iterative block space-time receiver structure for millimeter wave. (mmW) systems, to ... space equalizer for hybrid mmW massive MIMO systems. At the transmitter ...
2016 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)
Non Linear Space-Time Equalizer for Single-User Hybrid mmWave Massive MIMO Systems Roberto Magueta1, Daniel Castanheira1, Adão Silva1, Rui Dinis2, and Atílio Gameiro1. 1 DETI, Instituto de Telecomunicações,University of Aveiro, Portugal 2 Instituto de Telecomunicações,Faculdade de Ciências e Tecnologia, Univ. Nova de Lisboa, Portugal Abstract– The aim of this manuscript is to propose a new hybrid iterative block space-time receiver structure for millimeter wave (mmW) systems, to efficiently separate the spatial streams. We consider that both the transmitter and the received are equipped with a large antenna array and the number of radio frequency (RF) chains is lower that the number of antennas. The analog and digital parts of the equalizer are jointly optimized using as a metric the mean square error (MSE) between the transmitted data vector and its estimate after the digital equalizer. The specificities of the analog domain impose several constraints in the joint optimization. To efficiently deal with the constraints the analog part is selected from a dictionary based on the array response vectors. Our performance results have shown that the performance of the hybrid iterative space equalizer is close to the fully digital counterpart after only a few iterations. Moreover, it clearly outperforms the linear receivers recently proposed for hybrid mmW massive MIMO architectures. Index Terms—massive MIMO, mmWave communications, iterative block equalization, hybrid analog/digital architectures.
I. INTRODUCTION The global bandwidth shortage facing wireless carriers has motivated the exploration of the underutilized millimeter wave (mmW) frequency spectrum for future broadband cellular communication networks [1]. Therefore, several tens of GHz could become available for future wireless systems, offering multi Gb/s. Together with the access to more bandwidth, the deployment of a large number of antennas has been also considered an enabling technology for meeting the ever increasing demand of higher data rates in future wireless networks [2]. The use of massive MIMO (mMIMO) with mmW is very attractive, since the terminals can be equipped with large number of antennas [3]. MmW massive MIMO may exploit new and efficient spatial processing techniques such as beamforming/precoding and spatial multiplexing at the transmitter and/or receiver sides [4]. The system design at these techniques should follow different approaches than the ones adopted for lower frequencies counterparts, mainly due to the hardware limitations [5]. In fact, the high cost and power consumption of some mmW mixed-signal components, make it difficult to have a fully dedicated radio frequency (RF) chain for each antenna [6] as in conventional MIMO systems [7]. To overcome these limitations, hybrid analog/digital architectures, where some signal processing is This work was supported by the Portuguese Fundação para a Ciência e Tecnologia (FCT) PURE-5GNET (UID/EEA/50008/2013) project.
978-1-4673-8818-4/16/$31.00 ©2016 IEEE
done at the digital level and some left to the analog domain, have been discussed in [8]. Recently, some beamforming and/or combining/equalization schemes have been proposed for hybrid architectures [9]-[13]. A simple precoding scheme, only based on the knowledge of partial channel information at both terminals, in the form of angles of arrival (AoA) and departure (AoD), was proposed in [9]. The authors in [10] designed a joint digital and analog beamforming at the transmitter side, where first a set of fixed analog beamforming coefficients is selected and then a digital eigenmode based precoder is computed. However, a conventional MIMO receiver was assumed there. In [11], a hybrid spatially sparse precoding and combining approach was proposed for mmW massive MIMO systems. The spatial structure of mmW channels was exploited to formulate the single-user multi-stream precoding/combining scheme as a sparse reconstruction problem. A digitally assisted analog beamforming technique for mmW systems was considered in [12], where a digital beamsteering system using coarsely quantized signals assists the analog beamformer. In [13], a turbo-like beamforming was proposed to jointly compute the transmit and receive analog beamforming coefficients, but the digital processing part was not taken into account. Nonlinear equalizers were