Iterative Frequency-Domain Detection for IA-Precoded ... - IEEE Xplore

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 4, APRIL 2014

Iterative Frequency-Domain Detection for IA-Precoded MC-CDMA Systems Adão Silva, Member, IEEE, Sara Teodoro, Rui Dinis, Member, IEEE, and Atílio Gameiro

Abstract—Interference alignment (IA) is a promising technique that allows high capacity gains in interfering channels. On the other hand, iterative frequency-domain detection receivers based on the IB-DFE concept (Iterative Block Decision Feedback Equalization) can efficiently exploit the inherent space-frequency diversity of the MIMO MC-CDMA systems. In this paper we combine iterative IA precoding at the transmitter with IB-DFE based processing at the receiver for MC-CDMA systems. The receiver is designed in two steps: first a linear filter is used to mitigate the inter-user aligned interference, and then an iterative frequency-domain receiver is designed to efficiently separate the spatial streams in the presence of residual inter-user aligned interference at the output of the filter. The matrices for this nonlinear space-frequency equalizer are obtained by minimizing the overall mean square error (MSE) of all data streams at each subcarrier. Our receiver structure is explicitly designed taking into account the residual inter-user interference, allowing both an efficient separation of the spatial streams and a reduction in the number of iterations of the IA procedure. We also propose a simple, yet accurate analytical approach for obtaining the performance of the proposed receiver structure. Our scheme achieves the maximum degrees of freedom provided by the IA precoding, while allowing an almost optimum space-diversity gain, with performance close to the matched filter bound (MFB). Index Terms—Interference alignment, interference channels, iterative block equalization, MC-CDMA systems.

I. I NTRODUCTION HE increasing demand for wireless services has created the need for cost effective transmission techniques that can exploit scarce spectral resources efficiently. To achieve the high bit rates needed to meet the quality of service requirements of future multimedia applications, multi-carrier code division multiple access (MC-CDMA) has been considered as good air-interface candidate, especially for the downlink [1], [2]. This scheme combines efficiently orthogonal frequency division multiplex (OFDM) and CDMA. Therefore, MC-CDMA benefits from OFDM characteristics such as high spectral efficiency and robustness against multipath propagation, while CDMA allows a flexible multiple access with good interference properties [3], [4]. However, the user capacity of MC-CDMA system is essentially limited

T

Manuscript received August 29, 2013; revised November 30, 2013 and January 17, 2014. The editor coordinating the review of this paper and approving it for publication was S. Affes. A. Silva, S. Teodoro, and Atílio Gameiro are with DETI, Instituto de Telecomunicaçöes, University of Aveiro, Aveiro, Portugal (e-mail: {asilva, steodoro, atilio}@av.it.pt). R. Dinis is with Instituto de Telecomunicações, Faculdade de Ciências e Tecnologia, University Nova de Lisboa, Lisbon, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2014.022514.130681

by interference. This interference can be mitigated by employing precoding techniques [5], [6], iterative block decision feedback equalization (IB-DFE) based receivers [7], [8] and other efficient interference suppression techniques, proposed for different scenarios [9], [10], [11], [12]. Conventional frequency domain equalization (FDE) schemes employ a linear FDE optimized under the minimum mean square error (MMSE) criterion. However, the residual interference levels might still be too high, leading to performance that is still several dB from the matched filter bound (MFB) [13]. Nonlinear time-domain equalizers are known to outperform linear equalizers and conventional, time-domain DFEs are known to have good performancecomplexity tradeoffs [14]. For this reason, there has been significant interest in the design of nonlinear FDEs in general and frequency-domain FDEs in particular, with the IB-DFE being the most promising nonlinear FDE [13]. IB-DFE was originally proposed in [15] and was extended for a wide range of scenarios in the last 10 years, ranging from diversity scenarios [16], [17], MIMO systems [18] and MC-CDMA systems [19], [20], among many other. Essentially, the IB-DFE 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, although true turbo equalizers based on the IB-DFE concept can also be designed [21], [22]. An IB-DFE-based scheme specially designed for offset constellations (e.g. OQPSK, and OQAM) was also proposed in [23]. In the context of cooperative systems, an IB-DFE approach was derived to separate the quantized received signals from the different BSs [24]. Recently, analysis of interference channels has shown that each users’ capacity on an interference channel is one half the rate of its interference-free capacity in the high transmit power regime, regardless of the number of users [25]. One interesting scheme to efficiently eliminate the inter-user interference and achieve a linear capacity scaling is interference alignment (IA) [25]. This recent technique allows the transmitters to align in the unwanted users’ receive signals in any dimension, through the use of precoders. With this strategy more interferers can be completely cancelled than with other interference cancellation methods, thus achieving the maximum degrees of freedom (DoF) [26]. Applications of IA include cellular networks, twoway communication networks, cooperative communication networks, cognitive radio networks, etc. [27]. An explicit formulation of the precoding vectors achieving IA for time or frequency selectivity channels has been presented in [25]. A two-stage optimization of the precoding and decoding matrices in the 3-MIMO constant interference channels was proposed

