Low-Complexity Multiuser Detection in Massive Spatial Modulation MIMO Shengchu Wang Department of Electrical Engineering and Wireless and Mobile Communications R&D Center Tsinghua University, Beijing, China, 100084 Email:
[email protected] Abstract—In this paper, we research the multiuser detection (MUD) in a new massive Spatial Modulation Multiple Input Multiple Output (SM-MIMO) system, where the Base Station (BS) is equipped with massive antennas, and every User Equipment (UE) has multiple Transmit Antennas (TAs) but only one RadioFrequency (RF) chain. In the uplink, UEs transmit data to the BS over frequency selective channels by the Cyclic-Prefix SingleCarrier (CP-SC) SM. We construct a new Generalized Approximate Message Passing Detector (GAMPD), and analyzes its mean square error performance theoretically by the State Evolution (SE) tool. By exploiting both the prior probability distribution and sparsity of the transmitted signal, GAMPD shows superior detection performances and works well even when the number of TAs at the UEs is larger than the number of BS antennas. GAMPD calls for parallelized matrix-vector multiplication as the most complex operation, so it has low computational complexity and is suitable for hardware implementation. Simulation results show that GAMPD outperforms the linear detectors significantly, and is analyzed by SE successfully. In addition, compared to the classical massive MIMO, massive SM-MIMO shows better detection performance and higher spectral efficiency.
I. I NTRODUCTION Next-generation cellular communication system could equip massive antennas at the BS to serve many User Equipments (UEs) simultaneously [1], [2], [3], [4], [5]. Through harvesting extremely high Multiple-Input-Multiple-Output (MIMO) diversity and multiplexing gains [6], both the transmission rate and reliability can be improved remarkably, and the transmission power can be decreased significantly [1], [7]. In conventional massive MIMO [1], [2], [8], [9], UEs are assumed to have single Transmit Antenna (TA) connecting to single Radio-Frequency (RF) chain. [5], [12] introduced Spatial Modulation (SM) massive MIMO. Every UE still has single RF chain but multiple TAs. Consequently, except the Quadrature Amplitude Modulation (QAM) constellation, a new spatial constellation is introduced to improve uplink transmission rate without sacrificing the advantages of singleRF [5], [12], [13], [14]. Based on our survey, [5] pointed out the possibility of massive SM-MIMO but did not provide the implementation details. [12] researched massive SM-MIMO over flat0 This work was supported by National 973 project (NO. 2013CB329002), National 863 project (NO.2012AA011402), National Major Project (NO.2013ZX03004007-002), Program for New Century Excellent Talents in University (NCET), and Tsinghua-Qualcomm Joint Research Program (NCET-13-0321).
Yunzhou Li, Jing Wang and Ming Zhao Wireless and Mobile Communications R&D Center Tsinghua University, Beijing, China, 100084 Email: {liyunzhou, wangj}@tsinghua.edu.cn fading channels. Unfortunately, based on the physical channel measurements [15], [16], the frequency-selective case should be researched. Cyclic-Prefix Single-Carrier (CP-SC) SM has been proposed for single-use (SU) SM-MIMO over frequencyselective fading channels [17]. In this paper, it is extended to the MU case, and then we construct a new MU SC massive SM-MIMO system over frequency-selective fading channels (from then on, massive SM-MIMO indicates this new system). In the uplink, the key is how the BS fulfils the Multiuser Detection (MUD) to recover the data bits from different UEs. Unfortunately, almost all the existing detectors for SM-MIMO [5], [12], [14], [17], [19] are not suitable for massive SMMIMO. This is because their computational complexities are extremely high due to the curse of dimensionality of massive SM-MIMO. For example, Maximum Likelihood (ML) [14] and Sphere Decoding (SD) [19] have to partially or completely search over all the possible values of the transmitted signal whose number is enlarged disastrously in massive SM-MIMO. Message Passing Detector (MPD) [12] is specially designed for the flat-fading case. If it is applied to the frequencyselective case, MPD becomes very cumbersome because it has to maintain a huge number of messages. The stage-wised linear detectors for SM-MIMO [5], [13], [14] is extended to massive SM-MIMO. Unfortunately, these linear detectors will have to compute a huge matrix inversion for every input SC symbol. In this paper, we construct a new low-complexity GAMP Detector (GAMPD) based on the Generalized Approximate Message Passing (GAMP) algorithm in Compressed Sensing (CS) [18], [21]. We also develop State Evolution (SE) equation to predict the Mean Square Error (MSE) performance of GAMPD. GAMPD calls for the parallelized matrix-vector multiplication as its most complex operation, consequently, it has low complexity and is suitable for hardware implementation. Moreover, it utilizes both the sparsity (firstly pointed out by [20]) and prior probability distribution of the transmitted signal. Therefore, it shows superior detection performances, and can work well even when the total number of TAs at UEs exceeds the number of BS antennas. Simulation results show that GAMPD outperforms the linear detectors significantly, and SM is a potential technique for improving both the MUD performance and spectral efficiency of the massive antenna systems.
