Linear and Widely Linear Filtering Applied to ... - Semantic Scholar
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Linear and Widely Linear Filtering Applied to Iterative Detection of Generalized MIMO Signals Melanie Witzke Institute for Communications Engineering (LNT) Munich University of Technology (TUM) 80290 Munich, Germany Email: [email protected]
Abstract To suppress the co-antenna interference in multiple input multiple output (MIMO) systems, an iterative receiver with a linear detector for complex symbols is investigated. We show that the considered generalized MIMO system, i.e., a system that transmits complex conjugate repetitions in addition to the pure data, requires the application of a widely linear (WL) detector. A WL detector consists of four real filters which are represented by two complex filters for the received signal and its complex conjugate, respectively. Furthermore, we present approximations of the detector that significantly reduce computational complexity with only little loss in frame-error rate performance. Simulation results show that the proposed MIMO system achieves large gains over standard solutions.
R´esum´e Pour supprimer l’interf´erence entre les antennes d’un syst`eme a` entr´ees et sorties multiples (MIMO), un r´ecepteur it´eratif avec un d´etecteur lin´eaire pour des symboles complexes est examin´e. Nous montrons que pour le syst`eme tr`es g´en´eral que nous consid´erons, c’est-a-dire un syst`eme qui transmet les informations elles-mˆemes ainsi que des r´ep´etitions conjugu´ees, un d´etecteur lin´eaire au sens large (widely linear, WL) est exig´e. Le d´etecteur WL comprend quatre filtres r´eels, repr´esent´es par 2 filtres complexes, un pour le signal rec¸u et un autre pour le conjugu´e du signal rec¸u. En plus, nous pr´esentons des approximations du d´etecteur qui r´eduisent fortement la complexit´e avec seulement une petite perte en taux d’erreur de bloc. Des r´esultats de simulations montrent, que le syst`eme MIMO propos´e a une meilleure performance que les solutions standards. Index Terms Widely linear estimation, iterative linear detection, MIMO systems, second-order behavior
Biography - Melanie Witzke was born in Cologne, Germany, in 1974. She received the Dipl.-Ing. degree in communications engineering from the Aachen University of Technology, Aachen, Germany, in 1998. Since Spring 1999, she has been a Research Assistant at the Institute for Communications Engineering (LNT) at the Munich University of Technology (TUM), Munich, Germany. Her main research interests are coding and detection algorithms for MIMO systems.
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I. I NTRODUCTION A multiple input multiple output (MIMO) system has the potential for delivering large data rates. However, transmission over MIMO channels suffers from strong co-antenna interference. A promising approach to cope with this problem is to use an iterative receiver performing joint symbol detection and decoding. Although the a posteriori probability (APP) detector is optimal within such a structure, its prohibitive complexity demands alternatives. Approaches that reduce the brute-force full-APP search have been the focus of increased attention. In the following we review some of the major works. In [1], the transmit antennas are partitioned into groups. For each group, the transmitted bits are estimated by nulling the interference from the other antennas. Other approaches try to find a subset of log-likelihood contributions such as [2] which takes those sequences that lie within a hyper-sphere, [3] selects the subset via a tree-based algorithm and in [4] the LISt-Sequential detector applies sequential detection based on a stack that contains a limited number of hypotheses. Instead of identifying the most likely channel input sequence, another approach is to attempt to cancel and further suppress the interference with a linear detector. Aiming at a low-complexity solution, a fixed minimum mean squared error (MMSE) filter is used in [5]. Wang and Poor [6] were the first to propose an MMSE detector that adapts to the residual interference of its input signal within an iterative receiver. If the interference is removed perfectly (as might happen after some iterations), this detector achieves the APP detector’s performance. In [7] and further generalized in [8], the application of adaptive linear filtering to MIMO transmission was presented. In this article, we discuss how the linear detector has to be modified if generalized MIMO systems are considered. In contrast to [8], these MIMO transmitters also allow complex conjugate repetitions of the data symbols. Therefore, they require the application of a widely linear (WL) filter instead of a strictly linear filter. In [9], it has been shown that WL estimation is superior or at least identical in performance to strictly linear estimation. In contrast to a linear filter, the WL filter consists of two complex filters to process both the observation and its complex conjugate. This approach (named conjugate linear filtering) was already discussed in [10] for complex signals. Recently, WL estimation found some applications that confirm the promised gains. For instance, it was used in DS-CDMA systems employing BPSK [11]-[14], it was applied to blind multiuser detection [15] and it was integrated in an iterative multiuser detector for complex data [16]. Closely related with WL estimation is the theory of improper or non-circularly symmetric signals. The second-order properties of a non-circularly symmetric signal are not fully described by its covariance. We must also consider the correlation between the signal and its complex conjugate. A comprehensive treatment of the corresponding theory is given in [17]. Other articles have continued the discussion, e.g., [18]-[20]. In a detailed discussion we will show how the WL filter adapts to the different structures of the MIMO signal. In particular, we distinguish between MIMO systems that only transmit data and those that also allow complex conjugate repetitions. The former use the WL filter to exploit the non-circularity of the noise stemming from the residual interference. The noise becomes non-circular during the iterations. The latter require WL processing to account for the transmitted complex conjugate data. During the iterations, the non-circularity of the interference is also considered. The large computational complexity of the WL detector demands the use of simplifications. We present two approaches that significantly reduce computational complexity. Simulation results show a negligible loss in performance. We motivate the application of generalized MIMO signals with the theory of the linear dispersion codes [21]. Compared to purely linear space-time mappings, e.g., the V-BLAST transmitter [22], these codes do not only approach the MIMO channel capacity but also provide transmit diversity. By eliminating the interference, the iterative, interference cancelling receiver enables us to exploit this diversity. Apart from this, there has been an immense interest in designing space-time codes. For instance, in [23]-[26] and [3] turbo codes have been adapted to MIMO systems based on serial or parallel concatenation of different constituent codes. These so-called space-time turbo codes aim at exploiting the MIMO channel capacity whilst maximizing the diversity gain. However, the design of systems that achieve the best trade-off between rate and diversity remains an open research problem. Recently, an optimal trade-off curve has been presented in [27] showing that the maximum diversity and multiplexing gain cannot be achieved simultaneously. In Section II, we motivate the use of the WL filter. The system model and some basics about second-order theory are presented in Section III. We derive the WL detector in Section IV and discuss the properties of WL filtering
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considering MIMO signals in Section V. Two simplifications of the WL filter are given in Section VI. We briefly motivate and introduce the considered space-time mappings in Section VII. Section VIII shows simulation results and conclusions are drawn in Section IX. II. M OTIVATION Before going into details, we motivate the choice of a WL detector over a linear detector. A detailed discussion will follow in Section V. Let x be a complex symbol to be estimated in terms of an observation y that is a complex random vector. Furthermore, the vector x contains the symbols x. 1) Non-circular observation: If the complex vector w denotes an MMSE filter, the linear estimate is known to be x ˆ = wH y = wH yI + jwH yQ , where (·)H is the conjugate transpose. The subscripts ’I’ and ’Q’ refer to the in-phase and quadrature components of an equivalent baseband signal. Assume the observation y is a non-circular signal. Consequently, the covariances of the in-phase and the quadrature components of y are in general different. For instance, in iterative detection, the in-phase components of the data might have more reliable a priori values than the quadrature components. Then, a single complex MMSE filter cannot be optimal for detecting both y I and yQ . This problem can be solved by using two different complex filters g1 and g2 for yI and yQ , respectively. Such an approach gives the estimate x ˆ = g1H yI + jg2H yQ , = w1H y + w2H y∗ ,
(1)
where g1 = w1 + w2 , g2 = w1 − w2 and w1 , w2 are complex filters. The operator (·)∗ denotes the complex conjugate. It is worth mentioning that if g1 = g2 then w2 = 0 and we obtain the standard linear solution. Equ. (1) has the form of the WL estimate introduced in [9]. The WL filter is not linear in C but becomes linear in R 2 with four real-valued filters. For compatibility with the existing literature (e.g., [9], [14]), we continue the discussion based on the WL filter notation in (1). Furthermore, a comparison with the strictly linear filter characterized by w2 = 0 is more obvious. 2) Correlation of observation and conjugate data symbols: Assume that the relation between y and x conforms to y = H1 x + H2 x∗ , where H1 and H2 are matrices with complex entries. In this case y and x∗ are correlated. Obviously, a single linear filter could only exploit the information transmitted in x but not that transmitted in x ∗ . Thus, the processing of both the observation and its complex conjugate is essential and leads to the WL estimate of (1). III. S YSTEM M ODEL
AND
D EFINITIONS
A. MIMO system model We consider the transmission model as depicted in Fig. 1. It consists of a MIMO transmitter and an iterative receiver that are connected through a MIMO channel with nT transmit and nR receive antennas. 1) Transmitter: The binary data is encoded with a convolutional code of rate RC to a codeword c0L containing L · N · M bits, where L denotes the number of subblocks per codeword, N the number of complex symbols per subblock and 2M is the cardinality of the complex constellation. The random interleaver permutes c0L to cL . We T T T T partition cL into L subblocks cl = [cT l1 . . . clN ] , l = 1, . . . , L, with cln = [cln1 . . . clnM ] . The operator (·) denotes the transpose. Throughout the following derivations and also in Fig. 1, we only consider one such subblock and thus drop the subscript l to simplify the notation. The vector c of N · M coded bits is mapped to a complex symbol vector x = [x1 . . . xN ]T . The elements xn = map(cn ) are taken from a 2M -ary symmetric complex symbol alphabet A = {a1 . . . a2M }, where each ai corresponds to a bit pattern [bi1 . . . biM ], bij ∈ {0, 1}. The N symbols x are transmitted during nS channel uses over the MIMO channel. To achieve this, they are mapped to a length nT · nS vector s = G 1 x + G 2 x∗ (2) T T T of transmitted signals, where s = [sT 1 . . . snS ] , with sj = [sj1 . . . sjnT ] . The nT nS ×N complex-valued space-time (ST) mapping matrices G1 and G2 disperse the symbols x into space and time. They determine the rate of the ST
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data
Encoder
4
Space−Time Mapper
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Fig. 1.