considered in the last years to efficiently separate the spatial streams in the current MIMO based networks [14]. Iterative block decision feedback equalization (IB-DFE) approach are one of the most promising nonlinear equalization schemes [14]. IB-DFE was originally proposed in [15], and it can be regarded as a low complexity turbo equalizer implemented in the frequency-domain that does not require the channel decoder output in the feedback loop. In the last years, it has been extended to a wide range of scenarios, ranging from diversity scenarios, conventional MIMO, cooperative MIMO systems, among many others (see [14], [16]-[18] and references within). The IB-DFE principles can be used in mmW massive MIMO context to efficiently separate the spatial streams. However, as discussed, mmW massive MIMO brings new major challenges that prevent a direct plug and play of the iterative equalization based solutions developed for conventional fully digital MIMO systems. In this paper, we design an efficient iterative block space equalizer for hybrid mmW massive MIMO systems. At the transmitter side a space-time encoder structure is employed, before the digital and analog precoders, to 1) ensure that the transmit signal and consequently the noise plus interference, at
177
ISSN: 2157-023X
X Bit stream 1
Data Molulation
Bit stream 2
Data Molulation
(…) Bit stream Ns
(…)
Z
S
C
(…)
Time Precoder
Fd
ΠT
FT (…)
RF Chain 2
Fa
(…)
Analog Precoder
Digital Precoder
(…)
Interleaver
Ant. 1
RF Chain 1
Data Molulation
Ant. 2 (…) Ant. Nt
RF Chain NtR F Digital Part
Fig 1. Transmitter block diagram.
the receiver side, are Gaussian distributed (which simplifies the receiver optimization), 2) warrant that the signal to interference plus noise ratio (SINR) is independent of each spatial stream and time slot and 3) increase the inherent diversity of the mmW massive MIMO system. The analog and digital parts of the proposed hybrid equalizer are jointly optimized using as a metric the mean square error (MSE) between the transmitted data vector and its estimate after the digital equalizer. The specificities of the analog domain impose several constraints in the joint optimization. To efficiently deal with the constraints the analog part is selected from a dictionary based on the array response vectors. We show that the performance of the hybrid receiver tends to the performance of the digital one as the number of iterations increases. Moreover, for a single iteration our proposed iterative receiver structure reduces to the linear one proposed in [11]. The remainder of the paper is organized as follows: section II describes the hybrid mmW massive MIMO systems model. Section III, starts by briefly describing the iterative space-time receiver structure. Then, the fully digital equalizer is presented and finally the proposed hybrid space-time equalizer is derived in detail. Section IV presents the main performance results and the conclusions will be drawn in section V. Notations: Boldface capital letters denote matrices and boldface lowercase letters denote column vectors. The operations (.)T , (.) H , (.)* and tr (.) represent the transpose, the Hermitian transpose, the conjugate and the trace of a matrix. The operator sign( a ) represents the sign of real number a . For a complex
thermore, we assume a single-user mmW system with Nt transmit antennas and Nr receive antennas, where the transmitter sends Ns data streams to the receiver, per time-slot. The transmitter processing is decomposed into two parts, the digital baseband and the analog circuitry that are modeled mathematiRF
cally by precoder matrices Fa ∈ ℂNt × Nt
RF
and Fd ∈ ℂ Nt
× Ns
, re-
spectively. The digital part has NtRF transmit chains, with
N s ≤ NtRF ≤ Nt . Due to hardware constraints, the analog part is implemented using a matrix of analog phase shifters, which force all elements of matrix Fa to have equal norm (| Fa (i, l ) |2 = Nt−1 ) . The transmitter total power constraint is
|| X ||2F = N sT . The transmit signal is given by X = Fa Fd C ,
(1)
where C = [c1 ,…, cT ] ∈ ℂ Ns ×T denotes a codeword constructed by using a space-time block code (STBC) that can be mathematically described by (2) z t = Sft ,
ct = Π t z t ,
(3) T
where t = 1,…, T denotes the time index, ft ∈ ℂ denotes column t
of a T -point DFT matrix (FT = [f1 ,…, fT ]) , Ns ×Ns
where
Πt ∈ ℂ , t = 1,…, T is a random permutation matrix, known both at the transmitter and receiver sides and st , s , t ∈ {1,…, T } , with S = [ ss ,t ]1≤ s ≤ N s ,1≤t ≤T ∈ ℂ Ns ×T ,
ℜ(c) (ℑ(c)) represents the real part of c (imaginary part of c ).