The rest of the paper is organized as follows. Section II introduces the system model for massive SM-MIMO. In Section III, we construct the stage-wised linear detectors for massive SM-MIMO. Section IV is the core of this paper, where GAMPD is proposed and its MSE performance and computational complexity are analyzed. Finally, simulation results are presented in Section V and the paper is concluded by Section VI. √ Notation: i = −1 ; [N ] = 0, 1, ..., N − 1 for an integer N ; For a vector a and an integer set Ω, a(Ω) or aΩ is a sub-vector composed by the elements {ak }k∈Ω where ak is the kth element of a. For a matrix A, Ak is the kth column and AΩ is a sub-matrix composed by the columns {Ak }k∈Ω ; m : n indicate the integers increasing from m to n; IN is the identity matrix with dimension N × N ; 0N is the all-zeros ∗ vector with dimension N ; for a matrix or vector, () indicates T H the conjugate operation, (·) means the transpose, and (·) or † (·) is the complex conjugate transpose.
on the principles of SC-SM [17], it can transmit N log2 (M Nt ) binary-bits which are denoted as
II. S YSTEM M ODEL
. √ u suN −1 (nt ) ..su0 (nt )su1 (nt )... suN −1 (nt ) , ps |sN −L (nt )... {z } | {z }
In this section, Part A shows the uplink configuration of the MU SC massive SM-MIMO system over the frequencyselective fading channels. Part B and C introduces the data transmission at the UEs and signal reception at the BS, respectively. A. Uplink Configuration of Massive SM-MIMO At the transmitter sides, there exist U UEs. Every UE possesses Nt Transmit Antennas (TAs) but only one RF chain. At the receiver side, the BS is equipped with Nr antennas which have their own independent RF chains. hunr nt ∈ CL represents the multipath channel response between the nr th BS antenna and the nt th TA of the uth user. L is the number of channel taps. We consider the Rayleigh channel model where hunr nt ∼ CN (0L , 1/L IL ) with u ∈ [U ], nr ∈ [Nr ], and nt ∈ [Nt ]. B. Signal Transmission at UEs In this part, we introduce the transmitting operations at the UEs. In the uplink, UEs transmit data to the BS by the CP-SC SM technique [17]. In this paper, we assume that all UEs adopt the same uplink transmission power (ps ) and the same QAM modulation type (M -QAM ). The M -QAM alphabet is defined as [√ / ] A = A×{±(2kR +1)±i(2kI +1), kR ,kI ∈ M 2 }, (1) where M is the QAM modulation order and A is the power √ normalization factor (e.g., 1/ 10 for 16-QAM). Without loss of generality, we firstly consider the uth user in one SC symbol. One SC symbol contains the CP part and data part. The CP part is a simple repetition of the last L data samples of the data part. So we only focus on how to formulate the data part. The data part contains N data samples. Based
bu = {bu0 , bu1 , ..., buN −1 },
(2)
log2 (M Nt )
where bun ∈ {0, 1} with n ∈ [N ] is a binary subu vector. At the nth data sample, bu n is extracted from b . Then, based on the rules of SM [5], [13], [14], [17], the first log2 (Nt ) bits determines the index of the active TA (denoted as jnu ), and the remaining log2 (M ) bits are mapped as the QAM symbol qnu ∈ A transmitted by the active TA. Finally, bun is mapped as [ ]T 0 ... 0 qnu 0 ... 0 u sn = , (3) ↑ u jn
Nt ×1
which is the signal vector transmitted by the Nt TAs at the nth data sample (note that silent TAs transmit only 0s). After adding the Cyclic Prefix (CP), the SC symbol transmitted from the nt th TA of the uth UE is
