−4it is achieved using the ZF or MMSE detector12, focusing in that operation region, M-MIMO it is considered energy efficient. Besides, in that the lower bound given by (50) it is a good approximation.
Fig.6presents the average BER as a function of the ratio η = Lp/Ld for M-MIMO systems with PCSI and ICSI
N = 128 antennas, K = 16 UTs, L = 250, 1000, 10000,
µ = 1, Eb/N0=0 dB and 64-QAM. Asη decreases, the BER performance becomes closer to the PCSI performance and the loss in spectral efficiency becomes negligible as shown by (13). Hence, as the block length Ldincreases, the number of pilots Lp must increase too, but at a lesser rate. Observe in Fig.6 that there is an optimum number of pilot symbols that minimize the BER meeting the condition of the pilot matrix, that is orthogonality. In practice L can not grow
12Lower BER can be achieved performing channel coding.
FIGURE 6. Average BER as a function ofη for ZF detector in M-MIMO system with N = 128 antennas, K = 16 UTs,µ = 1 Eb/N0=0 dB and
64-QAM.
FIGURE 7.Average BER as a function of Eb/N0for ZF detector in M-MIMO
system with N = 128 antennas, K = 16 UTs, L = 250 symbols,µ = 1 and 64-QAM for different Lp.
unbounded and L = 10000 is an ideal scenario for time-invariant channels.
Fig.7shows the average BER as a function of Eb/N0for M-MIMO systems with N = 128 antennas, K = 16 UTs,
L = 250 symbols, Lp = 16, 32, 64, 128 pilot sym-bols, µ = 1 and 64-QAM. Observe that for low value of L there is a trade-off between spectral efficiency and BER, Furthermore, increasing the pilot symbols too much cause loss of performance in both BER and spectral efficiency. Thus, for BER equal to 10−6 and Lp = 16, it is observed a penalty of around 3.5 dB which is validated by using (53) and the spectral efficiency isε = 90.2 [bits/s/Hz] obtained by using (13). Increasing the number of pilot symbols to
Lp = 64, the penalty is minimized to approximately 2 dB at a cost of reduction in the spectral efficiency to ε = 76.43 [bits/s/Hz].
FIGURE 8. Average BER as a function of Eb/N0for ZF detector in a
unicellular M-MIMO system with N = 128 antennas at BS, K = 16 UTs, L = 250 symbols and 64-QAM modulation, for differentµ and Lp.
Fig.8shows the average BER as a function of Eb/N0for M-MIMO systems with N = 128 antennas, K = 16 UTs,
L = 250 symbols and 64-QAM for different Lp and µ. Observe that for Lp=64 andµ = 1, the penalty is minimized and the spectral efficiency isε = 71.4 [bits/s/Hz]. However, for Lp =16 andµ = 3.8, and the spectral efficiency is ε = 89.7 [bits/s/Hz]. Notice that in the second case the penalty at a lesser cost in spectral efficiency. This is explained, because more Espis being employed in order to improve the channel estimation quality. Note that increasing both Espand Lp, is not a good choice, as shown forµ = 2.3 and Lp =64, because the penalty increases, as most energy is expended in the chan-nel estimation and less energy is available for data transmis-sion. For choosing the ratio between, Lpandµ, that minimize the penalty and maximize the spectral efficiency use (53).
VIII. CONCLUSION
In this paper, closed-form and lower and upper bounds of the average bit error rate for MRC, ZF and MMSE detectors were obtained for unicellular M-MIMO systems for a time-invariant channel. The lower bounds for M-MIMO are sim-pler than the closed-form BER expressions and tight in the low Eb/N0region, while the asymptotic upper bound due to the great diversity is tight for Pb< 10−N in the high Eb/N0 region. Furthermore, an expression to evaluate the penalty in
Eb/N0due to the imperfect CSI was derived, showing that the penalty is independent of N in the asymptotic region. In the low Eb/N0region it was observed that there is a slight variation on the penalty which increases with N . Moreover, increasing the number of pilot symbols decreases the Eb/N0 penalty, but at a cost in the spectral efficiency. However, by using µ > 1, it is possible to maximize the spectral efficiency with a slight loss in the Eb/N0 penalty. Finally, as M-MIMO is a key technique of 5G systems along with low-density-parity-check-codes (LDPC) and polar codes, it is suggested to evaluate the impact of ICSI on the average BER
and the spectral efficiency in coded M-MIMO systems in future works.