APP Decoder
:
WL Detector
%8 9
MIMO transmission model – Mod`ele du syst`eme de transmission MIMO
mapping as RS = N/nS symbols/channel use and thus, the total rate of the system as R = RC · M · RS bits/channel use. Examples of ST mappings are given in Section VII. In the context of this article, we call (2) a widely linear ST mapping. If G2 = 0, we use the name linear ST mapping. The total average transmit energy per channel use is Es and since the channel is unknown at the transmitter, E{|sji |2 } = Es /nT . The operator E{·} denotes expectation. During nS channel uses, the total transmitted energy is (tr E{ssH }) = nS · Es and from this the average symbol energy can be determined as n S · Es , (3) Ex = H tr(G1 GH 1 + G 2 G2 ) where Ex = E{|xn |2 } and tr(·) computes the trace of a matrix. At each time instant, the nT symbols sj are simultaneously transmitted over the nR × nT MIMO channel H yielding the observation yj = Hsj + nj . Introducing the nR nS × nT nS block diagonal channel matrix HB = diag(H, . . . , H), we can summarize the nS observations yj in one vector y = HB s + n = H1 x + H2 x∗ + n,
(4)
where y = [y1T . . . ynTS ]T , with yj = [yj1 . . . yjnR ]T . The vector n, of length nR · nS , contains independent circular complex-valued white Gaussian noise signals of variance σn2 /2 per real dimension. N0 is the one-sided power spectral density of the channel noise. H1 and H2 denote effective channel matrices consisting of a concatenation of HB and the ST mapping matrices G1 and G2 , respectively. Note, that the ST mapping matrices transform the originally linear MIMO channel into a widely linear channel (see (4)). If H1 and H2 are decomposed in their in-phase and quadrature components, (4) can be represented by a twodimensional channel model nI HI,1 + HI,2 HQ,2 − HQ,1 xI yI (5) + = nQ HQ,2 + HQ,1 HI,1 − HI,2 xQ yQ | {z } HR
with two inputs and two outputs. Equ. (5) describes the well-known real-valued equivalent baseband channel. It is a generalized version of the textbook notation and consists of four different real-valued filters that connect the in-phase and quadrature components of the input and output signals. For H2 = 0, (5) is identical to standard notation with only two different filters.
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2) Channel model: We use the typical frequency non-selective, quasi-static, MIMO channel model, meaning that the channel is randomly independently chosen for each new transmitted block. The block length is identical to the codeword length. The channel coefficients which are the entries of H are modeled by independent zero-mean complex Gaussian random variables of unit variance. 3) Receiver: The receiver is assumed to have perfect channel knowledge and consists of a WL detector that is specified in Section IV and an a posteriori probability (APP) decoder which iteratively exchange soft information about the coded bits. The APP decoder operates according to the log-MAP algorithm [28]. We describe the soft information for the coded bits using log-likelihood ratios (LLRs) in the form of a vector T L(c) = L(c1 )T . . . L(cN )T , (6) T L(cn ) = L(cn1 ) . . . L(cnM ) , (7) P (cnj = 0) . (8) L(cnj ) = ln P (cnj = 1) Using channel and a priori information (which is not available in iteration 0), the detector computes extrinsic estimates Lde (c) for the coded bits. These deinterleaved estimates are then the input Lcc (c0 ) of the APP decoder. The decoder computes a posteriori information for the information bits and for the coded bits. The former is used to make decisions and the latter, Lc (c0 ), is fed back and interleaved giving Lda (c). From Lda (c) the extrinsic information Lda,e (c) is computed. The detector is then provided with the two types of a priori informations Lda (c) and Lda,e (c). The subscript ’e’ distinguishes the extrinsic from the a posteriori information and the subscript ’a’ denotes ’a priori’ information. Feeding back the full a posteriori information violates the turbo principle but is beneficial for soft interference cancellation in MIMO systems as shown in [8]. The additional extrinsic information is used for the soft-demapping. B. Second-order theory of complex signals Second-order statistics play a key role in the course of this contribution and are therefore briefly reviewed. We consider the two complex random vectors x and y. Neeser and Massey have shown [17], that the four covariance matrices arising between the real and imaginary parts of x and y can also be expressed by the covariance matrix Λxy = cov(x, y) = E{(x − E{x})(y − E{y})H } (9) as well as by the pseudo-covariance matrix ˜ xy = cov(x, y∗ ) = E{(x − E{x})(y − E{y})T }, Λ
(10)
also often referred to as the conjugate or complementary covariance matrix. For simplicity, the autocovariance ˜ y) ˜ x , respectively. λ(x, y) and λ(x, matrix and the pseudo-autocovariance matrix will be represented by Λx and Λ denote the covariance and the pseudo-covariance of the complex symbols x and y . Accordingly, we describe the ˜ . autocovariance and the pseudo-autocovariance by λ(x) and λ(x) ˜ x corresponds to a proper / circularly In [17], it is shown that a vanishing pseudo-autocovariance matrix Λ symmetric signal x. The pseudo-autocovariance matrix ˜ x = ΛxI − ΛxQ + j(ΛxI xQ + ΛT ) Λ xI xQ
(11)
is zero if the in-phase and quadrature components of x have identical autocovariance matrices and if their crosscovariance matrix is skew-symmetric. The pseudo-autocovariance ˜ λ(x) = λ(xI ) − λ(xQ ) + 2j λ(xI , xQ )
(12)
of a single complex component x vanishes if xI and xQ have identical covariances and xI and xQ are uncorrelated. ˘ = x − E{x} are identical to the Note, that the (pseudo-)autocovariances of a centered process x (pseudo-)autocovariances of the process itself Λx = cov(x, x) = cov(x − E{x}, x − E{x}) = Λx˘ , ˜ x = cov(x, x∗ ) = cov(x − E{x}, x∗ − E{x∗ }) = Λ ˜ x˘ . Λ
This relation can easily be shown and is frequently used in the following.
(13) (14)
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IV. W IDELY L INEAR (WL) D ETECTOR A. Soft estimates and soft (pseudo-)covariances For the computation of the WL detector we need the soft estimate x ¯n = E{xn |Lda (cn )}, covariance λ(xn ) = d ∗ d ˜ n ) = cov(xn , x |L (cn )) of a symbol xn = map(cn ), stemming cov(xn , xn |La (cn )) and pseudo-covariance λ(x n a M from the 2 -ary symbol alphabet A (see Section III). They are calculated from the a priori information L da (cn ) introduced in Section III as follows X x ¯n = ai P xn = ai |Lda (cn ) , (15) a ∈A
i X ai 2 P xn = ai |Lda (cn ) −|¯ x n |2 , λ(xn ) =
(16)
ai ∈A
˜ n) = λ(x
X
a2i P xn = ai |Lda (cn )
ai ∈A
−¯ x2n .
(17)
Independent bits can be assumed due to the interleaver, therefore P xn =
ai |Lda (cn )
=
M Y
j=1
P cnj = bij |Lda (cnj ) .
(18)
P (cnj ) itself is determined by the a priori LLR Lda (cnj ) according to the well-known relation P (c = 0) =
eL(c) , 1 + eL(c)
P (c = 1) =
1 . 1 + eL(c)
(19)
Note, that only the full a posteriori feedback Lda is involved in the evaluation of (15)-(17). B. Derivation of WL detector The WL detector of Fig. 1 is shown in more detail in Fig. 2. It is composed of three functional units which are interference cancellation, WL filtering and soft-demapping to compute the extrinsic estimates of the coded bits. In the following we let the subscripts k and m have fixed values. k denotes the k -th symbol of N symbols and m one bit out of a group of M bits.
w1,k
˘k y
y
(·)
∗
˘ k∗ y
xˆk
Lde (c)
w2,k
Recon− struction
Lda,e (c)
xk =0 ˙
x Fig. 2.
L(·)
Lda (c)
E{·}
Widely linear (WL) detector – D´etecteur lin´eaire au sens large
1) Interference cancellation: Keeping in mind that our overall aim is the estimation of the k -th complex symbol ¯ = [¯ x ˆk , we first consider soft interference cancellation. Using the vector x x1 . . . x ¯N ]T of soft estimates from (15), the interference-reduced signal is ¯∗ ¯ + H2 x ˘ k = y − H1 x . (20) y x ¯k =0 ˙
Forcing x ¯k to zero is necessary to ensure pure extrinsic information at the detector output [6], [8].