s ∈ {1,…, N s } denoting a complex data symbol chosen from a
The operator sign(.) is applied element-wise to matrices. Con-
QAM constellation with E[| ss ,t |2 ] = σ s2 , where
sider a vector a and a matrix A , then diag(a) and diag( A ) correspond to a diagonal matrix with diagonal entries equal to vector a and a diagonal matrix with entries equal to the diagonal entries of the matrix A , respectively. A ( j , l ) denotes the
For the sake of simplicity and, without loss of generality, in this work we consider only QPSK constellations. To compute codeword C we need to apply an FFT transform to the rows of the symbol matrix S (see (2)) and then permute each of the resulting T columns with a random permutation Π t , t = 1,…, T (see (3)). The received signal is given by
number
c ,
sign(c) = sign(ℜ(c)) + jsign(ℑ(c)) ,
element at row j and column l of the matrix A . I N is the identity matrix with size N × N .
Y = HX + N ,
where Y = [y1 ,…, yT ] ∈ ℂ
II. SYSTEM CHARACTERIZATION In this section, we present the mmW massive MIMO signal definition, the transmitter and receiver characterization. We consider a hybrid based architecture, as shown in Fig 1. Fur-
trix, X = [x1 ,…, xT ] ∈ ℂ
N = [n1 ,…, nT ] ∈ ℂ
178
Nr ×T
Nr ×T
Nt ×T
∑
Ns s =1
σ s2 = N s .
(4)
denotes the received signal mais the transmitted signal and
a zero mean Gaussian noise with vari-
Y
Ant. 1
Ant. 2
Wa(i ) (…)
Analog Equalizer
Ant. Nr
C
+
RF Chain 1
RF Chain 2
Wd( i )
(…)
Digital Equalizer
(i )
Z
(i )
(i ) Sɶ
ΠTH
+
Σ
(…)
(…)
DeInterleaver
FTH
(…)
+
RF Chain NrRF
(i ) Sɵ
Decoding
Time Equalizer
(…)
Decoding
Decoding _
_ (…)
C B (di )
( i−1)
Z
( i−1) Sɵ
( i−1)
FT
ΠT (…)
Interleaver
(…)
Delay
Delay
Time Precoder
(…) Delay Digital Part
Fig 2. Receiver block diagram. 2 n
ance σ . The channel matrix H ∈ ℂ
N r × Nt
is the sum of the
contribution of N cl clusters, each of which contribute Nray propagation paths which follows the clustered sparse mmW channel model discussed in [11]. It may be expressed as H = A r ΛAtH (5) where Λ is a diagonal matrix, with entries
( j, l )
correspond-
ing to the paths gains of the lth ray in the ith scattering cluster. At = [at (θ1,1t ),…, at (θ Nt cl ,Nray ))] , Ar = [ar (θ1,1r ),…, ar (θ Nr cl ,Nray ))] are the matrix of array response vectors at the transmitter and receiver, whereas θ jr,l and θ tj ,l are the azimuth angles of arrival and departure, respectively. The channel path gains and the angles are generated according to the random distributions discussed in [11]. We consider a block fading channel, i.e. the channel remains constant during a block, with size T , but it varies independently between blocks. At the receiver, we consider a hybrid iterative space-time decoder, as shown in Fig. 2. The received signal is firstly processed through the analog phase shifters, modeled by the RF
matrix Wa ∈ ℂNr
× Nr
, then follows the baseband processing,