APPENDIXES APPENDIX A
DISTRIBUTION OF bH AND eH
Since the channel matrix H entries are independent and iden-tically distributed (i.i.d.) complex Gaussian random variables with distribution hi,j∼CN
0, α2, it is easy to show that the mean of H is given by E {H} = 0 where 0 denotes the null matrix and the covariance matrix is given by:
Cov {H} = EnHHHo
=α2K IN. (54) Thereby, the channel matrix has Gaussian distribution, that is H ∼ CN0, α2K I
N
. As the noise W is Gaussian, the received pilot matrix (14) is also Gaussian because is the sum of two independent Gaussian matrices [28]. Therefore, the estimated channel matrix bH given in (17) that is function of Ypis also Gaussian with mean EbH = 0 and covariance matrix given by:
Cov b H =α2 α2 4|xp| 2L p α2 4|xp|2Lp+σ2 K IN =α2%KIN, (55) where% = α2|xp|2Lp α2|x p|2Lp+4σ2
is defined as the imperfection due to the channel estimation. Using that |xp|2 =
2Esp Ts ,σ 2 = N0 2Ts, Es = (1+µη)
µ Esp and Eb = Es/ log2M, the imperfec-tion due to the channel estimaimperfec-tion can be rewritten as (20). Thus, the estimated channel matrix has Gaussian distribution b
H ∼ CN0, α2%KINwith entries ˆh
i,jgiven by (18). The channel estimation error matrix eH = H − bH is the sum of two independent Gaussian random matrices. Therefore, is independent of H and bH and has the same distribution of them [28]. Thus, E
e
H = 0 is the mean and the covariance matrix is given by:
Cov e
H = Cov {H} − Cov bH
=α2(1 − %) KIN. (56) In this sense, the estimation error matrix has Gaussian distri-bution eH ∼ CN0, α2(1 − %) KINwith entries eh
i,jgiven by (19).
APPENDIX B
SNIR OF MRC DETECTOR
The SNIRγs conditioned on the kth user channel is given by [38]:
γs| ˆh k
= |Signalk| 2 VarnMAI + CEEI + Noise
ˆ hk o . (57) Notice that the elements in the denominator corresponds to variances of complex random variables.
From (25), the kth user signal power is given by: |Signalk|2= ˆ hHkhˆk xd,k 2 2 = ˆ hk 4|xd,k|2 4 . (58)
By considering that all UTs arrive with same power, the MAI variance is given by:
VarnMAI| ˆhk o =Var K X j=1 j6=k ˆ hHkhˆj xd,j 2 hˆk = α 2% 4 ˆ hk 2 |xd|2(K − 1) , (59) where was considered that all UT transmit with the same power. The variance of the interference due to the imperfect estimation is given by:
VarnCEEI| ˆhk o =Var K X j=1 ˆ hHkh˜j xd,j 2 hˆk = α 2(1 −%) 4 ˆ hk 2 |xd|2K. (60) Finally, the noise variance is given by:
VarnNoise| ˆhk o =VarnhˆHknk hˆk o = ˆ hk 2 σ2. (61)
By using (58)-(61) in (57), the SNIRγs| ˆh
k is finally given by (26).