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2) WL filtering: In order to further suppress residual interference, the signal is filtered. As will be shown in the following section, the application of the two complex filters w1,k and w2,k is in general beneficial. This yields the WL estimate H H ˘ k + w2,k ˘ k∗ , x ˆk = w1,k y y (21) ∗ ∗ where wf,k = [wf,k,1 . . . wf,k,n ]T , f = 1, 2. The two filters are chosen to minimize the mean squared error R nS E{|xk − x ˆk |2 |Lda (c)}. In [9] the solution to this problem is derived from the orthogonality principle giving the two filters h i ˜∗ ˜ y,k Λ∗−1 λ λ − Λ (22) w1,k = Γ−1 yx,k y,k yx,k , k i h −1 ˜∗ − Λ ˜ ∗ Λ−1 λyx,k , (23) w2,k = Γk∗ λ yx,k y,k y,k ˜ y,k Λ∗−1 Λ ˜ y,k = E{˘ ˜∗ ˘ kT }, λyx,k = E{˘ ˘ kH }, Λ yk x∗k } and where Γk = Λy,k − Λ yk y yk y y,k y,k . Λy,k = E{˘ ˜ yx,k = E{˘ λ yk xk } denote the covariance matrix, the pseudo-covariance matrix, the covariance vector and the pseudo-covariance vector, respectively. For the widely linear MIMO channel from (4), the evaluation of the filter components yields ∗ H H H 2 ˜ ˜∗ Λy,k = H1 Λx,k HH 1 + H2 Λx,k H2 + H1 Λx,k H2 + H2 Λx,k H1 + σn InR nS , ˜ y,k = H1 Λ ˜ x,k HT + H2 Λ ˜ ∗ HT + H1 Λx,k HT + H2 Λ∗ HT , Λ 1 2 x,k 2 x,k 1
λyx,k = Ex h1,k , ˜ yx,k = Ex h2,k . λ
(24) (25) (26) (27)
Ij is the j × j identity matrix, h1,k and h2,k are the k -th column of H1 and H2 , respectively, and Ex = E{|xk |2 }. Assuming independent symbols x we obtain the covariance matrix and the pseudo-covariance matrix as Λx,k = diag(λ(x1 ) . . . λ(xN )), ˜ 1 ) . . . λ(x ˜ N )), ˜ x,k = diag(λ(x Λ
with λ(xk ) = Ex , ˜ k ) = 0, with λ(x
(28) (29)
with the covariances and the pseudo-covariances from (16), (17) along the diagonal. The constraint from (20) yields ˜ k ) = 0 (see Section V-A). λ(xk ) = Ex and λ(x 3) Soft-demapping: The extrinsic LLR, i.e., the difference between the a posteriori and the a priori LLR, of the m-th coded bit of x ˆk is defined by the expression Lde (ckm ) = ln
= ln
p(ˆ xk |ckm = 0) p(ˆ xk |ckm = 1) P p(ˆ xk |xk = ai )
(30) M Q
P (ckj = bij |Lda,e (ckj ))
ai ∈A:ckm =0
j=1,j6=m
P
M Q
p(ˆ xk |xk = ai )
ai ∈A:ckm =1
P (ckj =
j=1,j6=m
, bij |Lda,e (ckj ))
where the product assumes independence of the bits within one symbol because of interleaving. The probability density function (pdf) p(ˆ xk |xk = ai ) is approximated by the Gaussian distribution and the probabilities P (ckj = bij |Lda,e (ckj )) are determined by Lda,e (ckj ) according to (19). In contrast to (15)-(17), the use of the extrinsic LLRs ˘ k the is mandatory here to guarantee pure extrinsic information in (30). Note, that in general for non-circular y estimate x ˆk is also non-circular. Thus, the two-dimensional Gaussian pdf p(ˆ xI,k , x ˆQ,k |xk = ai ) should be used. This 2 = cov(ˆ pdf is completely determined by the mean µki = E{ˆ xk |xk = ai } and the covariances σI,k xI,k , x ˆI,k |xk = ai ), 2 2 σQ,k = cov(ˆ xQ,k , x ˆQ,k |xk = ai ) and σIQ,k = cov(ˆ xI,k , x ˆQ,k |xk = ai ). Due to independence of the covariances on a certain symbol ai , the subscript ’i’ is neglected. Evaluating the expectation gives the mean of x ˆk given that the symbol ai was transmitted which can be written as H H H H µki = [w1,k (h1,k + h2,k ) + w2,k (h1,k + h2,k )∗ ] aI,i + j [w1,k (h1,k − h2,k ) − w2,k (h1,k − h2,k )∗ ] aQ,i . | {z } {z } | m1,k
m2,k
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Note, that for a non-circular x ˆk , the scaling factors m1,k and m2,k are different and complex-valued whereas for a circular signal they are identical and real-valued. If we express the in-phase and quadrature component of x ˆk as follows 1 1 H T ˘ k + g1,k ˘ k∗ ), (ˆ xk + x ˆ∗k ) = (g1,k x ˆI,k = y y 2 2 1 H 1 T ˘ k − g2,k ˘ k∗ ), xk − x ˆ∗k ) = −j (g2,k y y x ˆQ,k = −j (ˆ 2 2 ∗ and g ∗ with the two vectors g1,k = w1,k + w2,k 2,k = w1,k − w2,k , the covariances of the noise (interference and channel noise) can be derived as o 1 n H H ˜ ∗ 2 Λy,k g1,k + g1,k Λy,k g1,k , σI,k = < g1,k 2 n o 1 H H ˜ ∗ 2 (31) Λy,k g2,k − g2,k Λy,k g2,k , σQ,k = < g2,k 2 n o 1 2 H H ˜ ∗ σIQ,k = = −g1,k Λy,k g2,k + g1,k Λy,k g2,k . 2 ˜ y,k are from (24), (25), where now the k -th component of Λx,k and Λ ˜ x,k from (28), (29) takes on the Λy,k and Λ ˜ k ) = 0. value λ(xk ) = λ(x V. P ROPERTIES OF MIMO
SIGNALS AND OF
WL
FILTERING
After introducing the mathematics on widely linear estimation, we proceed to discuss the properties of the MIMO signals and show how the WL filter adapts to these signals. To provide an insight into the individual effects that influence the WL estimation, the interference reduced signal from (20) is rewritten as ˘ k = H1 (x − x ¯ ) + H2 (x − x ¯ )∗ + n y x ¯k =0 ˙
˘ k + H2 x ˘ ∗k + n, = H1 x
(32)
˘k = x − x ¯ , with x where x ¯k =0 ˙ . Fig. 3 shows the corresponding effective MIMO channel model which is depicted as a two-dimensional real-valued channel model (5). In this model, the effective channel input is the interference-
xI
˘ I,k x ¯I x x¯I,k =0 ˙ x¯Q,k =0 ˙ ¯Q x
xQ
˘ Q,k x
˘ I,k y
HI,1 + HI,2
nI HQ,2 + HQ,1 HQ,2 − HQ,1
nQ HI,1 − HI,2
˘ Q,k y
Fig. 3. Two-dimensional real-valued effective MIMO channel model with interference cancellation – Mod e` le e´ quivalent du canal MIMO avec suppression d’interf´erence
˜ x,k = Λ ˜ x˘ ,k ˘ k and the observation is y ˘ k . Therefore, the second order statistics Λx,k = Λx˘ ,k and Λ reduced signal x differ from those of the originally chosen modulation signal constellation at the transmitter due to the influence ˜ y,k = Λ ˜ y˘ ,k (see (24), (25)) of the output signal of the a priori information. The statistics Λy,k = Λy˘ ,k and Λ ˘ k depend not only on those of the input signal but also on the channel. Besides these parameters, the WL filter y ˜ yx,k = λ ˜ y˘ x,k . (22), (23) also depends on the covariance vector λyx,k = λy˘ x,k and the pseudo-covariance vector λ ∗ ˜ yx,k = 0) and if the ˘ k and xk are uncorrelated (λ Note, that the WL filter is identical to the strictly linear filter if y ˜ ˘ k is additionally circular (Λy,k = 0). observation y
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A. Second-order behavior of the effective data signal ˘ k to show In this section, we give an analysis of the second-order behavior of the effective data signal x how the statistics of the observation are affected. According to (28), (29), the covariance matrix Λ x,k and the ˜ n ) along the diagonal. These (pseudo-)covariances depend ˜ x,k have λ(xn ) and λ(x pseudo-covariance matrix Λ on the a priori information as defined in (16), (17). The a priori information causes the signal points of the complex constellation to be unequally probable and thus disables a general prediction of the second-order behavior. Nevertheless, for the considered symmetric complex modulation scheme with equally likely signal points, the special cases of zero or perfect a priori information can be treated easily. 1) Zero a priori information All signal points are uniformly distributed, i.e., P (xn = ai |Lda (cn ) = 0) = 2−M for all ai ∈ A. The soft P −M estimate is x ¯n = 2 ai ∈A ai = 0 and (16), ( 17) we can write 1 X 2 λ(xn ) = M ai = E x , (33) 2 ai ∈A X ˜ n) = 1 (34) a2I,i − a2Q,i + 2jaI,i aQ,i = 0. λ(x M 2 ai ∈A
˜ x,k = 0. Consequently, Λx,k = Ex IN and Λ 2) Perfect a priori information In this case, the transmitted symbol aj is perfectly known, i.e., P (xn = aj |Lda (cn )) = 1 and P (xn = ai |Lda (cn )) = 0 for ai ∈ A, ai 6= aj . The soft estimate is x ¯n = aj and the (pseudo-)covariances 2 2 λ(xn ) = aj − x ¯n = 0, (35) 2 2 ˜ λ(xn ) = a − x ¯ = 0 (36) j
n
˜ x,k = 0. Ex appears at the k -th position of the diagonal since xk cause Λx,k = diag(0, . . . , Ex , . . . , 0) and Λ is never affected by a priori information to obtain pure extrinsic information at the detector output (see (20)). Note, that the data signal xk which is the focus of the estimation is always circular. On the other hand the remaining symbols x ˘n , n = 1, . . . , N, n 6= k that produce the interference are circular for zero or perfect a priori information and generally non-circular during the iterations.
B. Linear channel The linear channel is characterized by H2 = 0 following from G2 = 0 associated with linear ST mappings. In this case, the (pseudo-)covariance matrices (24)-(27) simplify to 2 Λy,k = H1 Λx,k HH 1 + σ n In S n R , ˜ y,k = H1 Λ ˜ x,k HT , Λ 1
λyx,k = Ex h1,k , ˜ yx,k = 0. λ
(37) (38) (39) (40)
˜ yx,k = 0 indicates that y ˘ k and x∗k are uncorrelated. λ 1) Non-circular observation According to (38), the circular-property of the channel output signal depends on that of the input signal. A ˘ k forces y ˘ k to become non-circular, too. If so, and with (40) the WL filter (22), (23) is of the non-circular x form ˜ y,k Λ∗−1 Λ ˜ ∗ −1 λyx,k , w1,k = Λy,k − Λ (41) y,k y,k ∗−1 ∗ −1 −1 ∗ ∗ ˜ ˜ Λ λyx,k . ˜ y,k Λ Λ Λ w2,k = − Λy,k − Λ (42) y,k y,k y,k y,k ˜ y,k . ˘ k by Λ It exploits the non-circularity of the data x
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2) Circular observation ˜ x,k = 0 the observation y ˜ y,k = 0 and (40), the WL filter (22), (23) reduces ˘ k is now circular. With Λ For Λ to w1,k = Λ−1 y,k λyx,k
(43)
2 −1 = [H1 Λx,k HH 1 + σn InS nR ] Ex h1,k ,
w2,k = 0.
(44)
w1,k is the familiar adaptive linear MMSE filter [6], [8]. Hence, for a linear channel with a circular output, ˜ yx,k = 0) is referred to as joint ˜ y,k = λ the presently used linear approach is optimal. In [9], this situation (Λ ˘ k and xk . circularity of y If the ST mapping is linear, the upper and lower performance bounds determined by zero or perfect a priori information (see Section V-A) cannot be improved with the WL filter compared to the linear filter. However, we expect improved performance during the iterations.
C. Widely linear channel The widely linear channel is characterized by H2 6= 0 following from G2 6= 0 associated with widely linear ST mappings. In this case, the (pseudo-)covariance matrices (24)-(27) apply directly. Such a channel causes correlations ˜ yx,k 6= 0. ˘ k and x∗k since λ between y 1) Non-circular observation ˜ y,k from (25) clarifies that the given widely linear channel as well as a non-circular channel Investigating Λ input generally cause a non-circular observation. Even for a circular channel input, the pseudo-covariance ˜ y,k = H1 Λx,k HT + H2 Λ∗ HT generally does not disappear. Thus, the WL filter does not simplify. matrix Λ 2 x,k 1 ˜ x,k 6= 0) by both It rather retains the structure according to (22), (23) and exploits the non-circular data (Λ ˜ Λy,k and Λy,k (see (24), (25)). 2) Circular observation ˜ y,k to zero in (22), (23) we obtain the WL filter as If we assume a circular observation and set Λ w1,k = Λ−1 y,k λyx,k ,
∗
∗ ˜ w2,k = Λy,k λyx,k . −1
(45)
Note, that w1,k is identical to the standard linear filter of (43). The following example shows that a circular observation can occur in real systems. Example: An important special case is the absence of a priori information since it reflects the situation of MMSE ˜ y,k = estimation with constant filters. With (33), (34), the pseudo-covariance matrix can be written as Λ T T ∗ ∗ Ex (H1 H2 + H2 H1 ). If for instance the matrix pairs H1 , H2 and H2 , H1 have orthogonal row vectors, then ˜ y,k = 0. Examples of ST mappings that achieve such properties of the channel both pairs vanish yielding Λ matrices are EMSSTC (nT = nR = 4) and the Alamouti scheme (nT = 2, nR = 1). Interestingly, if a widely linear channel is used the WL filter does not reduce to a linear filter. This was already ˜ yx,k 6= 0) ˘ k , xk (λyx,k 6= 0) but also y ˘ k , x∗k (λ discussed in the beginning of this article in Section II. Since not only y ∗ ˘ k and y ˘ k is required. are correlated, the processing of both y VI. S IMPLIFIED WL
FILTER
Unfortunately, the symbol-wise computation of the WL filter (22), (23) causes a high computational complexity. Hence, approximations that aim at reducing the computational load are investigated. A. Approximated WL filter ˜ y,k as well as Λ ˜ x,k to zero, we neglect the non-circularity of the observation y ˘ k and also the nonIf we set Λ ˘ k . Hence, the filters are the strongly simplified ones of (45). Furthermore, (24) circularity of the data signal x 2 becomes Λy,k = H1 Λx,k HH + H2 Λ∗x,k HH 1 2 + σ n In R n S .