ing the signal output of the feedback path from the filtered received signal Wd Wa Y . The analog and digital forward matrices and the digital feedback matrix are designed in the following sections. III. ITERATIVE SPACE-T IME RECEIVER DESIGN In this section, we derive the proposed hybrid iterative block space-time feedback equalizer. We start by designing the fully digital receiver, that can serve as lower bound for the hybrid one and then a detailed formulation of the proposed iterative approach is presented. In this manuscript, we assume a decoupled joint transmitterreceiver optimization problem and focus on the design of the hybrid equalizer. Note that the joint transmitter-receiver design is a very hard task as discussed in [11] due to the coupled nature of the transmitter precoder and receiver equalizer. A. Description of Iterative Receiver A block diagram of the proposed receiver is presented in Fig. 2. At the ith iteration the received signal at the tth time slot, after the de-interleaver, is given by zɶ t(i ) = ΠtH ( Wd(i,t) Wa(,it) y t − B (di,)t Πt zˆ t( i−1) ) , (8)
composed of NrRF processing chains. All elements of the 2
−1 r
matrix Wa must have equal norm (| Wa ( j, l ) | = N ) . Specifically, the baseband processing includes a digital feedback closed-loop comprising a forward and a feedback path. For the forward path the signal first passes through a linear filter
Wd ∈ ℂ
Ns × NrRF
, then follows the decoding of the STBC (demodulation included). In the feedback path, the data recovered in the forward path is first modulated and encoded using the Ns ×Ns
. STBC, then it passes through the feedback matrix B d ∈ ℂ The encoding of the STBC follows (2) and (3), and its decoding obeys ɶ = [Π H cɶ ,…, Π H cɶ ] , Z (6) 1 1 T T
ɶ H. (7) Sɶ = ZF T The feedback and feedforward paths are combined by subtract-
ˆ (i −1) = Sˆ (i −1) F , Z T where Wa(,it) ∈ ℂ
N rRF × N r
(9)
denotes the analog part of the feedforRF
ward matrix, W ∈ ℂ N s × Nr the digital part of the feedforward (i ) d ,t
N ×N matrix, B(di,)t ∈ ℂ s s the feedback matrix, Πt ∈ ℂ Ns ×Ns the
interleaver,
Π tH ∈ ℂ N s × Ns
the
de-interleaver
matrix
and
ˆ Z = [zˆ ,…, zˆ ] ∈ ℂ is the DFT of the detector output ˆ ( i ) = [Π zˆ (i ) ,…, Π zˆ (i ) ] is the hard estimate Sˆ ( i −1) . The matrix C 1 1 T T ( i −1)
( i −1) 1
( i −1) T
Ns ×T
( )
of the transmitted codeword C and Sˆ ( i ) = sign Sɶ ( i )
the hard
decision associated to QPSK data symbols S , at iteration i. From the central limit theorem the entries of vector z t ,
t ∈ {1,…, T } are Gaussian distributed, then as the input-output
179
relationship between variables z t and zˆ t( i ) , t ∈ {1,…, T } is memoryless, follows zˆ t(i ) = Ψ(i ) z t + εˆt(i ) , t ∈{1,…, T } , (10) where Ψ (i ) is a diagonal matrix given by
(
)
Ψ (i ) = diag ψ 1( i ) ,…,ψ s( i ) ,…,ψ N( is) ,
ψ
(i ) s
=
E zˆ t( i ) ( s)z*t ( s) E | z t ( s) |2
(11)
, s ∈{1,…, N s } ,
(12)
and εˆt( i ) is a zero mean error vector uncorrelated with z t ,
(
t ∈ {1,…, T } , with E εˆt( i )εˆt(i ) = I Ns − Ψ ( i ) H
2
)σ
2 s
C. Design of Hybrid Iterative Space-time Receiver In this section, we design the proposed hybrid iterative block space-time receiver. Clearly, the previous optimization problem of (15) does not take into account the analog domain constraints. Let us denote by Wa the set of feasible RF precoders, i.e. the set of Nt × NtRF matrices with constant-magnitude entries, then the reformulated optimization problem for the hybrid iterative equalizer is as follows Π Π ( Wa(i,t) )Πopt , ( Wd( i,)t )opt , (B(di,)t )opt = arg min MSE (t i )
(
)
T
s .t. ∑ diag(( Wd(i,t) )Π ( Wa(,it) )Π H tΠ ) = TI N s
. Let us de-
( Wa(,it) )Π ∈ Wa .
