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Generalized channel estimation and user detection for massiveconnectivity with mixed-ADC massive MIMO
Liu, T., Jin, S., Wen, C-K., Matthaiou, M., & You, X. (2019). Generalized channel estimation and user detectionfor massive connectivity with mixed-ADC massive MIMO. IEEE Transactions on Wireless Communications.https://doi.org/10.1109/TWC.2019.2912370
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Download date:21. Jun. 2022
1
Generalized Channel Estimation and User Detectionfor Massive Connectivity With Mixed-ADC
Massive MIMOTing Liu, Shi Jin, Chao-Kai Wen, Michail Matthaiou, and Xiaohu You
Abstract—This paper aims to provide a partial-DFT pilotsequence assisted joint channel estimation and user activity de-tection scheme for massive connectivity, in which a large numberof devices with sporadic transmission communicate with a basestation (BS) in the uplink. The joint channel estimation and devicedetection problem can be formulated as a compressed sensingsingle measurement vector (SMV) or multiple measurementvector (MMV) problem depending on whether the BS is equippedwith single or large number of antennas. Due to high hardwarecost and power consumption in massive multiple-input multiple-output (MIMO) systems, a mixed analog-to-digital converter(ADC) architecture is considered. In order to accommodate alarge number of simultaneously transmitting devices, the jointchannel estimation and active user detection are formulated asa MMV problem for the massive connectivity scenario; andthe proposed GTurbo-MMV algorithm can precisely estimatethe channel state information (CSI) and detect active deviceswith relatively low overhead. Furthermore, we study the stateevolution (SE) for the MMV problem to obtain achievable boundson channel estimation and device detection performance, in whichboth the missing and false detection probabilities can be madetend to zero in the massive MIMO regime. Simulation resultsconfirm the theoretical accuracy of our analysis.
Index Terms—Channel estimation, GTurbo-MMV, mixed-ADCarchitecture, partial-DFT pilots, sporadic transmission, stateevolution (SE), user activity detection.
I. INTRODUCTION
Massive machine-type communication (mMTC) is a keytechnology for 5G mobile communication networks [1], whichis expected to support different applications, as shown in Fig.1. In such a massive device connectivity scenario for 5G, theuse of massive MIMO is required,1 in which a BS equippedwith a large number of antennas can serve a massive number
The work of S. Jin was supported in part by the National ScienceFoundation (NSFC) for Distinguished Young Scholars of China with Grant61625106 and the NSFC with Grant 61531011. The work of C.-K. Wen wassupported in part by the Ministry of Science and Technology of Taiwan underGrants MOST 107-2221-E-110-026. The work of M. Matthaiou was supportedby EPSRC, UK, under grant EP/P000673/1.
T. Liu, S. Jin and X. You are with the National Mobile CommunicationsResearch Laboratory, Southeast University, Nanjing 210096, P. R. China.(email: [email protected]; [email protected]; [email protected]).
C.-K. Wen is with the Institute of Communications Engineering,National Sun Yat-sen University, Kaohsiung 804, Taiwan. (e-mail:[email protected]).
M. Matthaiou is with the Institute of Electronics, Communications andInformation Technology (ECIT), Queen’s University Belfast, Belfast, BT39DT, U.K. (email: [email protected]).
1Massive MIMO networks are naturally suited to support mMTC, due totheir: (a) unprecedented processing gains; (b) improved contention in resourceaccess [4]; (c) effective improvement in the successful probability of activedevice detection [13].
Fig. 1. Massive device communication, where only a fraction ofdevices are active at each time slot.
of devices. The massive connectivity of mMTC aims at sup-porting only a small fraction of active potential devices out ofan extended number of devices to communicate with the basestation (BS); a technology known as massive sporadic trans-mission [2–4]. For such a scheme, small packets may causegreat burden by increasing the ratio of signaling overheadto useful payload under current media access control (MAC)protocols [5] where a user’s transmission commences uponthe reception of uplink grant in the physical downlink controlchannel. Therefore, contention based uplink transmission hasbeen introduced by using identical signature sequences foreach active user [6], which remarkably reduces the significantoverhead caused by the large number of sporadically activeusers. However, this technique is limited by the resolution ofcontention and the collision caused by the large number ofonline devices, and it is impractical to assign orthogonal pilotsequences to all potential terminal devices. Fortunately, grant-free transmission scheme with non-orthogonal pilot sequenceshas been introduced to avoid the overhead in signaling ofactivity, and reliable communication can be achieved afterjoint channel estimation and activity detection [7–9].
A. Related Work
There are many reported attempts on addressing massivesporadic transmission problems by using tools of compressedsensing (CS). In [8], Xu et al. considered the case when the BShas perfect knowledge of the large scale fading parameters ofdifferent online users and modified the Bayesian CS algorithmto detect active users and estimate their channel responses. Byfurther exploiting the sparsity of user activity, [10] proposeda sparsity-based detector by maximizing the posterior prob-
2
ability with finite alphabet restrictions. Moreover, [11] per-formed joint information decoding without prior knowledge ofchannel state information (CSI) based on a greedy algorithm.Recently, in [12, 13], an approximate message passing (AMP)[14] based algorithm has been introduced to massive spo-radic transmission where an independent identically distributed(i.i.d.) Gaussian pilot sequence was assigned to each user,when the BS is equipped with single antenna and multipleantennas, which transforms the problem to single measurementvector (SMV) based AMP and vector AMP for multiplemeasurement vector (MMV) (VAMP-MMV) [15], respective-ly. The statistical characteristics of channel estimation anduser activity detection were studied via the so called stateevolution (SE). From an information theory perspective, anoverall achievable rate was characterized in closed-form withmaximal-ratio combing (MRC) and minimum mean square er-ror (MMSE) beamforming at the BS for massive connectivitywith massive MIMO in [16].
A common limitation of the massive MTC (mMTC) withmassive MIMO is that a large number of RF chains usedfor antennas at the BS would significantly complicate thehardware design for massive connectivity. Such systems re-quire an ADC unit for each receiver antenna. However, theimplementation cost and power consumption are exponentiallyincreased by using high-resolution ADCs. Nevertheless, thisproblem can be resolved by using low-resolution ADCs (e.g.,1-3 bits) to replace high-resolution ADCs, resulting quan-tized massive MIMO systems [17–19]. Unfortunately, low-resolution ADC systems face formidable challenges, such asdata rate loss, error floor, and high pilot overhead caused bythe nonlinear characteristics of the low-resolution quantization[20–22]. Motivated by the above concerns, a mixed-ADC ar-chitecture was proposed for massive MIMO in [23, 24], wherea few number of antennas are equipped with high-resolutionADCs while others are connected with low-resolution ADCs.Particularly, the reserved high-resolution ADCs can be utilizedto perform accurate channel estimation in a round-robin man-ner with reduced-complexity algorithm and affordable pilotoverhead. In [25], the mixed-ADC architecture was proposedthat can help maintain the promised detection performancewhile significantly reducing the computational complexity.Moreover, this architecture has inherent practical relevancefor massive MIMO, while still offering a large proportionof the channel capacity of the ideal ADC architecture. Thespectral and energy efficiency of the mixed-ADC frameworkhave been analyzed in [26], showing a better balance betweenenergy consumption and throughput compared to the purelow-resolution and ideal high-resolution ADC architectures. Inthis process, the mixed-ADC architecture is practically usefulbecause it provides comparable performance with the idealinfinite-resolution ADC topology, while the signal processingcomplexity and power consumption can be reduced at thesame time. On these grounds, the Bayesian optimal datadetector in hybrid broadband mmWave systems with mixed-ADC architecture was investigated in [27]. We also note thatthe computational complexity of massive connectivity regimeis still an open problem. This paper shows that adopting thediscrete Fourier transformation (DFT) pilot matrix as a pilot
sequence pool can significantly reduce the computational com-plexity of massive MIMO supporting massive connectivity.