CP
(xnut )T
(4) where xunt is the data part. Finally, all other UEs can be discussed similar as the uth UE. C. Signal Receiving at the BS This part introduces the signal reception at the BS. Similar as Part B, at first, only the uth UE is focused on. At the receiver side, after removing the CP, the received signal at the nr th BS antenna is √ ynur = ps Hunr xu , (5) ] [ where Hunr = Hunr 0 Hunr 1 ... Hunr (Nt −1) , and xu = ]T [ T T T (xu0 ) (xu1 ) ... (xu(Nt −1) ) is the signal vector from the Nt TAs of the uth user. Hunr nt = toep(hunr nt , N ) (with nr ∈ [Nr ] and nt ∈ [Nt ]) is a N × N -sized Toeplitz matrix. For a vector c ∈ CL ,toep(c, N ) is defined as c0 0 ... cL−1 ... c1 c1 c0 ... 0 ... c2 c2 c1 c 0 ... c3 0 toep(c, N ) = . (6) ... ... ... ... ... ... ... cL−1 ... c1 c0 0 ... ... cL−1 ... c1 c0 N ×N The Toeplitz structure of (6) comes from the fact that data transmission over the multipath channel is equivalent to a circulant convolution after adding CP at the transmitter and removing CP at the receiver [6]. Based on (3), (4) and (5), xu and sun satisfy xu (Ψn ) = sun , (7) where u ∈ [U ], n ∈ [N ], Ψn = {nt N + n}nt ∈[Nt ] . Ωu = {jnu N + n}n∈[N ] contains the positions of the N non-zeros in xu
Next, we discuss the multiuser case. Now, the received signal at the nr th BS antenna is a noisy combination of {ynur }u∈[U ] . It is ∑ √ ynr = ynur + wnr = ps Hnr x + wnr , (8) u∈[U ] [ 0 ] 1 U −1 H = = where Hnr [ ] nr Hnr ... Hnr , x 2 0 T 1 T U −1 T , and w ∼ CN (0, σ IN ) (x ) (x ) · · · (x ) nr 2 is the Gaussian noise with variance σ . By stacking the equation (8) over nr , the received signal at the BS can be expressed as √ y = ps Hx + w, (9) [ T T ] [ ]T T T where y = y0 y1 ... yN , H = HT0 HT1 ... HTNr −1 r −1 [ ]T T and w = w0T w1T ... wN . (5), (8) and (9) show r −1 u∈[U ] how H is constructed by {hunr nt }nr ∈[Nr ], nt ∈[Nt ] . Ω = {uNt N + Ωu }u∈[U ] contains the positions of the U N nonzero entries of x. Finally, similar as the convention in CS [21], the Signalto-Noise Ratio (SNR) during MUD is defined as SN R = 10 lg(U ps /σ 2 ). Until now, the MUD problem can be formulated as how to estimate the transmitted signal x based on (9) and then recover the bit-streams {bu }u∈[U ] transmitted from the U UEs. This problem is the core topic of the following Section III and IV. III. S TAGE - WISED L INEAR D ETECTOR FOR M ASSIVE SM-MIMO Similar as [5], [13], [14] (where flat-fading cases were discussed), we design stage-wised linear detectors for massive SM-MIMO over frequency-selective fading channels. These detectors contain three stages as follows. i) Support Detection The target at this stage is estimating Ω which contains the positions of the non-zero elements in x. At first, we need to acquire a crude estimation of x, which can be estimated by the Matching Filter (MF) detector as x ˆ= √
1 HH y ps Nr
(10)
or by the Minimum Mean Square Error (MMSE) estimator as 1 σ2 x ˆ = √ (HH H + IU Nt N )−1 HH y. ps ps
ˆjnu = arg max {ˆ xu (Ψn )} (Ψn is defined in (7)), and then nt ∈[Nt ]
ˆ u = {ˆjnu N + n} Ωu is approximated as Ω n∈[N ] . Finally, Ω u ˆ ˆ is estimated as Ω = {uNt N + Ω }u∈[U ] . ii) Signal Reconstruction ˆ (9) is shrunk as Based on Ω, √ y = ps HΩˆ xΩˆ + n. (13) Then, the MMSE estimation of xΩˆ is 1 σ2 x ˆΩˆ = √ (HH IU N )−1 HH ˆ + ˆ HΩ ˆ y, Ω Ω ps ps
(14)
which is the final estimation of xΩ ∈ CU N . iii) Bit Recovery { }u∈[U ] Based on ˆjnu and the QAM hard-decisions of n∈[N ]