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As detailed in Section V-C, the approximation is optimal for zero a priori information and some specific ST ˜ y,k as well as mappings. In general, the approach is justified, because for the considered system the entries of Λ ˜ x,k take on values close to zero. those of Λ By saving a matrix inversion as well as several multiplications, this approximation achieves a significant reduction in computational complexity. Note, that for linear ST mappings the approximation is simply the linear filter of (43). B. Averaged WL filter In [29], the use of averaged covariances for the linear filter has been proposed. This approach can be adapted to the WL filter as well. ˜ x,kl from (28), Recall that ’l’ denotes one of L subblocks forming a codeword (see Section III). If so, Λx,kl and Λ (29) are the corresponding covariance and pseudo-covariance matrices, respectively. Assuming identical covariances among the subblocks, Λx,kl can be approximated by its average L
X ¯ x,k = 1 Λx,kl . Λ L
(46)
l=1
¯˜ ˜ x,kl is replaced by Λ Λ x,k = 0, since the pseudo-covariances are zero-mean within the considered system model. Thus, the non-circularity of the data signals is not taken into account. Therefore, the averaged version of the WL and the linear filter are identical for linear channels (see Section V-B). The approach reduces the number of necessary filter calculations of (22), (23) by a factor of L, resulting in just N filter computations per iteration. Furthermore, to reduce the computational complexity of the LLR-calculation, the same approximation is applied to the LLRs, too.
VII. S PACE -T IME
MAPPING
In Section III-A, we introduced the widely linear ST mapping (2) but did not yet explain the reasons behind the use of such a mapping. The answer is that it provides more degrees of freedom for the design of ST codes compared to a purely linear mapping. Linear dispersion (LD) codes [21] are able to successfully exploit these extra degrees of freedom leading to a superiority over many known ST mappings. For an arbitrary number of transmit and receive antennas, they approach the MIMO channel capacity and further provide transmit diversity. If G1 and G2 describe a LD code, the matrices are designed to maximize the average mutual information h i E 1 x T E log2 det I2nS nR + HR HR (47) C¯S = max G1 ,G2 2nS N0 (in bits/channel use) between the transmitted and received signal. The scaling factor 1/(2 · n S ) considers the nS channel uses as well as the real-valued channel matrix HR of (5). C¯S is upper bounded by the capacity of the MIMO channel. With N chosen equal to min{nT , nR } · nS , these codes achieve the maximum rate. Furthermore, G1 and G2 have to satisfy an energy constraint which is chosen such that it guarantees not only a total transmit energy of s equal to nS · ES but also a dispersion of the data symbols with identical energy in all spatial and temporal dimensions and thus provides transmit diversity [21]. Writing down optimized ST mapping matrices would not provide much insight. However, to give the reader an idea about a possible structure of a certain LD code, we explain the Enhanced Multi-Stratum Space-Time Code (EMSSTC) as an example. The structure is based on MSSTC [30], but enhanced by the use of a quasi-orthogonal Space-Time Block-Code (STBC) [31]. T T To obtain the EMSSTC, the N data symbols x = [x(1) . . . x(nL ) ]T are split up into nL groups of nS symbols x(i) , i.e., N = nL · nS . The transmitted signal is then given by s=
nL X
D(i) [G1,STBC x(i) + G2,STBC x(i) ]. ∗
(48)
i=1
The nT nS × nS matrices G1,STBC and G2,STBC generate the quasi-orthogonal STBC [31] on each of the nL layers and disperse the symbols into space and time. The block diagonal matrix D(i) = diag(di1 InT , . . . , dinS InT ) modifies
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STBC+Hadamard
(+) x1 (+)x2 (+)-x∗2 (+)x∗1
x2 x1 x
STBC+Hadamard
(+) x3 (+)x4 (−)-x∗4 (−)x∗3
x4 x3
Fig. 4.
(−x∗2 + x∗4 )(x1 + x3 )
(x∗1 − x∗3 )(x2 + x4 )
Example of EMSSTC for nT = nR = nS = 2, Ex = Es /4 – Exemple de code EMSSTC pour nT = nR = nS = 2, Ex = Es /4
the signs of the symbols according to the entries dij of an nS × nS Hadamard matrix and nL ≤ nS . To obtain the maximum rate, we choose N = min{nT , nR } · nS . Fig. 4 illustrates an example of EMSSTC for nT = nR = 2. nS = 2 symbols are encoded with the Alamouti scheme [32] on a per layer basis. After multiplication with the entries of the Hadamard matrix, superposition yields the transmitted signal. Note that, if we apply an interference cancelling receiver to eliminate the interference caused for example by the symbols x3 and x4 of the second layer, we are left with the standard Alamouti scheme where we can now exploit the full transmit diversity of two. The iterative receiver has the potential to suppress the interference. This motivates the use of ST mappings that provide transmit diversity. The 10%-outage capacities of different ST mappings can be determined by Monte-Carlo simulations (see Fig. 5). We have chosen the outage instead of the average capacity (47), because it is more meaningful for the quasi-static channel model. Comparisons of EMSSTC with strictly linear ST mappings are given. For nT = nR = 4, both the 15
nT=4,nR=2
n =n =4 R
10
C
10
in bits/channel use
T
5
MIMO channel EMSSTC V−BLAST SWITCH
0
0
5
10
ES/N0 in dB
15
20
25
Fig. 5. 10% outage capacities of different ST mappings and MIMO channels – Capacit´e au taux de perte de 10% pour diff´erents codes spatio-temporels
linear V-BLAST transmitter [22] and EMSSTC achieve the channel capacity. In contrast, for n T = 4, nR = 2, the capacities of all mappings are lower than the channel capacity. However, EMSSTC performs better than the linear switched diversity. For switched diversity, a pair of 2 out of 4 antennas is used per channel use. The antenna pairs
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are selected in a circular fashion during four channel uses. EMSSTC benefits in terms of transmit diversity by simultaneously using all antennas. VIII. S IMULATION R ESULTS In all simulations a data frame consists of 4000 information bits which are first encoded with a convolutional code (G(D) = [1 + D 3 + D4 , 1 + D + D 3 + D4 ]) and then randomly interleaved. The coded bits are modulated to complex symbols using Gray mapping. As in the literature we assumed the 1-dimensional Gaussian pdf for the soft-demapping when using the linear filter. In combination with the WL filter the two-dimensional pdf was applied according to (30). For all simulations we used the quasi-static MIMO channel model as introduced in Section III-A.2. Fig. 6 and Fig. 7 depict the frame-error rate (FER) performance over Es /N0 of the iterative receiver of iterations 0, 1, 2 and 5 for V-BLAST and of iterations 0, 1, 2 and 8 for EMSSTC. The curves are compared with the possible performance of interference-free transmission obtained by providing the detector with perfect a priori information. The V-BLAST curves converge after approximately 5 iterations and EMSSTC after 8 iterations. As simulation example we have chosen a MIMO channel with nT = nR = 4. Thus, V-BLAST transmits N = 4 symbols during nS = 1 channel use whereas for EMSSTC N = 16 and nS = 4. In both cases 16-QAM is used yielding a total rate of R = 8 bits/channel use. 0
10
It. 1
It. 2 −1
FER
10
It. 5
WL filter linear filter avrg. linear filter int.−free transmission
−2
10
−3
10
8
9
10
11
ES/N0 in dB
12
13
14
Fig. 6. V-BLAST: Performance comparison of the iterative receiver with different filters, 16-QAM, n T = nR = 4 – V-BLAST: Comparaison des performances du r´ecepteur it´eratif pour des filtres diff´erents, 16-QAM, nT = nR = 4
Fig. 6 shows how the WL detector’s performance compares to that of the standard linear detector for V-BLAST. According to Section V-B, the result of both detectors is identical for zero or perfect a priori information, i.e., in iteration 0 and in case of interference-free transmission. The gain of exploiting the non-circularity of the data with the WL detector is small in iterations 1 and 2 and afterwards hardly visible. Applying averaged covariance matrices reduces the number of necessary filter calculations by a factor of L ≈ 500 and results in a negligible performance degradation. Fig. 7 shows results for EMSSTC where we compare the full WL filter, the averaged WL filter (see Section VI-B) and the approximated WL filter (see Section VI-A). Both simplified WL filters use averaged (pseudo-)covariances. The approximated filter is optimal in iteration 0, since the data as well as the observation are circular. For the subsequent iterations a performance degradation is visible corresponding to the degree of simplifications used in
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0
10
It. 1
It. 2
−1
FER
10
It. 8
−2
10
WL filter WL filter + averaging approx. WL filter + averaging int.−free transmission
−3
10
8
9
10
11
12
E /N in dB S
13
14
15
0
Fig. 7. EMSSTC: Performance comparison of the iterative receiver with different detectors, 16-QAM, n T = nR = 4 – EMSSTC: Comparaison des performances du r´ecepteur it´eratif pour des d´etecteurs diff´erents, 16-QAM, nT = nR = 4
