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Massive-MIMO Iterative Channel Estimation and
Decoding (MICED) in the Uplink Daniel Verenzuela, Emil Björnson, Xiaojie Wang, Maximilian Arnold and Stephan ten
Brink
The self-archived postprint version of this journal article is available at Linköping
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N.B.: When citing this work, cite the original publication. Verenzuela, D., Björnson, E., Wang, X., Arnold, M., Brink, S. t., (2019), Massive-MIMO Iterative Channel Estimation and Decoding (MICED) in the Uplink, IEEE Transactions on Communications, , 1-1. https://doi.org/10.1109/TCOMM.2019.2947906
Original publication available at: https://doi.org/10.1109/TCOMM.2019.2947906 Copyright: Institute of Electrical and Electronics Engineers (IEEE) http://www.ieee.org/index.html ©2019 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
1
Massive-MIMO Iterative Channel Estimation andDecoding (MICED) in the Uplink
Daniel Verenzuela, Student Member, IEEE, Emil Björnson, Senior Member, IEEE,Xiaojie Wang, Student Member, IEEE, Maximilian Arnold, Student Member, IEEE,
and Stephan ten Brink, Senior Member, IEEE
Abstract—Massive MIMO uses a large number of antennas toincrease the spectral efficiency (SE) through spatial multiplexingof users, which requires accurate channel state information. It isoften assumed that regular pilots (RP), where a fraction of thetime-frequency resources is reserved for pilots, suffices to providehigh SE. However, the SE is limited by the pilot overhead andpilot contamination. An alternative is superimposed pilots (SP)where all resources are used for pilots and data. This removesthe pilot overhead and reduces pilot contamination by usinglonger pilots. However, SP suffers from data interference thatreduces the SE gains. This paper proposes the Massive-MIMOIterative Channel Estimation and Decoding (MICED) algorithmwhere partially decoded data is used as side-information toimprove the channel estimation and increase SE. We show thatusers with precise data estimates can help users with poor dataestimates to decode. Numerical results with QPSK modulationand LDPC codes show that the MICED algorithm increases theSE and reduces the block-error-rate with RP and SP comparedto conventional methods. The MICED algorithm with SP deliversthe highest SE and it is especially effective in scenarios with shortcoherence blocks like high mobility or high frequencies.
I. INTRODUCTION
Next generation wireless networks need to accommodate alarge amount of data and number of devices while fulfilling avariety of requirements like high data rates and low energyconsumption. Massive MIMO is a multiuser multiple-inputmultiple-output (MIMO) technology able to serve severaluser equipments (UEs) on the same time-frequency resourcesby means of spatial multiplexing in which the base station(BS) utilizes a large number of antennas. This technologyhas received large attention from both academia and industryfor its ability to greatly increase the spectral efficiency (SE)compared to current cellular networks [1]–[3].
To enable the spatial multiplexing of UEs in MassiveMIMO, the signals from all BS antennas are processed co-herently for which accurate channel state information (CSI)is required. A standard approach for channel estimation,
D. Verenzuela and E. Björnson are with the Department of ElectricalEngineering (ISY), Linköping University, Linköping, SE-58183 Sweden (e-mail: [email protected]; [email protected]).
X. Wang, M. Arnold and S. ten Brink are with the Institute of Telecom-munications, University of Stuttgart, Stuttgart 70659, Germany (e-mail:[email protected]; [email protected];[email protected]).
This paper has received funding from ELLIIT and the Swedish Foundationfor Strategic Research (SSF).
The simulations were performed on resources provided by LinköpingUniversity (LiU) at National Supercomputer Centre (NSC). We thank PeterKjellström, Mats Kronberg, Peter Münger, and Kent Engström at the NSC fortheir assistance with technical support.
called regular pilots (RP), is to reserve some time-frequencyresources for UEs to send known orthogonal signals in theuplink (UL), called pilots. This method can provide channelestimates of sufficient quality at the expense of having apilot overhead, due to the fact that not all time-frequencyresources can be used for data transmission. Typically, theSE has the form SE = prelog × log2 (1 + SINR) where SINRstands for the signal-to-interference plus noise ratio, and prelogrefers to the fraction of time-frequency resources used for datatransmission. Having a pilot overhead means that prelog < 1,and as more pilots are used the SE decreases linearly with theprelog. On the other hand, using more pilot symbols results inbetter channel estimates and thereby higher SINR. Thus, thereis a non-trivial trade-off between channel estimation qualityand pilot overhead to obtain the highest SE [4].
In a multicell Massive MIMO system, many UEs areexpected to be active which means that there are not enoughtime-frequency resources to assign orthogonal pilot signals toeach UE. Thus, some pilots would need to be reused causinginterference in the channel estimation process which, in turn,results in lower coherent MIMO signal gain and the presenceof coherent interference that reduces the SE. This phenomenonis known as pilot contamination [2], [3].
The problem of pilot contamination has been well studied inthe Massive MIMO literature resulting in many methods forits mitigation. For instance, (semi) blind channel estimationmethods making use of angle domain representation andamplitude interference rejection have been proposed in [5]–[8].Other approaches in [3], [9]–[11], exploit the structure of thespatial correlation matrices to mitigate pilot contamination.In particular, [11] has shown that the capacity of multicellMassive MIMO systems grows without bound with the numberof BS antennas if the correlation matrices are known andadvanced processing is used. A simpler method to reducethe pilot contamination is to increase the pilot overhead toafford longer pilots that are reused more sparsely in the spatialdomain by introducing a pilot reuse factor [4], [12]–[14].
In the aforementioned methods, the transmission of pilotsand data is done on disjoint time-frequency resources. Analternative approach is to send a superposition of pilotsand data to support longer pilot sequences and eliminatethe pilot overhead. This is called superimposed pilot (SP)transmission [15], [16]. In Massive MIMO, the SP methodhas been proposed to mitigate the pilot contamination effectby allowing the use of longer pilots that can support sparserpilot reuse [17], [18]. However, when pilot and data symbols
2
are superimposed, there is interference from data symbols inthe channel estimation process. This interference reduces thecoherent gain and creates coherent interference that limits theSE gains of SP over RP [19].
In summary, RP channel estimation provides sufficientlyaccurate CSI to obtain high SINR, which translates into highSE in Massive MIMO. However, this comes at the expense ofhaving a pilot overhead, which in turn, limits the maximumachievable SE through the prelog factor. On the other hand, SPchannel estimation gives comparable SE to that of RP and it isinstead limited by the data interference. Thus, a potential wayto further improve the SE in Massive MIMO is to use SP withdata-aided channel estimation to reduce the data interference.
A. Contributions
This paper evaluates the potential improvements of data-aided channel estimation in the UL of multicell MassiveMIMO systems with RP and SP methods while consideringthe effect of channel coding and spatially correlated fadingamong BS antennas.
The idea of using partially decoded data to improve channelestimation with SP has been proposed a couple of decadesago for single antenna systems [15]. Extensions to single-userpoint-to-point MIMO systems [20], [21] show an improvementin terms of bit-error-rate (BER) with SP, which translatesinto higher SE compared to RP since SP removes the pilotoverhead. In the case of Massive multiuser MIMO, [17], [18]depict that iterative data-aided channel estimation with SP hasthe potential to increase the SE compared to RP systems.However, this was only shown for uncoded data estimateswhere the effect of channel coding was not considered. Re-cently, [22] used partially decoded data to improve channelestimation in the UL of a multicell single-input multiple-output (SIMO) system (only one UE served per cell) with RP,while considering i.i.d. Rayleigh fading and maximum ratio(MR) combining (also known as maximal ratio combining[23]). The results show that the pilot contamination effectcan be reduced by means of iterative data-aided channelestimation which in turn reduces the BER. However, in [22]the main analysis considers a single UE per cell, disregardingthe effect of inter-user interference.1 An extension to theMassive multiuser MIMO case was done in [24] showingthat SP outperforms RP in scenarios with high mobility andhigh number of spatially multiplexed UEs. However, in [24]although the paper considers a multi-cell setup, the authorsapproximate the intercell interference as i.i.d. Gaussian noise,which removes all the structure that intercell interference hasand effectively reduces the model to a single-cell setup. Thus,the effect of data-aided channel estimation considering channelcoding in a multicell Massive MIMO system remains to beinvestigated. In addition, theoretical analysis on the impact ofspatial correlation, inter-user and intercell interference in theaforementioned system is missing in the literature.
In this article, data-aided channel estimation refers to theuse of partially decoded bits as side information to increase the
1Here, the term inter-user refers to UEs that are served by the same BSvia spatial multiplexing.
channel estimation quality. That is, initial channel estimatesfrom pilots are used to obtain soft data estimates which,in turn, are utilized to revise the channel estimates. Thisprocess is done in an iterative form using previous soft dataestimates to update the channel estimates. The revised data-aided channel estimates are then used to decode the datasymbols to reduce errors.
The Massive-MIMO Iterative Channel Estimation and De-coding (MICED) algorithm is proposed to harvest the benefitsof data-aided channel estimation in multicell Massive MIMOsystems. To obtain insights into the benefits of the MICEDalgorithm, closed-form expressions for the error correlationmatrices of data-aided channel estimates are computed assum-ing Gaussian data symbols. These expressions are analyzedto indicate how the mean squared error (MSE) of data-aidedchannel estimates behaves in terms of the data estimationquality and number of time-frequency resources. Note that incontrast to [5]–[8] the MICED algorithm does not rely onasymptotic results, angle domain representations, or separa-bility of power levels between UEs, which are conditions thatmight not be satisfied in practice. For example, having similarreceived power levels between UEs in the UL is often desiredto mitigate the near-far effect of pathloss and to have signalswith a low dynamic range which is important for the use oflow-resolution analog-to-digital converters in the BSs [25].