APPENDIX C
SNIR OF ZF DETECTOR
Before deriving the SNIR of ZF detector, some properties of the pseudo inverse matrix are presented. The pseudo-inverse matrix bH+ = b HH b H−1 b
HH has covariance matrix given by [34]: CovHb + = En b H+bH+H o = 1 α2(N − K )IK, for N ≥ K − 1. (62) The variance of the kth row vector ˆh+k is given by:
Var{ ˆh+k} =E ˆ h+k 2 =Cov b H+ kk. (63)
Furthermore, accordingly with [33], 1
k ˆh+kk2 follows a
chi-square distribution with 2(N − K + 1) degrees of freedom. The SNIR conditioned on the kth user channel vectorγs| ˆh
k is given by:
γs| ˆhk = |Signalk| 2 VarnCEEI + Noise
ˆ hk
o , (64)
where the kth user signal power is given by: |Signalk|2= |xd,k|
2
2 . (65)
The variance of the interference due to imperfect channel estimation is given by:
Var n CEEI| ˆhk o =Var K X j=1 ˆ h+kh˜j xd,j 2 hˆk = α 2(1 −%) 4 ˆ h+k 2 |xd|2K. (66) Finally, the noise variance is:
VarnNoise| ˆhk o =Varnhˆ+knk| ˆh+k o =σ2 ˆ h+k 2 . (67)
By using (65)-(67) in (64), the SNIRγ
s| ˆhkis finally written as (32).
APPENDIX D
SNIR OF MMSE DETECTOR
The compensation channel vector ak, is given by [39]: ak = ˆhHk R(k)ydyd −1 , (68) where R(k)ydyd = E n y d,ky H d,k o with y
d,k being the received signal vector ydwithout the kth user entry. The matrix R(k)ydyd can be rewritten as:
R(k)y dyd= |xd|2 4 bHkbH H k + " α2 4 (1 −%)|xd| 2K +σ2 # IN, =|xd| 4 4 U HDU+"α2 4 (1 −%)|xd| 2K +σ2 # IN, (69) where bHk is the channel matrix bH without the kth column vector and bHkbH
H
k = UHDU is the eigen-factorization [34], where U is an orthonormal matrix that contains the eigen-vectors and D = diag[λ1 λ2 · · · λK −1
N −K +1 z }| {
0 · · · 0] is a diag-onal matrix that contains the eigenvalues. Notice that there are K − 1 random eigenvalues and N − K + 1 null eigenvalues [39].
The SNIR of the MMSE detectorγs| ˆh
k is given by: γs| ˆhk = |Signalk|
2 VarnMAI + CEEI + Noise
ˆ hko . (70)
By using (68) in (38), it is possible to show that the kth user signal power is given by:
|Signalk|2=|xd,k| 2 4 ˆ hHk R(k)yy−1hˆkhˆHk R(k)yy−1hˆk, (71) and the variances on the denominator:
VarnMAI + CEEI + Noise ˆ hk o = ˆhHk R(k)yy −1 ˆ hk. (72)
By replacing (71) and (72) in (70), the SNIR can be rewrit-ten as: γs| ˆh k = |xd,k|2 4 ˆ hHk R(k)yy−1hˆk. (73) By using (69) in (73) and by considering that the product by an orthonormal matrix does not change the statistics of a random matrix, after some algebraic manipulation, the con-ditioned SNIR can be rewritten as:
γs| ˆhk =PN j=1
|xd,k|2| ˆhj,k|2 |xd|2λj+α2(1−%)|xd|2K +4σ2.
(74) Finally, the SNIR of the MMSE detector in (39) is (74) split into two summations.
APPENDIX E PENALTY
Evaluating the SNR penalty using the exact average BER expression it is a quite complex task due to the presence of summations. For easiest approach, we evaluate the penalty on the asymptotic region by deriving an approximated average BER for high SNR.
For ZF detector, the solution of the integral expression given in (45) for high SNR, can be approximated by: I(νi) ≈ N − K + 1 4νiγs N −K +1 2(N − K + 1) − 1 N − K +1 (75) where it has been employed series expansion over p, that is13:
p = 1 2 1 − s γs (N − K + 1)νi+γs ! p = 1 2 1 − s 1 1 +(N −K +1)νi γs ! p ' N − K +1 4νiγs . (76)
Besides, that 1 − p ' 1 and that PN −Kj=0 N −K +jj =
2(N −K +1)−1 N −K +1
.
By replacing (75) in (42) and taking the first term of the sum over i, the approximated average BER on the asymptotic region14for the ZF detector is given by:
Pb≈ 2( √ M −1) √ Mlog2(M ) 2(N − K + 1) − 1 N − K +1 × (M − 1) 6 (N − K + 1) γs N −K +1 . (77) Considering ICSI in (77), the SNIR is given by (33) and γs = γs,I. Thus, the asymptotic average BER for ICSI is given by (51).