B. Main Contributions
This paper considers the uplink of a single-cell massive con-nectivity scenario, where a plethora of online users or devicesare served by multiple antennas at the BS. We emphasize thatmassive MIMO has been widely studied in the literature [28–30]. However, the biggest body of related research is focusedon the regime of small number of online users as compared tothe number of BS antennas. In contrast, this paper articulatesthat the channel estimation and user detection can be jointlyachieved under the mixed-ADC architecture, when a massivenumber (larger than the number of BS antennas) of devicesis deployed. Due to the sporadic characteristic of massiveconnectivity, principles and techniques from the CS literature[31] can be leveraged for activity detection and channelestimation. In our previous study, the generalized turbo signalrecovery algorithm (GTurbo-SR) was applied in solving CSproblems [32] under non-linear measurements with a partialDFT pilot matrix. The partial DFT pilot matrix not onlyreduces the hardware cost and implementation complexity, butalso provides stronger guarantee of convergence compared tothe AMP algorithm [33–35]. Our specific contributions are thefollowing:
1. For the case where the BS is equipped with a singleantenna, we first propose the GTurbo-SMV algorithm toinvestigate the signal detection for the pure low-resolutionmeasurement and analyze the scalar-wise SE [32]. To utilizethe extra degrees of freedom (DoF) provided by multipleantennas, we extend the GTurbo-SMV to GTurbo-MMV. Byfully exploiting the joint sparsity property of the channelresponse, the GTurbo-MMV algorithm can be proved to becapable of identifying active users with enhanced accuracy.We also analyze the SE for GTurbo-MMV in the large systemregime, which provides valuable insights into the channelestimation of our algorithm, and further asymptotic activitydetection accuracy conclusion is derived.
2. A mixed ADC architecture can assist in channel esti-mation and active device detection due to the available high-resolution ADCs at the existing BS. What is more, a mixedADC architecture is economically beneficial because it canbe implemented by deploying some low-resolution ADCs atthe BS. In this paper, GTurbo-MMV can be shown to becapable of achieving strong performance and strong robustnessin solving user detection and channel estimation problems witha mixed-ADC architecture. On the other hand, we also derivetheoretical bounds on the channel estimation mean square error(MSE) by using our previous SE analysis.
3. Instead of using the traditional i.i.d. Gaussian sequences(non-orthogonal sequences) or some other orthogonal se-quences as pilots, a row-orthogonal matrix, e.g., a partialDFT pilot matrix, is allocated to different online users toreduce the cost of pilot overhead. Most importantly, the DFTpilot matrix is not required to be stored, which reduces thebuffering requirements to a large extent. Moreover, with theimplementation of the Cooley-Tukey FFT algorithm, channel
3
Fig. 2. MTC communication between the BS and N online devices.
estimation and user activity detection can be implemented withlow complexity.
The rest of this paper is organized as follows. The problemunder investigation is formulated in Section II, along with thepilot allocation scheme. In Section III, we first introduce theGTurbo-MMV algorithm for user activity detection and chan-nel estimation, and then use our SE analysis to derive severalpropositions and theorems. Section IV presents the numericalresults of our proposed algorithm. The paper concludes inSection V.
Notations: A scalar variable is denoted by lowercase normalletter a. Vectors and matrices are denoted by bold lowercasea and uppercase letters A, respectively. A K × K identitymatrix is denoted as IK . For a given vector a, its conjugatetranspose is denoted as aH, and aT denotes its transpose;‖a‖2 denotes the Euclidean norm of vector a; ‖A‖F is theFrobenius norm of matrix A. We let ai denote the i-th columnof A, a[i] denotes the i-th row of A while ai,j represents the(i, j) entity of A. For a given square matrix A, diag (A)would get all elements in its diagonal and form a new vectorin original order. For a vector a, diag (a) would constructa diagonal matrix with a the diagonal elements. The italicI denotes the set and |I| is its cardinality. A random vectorx drawn from the complex Gaussian distribution of mean aand covariance A is denoted as CN (x; a,A) with probabilitydensity function e−(x−a)HA−1(x−a)/ det(πA). The discreteset with B-bit quantizer is RB, and the set of natural numberswithout zero is R+.
II. SYSTEM MODEL
We consider the uplink massive sporadic transmission of asingle-cell wireless cellular system, as shown in Fig. 2, whichconsists of N online devices with single-antenna and the oneBS has MR antennas. Under the contention-based transmissionscheme, user terminals are independent to each other andacross each frame, and assumed to have the same probabilitypa to transmit data. An indicator function βn is used to indicatethe activity state of online user n, i.e., P(βn = 1) = pa, andP(βn = 0) = 1−pa. Therefore, in massive sporadic transmis-sion where pa 1, N 1, the number of active users withina frame is a Poisson random variable Na (Na =
∑Nn=1 βn).
To efficiently recover the information transmitted by the active
users, pilot-based channel estimation and active user detectionhave to be executed within each frame.
Define sn ∈ CL to be the pilot sequence of user n, whereL is the length of pilot sequence. Each pilot sequence sn isextracted from the sequence pool S,
S = ΥF, (1)
where S = [s1, s2, . . . sN ] ∈ CL×N , Υ ∈ CL×N is a row-selection matrix with measurement ratio α = L/N , and F ∈CN×N is a unitary DFT matrix.
Then, the received signal at the BS can be given by
YT = SHT + ZT, (2)
where YT = [y1,y2, . . . ,yMR] ∈ CL×MR , H =
[h1,h2, . . . ,hN ] with hn ∈ CMR representing the channel re-sponse of the n-th user. In this paper, we assume that a block-fading is adopted, and all channels follow the independent flatfading within a coherence block, where hn’s remain constant,but vary independently from block to block; Z ∈ CMR×L
denotes the complex white Gaussian noise with element-wisevariance σ2. Denote INa as the active users’ index, whichis the subset of online user set IN = 1, 2, . . . , N with|INa | = Na, i.e., hn = 0 for n /∈ INa . We assume that hnis i.i.d., and can be seen as drawn from the joint Bernoulli-Gaussian (BG) distribution
P(hn) = (1− pa)δ(hn) + pa CN (hn; 0, µnI), (3)
where δ(·) represents the Dirac delta for a vector, and µn is thelarge-scale fading factor of user n. The large-scale componentsare assumed to be known at the BS.
By taking the distribution of hn in (3), HT can be charac-terized with joint sparsity, which makes (2) a MMV problemwithin the field of CS [31]. To further simplify our systemmodel, we denote
Ξ = FHT, (4)
where Ξ =[ξ1, ξ2, . . . ξMR
]. Then, Eq. (2) can be equiva-
lently expressed as
YT = ΥΞ + ZT. (5)
III. CHANNEL ESTIMATION AND USER DETECTION
In this section, we first discuss the GTurbo algorithmfor SMV. Based on the joint sparsity of HT in (2), wethen propose a GTurbo algorithm for MMV (GTurbo-MMV),which iterates the extrinsic information for each antenna andcombines them within the joint estimator of HT, to estimatethe channel responses and detect user activity.
A. Bayesian Detector Formulation
A main point of this paper is to exploit the massive con-nectivity by adopting the Bayesian inference under a mixed-ADC massive MIMO architecture. We assume that someof the BS antennas adopt low-resolution ADCs (e.g., 1-3bits) while the remaining BS antennas adopt high-resolutionADCs (i.e., infinite-resolution ADCs). The complex-valued
4
quantizer Qc(·) consists of two real-valued quantizers Q, andthe resulting quantized signal is expressed as
Y = Qc(YT)
∆= Q(YT
R) + jQ(YTI ), (6)
where YTR and YT
I are the real and imaginary part of YT
respectively.The quantized output of each element ym,l is assigned the
value ym,l = yR,(m,l)+jyI,(m,l) by judging whether each part,yR,(m,l) or yI,(m,l), falls into the interval (ylowR,(m,l), y
upR,(m,l)] or
(ylowI,(m,l), yupI,(m,l)]. For a typical uniform B-bit quantizer with
quantization step size ∆, the quantized output is given by
yR,(m,l),yI,(m,l)
∈(−1
2+ b
)∆; b = −2B
2+ 1, · · · , 2B
2
, RB,
(7)
where m = 1, 2, . . .MR, l = 1, 2, . . . L, and B is the numberof quantization bits. The associated lower and upper thresholdsof yR,(m,l) are respectively expressed as2
ylowR,(m,l) =
yR,(m,l) − ∆
2 , for yR,(m,l) ≥ −(
2B
2 − 1)
∆,
−∞, otherwise,
yupR,(m,l) =
yR,(m,l) + ∆
2 , for yR,(m,l) ≤(
2B
2 − 1)
∆,
∞, otherwise.(8)
For linear measurements in (5) and nonlinear (quantized)measurements in (6), we both consider the Bayesian optimalinference to estimate HT and Ξ from YT or Y in terms ofMSE. Toward this end, the posterior joint distribution undernonlinear measurements can be divided into
P(Y|Ξ)=
MR∏m=1
L∏l=1
P( ym,l| ξΘ(l),m), (9)
where Θ (l) is a mapping function to map the selected rowindex to the original row number representing as the con-sequence of Υ. The right hand side of (9) results from theproduct of real and imaginary counterparts is given by
P(ym,l|ξΘ(l),m
)=
(1√πσ2
∫ yupR,(m,l)
ylowR,(m,l)
e−(y−ξR,(Θ(l),m))
σ2 dy
)
×
(1√πσ2
∫ yupI,(m,l)
ylowI,(m,l)
e−(y−ξI,(Θ(l),m))
σ2 dy
),
(10)
where yR,(m,l) ∈ (ylowR,(m,l), yupR,(m,l)] and yI,(m,l) ∈
(ylowI,(m,l), yupI,(m,l)]. According to the cumulative Gaussian dis-
tribution function, we obtain
P(ym,l|ξΘ(l),m
)= Ψ
(yR,(m,l); ξR,(Θ(l),m),
σ2
2
)×Ψ
(yI,(m,l); ξI,(Θ(l),m),
σ2
2
), (11)
2The analysis of yI,(m,l) is identical.
with
Ψ(y; ξ, c2
) ∆= Φ
(yup − ξ
c
)− Φ
(ylow − ξ
c
), (12)
where Φ(·) denotes the cumulative Gaussian distribution func-tion.