To evaluate the SE, MR and single-cell MMSE (S-MMSE)combining are assumed to assess the differences betweenmaximizing coherent combination and suppressing inter-userinterference through linear signal processing. The S-MMSEmethod is based on [3, Ch. 4] while considering no CSIexchange among BSs.
The MICED algorithm is also implemented with finite-alphabet modulated symbols indicating how the redundancyof channel coding can be used to obtain estimates of complexmodulated data symbols. Finally, numerical analysis withquadrature phase shift keying (QPSK) modulation and low-density parity check (LDPC) codes is used to evaluate theperformance of the MICED algorithm compared to conven-tional pilot-based channel estimation in terms of the block-error-rate (BLER) and the achievable SE. The results showthat the MICED algorithm increases the SE and reduces theBLER compared to pilot-based channel estimation with bothRP and SP. The highest SE is found when implementing theMICED algorithm with SP since there is no pilot overhead andthe data interference is mitigated by the data-aided channelestimation process. The use of SP with the MICED algorithmincreases the SE in scenarios with high mobility and highcarrier frequencies. In addition, it allows for aggressive spatialmultiplexing that can enable other services such as machinetype communications.
Notation
Bold lower and upper case letters denote column vectors andmatrices respectively. The trace, matrix inversion, transpose,conjugate, and conjugate transpose operations are denoted astr(·), (·)−1, (·)) , (·)∗, (·)� respectively. The sets of natural,real, and complex numbers are denoted as N, R, and C
3
respectively. The element on the 8Cℎ row and 9 Cℎ column of amatrix X is denoted as [X]8 9 , the 9 Cℎ column of X is denotedas [X] 9 . The matrix composed of the first = columns of Xis denoted as [X]1:=. The 9 Cℎ element of vector x is denotedas [x] 9 . The identity matrix of size # is denoted as I# . ForA, B ∈ C#×# , the ordering notations A � B and A � Bindicate that A − B is a positive semi-definite and definitematrix, respectively.
II. SYSTEM MODEL
Consider the UL of a multicell Massive MIMO systemwhere each base station (BS) has " antennas and serves single-antenna user equipments (UEs) via spatial multiplexing.The BS serving the UEs in cell ; ∈ Φ is denoted as BS; whereΦ is a set containing all the cell indices. The UE : in cell ;is denoted as UE;: . A standard block fading channel modelis assumed where the channel is considered static over a timeperiod of )2 [s] and frequency-flat within a bandwidth of �2[Hz] [2]. The total system bandwidth is �W and is equallydivided between all coherence blocks, such that �W/�2 isan integer.2 The time-frequency block in which the channelis considered time-invariant and frequency-flat is called acoherence block and it is comprised of g2 = )2�2 complexsamples. The block fading model assumes that the channel isconstant within a coherence block and changes independentlyfrom one coherence block to another to account for the effectof frequency selectivity and time variations [2].
The communication channel is modeled as a random vari-able that has an independent realization in each coherenceblock. Let h;;: ∼ CN(0,R;;: ) be the channel between BS;and UE;: where R;;: is the spatial correlation matrix andV;;: = tr(R;;: )/" is the average channel gain. The receivedsignal at BS; in the UL is
Y; =∑ℓ∈Φ
∑:′=1
h;ℓ:′︸︷︷︸"×1
x)ℓ:′︸︷︷︸1×g2
+N ∈ C"×g2 (1)
where N = [n1, . . . , ng2 ] is the thermal noise with i.i.d.columns distributed as n 9 ∼ CN(0, f2I" ) with f2 being theaverage noise energy per symbol. The signal transmitted fromUE;: is
x;: =[√@RP;:5);:
√dRP;:
s);:
] )with RP√
@SP;:>;: +
√dSP;:
s;: with SP
where @RP;:
, dRP;:
, @SP;:
, and dSP;:
are the pilot and data energyper symbol with RP, and SP respectively. Note that with SP,the available transmission energy per symbol r;: is dividedbetween pilot and data symbols such that @SP
;:+ dSP
;:= r;: . On
the other hand, with RP, the same energy per symbol is usedfor pilot and data symbols since they are transmitted disjointly.Thus,
@RP;: = dRP
;: = r;: with RP,@SP;: = Δ ;: r;: , d
SP;: = (1 − Δ ;: )r;: with SP,
2This can be accomplished, for example, by utilizing orthogonal frequencydivision multiplexing (OFDM) modulation [2].
g3
g3 g3g?
g? g?
codeword 1
g3
g3 g3g?
g? g?
codeword 2...
. . .
· · ·...
. . .
· · ·
g2
g2 g2
codeword 1
g2
g2 g2
codeword 2...
. . .
· · ·
...
. . .
· · ·
· · ·
· · ·
RP: Pilot Data
SP: Pilot + data
Figure 1: Time-frequency resource allocation with RP and SPover codewords of equal length.
where Δ ;: ∈ [0, 1] is the proportion of power used for pilots.The pilot symbols are given by 5;: ∈ Cg? and >;: ∈ Cg2 withRP and SP respectively. The UL data symbols are denotedas s;: ∈ Cg3 with both RP and SP. In the case of RP, thepilot and data symbols are sent disjointly, thus g? ≤ g2 ULsamples of the coherent block are used for pilots and g3 =
g2 − g? for UL data symbols. With SP, all UL symbols areused for data and pilots, thus g3 = g2 . Figure 1 illustrates theallocation of samples in the coherence block correspondingto the complex symbols transmitted in codewords of equalsize. Note that since RP has a pilot overhead, it needs morecoherence blocks to transmit a full codeword compared to SP.Section V explains in more detail how the information bitsare mapped into the complex symbols transmitted over thechannel within coherence blocks.
In this article, all deterministic quantities (e.g., transmis-sion powers, spatial correlation matrices, etc.) are consideredknown. Since they are deterministic, the signaling overheadfor estimating them is negligible. In practice, they can beestimated by aggregating observations from several coherenceblocks [26].
Pilot-based Channel Estimation
In this section, standard pilot-based channel estimation withRP and SP is described. Consider the channel realizations tobe estimated based on the UL pilot symbols. Let Ug? be aset of g? mutually orthogonal pilot sequences with elementshaving unit modulus such that for 50, 51 ∈ Ug? , | [50] 9 | =| [51] 9 | = 1∀ 9 ∈ {1, . . . , g?}, 5�0 51 = 0 if 0 ≠ 1, and 5�0 51 =
g? if 0 = 1. The choice of pilot sequences having elementswith unit modulus is done to have equal energy per symbols,more information on how to generate such sequences can befound in [3, Sec. 3.1.1]. The pilot sequence used by UE;: is5;: ∈ Ug? and >;: ∈ Ug2 with RP and SP respectively. In alarge multicell network, the total number of UEs is larger thanthe available pilots, which means that these pilots need to bereused among cells. The sets containing all UEs that share thesame pilot as UE;: (itself included) are defined as
PRP;: =
{(ℓ, : ′) : 5�ℓ:′5;: ≠ 0
}, with RP, (2)
PSP;: =
{(ℓ, : ′) : >�ℓ:′>;: ≠ 0
}, with SP. (3)
Note that since with SP the pilots are much longer than withRP, the number of elements in PSP
;:is far less than that of PRP
;:.
4
For example, if g2 = 100 and g? = 10 there will be ten timesless UEs sharing pilots with SP than with RP.
To estimate the channel from UE;: at BS; , the receivedUL pilot signal is multiplied with the pilot sequence of UE;: ,which is equivalent to a de-spreading operation, to obtain theobservations
zRP;;: = [Y;]1:g?
5∗;:
g?
√@RP;:
= h;;: +∑
(ℓ,:′) ∈PRP;:\(;,:)
h;ℓ:′
√@RPℓ:′
@RP;:︸ ︷︷ ︸
Pilot contamination
+ n√g?@
RP;:︸ ︷︷ ︸
Noise
(4)
zSP;;: = Y;
>∗;:
g2
√@SP;:
= h;;: +∑
(ℓ,:′) ∈PSP;:\(;,:)
h;ℓ:′
√@SPℓ:′
@SP;:︸ ︷︷ ︸
Pilot contamination
+∑ℓ∈Φ
∑:′=1
h;ℓ:′
√dSPℓ:′
@SP;:
s)ℓ:′>
∗;:
g2︸ ︷︷ ︸Data interference
+N>∗
;:
g2
√@SP;:︸ ︷︷ ︸
Noise
, (5)
where n = [N]1:g?5∗;:/√g? ∼ CN(0, f2I" ) is the equivalent
noise after the de-spreading operation with RP. The linearminimum mean-squared-error (LMMSE) channel estimate ofh;;: is summarized in the following lemma.
Lemma 1. Based on the observations zRP;;:
and zSP;;:
, theLMMSE estimates of h;;: are
hRP;;: = R;;:RP
;;:
−1zRP;;: (6)
hSP;;: = R;;:SP
;;:
−1zSP;;: (7)
where
RP;;: =
∑(ℓ,:′) ∈PRP
;:
R;ℓ:′@RPℓ:′
@RP;:︸ ︷︷ ︸
Pilot contamination
+ f2
@RP;:g?
I"︸ ︷︷ ︸Noise
, (8)
SP;;: =
∑(ℓ,:′) ∈PSP
;:
R;ℓ:′@SPℓ:′
@SP;:︸ ︷︷ ︸
Pilot contamination
+ 1g2
( ∑ℓ∈Φ
∑:′=1
R;ℓ:′dSPℓ:′
@SP;:︸ ︷︷ ︸
Data interference
+ f2
@SP;:
I"︸ ︷︷ ︸Noise
).