c 2014 IEEE 0090-6778/14$31.00

SILVA et al.: ITERATIVE FREQUENCY-DOMAIN DETECTION FOR IA-PRECODED MC-CDMA SYSTEMS

in [28]. As closed-form solution for constant channels is still unknown except with 3 users, iterative algorithms [29]–[31] have been proposed for an arbitrary number of transmitterreceiver pairs. In [29] some examples of iterative algorithms were presented, where advantage of the reciprocity of wireless networks to achieve interference alignment with only local channel state information (CSI) knowledge at each node is taken. An MMSE-based iterative IA scheme was proposed in [30]. Several iterative linear precoding designs using alternating minimization were proposed in [31]. Most of these iterative algorithms require significant number of iteration to align the inter-user interference and exchange of information between the transmitter-receiver pairs at each step [31]. An overview of practical issues including performance in realistic propagation environments, the role of CSI at transmitter, and the practicality of IA in large networks was given in [32]. In this paper we consider MIMO MC-CDMA systems with iterative IA precoding at transmitter and iterative frequencydomain receivers based on the IB-DFE concept. To the best of our knowledge joint IA-precoding and IB-DFE based equalizer for MC-CDMA systems has not been addressed in the literature. The main motivation to consider this combination, is that IA based techniques achieve the maximum DoF in MIMO interference channels. However, they cannot by themselves efficiently exploit the space-frequency diversity inherent of the MIMO MC-CDMA systems. On the other hand, IB-DFE based receivers are well known to be one of the most efficient techniques to exploit this space-frequency diversity. Therefore, this combination allows us to design a system that is able to achieve maximum DoF (number of spatial stream per subcarrier) and exploit the high diversity order inherent to these systems. In the proposed scheme the chips are IA-precoded instead of the data symbols as in narrowband or OFDMbased systems. The proposed receiver structure is designed in two steps: first a linear filter is used to mitigate the interuser aligned interference, and then an IB-DFE based equalizer is employed to efficiently separate the spatial streams. The matrices for this non-linear space-frequency equalizer are obtained by minimizing the overall mean square error (MSE) of all data streams at each subcarrier. In the design of the equalizer, we explicitly take into account the residual interuser interference, allowing not only an efficient separation of the spatial streams but also a reduction the number of iteration in the IA procedure, and thus also reducing the information needed to be exchanged between the different transmitter-receiver pairs. Furthermore, we propose a simple, yet accurate analytical approach for obtaining the performance of the proposed receiver structure. The remainder of the paper is organized as follows: Section II presents the MIMO IA-precoded MC-CDMA system model. Section III, starts by briefly presenting the iterative IA-precoding considered in this paper. Then, the proposed receiver structure is presented in detail and an analytical approach for obtaining the performance is discussed. Section IV presents the main performance results, both numerical and analytical. The conclusions will be drawn in Section V. Notation: Throughout this paper, we will use the following notations. Lowercase letters, boldface lowercase letters and boldface uppercase letters are used for scalars, vectors and

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matrices, respectively. (.)H , (.)T , and (.)∗ represents the complex conjugate transpose, transpose, and complex conjugate operators, E [.] represents the expectation operator, IN is the identity matrix of size N · N , CN (., .) denotes a circular symmetric complex Gaussian vector, {α} represents a LP (A) is length block, tr(A) is the trace of matrix A, and vmin the matrix whose columns are the eigenvectors corresponding to the P smallest eigenvalues of matrix A. II. S YSTEM M ODEL We consider a K-user MIMO interference channel with constant coefficients on a per-subcarrier basis. It comprises K transmitter-receiver pairs sharing the same physical channel, where a given transmitter only intends to have its Pk spatial data symbols on each subcarrier decoded by a single receiver. Without loss of generality, we consider a symmetric case where all transmitters and receivers have M antennas, and Pk = P ∀k, which is denoted by an (M, M, K) interference channel with P data symbols persubcarrier. It has been shown that for K ≤ 3, the number of spatial DoF achievable in an (M, M, K) interference channel is KM/2, while for K > 3 the overall spatial K Pk < 2M − 1 [33]. DoF achievable is only Pt = k=1