The Bayesian estimation of HT is a CS problem due tothe joint sparsity of channel response. Further, as the sparsitypattern is sensed at BS with multiple antennas, this becomes aMMV compressed sensing setup. In our conference paper [32],an innovative GTurbo signal recovery (GTurbo-SR) algorithmwas proposed as an iterative scheme to recover CSI fromquantized measurements. We will briefly outline the GTurbo-SR algorithm and extend to the MMV scenario with mixed-ADC architectures in the following subsections.
B. GTurbo-SMV
GTurbo-SR algorithm was proposed by [32] as an iterativealgorithm for CS. We first introduce the GTurbo algorithm forthe SMV problem. The basic idea in using GTurbo algorithmis to perform signal recovery based on the turbo principle initerative decoding within each iteration, which is shown inFig. 3, where y, y, ξ and h are the vector representations ofY, Y, Ξ, and H when the BS has only one antenna. Duringeach iteration of Algorithm 1, module A estimates ξ based onthe observation of y or y through posterior estimation, whilemodule B estimates the channel response hpost in the samemanner. According to the turbo principle, the extrinsic outputof one module is fed back to the other module as the prioriinput and the two modules are executed iteratively until theymeet the convergence criteria. Finally, in the last iteration, theoutput of module B is considered as the channel estimation ofh. The expectations and variances of lines 1 and 2, as well aslines 8 and 9 are taken with respect to (w.r.t.) the followingposterior probabilities,
P(ξn |yn ) = P(yn|ξn )P(ξn)∫P(yn|ξn )P(ξn)dξn
, linear,
P(ξn |yn ) = P(yn|ξn )P(ξn)∫P(yn|ξn )P(ξn)dξn
, nonlinear,(13a)
P(hn|hpriB,n) =
CN (hn;hpriB,n, v
priB )P(hn)∫
CN (hn;hpriB,n, v
priB )P(hn)dhn
, (13b)
where P(ξn) = CN (ξn; ξpriA,n, v
priA ), P(yn |ξn ) =
CN (yn; ξn, σ2), and P(yn |ξn ) is defined in (11). The
elements of h are drawn i.i.d. from the BG distribution, i.e.,P(hn) = (1 − pa)δ(hn) + paCN (hn; 0, 1). Please refer to[32] for the explicit expressions of lines 1 and 2, lines 8 and9.
C. GTurbo-MMV for massive connectivity with mixed-ADCarchitectures
We now extend the GTurbo-SMV algorithm to the GTurbo-MMV algorithm for the massive connectivity problem in themultiple-antenna case with mixed-ADC topologies. Comparedwith SMV, the major difference of the MMV problem in (2)mainly comes from the joint sparse property of HT. To applythe GTurbo scheme in the MMV scenario, we first derive the
5
Fig. 3. Block diagram of the GTurbo-SMV algorithm.
Algorithm 1: GTurbo-SMVinput : Observations y or y, pilot matrix F, likelihood P(y |ξ ) or
P(y |ξ ), and prior distributions P(h)
output : Recovered channel h
initialize : Current iteration t← 1, ξpriA ← 0, vpriA ← E[|h|2]
while t ≤ T do1) Output steps:
Compute the posterior mean and variance of ξ1 ξpost ← E
[ξ∣∣∣ξpriA , vpriA
];
2 vpostξ ← Var
[ξ∣∣∣ξpriA , v
priA
];
2) Estimate h via FHξ and its corresponding variance3 hpost
A ← FHξpost;
4 vpostA ← 1N
N∑n=1
vpostξ,n ;
Compute the extrinsic mean and variance of h5 hpri
A ← FHξpriA ;
6 vpriB ← vextA ←(
1
vpostA
− 1
vpriA
)−1
;
7 hpriB ← hext
A ← vextA
(h
postA
vpostA
− hpriA
vpriA
);
3) Input steps:Compute the posterior mean and variance of h
8 hpost ← E[h∣∣hpri
B , vpriB
];
9 vposth ← Var
[h∣∣hpri
B , vpriB
];
4) Estimate ξ via Fh and its corresponding variance10 ξpostB ← Fhpost;
11 vpostB ← 1N
N∑n=1
vposth,n ;
Compute the extrinsic mean and variance of Ξ12 ξpriB ← Fhpri
B ;
13 vpriA ← vextB ←(
1
vpostB
− 1
vpriB
)−1
;
14 ξpriA ← ξextB ← vextB
(ξpostB
vpostB
− ξpriB
vpriB
);
Update for the next iteration:
15 t← t+ 1
estimator of a jointly sparse vector by considering an arbitraryAWGN channel
r = h + υ, (14)
where υ ∼ CN (υ; 0,V) with V a real-diagonal covariancematrix, and h follows the Bernoulli-Gaussian distributiongiven by
P(h) = (1− pa)δ(h) + paCN (h; 0, µI). (15)
Proposition 1: The explicit expressions of the posteriormean and variance as well as the corresponding mse of model(14) are respectively given by
E [h |r ] =paP(r)
CN (r; 0, µI + V)
(I +
1
µV
)−1
r,
Var [h |r ] =paP(r)
CN (r; 0, µI + V)[(ΛV−1r)(ΛV−1r)H + Λ]
− E [h |r ]E[h |r ]H,
mse = paµI− pa2(ΛV−1
)×[∫
rrHCN (r; 0, µI + V)
pa + (1− pa)CN (r; 0,V)/CN (r; 0, µI + V)dr
]×(ΛV−1
)H, (16)
where Λ = (µ−1I + (V)−1)−1, P(r) =(1 − pa)CN (0; r,V) + paCN (r; 0, µI + V), andmse , E [Var[h |r ]].
Proof: Please refer to Appendix A.
From Proposition 1, we can then construct the GTurbo algo-rithm for MMV as in Fig. 4. The overall workflow of GTurbo-MMV is to make a row-vector-wise MMSE estimation ofH, and maximum posterior probability (MAP) estimation ofactive user set INa in module B within each iteration. Giventhe MR pairs of extrinsic information Hext
A→B, vextA→B that are
calculated in parallel on each antenna in module A, module Bthen estimates the MMSE of H, and then Ξext
B→A, vextB→A are
fed back to module A as the priori inputs, which completesthe entire loop of one iteration. The reason why GTurbo-MMVwill outperform MR parallel GTurbo-SMV is that GTurbo-MMV takes full knowledge of the joint sparsity of HT, andthen provides additional information through the posterioriestimation. The proposed GTurbo-MMV is summarized inAlgorithm 2, and we omit the superscript t at each iterationfor brevity in this subsection.