fine the vector Γ (i ) = T −1 ∑ t =1 ( Π tH Wd( i,t) Wa(i,t) HFa Fd Π t ) , that T
corresponds to the equivalent overall channel from signal st to
sɶt(i ) , then
sɶt(i ) = Γ(i )st + eɶ t(i ) ,
(13)
where eɶ t(i ) = sɶt( i ) − Γ( i )st denotes an overall error that includes
Due to the digital nature of the feedback equalizer (B(di,)t )Π and since the new constraint does not impose any restriction on this matrix, the feedback equalizer for the hybrid iterative equalizer is similar to the fully digital iterative equalizer discussed in the previous section, and thus given by
(
Π Π Π = ( Wd( i,t) )opt HtΠ − I Ns (B(di,)t )opt ( Wa(,it) )opt
both the channel noise and the residual ISI. The vector εˆt( i ) is a zero mean error vector uncorrelated with z t , then it can be proven that the average error (εɶt(i ) ) power is given by
MSE
= (W
(i ) t
(i ) Π d ,t
) H − Γ − (B ) Ψ
+ (B (di,)t )Π (I N s − | Ψ ( i −1) |2 )1/ 2
2 F
(
( i −1) 2 F
σ
(14)
2 s
(W
σ s2 + ( Wad(i ),t )
Π 2 F
σ n2 .
) , (B (di,)t )Πopt ) = arg min MSE t( i ) (i ) Π ad ,t
) H tΠ ) = TI N s .
(15)
B d ,t . The solution to the optimization problem (15) is −1
Π ( Wad( i ),t )opt = Ω ( R t( i −1) ) (H tΠ ) H ,
(B )
(
2
, (22) F
(i ) Π ad ,t opt
)
(23)
= (I N s − | Ψ (i −1) |2 )Ω −1 ( Wad(i ),t )Πopt ,
(24)
of the optimum fully digital feedforward matrix and the correlation of the ISI plus channel noise. Due to the non-convex nature of the feasible set Wa , an analytical solution to the problem (20) is difficult to obtain, if not impossible. Nevertheless, we find an approximate solution to RF problem (20) by assuming that the matrix ( Wa(,it) )Π is a N r
RF a N r sparse linear combination of the columns of matrix RF
t =1
In this case, the number of receiver RF chains is equal to the number of receiver antennas, and thus we only have a digital linear feedforward filter referred as Wad ,t and a feedback filter
(i ) Π d ,t opt
1/2
)
r ,u sparse linear combination of vectors a r ,u (θ j ,l ) or equivalently
T
∑ diag((W
(21)
(i ) ɶ (i −1) denote a non-normalized version where ( W ad ,t ) Πopt and R t
(i ) Π ad , t opt
s .t.
.
ɶ (i −1) = H Π (I − | Ψ (i −1) |2 )(H Π ) H + σ 2σ −2 I , R t t Ns t n s Nr
B. Design of Digital Iterative Space-time Receiver Firstly, we design the fully digital iterative space-time receiver based on the IB-DFE principles. The performance of this approach can be regarded as a lower bound for the proposed hybrid iterative block equalizer designed in the next section. The equalizer is designed by minimizing the MSE
( (W
)(
(i ) (i ) ɶ (i −1) MSEt = ( Wd(i,t) )Π ( Wa(i,t) )Π − ( W ad ,t )Πopt R t
== E[|| zɶ t( i ) − z t ||2 ] = E[|| εɶt(i ) ||2 ] Π t
( i −1) H
)( Ψ )
From (14) and (21), the MSE expression simplifies to is equal (up to a constant) to
(i ) t
(i ) Π ad , t
(20)
t =1
(i ) Π ad ,t opt
Π t
= ( W ) H − I Ns
( i −1) H
)(Ψ )
,
( (Wɺɺɺ
(i )
(i ) Π d ,t
= arg min
Π ) A rH − ( W ad ,t )opt
) ( Rɶ
( i −1) 1/2 t
)
T
(17)
s.t.
ɺɺɺ ( i ) )Π A H H Π ) = TI ∑ diag((W d ,t r t Ns
2 F
(25)
t =1
−1
(18)
Rt(i −1) = (HtΠ ) H HtΠ (I Ns − | Ψ(i −1) |2 ) + σ n2σ s−2I Ns .