(9)
The channel estimates are uncorrelated to the channel estima-tion errors which are defined as
hRP;;: = h;;: − hRP
;;: , (10)
hSP;;: = h;;: − hSP
;;: , (11)
with error correlation matrices given by
CRP;;: = R;;: − R;;:RP
;;:
−1R;;: , (12)
CSP;;: = R;;: − R;;:SP
;;:
−1R;;: . (13)
Proof: It follows from employing standard LMMSE esti-mation techniques to the problem at hand where the LMMSEestimate of a vector h from an observation z is given by
E{zh�
} (E
{zz�
} ) −1 z (assuming all random variables havezero mean) [3, Ch. 3], [27, Ch. 15].
Notice that the MSE of the channel estimates, that istr(CRP
;;:)/" and tr(CSP
;;:)/" with RP and SP respectively,3 can
be reduced by increasing the pilot length. Having more pilotsymbols would lower the number of shared pilots (i.e., thenumber of elements in PRP
;:and PSP
;:) and also decrease the
effect of noise. However, the pilot length (being g? ≤ g2with RP and g2 with SP) is ultimately limited by the sizeof the coherence block g2 which is set by physical propertiesof the channels and cannot be made arbitrarily large. Thus,the estimation errors cannot be alleviated and interferencemanagement would be a potential way to improve the channelestimation quality.
III. UPLINK COMBINING AND ACHIEVABLE SE
This section introduces the process of coherently combiningsignals in Massive MIMO with RP and SP, as well as, thedefinition of an achievable SE for performance evaluation.Linear signal processing is assumed where the combiningvector for UE;: is defined as
v;: =
h;;: MR( ∑:′=1
r;:′(h;;:′ h�;;:′+C;;:′
)+f2I"
)−1h;;: r;: S-MMSE
(14)
where the superscripts indicating RP and SP are dropped toshow that these combining methods can be applied with RPand SP alike. MR combining aims at maximizing the receivedpower from UE;: , whereas S-MMSE balances interferencesuppression and signal amplification while only using CSIavailable at BS; . This means, that S-MMSE only relies onCSI obtained from UL pilots and does not require sharingCSI among BSs.
After combining the received signal from all BS antennas,the following observations of the data symbols within onecoherence block are obtained, with RP and SP respectively:
yRP)
;:︸︷︷︸1×g3
= v�;:︸︷︷︸1×"
[Y;]g?+1:g2︸ ︷︷ ︸"×g3
(15)
ySP)
;:︸︷︷︸1×g2
= v�;:︸︷︷︸1×"
Y;︸︷︷︸"×g2
. (16)
To compute an achievable SE, which is a rigorous lowerbound on the ergodic capacity, the data symbols are assumedas i.i.d. s;: ∼ CN(0, Ig3 ), recall that g3 = g2 − g? with RPand g3 = g2 with SP. Since the data symbols are i.i.d. and thechannel is memoryless, it is enough to focus on one arbitrarydata symbol, denoted as B;: , taken from s;: . The correspondingdata observation taken from (15) with RP, or (16) with SP isdenoted as H;: , and can be expressed as
H;: = E{H;: B
∗;:
}︸ ︷︷ ︸channel gain
B;: + H;: − E{H;: B
∗;:
}B;:︸ ︷︷ ︸
effective noise
(17)
3Recall that the MSE of h;;: is defined as E{ ‖h;;: − h;;: ‖2 }/" .
5
where adding and subtracting the first term in (17) results in anequivalent single-input single-output (SISO) system with de-terministic known channel gain and uncorrelated non-Gaussianeffective noise. Then, a lower bound on the ergodic capacityis obtained by considering the effective noise to be Gaussiansince the Gaussian distribution maximizes the entropy, andtherefore, corresponds to the worst-case distribution for theeffective noise. This bounding technique is sometimes calledthe “use-and-then-forget” bound and it is commonly usedin Massive MIMO literature [2], [3], [28]. The name “use-and-then-forget” follows from using the CSI to construct thecombining vector v;: but then dismissing it in (17) to obtainthe lower bound on the ergodic capacity. Thus, an achievableSE is given by
SE;: =g3
g2log2
(1 +
��E {H;: B
∗;:
} ��2Var
{H;: − E
{H;: B
∗;:
}B;:
} ), (18)
the superscripts for RP and SP are removed since the samebound can be applied in both cases.
IV. MICED - MASSIVE-MIMO ITERATIVE CHANNELESTIMATION AND DECODING
In this section, the proposed MICED algorithm is defined,explained and analyzed in terms of channel estimation quality,and SE, while discussing its feasibility in terms of computa-tional complexity. First, the basis of the MICED algorithmis illustrated in Algorithm 1. Second, a detailed analysis ofthe data-aided channel estimation process is given. Third,the computational complexity of the MICED algorithm isdiscussed. Fourth, numerical examples with Gaussian datasymbols illustrate the possible gains of the MICED algorithmin terms of MSE of channel estimates and achievable SE.
A. Basis of the MICED algorithm
The main principle of using data estimates to perform chan-nel estimation is to spread the interference effect between UEs.That is, to trade the main interfering sources in the channelestimation for reduced interference coming from all cells. Themain source of interference in the channel estimation with RPis pilot contamination, and the use of data estimates reducesthis effect. On the other hand, with SP the pilot contaminationis reduced by having longer pilots and the main interference inthe channel estimation is due to data symbols, thus the aim ofusing data estimates in this case is to reduce the data intracell(same cell as UE;: ) interference.
At the start of the MICED algorithm, pilot-based channelestimation is performed once (at iteration 8 = 0) as shownin lines 3-5 of Algorithm 1. Afterwards, linear combiningis performed followed by data decoding (see lines 6-11 ofAlgorithm 1). This initial decoding procedure provides softestimates of the data symbols which are then used to improvethe quality of channel estimates with the aim of achievinglower data decoding errors in the next iterations.
Once the received signals have been linearly combined, thedata observations in (15) and (16) are used to detect whichsymbols were sent. Due to the effects of interference and noise,some symbols may be detected erroneously leading to a failure
Algorithm 1 MICED basic algorithm. The following abbre-viations are used: calculate (cal.), and observation (obs.)
1: Initialize:2: set 8 = 0 and define 8max ∈ N3: Pilot-based channel estimation for UE;: ∀: ∈ {1, . . . , }4: cal. channel obs. z(0)
;;:⊲ use (4) with RP and (5) with SP
5: cal. channel estimates h(0);;:
⊲ use Lemma 16: Linear combining7: cal. combining vector v(0)
;:⊲ use (14)
8: cal. data obs. y(0);:
⊲ use (15) with RP and (16) with SP9: Data decoding for UE;: ∀: ∈ {1, . . . , } ⊲ See Section V for
details10: decode data and extract data estimates s(0)
;:
11: set 8 = 112: while 8 ≤ 8max or at least one UE has decoding errors do13: Data-based channel estimation for UE;: ∀: ∈ {1, . . . , }14: cal. channel obs. z(8)
;;:⊲ use (22) with s(8−1)
;:
15: cal. channel estimates h(8);;:
⊲ use Lemma 1 with z(8);;:
16: Linear combining17: cal. combining vector v(8)
;:⊲ use (14) with h(8)
;;:
18: cal. data obs. y(8);:
⊲ use (19) with RP, (20) with SP19: Data decoding for UE;: ∀: ∈ {1, . . . , } ⊲ See
Section V for details20: decode data and extract data estimates s(8)
;:
21: set 8 = 8 + 122: end while
in retrieving the desired information. In practice, the redundantinformation in the channel code is used to detect when thedecoding procedure fails, for instance cyclic redundancy checkcodes are often used for this purpose. In such cases, thedata observations in (15) and (16) can also be used to obtainestimates of the data symbols and, in turn, use those to improvethe channel estimates as shown in lines 13-15 of Algorithm 1.These improved channel estimates can then be used to performlinear combining again (see lines 16-18 of Algorithm 1), andobtain updated data observations as follows:
yRP (8))
;:= vRP (8)�
;:
([Y;]g?+1:g2 −
∑:′=1:′≠:
hRP (8);;:′
√dRP;:′ s
RP (8−1))
;:′
)(19)
ySP (8))
;:= vSP (8)�
;:
(Y; −
∑:′=1
hSP (8);;:′
√@SP;:′>
);:′
− ∑:′=1:′≠:
hSP (8);;:′
√dSP;:′ s
SP (8−1))
;:′
)(20)
with RP and SP respectively. Note that updated channel esti-mates and previous data estimates are also used to subtract theintracell interference. Thus, the updated data observations maycontain less interference and, in turn, lead to fewer decodingerrors. This procedure is done iteratively to improve data andchannel estimates in each iteration as shown in Figure. 2. Eachiteration starts with the channel estimation, followed by linearcombining, and finishing with the data decoding. The MICED
6
(8);;:
= R;;: + ∑:′=1
R;;:′E{���u(8)�
;:x(8);:′
���2}︸ ︷︷ ︸Intracell interference
+∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E{���u(8)�
;:xℓ:′
���2}︸ ︷︷ ︸Intercell interference
+f2E{ u(8)
;:
2}
I"︸ ︷︷ ︸Noise
(21)
Channelestimation
Linearcombining Decoder
Y1,1:gD;
Y",1:gD;
...
h(8);;1 h(8)
;; .....
y(8);;1
y(8);;
...
b(8);1
b(8);
...
s(8);;
s(8);;1 .....
Figure 2: Block diagram of iterative receiver. The notation ·(8)is used to represent the values of variables at the 8Cℎ iterationof the MICED algorithm.
algorithm ends when the maximum number of iterations isreached or the data from all UEs is successfully decoded.
Note that the MICED algorithm is analyzed using anarbitrary set of coherence blocks since the channel realizationsare considered independent across blocks. In real propagationchannels, the coherence blocks that are close to each other(either in time or frequency) exhibit some degree of correlationwhich can also be exploited to improve channel the estimation[29]. However, this analysis falls outside the scope of thisarticle and it is therefore left for future work.