x ˆΩˆ , hard decisions for {bu }u∈[U ] are obtained by reverting the mapping process from {bu }u∈[U ] to x shown by (2) ∼ (7). However, the linear detectors have several drawbacks. Firstly, at the stage of support detection, MF (10) has limited ability to remove the MU Interference (MUI) [1], [22]. Although MMSE (11) outperforms MF significantly, it requires to pre-compute N U Nt × U Nt -sized matrix inversions, which could be problematic in hardware implementation when U Nt is large (e.g., 16) [8]. Secondly, at the stage of signal reconstruction, (14) has to directly compute a massive U N × U N -sized matrix inversion, because HΩˆ has no special structure. To make matters worse, this matrix inversion has to be recomputed for every input SC symbol whose support sets Ω keeps changing (so does its estimation ˆ Consequently, (14) is difficult if not impossible to be Ω). implemented on the real-time systems. In summary, the linear detectors are not suitable for massive MIMO. Fortunately, a new low-complexity detector has been constructed in the next section. IV. G ENERALIZED A PPROXIMATE M ESSAGE PASSING D ETECTOR FOR M ASSIVE SM-MIMO In Part A, GAMPD is constructed to solve the MUD problem (9). Then, its MSE performance and computational complexity are analyzed in Part B and C, respectively. A. Generalized Approximate Message Passing Detector
(11)
In (11), direct matrix inversion is cumbersome and unnecessary. Based on the Toeplitz-like structure of H and the principles of MIMO-Orthogonal Frequency Division Multiplexing (MIMO-OFDM), it can be decomposed as N U Nt ×U Nt sized matrix inversions (please refer to [6], [22] for more details). Similar as x in (8), x ˆ is rewritten as [ ]T T T T x ˆ = (ˆ , (12) x0 ) (ˆ x1 ) ... (ˆ xU −1 ) where x ˆu = x ˆ(uN : uN + N − 1) is the estimation of u x . Then, based on [13], [14], jnu in (3) is estimated as
Firstly, the entries of x are treated as complex random −1 variables. Based on the mapping processes from {bu }U u=0 to x in Section II-B, they are independent and identically distributed (i.i.d.) as ∑ 1 1 p(x) = (1 − )δ(x) + δ(x − Ak ), (15) Nt Nt M k∈[M ]
where δ(·) is the Dirac function, and Ak is the k th element of QAM alphabet A (1). (15) is a Bernoulli-Discrete (BD) probability distribution function (pdf). In (15), (1 − 1/Nt ) and 1/Nt are the zero and non-zero probabilities of any element in x. This is because, at every data sample, every
UE randomly activates only one TA among its Nt TAs with equal-probability. The two δ(·)s comes from the fact that inactive TAs transmit zeros and the active TA transmit a QAM symbol chosen from A uniformly. the MMSE estimation of the elements of x are { Secondly, } ∑ x ˆn = xn ∈A xn p(xn |y) n∈[U N N ] , where p(xn |y) is the t posterior marginal pdf of xn . p(xn |y) can be approximately calculated by the Loopy-Belief Propagation (LBP). However, [18], [22], [23], [24] have pointed out that LBP cannot applied directly due to its high-dimensional integrations and large-number of messages. Fortunately, based on the Central Limit Theorem (CLT) and a series of Taylor expansions [18], we have simplified LBP for massive SM-MIMO as a new GAMP Detector (GAMPD). The simplification procedures mainly follow the derivations in the Appendix C-D of [18], so they are omitted here. Finally, GAMPD is listed as follows. 1) Initialization: Set the iteration number t=0. With m ∈ [Nr N ] and n ∈ [U Nt N ] , x ˆn (0) = 0 and τnx (0) = 1/Nt represent the initial mean and variance of xn , respectively. We set {um (−1)}m∈[Nr N ] = 0, and the termination parameter tol is a small positive number. 2) Iteration: Compute: ∑ ps τ p (t) = τnx (t), (16) N n∈[U Nt N ]