(19)
)
term representation over the dictionary A r . Therefore, optimization problem can be approximated as follows ɺɺɺ (i ) )Π (W d ,t opt
(16)
−1 T Ω = T ∑ diag ( R t( i −1) ) (HtΠ ) H HtΠ , t =1
(
(i ) Π A r = [ A r ,1 , … , A t ,U ] . We may say that ( Wa ,t ) has a N r
ɺɺɺ (i ) )Π ) H ( W ɺɺɺ (i ) )Π ) = N RF , diag((( W d ,t d ,t t 0
ɺɺɺ (i ) ) Π ) H ( W ɺɺɺ ( i ) ) Π ) = N RF represents the sparsiwhere diag((( W d ,t d ,t t
180
0
ty constraint and enforces that only NrRF columns of matrix ɺɺɺ ( i ) )Π are non-zero. The optimum digital feedforward matrix (W d ,t
ɺɺɺ (i ) )( i ) by removing the zero ( Wd(i,t) )Πopt is obtained from ( W d ,t opt columns and the optimum analogue feedforward matrix ( Wa(,it) )Πopt is obtained from A rH by selecting the rows corre-
ous steps iterate on the updated residue value to obtain the NrRF index set to index the dictionary A r . The proposed iterative hybrid space-time equalizer is identical to the equalizer proposed in [11] when the block length is equal to one (T = 1) and for iteration one (i = 1) . IV. PERFORMANCE RESULTS
ɺɺɺ ( i ) )Π . sponding to the non-zero columns of ( W d ,t From optimality condition (associated Lagrangian equal to (i ) zero) we obtain Wres ,t that is the residue matrix that is given by
(
(i )
(i ) Π ɺɺɺ (i ) Π H Wres , t = ( Wd , t ) A r − ( W ad , t ) opt
) ( Rɶ
( i −1) t
)+U
d
(H tΠ ) H . (26) (i )
ɶ (i −1) , R(i−1) , ( W ad ,t ) Π and From the definition of matrices R t t opt ( Wad(i ),t )Πopt equation (26) simplifies to (i ) ɺɺɺ (i ) Π H ɶ (i−1) − Ω (HΠ ) H , Wres ,t = ( Wd ,t ) A r Rt d t
where Ωd = I Ns − | Ψ
(27)
(i −1) 2
| +Ud denotes a redefined Lagrangian
multipliers matrix, that must be selected so that the constraint of the optimization problem (25) is respected. The matrix U d = diag( µ1 ,…, µ N s ) is a diagonal matrix where
µs , s ∈{1,…, N s } are the Lagrange multipliers. To enforce the sparsity constraint, the best columns of the dictionary A r are selected using an iterative greedy method.
In this section, we access the performance of the proposed hybrid space-time iterative equalizer. We consider a clustered channel model with N cl = 8 clusters, each with Nray = 10 rays, with Laplacian distributed azimuth angles of arrival and departure . The average power of all N cl clusters is the same and the angle spread at both the transmitter and receiver is set to 8 degrees. We assume that the transmitter’s sector angle is 60º wide in the azimuth domain and the receiver antenna array has omnidirectional antenna elements. The antenna element spacing is assumed to be half-wavelength. The channel remains constant during a block, with size T = 32 , and takes independent values between blocks. In the following we present results for the hybrid based precoders (Fa and Fd ) derived in [11]. We present results for two different scenarios. For the scenario 1, the parameters are N r = 32 , N t = 128 , N s = 8 ,
N rRF = NtRF = 8 and for scenario 2 the parameters are N r = 64 ,
columns of the matrix A r (one column per iteration) to form
Nt = 256 , N s = 16 , N rRF = NtRF = 16 , i.e. two times times higher than scenario 1. For these two scenarios, we present results for iteration 1, 2 and 4 of the derived digital and hybrid iterative space-time equalizers, which are referred as digital and hybrid, in the following. The performance metric considered is the BER, which is presented as a function of the Eb / N 0 , with Eb denoting the
the analog feedforward equalizer matrix ( Wa(,it) )Πopt we obtain
average bit energy and N 0 denoting the one-sided noise power
At each iteration the column of A r that is most correlated with (i ) the actual value of the residue Wres ,t is selected. In the first