B. Analysis of data-aided channel estimation with Gaussiansymbols
In practical implementations, the transmitted information isencoded into bits that are then modulated into a finite alphabetof complex symbols. Therefore, the detection is done basedon bits rather than complex symbols, which is enclosed withinthe decoder stage shown in Figure 2 where b;;: representsthe hard bit estimates from UE;: ∀: ∈ {1, . . . , } at theoutput of the decoder. In this section, the data symbols areconsidered as i.i.d. s;: ∼ CN(0, Ig3 ) for analytical tractability.This assumption yields theoretical results that give insightsinto the gains in channel estimation quality that the MICEDalgorithm can offer. In later sections, the analysis will beextended towards finite-alphabet symbols and the Gaussianassumption will be dropped.
At the 8Cℎ iteration of the receiving algorithm, assume thatthe MMSE data estimate4 of B;: is B (8)
;:∼ CN(0, f (8)2
;:) such
that B;: = B(8);:
+ B (8);:
, where B (8);:
is the estimation error that isuncorrelated (i.e., E{B (8)
;:B(8)∗
;:} = 0) and independent of the data
estimates.
4These data estimates are obtained from the decoding procedure. Section Vexplains in detail how to perform this in practical implementations with finite-alphabet symbols.
Based on the estimates of the data from UEs within thesame cell, the BS can obtain an estimate of the transmittedsignal from UEs in cell ; as
x(8);:
=
[√@RP;:5);:
√dRP;:(s(8);:))
] )with RP,√
@SP;:>;: +
√dSP;:
s(8);:
with SP,
where x(8);:
= x;: − x(8);:
is the signal estimation error at the 8Cℎ
iteration.
By collecting these signal estimates from the UEs servedby BS; within an arbitrary coherence block, and stackingthem into the matrix X(8)
;= [x(8)
;1 , . . . , x(8); ] ∈ Cg3× , a new
observation of the channel h;;: can be obtained by projecting
the received signal in (1) with u(8);:
=
[X(8);
(X(8)�
;X(8);
) −1]:
(note that g3 > is assumed), which yields
z(8);;:
= Y;u(8)∗
;:(22)
with correlation matrix (i.e., (8);;:
= E{z(8);;:
z(8)�
;;:}) given in (21)
at the top of the page. The superscripts denoting RP and SP areremoved to indicate that the correlation matrix in both caseshas the same formulation, thus the difference lies in what goesinto the expectations.
The use of u(8);:
to obtain the channel observation z(8);;:
isaimed at reducing the intracell interference in the channelestimation process. Thus, the more accurate the data estimatesare, the less intracell interference will be present in the data-aided channel estimates.
Remark 1. The quality of data estimates differs among UEsdue to the large-scale fading, transmission power, and in-terference conditions. Thus, a particular case of interest forthe MICED algorithm is when some UEs have high dataestimation quality and others not. In this case, the high qualitydata estimates can be used to improved the data decoding ofthe UEs with low data estimation quality.
Notice that the expectations in (21) are non-trivial to com-pute since they involve inverse moments of non-central com-plex Wishart matrices. To obtain insights into the performanceand behavior of data-aided channel estimation, the followingtheorem introduces closed-form expressions that bound thecorrelation matrices in (21) in the positive semi-definite sense.
7
RP (8);;: =
∑:′=1:′≠:
R;;:′dRP;:′
(1 − f (8)2
;:′
)+ ∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dRPℓ:′
(@RP;:)2g2
?
dRP;:f
(8)2
;:(g3 − )
+ 2@RP;: g? + d
RP;: f
(8)2
;:g3
+
∑(ℓ,:′) ∈PRP
;:\(;,:)
R;ℓ:′@RPℓ:′
@RP;: +
2g3dRP;:f
(8)2
;:
g?+(dRP;:f
(8)2
;:)2g3 (g3 + 1)@RP;:g2?
+ f2I"@RP;:g? + dRP
;:f
(8)2
;:g3
(25)
SP (8);;: =
∑:′=1:′≠:
R;;:′dSP;:′
(1 − f (8)2
;:′
)+ ∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dSPℓ:′
g2
(@SP;:+ dSP
;:f
(8)2
;:
) +
∑(ℓ,:′) ∈PSP
;:\(;,:)
R;ℓ:′@SPℓ:′
(@SP;:+ dSP
;:f
(8)2
;:)2
@SP;:
+dSP;:f
(8)2
;:(2@SP
;:+ dSP
;:f
(8)2
;:)
@SP;:g2
+ f2I"g2
(@SP;:+ dSP
;:f
(8)2
;:
)(26)
Theorem 1. Given (8);;:
in (21), it holds that
RP (8);;:
� R;;:
+ R;;:dRP;:
(1 − f (8)2
;:
)(@RP;:)2g2
?
dRP;:f
(8)2
;:(g3 − )
+ 2@RP;: g? + d
RP;: f
(8)2
;:g3
+ RP (8);;:
(23)
SP (8);;:
� R;;: + R;;:dSP;:
(1 − f (8)2
;:
)g2
(@SP;:+ dSP
;:f
(8)2
;:
) + SP (8);;: (24)
where RP (8);;: and SP (8)
;;: are given in (25) and (26) respectively,at the top of next page.
Proof: The proof is shown in Appendix A.Recall that the MSE of the channel estimates is given by
tr(C;;: )/" where C;;: is the correlation matrix of the channelestimation error that follows the same formulation as in (12)with RP and (13) with SP. Hence, a lower bound on the MSEof data-aided channel estimates can be obtained by replacingthe correlation matrix of channel observations in (8) and (9)with those in the right-hand-side of (23) and (24) respectively.This lower bound is then characterized by the behavior of (8)
;;: ,defined in (25) and (26) with RP and SP respectively, in thesubspace spanned by R;;: (see (12) and (13)). In addition,notice that since linear combining is considered for datadetection, the terms in (25) and (26) will combine coherently.Thus, reducing tr( (8)
;;: ) leads to lower MSE of the data-aidedchannel estimates and coherent interference.
To obtain better insights into the benefits that the MICEDalgorithm may bring, the influence of the data estimationquality and number data symbols for each term in (25) and(26) is analyzed in detail. First, notice that tr( (8)
;;: ) is adecreasing function of f (8)
;:′ for : ′ ≠ : (see the first term in(25) and (26)). This means that the influence of the intracellinterference decreases with the quality of the data estimates.Second, by inspecting the derivative with respect to f
(8)2
;:of
the denominator in the first term of (25) (note that this termis a scalar) it can be shown that for
f(8)2
;:≥
@RP;:g?
dRP;:
√g3 (g3 − )
(27)
the term tr(RP (8);;: ) is a decreasing function of f (8)2
;:. Moreover,
tr(SP (8);;: ) is also a decreasing function of f (8)2
;:. Thus, the
higher quality the data estimates have, the lower influencethe interference has on the data-aided channel estimates. Thisconfirms the intuition provided in Remark 1 that accuratedata estimates of UEs within a given cell can be useful toimprove the data decoding of other UEs in the same cell.Third, consider the influence of g3 with RP, by inspecting thedenominator of the first term in (25) it follows that for
g3 ≥@RP;:g?
dRP;:f
(8)2
;:
+ (28)
the term tr(RP (8);;: ) is a decreasing function of g3 since the
second term in (25) also decreases with g3 . The condition (28)can be interpreted as the minimum value of g3 from whichit is feasible to implement the MICED algorithm with RP.Moreover, when the number of data symbols increases beyondthis condition, the interference effect is reduced. On the otherhand, with SP, the trace of the first and last terms in (26) isa decreasing function of g2 , whereas, the trace of the secondterm in (26) has a more involved dependency on g2 since thenumber of elements in PSP
;:decreases with g2 . This means
that the data interference and noise are reduced with higherg2 while the effect of pilot contamination, in turn, is reducedby having more sparse pilot reuse factors as g2 increases.
In the case of RP, comparing (8) with (23) and (25) showsthat by using the data estimates the pilot contamination istraded for interference from all UEs that decreases with thequality of data estimates and number of data symbols, whichin turn might be substantially smaller. Whereas with SP,comparing (9) with (24) and (26) shows that the intracellinterference from data symbols decreases with the quality ofdata estimates and the size of the coherence block.
In summary, the purpose of utilizing data estimates to revisethe channel estimation is to trade a few terms that cause highinterference with many terms that cause low interference.
C. Computational complexity
In the past few years, several real-time testbeds for MassiveMIMO have been built to evaluate its performance in realpropagation scenarios [30]. In particular, the Lund University
8
Table I: Simulation parameters.
Parameter ValueSystem bandwidth �W = 20 [MHz]
Maximum transmissionpower per UE 10 log10 (rMAX�W) = 20 [dBm]
Proportion of pilot powerwith SP Δ = 0.3
Noise power 10 log10
(f2�W
)= −94 [dBm]
Inter-BS distance 0.15 [km]Pathloss exponent U = 3.76Pathloss at 1 km l = 148.1 [dB]
Shadow fading std.deviation fsf = 10 [dB]
Angular std. deviation fang = 10◦
Massive MIMO testbed (LuMaMi) [31] runs a real-time Mas-sive MIMO system with RP, " = 100, and = 12, usinga 20 MHz bandwidth with 1200 subcarriers and an OFDMsymbol length of 71.4 `s. In the LuMaMi testbed, the maincontributor to the usage of the processing resources is theQR-decomposition which is used to invert the Gramian matrix(e.g., (H� H)−1 where H is a "× channel estimates matrix).This matrix invertion is employed for interference suppressiontechniques in the spatial domain like zero-forcing (ZF) orregularized ZF (RZF). However, the latency evaluation in [32]shows that the overall time for performing UL channel estima-tion and transmitting precoded signals in the downlink5 (calledprecoding turnaround time) is 132 `s. Furthermore, the largestcontributor to the latency is OFDM modulation/demodulationwhereas the impact of channel estimation and precoding isnegligible in comparison.
