Hybrid precoded index modulation in downlink mmWave MU-MIMO systems ´ Alvaro Javier Ortega and Raimundo Sampaio-Neto
Rodrigo Pereira David
PUC-Rio Rio de Janeiro, Brazil Email: (javier.ortega - raimundo)@cetuc.puc-rio.br
Abstract—We explore the problem of downlink spatial modulation in mmWave MU-MIMO systems, where the spatial information is conveyed by the indices of the receiving inphase/quadrature streams. Four hybrid precoder/combiner designs are described: two already proposed in the literature, HBA-MMSE – based on the MMSE precoder for the digital part of the hybrid beamforming (HB) and on the singular value decomposition of the channel for the HB’ analog part –, and HBA∗ -BD – it performs an optimization on the equivalent baseband channel and applies block diagonalization precoder to HB digital part –. The two others, examined here and referred to as HB-ABD and HB-A∗ -MMSE, are modified versions of the above HBs obtained by interchanging their digital part designs. Simulation results indicate: (i) the advantages of using MMSE precoder as HB digital part; (ii) significant bit error rate performance improvement can be obtained with the proposed HB-A∗ -MMSE in a multiuser scenario; (iii) performing the hybrid precoded streaming spatial modulation in downlink mmWave MU-MIMO system is a good strategy for reducing the required transmit energy in the base station; and (iv) a simple detection approach can be used for data estimation when the number of users is not large.
I. I NTRODUCTION Wireless communications systems exploiting millimeter wave (mmWave) frequencies are thought to be a core technology that will enable the deployment of the fifth generation (5G) cellular system. Broadband mmWave systems promise significant increase in the data rates due to the extremely wide bandwidths available in the mmWave spectrum. The adverse channel conditions at mmWave frequencies make the communication a hard challenge, however it can be compensate by means of using a large number of antennas that results in large array gain. The digital beamforming techniques cannot be implemented easily due to their high cost and power consumption, therefore analog and HB techniques have been proposed [1], [2]. The concepts of the spatial modulation (SM) family addressed to mmWave systems has recently attracted substantial research interest. We have principally two members of this family: the generalized SM (GSM) and the generalized precoding SM (GPSM). For both, two data streams are transmitted in parallel: one in the in-phase/quadrature (IQ) domain by using conventional signal constellations, and the other in the spatial domain where the information is transmitted via the activated
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transmitting (receiving) antenna indices in the GSM (GPSM) case [3]–[6] ( [7]). We explore the problem of downlink SM in mmWave MUMIMO systems, where the spatial information is conveyed by the receiving IQ stream indices. The hybrid processing considers channel side information (CSI) knowledge. Four hybrid precoder/combiner designs are described: HB-A-MMSE [8], its authors were the pioneers in defining the equivalente baseband channel concept, their work is based on the classical MMSE and ZF precoders; HB-A∗ -BD [9], it performs an optimization on the equivalent baseband channel and applies block diagonalization (BD) to hybrid precoder’s digital part. The two others, proposed here and referred to as HB-A-BD and HB-A∗ -MMSE, are modified versions of the above HBs obtained by interchanging their digital part designs. Numerical results in terms of BER performance are provided, where five data detectors are considered for the stream estimations. The remainder of this paper is organized as follows: Section II and Section III describe the system model and channel model, respectively; Section IV presents four HB designs within which two are proposed. Section V describes five different sub-optimal detector to provide the symbol vector estimation when SM is used. Section VI and VII are dedicated for the simulations results and conclusions, respectively. The following notation is used throughout the paper: C denotes the field of complex numbers; N denotes the set of natural numbers; Ba denotes the field of base-a numbers, where a is prime number; A is a set; A is a matrix; a is a vector; a is a scalar; Aa,b , Aa,: , A:,b , denote the (a, b)-th entry, a-th row, and b-th column of the matrix A, respectively; 1a,b is the axb all ones matrix; I(N ) is the N xN identity matrix; k . kp is the p-norm; | . | represents the cardinal function; ⊗ is the Kronecker product; (.)T and (.)H denote the transpose and conjugate transpose, respectively; E[.] is the expectation operator; CN (m, σ 2 ) denotes a complex Gaussian random variable with mean m and variance σ 2 ; and the function Ψ(A) returns the entries of the matrix A ∈ Cn×m with magnitude Ai,j equal to 1, i.e., (Ψ(A))i,j = kAi,j k , i = 1, ..., n, j = 1, ....m II. S YSTEM MODEL We consider downlink mmWave MU-MIMO systems using HB in the base station (BS) and each mobile station (MS).