the filters. However, the reduction of necessary filter computations by a factor of L ≈ 125 as well as the reduction due to the approximations justify the performance loss. A comparison with the linear detector is not meaningful for EMSSTC as it would ignore the information transmitted in x∗ resulting in a FER of 1 over the whole SNR range. Our results enable us to make a performance comparison between the widely linear and the linear ST mappings which we are interested in because the former requires the more complex WL filter. Recall from Fig. 5 that both V-BLAST and EMSSTC achieve the channel capacity for nT = nR = 4. Nevertheless, the previously discussed plots show that EMSSTC is superior in terms of FER. For instance a FER of 10−2 is achieved at an SNR of approximately 13.1 dB with EMSSTC using WL filtering and at 13.7 dB with V-BLAST using linear filtering. Fig. 8 depicts iterations 0 and 8 of EMSSTC and switched diversity (see Section VII) for n T = 4, nR = 2, N = 8 and nS = 4. With 8-PSK, the total rate is R = 3 bits/channel use. Aiming at a complexity reduction we used averaged (pseudo-)covariances for both the WL filter (EMSSTC) and the linear filter (switched diversity). The results justify the approximation since both systems almost achieve their possible performance of interference-free transmission. Thus, applying a symbol-wise calculated WL filter could only lead to a benefit of a portion of a dB in both cases. Identical to the capacities of Fig. 5, EMSSTC outperforms switched diversity. For a FER of 10 −2 the gain of EMSSTC is approximately 1.8 dB in iteration 8 over the linear ST mapping. This shows that the transmit diversity provided by EMSSTC is nicely exploited with the iterative receiver. IX. C ONCLUSION We have discussed how a widely linear (WL) detector can be applied to iterative detection of MIMO signals. For linear space-time (ST) mappings the WL filter differs from the linear filter since it additionally exploits the non-circularity of the data signal arising inherently within an iterative receiver. However, simulations with VBLAST have shown only small deviations in performance between both approaches. In contrast, for widely linear ST mappings we have explained that WL filtering is necessary to consider the transmitted complex conjugates of the data. For cases where using the full WL filter is too computationally demanding, we have presented approximations in order to reduce the computational load. For EMSSTC these approximations have shown only little degradation
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0
10
It. 0
It. 8
−1
FER
10
It. 8
−2
10
EMSSTC, avrg. WL filter EMSSTC, int.−free transmission SWITCH, avrg. linear filter SWITCH, int.−free transmission
−3
10
5
6
7
8
9
10
E /N in dB S
11
12
13
14
0
Fig. 8. Performance comparison of different ST mappings, 8-PSK, n T = 4, nR = 2 – Comparaison des performances de codeurs spatio-temporels diff´erents, 8-PSK, nT = 4, nR = 2
in performance with significant reduction of computational complexity. Furthermore, comparative results of different ST mappings have shown remarkable gains in favor of the widely linear mappings compared to the linear mappings. This justifies the larger complexity of the required WL approach. R EFERENCES [1] A. Stefanov and T. M. Duman, “Turbo-coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches, and practical considerations,” IEEE J. Select. Areas Commun., vol. 19, no. 5, pp. 958–968, May 2001. [2] B. Hochwald and S. ten Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389–399, Mar. 2003. [3] A. Reial and S. G. Wilson, “Concatenated space-time coding,” in Proc. 34th Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA, USA, vol. 1. IEEE, Oct. 2000, pp. 710–715. [4] S. B¨aro, J. Hagenauer, and M. Witzke, “Iterative detection of MIMO transmission using a list-sequential (LISS) detector,” in Proc. ICC, vol. 4. Anchorage (AK), USA: IEEE, May 2003, pp. 2653–2657. [5] E. Biglieri, A. Nordio, and G. Taricco, “Suboptimum receiver interfaces and space-time codes,” IEEE Trans. Signal Processing, vol. 51, no. 11, pp. 2720–2728, Nov. 2003. [6] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, July 1999. [7] H. E. Gamal and A. R. Hammons, “A new approach to layered space-time coding and signal processing,” IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2321–2334, Sept. 2001. [8] M. Witzke, S. B¨aro, F. Schreckenbach, and J. Hagenauer, “Iterative detection of MIMO signals with linear detectors,” in Proc. 36th Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA, USA, vol. 1. IEEE, Nov. 2002, pp. 289–293. [9] B. Picinbono and P. Chevalier, “Widely linear estimation with complex data,” IEEE Trans. Signal Processing, vol. 43, no. 8, pp. 2030–2033, Aug. 1995. [10] W. M. Brown and R. B. Crane, “Conjugate linear filtering,” IEEE Trans. Inform. Theory, vol. 15, no. 4, pp. 462–465, July 1969. [11] Y. C. Yoon and H. Leib, “Maximizing SNR in improper complex noise and applications to CDMA,” IEEE Commun. Lett., vol. 1, no. 1, pp. 5–8, Jan. 1997. [12] S. Buzzi, M. Lops, and A. M. Tulino, “A new family of MMSE multiuser receivers for interference suppression in DS/CDMA systems employing BPSK modulation,” IEEE Trans. Commun., vol. 49, no. 1, pp. 154–167, Jan. 2001. [13] A. M. Tulino and S. Verd´u, “Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading,” IEEE J. Select. Areas Commun., vol. 19, no. 8, pp. 1544–1555, Aug. 2001. [14] R. Schober, W. H. Gerstacker, and L. H.-J. Lampe, “A widely linear DF-MMSE receiver for DS-CDMA with BPSK modulation,” in Proc. GLOBECOM, San Francisco, CA, USA, vol. 3. IEEE, Dec. 2003, pp. 1548–1552.
SUBMITTED TO ANNALS OF TELECOMMUNICATIONS
16
[15] G. Gelli, L. Paura, and A. R. P. Ragozini, “Blind widely linear multiuser detection,” IEEE Commun. Lett., vol. 4, no. 6, pp. 187–189, June 2000. [16] A. Lampe, R. Schober, W. Gerstacker, and J. Huber, “A novel iterative multiuser detector for complex modulation schemes,” IEEE J. Select. Areas Commun., vol. 20, no. 2, pp. 339–350, Feb. 2002. [17] F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inform. Theory, vol. 39, no. 4, pp. 1293–1302, July 1993. [18] B. Picinbono, “Second-order complex random vectors and normal distributions,” IEEE Trans. Signal Processing, vol. 44, no. 10, pp. 2637–2640, Oct. 1996. [19] B. Picinbono and P. Bondon, “Second-order statistics of complex signals,” IEEE Trans. Signal Processing, vol. 45, no. 2, pp. 411–420, Feb. 1997. [20] P. J. Schreier and L. L. Scharf, “Second-order analysis of improper complex random vectors and processes,” IEEE Trans. Signal Processing, vol. 51, no. 3, pp. 714–725, Mar. 2003. [21] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inform. Theory, vol. 48, no. 7, pp. 1804–1824, July 2002. [22] G. D. Golden, C. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electronics Letters, vol. 35, no. 1, pp. 14–16, Jan. 1999. [23] K. R. Narayanan, “Turbo decoding of concatenated space-time codes,” in Proc. 37th Annu. Allerton Conf. on Communication, Control, and Computing, Monticello, IL, USA, Sept. 1999. [24] Y. Liu and M. P. Fitz, “Space-time turbo codes,” in Proc. 37th Annu. Allerton Conf. on Communication, Control, and Computing, Monticello, IL, USA, Sept. 1999. [25] A. Stefanov and T. M. Duman, “Turbo coded modulation for wireless communications with antenna diversity,” in Proc. Veh. Techn. Conference (VTC1999-Fall), Amsterdam, The Netherlands, vol. 3. IEEE, Sept. 1999, pp. 1565–1569. [26] X. Lin and R. S. Blum, “Improved space-time codes using serial concatenation,” IEEE Commun. Lett., vol. 4, no. 7, pp. 221–223, July 2000. [27] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inform. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003. [28] P. Robertson, E. Villebrun, and P. Ho¨ her, “A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain,” in Proc. ICC, Seattle, USA, vol. 2. IEEE, June 1995, pp. 1009–1013. [29] M. T¨uchler, A. Singer, and R. K¨otter, “Minimum mean squared error equalization using a-priori information,” IEEE Trans. Signal Processing, vol. 50, no. 3, pp. 673–683, Mar. 2002. [30] U. Wachsmann, J. Thielecke, and H. Schotten, “Exploiting the data-rate potential of MIMO channels: Multi-stratum space-time coding,” in Proc. Veh. Techn. Conference (VTC2001-Spring), Rhodos, Greece, vol. 1. IEEE, May 2001, pp. 199–203. [31] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, pp. 1–4, Jan. 2001. [32] S. Alamouti, “A simple transmitter diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451–1458, 1998.
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Linear and Widely Linear Filtering Applied to Iterative Detection of Generalized MIMO Signals Melanie Witzke Institute for Communications Engineering (LNT) Munich University of Technology (TUM) 80290 Munich, Germany Email: [email protected]
Abstract To suppress the co-antenna interference in multiple input multiple output (MIMO) systems, an iterative receiver with a linear detector for complex symbols is investigated. We show that the considered generalized MIMO system, i.e., a system that transmits complex conjugate repetitions in addition to the pure data, requires the application of a widely linear (WL) detector. A WL detector consists of four real filters which are represented by two complex filters for the received signal and its complex conjugate, respectively. Furthermore, we present approximations of the detector that significantly reduce computational complexity with only little loss in frame-error rate performance. Simulation results show that the proposed MIMO system achieves large gains over standard solutions.
R´esum´e Pour supprimer l’interf´erence entre les antennes d’un syst`eme a` entr´ees et sorties multiples (MIMO), un r´ecepteur it´eratif avec un d´etecteur lin´eaire pour des symboles complexes est examin´e. Nous montrons que pour le syst`eme tr`es g´en´eral que nous consid´erons, c’est-a-dire un syst`eme qui transmet les informations elles-mˆemes ainsi que des r´ep´etitions conjugu´ees, un d´etecteur lin´eaire au sens large (widely linear, WL) est exig´e. Le d´etecteur WL comprend quatre filtres r´eels, repr´esent´es par 2 filtres complexes, un pour le signal rec¸u et un autre pour le conjugu´e du signal rec¸u. En plus, nous pr´esentons des approximations du d´etecteur qui r´eduisent fortement la complexit´e avec seulement une petite perte en taux d’erreur de bloc. Des r´esultats de simulations montrent, que le syst`eme MIMO propos´e a une meilleure performance que les solutions standards. Index Terms Widely linear estimation, iterative linear detection, MIMO systems, second-order behavior
Biography - Melanie Witzke was born in Cologne, Germany, in 1974. She received the Dipl.-Ing. degree in communications engineering from the Aachen University of Technology, Aachen, Germany, in 1998. Since Spring 1999, she has been a Research Assistant at the Institute for Communications Engineering (LNT) at the Munich University of Technology (TUM), Munich, Germany. Her main research interests are coding and detection algorithms for MIMO systems.