Therefore, our system achieves KM/2 and Pt spatial DoF per-subcarrier for K ≤ 3 and K > 3, respectively. Fig. 1 shows the proposed kth MC-CDMA based transmitter. As can be seen, each one of the P L-length data symbols blocks, {dk,p,l ; k = 1, . . . K, p = 1, . . . , P, l = 0, . . . , L − 1}, where the constellation symbol dk,p,l (with E |dk,p,l |2 = σd2 ) is selected from the data according to given mapping rule, is spread into L chips using orthogonal Walsh-Hadamard codes, leading to the block {sk,p,l ; p = 1, . . . , P, l = 0, . . . , L − 1}. Then, a set of P chips (one of each block) is weighted by an IA-precoding matrix. Note that here the IA-precoding is applied on a chip level instead of data level as in the conventional IA systems. The signal after the IA precoding at the kth transmitter subcarrier l can be written as xk,l = Wk,l sk,l ,

(1)

where Wk,l ∈ C M×P is the linear precoding matrix computed at the kth transmitter on subcarrier l, constrained to Wk,l 2F ≤ Tp and Tp is the transmit power at the transmitT ters, with sk,l = [sk,1,l · · · sk,P,l ] . Finally, the precoding signals are mapped into the OFDM symbol and the cyclic prefix (CP) is inserted. The received frequency-domain signal (i.e., after cyclic prefix removal and FFT operation) for the kth receiver and the lth subcarrier is given by yk,l = Hk,k,l Wk,l sk,l +

K

Hk,j,l Wj,l sj,l + nk,l ,

(2)

j=1 j=k

provided that the cyclic prefix is long enough to account for different overall channel impulse responses between the transmitters and the receivers (i.e., including transmit and receive filters, multipath propagation effects and differences in the time-of-arrival for different transmitter-to-receiver links).

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{d }

sk ,1,1

k ,1,l

{d }

s/p

Spreading (L)

sk ,1, L

. . .

k , P ,l

s/p

. . .

sk , P ,1 Spreading (L)

. . .

sk , P , L

IA Preco ding

xk ,1,1 xk ,M ,1

IFFT + CP

. . . IA Preco ding

IFFT + CP

FFT CP

Antenna M Antenna M

FFT CP

{y }

{y }

k ,1,l

k ,1,l

. . .

. . .

xk ,1, L xk ,M , L

Antenna 1 Antenna 1

Linear IA Filter

k , P ,l

k , M ,l

CSI

√ (iid) The size-M × M matrix Hk,j,l = αk,j Hk,j,l denotes the overall channel between the transmitter j and receiver k on (iid) subcarrier l, where Hk,j,l contains the fast fading coefficients which is assumed to have i.i.d CN (0, 1) entries (independent, identically distributed complex normal random variables) and αk,j represents the long term channel power on the same link. nk,l is the additive white Gaussian noise (AWGN) vector at receiver k on subcarrier l, i.e., nk,l ∼ CN 0, σn2 IM . As can be seen from Fig. 2, the detection procedure is done in two steps: first a linear filter is used to mitigate the aligned user’s interference. Thus, the signal after the filtering process is given by yk,l

= ΦH k,l Hk,k,l Wk,l sk,l +ΦH k,l

K

Hk,j,l Wj,l sj,l + Φk,l nk,l

(3)

j=1 j=k

where Φk,l ∈ C M×P denotes the linear receiving filter. Second, a non-linear equalizer based on IB-DFE principle is designed to efficiently separate the spatial P L-length data block, considering the equivalent channels provided by the first block. III. I TERATIVE E QUALIZER D ESIGN In this section we start by briefly reviewing the iterative minimum interference leakage (IL) IA precoding algorithm [31]. We only selected this iterative IA for simplicity, but our iterative second step equalizer structure can be easily extended considering other iterative IA schemes such as the ones of [30], [31]. Then, the proposed non-linear iterative space-frequency equalizer is presented in detail. A. IA Precoders In constant MIMO interference channel, closed-form solutions have been found in only a few specific cases [31] (e.g. K ≤ 3), thus we consider an iterative approach for a general case. The aim is to precode the signal at transmitter j in such a way that the interference caused by that transmitter in receiver k, with k = j, is nearly orthogonal to a subspace of its receive space. This subspace, with orthonormal basis Φk,l , and precoders are jointly designed to optimize an appropriate cost function. The iterative interference leakage algorithm minimizes the total IL that remains at each receiver after attempting

Fig. 2.

IB-DFE Based Equalizer

{y }

{y }

CSI from Other Tx

Fig. 1. Proposed MC-CDMA transmitter with interference alignment precoding.

. . .