For each m ∈ [Nt N ], calculate ∑ √ pˆm (t) = ps Hmn x ˆn (t) − τ p (t)um (t − 1) (17) n∈[U Nt N ]
um (t) = (ym − pˆm (t))/(τ p (t) + σ ˜ 2 ), / where σ ˜ 2 = σ 2 U Nt . Compute: τ r (t) =
1 (τ p (t) + σ ˜ 2 ). Nr ps
For each n ∈ [U Nt N ], calculate ∑ √ rˆn (t) = x ˆn (t) + ps τ r (t)
∗ Hmn um (t)
(18)
(19)
(20)
m∈[Nt N ]
x ˆn (t + 1) = EXP(xn |ˆ rn (t), τ r (t))
(21)
τnx (t + 1) = VAR(xn |ˆ rn (t), τ r (t))
(22)
(21) and (22) are the expectation and variance of xn at the (t+ 1)th iteration which are evaluated with respect to p(xn |ˆ rn ) ∝ CN (xn ; rˆn , τ r )p(xn ). p(xn ) is the BD pdf (15). 3) Termination Judgment: If the termination condition ||ˆ x(t + 1) − x ˆ(t)||22 < tol (23) ||ˆ x(t + 1)||22 is satisfied (the vector x ˆ(t + 1) contains x ˆn (t + 1) as its nth element, and x ˆ(t) is defined similarly), set G = t + 1 and go to step 4); otherwise, set t = t + 1 and go back to step 2) for
the next iteration. 4) Bit-Recovery: Set G = t (G records the number of iterations before (23) is satisfied); output x ˆ(G) as the final estimation of x; Similar as the second and third stages of the linear detectors, ˆ approximate xΩ we can obtain the estimation of Ω (i.e. Ω), ˆu } as x ˆ(G)Ωˆ , and recover the bit hard decisions of {b u∈[U ] ˆ and x based on Ω ˆ(G)Ωˆ . Next, we present more details on the implementation and derivation of GAMPD. Firstly, by substituting (15) into (21) and (22), we have ∑ x ˆn (t + 1) = Ak pk (24) k∈[M ]
and τnx (t + 1) =
∑ k∈[M ]
|Ak |2 pk − |ˆ xn (t + 1)|2
(25)
In (24) and (25), pk = ϕ(Ak ; rˆn (t), τ r (t))/M Nt P , where ϕ(s; rˆn (t), τ r (t)) = exp(−|s − rˆn (t)|2 /τ r (t) ) is a complex Gaussian ∑ functionand P = (1 − 1/Nt )ϕ(0; rˆn (t), τ r (t)) + 1/M Nt ˆn (t), τ r (t)) is the probability nork∈[M ] ϕ(Ak ; r malization factor. In the derivation of (24) ∼ (25), we have exploited the property that p(xn |ˆ rn ) ∝ CN (xn ; rˆn , τ r )p(xn ) in (21) and (22) is a BD pdf because it is the multiplication result between a Gaussian pdf CN (xn ; rˆn , τ r ) and a BD pdf p(xn ). Secondly, in (16) and (19), we have exploited the approximation |Hmn |2 ≈ 1/N to remove two potential matrix-vector multiplications involving the matrix |H|2 whose (m, n)th element is |Hmn |2 . We have this approximation because the ℓ2 -norm of every row of H is U Nt due to the channel model in Section II-A. This simple trick has been widely used to simplify the AMP-like algorithms in [18]. Finally, GAMPD exploits the signal prior BD pdf (15) in (21) and (22). Similar as the popular Bernoulli-Gaussian (BG) pdf in CS [21], [25], BD pdf is also a sparsity-inducing distribution. Consequently, GAMPD have actually exploited both the sparsity and prior pdf of the transmitted signal x. In contrast, the linear detectors fail to exploit these important information of x. This is why they are inferior to GAMPD (see the simulation results in Section V-A). B. State Evolution for GAMPD The Mean Square Error (MSE) between the estimated signal at the tth iteration of GAMPD (ˆ x(t)) and the true signal (x) is defined as eGAMPD (t) =
||ˆ x(t) − x||22 , U Nt N
(26)