To implement the MICED algorithm, the additional compu-tational complexity comes from re-estimating the channel, per-forming the linear combining, and decoding the data in eachiteration. Moreover, to obtain the data-aided channel estimatesanother × matrix inversion needs to be made correspondingto the Gramian of signal estimates (i.e., (X(8)�
;X(8);)−1 see
Section IV-B). This would for sure add an important burden tothe signal processing. However, this can be addressed throughparallel computing techniques similar to the ones used in theLuMaMi testbed [31], [32] where even if the extra processingduplicates the delay, it would still be less than 285 `s whichis their constraint for the precoding turnaround time. Thus,based on the existing developments in digital signal processingapplied to Massive MIMO systems [30], [31] it is indeedpossible to implement the MICED algorithm in practice forat least a few tens of iterations.
D. Numerical example with Gaussian symbols
To illustrate the possible gains of the MICED algorithm,numerical results considering Gaussian data symbols are pre-sented in Figures 3 and 4. The simulation setup is based on ahexagonal cell grid with UEs uniformly distributed in eachcell, and large-scale fading modeled as V;;: = l−13−U
;;:�;;: .
The term l is the fixed pathloss at a reference distance of 1 km
5Note that this time includes the computation of matrix inversions toperform ZF or RZF in the downlink.
to account for propagation effects independent of the distance,for example, antenna gains, and wall penetration losses. Thedistance between UE;: and BS; is denoted by 3;;: , and theshadow fading is defined by 10 log10 (�;;: ) ∼ N (0, f2
sf).6 The
spatial correlation matrices are computed based on the Gaus-sian local scattering model with angular standard deviationfang defined in [3, Ch. 2]. Statistical channel inversion powercontrol is considered such that r;: = min{r/V;;: , rMAX} wherer is a design parameter to set the average transmission energyper symbol and rMAX is the maximum transmission energy persymbol for each UE. In the case of SP, the proportion betweenpilot and data power is fixed as Δ ;: = Δ .7 A summary of themain simulation parameters are given in Table I. To calculatethe SE per UE in Figure 4, the achievable SE in Section IIIis used.
Figure 3 depicts the MSE of the channel estimates versusthe data estimation quality and size of the coherence block.Note that with RP g3 = g2 − g? and with SP g3 = g2 , thusg2 is always proportional to the number of transmitted datasymbols. Figure 3a shows that with RP having g? = , andSP the MSE of the channel estimates is a decreasing functionof the data estimation quality. Moreover, with relatively lowvalues of data estimation quality (i.e., f2
EST > 0.2), the MSEof the channel estimates improves with respect to their initialvalue with pilot-based channel estimation only. In practice,one might have good data estimation quality for some UEsand utilize that to reduce the MSE for the channels to otherUEs. Figure 3a also shows the MSE of the channel estimateswith RP having g? = 3 , that is a pilot reuse of 3, which is astandard approach to mitigate the effect of pilot contamination.The same channel quality as in that case can be achieved bythe MICED algorithm when the data estimation quality is highenough.
The performance of the MICED algorithm depends on thesize of the coherence block. In Figure 3b, as g2 increaseswith RP, the use of data-aided channel estimation continuouslydecreases the MSE and the improvement over pilot-basedchannel estimation increases accordingly which is a resultfrom having more observations. On the other hand, withSP, the MSE of the channel estimates with both pilot-basedand data-aided methods decreases at the same pace with g2since the number of observations is the same in both cases.However, the data-aided channel estimation reduces the MSEcompared to pilot-based channel estimation. In addition, whenthe coherence block is large enough, the data-aided channelestimation quality is higher compared to the standard pilot-based approach with RP and pilot reuse 3.
Figure 4a shows that when the quality of the data estimatesis very low, the use of the MICED algorithm provides littleor no improvement in terms of SE per UE since the use ofcorrupted data estimates fails to reduce the intracell interfer-ence. However, when the data estimation quality is above acertain value, the SE per UE becomes an increasing function
6This stands in contrast to the simulation setup in [24] where the large-scalefading and intercell interference are fixed.
7Note that Δ has been selected to maximize the SE with pilot only channelestimation based on numerical results that are omitted in this paper for brevity.See [33] for more details on power control optimization with SP.
9
RP pilot g? = RP MICED g? =
RP pilot g? = 3 RP MICED g? = 3 SP pilot SP MICED
0 0.2 0.4 0.6 0.8 110−2
10−1
Pilot-only
MICED
Data estimation quality (f2EST)
MSE
ofch
anne
les
t.
(a) MSE of channel estimates vs f2EST such that f (8)2
;:= f2
EST∀: ∈ {1, . . . , }, and g2 = 190.
100 200 300 400 50010−2
10−1Pilot-only
MICED
Size of coherence block (g2)
MSE
ofch
anne
les
t.
(b) MSE of channel estimates vs g2 withf
(8)2;:
= 0.6 ∀: ∈ {1, . . . , }.
Figure 3: MSE of channel estimates per UE versus f2EST and
g2 for r = f2 (SNR = 0 dB), " = 100, and = 10. Themarkers correspond to Monte-Carlo simulations and the linesto the closed-form expressions in Lemma 1 and Theorem 1(recall that the latter are lower bounds).
of the data estimation quality and improves with respect totheir initial value with pilot-based channel estimation only.Figure 4b shows that the SE per UE is an increasing functionof g2 for all methods. In the case with RP, the gap between SEwith pilot-based and data-aided channel estimation increaseswith g2 since the MICED algorithm utilizes more observationsfor data-aided channel estimation as g2 increases, whereas, thenumber of observations with pilot-based channel estimationremain the same. In addition, with RP the benefit of theMICED algorithm is lower when using S-MMSE processingwhich is a consequence of the additional interference addedin the data-aided channel estimates making the interferencesuppression less accurate. In the case of SP, the gap betweenSE with pilot-based and data-aided estimation decreases withg2 , which means that the MICED algorithm provides more
benefits when the size of the coherence block is short.In summary, when the quality of the data estimates is high
enough, the MICED algorithm can lower the MSE of thechannel estimation, and in turn, increase the SE per UE. Bycomparing the SE with RP and SP, the former provides higherSE in most cases except when the number of samples in thecoherence block is low and data-aided channel estimation isused. Thus, the MICED algorithm with RP is more beneficialin low mobility scenarios or low carrier frequencies withlong coherence blocks, while the MICED algorithm withSP performs best in high mobility scenarios or high carrierfrequencies where the size of the coherence block is short.In addition, notice that in Figure 4b there is a cross pointbetween RP and SP using the MICED algorithm, and thispoint depends on the number of multiplexed UEs . Notethat when increases and g2 remains fixed, the pilot overheadwith RP limits the SE making SP the preferred choice. Theaforementioned cross point between RP and SP can also beobserved in Figure 4c where the sum SE per cell is plottedversus with MR. The same behavior is found with S-MMSEbut the plots are omitted for ease of illustration.
Remark 2. The cross point between the SE with RP and SPwith respect to g2 (see in Figure 4b), and (see Figure 4c)indicates that the MICED algorithm with SP also has thepossibility to utilize more aggressive spatial multiplexing thatnot only increases SE but also facilitates the implementation ofmachine type communication systems where many UEs needto be served.
Figure 4d depicts the SE per UE versus the numberof BS antennas. In the case with RP, the benefit of theMICED algorithm is higher when using MR processing andit becomes less significant for S-MMSE when the numberof BS antennas grows large. This indicates that due to theadditional interference in the data-aided channel estimationwith RP the interference suppression capabilities of S-MMSEare less effective compared to performing MR combiningand then subracting the estimated intracell interference asshown in (19). On the other hand, with SP, the benefit ofusing the MICED algorithm compared to pilot-based channelestimations grows with the number of BS antennas and it ishigher for S-MMSE processing. Here, the pilot-based channelestimates have interference from all UEs and the MICEDalgorithm reduces the intracell interference, which in turn,enhances the interference suppression of S-MMSE.
V. FINITE ALPHABET SYMBOLS
The MICED algorithm was introduced in Section IV andevaluated under the assumption of Gaussian data symbols toperform a tractable theoretical analysis. This gave key insightsinto the cases where the MICED algorithm can providegains compared to pilot-based channel estimation in termsof channel estimation quality and SE. In this section, theimplementation of the MICED algorithm with finite-alphabetmodulation is described and evaluated in terms of achievableSE and BLER to further assess its potential benefits in realsystems.
10
RP pilot S-MMSE SP pilot S-MMSE RP MICED S-MMSE SP MICED S-MMSE
RP pilot MR SP pilot MR RP MICED MR SP MICED MR
0 0.2 0.4 0.6 0.8 1
1.5
2
2.5
3
3.5S-MMSE
MR
Data estimation quality (f2EST)
SEpe
rU
E[b
it/s/
Hz]
(a) SE per UE vs f2EST , " = 100, = 10, and g2 = 190.
50 100 150 200 250 300 350 400
1.5
2
2.5
3
3.5S-MMSE
MR
Size of coherence block (g2)
SEpe
rU
E[b
it/s/
Hz]
(b) SE per UE vs g2 with f2EST = 0.6, " = 100, and = 10.
10 20 30 40 50 6020
30
40
50
60
MR
Number of UEs per cell ( )
Sum
SEpe
rce
ll[b
it/s/
Hz]
(c) Sum SE per cell vs with f2EST = 0.6, " = 100, and g2 = 190.
50 100 150 200 250 300 350 400
1.5
2
2.5
3
3.5
4
4.5S-MMSE
MR
Number of BS antennas (")
SEpe
rU
E[b
it/s/
Hz]
(d) SE per UE vs " with f2EST = 0.6, = 10, and g2 = 190.