In Algorithm 2, step 1) contains MR parallel runningstages. Each stage computes one pair of extrinsic information,hpri
B [m] ∈ C1×N and vpriB,m, in the manner of calculation
of lines 1 to 7 in Algorithm 1. Subsequently, the channelestimation part in module B computes the posterior mean andvariance. The explicit expressions of MMSE estimation hpost
B,n
and its corresponding variance vpostB,n in lines 2-3 of Algorithm
2 are directly taken from (16) in Proposition 1 and can be
6
Fig. 4. Block diagram of the GTurbo-MMV algorithm.
given as
E[hn
∣∣∣hpriB,n,v
priB
]=
pa
P(hpriB,n)CN
(hpri
B,n; 0, µnI + Vpri)
×(
I +1
µnVpri
)−1
hpriB,n, (17)
Var[hn
∣∣∣hpriB,n,v
priB
]=
pa
P(hpriB,n)CN (hpri
B,n; 0, µnI + Vpri)
×
Λ(Vpri)−1
hpriB,n(hpri
B,n)H
((Vpri)−1
)H
ΛH + Λ
− E
[hn
∣∣∣hpriB,n,v
priB
]E[hn
∣∣∣hpriB,n,v
priB
]H, (18)
where Vpri = diag(vpriB ), P(hpri
B,n) = (1 −pa)CN (0; hpri
B,n,Vpri) + paCN (hpri
B,n; 0, µnI + Vpri),and Λ = (µ−1
n I + (Vpri)−1)−1.In line 4 of Algorithm 2, we employ the MAP method to
detect the user activity. The activity of user n, i.e., βn, isinferred from the ratio between P(βn = 1|hpri
B,n) and P(βn =
0|hpriB,n), that is,
P(βn = 1|hpriB,n)
P(βn = 0|hpriB,n)
=P(hpri
B,n|βn = 1)P(βn = 1)
P(hpriB,n|βn = 0)P(βn = 0)
, (19)
where P(hpriB,n |βn = 1) = CN (hpri
B,n; 0, µnI+Vpri) andP(hpri
n |βn = 0) = CN (hprin ; 0,Vpri) for active devices and
sleeping devices. Therefore, βn can be judged according tothe following criterion:
βn =
1, if P(βn = 1|hpri
B,n) > P(βn = 0|hpriB,n),
0, otherwise.(20)
Consequently, INa in line 4 of GTurbo-MMV can be obtained.After the processing of module B, step 3) of Algorithm 2evaluates the extrinsic information of Ξ based on the lineartransform Ξ = FHT, and then feds them back to module Aas the priori information of next iteration.
IV. PERFORMANCE ANALYSIS OF GTURBO-MMV
We now analyze the performance when a large numberof online users are served by the BS, that is the so calledSE [35, 36]. The SE analysis can be extended to MMV casewhen the unknown matrix is jointly sparse. In this section,we first use SE to derive achievable bounds on the channelestimation accuracy, then apply these bounds to determine
Algorithm 2: GTurbo-MMVinput : Observations Y or Y, pilot matrix F, number of online
users N , activity possibility pa, number of BS antennas MR
and noise variance σ2
initialize : Current iteration t← 1, index set Λ0 = φ, ΞpriA ← 0,
vpriA ← E[|hi,i|2
]while t ≤ T do
1) Compute the posterior information of Ξ and the extrinsicinformation of H for M BS antennas
for m = 1 to MR
1 Perform line 1 to line 7 in GTurbo-SMV to obtain ξpostm andvpostξ,m , hpost
A [m] and vpostA,m , hpriB [m] and vpriB,m;
end for
2) Channel estimation with joint sparsity for N online usersfor n = 1 to N
compute the posterior mean and variance of hn2 hpost
B,n ← E[hn|hpriB,n,v
priB ];
3 vpostB,n ← diag(Var[hn|hpri
B,n,vpriB ]);
end forUser Activity Detection
4 Estimate active users’ index INa based on P(βn|hpriB,n);
3) Estimate Ξ and its corresponding extrinsic information forMR BS antennas
for m = 1 to MR
5 Perform line 10 to line 14 in GTurbo-SMV to obtain ξpostB,m
and vpostB,m, ξpriA,m and vpriA,m;end for
Update for the next iteration:
6 t← t+ 1;
output : Active users’ index INa = Λk and their channelestimations Hpost
INa
the asymptotic performance of user detection when massiveantennas are deployed at the BS.
A. SE Analysis of Channel Estimation
The performance of channel estimation can be analyzed bythe SE equations, and we first focus on a general SE trajectoryof GTurbo-MMV in Proposition 2.
Proposition 2: The SE of the GTurbo-MMV algorithm is
7
characterized by
ϑt =[f1
(vt1,1), f2
(vt2,2), · · · , fMR
(vtMR,MR
)],
Ωt+1 = diag
(1
(αϑt1)−1 − vt1,1
, · · · , 1
(αϑtMR)−1 − vtMR,MR
),
Vt+1 =
((1
N
∑N
n=1msen(Ωt+1, µn)
)−1
−Ωt+1
)−1
,
(21)
where the initial values are v01,1 = v0
2,2 = · · · v0MR,MR
=
E[|hi,i|2
], and
fm(vt) =1
vt+σ2 , ∆m = 0,∑y∈RB
∫ [Ψ′(y;√
(vh−vt)/2ξ,(σ2+vt)/2)]2
Ψ(y;√
(vh−vt)/2ξ,(σ2+vt)/2)Dξ, ∆m > 0,
(22)
with Dξ = 1√2πe−
ξ2
2 dξ. When ∆m > 0, the definition of Ψ(·)is given by (12), and Ψ′(·) of fm(vt) is expressed as
Ψ′(y; z, c2) ,∂Ψ(y; z, c2)
∂z= −e
− (yup−z)2
2c2 − e−(ylow−z)2
2c2
√2πc2
.
(23)The iteration index and measurement ratio are defined as t andα respectively, and msen(·, ·) is the mse of device n.
Proof: The SE of the GTurbo-SMV algorithm is character-ized by [32],
ϑt = fm(vt),
ηt+1 =1
(αϑt)−1 − vt,
vt+1 =
(1
mse(ηt+1)− ηt+1
)−1
, (24)
with the initialization v0 = E[|hi,i|2
].
By inspection of step 1) and step 4) in Algorithm 2, thefirst two equations in (24) can be extended to the multiplemeasurement case by calculating them in parallel based onthe antennas’ different quantization resolutions. That is,
ϑt = [ϑt1, · · · , ϑtMR]
=[f1
(vt1,1), f2
(vt2,2), · · · , fMR
(vtMR,MR
)],
Ωt+1 = diag
(1
(αϑt1)−1 − vt1,1
, · · · , 1
(αϑtMR)−1 − vtMR,MR
),
(25)
and msen(Ωt+1, µn) in the last equation of (24) for each usern is shown in (26) at the top of the next page, with (Ωt+1)−1
being a real diagonal matrix. Therefore, the SE of GTurbo-MMV are extensions of (24) by replacing scalar values withvectors and matrices in (25) and (26).
However, the integral part of (26) is computational chal-lenging since the diagonal entries of Ωt+1 are not identical.From [24–26], we know that the MMSE of mixed-ADCarchitecture can be modeled as linear combinations of MMSEsfrom different estimators, where the antennas with distinct
quantization accuracy contribute relative portions to the overallperformance. This observation motivates us to use the samequantization model within (22) to derive bounds for our stateevolution analysis, which results in Proposition 3 as follows.
Proposition 3: Define flower (vt) = f∆minm
(vt),fupper (vt) = f∆max
m(vt). The SE of the GTurbo-MMV
algorithm in channel estimation is bounded by
ϑt = fbound(vt),
ηt+1 =1
(αϑt)−1 − vt,
vt+1 =
(1
1N
∑Nn=1 msen(ηt+1, µn)
− ηt+1
)−1
, (27)
where msen(ηt+1, µn) in (27) is given by
msen(ηt+1, µn) = paµn
−
(paη
t+1
1/µn+ηt+1
)2
MR
∫ rHrg(r, µn + 1
ηt+1
)pa + (1− pa)
g(r, 1
ηt+1
)g(r,µn+ 1
ηt+1
)dr,
(28)
with g (a, b) = CN (a; 0, bI). The lower bound of channelestimation is calculated by setting fbound (vt) = flower (vt),with upper bound fbound (vt) = fupper (vt).
Proof: Please refer to Appendix B.Due to the bit-resolution B in ϑt, msen(ηt+1, µn) can be
considered as msen(B, µn). This relationship exemplifies thefundamental tradeoff between hardware cost/implementationcomplexity and system performance. In other words, the bigpicture is that wireless system designers should not sim-ply strike to maximize the system throughput, but optimizehardware cost scaling to achieve specific throughput targets.Observing that the asymptotic behavior of channel estimationin Algorithm 2 can be characterized by MMV equations of SE,thus a large class of performance metrics (i.e., MSE or BER)can be computed easily. Moreover, the analytical frameworkcan be applied to arbitrary postulated measures by changingthe prior distribution of channel response.
B. Asymptotic Performance of Active Device Detection
In this subsection, we first bound the probability of theGTurbo-MMV-based active device detection in terms of themissing and false detection in Theorem 1. Then, further letMR → ∞ for an insightful study in the massive MIMOregime.