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I. I NTRODUCTION A multiple input multiple output (MIMO) system has the potential for delivering large data rates. However, transmission over MIMO channels suffers from strong co-antenna interference. A promising approach to cope with this problem is to use an iterative receiver performing joint symbol detection and decoding. Although the a posteriori probability (APP) detector is optimal within such a structure, its prohibitive complexity demands alternatives. Approaches that reduce the brute-force full-APP search have been the focus of increased attention. In the following we review some of the major works. In [1], the transmit antennas are partitioned into groups. For each group, the transmitted bits are estimated by nulling the interference from the other antennas. Other approaches try to find a subset of log-likelihood contributions such as [2] which takes those sequences that lie within a hyper-sphere, [3] selects the subset via a tree-based algorithm and in [4] the LISt-Sequential detector applies sequential detection based on a stack that contains a limited number of hypotheses. Instead of identifying the most likely channel input sequence, another approach is to attempt to cancel and further suppress the interference with a linear detector. Aiming at a low-complexity solution, a fixed minimum mean squared error (MMSE) filter is used in [5]. Wang and Poor [6] were the first to propose an MMSE detector that adapts to the residual interference of its input signal within an iterative receiver. If the interference is removed perfectly (as might happen after some iterations), this detector achieves the APP detector’s performance. In [7] and further generalized in [8], the application of adaptive linear filtering to MIMO transmission was presented. In this article, we discuss how the linear detector has to be modified if generalized MIMO systems are considered. In contrast to [8], these MIMO transmitters also allow complex conjugate repetitions of the data symbols. Therefore, they require the application of a widely linear (WL) filter instead of a strictly linear filter. In [9], it has been shown that WL estimation is superior or at least identical in performance to strictly linear estimation. In contrast to a linear filter, the WL filter consists of two complex filters to process both the observation and its complex conjugate. This approach (named conjugate linear filtering) was already discussed in [10] for complex signals. Recently, WL estimation found some applications that confirm the promised gains. For instance, it was used in DS-CDMA systems employing BPSK [11]-[14], it was applied to blind multiuser detection [15] and it was integrated in an iterative multiuser detector for complex data [16]. Closely related with WL estimation is the theory of improper or non-circularly symmetric signals. The second-order properties of a non-circularly symmetric signal are not fully described by its covariance. We must also consider the correlation between the signal and its complex conjugate. A comprehensive treatment of the corresponding theory is given in [17]. Other articles have continued the discussion, e.g., [18]-[20]. In a detailed discussion we will show how the WL filter adapts to the different structures of the MIMO signal. In particular, we distinguish between MIMO systems that only transmit data and those that also allow complex conjugate repetitions. The former use the WL filter to exploit the non-circularity of the noise stemming from the residual interference. The noise becomes non-circular during the iterations. The latter require WL processing to account for the transmitted complex conjugate data. During the iterations, the non-circularity of the interference is also considered. The large computational complexity of the WL detector demands the use of simplifications. We present two approaches that significantly reduce computational complexity. Simulation results show a negligible loss in performance. We motivate the application of generalized MIMO signals with the theory of the linear dispersion codes [21]. Compared to purely linear space-time mappings, e.g., the V-BLAST transmitter [22], these codes do not only approach the MIMO channel capacity but also provide transmit diversity. By eliminating the interference, the iterative, interference cancelling receiver enables us to exploit this diversity. Apart from this, there has been an immense interest in designing space-time codes. For instance, in [23]-[26] and [3] turbo codes have been adapted to MIMO systems based on serial or parallel concatenation of different constituent codes. These so-called space-time turbo codes aim at exploiting the MIMO channel capacity whilst maximizing the diversity gain. However, the design of systems that achieve the best trade-off between rate and diversity remains an open research problem. Recently, an optimal trade-off curve has been presented in [27] showing that the maximum diversity and multiplexing gain cannot be achieved simultaneously. In Section II, we motivate the use of the WL filter. The system model and some basics about second-order theory are presented in Section III. We derive the WL detector in Section IV and discuss the properties of WL filtering
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considering MIMO signals in Section V. Two simplifications of the WL filter are given in Section VI. We briefly motivate and introduce the considered space-time mappings in Section VII. Section VIII shows simulation results and conclusions are drawn in Section IX. II. M OTIVATION Before going into details, we motivate the choice of a WL detector over a linear detector. A detailed discussion will follow in Section V. Let x be a complex symbol to be estimated in terms of an observation y that is a complex random vector. Furthermore, the vector x contains the symbols x. 1) Non-circular observation: If the complex vector w denotes an MMSE filter, the linear estimate is known to be x ˆ = wH y = wH yI + jwH yQ , where (·)H is the conjugate transpose. The subscripts ’I’ and ’Q’ refer to the in-phase and quadrature components of an equivalent baseband signal. Assume the observation y is a non-circular signal. Consequently, the covariances of the in-phase and the quadrature components of y are in general different. For instance, in iterative detection, the in-phase components of the data might have more reliable a priori values than the quadrature components. Then, a single complex MMSE filter cannot be optimal for detecting both y I and yQ . This problem can be solved by using two different complex filters g1 and g2 for yI and yQ , respectively. Such an approach gives the estimate x ˆ = g1H yI + jg2H yQ , = w1H y + w2H y∗ ,
(1)
where g1 = w1 + w2 , g2 = w1 − w2 and w1 , w2 are complex filters. The operator (·)∗ denotes the complex conjugate. It is worth mentioning that if g1 = g2 then w2 = 0 and we obtain the standard linear solution. Equ. (1) has the form of the WL estimate introduced in [9]. The WL filter is not linear in C but becomes linear in R 2 with four real-valued filters. For compatibility with the existing literature (e.g., [9], [14]), we continue the discussion based on the WL filter notation in (1). Furthermore, a comparison with the strictly linear filter characterized by w2 = 0 is more obvious. 2) Correlation of observation and conjugate data symbols: Assume that the relation between y and x conforms to y = H1 x + H2 x∗ , where H1 and H2 are matrices with complex entries. In this case y and x∗ are correlated. Obviously, a single linear filter could only exploit the information transmitted in x but not that transmitted in x ∗ . Thus, the processing of both the observation and its complex conjugate is essential and leads to the WL estimate of (1). III. S YSTEM M ODEL
AND
D EFINITIONS
A. MIMO system model We consider the transmission model as depicted in Fig. 1. It consists of a MIMO transmitter and an iterative receiver that are connected through a MIMO channel with nT transmit and nR receive antennas. 1) Transmitter: The binary data is encoded with a convolutional code of rate RC to a codeword c0L containing L · N · M bits, where L denotes the number of subblocks per codeword, N the number of complex symbols per subblock and 2M is the cardinality of the complex constellation. The random interleaver permutes c0L to cL . We T T T T partition cL into L subblocks cl = [cT l1 . . . clN ] , l = 1, . . . , L, with cln = [cln1 . . . clnM ] . The operator (·) denotes the transpose. Throughout the following derivations and also in Fig. 1, we only consider one such subblock and thus drop the subscript l to simplify the notation. The vector c of N · M coded bits is mapped to a complex symbol vector x = [x1 . . . xN ]T . The elements xn = map(cn ) are taken from a 2M -ary symmetric complex symbol alphabet A = {a1 . . . a2M }, where each ai corresponds to a bit pattern [bi1 . . . biM ], bij ∈ {0, 1}. The N symbols x are transmitted during nS channel uses over the MIMO channel. To achieve this, they are mapped to a length nT · nS vector s = G 1 x + G 2 x∗ (2) T T T of transmitted signals, where s = [sT 1 . . . snS ] , with sj = [sj1 . . . sjnT ] . The nT nS ×N complex-valued space-time (ST) mapping matrices G1 and G2 disperse the symbols x into space and time. They determine the rate of the ST
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data
Encoder
4
Space−Time Mapper
; =/;@?
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23!./4/./)65.7#%$- "./$0$1#& data estimate
Fig. 1.
APP Decoder
:
WL Detector
%8 9
MIMO transmission model – Mod`ele du syst`eme de transmission MIMO
mapping as RS = N/nS symbols/channel use and thus, the total rate of the system as R = RC · M · RS bits/channel use. Examples of ST mappings are given in Section VII. In the context of this article, we call (2) a widely linear ST mapping. If G2 = 0, we use the name linear ST mapping. The total average transmit energy per channel use is Es and since the channel is unknown at the transmitter, E{|sji |2 } = Es /nT . The operator E{·} denotes expectation. During nS channel uses, the total transmitted energy is (tr E{ssH }) = nS · Es and from this the average symbol energy can be determined as n S · Es , (3) Ex = H tr(G1 GH 1 + G 2 G2 ) where Ex = E{|xn |2 } and tr(·) computes the trace of a matrix. At each time instant, the nT symbols sj are simultaneously transmitted over the nR × nT MIMO channel H yielding the observation yj = Hsj + nj . Introducing the nR nS × nT nS block diagonal channel matrix HB = diag(H, . . . , H), we can summarize the nS observations yj in one vector y = HB s + n = H1 x + H2 x∗ + n,
(4)
where y = [y1T . . . ynTS ]T , with yj = [yj1 . . . yjnR ]T . The vector n, of length nR · nS , contains independent circular complex-valued white Gaussian noise signals of variance σn2 /2 per real dimension. N0 is the one-sided power spectral density of the channel noise. H1 and H2 denote effective channel matrices consisting of a concatenation of HB and the ST mapping matrices G1 and G2 , respectively. Note, that the ST mapping matrices transform the originally linear MIMO channel into a widely linear channel (see (4)). If H1 and H2 are decomposed in their in-phase and quadrature components, (4) can be represented by a twodimensional channel model nI HI,1 + HI,2 HQ,2 − HQ,1 xI yI (5) + = nQ HQ,2 + HQ,1 HI,1 − HI,2 xQ yQ | {z } HR
with two inputs and two outputs. Equ. (5) describes the well-known real-valued equivalent baseband channel. It is a generalized version of the textbook notation and consists of four different real-valued filters that connect the in-phase and quadrature components of the input and output signals. For H2 = 0, (5) is identical to standard notation with only two different filters.