iteration, (i ) res ,t
W
the
= −( W
residue is set to the trivial value ɶ (i −1) . Then, after identifying a set of ) R t
(i ) Π ad ,t opt
(
)
the optimum digital feedforward equalizer matrix ( Wd(i,t) )Πopt
2 2 spectral density. We consider σ 1 = …, = σ Ns = 1 and then the
using the orthogonality condition. Let ( Wd( i,t) )Π and ( Wa(i,t) )Πopt denote the digital and analog parts of the feedforward matrix, restricted to the selected indiɺɺɺ (i ) )Π = [( W(i ) )Π , 0] , if the selected indices were the ces, i.e. ( W d ,t ' d ,t ' first. It can be proven that the digital feedforward matrix is
( Wd(i,t) )Πopt = Ωd ( ( Wa(,it) )Πopt H tΠ )
H
( i −1) −1 d ,t
(R )
,
(28)
Π ɶ (i −1) Π H Rt (( Wa(,it) )opt ) and to respect the where R (di,−t1) = ( Wa(,it) )opt
constraint of problem (25) Ω d is given by H −1 T Ω d = T ∑ diag ( ( Wa(,it) )Πopt H tΠ ) ( R (di,−t1) ) t =1
(
(i ) Π a ,t opt
×( W ) H
Π t
))
−1
(29)
,
After obtaining the optimum value of the digital feedforward matrix ( Wd( i,t) )Π the residue matrix (26) is updated. The previ-
average Eb / N 0 is identical for all streams s ∈{1,…, N s } . From Figs. 3 and 4 we can see the performance improves as the number of iterations increases as expected. Furthermore, the proposed hybrid equalizer is quite close to the digital counterpart for the 2-4th iterations. From these results, we verify that the gaps from the 1st to the 2nd iteration are much higher than from the 2nd iteration to the 4th.This larger gap is mainly due to the removal of the residual ISI which enables the added benefit of a larger diversity. From the 2nd to the 4nd iteration there is also a benefit from ISI removal, but the gains are smaller since most of the ISI is removed in the 4nd iteration. The main difference is the smaller gaps for scenario 2 between the digital and hybrid equalizers achieved mainly due to the higher diversity and array gain provided by the larger dimension of the antenna arrays, both at the transmitter and receiver sides. At iteration 4, the BER target of 10 −3 is achieved for an Eb / N 0 of -15.4 and -20.0dB, respectively. This corresponds
181
-1
-2
-2
10
-3
10
-4
10
-5
-5
-20
-16
-12 -8 -4 Eb/N0 (dB)
0
4
10 -26
8
Fig 3. Performance of the proposed hybrid equalizer for scenario 1. [6] [7]
V. CONCLUSION
[8]
In this manuscript, we proposed a new hybrid iterative space-time receiver structure for mmW massive MIMO systems. The analog and digital parts of the hybrid iterative equalizer were designed jointly using as a metric the MSE. The analog part was selected from a dictionary, which efficiently models the specific hardware limitation inherent to the analog domain processing in the joint optimization problem. A spacetime encoder was used, before the analog precoders, to ensure transmit Gaussian based signals, which allowed to simplify the receiver optimization and to increase the system diversity. The results have shown that the proposed hybrid iterative space-time receiver is quite efficient to separate the spatial streams, while allowing a performance close to the digital counterpart with a very few number of iterations, showing that the dictionary based approximation made for the analog part of the feedforward matrix is quite accurate. Therefore, we can clearly argue that the proposed receiver structure is quite interesting for practical mmW massive MIMO based systems, where the number of RF chains must be lower than the number of transmit and receive antennas.
[9] [10]
[11] [12] [13] [14]
[15] [16] [17]
REFERENCES
[2]
[3] [4] [5]
-22
-18
-14 -10 -6 Eb/N0 (dB)
-2
2
6
Fig 4. Performance of the proposed hybrid equalizer for scenario 2.
to a gain of around 4.6dB from scaling the parameters by a factor of 2.