Theorem 1: In the large system regime, within each it-eration t in GTurbo-MMV, the probabilities of missingPtMissing,n and false PtFalse,n detection of user n are boundedseparately,
R1,n
(MR, r
t1,n(κtlower)
)≤ PtMissing,n
≤ R1,n
(MR, r
t1,n(κtupper)
),
R2,n
(MR, r
t2,n(κtlower)
)≤ PtFalse,n
≤ R2,n
(MR, r
t2,n(κtupper)
), (29)
8
msen(Ωt+1, µn) = paµnI− pa2
((1
µnI + Ωt+1
)−1
Ωt+1
)[∫
rrHCN(r; 0, µnI + (Ωt+1)−1
)pa + (1− pa)CN (r; 0, (Ωt+1)−1) /CN (r; 0, µnI + (Ωt+1)−1)
dr
](Ωt+1
(1
µnI + Ωt+1
)−1). (26)
where PtMissing,n , P(βtn 6= 1|βn = 1), PtFalse,n , P(βtn =
1|βn 6= 1); κtlower = (αfupper(vt−1
))−1 − vt−1, κtupper =
(αflower(vt−1
))−1 − vt−1 with fupper, flower, vt−1 defined
in Proposition 3;
R1,n
(MR, r
t1,n(κt)
)= 1−
Γ(MR, r
t1,n(κt)MR
)Γ (MR)
,
R2,n
(MR, r
t2,n(κt)
)=
Γ(MR, r
t2,n(κt)MR
)Γ (MR)
, (30)
with the upper incomplete gamma function Γ(·, ·), gammafunction Γ(·); and
rt1,n(κt)
=κt
µn
(1
MRln
1− papa
+ lnµn + κt
κt
),
rt2,n(κt)
= (1 + κt/µn)
(1
MRln
1− papa
+ lnµn + κt
κt
).
(31)
Proof: Please refer to Appendix C.Based on Theorem 1, we can further deduce an asymptotic
user detection performance theorem when MR grows large.Corollary 1: When the number of BS antennas MR →∞,
we have
limMR→∞
PtMissing,n = limMR→∞
PtFalse,n = 0. (32)
Proof: Please refer to Appendix D.It is noted that Theorem 1 demonstrates that the device
detection error probability approaches zero as MR → ∞ forarbitrary t. Interestingly, we found that Theorem 1 would holdtrue even if the SNR is relative small. That is to say, byleveraging the properties of massive MIMO, our algorithmcan exactly detect all active users in massive connectivitywith probability 1. This advantage mainly comes from thediversity and processing gains provided by massive antennasat the BS. Another remarkable property of Theorem 1 is thatthe accurate device detection is guaranteed as long as MR islarge, regardless of the values of N,Na, L.
C. Computational Complexity Analysis
The vector-wise manner at each iteration of GTurbo-MMValgorithm significantly reduces the computational complexi-ty. Besides, the DFT operation outputs do not need to bestored, which is hardware-cost friendly to MMV in massiveconnectivity. The computational complexity of the GTurbo-MMV algorithm mainly comes from the calculation of theDFT and IDFT of size N , which is O(MRN logN) ateach iteration. Besides, vector-wise operations of module Aare O(MRL) and vector multiplications of module B areO(MR
2N) in Algorithm 2 at each iteration. As a result,
the overall computational complexity of the GTurbo-MMValgorithm is O [TMR (a4N log(N) + b4L+ c4MRN)] withconstants a4, b4, c4 and T iterations (see Table I). It canbe shown in the following analysis that the convergent MSEperformance is normally achieved within few iterations (7-10).Since the first term contributes more to the overall compu-tational complexity, we can conclude that the computationalcomplexity of GTurbo-MMV is lower than that of AMP-MMVdue to log(N) < L.
V. NUMERICAL RESULTS
In this section, we will evaluate the performance of ourproposed GTurbo-MMV in channel estimation and user activ-ity detection. We use missing, false and successful probabilityof user detection (MPUD, FPUD and SPUD) to measure theaccuracy in user activity detection. MPUD or FPUD is definedto be the mean ratio of missing or false detected active usersto the number of active or sleeping users, and SPUD is definedto be the possibility of exact recovery of active users’ indices.Meanwhile, the performance of channel estimation is evaluatedby the estimation MSE, defined as,
MSE∆= ||H−H||2F/NMR. (33)
The SNR of this system is defined as SNR = 1/σ2. Thepossibility pa that each N = 2000 online users transmit datawithin a frame is set to be 0.05, and L = 200 normalizedpartial DFT matrix is used as pilot sequence for each onlineuser. Let D = [d1, ..., dN ], and dn denotes the distancebetween device n and the BS. It is assumed that D arerandomly distributed in the regime [A1, A2]km, and thepath loss model of the wireless channel for device n canbe expressed as µn = −128.1 − 36.7log10(dn) in dB withA1 = 0.05, A2 = 0.1. We assume that an ideal power controlis performed to obtain the same large-scale fading factors fromFig. 5 to Fig. 11, while Fig. 12 demonstrates the performancewith different large-scale components.
We first consider the iterative performance of our proposedGTurbo-MMV by replacing some low-resolution ADCs withhigh-resolution ADCs in Fig. 5. Here, M∞R represents thenumber of high-resolution ADCs, and MR − M∞R = MB
R
is the number of B-bit ADCs. We set MR = 16, and SNR= 20dB. As we can see in Fig. 5, the channel estimationand active device detection can converge in no more than7 iterations for linear measurements and different mixed-ADC architectures. This means that our proposed algorithmconverges fast under linear and nonlinear scenarios. As forthe nonlinear measurements, both mixed 2-bit architecture andmixed 3-bit architecture are considered in Fig. 5. It shouldbe noted that the performance loss in typical massive MIMO
9
TABLE I. Computational Complexity AnalysisMethod SMV MMVAMP O [T (a1LN + b1L+ c1N)] O [TMR (a2LN + b2L+ c2MRN)]
GTurbo O [T (a3N log(N) + b3L+ c3N)] O [TMR (a4N log(N) + b4L+ c4MRN)]
1 2 3 4 5 6 7 8 9 10
Iterations
-21.5
-21
-20.5
-20
-19.5
-19
-18.5
-18
-17.5
-17
-16.5
MS
E/d
B
MR
=6
MR
=8
MR
=10
MR
=6
MR
=8
MR
=10
Linear GTurbo-MMVMixed 2-bit architecture
Mixed 3-bit architecture
(a) Channel Estimation
1 2 3 4 5 6 7 8 9 10
Iterations
10-4
10-3
10-2
10-1
100
Succ
ess
ful a
nd M
issi
ng P
robabili
ty o
f A
ctiv
e U
ser
Dete
ctio
n
MR
=6,Mixed 2-bit
MR
=8,Mixed 2-bit
MR
=10,Mixed 2-bit
MR
=6,Mixed 3-bit
MR
=8,Mixed 3-bit
MR
=10,Mixed 3-bit
Linear GTurbo-MMV
SPUD
MPUD
(b) Active User Detection
Fig. 5. Iterative performance of the proposed algorithm under d-ifferent measurement architecture with pa = 0.05, online usersN = 2000, SNR = 20dB. The length of pilot sequence is L = 200.
systems is mainly caused by low-resolution ADCs, whichcan be compensated by employing high-resolution ADCs atthe BS to form a mixed-ADC architecture. It can be seenthat the performance of channel estimation and active userdetection is relatively poor in Fig. 5 (a) and Fig. 5 (b) when6 high-resolution ADCs are installed under the mixed 2-bitarchitecture, and if we further increase M∞R to 10, then theMSE and the MPUD under the mixed 2-bit architecture willbe improved and will approach that of linear measurement.Moreover, when it comes to the mixed 3-bit architecture, theperformance of both MSEs and MPUDs are much better thanthat of the mixed 2-bit architecture in the same scenario. Wealso observe that MPUDs in Fig. 5 (b) improve significantlyif 8 or more high-resolution ADCs are deployed at the BS.For example, the SPUD is about 0.90, and the MPUD isapproximately 0.0006 under the mixed 3-bit architecture whenM∞R = 10, let alone the scenario of M∞R > 10 or linearmeasurements. There is a point that should be mentioned herethat the performances of channel estimation and active userdetection are almost identical to that of linear measurementsif the number of high-resolution ADCs increases to 12 or evenlarger with a mixed 3-bit architecture.