Eq. Channels

General overview of the proposed MC-CDMA receiver structure.

to cancel the aligned interference by left multiplication with ΦH k,l for each user k (see (3) ). The global function to optimize is K K H Φk,l Hk,j,l Wj,l 2 (4) IL = F k=1 j=1 j=k

which is usually denoted by IL. The optimization problem can be formulated as (5) at the top of the next page. A simple approach to solve this problem is to use an alternating minimization procedure [31]. This algorithm takes the following iterative form: 1) Define an arbitrary orthogonal basis Φk,l for each receiver subspace on each subcarrier 2) Find the precoder matrix Wj,l such that each node has maximum squared Euclidean distance between it and the subspace spanned by the columns of each Φk,l , by using ⎛ ⎞ K ⎟ P ⎜ H Wj,l = νmin HH (6) ⎝ k,j,l Φk,l Φk,l Hk,j,l ⎠ k=1 k=j

3) Update the receiver orthonormal subspaces according to ⎛ ⎞ K ⎜ ⎟ P ⎜ H ⎟ Φk,l = νmin Hk,j,l Φj,l Wj,l HH k,j,l ⎠ ⎝

(7)

j=1 j=k

4) Repeat steps 2 and 3 until convergence. This can be carried out until IL (t) < ε if feasibility conditions are met, or | IL (t − 1) − IL (t)| < ε otherwise, for an arbitrary threshold ε. The index t refers to the iteration number and it was dropped in the above equations for simplicity. The subspace Φk,l is reserved for each k’s signal (i.e., for each user) on each subcarrier, thus the IA at receiver k on subcarrier l is ideally orthogonal to this subspace. Then, each receiver must still separate the desired spatial streams after the interference aligned has been mitigated with left multiplication of ΦH k,l . A standard MMSE linear equalizer can be employed for this purpose. Therefore, in the conventional receiver a linear equalizer Gk,l is formed by multiplying ΦH k,l and the linear spatial equalizer Gk,l , which neglects the residual interuser aligned interference and equalizes only the desired signal,


min IL ({Wj,l } , {Φk,l }) s.t.

{s%( ) } i k ,1,l

y k ,l

x

(i )

+

∑

s% k ,l S/P

-

{s%( ) } i k , P ,l

Despreading (L) . . .

Despreading (L)

{d% ( ) } i k ,1,l

Soft Demod.

{d% ( ) } i k , P ,l

Soft Demod.

{d ( ) } i k ,1,l

{d ( ) } i k , P ,l

(i )

Fk ,l

{s ( ) } i −1 k ,1,l

x

sk(,il−1)

P/S

{s } ( i −1)

k , P ,l

Spreading (L) . . .

Spreading (L)

{d ( )}

Delay

It is well known that for MC-CDMA based systems, linear equalization is not efficient to separate the spatial streams due to the residual inter-carrier interference (ICI). In the context of IA based systems, it also neglects the residual interuser aligned interference. Therefore, a conventional equalizer design may not be the best strategy. As mentioned, the subspace Φk,l should be ideally orthogonal to interference aligned subspace, but full orthogonality may be not possible for some practical scenarios [32] and/or requires a significant number of iterations [31]. Thus, we design a new non-linear receiver structure based on IB-DFE principles which also takes into account the residual inter-user interference so that perfect alignment constraint can be relaxed. This receiver is quite efficient to separate the P L-length data symbol blocks, even for the case where the system suffers from residual inter-user interference, and thus the full orthogonality requirement can be relaxed. Fig. 3 shows the main blocks of the IB-DFE based procedure. For each iteration we detect all P L-length data blocks of the kth receiver, in a parallel way, using the most updated estimated of the transmit data symbols to cancel the residual interference, which it could not be cancelled in the first equalizer block. Thus, our receiver can be regarded as an iterative parallel interference cancellation (PIC). However, as with conventional IB-DFE based receivers, we take into account the reliability of the block data estimates for each detection procedure. At the ith iteration, the signal at kth receiver on lth subcarrier, before the despreading operation is given by, (i)T (i−1)

(i)

,

j=1 j=k

+ ΦH k,l nk,l ,

(9)

noise

Heq k,j,l

∈ C

P ×P

represents the equivalent channels T (i−1) (i−1) (i−1) after the IA procedure, and sk,l = sk,1,l · · · sk,P,l , (i−1) where the block sk,1,l ; l = 0, . . . , L − 1 is the spreading of the pth de-spreading block average values conditioned to the (i−1) detector output dk,p,l ; l = 0, . . . , L − 1 for each iteration where

k , P ,l

B. Iterative Equalizer Design

(i)T

Heq k,j,l sj,l

res. inter-user aligned interf.

( i −1)

so that Gk,l = Gk,l ΦH sk,l = Gk,l yk,l is k,l . Then the vector the estimate of the original transmit vector sk,l .

(i)

K

yk,l = Heq k,k,l sk,l + desired signal

Iterative receiver PIC equalizer based on IB-DFE principle.

sk,l = Fk,l yk,l − Bk,l sk,l

(5)

ΦH k,l Hk,j,l Wj,l , (3) can be rewritten as

Delay

B(ki,)l

Fig. 3.