Figure 4: Sum SE per cell versus , and SE per UE versus f2EST, g2 and " for r = f2 (SNR = 0 dB). The data estimation
quality is selected such that f (8)2
;:= f2
EST ∀: ∈ {1, . . . , }.
Encoder Modulator
. . . #2 coh.block
#1 coh.block
Informationbits b
Codewordbits
Complex
symbols
Figure 5: Commun. model over several coherence blocks.
In practical implementations, the information bits sent overa communication system are encoded into finite length code-words by using a predefined channel code. This procedureadds redundant information to combat the errors introducedby the variations of the channel. The bits that make up thecodewords are then modulated into complex symbols which inturn are transmitted over the channel. Since typical codewordsare made up of long sequences of bits, the resulting numberof modulated symbols tends to span several coherence blocks,as illustrated in Figure 5. Moreover, the pilot symbols are
inserted into each coherence block along with the modulateddata symbols, as shown in Figure 1, resulting in differentnumber of coherence blocks that contain a full codeword withRP or SP. To successfully decode the received bits and retrievethe information bits at the receiver, the complex symbolscontaining the bits that form a full codeword need to bereceived. Based on the observations of the complex symbolsobtained at the receiver, log-likelihood ratios (LLR) for eachbit in the codeword are computed, and then fed into thedecoding algorithm.
Let #<>3 be the size of the alphabet used by the modulationscheme, and #1 = log2 (#<>3) the number of bits per complexmodulation symbol. Denote by {1;:1, . . . , 1;:#1
} a set of bitstransmitted by UE;: , and modulated into an arbitrary complexsymbol represented by B;: = [s;: ] 9 for any 9 ∈ {1, . . . , g3} ina given coherence block. Then, the corresponding observationH;: = [y;: ] 9 obtained from (15) with RP, and (16) with SP,is given by (30) and (31) respectively, at the top of the nextpage. Notice that to fully utilize the side information of thechannel and data estimates, an estimate of the received intra-
11
(yRP;: )
) = v�;: h;;:√dRP;:
︸ ︷︷ ︸= 6RP
;:(known equivalent channel)
s);: + v�;:
(h;;:
√dRP;:
s);: +∑ℓ∈Φ
∑:′=1
(ℓ,:′)≠(;,:)
h;ℓ:′√dRPℓ:′s
)ℓ:′ −
∑:′≠:
h;;:′√dRP;:′ s
);:′ + [N]g?+1:g2
)︸ ︷︷ ︸
=nRP;:
(effective noise)
(30)
(ySP;: )
) = v�;: h;;:√dSP;:
︸ ︷︷ ︸= 6SP
;:(known equivalent channel)
s);: + v�;:
(h;;:x);: +
∑ℓ∈Φ
∑:′=1
(ℓ,:′)≠(;,:)
h;ℓ:′x)ℓ:′ −∑:′≠:
h;;:′ x);:′ + N
)︸ ︷︷ ︸
=nSP;:
(effective noise)
(31)
cell interference is subtracted, see third term of the effectivenoise in (30) and (31). The LLR of an arbitrary bit 1;:= for= ∈ {1, . . . , #1} is given by
LLR;:== log(Pr (1;:= = 0| H;: )Pr (1;:= = 1| H;: )
)= log
©«∑ #<>3
28′=1 Pr
(H;: |1;:= ∈ H8′,1;:==0
)Pr
(1;:= ∈ H8′,1;:==0
)∑ #<>3
28′=1 Pr
(H;: |1;:= ∈ H8′,1;:==1
)Pr
(1;:= ∈ H8′,1;:==1
) ª®®¬(29)
where the set H8′,1;:==1 = {1;:1, . . . , 1;:= = 1, . . . , 1;:#1}
is defined as one of the #<>3/2 possible sets of modulatedbits where 1;:= = 1 and 1;:=′ ∈ {0, 1} for =′ ≠ =. Aftercomputing the LLRs for each bit, the decoder utilizes theredundancy in the channel code to correct errors, and obtainthe maximum likelihood estimate of the originally transmittedbits. Notice that due to interference and noise, this procedure isnot always perfect leading to errors that cannot be corrected bythe decoder. For example, the decoder might be able to decodethe signal from UEs that are close to the BS (since they wouldhave a high channel gain with respect to the interference andnoise), but not for cell-edge UEs that are more susceptibleto interference. Similarly, UEs that are subject to strong pilotcontamination are more likely to get decoding errors.
The performance of the decoder depends on the effectiveSNR between the power of the equivalent channel and theeffective noise (see (30) with RP, and (31) with SP), thatis denoted as SNREFF
;:= E{|6;: |2}/E{|n;: |2}. At this stage,
the interference mitigation processing has already been doneand therefore the effective noise is treated as a noise ratherthan interference, even though it is made up of interferenceterms as well as noise. The relation between SNREFF
;:and
the SNR required to successfully decode the informationdetermines how the decoder performs. Notice, that SNREFF
;:
depends on the channel estimation accuracy and the linearcombining strategy, as well as the interference level and pilotcontamination effect. If SNREFF
;:is too low, the decoder will
fail and there will be some erroneous bits. However, the LLRsat the output of the decoder could still be used, as sideinformation, to estimate the complex modulated data symbolsthat were sent. More importantly, if the information of someother UEs is decoded successfully, then, perfect knowledge oftheir complex modulated data symbols will be available.
The data estimates are obtained from LLRs at the output ofthe decoder which, in turn, require the data observations (see(30) and (31)) corresponding to all bits in a full codeword.Thus, let #CW be the number of bits that make a full code-word, and denote by y;: ∈ C#CW/#1 the observations of thecorresponding complex symbols obtained by staking severalinstances of (30) and (31), with RP and SP respectively.8
To obtain the estimates of the complex data symbols, theLLRs of each bit (after the decoding procedure) are mappedinto complex symbols based on the modulation scheme used.Assume that the complex symbol alphabet is given by the setA = {B(1), . . . , B(#<>3)} where each symbol maps #1 bitssuch that B 9 = {1 9 (1), . . . , 1 9 (#1)} corresponds to the set ofbits mapped into a given symbol B( 9). At the 8Cℎ iteration theMMSE estimate of an arbitrary data symbol B;: is
B(8);:
= E{B;: |y(8);:} =
#<>3∑9=1
B( 9)Pr(B;: = B( 9) |y(8);:)
=
#<>3∑9=1
B( 9)#1∏==1
Pr(1;:= = 1 9 (=) |y(8);:)
where the conditional probabilities of each bit are given by
Pr(1;:= = 0|y(8);:) = 1
1 + exp(−LLRD (8);:=
),
Pr(1;:= = 1|y(8);:) = 1
1 + exp(LLRD (8);:=
), (32)
and LLRD (8);:=
represents the LLR after the decoding procedurefor the 8Cℎ iteration. Then, the variance of the data estimates isgiven by f (8)2
;:= E
{| B (8);:|2}
, which can be estimated by takinga sample mean for all estimated symbols in a codeword.
It is worth mentioning that the MICED algorithm does notdepend on the channel code being used, and thus, it can beimplemented with any state of the art decoder. This stands incontrast to [24] which focuses on the design of a forward-error-correction channel code.
A. Numerical examples with finite-alphabet symbols
To illustrate the gains of the MICED algorithm in a practicalsystem, the same simulation setup as in Section IV-D is usedbut now considering LDPC codes and QPSK modulation. The
8Recall that one codeword spans several coherence blocks, see Figure 5.
12
0 2 4 6 8 10 12 14
0.05
0.1
0.15
0.2
RP S-MMSE pilot reuse-3
Number of iterations (8)
BL
ER
SP MR SP S-MMSE
RP MR RP S-MMSE
(a) BLER vs 8 with " = 100, and r = f2 (SNR = 0 dB).
−25 −20 −15 −10 −5 0 5 10
10−1
100
RP S-MMSE pilot reuse-3
SNR (r/f2) [dB]
BL
ER
SP MR pilotRP MR pilot
SP MR MICED
RP MR MICED
(b) BLER vs SNR with 8 = 8, and " = 100.
100 200 300 400
10−2
10−1
100
25
RP S-MMSE pilot reuse-3
Number of BS antennas (")
BL
ER
RP MR pilotRP S-MMSE pilot
RP S-MMSE MICED
RP MR MICED
(c) BLER vs " with RP, 8 = 8, and r = f2 (SNR = 0 dB).
100 200 300 400
10−2
10−1
100
25
RP S-MMSE pilot reuse-3
Number of BS antennas (")
BL
ER
SP MR pilotSP S-MMSE pilot
SP S-MMSE MICED
SP MR MICED
(d) BLER vs " with SP, 8 = 8, and r = f2 (SNR = 0 dB).
Figure 6: BLER versus number of iterations, SNR, and number of BS antennas with g2 = 200, = 10, and '23 = 1/2. TheBLER with RP, pilot-based channel estimation, S-MMSE combining, and pilot reuse 3 is included as a benchmark.
choice of parity check matrix is done following the new radio(NR) 3GPP specifications [34]. Two code rates are evaluated:'23 ∈ {1/2, 3/4} with a codeword length of 3840 and 3888bits respectively. Note that for these simulations, the channelerror correlation matrices in (21) are computed numerically.This is done to obtain more accurate results compared tousing the closed-form expressions Theorem 1, given that thedata symbol distribution is no longer Gaussian. To evaluatethe reliability of the transmitted data, the BLER is calculatedassuming that each codeword corresponds to one block. Theresults of implementing the MICED algorithm with RP assumepilot reuse one (g? = ). In the case of SP, the pilot reuse isthe closest to g2/ that is allowed in an hexagonal grid. Toestablish a benchmark, the performance of standard pilot-basedchannel estimation with RP is included with pilot reuse 1 and3.