10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15
SNR/dB
-18
-17.5
-17
-16.5
-16
-15.5
-15
MS
E/d
B
MR
=24,Mixed 2-bit
MR
=32,Mixed 2-bit
MR
=40,Mixed 2-bit
MR
=24,Mixed 3-bit
MR
=32,Mixed 3-bit
MR
=40,Mixed 3-bit
MR
=64,Linear GTurbo-MMV
11 11.2 11.4 11.6-16.2
-16
-15.8
-15.6
(a) MR=64
14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19
SNR/dB
-20.5
-20
-19.5
-19
-18.5
-18
-17.5
-17
MS
E/d
B
MR
=12,Mixed 2-bit
MR
=16,Mixed 2-bit
MR
=20,Mixed 2-bit
MR
=12,Mixed 3-bit
MR
=16,Mixed 3-bit
MR
=20,Mixed 3-bit
MR
=32,Linear GTurbo-MMV
(b) MR=32
17 18 19 20 21 22
SNR/dB
-23
-22.5
-22
-21.5
-21
-20.5
-20
-19.5
-19
-18.5
-18
MS
E/d
B
MR
=6,Mixed 2-bit
MR
=8,Mixed 2-bit
MR
=10,Mixed 2-bit
MR
=6,Mixed 3-bit
MR
=8,Mixed 3-bit
MR
=10,Mixed 3-bit
MR
=16,Linear GTurbo-MMV
(c) MR=16
Fig. 6. MSEs over different measurement architectures and BSantennas against SNRs when pa of each N = 2000 online usersis 0.05 and the pilot length L = 200.
Fig. 6 demonstrates the performance of the GTurbo-MMValgorithm in terms of channel estimation with different numberof BS antennas under mixed 2-bit and mixed 3-bit architec-tures. As expected, GTurbo-MMV has its performance gainsin terms of channel estimation with the growth of SNR,and MSEs are generally improved by increasing M∞R or theresolution of ADCs. Moreover, when SNR=11dB, the MSE ofmixed 3-bit architecture with M∞R = 40 only exhibits 0.05dBloss when compared with linear measurement in Fig. 6 (a).Particularly, the gaps of MSE with mixed 3-bit architectureare not negligible when M∞R varies from 24 to 40 in Fig. 6
10
10 12 14 16 18 20 22
SNR/dB
10-4
10-3
10-2
10-1
100
Suc
cess
ful P
roba
bilit
y of
Act
ive
Dev
ice
Det
ectio
n
MR
=6,MR
=12,MR
=24,Mixed 2-bit
MR
=8,MR
=16,MR
=32,Mixed 2-bit
MR
=10,MR
=20,MR
=40,Mixed 2-bit
MR
=6,MR
=12,MR
=24,Mixed 3-bit
MR
=8,MR
=16,MR
=32,Mixed 3-bit
MR
=10,MR
=20,MR
=40,Mixed 3-bit
MR
=16,MR
=32,MR
=64,Linear GTurbo-MMV
MR=32 MR=16MR=64
Fig. 7. SPUDs versus SNRs under different measurement architec-tures with different number of high resolution ADCs when pa of eachN = 2000 online users is 0.05 and the pilot length L = 200.
10 12 14 16 18 20 22
SNR/dB
10-6
10-5
10-4
10-3
10-2
10-1
Mis
sing
Pro
babi
lity
of A
ctiv
e U
ser
Det
ectio
n
MR
=6,MR
=12,MR
=24,Mixed 3-bit
MR
=8,MR
=16,MR
=32,Mixed 3-bit
MR
=10,MR
=20,MR
=40,Mixed 3-bit
MR
=16,MR
=32,MR
=64,Linear GTurbo-MMV
MR
=16,MR
=32,MR
=64,Linear AMP-MMV(partial DFT)
MR
=16,MR
=32,MR
=64,Linear AMP-MMV(i.i.d. Gaussian)
MR=16MR=32MR=64
Fig. 8. Missing detection performance comparisons of GTurbo-MMVand AMP-MMV when pa of each N = 2000 online users is 0.05and the pilot length L = 200.
(a), from 12 to 20 in Fig. 6 (b), and from 6 to 10 in Fig. 6(c), respectively. The point is that GTurbo-MMV can achievenearly linear channel estimation performance with less high-resolution ADCs at the BS under the mixed 3-bit architecture,which is more pronounced when MR = 64.
It can be observed in Fig. 7 that for all values of SNR,when the BS is equipped with MR = 64 antennas, theSPUDs are significantly higher than those in the cases whenthere are MR = 16 and MR = 32 antennas at the BS.Further, the values of SPUD can approach 1 when the SNRis around 14dB with 64 antennas at the BS, or 18dB with32 BS antennas and 22dB with 16 BS antennas for linearand nonlinear measurements, respectively. On the other hand,all SPUDs with mixed 3-bit architecture are better than thatwith mixed 2-bit architecture irrespective of the value of MR.Consequently, it is indicated that the GTurbo-MMV grant-free access technique can estimate CSI and detect deviceactivity with extremely high accuracy by increasing M∞R orthe quantization bits of low-resolution ADCs in massive MTCconnectivity systems.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Online Users
0.88
0.9
0.92
0.94
0.96
0.98
1
Succ
ess
ful P
robabili
ty o
f A
ctiv
e U
ser
Dete
ctio
n MR
=8,L=120,Mixed 2-bit
MR
=10,L=120,Mixed 2-bit
MR
=8,L=120,Mixed 3-bit
Linear GTurbo-MMV
(a)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Online Users
10-6
10-5
10-4
10-3
10-2
Fals
e a
nd M
issi
ng P
robabili
ty o
f A
ctiv
e U
ser
Dete
ctio
n
PM
,MR
=8,Mixed 2-bit
PF
,MR
=8,Mixed 2-bit
PM
,MR
=10,Mixed 2-bit
PF
,MR
=10,Mixed 2-bit
PM
,MR
=8,Mixed 3-bit
PF
,MR
=8,Mixed 3-bit
PM
,Linear GTurbo-MMV
PF
,Linear GTurbo-MMV
(b)
Fig. 9. Successful and error (includes missing and false) probabilityof active user detection performances for different online users underthe nonlinear and linear measurements with active access probabilitypa = 0.01 among N = 2000 online users. The length of pilotsequence is L = 120 and SNR = 20dB.
Fig. 8 investigates the missing detection performance whenthere are different numbers of antennas at the BS for linearmeasurement and mixed 3-bit architecture, respectively. TheAMP-MMV algorithm for linear measurement with partialDFT and i.i.d. Gaussian pilot sequences are also illustratedin Fig. 8 as comparison. Here, the variance of each entryin the i.i.d. Gaussian pilots is normalized to 1/N for a faircomparison. We observe that the MPUDs of linear AMP-MMV with different pilot sequences are poorer than thatof linear GTurbo-MMV when there are MR = 16, 32, 64antennas at the BS. The performance difference between linearAMP-MMV with i.i.d. Gaussian pilots and partial DFT pilotsindicates that the SE of linear AMP-MMV cannot be appliedto partial DFT matrices directly. This is mainly because AMPor AMP-MMV is developed for i.i.d. Gaussian matrices [33].Furthermore, when MR = 64, all MPUDs are much lower thanthose when MR = 16 or MR = 32 in the same regime of SNR,which indicates that the performance of channel estimation andactive device detection can be improved significantly when alarge number of BS antennas is deployed at the BS.
We also consider the performance of GTurbo-MMV algo-rithm in terms of successful detection and error detection withdifferent online users in Fig. 9. The number of BS antennas isMR = 16. The possibility pa is set to be 0.01 and L = 120,but it has to be greater than the number of active devices toensure no performance impact, or else a larger number of BS
11
1 2 3 4 5 6 7 8 9 10
Iterations
-19.5
-19
-18.5
-18
-17.5
-17
-16.5
-16
-15.5
MS
E/d
B
SimSE
MR =0,6,8,10,16
Lower bound
Upper bound
(a) Channel estimation
1 2 3 4 5 6 7 8 9 10
Iterations
10-4
10-3
10-2
10-1
100
Err
or
Pro
babili
ty o
f A
ctiv
e U
ser
Dete
ctio
n
PM
,Sim
PM
,SE
PF
,Sim
PF
,SEMR =0,6,8,10,16
MR =0,6,8,10,16Lower bound
Upper bound
Upper bound
Lower bound
(b) Active user detection
Fig. 10. Performance bound of GTurbo-MMV with different numberof high-resolution ADCs, SNR = 17dB, the number of active users is100 for each online users N = 2000, and the pilot length is L = 200.
antennas are needed. As we can see in Fig. 9 (a) and Fig. 9 (b),both relatively high SPUDs and relatively low MPUDs can beachieved in a wide range of online devices N . Moreover, Fig.9 (a) demonstrates that the SPUD of a mixed 2-bit architecturecan be improved when M∞R is increased from 8 to 10. It is alsoobserved that the SPUD with M∞R = 8 under the mixed 3-bitarchitecture is higher than that of the mixed 2-bit frameworkwith 10 high-resolution ADCs. Fig. 9 (b) depicts that theFPUD can approach zero when the number of online usersis no more than 900 for both linear and nonlinear scenarios.It is indicated in Fig. 9 (b) that the FPUDs and MPUDs withmixed 3-bit architecture are lower than that of the mixed 2-bitarchitecture even though the number of high-resolution ADCsof mixed 3-bit architecture is smaller than that of the mixed2-bit framework, which agrees with the analysis of SPUDs inFig. 9 (a).