H Wj,l Wj,l = Tp Ip , j ∈ {1, . . . , K} H Φk,l Φk,l = Ip , k ∈ {1, . . . , K}

i −1 k ,1,l

{d }

1243

(8)

where Fk,l ∈ C P ×P denoting the feedforward matrix and (i) Bk,l ∈ C P ×P is the feedback matrix. Setting Heq k,j,l =

(i−1)

i. For normalized QPSK constellations (i.e. dk,p,l = ±1 ± j) the average values are given by (see [19] for details), (i)Re Lk,p,l LiIm (i) k,p,l + j tanh , (10) dk,p,l = tanh 2 2 where

⎧ (i)Re ⎪ ⎨ Lk,p,l = ⎪ ⎩ LiIm k,p,l =

and (i)2

σk,p,l =

2

(i)2 σk,p,l

2 (i)2 σk,p,l

(i) Re dk,p,l (i) Im dk,p,l

L−1 1 (i) (i) 2 dk,p,l − dk,p,l . 2L

(11)

(12)

l =0

The extension to other constellations is considered in [34], [35]. The harddecision to the symboldk,p,l are (i) associated (i) (i) + jsign Im dk,p,l . It can dk,p,l = sign Re dk,p,l (i−1)

(i−1)2

(i−1)2

be shown that sk,l ≈ Ψk sk,l + Ψk Δk,l , where T the error Δk,l = [Δk,1,l · · · Δk,P,l ] has zero mean and (i) (i) Ψik = diag ψk,1 , . . . , ψk,P , with correlation coefficients defined as (i) E dk,p,l d∗k,p,l (i) , ψk,p = (13) 2 E |dk,p,l | being a measure of the pth L-length block estimates reliability associated to the ith iteration, which can be approximately given by L−1 1 (i)Re (i) iIm ψk,p,l + ψk,p,l , (14) ψk,p ≈ 2L l=0

with

⎧ ! (i)Re " Lk,p,l ⎪ (i)Re ⎪ ⎨ ψk,p,l = tanh 2 ! iIm " . L ⎪ | (i)Im k,p,l | ⎪ ⎩ ψk,p,l = tanh 2

(15)

For a given iteration and at each receiver, the iterative (i) non-linear equalizer is characterized by the coefficients Fk,l (i) and Bk,l . These coefficients are computed to minimize the average bit error rate (BER) of all P streams and for a QPSK

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constellation with Gray mapping the BER of the kth receiver, can be approximately given ⎞ ⎛ # ⎟ ⎜$ P ⎟ ⎜ (i) (16) BERk ≈ Q ⎜$ ⎟, $ L−1 ⎝% 1 (i) ⎠ MSE k,l L

(i)

Fk,l

j=1,j=k

and

l=0

where Q (x) denotes the well-known Gaussian function and (i) MSEk,l is the overall mean square error of the spreading samples given by, & 2 ' (i) (i) MSEk,l = E sk,l − sk,l & ! "' H (i) (i) = E tr sk,l − sk,l sk,l − sk,l .(17) After some mathematical manipulations, it can be shown that (17) can be rewritten as (18) at the top of the next page. The different correlation matrices of (18) are given by, ⎧ yk,l ) ( eq ⎪ Rk,l = E y∗k,l yTk,l = HeqH ⎪ k,k,l Rd Hk,k,l ⎪ ⎪ K ⎪ ⎪ eq ⎪ + HeqH ⎪ k,j,l Rd Hk,j,l + Rn ⎪ ⎪ j=1,j = k ⎪ ⎪ ⎨ (i−1)sk ,sk (i−1)∗ (i−1)T (i−1)2 = Ψk Rk = E sk sk Rd ) ( ∗ ⎪ yk,l sk eqH ⎪ ⎪ Rk,l = E yk,lsk = Hk,k,l ⎪ Rd ⎪ ⎪ (i−1)sk ,sk (i−1)2 ⎪ (i−1)∗ ⎪ Rk =E s sk = Ψk Rd ⎪ ⎪ ⎪ 2 ⎪ (i−1)s ,y k (i−1)∗ (i−1) k,l ⎩ R = E sk yk,l = Ψk Rd Heq k,l k,k,l (19) 2 2 with Rd = σd2 IP and Rn = ΦH Φ σ = σ I , being the k,l P n n k,l correlation matrices of data symbols and residual noise. From (15) we can see that to minimize the average BER at each receiver, we need to minimize the overall MSE at each subcarrier. The optimization problem can be formulated as, (i)

min MSEk,l s.t.