The hybrid beamformer in the BS can be represented by the product between the RF beamformer, FRF ∈ CNt ×NRFt , and the baseband beamformer, FBB ∈ CNRFt ×KNs . There are K users equipped with Nr antennas and NRFr RF chains. Each user receives Ns streams, with Na (1 ≤ Na ≤ Ns ≤ NRFr ) active streams. Therefore, the user k’s symbol vector, sk ∈ CNs ×1 , has Na IQ symbols and Ns −Na zeros. Thus, the users have two information streams, the first one associated with the Na selected IQ symbols and the other related with their positions a.k.a. spatial stream information. The total date rate Ns of the system is RT = K(log2 N + N log a 2 M ), where M is a the constellation size. Note that we do not restrict the number of streams combinations to an integer power of two, because Ns the spatial rate log2 N can be achieved using fractional bits a transmission or bit-padding methods [4]. The user k’s signal vector, sk , is given by
The SNR is defined as follows SN R
= =
E[kFRF FBB sk2 ] σn2 2 KNa ET = 2 Ns σn2 σn
(4)
where ET = KNa2 /Ns represents the total energy at the BS for the transmission. III. C HANNEL MODEL The mmWave channel can be described as follows [11] s Hk
=
Np Nt Nr X r αk,p dNr (h(φrk,p ), v (θk,p )) Np p=1 t dNt (h(φtk,p ), v (θk,p ))H (5)
sk = Φik dk
s×Na with Φik = I(Ns ):,cik ∈ BN 2
(1)
where the components of dk ∈ QNa ×1 are the user k’s IQ symbols which belong to constellation Q and satisfy E[dk dH k ] = I(Na ), Φik is a submatrix of I(Ns ) obtained by the selection of its columns according to the indexes from s×Nc Ns cik , cik = C:,ik , where C ∈ BN , with Nc = N , is 2 a a position pattern matrix whose construction is done using the minimum Hamming distance criterion between adjacent columns, where each column represents a possible spatial symbol. The index ik ∈ {1, ..., Nc } is the spatial symbol label of the user k. The ik selection is giving by the fractional bits transmission algorithm described in [10]. The BS has Nt antennas and sends K(Na + 1) streams simultaneously using NRFt RF chains, where NRFt satisfies KNa ≤ KNs ≤ NRFt ≤ Nt . If NRFt is equal to Nt , the BS performs digital beamformer [11]. The power normalization is satisfied such that k FRF FBB k2F = KNa . Then the received signal by the user k, yk ∈ CNr ×1 , is expressed as yk = Hk FRF FBB s + nk
(2)
where Hk ∈ CNr ×Nt denote the channel matrix from the BS to the user k satisfying E[k Hk k2F ] = Nt Nr ; nk ∈ CN r×1 is the complex Gaussian noise vector with zero-mean and covariance matrix σn2 I(Nr ), i.e., CN (0, σn2 I(Nr )); s ∈ CNS ×1 is the data stream vector which is expressed as concatenation T of each user’s stream vector such that s = sT1 , sT2 , ..., sTK . The receiver uses its NRFr RF chains and analog phase shifters to obtain the processed received signal y ˜k =
H H WBB WRF Hk FRF FBB s k k
+
H H WRF n WBB k k k
(3)
where WRFk ∈ CNr ×NRFr is the RF combining matrix and WBBk ∈ CNRFr ×Ns denotes the baseband combining matrix of the user k. Similarly to the RF precoder, WRFk is impleH mented using phase shifters and therefore (WRFk WRF ) = k l,l −1 Nr [1].