Next, we discuss the simulation results and analytical solu-tions in Fig. 10. As for the mixed 3-bit architecture with differ-ent number of high-resolution ADCs, the performance curvesof channel estimation and active device detection are betweenthe corresponding lower bounds (i.e., linear measurementswhen MR = M∞R ) and upper bounds (i.e., 3-bit nonlinearmeasurements when MR = M3
R). It can be inferred thatthe performance gap will decrease with the increase of M∞Rand will eventually be eliminated within the limit of lowerbound. Therefore, the mixed architecture can help maintainthe promised performance while significantly reducing thecomputational complexity.
1 2 3 4 5 6 7 8 9 10
Iterations
-19.5
-19
-18.5
-18
-17.5
-17
-16.5
-16
-15.5
MS
E/d
B
SimSE
SNR=17dB
MR=16
MR=64
SNR=13dB,MR=64
SNR=15dB,MR=32
Fig. 11. SE of GTurbo-MMV when there are different number ofantennas at the BS, online devices are N = 2000, active devices are100, and the pilot length is L = 200.
Fig. 11 illustrates the analytical performance of channelestimation when the number of BS antennas varies from16 to 64. It can be shown in Fig. 10 and Fig. 11 that allMSEs and error probabilities can converge in no more than 6iterations. Simulations are conducted to verify the accuracyof analytical results. In particular, we compare the SE ofMMV in (27) and (28) with those obtained by simulations.Especially, it can also be observed that the numerical resultscan match the SE equations characterized in Proposition 2very well at a relatively low SNR when there are MR = 64antennas at the BS, which means that the simulation resultsof channel estimation obtained from GTurbo-MMV algorithmcan perfectly match those predicted by the equations of SEwhen there are a large number of antennas at the BS.
Finally, Fig. 12 studies the scenario where power controlis not performed in order to reduce the power consumption.We observe that a low missing probability of active devicedetection can still be achieved when MR = 16 under the mixed3-bit architecture. However, the signals from a long-distancedevice will be dominated to a large extent by the other signalswith strong strength if the interval of distance is large, such asA1 = 0.05, A2 = 1. Fortunately, power control can be appliedto make the effective path loss factors identical across differentdevices. Consequently, both scenarios can achieve satisfactoryperformance.
VI. CONCLUSION
In this paper, we presented a new scheme in allocating non-orthogonal pilot sequences to different online users in massive,small-packet sporadic transmission, which can offer potentialsolution to the emerging mMTC applications. To exactlyidentify active users and estimate their channel responses, wethen proposed a GTurbo-MMV algorithm to perform jointchannel estimation and user activity detection for linear andmixed ADC architectures. Our simulation results demonstrateda nearly linear performance of our proposed algorithm inchannel estimation for nonlinear measurements, along withan approximately accurate active user performance even whenthe number of online users was far greater than the pilot
12
20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25
SNR/dB
10-6
10-5
10-4
10-3
10-2
UD
MR
Linear measurement
MR
=10,Mixed 3-bit
MR
=8,Mixed 3-bit
MR
=6,Mixed 3-bit
Fig. 12. The missing detection performance in the scenario of large-scale fading when MR = 16.
length. Analytical results in terms of SE were provided toprecisely describe the asymptotic behavior of the GTurbo-MMV algorithm.
APPENDIX
A. Proof of Proposition 1
The posterior probability P (h |r ) of (14) can be computedfollowing the Bayes’ rule,
P (h |r ) =P (r |h )P(h)∫P(r |h )P(h)dh
, (34)
where P(r |h ) = CN (h; r,V) and P(h) is defined in (15).Then, based on (34), we have the posterior mean E [h |r ]
and its corresponding variance Var [h |r ],
E [h |r ] =
∫hP(r |h )P(h)dh
P(r)
=paP(r)
CN (r; 0, µI + V)
(I +
1
µV
)−1
r,
Var [h |r ] = E[hhH |r
]− E [h |r ]E[h |r ]
H
=paP(r)
CN (r; 0, µI + V)[(ΛV−1r)(ΛV−1r)H + Λ]
− E [h |r ]E[h |r ]H, (35)
where Λ = (µ−1I + (V)−1)−1 and P(r) = (1 −pa)CN (0; r,V) + paCN (r; 0, µI + V).
To further derive the mse of estimator (35), we rewriteVar[h |r ] as
Var[h |r ] =A2
P(r)− A1
P(r)
(A1
P(r)
)H
, (36)
where A1 =∫
hP(r |h )P(h)dh and A2 =∫hhHP(r |h )P(h)dh for the notation convenience. Then,
E [Var[h |r ]] is given by
E [Var[h |r ]] =
∫Var[h |r ]P(r)dr =
∫A2dr−
∫A1A
H1
P(r)dr,
(37)
as for the first part in (37), we have∫A2dr = pa
∫hhHCN
(h; ΛV−1r,Λ
)CN (r; 0, µI + V)dr
= pa(ΛV−1
)(µI + V)
(ΛV−1
)H+ paΛ
= paµI, (38)
while the second part is shown in (39) at the top of thenext page. By substituting (38) and (39) into (37), we haveE [Var[h |r ]] in (40) at the top of the next page. Consequently,(35) and (37) are equations given in Proposition 1.
B. Proof of Proposition 3
We first observe that when each antenna at the BS has thesame quantization model, the fm (vt) in (22) are then identicalfor all m = 1, 2, . . .MR, which results the same diagonalelements of Ωt+1.
Consider Ωt+1 = θt+1I with identical diagonal entries,msen(Ωt+1, µn) in (26) is then simplified to (41) at the top ofthe next page, where msen(Ωt+1, µn) has the same diagonalentries due to symmetry property. And the (i, i)th elementof msen(Ωt+1, µn) can be further simplified by using thematrix’s trace property,
msen(θt+1, µn) =paµn −
(paθ
t+1
1/µn+θt+1
)2
M
×∫
rHrg(r, µn + 1
θt+1
)pa + (1− pa)
g(r, 1
θt+1 )g(r,µn+ 1
θt+1 )
dr, (42)
where g (a, b) = CN (a; 0, bI). Note that the integral can beobtained through spherical coordinates instead of Cartesiancoordinates. This completes the proof for Proposition 3.
C. Proof of Theorem 1
To complete the proof of Theorem 1, we first consider aspecial case in channel (14) when V = vI. Define an indicatorfunction ∇(·) as
∇(h) =
1, h 6= 0,0, h = 0.
(43)
Under this circumstance, the MAP estimator of h (i.e., h) canbe simplified in Proposition 4.
Proposition 4: By specifying V = vI in (14), the MAPinference of ∇(h) can be simplified to a threshold detector,i.e.,
∇(h) =
1, ||h||22 > ε,0, otherwise,
(44)
with M being the dimension of h, ε =v(µ+v)µ
(ln 1−pa
pa+M ln µ+v
v
). Consequently, the error
13
∫A1A
H1
P(r)dr = pa
2
∫CN (r; 0, µI + V)
2(ΛV−1r)(ΛV−1r)
H
P(r)dr
= pa2(ΛV−1
) [∫ rrHCN (r; 0, µI + V)
pa + (1− pa)CN (r; 0,V)/CN (r; 0, µI + V)dr
] (ΛV−1
)H, (39)
E [Var[h |r ]] = paµI− pa2(ΛV−1
) [∫ rrHCN (r; 0, µI + V)
pa + (1− pa)CN (r; 0,V)/CN (r; 0, µI + V)dr
] (ΛV−1
)H, (40)
msen(Ωt+1, µn) = paµnI− (paθt+1)2
(1/µn + θt+1)2
∫rrHCN
(r; 0,
(µn + 1
θt+1
)I)
pa + (1− pa)CN(r; 0, 1
θt+1 I)/CN
(r; 0,
(µn + 1
θt+1
)I)dr. (41)
probabilities are
P(∇(h) = 0|∇(h) = 1)
= 1−Γ(M, vµ
(ln 1−pa
pa+M ln µ+v
v
))Γ(M)
,
P(∇(h) = 1|∇(h) = 0)
=Γ(M, (1 + v/µ)
(ln 1−pa
pa+M ln µ+v
v
))Γ(M)
. (45)
Proof: Threshold ε in (44) can be obtained by calculat-ing ||h||22 in equation P(h|∇(h) = 0)P(∇(h) = 0) =P(h|∇(h) = 1)P(∇(h) = 1).