(i)

(i)

Fk,l ,Bk,l

L−1 1 (i)T eqT tr Fk,l Hk,k,l = P. L

(20)

l=0

We use the Karush-Kuhn-Tucker (KKT) [36] conditions to solve the optimization at each step with all but one variable fixed. The Lagrangian associated with this problem can be written by (i) (i) (i) L Fk,l , Bk,l , μk L−1 1 (i)T eqT (i) (i) = MSEk,l − μk tr Fk,l Hk,k,l − P , (21) L l=0

where μk is the Lagrangian multiplier [37]. The KKT conditions are, ⎧ (i) (i) (i) ⎪ ∇ Fk,l , Bk,l , μk = 0 (i) L ⎪ F ⎪ k,l ⎪ ⎨ (i) (i) (i) ∇B(i) L Fk,l , Bk,l , μk = 0 (22) k,l ⎪ ⎪ L−1 ⎪ (i)T ⎪ ⎩ L1 tr Fk,l HeqT k,k,l − P = 0 l=0

After lengthy mathematical manipulation we obtain the feedforward and feedback matrices, given by

⎞−1 (i−1)2 Heq HeqH k,k,l IP − Ψk k,k,l ⎟ ⎜ (i) eq K =⎝ 2 ⎠ Hk,k,l Ωk , σn eq + HeqH R H + I P d k,j,l k,j,l σ2 ⎛

with

(i)

d

(23) (i)

Bk,l = Heq k,k,l Fk,l − IP ,

(24)

(i) μ (i) (i−1)2 Ωk = IP − Ψk − 2k IP . σd L

(25)

The Lagrangian multiplier is selected, at each iteration i, L−1 (i)T eqT to ensure that the constraint L1 tr Fk,l Hk,k,l = P is l=0

fulfilled. It should be pointed out that for the first iteration (0) (i = 1), Ψk is a null matrix and sk,l is a null vector. Considering the proposed receiver scheme of Fig. 2, where the standard MMSE/ZF equalizer is replaced by our proposed IB-DFE equalizer, we slightly increase the complexity of this block. However, it should be emphasized that the main (i) complexity issues come from the computation of matrix Fk,l , and to compute it we only need to invert a P × P matrix on each subcarrier, irrespective to the number of antennas and users. Although, the complexity of the equalizer block is increased, the complexity to compute the joint precoders and filters (iterative IA procedure) can be significantly reduced (the fully orthogonally constraint can be relaxed) since our equalizer is designed to deal with the residual inter-user aligned interference. As a consequence of this reduction in the number of iterations, the information needed to be exchanged between the different transmitter-receiver pairs is reduced. IV. P ERFORMANCE R ESULTS In this section we present a set of performances results, analytical and numerical, for the proposed receiver structure (IA equalizer with IB-DFE based equalizer). We consider three scenarios: • 3 (K = 3) transmitter-receiver pairs, each equipped with 2 (M = 2) and transmitting simultaneously 1 (P = 1) L-length data block (with L = 128), referred as (2, 2, 3) interference channel with P = 1. • 3 (K = 3) transmitter-receiver pairs, each equipped with 4 (M = 4) and transmitting simultaneously 2 (P = 2) L-length data block, referred as (4, 4, 3) interference channel with P = 2. • 4 (K = 4) transmitter-receiver pairs, each equipped with 5 (M = 5) and transmitting simultaneously 2 (P = 2) L-length data block, referred as (5, 5, 4) interference channel with P = 2. The FFT size is set to 128 and a QPSK constellation under Gray mapping rule is considered. The channels between each transmitter and receiver pair are uncorrelated and severely time-dispersive, each one with rich multipath propagation and uncorrelated Rayleigh fading for different multipath components. Also, we assume perfect channel state information and synchronization. Our performance results are presented in terms of the average bit error rate (BER) as a function of Eb /N0 , with Eb denoting the average bit energy and N0


(i)H yk,l (i) (i)H (i−1)sk ,sk (i) tr Fk,l Rk,l Fk,l + tr Bk,l Rk Bk,l (i)H yk,l ,sk (i)H (i−1)sk ,sk + 2tr Re Bk,l Rk +P σd2 − 2tr Re Fk,l Rk,l (i)H (i−1)sk ,yk,l (i) Fk,l −2tr Re Bk,l Rk