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2) Channel model: We use the typical frequency non-selective, quasi-static, MIMO channel model, meaning that the channel is randomly independently chosen for each new transmitted block. The block length is identical to the codeword length. The channel coefficients which are the entries of H are modeled by independent zero-mean complex Gaussian random variables of unit variance. 3) Receiver: The receiver is assumed to have perfect channel knowledge and consists of a WL detector that is specified in Section IV and an a posteriori probability (APP) decoder which iteratively exchange soft information about the coded bits. The APP decoder operates according to the log-MAP algorithm [28]. We describe the soft information for the coded bits using log-likelihood ratios (LLRs) in the form of a vector T L(c) = L(c1 )T . . . L(cN )T , (6) T L(cn ) = L(cn1 ) . . . L(cnM ) , (7) P (cnj = 0) . (8) L(cnj ) = ln P (cnj = 1) Using channel and a priori information (which is not available in iteration 0), the detector computes extrinsic estimates Lde (c) for the coded bits. These deinterleaved estimates are then the input Lcc (c0 ) of the APP decoder. The decoder computes a posteriori information for the information bits and for the coded bits. The former is used to make decisions and the latter, Lc (c0 ), is fed back and interleaved giving Lda (c). From Lda (c) the extrinsic information Lda,e (c) is computed. The detector is then provided with the two types of a priori informations Lda (c) and Lda,e (c). The subscript ’e’ distinguishes the extrinsic from the a posteriori information and the subscript ’a’ denotes ’a priori’ information. Feeding back the full a posteriori information violates the turbo principle but is beneficial for soft interference cancellation in MIMO systems as shown in [8]. The additional extrinsic information is used for the soft-demapping. B. Second-order theory of complex signals Second-order statistics play a key role in the course of this contribution and are therefore briefly reviewed. We consider the two complex random vectors x and y. Neeser and Massey have shown [17], that the four covariance matrices arising between the real and imaginary parts of x and y can also be expressed by the covariance matrix Λxy = cov(x, y) = E{(x − E{x})(y − E{y})H } (9) as well as by the pseudo-covariance matrix ˜ xy = cov(x, y∗ ) = E{(x − E{x})(y − E{y})T }, Λ
(10)
also often referred to as the conjugate or complementary covariance matrix. For simplicity, the autocovariance ˜ y) ˜ x , respectively. λ(x, y) and λ(x, matrix and the pseudo-autocovariance matrix will be represented by Λx and Λ denote the covariance and the pseudo-covariance of the complex symbols x and y . Accordingly, we describe the ˜ . autocovariance and the pseudo-autocovariance by λ(x) and λ(x) ˜ x corresponds to a proper / circularly In [17], it is shown that a vanishing pseudo-autocovariance matrix Λ symmetric signal x. The pseudo-autocovariance matrix ˜ x = ΛxI − ΛxQ + j(ΛxI xQ + ΛT ) Λ xI xQ
(11)
is zero if the in-phase and quadrature components of x have identical autocovariance matrices and if their crosscovariance matrix is skew-symmetric. The pseudo-autocovariance ˜ λ(x) = λ(xI ) − λ(xQ ) + 2j λ(xI , xQ )
(12)
of a single complex component x vanishes if xI and xQ have identical covariances and xI and xQ are uncorrelated. ˘ = x − E{x} are identical to the Note, that the (pseudo-)autocovariances of a centered process x (pseudo-)autocovariances of the process itself Λx = cov(x, x) = cov(x − E{x}, x − E{x}) = Λx˘ , ˜ x = cov(x, x∗ ) = cov(x − E{x}, x∗ − E{x∗ }) = Λ ˜ x˘ . Λ
This relation can easily be shown and is frequently used in the following.
(13) (14)
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IV. W IDELY L INEAR (WL) D ETECTOR A. Soft estimates and soft (pseudo-)covariances For the computation of the WL detector we need the soft estimate x ¯n = E{xn |Lda (cn )}, covariance λ(xn ) = d ∗ d ˜ n ) = cov(xn , x |L (cn )) of a symbol xn = map(cn ), stemming cov(xn , xn |La (cn )) and pseudo-covariance λ(x n a M from the 2 -ary symbol alphabet A (see Section III). They are calculated from the a priori information L da (cn ) introduced in Section III as follows X x ¯n = ai P xn = ai |Lda (cn ) , (15) a ∈A
i X ai 2 P xn = ai |Lda (cn ) −|¯ x n |2 , λ(xn ) =
(16)
ai ∈A
˜ n) = λ(x
X
a2i P xn = ai |Lda (cn )
ai ∈A
−¯ x2n .
(17)
Independent bits can be assumed due to the interleaver, therefore P xn =
ai |Lda (cn )
=
M Y
j=1
P cnj = bij |Lda (cnj ) .
(18)
P (cnj ) itself is determined by the a priori LLR Lda (cnj ) according to the well-known relation P (c = 0) =
eL(c) , 1 + eL(c)
P (c = 1) =
1 . 1 + eL(c)
(19)
Note, that only the full a posteriori feedback Lda is involved in the evaluation of (15)-(17). B. Derivation of WL detector The WL detector of Fig. 1 is shown in more detail in Fig. 2. It is composed of three functional units which are interference cancellation, WL filtering and soft-demapping to compute the extrinsic estimates of the coded bits. In the following we let the subscripts k and m have fixed values. k denotes the k -th symbol of N symbols and m one bit out of a group of M bits.
w1,k
˘k y
y
(·)
∗
˘ k∗ y
xˆk
Lde (c)
w2,k
Recon− struction
Lda,e (c)
xk =0 ˙
x Fig. 2.
L(·)
Lda (c)
E{·}
Widely linear (WL) detector – D´etecteur lin´eaire au sens large
1) Interference cancellation: Keeping in mind that our overall aim is the estimation of the k -th complex symbol ¯ = [¯ x ˆk , we first consider soft interference cancellation. Using the vector x x1 . . . x ¯N ]T of soft estimates from (15), the interference-reduced signal is ¯∗ ¯ + H2 x ˘ k = y − H1 x . (20) y x ¯k =0 ˙
Forcing x ¯k to zero is necessary to ensure pure extrinsic information at the detector output [6], [8].
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2) WL filtering: In order to further suppress residual interference, the signal is filtered. As will be shown in the following section, the application of the two complex filters w1,k and w2,k is in general beneficial. This yields the WL estimate H H ˘ k + w2,k ˘ k∗ , x ˆk = w1,k y y (21) ∗ ∗ where wf,k = [wf,k,1 . . . wf,k,n ]T , f = 1, 2. The two filters are chosen to minimize the mean squared error R nS E{|xk − x ˆk |2 |Lda (c)}. In [9] the solution to this problem is derived from the orthogonality principle giving the two filters h i ˜∗ ˜ y,k Λ∗−1 λ λ − Λ (22) w1,k = Γ−1 yx,k y,k yx,k , k i h −1 ˜∗ − Λ ˜ ∗ Λ−1 λyx,k , (23) w2,k = Γk∗ λ yx,k y,k y,k ˜ y,k Λ∗−1 Λ ˜ y,k = E{˘ ˜∗ ˘ kT }, λyx,k = E{˘ ˘ kH }, Λ yk x∗k } and where Γk = Λy,k − Λ yk y yk y y,k y,k . Λy,k = E{˘ ˜ yx,k = E{˘ λ yk xk } denote the covariance matrix, the pseudo-covariance matrix, the covariance vector and the pseudo-covariance vector, respectively. For the widely linear MIMO channel from (4), the evaluation of the filter components yields ∗ H H H 2 ˜ ˜∗ Λy,k = H1 Λx,k HH 1 + H2 Λx,k H2 + H1 Λx,k H2 + H2 Λx,k H1 + σn InR nS , ˜ y,k = H1 Λ ˜ x,k HT + H2 Λ ˜ ∗ HT + H1 Λx,k HT + H2 Λ∗ HT , Λ 1 2 x,k 2 x,k 1
λyx,k = Ex h1,k , ˜ yx,k = Ex h2,k . λ
(24) (25) (26) (27)
Ij is the j × j identity matrix, h1,k and h2,k are the k -th column of H1 and H2 , respectively, and Ex = E{|xk |2 }. Assuming independent symbols x we obtain the covariance matrix and the pseudo-covariance matrix as Λx,k = diag(λ(x1 ) . . . λ(xN )), ˜ 1 ) . . . λ(x ˜ N )), ˜ x,k = diag(λ(x Λ
with λ(xk ) = Ex , ˜ k ) = 0, with λ(x
(28) (29)
with the covariances and the pseudo-covariances from (16), (17) along the diagonal. The constraint from (20) yields ˜ k ) = 0 (see Section V-A). λ(xk ) = Ex and λ(x 3) Soft-demapping: The extrinsic LLR, i.e., the difference between the a posteriori and the a priori LLR, of the m-th coded bit of x ˆk is defined by the expression Lde (ckm ) = ln
= ln
p(ˆ xk |ckm = 0) p(ˆ xk |ckm = 1) P p(ˆ xk |xk = ai )
(30) M Q
P (ckj = bij |Lda,e (ckj ))
ai ∈A:ckm =0
j=1,j6=m
P
M Q
p(ˆ xk |xk = ai )
ai ∈A:ckm =1
P (ckj =
j=1,j6=m
, bij |Lda,e (ckj ))
where the product assumes independence of the bits within one symbol because of interleaving. The probability density function (pdf) p(ˆ xk |xk = ai ) is approximated by the Gaussian distribution and the probabilities P (ckj = bij |Lda,e (ckj )) are determined by Lda,e (ckj ) according to (19). In contrast to (15)-(17), the use of the extrinsic LLRs ˘ k the is mandatory here to guarantee pure extrinsic information in (30). Note, that in general for non-circular y estimate x ˆk is also non-circular. Thus, the two-dimensional Gaussian pdf p(ˆ xI,k , x ˆQ,k |xk = ai ) should be used. This 2 = cov(ˆ pdf is completely determined by the mean µki = E{ˆ xk |xk = ai } and the covariances σI,k xI,k , x ˆI,k |xk = ai ), 2 2 σQ,k = cov(ˆ xQ,k , x ˆQ,k |xk = ai ) and σIQ,k = cov(ˆ xI,k , x ˆQ,k |xk = ai ). Due to independence of the covariances on a certain symbol ai , the subscript ’i’ is neglected. Evaluating the expectation gives the mean of x ˆk given that the symbol ai was transmitted which can be written as H H H H µki = [w1,k (h1,k + h2,k ) + w2,k (h1,k + h2,k )∗ ] aI,i + j [w1,k (h1,k − h2,k ) − w2,k (h1,k − h2,k )∗ ] aQ,i . | {z } {z } | m1,k
m2,k
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Note, that for a non-circular x ˆk , the scaling factors m1,k and m2,k are different and complex-valued whereas for a circular signal they are identical and real-valued. If we express the in-phase and quadrature component of x ˆk as follows 1 1 H T ˘ k + g1,k ˘ k∗ ), (ˆ xk + x ˆ∗k ) = (g1,k x ˆI,k = y y 2 2 1 H 1 T ˘ k − g2,k ˘ k∗ ), xk − x ˆ∗k ) = −j (g2,k y y x ˆQ,k = −j (ˆ 2 2 ∗ and g ∗ with the two vectors g1,k = w1,k + w2,k 2,k = w1,k − w2,k , the covariances of the noise (interference and channel noise) can be derived as o 1 n H H ˜ ∗ 2 Λy,k g1,k + g1,k Λy,k g1,k , σI,k = < g1,k 2 n o 1 H H ˜ ∗ 2 (31) Λy,k g2,k − g2,k Λy,k g2,k , σQ,k = < g2,k 2 n o 1 2 H H ˜ ∗ σIQ,k = = −g1,k Λy,k g2,k + g1,k Λy,k g2,k . 2 ˜ y,k are from (24), (25), where now the k -th component of Λx,k and Λ ˜ x,k from (28), (29) takes on the Λy,k and Λ ˜ k ) = 0. value λ(xk ) = λ(x V. P ROPERTIES OF MIMO