Since ||h||22 can be modeled as a chi-square distributedrandom variable with 2M degrees of freedom and P(||h||22 ≤x) = 1− Γ(M,x/2)/Γ(M), it consequently follows that,
P(∇(h) = 0|∇(h) = 1) = P
(||h||22 ≤
2ε
µ+ v
)= 1− Γ(M, r1M)
Γ(M),
P(∇(h) = 1|∇(h) = 0) = P
(||h||22 ≥
2ε
µ+ v
)=
Γ(M, r2M)
Γ(M), (46)
where
r1 =ε
M(µ+ v)=v
µ
(1
Mln
1− papa
+ lnµ+ v
v
),
r2 =ε
Mv= (1 + v/µ)
(1
Mln
1− papa
+ lnµ+ v
v
). (47)
Based on Proposition 3, the missing P tMissing,n and falseP tFalse,n probabilities can be bounded separately by evaluating(45) on either upper or lower bounds in (27), which completesthe proof.
D. Proof of Corollary 1Lemma 1: For a ∈ R+,
limMR→∞
Γ(MR, aMR)
Γ(MR)= 1− lim
MR→∞e−aMR
∞∑k=MR
(aMR)k
k!
=
1, a ∈ (0, 1),0, a ∈ (1,∞).
(48)
Proof: Note that e−aMR
∞∑k=MR
(aMR)k
k! is the tail probability,
i.e., P(X ≥MR) of aMR-mean Poisson random variable X .The bounds are derived by using Chernoff-bound technique[38],
P(X ≥MR) ≤ (ae1−a)MR , a ∈ (0, 1),
P(X ≥MR) > P(X > MR) ≥ 1− (ae1−a)MR , a ∈ (1,∞),(49)
where we denote f(a)∆= ae(1−a), and it can be proved that
f(a) ∈ (0, 1] for a ∈ R+ and achieves the maximum onlywhen a = 1. Then, we can obtain
limMR→∞
Γ(MR, aMR)
Γ(MR)= 1− lim
MR→∞P(X ≥MR)
=
1, a ∈ (0, 1),0, a ∈ (1,∞).
(50)
To complete the proof for Corollary 1, we can conclude from(28) that,
paµn > msen(ηt, µn
)>paµn −
(paη
t
1/µn+ηt
)2
MR
×∫
rHrg(r, µn + ηt
)dr = pac
t,
(51)
where ct = µn − pa(µnηt)
2(µn+ηt)
(1+µnηt)2 . Also, based on the
monotonically decreasing property of function fm(vt−1
)in
(22) when ∆m is fixed, κt = (αfm(vt−1
))−1 − vt−1 can be
proven to be monotonically increasing, which yields
(αfm (paµn))−1 − paµn > κt >
(αfm
(pac
t))−1 − pact.
(52)Next, combined with Lemma 1 and κt/µn ln(1 +µn/κ
t) < 1,(1 + κt/µn) ln(1 + µn/κ
t) > 1 when κt > 0, we have
limMR→∞
R1,n
(MR, r
t1,n(κt)
)= limMR→∞
1−Γ(MR, r
t1,n(κt)MR
)Γ (MR)
= 0,
limMR→∞
R2,n
(MR, r
t2,n(κt)
)= limMR→∞
Γ(MR, r
t2,n(κt)MR
)Γ (MR)
= 0. (53)
14
Consequently, (32) can be proved by applying the squeezetheorem on equations (29).
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Ting Liu (S’15) received the B.S. degree in elec-tronic information engineering and M.S. degree insignal and information processing from NanjingUniversity of Information Science and Technology,Nanjing, China, in 2011 and 2014. She is currentlyworking toward the Ph.D. degree in information andcommunications engineering from Southeast Univer-sity, Nanjing, China. Her research interests includemassive MIMO wireless communications, detectionand estimation theory, and compressed sensing.
15
Shi Jin (S’06-M’07-SM’17) received the B.S. de-gree in communications engineering from GuilinUniversity of Electronic Technology, Guilin, China,in 1996, the M.S. degree from Nanjing University ofPosts and Telecommunications, Nanjing, China, in2003, and the Ph.D. degree in information and com-munications engineering from the Southeast Univer-sity, Nanjing, in 2007. From June 2007 to October2009, he was a Research Fellow with the AdastralPark Research Campus, University College London,London, U.K. He is currently with the faculty of the
National Mobile Communications Research Laboratory, Southeast University.His research interests include space time wireless communications, randommatrix theory, and information theory. He serves as an Associate Editor for theIEEE Transactions on Wireless Communications, and IEEE CommunicationsLetters, and IET Communications. Dr. Jin and his co-authors have beenawarded the 2011 IEEE Communications Society Stephen O. Rice Prize PaperAward in the field of communication theory and a 2010 Young Author BestPaper Award by the IEEE Signal Processing Society.
Chao-Kai Wen (S’00–M’04) received the Ph.D.degree from the Institute of Communications En-gineering, National Tsing Hua University, Taiwan,in 2004. He was with Industrial Technology Re-search Institute, Hsinchu, Taiwan and MediaTekInc., Hsinchu, Taiwan, from 2004 to 2009, where hewas engaged in broadband digital transceiver design.Since 2009, he jointed the Institute of Communica-tions Engineering, National Sun Yat-sen University,Kaohsiung, Taiwan, where he is currently Professor.His research interests center around the optimization
in wireless multimedia networks.
Michail Matthaiou (S’05–M’08–SM’13) was bornin Thessaloniki, Greece in 1981. He obtained theDiploma degree (5 years) in Electrical and Com-puter Engineering from the Aristotle University ofThessaloniki, Greece in 2004. He then received theM.Sc. (with distinction) in Communication Systemsand Signal Processing from the University of Bristol,U.K. and Ph.D. degrees from the University ofEdinburgh, U.K. in 2005 and 2008, respectively.From September 2008 through May 2010, he waswith the Institute for Circuit Theory and Signal
Processing, Munich University of Technology (TUM), Germany working as aPostdoctoral Research Associate. He is currently a Reader (equivalent to As-sociate Professor) in Multiple-Antenna Systems at Queen’s University Belfast,U.K. after holding an Assistant Professor position at Chalmers University ofTechnology, Sweden. His research interests span signal processing for wirelesscommunications, massive MIMO, hardware-constrained communications, andperformance analysis of fading channels.
Dr. Matthaiou and his coauthors received the 2017 IEEE CommunicationsSociety Leonard G. Abraham Prize. He is currently awarded the prestigious2018/2019 Royal Academy of Engineering/The Leverhulme Trust SeniorResearch Fellowship and will receive the 2019 EURASIP Early CareerAward. His team was the Grand Winner of the 2019 Mobile World CongressChallenge. He was the recipient of the 2011 IEEE ComSoc Best YoungResearcher Award for the Europe, Middle East and Africa Region and a co-recipient of the 2006 IEEE Communications Chapter Project Prize for thebest M.Sc. dissertation in the area of communications. He has co-authoredpapers that received best paper awards at the 2018 IEEE WCSP and 2014IEEE ICC and was an Exemplary Reviewer for IEEE COMMUNICATIONSLETTERS for 2010. In 2014, he received the Research Fund for InternationalYoung Scientists from the National Natural Science Foundation of China. Inthe past, he was an Associate Editor for the IEEE TRANSACTIONS ON COM-MUNICATIONS, Associate Editor/Senior Editor for IEEE COMMUNICATIONSLETTERS and was the Lead Guest Editor of the special issue on “Large-scalemultiple antenna wireless systems” of the IEEE JOURNAL ON SELECTEDAREAS IN COMMUNICATIONS.
Xiaohu You (F’11) was born in 1962. He receivedthe master’s and Ph.D. degrees in electrical engineer-ing from Southeast University, Nanjing, China, in1985 and 1988, respectively. From 1999 to 2002, hewas the Principal Expert of C3G Project, responsiblefor organizing China’s 3G mobile communicationsresearch and development activities. From 2001 to2006, he was the Principal Expert of the ChinaNational 863 5G Project. Since 1990, he has beenwith the National Mobile Communications ResearchLaboratory, Southeast University, where he is cur-
rently the Director and a Professor. He has contributed over 100 IEEEjournal papers and two books in the areas of adaptive signal processing, andneural networks and their applications to communication systems. His researchinterests include mobile communication systems, and signal processing andits applications. Dr. You was a recipient of the National First Class InventionPrize in 2011. He served as the General Chair of the IEEE WCNC 2013 andthe IEEE VTC 2016. He is a Secretary General of the Future Forum and theVice Chair of the China IMT-2020 Promotion Group and the China NationalMega Project on New Generation Mobile Network.