denoting the one-sided noise power spectral density. In all scenarios we present the theoretical and simulation average BER performances for the proposed receiver structure. For the sake of comparisons we also include the matched filter bound (MFB). Figs. 4, 5 and 6 show the performance results for (2, 2, 3) with P = 1, (4, 4, 3) with P = 2 and (5, 5, 4) with P = 2, respectively. In all these figures, the number of iteration for the IA procedure was set to 100, this number was found enough to have free inter-user aligned interference. We present results for 1, 2 and 4 iterations of the IB-DFE based equalizer. Starting by analysing the results presented in Fig. 4, it is clear that the proposed analytical approach is very precise for the first iteration, where the IB-DFE based equalizer reduces to a linear MMSE-based frequency domain equalizer, since (0) Ψk is a null matrix and sk,l is a null vector. Although there is a small difference between theoretical and simulated results for the subsequent iterations, our analytical approach is still very accurate, with differences of just a few tenths of dB. The difference is slightly higher for the second iteration, decreasing as we increase the number of iterations. This behavior is a consequence of the accuracy of the Gaussian approximation that is behind our theoretical results. For a severely timedispersive channel with rich multipath propagation the ISI is high, which validates the Gaussian approximation of the residual interference after the first iteration. For the second iteration we reduce significantly the ISI, which makes the Gaussian approximation less accurate. However, as we increase the number of iterations we remove almost entirely the ISI (especially for large Eb /N0 and, consequently, small BER) and we converge to a noise-only scenario where, once again, the Gaussian approximation is valid. Errors in the computation of the reliability of the estimates employed in the feedback loop (which are only accurate for the first iteration (zero reliability) and when we have very low BER (which means that the estimates are accurate, i.e., its reliability is 1)) also contribute to higher differences between theoretical and simulated results especially at the second iteration. As expected, the BER performance improves with the iterations and its can be observed that for the 4th iteration the performance is close the one obtained by the MF. From Fig. 5, we can also observe that the BER performance approaches the MFB after just a few iterations (typically 3 or 4 iterations). We can see that the performance gap between the first iteration (linear MMSE-based equalizer) and the fourth one is higher than in the scenario of Fig. 4. Note that for this scenario the IB-DFE based equalizer must deal with interblock interference, since 2 128-length blocks are transmitted simultaneously by each transmitter, and residual inter-carrier interference. Therefore, the proposed scheme is quite efficient

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to separate the spatial streams and achieve the high diversity order inherent to this scenario, with only a few iterations. From Fig. 6 we basically can arrive at the same conclusions as for the results obtained in the previous scenario. Increasing the number of users to 4 with P = 2, we also need to increase the number of antennas to 5, in order to fulfil the feasibility conditions for the alignment as discussed in Section II. This additional number of antennas is used to increase the overall DoF keeping the inherent diversity at the same level as the previous scenario (K = 3), and thus the performance is basically the same.

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The results of Figs. 7 and 8 were obtained for the same parameters of the Figs. 5 and 6, respectively, but now the number of iteration for the IA procedure was set to 10. We also present results for 8 iterations of the IB-DFE based equalizer. In this case the inter-user aligned interference cannot be neglected and may have a significant impact on the system performance. From these figures, we can see that the analytical approach proposed for the receiver structure is accurate, as in the previous scenarios. It can be seen that the performance degradation for the first iteration is significant when compared with Figs. 5 and 6 (about 2dB for BER= 10−2 ). However, for the fourth iteration the performance degradation regarding to the Figs. 5 and 6 is much lower (approximately 0.7 dB for both). This means that our proposed scheme is able to efficient mitigate the residual inter-user interference. Hereinafter, the curves for the analytical approach are not added to the figures only for clarity. Fig. 9 presents results for (4, 4, 3) with P = 2, 1 and 4 iterations for the IB-DFE based equalizer and several iteration for the IA procedure, 5, 10, 20,

50 and 100. When we have a single equalizer iteration there is a significant performance degradation from 100 to 5 IA iterations, since the residual inter-user interference increases as the number of IA iterations decreases. However, for the fourth equalizer iterations we can see that the performance degradation from 100 to 5 IA iterations is much lower. In fact, we have almost the same performance with 100 and 10 to 20 IA iterations, which means we can reduce significantly the number of IA iterations. Thus also reducing the information needed to be exchanged between the different transmitterreceiver pairs. The results of Figs. 10, 11 were obtained for the same parameters of the Figs. 4 and 5. The only exception was the number IA iteration set to 10. We compare the performance results of our propose scheme (referred in these curves as Robust) against the case where the residual aligned interference was ignored in the equalizer design, i.e., the term K eq HeqH k,j,l Rd Hk,j,l in (23) is not considered (referred in j=1,j=k

these curves as Non-Robust). From these figures we can see


for obtaining the performance of our receiver structure. The results have shown that the proposed receiver structure is robust to the residual inter-user aligned interference and quite efficient to separate the spatial streams, while allowing a close-to-optimum space-diversity gain, with performance close to the MFB with only a few iterations of the equalizer. The proposed robust residual inter-user aligned interference allows significant reduction in the number of iterations of the IA process, and thus reducing the information needed to be exchanged between the different transmitter-receiver pairs. To conclude we can clearly state that the proposed receiver structure is an excellent choice for the IA-precoded MCCDMA based systems.