where Np is the number of multi-path components in the channel; αk,p v CN (0, 1) is the complex gain of the p-th multi-path component in the channel for the k-th user, whereas r t φrk,p (θk,p ) and φtk,p (θk,p ) are its azimuth (elevation) angles of arrival and departure, respectively [1]. We consider the use of an uniform planar array (UPA) formed by Nt = Nth Ntv (Nr = Nrh Nrv ) antennas, Nth (Nrh ) antennas in the horizontal side and Ntv (Nrv ) antennas in the vertical side, with the antenna spacing of half wave length at the transmitter (receiver) [2], whose response is given by: dNt (h(φ), v (θ)) = dNth (h(φ)) ⊗ dNtv (v (θ))
(6)
with h(φ) = πcos(φ)sin(θ); v (θ) = πcos(θ); and iT 1 h jψ 1, e , ..., ej(M −1)ψ ∈ CM ×1 dM (ψ) = √ M IV. H YBRID DESIGNING APPROACHES
(7)
A. HB-A-MMSE [8] algorithm The design of the HB in [8] is based on the channel knowledge of each user, the analog combiner for each user is independently designed based on the singular value decomposition (SVD), while the analog precoder is obtained by the conjugate transposition to maximize the effective channel gain. Then, with the resulting effective channel, low dimensional baseband precoders can be efficiently applied, e.g., MMSE or ZF filter. The Algorithm 1 resumes the steps for the designing [8]. B. HB-A∗ -BD [9] algorithm The design problem of HB for massive MU-MIMO mmWave systems in [9] was inspired by [8]. In [9], the authors focused on the design problem of equivalent channel, i.e., the third step in the Algorithm 1, and eliminated the interference by the baseband block diagonalization (BD). The digital algorithm for MU-MIMO systems [12] is summarized in Appendix A (Algorithm 3). For the analog precoder and combiner design, the authors proposed an iterative algorithm
Algorithm 1 HB - [8] algorithm Description of the inputs and outputs Inputs: Hk , k = 1, ..., K Output: FRF , FBB , WRFk , WBBk = I(Ns ) 2: Compute the analog beamforming precoder and combiner of each user WRFk = N1r Ψ Uk:,1:Ns ,where Hk = Uk Σk VkH FRFk = N1t Ψ HH k WRFk FRF = FRF1 FRF2 · · · FRFK 3: Compute the equivalent baseband channel ˜ k = WH Hk FRF H TRFk T ˜ = H ˜ ˜ ˜T T H H ··· H 1 2 K 4: Compute the digital beamforming precoder ˜ H (H ˜H ˜ H + KNs σ 2 I(KNs )) FBB = H n 1:
that seeks to maximize the system capacity which is summarized in the Algorithm 2. Algorithm 2 Analog beamforming design - [9] algorithm Description of the inputs and outputs Inputs: Hk , k = 1, ..., K, Output: FRF , WRFk 2: Definitions T H = HT1 HT2 · · · HTK WRF = blkdiag(WRF1 , ..., WRFK ) ¯ Ak = Hk HH k , and WRFkj is the submatrix of WRFk with the j-th column vector removed ¯ H j Ak W ¯ j Cj = W
•
V. D ETECTION APPROACHES When SM modulation is used the components of the estimated symbol vector of user k, ˆ sk , can not be detected individually and joint Maximum-Likelihood (ML) detection would provide the optimum detection procedure. Since the ML data detection requires full knowledge of the statistics of the inter-user interference, this section presents five sub-optimal ˆ k , ˆik ) requiring different levels approaches to obtain ˆ sk ≡ (d of parameter knowledge (or estimation) as follows: D1 Minimum distance detection (MDD) ˆ k , ˆik = arg min k y ˜ k − Ak Φi d k2 d (8) i ∈ [1, ..., Nc ] d ∈ QNa ×1
1:
RFk
D2
H H where Ak = WBB WRF Hk FHBk , and FHBk is k k the, unknown to the receiver, submatrix of FRF FBB corresponding to the hybrid precoder of user k. Approximate MDD (assumes that Ak ≈ I(Ns )) ˆ k , ˆik = arg min k y ˜ k − Φi d k2 d (9) i ∈ [1, ..., Nc ] d ∈ QNa ×1
RFk
−1 ¯ H ¯ Gj = Ak − Ak W RFkj Cj WRF j Ak k 3: Compute the analog beamforming combiner of each user (0) WRFk = √1N 1Nr ×Ns r while WRFk does not converge for j = 1, ..., Ns Compute Cj and Gj for i = 1, ..., Nr P (l) (l−1) 1 Ψ (WRFk )i,j = √N (G ) (W ) j i,j l,j RF l6 = i k r end end end (l) return WRFk = WRFk 4: For computing the analog beamforming precoder execute the steps 2 and 3 with the next changes H Ak := A = HH WRF WRF H, WRFk := FRF , Nr := Nt , and Ns := KNs
BD where the inputs are users’ effective channels, Hef fk , and outputs are WBBk and FBB . HB-A∗ -MMSE: it uses Algorithm 2 to obtain the effective channel matrix Hef f and applies it the MMSE filter, i.e., the step 4 of the Algorithm 1, with this approach the hybrid combiner complexity decreases in relation with BD because WBBk = INs , this means that just analog beamforming is needed.