Figure 6 depicts the BLER versus the number of iterations,SNR, and the number of BS antennas with a code rate'23 = 1/2. Figure 7 shows the achievable SE versus the size
of the coherence block and the number of BS antennas witha code rate of '23 = 3/4. Note that when the BLER is low(e.g. 10−2) the resulting changes in the achievable SE are verysmall. Thus, different code rates for BLER and achievable SEcurves are chosen to observe the potential difference betweenthe evaluated methods within a few hundred BS antennas.The achievable SE at the 8Cℎ iteration is obtained from themutual information between the input bits, and the soft symbolestimates at the output of the decoder. The number of encodedbits in a codeword is denoted as #ENC, and these encoded bitsare staked into the vector b;: such that [b;: ]< ∈ {0, 1} for< ∈ {1, . . . , #ENC}. Assuming that the bits at the output of thedecoder are independent, then the achievable SE is
R;: =g3#1'23
g2#ENC
#ENC∑<=1
(1
+E{ 1∑3=0
Pr([b;: ]< = 3 |y(8)
;:
)log2
(Pr
([b;: ]< = 3 |y(8)
;:
) ) })(33)
13
50 100 150 200 250 3000.8
1
1.2
1.4
S-MMSE pilot reuse-3
Size of coherence block (g2)
Ach
ieva
ble
SE[b
it/s/
Hz]
RP S-MMSE MICED
RP MR MICED
RP S-MMSE pilotRP MR pilot
(a) Achievable SE vs g2 with RP and " = 100.
50 100 150 200 250 3000.8
1
1.2
1.4
S-MMSE pilot reuse-3
Size of coherence block (g2)
Ach
ieva
ble
SE[b
it/s/
Hz]
SP S-MMSE MICED
SP MR MICED
SP S-MMSE pilotSP MR pilot
(b) Achievable SE vs g2 with SP and " = 100.
50 100 150 200
0.8
1
1.2
1.4
S-MMSE pilot reuse-3
Number of BS antennas (")
Ach
ieva
ble
SE[b
it/s/
Hz]
RP S-MMSE MICED
RP MR MICED
RP S-MMSE pilotRP MR pilot
(c) Achievable rate vs " with RP and g2 = 200.
50 100 150 200
0.8
1
1.2
1.4
S-MMSE pilot reuse-3
Number of BS antennas (")
Ach
ieva
ble
SE[b
it/s/
Hz]
SP S-MMSE MICED
SP MR MICED
SP S-MMSE pilotSP MR pilot
(d) Achievable rate vs " with SP and g2 = 200.
Figure 7: Achievable SE versus coherence block size and number of BS antennas with 8 = 8, = 10, r = f2 (SNR = 0dB), and '23 = 3/4. The SE with RP, pilot-based channel estimation, S-MMSE combining, and pilot reuse 3 is included as abenchmark.
where the probabilities in (33) are obtained as in (32). Notethat the achievable SE in (33) accounts the overhead fromcoding and using dedicated pilot symbols in the case of RP.
Figure 6a depicts the BLER versus the number of iterations.It can be seen after 8 iterations the results stabilize andtherefore that is the number of iterations selected for therest of the figures. In addition, as the iterations progress theMICED algorithm decreases the BLER with MR combiningfurther than with S-MMSE when compared to its initial stateat 8 = 0. Figure 6b shows the BLER versus the averageSNR per symbol indicating that the benefits of the MICEDalgorithm are achieve for both low and high SNR regimes. InFigures 6c and 6d the BLER is shown as a function of thenumber of BS antennas. It can also be seen that comparedto pilot-based channel estimation, the use of the MICEDalgorithm is more beneficial for MR combining and whenthe number of antennas grows it outperforms S-MMSE. Thiseffect is due to the presence of poor quality data estimatesin the data-aided channel estimation, which in turn, make theinterference suppression capabilities of S-MMSE less effective
when compared to subtracting the estimated intracell interfer-ence (see third term of the effective noise in (30) and (31)).Moreover, when using the MICED algorithm with RP, theBLER is lower compared to S-MMSE combining with pilotreuse 3 which means that greater reliability can be achieveddespite the 3 times lower pilot overhead. By comparing theBLER with RP and SP, it can be seen that RP provides lowerBLER than SP, however for practical number of BS antennas(e.g. " = 100) the difference between RP and SP is rathersmall.
Figure 7 shows the achievable SE as a function of thecoherence block size and number of BS antennas. In thiscase, the use of the MICED algorithm provides the greatestgains for MR combining which is in line with the resultsin Section IV-D. Furthermore, the SE with the MICED al-gorithm for MR and S-MMSE is very close which meansthat the benefit of the interference suppression with S-MMSEis comparable to removing the intracell interference withMR (see third term of the effective noise in (30) and (31)).The reason for this behavior, is the presence of inaccurate
14
data estimates in the data-aided channel estimation process.Therefore, the performance of the MICED algorithm with S-MMSE combining can be further improved by controlling theuse of data estimates based on their accuracy.
In addition, Figure 7 shows that the MICED algorithm withSP achieves higher SE than RP since it does not have a pilotoverhead (i.e., all symbols in the coherence block are used fordata). The most benefit of the MICED algorithm is obtainedfor small coherence block size which corresponds to highmobility scenarios or higher carrier frequencies.
VI. CONCLUSION
This article evaluates the use of iterative data-aided chan-nel estimation in multicell Massive MIMO systems, wherethe partially decoded bits are used to improve the channelestimates and reduce the decoding errors at the receiver. TheMICED algorithm is proposed and analyzed with RP and SPtransmission methods along with MR and S-MMSE processingassuming spatially correlated channels. The results show thatthe MICED algorithm increases the SE and reduces the BLERcompared to pilot-based channel estimation with both RP andSP. The highest SE is found when implementing the MICEDalgorithm with SP since the cost of the pilot overhead is re-moved and the data interference is mitigated by the data-aidedchannel estimation process. The MICED algorithm with SP ismost beneficial in high mobility or high carrier frequenciesscenarios with small coherence block size, outperforming RPin terms of SE. In addition, the MICED algorithm with SPallows for aggressive spatial multiplexing, increasing SE andfacilitating implementation of other technologies like machinetype communication.
The quality of data estimates plays a key roles when usinglinear combining that has interference suppression like S-MMSE. Thus, further improvements of the MICED algorithmcan be attained when adding control mechanisms for the useof data estimates based on their quality.
APPENDIX APROOF OF DATA-AIDED CORRELATION MATRICES
Consider a square matrix A that is positive semi-definiteand constants 0, 1 ∈ R such that 0 ≥ 1 ≥ 0, then itfollows that A0 � A1. Note that all correlation matrices arepositive semi-definite by definition, and since the expectationsin (21) are scalar quantities, taking lower bounds on theexpectations would result in correlation matrices that fulfillTheorem 1. It is also worth mentioning that by assumingcircularly symmetric complex Gaussian symbols (i.e., s;: ∼CN(0, Ig3 )) the resulting MMSE data estimate and its errorare statistically independent [2], [3]. To obtain the closed-form expression in (23) and (25) with RP, the terms in (21)are analyzed separately, let Dd = diag(dRP
;1 , . . . , dRP; ) and
D@d = diag(@RP;1 /d
RP;1 , . . . , @
RP; /dRP; ). Then, the calculations in
(34) at the top of the page hold, where (a1) follows from theindependence between data estimates and errors (i.e., u(8)
;:and
x(8);:′ are independent). The second equality (a2) is obtained by
expanding the Gramian of X(8);
and applying known propertiesof the matrix inverse operator. Note that T(8)
;is Hermitian
and positive semi-definite, thus from the result in Lemma 2of Appendix B, it holds that [(T(8)
;)−1]:: ≥ 1/[T(8)
;]:: . In
addition, the diagonal elements of S(8)�
;S(8);
are independentand have a chi-squared distribution. Thus, (a3) holds for j0 ∼0.5j2
2(g3+1− ) and j1 ∼ 0.5j22g3 [2, App. B]. Then, a final
lower bound on a(8);:
is obtained based on Jensen’s inequalitysince E{1/G} ≥ 1/E{G}, which yields the expression of thesecond term in (23) and first term in (25).
For the second term in (21) (intercell interference), note thatthe data symbols between UEs are independent, thus, sℓ:′ andu(8);:
are independent for ℓ ≠ ;. Let PRP;
= [5;1, . . . , 5; ], thenthe calculations in (35) and (36) at the top of the page hold.
Notice that interference from pilot symbols (see the firstterm in (35)) is only non-zero if UEℓ:′ shares a pilot witha UE in cell ;, and in particular, the largest interference willcome from UEs sharing the same pilots as UE;: . Thus, (b1)holds by discarding the pilot interference that does not comefrom UEs sharing the same pilots as UE;: (see the first termof (36)) and by applying the same method as in (34) for thedata interference (see the second term of (36)). Then, (b2)holds by applying Lemma 2 in Appendix B and taking theexpectation over the inverse of the diagonal elements. Finally,by applying Jensen’s inequality the expression for the pilotintercell interference (see second term in (25)) is found. Forthe third term in (21) (noise), the following result holds
E{ u(8)
;:
2}= E
{ [ (X(8)�
;X(8);
) −1]::
}≥ E
{(@RP;: g? + d
RP;: f
(8)2
;:j1
) −1},
then, by means of the Jensen’s inequality the third term in(25) is obtained.