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ACKNOWLEDGEMENT Fig. 10. Performance of the proposed receiver structure (Rob.) against the non-robust approach (Non-Rob.) for (2, 2, 3) with P = 1 and 10 IA iterations.

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This work was supported by the Portuguese Fundação para a Ciência e Tecnologia (FCT) COPWIN (PTDC/EEI-TEL/1417/2012), CROWN (PTDC/EEATEL/115828/2009), ADIN (PTDC/EEI-TEL/2990/2012) and PEst-OE/EEI/LA0008/2013 projects. The authors would also like to thank the Editor and the reviewers for a careful reading of the manuscript and insightful suggestions.

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the significant performance impact that the residual inter-user aligned interference may cause if not taking into account on the design of the second equalizer, mainly for lower BER values. These results clearly show the robustness of our proposed receiver structure to the residual inter-user aligned interference. V. C ONCLUSIONS In this paper we considered IA precoding, on a persubcarrier basis, at the transmitter with IB-DFE based processing at the receiver for MIMO MC-CDMA systems. The proposed receiver structure was designed in two steps: first a linear filter was considered to mitigate the inter-user aligned interference, and then an iterative frequency-domain equalizer was designed to efficiently separate the spatial streams. The matrices were obtained by minimizing the overall MSE of all data streams at each subcarrier. In the design of the proposed equalizer, we explicitly took into account the residual interuser aligned interference, and thus the full orthogonality constraint of the linear filter to the IA subspace could be relaxed. We also proposed a simple, yet accurate analytical approach

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Adão Silva received the M.Sc. and Ph.D. degrees in electronics and telecommunications from the University of Aveiro in 2002 and 2007, respectively. He is currently an Assistant Professor in the Department of Electronics, Telecommunications and Informatics of the University of Aveiro, and a researcher at the Instituto de Telecomunicações. He has participated in several national and European projects, namely ASILUM, MATRICE, and 4MORE within the ICT programme, and the FUTON and CODIV projects with the FP7 ICT. He has led several research projects in the broadband wireless communications area at the national level. He has been a member of the TPC of several international conferences. His research interests include multiuser MIMO, multicarrier based systems, cooperative networks, precoding, and multiuser detection. Sara Teodoro received the Licenciatura degree (five years course) in electrical engineering in 2004 from the University of Coimbra, and the Ph.D. degree in electronics and telecommunications in 2011 from the University of Aveiro. She is currently a Postdoctoral researcher at the Instituto de Telecomunicações in the University of Aveiro. Her research interests are in wireless communications, particularly in cooperative systems, multi antenna systems, and OFDM technology. Her recent work is on interference cancellation methods and limited feedback techniques applied to multi-antenna systems. Rui Dinis received the Ph.D. degree from the Instituto Superior Técnico (IST), Technical University of Lisbon, Portugal, in 2001 and the Habilitation in Telecommunications from the Faculdade de Ciências e Tecnologia (FCT), Universidade Nova de Lisboa (UNL), in 2010. From 2001 to 2008, he was a Professor at IST. Currently, he is an associated professor at FCT-UNL. During 2003, he was an invited professor at Carleton University, Ottawa, Canada. He was a researcher at CAPS (Centro de Análise e Processamento de Sinal), IST, from 1992 to 2005 and a researcher at ISR (Instituto de Sistemas e Robótica) from 2005 to 2008. Since 2009, he has been a researcher at IT (Instituto de Telecomunicações). Rui Dinis is an editor for the IEEE T RANSACTIONS ON C OMMUNICATIONS (Transmission Systems - Frequency-Domain Processing and Equalization) and guest editor for Elsevier Physical Communication (Special Issue on Broadband Single-Carrier Transmission Techniques). He has been actively involved in several national and international research projects in the broadband wireless communications area. His research interests include modulation, equalization, channel estimation, and synchronization. Atílio Gameiro received his Licenciatura and his Ph.D. from the University of Aveiro in 1985 and 1993, respectively. He is currently an Associate Professor in the Department of Electronics and Telecom. of the University of Aveiro, and a researcher at the Instituto de Telecomunicações, Pólo de Aveiro, where he is head of the group. His industrial experience includes a period of one year at BT Labs and one year at NKT Elektronik. His main research interests lie in signal processing techniques for digital communications and communication protocols, and within this research line, he has done work for optical and mobile communications, either at the theoretical and experimental level, and has published over 200 technical papers in international journals and conferences. His current research activities involve space-time-frequency algorithms for broadband wireless systems and cross-layer design. He has been involved and has led IT or University of Aveiro participation on more than 20 national and European projects.