D3 Noise whitening operation followed by MDD ˆ k , ˆik = arg min k K−1/2 (˜ d yk − Ak Φi d) k2 k i ∈ [1, ..., Nc ] d ∈ QNa ×1
D4
(10) H H W W where Kk = σn2 WBB W . RF BB k k RF k k Noise whitening operation followed by approximate MDD ˆ k , ˆik = arg min k K−1/2 (˜ d yk − Φi d) k2 (11) k i ∈ [1, ..., Nc ] d ∈ QNa ×1
D5 Noise and interference whitening operation followed by MDD Consider equation (10) but this time with Kk = H H σn2 WBB WRF WRFk WBBk + E[lk lH k ], where lk repk k resents the interuser interference and it is giving by X H H lk = WBB WRF Hk FHBj sj (12) k k j6=k
C. Proposals for the HB designing From the above HB algorithms we propose two different HB designs interchanging its digital beamforming parts from one to another as follows: • HB-A-BD: it uses Algorithm 1 to obtain the analog beamforming part for both the precoder, FRF , and combiner, WRFk . The digital beamforming part is computed using
VI. N UMERICAL RESULTS This section discusses the viability of using SM addressed to switch ON/OFF the streams on downlink MU-MIMO mmWave systems using HB in terms of BER performance. The users’ channels are generated with Np = 10 multi-paths components, the azimuth and elevation departure angles values are given by a random variable with uniform distribution in
the interval of (0, 2π) and (0, π), respectively. The UPAs have square formats for both transmitter √ and receivers, i.e., √ Nth = Ntv = Nt and Nrh = Nrv = Nr . For figures 1, 2 and 3, the BS has Nt = 64 antennas and sends Na = 1 active stream from Ns = 2 streams to K = 1, 4 and 8 users, respectively. The users are equipped with Nr = 4 and NRFr = 2. These figures show a BER performance comparison between HB-A∗ -MMSE and HB-A-MMSE using the five detectors described in the Section V. It can be observed the HB-A∗ -MMSE benefits for multi-user scenarios, K > 1, nevertheless, for the single-user case the HB-A-MMSE is a better option, because this algorithm can address the beam steering beamforming in a better way in the absence of interuser interference. In practice D5 is unrealizable, therefore its performance just represents a benchmark. In addition, D1 and D3 both require the knowledge of Ak and, therefore, the knowledge (or estimation) of FHBk . However, D2 and D4 can be implemented without any estimation procedure which improves the spectral efficiency, and for D2 no matrix inversion is needed. Figures 2 and 3 evidence that the BER performance difference between D2 and D5 is only about 1.5 dB. In Figure 2 the HB-A∗ MMSE performance without spatial modulation, Na = Ns , using D5 is added. Comparing this curve with its SM counterpart, it is observed that the use of hybridly precoded streaming SM gives about 3.5 dB of gain in performance at the cost however of transmitting 75% of the data information per channel use. Figure 4 shows the BER performance comparison between the BD hybrid approaches for the same simulation setting above when D5 is used, which represents the best achieved performance. It can be noted that even using D5 , the performances of the BD hybrid approaches are far away from the MMSE hybrid approaches. For Figure 5, the BS has Nt = 64 and NRFt = 16, it sends Na = 2 active streams from Ns = 4 streams to K = 4 users. The users are equipped with Nr = 16 and NRFr = 4. This figure shows a BER performance comparison between HB-A∗ MMSE and HB-A-MMSE using the five detectors described before. It can be observed that the HB-A∗ -MMSE benefits for multi-user remain, however, when the number of streams, Ns , increases the performance difference between D2 and D5 increase as well. This means that a more sophisticated detector as D3 is recommended. VII. C ONCLUSIONS We explored downlink SM in mmWave MU-MIMO systems, where the spatial information is conveyed by the indices of the receiving IQ streams. The hybrid processing was performed assuming channel side information (CSI) knowledge. Four hybrid precoder/combiner designs were described: HBA-MMSE [8], that is based on the classical MMSE and ZF precoders for the HB digital part and SVD of the channel for HB analog part; HB-A∗ -BD [9], that performs an optimization on the equivalent baseband channel and applies BD to HB digital part. The two others, proposed here, HB-A-BD and HBA∗ -MMSE, are modifed versions of the first two HBs obtained
Fig. 1: Nt = 64, Nr = 4, K = 1, Ns = 2, Na = 1
Fig. 2: Nt = 64, Nr = 4, K = 4, Ns = 2, Na = 1
by interchanging their digital part designs. The simulation results indicate the advantages of using MMSE precoder as HB digital part and that significant BER performance improvement can be obtained through our proposal HB-A∗ -MMSE. Performing the hybrid precoded streaming SM in downlink mmWave MU-MIMO system is a good strategy for reducing the required transmit energy in the BS. Furthermore, a simple detector scheme can be used to provide data estimation when the number of users is not large. R EFERENCES [1] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave mimo systems,” IEEE transactions on wireless communications, vol. 13, no. 3, pp. 1499–1513, 2014. [2] J. Song, J. Choi, and D. J. Love, “Common codebook millimeter wave beam design: Designing beams for both sounding and communication with uniform planar arrays,” IEEE Transactions on Communications, vol. 65, no. 4, pp. 1859–1872, 2017.