In the case of the expression in (24) and (26) with SP,notice that x(8)
;:′ =
√dSP;:′ s
(8);:′ , then, because of independence
between data estimates and error, the result in (37) holds.Here, (c1) holds by taking the inverse of diagonal elementsbased on Lemma 2 in Appendix B, and the expressions in thesecond term of (24) and first term in (26) follow from Jensen’sinequality. Notice that the term b(8)
;:in (37) is also used to
calculate the noise influence (i.e., the last term in (21)). Forthe intercell interference, let D@ = diag(@SP
;1 , . . . , @SP; ), then
the result in (38) holds where (d1) follows from discardingthe cross products between pilot and data symbols (see thefirst term in (38)), and (d2) holds by discarding the pilotinterference that is caused by UEs that do not share the samepilot as UE;: . Then similarly to the result in (37), by applyingJensen’s inequality to the expectation of the inverse diagonalelements of the Gramian of X(8)
;, the expression in the second
term of (26) is found.
APPENDIX B
Lemma 2. Let A ∈ C#×# be Hermitian (i.e., A = A� ) andpositive semi-definite, then it holds that[
(A)−1]::
≥ 1[A] ::
. (39)
15
∑:′=1
R;;:′E{��(u(8)
;:)� x(8)
;:′
��2} (a1)=
∑:′=1
R;;:′dRP;:′ (1 − f (8)2
;:′ )E{ [S(8)
;D
12d
(X(8)�
;X(8);
) −1]:
2}
(a2)=
∑:′=1
R;;:′dRP;:′ (1 − f (8)2
;:′ )2@RP;:g?
E
{ [ (g?
2D@d
(S(8)�
;S(8);
) −1+ I + 1
2g?D−1@dS
(8)�
;S(8);︸ ︷︷ ︸
=T(8);
) −1]::
}
(a3)�
∑:′=1
R;;:′dRP;:′ (1 − f (8)2
;:′ )1
2@RP;:g?E
©«@RP;:g?
2dRP;:f
(8)2
;:j0
+ 1 +dRP;:f
(8)2
;:
2@RP;:g?
j1ª®¬−1︸ ︷︷ ︸
=a(8);:
(34)
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E{���u(8)�
;:xℓ:′
���2} =∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E�����(√@RP
ℓ:′
[5ℓ:′
0
]+
√dRPℓ:′
[0
sℓ:′
] ) �u(8);:
�����2=
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E
{@RPℓ:′
���� [5�ℓ:′PRP; D
12@
(X(8)�
;X(8);
) −1]:
����2 + dRPℓ:′
[S(8);
D12d
(X(8)�
;X(8);
) −1]:
2}(35)
(b1)�
∑(ℓ,:′) ∈PRP
;:
R;ℓ:′@RPℓ:′@
RP;: g
2?E
{����[ (X(8)�
;X(8);
) −1]::
����2} +∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dRPℓ:′E
{ [ (T(8);
) −1]::
}2@RP;:g?
(36)
(b2)�
∑(ℓ,:′) ∈PRP
;:
R;ℓ:′@RPℓ:′@
RP;: g
2?E
{(@RP;: g? + d
RP;: f
(8)2
;:j1
) −2}+
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dRPℓ:′a
(8);:.
∑:′=1
R;;:′E{��(u(8)
;:)� x(8)
;:′
��2} =
∑:′=1
R;;:′dSP;:′
(1 − f (8)2
;:′
)E
{ [ (X(8)�
;X(8);
) −1]::
}(c1)�
∑:′=1
R;;:′dSP;:′
(1 − f (8)2
;:′
)E
{(@SP;:g2 + d
SP;:
s(8);:
2 + 2√@SP;:dSP;:<
{>�;: s
(8);:
} ) −1}︸ ︷︷ ︸
=b(8);:
. (37)
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E{���u(8)�
;:xℓ:′
���2} =∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E�����(√@SP
ℓ:′>ℓ:′ +√dSPℓ:′sℓ:′
) �u(8);:
�����2(d1)�
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′E
{@SPℓ:′
���� [>�ℓ:′PSP; D
12@
(X(8)�
;X(8);
) −1]:
����2 + dSPℓ:′
[ (X(8)�
;X(8);
) −1]::
}(38)
(d2)�
∑(ℓ,:′) ∈PSP
;:
R;ℓ:′@SPℓ:′@
SP;:g
22E
{����[ (X(8)�
;X(8);
) −1]::
����2} +∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dSPℓ:′b
(8);:
(d3)�
∑(ℓ,:′) ∈PSP
;:
R;ℓ:′@SPℓ:′@
SP;:g
22E
{(@SP;:g2 + d
SP;:
s(8);:
2+ 2√@SP;:dSP;:<
{>�;: s
(8);:
} )−2}+
∑ℓ∈Φ\;
∑:′=1
R;ℓ:′dSPℓ:′b
(8);:
Proof: The matrix A can be expressed as A =[0 b�
b C]
where 0 ∈ R, b ∈ C(#−1)×1, and C ∈ C(#−1)×(#−1) . Byapplying results from the inverse of a partitioned matrix and
the ShermanMorrisonWoodbury formula [35, Ch. 0] it followsthat [
(A)−1]11 =
(0 − b�C−1b
) −1
16
=10+ 102 b�
(C − bb�
0
) −1
b(e1)≥ 1
0
where (e1) holds by discarding the second positive term sinceC − bb�
0� 0 given that A is Hermitian and positive semi-
definite [35, Ch. 7]. Let � ∈ R#×# be a permutation matrixthat moves the : Cℎ row of a matrix towards the first positionsuch that [A] :: =
[�A��
]11 and �−1 = �� , it follows that[
A−1]::
=[�A−1��
]11 =
[ (�A��
) −1]
11
≥ 1[�A��
]11
=1
[A] ::.
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Daniel Verenzuela received the M.Sc. degreein Electrical Engineering with the double degreeMERIT program from the Polytechnic of Turin(PdT), Turin, Italy and the Royal Institute of Tech-nology (KTH), Stockholm, Sweden, in 2014, andthe Licentiate of Engineering degree from LinköpingUniversity (LiU), Linköping, Sweden, in 2018. Cur-rently, he is pursuing his Ph.D. degree at the Divisionof Communication Systems in the Department ofElectrical Engineering (ISY) of Linköping Univer-sity, Linköping, Sweden. His main research interest
lies within the area of signal processing in multi-user Massive MIMO systemswith emphasis in energy efficiency design.
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Emil Björnson (S’07-M’12-SM’17) received theM.S. degree in engineering mathematics from LundUniversity, Sweden, in 2007, and the Ph.D. degreein telecommunications from the KTH Royal Instituteof Technology, Sweden, in 2011. From 2012 to2014, he held a joint post-doctoral position at theAlcatel-Lucent Chair on Flexible Radio, SUPELEC,France, and the KTH Royal Institute of Technology.He joined Linköping University, Sweden, in 2014,where he is currently an Associate Professor and aDocent with the Division of Communication Sys-
tems.He has authored the textbooks Optimal Resource Allocation in Coordinated
Multi-Cell Systems (2013) and Massive MIMO Networks: Spectral, Energy,and Hardware Efficiency (2017). He is dedicated to reproducible research andhas made a large amount of simulation code publicly available. He performsresearch on MIMO communications, radio resource allocation, machine learn-ing for communications, and energy efficiency. Since 2017, he has been on theEditorial Board of the IEEE TRANSACTIONS ON COMMUNICATIONSand the IEEE TRANSACTIONS ON GREEN COMMUNICATIONS ANDNETWORKING since 2016.
He has performed MIMO research for over ten years and has filed more thanten MIMO related patent applications. He has received the 2014 OutstandingYoung Researcher Award from IEEE ComSoc EMEA, the 2015 IngvarCarlsson Award, the 2016 Best Ph.D. Award from EURASIP, the 2018 IEEEMarconi Prize Paper Award in Wireless Communications, the 2019 EURASIPEarly Career Award, and the 2019 IEEE Communications Society Fred W.Ellersick Prize. He also co-authored papers that received Best Paper Awards atthe conferences, including WCSP 2009, the IEEE CAMSAP 2011, the IEEEWCNC 2014, the IEEE ICC 2015, WCSP 2017, and the IEEE SAM 2014.
Xiaojie Wang received his B.Sc. and M.Sc. (withdistinction) degrees, both with a major in electricalengineering, from the University of Stuttgart, Ger-many, in 2012 and 2014 respectively. Currently, heis working as a research staff at the Institute ofTelecommunications, University of Stuttgart, Ger-many, pursuing the doctoral degree in the mean-time. His research interests include multiuser detec-tion/multiple access techniques, advanced waveformand air interface design and estimation/detectiontheory.
Maximilian Arnold received the B.Sc. (2014) andM.Sc (2016) from Stuttgart University in electri-cal engineering. In 2016 he joined the institute oftelecommunication as a PHD student. He receiveda best student paper award on ISWCS2017 for hiswork in Massive MIMO measurements. His currentresearch topics are development of prototypes forMassive MIMO and Deep Learning techniques ap-plied to Massive MIMO.
Stephan ten Brink (M97SM11) has been a facultymember at the University of Stuttgart, Germany,since July 2013, where he is head of the Instituteof Telecommunications.
From 1995 to 1997 and 2000 to 2003, Dr. tenBrink was with Bell Laboratories in Holmdel, NewJersey, conducting research on multiple antenna sys-tems. From July 2003 to March 2010, he was withRealtek Semiconductor Corp., Irvine, California, asDirector of the wireless ASIC department, develop-ing WLAN and UWB single chip MAC/PHY CMOS
solutions. In April 2010 he returned to Bell Laboratories as Department Headof the Wireless Physical Layer Research Department in Stuttgart, Germany.
Dr. ten Brink is a recipient and co-recipient of several awards, includingthe IEEE Stephen O. Rice Paper Prize, the IEEE Communications SocietyLeonard G. Abraham Prize for contributions to channel coding and signaldetection for multiple-antenna systems.
He is best known for his work on iterative decoding (EXIT charts) andMIMO communications (soft sphere detection, massive MIMO).