[1]
-3
10
-4
10
10 -24
Digital Hybrid Iteration 1 Iteration 2 Iteration 4
10
BER
10 BER
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Digital Hybrid Iteration 1 Iteration 2 Iteration 4
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S. Rangan, T. S. Rappaport, and E. Erkip, "Millimeter-wave cellular wireless networks: potentials and challenges", Proceedings of the IEEE, vol. 102, no. 3, pp. 366-385, March 2014. F. Rusek, D. Persson, B. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, "Scaling up MIMO: opportunities and challenges with very large arrays,'' IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40-60, Jan. 2013. A. Swindlehurts, E. Ayanoglu, P. Heydari, F. Capolino, “Millimeterwave massive MIMO: The next wireless revolution?,” IEEE Commun. Mag., vol. 52, no. 9, pp. 52–62, Sep. 2014. W. Roh et. al., "Millimeter-wave beamforming as an enabling technology for 5G Cellular communications: theoretical feasibility and prototype results," IEEE Commun. Mag., vol. 52, no. 2, 106-113, 2014. T. Rappaport et. al., Millimeter wave wireless communications, Prentice Hall, 2014.
[18]
182
T. S. Rappaport, J. N. Murdock, and F. Gutierrez, “State of art in 60 GHz integrated circuits and systems for wireless communications,” Proceedings of the IEEE, vol. 99, no. 8, pp. 1390-1436, 2011. M. Vu, and A. Paulraj,” MIMO wireless linear precoding,” IEEE Signal Processing Mag., vol. 24, no. 5, pp. 86-105, 2007. S. Han, Chih-Li I, Z. Xu, andC. Rowell, “Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G,”, IEEE Commun. Mag., vol. 53, no. 1, pp. 186–194, 2015. A. Alkhateeb, O. El Ayach, G. Leus, and R. Heath, Jr, “Hybrid precoding for millimeter wave cellular systems with partial channel knowledge,”, in Proc. Information Theory and Applications Workshop (ITA), 2013. T. Obara, S. Suyama, J. Shen, and Y. Okumura, “Joint fixed beamforming and eigenmode precoding for super high bit rate massive MIMO systems using higher frequency bands”, in Proc. IEEE PIMRC, 2014. O. Ayach, S. Rajagopal, S. Surra, Z. Piand, R. Heath, Jr., “Spatially Sparse Precoding in millimeter wave MIMO systems”, IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1499–1513, Mar. 2014. A. Kokkeler, and G. Smit, “Digitally assisted analog beamforming for millimeter-wave communications,” in Proc. ICC’15-Workshop on 5G & Beyond – Enabling Technologies and Applications, 2015. X. Gao, L. Dai, C. Yuen, and Z. Wang, “Turbo-like beamforming based on Tabu search algorithm for millimeter-wave massive MIMO systems”, IEEE Trans. Veh. Technol., Jul. 2015 (online). N. Benvenuto, R. Dinis, D. Falconer, and S. Tomasin, “Single carrier modulation with non linear frequency domain equalization: An idea whose time has come - Again,” Proceedings of the IEEE, vol. 98, no. 1, pp. 69-96, Jan. 2010. N. Benvenuto, and S. Tomasin, “Block iterative DFE for single carrier modulation,” Electron. Lett., vol. 39, issue 19, pp.1144-1145, Sep. 2002. R. Kalbasi, D. Falconer, A. Banihashemi and R. Dinis, "A comparison of frequency domain block MIMO transmission systems," IEEE Trans. Veh. Technol., vol. 58, no. 1, pp. 165-175, Jan. 2009. D. Castanheira, A. Silva, R. Dinis, and A. Gameiro, “Efficient transmitter and receiver designs for SC-FDMA based heterogeneous networks,” IEEE Trans. Commun., vol. 63, no. 7, pp. 2500 - 2510, 2015. P. Silva, R. Dinis, Frequency-Domain Multiuser Detection for CDMA Systems, River Publishers – Series in Communications, 2012