Furthermore, the expression (77) is valid for PCSI, by con-sidering that the SNIR isγs=γs,P, which is given by:
γs,P=α2(N − K + 1) log2(M )
Eb
N0.
(78) 13We have used the first two terms of the Maclaurin seriesq 1
1+x =1 − x 2+ 3x2 8 − 5x3 16 + · · ·.
14The BER on the asymptotic region is considered an upper bound
where it has been used% = 1 and η = 0 in (33). Thus, using (78) in (77), the asymptotic average BER for PCSI is given by (52).
On the asymptotic region considering that Pb,I = Pb,P, using (51) and (52) it is possible to show thatγs,P = γs,I, which can be written as:
α2(N − K + 1) log 2(M ) Eb N0 P =(N − K + 1)% × K(1 −%) + (1 +µη) α2log 2(M ) Eb N0 I −1 , (79) by using (20) in (79) and after some mathematical manipula-tion, it can be rewritten as:
Eb N0 P = Eb N0 I (1 +µη) " 1 +LpKµ+ (1+µη) α2LpµEb N0 I # . (80)
The dB representation of (80) is given by: Eb N0 P,dB ' Eb N0 I,dB −10 log10(1 +µη) −10 log10 1 + K Lpµ . (81) where it has been considered that (1+µη)
α2L pµ Eb N0 I '0 due to the high SNR analyses.
Finally, by rewriting (81), the penalty is given by (53).
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CARLOS DANIEL ALTAMIRANO received the degree in electronic and telecommunications engi-neering from the Universidad de las Fuerzas Armadas-ESPE, Sangolquí, Ecuador, in 2008, and the M.Sc. degree in electrical engineering from the University of Campinas (UNICAMP), Brazil, in 2011, where he is currently pursuing the Ph.D. degree. He is currently a Full Professor with the Universidad de las Fuerzas Armadas-ESPE. His current research interests include digital com-munications with specific emphasis on multiple access techniques, fad-ing channels, MIMO, error control codfad-ing, software defined radio, and 5G technologies.
JUAN MINANGO received the B.Sc. degree in electronics and telecommunications engineering from the Universidad de las Fuerzas Armadas-ESPE, Sangolquí, Ecuador, in 2011, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Campinas (UNICAMP), Brazil, in 2014 and 2019, respectively. In 2019, he was a Visiting Researcher with Laval Uni-versity, Canada. He is currently an Independent Researcher and a Scholarship Holder from the Institute for the Promotion of the Human Talent (IFTH)-SENESCYT. His research interests include digital communications with specific of digital communications, mobile communications, and MIMO.
HENRY CARVAJAL MORA received the B.Sc. degree in electronics and telecommunications engineering from the Universidad de las Fuerzas Armadas-ESPE, Sangolquí, Ecuador, in 2009, and the M.Sc. and Ph.D. degrees from the School of Electrical and Computer Engineering (FEEC), University of Campinas (UNICAMP), Brazil, in 2014 and 2018, respectively, all in electrical engineering. He was ranked first in his undergrad-uate program. In 2018, he was the Director of the technology transfer area with the Secretaria Nacional de Educación Superior, Ciencia, Tecnología e Innovación (SENESCYT), Ecuador. He is currently a Full Professor with the Universidad de las Américas (UDLA), Quito, Ecuador. His current research interests include diversity-combining systems, multiple access systems, multiuser detection, MIMO systems, and 5G communications systems.
CELSO DE ALMEIDA received the degree in electrical engineering, and the M.Sc. and Ph.D. degrees from the University of Campinas (UNI-CAMP), Brazil, in 1980, 1983 and 1990, respec-tively. He was an Electrical Engineer in optical communications with the industry, from 1982 to 1990. He joined the Faculty of Electrical Engi-neering, UNICAMP, in 1990, where he is cur-rently a Full Professor. His current research inter-est include cellular systems, CDMA, OFDMA, multiuser detection, antenna arrays, MIMO systems, wireless communica-tions, cryptography, and error correction codes.