Fig. 3: Nt = 64, Nr = 4, K = 8, Ns = 2, Na = 1
[3] L. He, J. Wang, and J. Song, “On generalized spatial modulation aided millimeter wave mimo: Spectral efficiency analysis and hybrid precoder design,” IEEE Transactions on Wireless Communications, vol. 16, no. 11, pp. 7658–7671, 2017. [4] P. Liu, M. Di Renzo, and A. Springer, “Line-of-sight spatial modulation for indoor mmwave communication at 60 ghz,” IEEE Transactions on Wireless Communications, vol. 15, no. 11, pp. 7373–7389, 2016. [5] ——, “Variable-nu generalized spatial modulation for indoor los mmwave communication: Performance optimization and novel switching structure,” IEEE Transactions on Communications, vol. 65, no. 6, pp. 2625–2640, 2017. [6] L. He, J. Wang, and J. Song, “Spectral-efficient analog precoding for generalized spatial modulation aided mmwave mimo,” IEEE Transactions on Vehicular Technology, vol. 66, no. 10, pp. 9598–9602, 2017. [7] N. S. Perovi´c, P. Liu, M. Di Renzo, and A. Springer, “Receive spatial modulation for los mmwave communications based on tx beamforming,” IEEE Communications Letters, vol. 21, no. 4, pp. 921–924, 2017. [8] A. Li and C. Masouros, “Hybrid precoding and combining design for millimeter-wave multi-user mimo based on svd,” in Communications (ICC), 2017 IEEE International Conference on. IEEE, 2017, pp. 1–6. [9] C. Hu, J. Liu, X. Liao, Y. Liu, and J. Wang, “A novel equivalent baseband channel of hybrid beamforming in massive multiuser mimo systems,” IEEE Commun. Lett., vol. PP, no. 99, pp. 1–1, 2017. [10] N. Serafimovski, M. Di Renzo, S. Sinanovic, R. Mesleh, and H. Haas, “Fractional bit encoded spatial modulation (fbe-sm),” IEEE Communications Letters, vol. 14, no. 5, pp. 429–431, 2010. [11] G. Kwon and H. Park, “A joint scheduling and millimeter wave hybrid beamforming system with partial side information,” in Communications (ICC), 2016 IEEE International Conference on. IEEE, 2016, pp. 1–6. [12] K. Zu, R. C. de Lamare, and M. Haardt, “Generalized design of lowcomplexity block diagonalization type precoding algorithms for multiuser mimo systems,” IEEE Transactions on Communications, vol. 61, no. 10, pp. 4232–4242, 2013.
A PPENDIX A MU-MIMO B LOCK D IAGONALIZATION ALGORITHM Algorithm 3 Review of the BD precoder algorithm Description of the inputs and outputs Inputs: Hk , k = 1, ..., K Output: Fk = Fak Fbk and Wk 2: Compute the null space to avoid the multiuser interference ¯ k = HT1 · · · HT HT · · · HT T ∈ C(K−1)Nr ×Nt H K k−1 k+1 h iH ¯k = U ¯ kΣ ¯ kV ¯H = U ¯ kΣ ¯k V ¯ (1) V ¯ (0) H i k k ¯ (0) ∈ CN t×Ns Fak = V k ¯ k Fa = 0 Note that ∀k ∈ (1, ..., K) H k 3: Compute the precoder’s second part to improve the energy signal as follows h i 1:
Fig. 4: Nt = 64, Nr = 4, Ns = 2, Na = 1, D5
(1)
(0)
Hk Fak = Uk Σk VkH = Uk Σk Vk Vk (1)
H
Fbk = Vk Λk (1) where Vk ∈ CNs ×Ns and Λk is the user k’s power loading matrix that depends on the optimization criterion, e.g., waterfilling. 4: The user k’s decoding matrix is obtained as Wk = Uk
Fig. 5: Nt = 64, Nr = 16, K = 4, Ns = 4, Na = 2