where t starts from 0, and x ˆ(0) is initialized as 0U Nt N . Moreover, x is a sparse vector and contains U N unit-power M -QAM symbols. Consequently, eGAMPD (0) ≈ 1/Nt . For the AMP-like algorithms [18], [21], [22], the MSEs at two adjacent iterations can be tracked by a simple onedimensional iteration named as State Evolution (SE). GAMPD
0
10
(27)
where eSE (0) = 1/Nt , VAR(·) is defined in (22), β = 2 Nr /U Nt , σSE = σ 2 /Nr ps and the expectation E(·) is taken over the scalar random variable r = x + w where x ∼ pX (x) 2 (15) and w ∼ CN (0, eSE (t)/β + σSE ). Based on the principles of SE [18], [22], {eSE (t + 1)}t are almost equal to {eGAMPD (t + 1)}t . In other words, the MSE performance of GAMPD is forecasted by SE without resorting to cumbersome Monte-Carlo simulations.
−1
10
−2
10
BER
can also be analyzed by SE, and its SE equation is 1 2 )), eSE (t + 1) = E(VAR(x|r, eSE (t) + σSE β
−3
10
MF U=8 MMSE U=8 MMSE U=16 MMSE U=24 GAMPD U=8 GAMPD U=16 GAMPD U=24
−4
10
−5
C. Computational Complexity of GAMPD In this part, we analyze the computational complexity of GAMPD. Based on Section IV-A, the two matrix-vector multiplications in (17) and (20) determine the complexity of every single iteration in GAMPD. They can be accelerated by exploiting the Toeplitz-like structure of H, so their total complexity is o(N U Nt Nr ). Therefore, the complexity of GAMPD is o(GN U Nt Nr ). Moreover, G is only influenced by α = Nr /U once Nt , M -QAM, and SN R are fixed. In other words, G is almost unchanged if we fix α and scale up Nr and U simultaneously. Therefore, under an arbitrary fixed α, the complexity of GAMPD is o(N U Nt Nr ). For ease of comparison, we also present the computational complexity of the stage-wised linear detectors in Section III. At the stage of support detection, the complexities of MF (10) and MMSE (11) are o(N U Nt Nr ) and o(N U 3 Nt3 ), respectively. At the stage of signal reconstruction, the complexity of MMSE (14) is o(N 3 U 2 Nr ). Consequently, the total complexity of the linear detectors is o(N 3 U 2 Nr ) which is much higher than o(N U Nt Nr ) of GAMPD. V. S IMULATION In Part A, GAMPD is compared to the linear detectors. In Part B, massive SM-MIMO is compared to the classical massive MIMO [1], [2]. For clarity, the common system parameters are summarized as follows. In one SC symbol, the CP length is equal to the number of the channel taps which is L = 40, and the length of the data part is N = 256. The noise variance σ 2 in (9) is fixed as 1. Therefore, based on the definition of SN R in Section IIC, the signal transmission power ps is 10SN R/10 /U . tol, the termination parameter of GAMPD, is fixed as 10−3 . The uncoded Bit Error Rate (BER) is obtained by averaging across all the UEs. A. Comparison between GAMPD and the Linear detectors In this part, we compare GAMPD with the linear detectors. In order to ensure the matrix inversion in (14) can be handled, we set the parameters to be not so large as: Nr =64, Nt =4, and 16-QAM. U is chosen as 8, 16 and 24, which correspond to the three cases that the total number of UE TAs (U Nt ) is smaller than, equal to, and larger than Nr . Figure 1 shows that MF performs poorly even when U is 8 (i.e. U Nt < Nr ). This is because MF has limited
10
−6
10
0
5
10
15 SNR dB
20
25
30
Fig. 1. The un-coded BER performance of GAMPD (red curves), MMSE (blue curves),and MF (black curve) under different SNRs; Nr = 64, Nt = 4, 16-QAM, and U has three possible values 8, 16 and 24 which correspond to the three cases that U Nt ≪ Nr , U Nt ≈ Nr , and U Nt > Nr .
ability to remove MUI [1], [22]. Therefore, from then on, we focus only on GAMPD and MMSE. In general, GAMPD outperforms MMSE significantly. With the increase of U , the BER performance of GAMPD degrades gracefully, but the performance of MMSE deteriorates quickly. The advantages of GAMPD over MMSE are enhanced with the increase of U . For example, at BER 10−2 , the SNR gain is 2dB when U is 8, and is enlarged as 6.5dB when U is 16. When U is 24, MMSE fail to reach BER 10−2 , and is trapped at the “error floor” 10−1.3 once SN R exceeds 20dB. In contrast, GAMPD still works very well, and its BER approaches to 0 with the increase of SN R. The reason for this is that, when U = 24, U Nt > Nr and the linear equation (9) becomes to be underdetermined. Consequently, (11) in MMSE is not a reliable estimation of x for support detection. In contrast, GAMPD cleverly handles this underdetermined problem by exploiting the sparsity of x (see Section IV). This exactly the main story of CS [21]. B. Massive SM-MIMO v.s. Massive MIMO [1], [2] In classical massive MIMO [1], [2], every UEs has only one TA, i.e., Nt = 1. In order to validate the possible advantages of SM, this part compares massive SM-MIMO to massive MIMO. In our simulation, we fix Nr = 128 and U = 24. Three possible values of (Nt , M -QAM) are (1, 16-QAM), (4, 4-QAM) and (16, 4-QAM), among which the first one corresponds to massive MIMO, and the latter two correspond to massive SM-MIMO. Based on Figure 2, we obtain three important observations. Firstly, although the system throughputs are the same under (1, 16-QAM) and (4, 4-QAM), MPDQD with (4, 4-QAM) outperforms MPDQD with (1, 16-QAM) significantly. For example, the performance gain is 3.7dB at BER 10−2 . This indicates that the detection performance of massive SM-MIMO is better than massive MIMO under the same throughput
0
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−1
10
−2
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−3
BER
10
−4
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−5
10
GAMPD (1,16−QAM) GAMPD (4,4−QAM) GAMPD (16,4−QAM)
−6
10
−7
10
0
2
4
6
8 SNR dB
10
12
14
16
Fig. 2. Comparison between massive SM-MIMO and conventional massive MIMO [1], [2]; Nr = 128, U = 24, and the three possible values of (Nt , M -QAM) are shown in the legend.
constraint. Second, the spectral efficiency of massive SMMIMO with (16, 4-QAM) is higher than massive MIMO with (1, 16-QAM). At the same time, GAMPD with (16,4-QAM) outperforms GAMPD with (1,16-QAM) when SN R is larger than 9.5dB. Low SN R (e.g., less than 9.5dB) make SMMIMO to be inferior to MIMO. Fortunately, we are not interested in these operating points because their corresponding BERs are too high for reliable communication. For example, once SN R is lower than 9.5dB, the BERs are higher than 10−1.7 . Consequently, the above analyses indicate that SM can improved the spectral efficiency without sacrificing the system performances. In summary, SM is a potential technique for improving both the detection performances and spectral efficiency of the massive antenna systems. VI.
CONCLUSION
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