SIGNALS AND OF
WL
FILTERING
After introducing the mathematics on widely linear estimation, we proceed to discuss the properties of the MIMO signals and show how the WL filter adapts to these signals. To provide an insight into the individual effects that influence the WL estimation, the interference reduced signal from (20) is rewritten as ˘ k = H1 (x − x ¯ ) + H2 (x − x ¯ )∗ + n y x ¯k =0 ˙
˘ k + H2 x ˘ ∗k + n, = H1 x
(32)
˘k = x − x ¯ , with x where x ¯k =0 ˙ . Fig. 3 shows the corresponding effective MIMO channel model which is depicted as a two-dimensional real-valued channel model (5). In this model, the effective channel input is the interference-
xI
˘ I,k x ¯I x x¯I,k =0 ˙ x¯Q,k =0 ˙ ¯Q x
xQ
˘ Q,k x
˘ I,k y
HI,1 + HI,2
nI HQ,2 + HQ,1 HQ,2 − HQ,1
nQ HI,1 − HI,2
˘ Q,k y
Fig. 3. Two-dimensional real-valued effective MIMO channel model with interference cancellation – Mod e` le e´ quivalent du canal MIMO avec suppression d’interf´erence
˜ x,k = Λ ˜ x˘ ,k ˘ k and the observation is y ˘ k . Therefore, the second order statistics Λx,k = Λx˘ ,k and Λ reduced signal x differ from those of the originally chosen modulation signal constellation at the transmitter due to the influence ˜ y,k = Λ ˜ y˘ ,k (see (24), (25)) of the output signal of the a priori information. The statistics Λy,k = Λy˘ ,k and Λ ˘ k depend not only on those of the input signal but also on the channel. Besides these parameters, the WL filter y ˜ yx,k = λ ˜ y˘ x,k . (22), (23) also depends on the covariance vector λyx,k = λy˘ x,k and the pseudo-covariance vector λ ∗ ˜ yx,k = 0) and if the ˘ k and xk are uncorrelated (λ Note, that the WL filter is identical to the strictly linear filter if y ˜ ˘ k is additionally circular (Λy,k = 0). observation y
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A. Second-order behavior of the effective data signal ˘ k to show In this section, we give an analysis of the second-order behavior of the effective data signal x how the statistics of the observation are affected. According to (28), (29), the covariance matrix Λ x,k and the ˜ n ) along the diagonal. These (pseudo-)covariances depend ˜ x,k have λ(xn ) and λ(x pseudo-covariance matrix Λ on the a priori information as defined in (16), (17). The a priori information causes the signal points of the complex constellation to be unequally probable and thus disables a general prediction of the second-order behavior. Nevertheless, for the considered symmetric complex modulation scheme with equally likely signal points, the special cases of zero or perfect a priori information can be treated easily. 1) Zero a priori information All signal points are uniformly distributed, i.e., P (xn = ai |Lda (cn ) = 0) = 2−M for all ai ∈ A. The soft P −M estimate is x ¯n = 2 ai ∈A ai = 0 and (16), ( 17) we can write 1 X 2 λ(xn ) = M ai = E x , (33) 2 ai ∈A X ˜ n) = 1 (34) a2I,i − a2Q,i + 2jaI,i aQ,i = 0. λ(x M 2 ai ∈A
˜ x,k = 0. Consequently, Λx,k = Ex IN and Λ 2) Perfect a priori information In this case, the transmitted symbol aj is perfectly known, i.e., P (xn = aj |Lda (cn )) = 1 and P (xn = ai |Lda (cn )) = 0 for ai ∈ A, ai 6= aj . The soft estimate is x ¯n = aj and the (pseudo-)covariances 2 2 λ(xn ) = aj − x ¯n = 0, (35) 2 2 ˜ λ(xn ) = a − x ¯ = 0 (36) j
n
˜ x,k = 0. Ex appears at the k -th position of the diagonal since xk cause Λx,k = diag(0, . . . , Ex , . . . , 0) and Λ is never affected by a priori information to obtain pure extrinsic information at the detector output (see (20)). Note, that the data signal xk which is the focus of the estimation is always circular. On the other hand the remaining symbols x ˘n , n = 1, . . . , N, n 6= k that produce the interference are circular for zero or perfect a priori information and generally non-circular during the iterations.
B. Linear channel The linear channel is characterized by H2 = 0 following from G2 = 0 associated with linear ST mappings. In this case, the (pseudo-)covariance matrices (24)-(27) simplify to 2 Λy,k = H1 Λx,k HH 1 + σ n In S n R , ˜ y,k = H1 Λ ˜ x,k HT , Λ 1
λyx,k = Ex h1,k , ˜ yx,k = 0. λ
(37) (38) (39) (40)
˜ yx,k = 0 indicates that y ˘ k and x∗k are uncorrelated. λ 1) Non-circular observation According to (38), the circular-property of the channel output signal depends on that of the input signal. A ˘ k forces y ˘ k to become non-circular, too. If so, and with (40) the WL filter (22), (23) is of the non-circular x form ˜ y,k Λ∗−1 Λ ˜ ∗ −1 λyx,k , w1,k = Λy,k − Λ (41) y,k y,k ∗−1 ∗ −1 −1 ∗ ∗ ˜ ˜ Λ λyx,k . ˜ y,k Λ Λ Λ w2,k = − Λy,k − Λ (42) y,k y,k y,k y,k ˜ y,k . ˘ k by Λ It exploits the non-circularity of the data x
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2) Circular observation ˜ x,k = 0 the observation y ˜ y,k = 0 and (40), the WL filter (22), (23) reduces ˘ k is now circular. With Λ For Λ to w1,k = Λ−1 y,k λyx,k
(43)
2 −1 = [H1 Λx,k HH 1 + σn InS nR ] Ex h1,k ,
w2,k = 0.
(44)
w1,k is the familiar adaptive linear MMSE filter [6], [8]. Hence, for a linear channel with a circular output, ˜ yx,k = 0) is referred to as joint ˜ y,k = λ the presently used linear approach is optimal. In [9], this situation (Λ ˘ k and xk . circularity of y If the ST mapping is linear, the upper and lower performance bounds determined by zero or perfect a priori information (see Section V-A) cannot be improved with the WL filter compared to the linear filter. However, we expect improved performance during the iterations.
C. Widely linear channel The widely linear channel is characterized by H2 6= 0 following from G2 6= 0 associated with widely linear ST mappings. In this case, the (pseudo-)covariance matrices (24)-(27) apply directly. Such a channel causes correlations ˜ yx,k 6= 0. ˘ k and x∗k since λ between y 1) Non-circular observation ˜ y,k from (25) clarifies that the given widely linear channel as well as a non-circular channel Investigating Λ input generally cause a non-circular observation. Even for a circular channel input, the pseudo-covariance ˜ y,k = H1 Λx,k HT + H2 Λ∗ HT generally does not disappear. Thus, the WL filter does not simplify. matrix Λ 2 x,k 1 ˜ x,k 6= 0) by both It rather retains the structure according to (22), (23) and exploits the non-circular data (Λ ˜ Λy,k and Λy,k (see (24), (25)). 2) Circular observation ˜ y,k to zero in (22), (23) we obtain the WL filter as If we assume a circular observation and set Λ w1,k = Λ−1 y,k λyx,k ,
∗
∗ ˜ w2,k = Λy,k λyx,k . −1
(45)
Note, that w1,k is identical to the standard linear filter of (43). The following example shows that a circular observation can occur in real systems. Example: An important special case is the absence of a priori information since it reflects the situation of MMSE ˜ y,k = estimation with constant filters. With (33), (34), the pseudo-covariance matrix can be written as Λ T T ∗ ∗ Ex (H1 H2 + H2 H1 ). If for instance the matrix pairs H1 , H2 and H2 , H1 have orthogonal row vectors, then ˜ y,k = 0. Examples of ST mappings that achieve such properties of the channel both pairs vanish yielding Λ matrices are EMSSTC (nT = nR = 4) and the Alamouti scheme (nT = 2, nR = 1). Interestingly, if a widely linear channel is used the WL filter does not reduce to a linear filter. This was already ˜ yx,k 6= 0) ˘ k , xk (λyx,k 6= 0) but also y ˘ k , x∗k (λ discussed in the beginning of this article in Section II. Since not only y ∗ ˘ k and y ˘ k is required. are correlated, the processing of both y VI. S IMPLIFIED WL
FILTER
Unfortunately, the symbol-wise computation of the WL filter (22), (23) causes a high computational complexity. Hence, approximations that aim at reducing the computational load are investigated. A. Approximated WL filter ˜ y,k as well as Λ ˜ x,k to zero, we neglect the non-circularity of the observation y ˘ k and also the nonIf we set Λ ˘ k . Hence, the filters are the strongly simplified ones of (45). Furthermore, (24) circularity of the data signal x 2 becomes Λy,k = H1 Λx,k HH + H2 Λ∗x,k HH 1 2 + σ n In R n S .
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As detailed in Section V-C, the approximation is optimal for zero a priori information and some specific ST ˜ y,k as well as mappings. In general, the approach is justified, because for the considered system the entries of Λ ˜ x,k take on values close to zero. those of Λ By saving a matrix inversion as well as several multiplications, this approximation achieves a significant reduction in computational complexity. Note, that for linear ST mappings the approximation is simply the linear filter of (43). B. Averaged WL filter In [29], the use of averaged covariances for the linear filter has been proposed. This approach can be adapted to the WL filter as well. ˜ x,kl from (28), Recall that ’l’ denotes one of L subblocks forming a codeword (see Section III). If so, Λx,kl and Λ (29) are the corresponding covariance and pseudo-covariance matrices, respectively. Assuming identical covariances among the subblocks, Λx,kl can be approximated by its average L
X ¯ x,k = 1 Λx,kl . Λ L
(46)
l=1
¯˜ ˜ x,kl is replaced by Λ Λ x,k = 0, since the pseudo-covariances are zero-mean within the considered system model. Thus, the non-circularity of the data signals is not taken into account. Therefore, the averaged version of the WL and the linear filter are identical for linear channels (see Section V-B). The approach reduces the number of necessary filter calculations of (22), (23) by a factor of L, resulting in just N filter computations per iteration. Furthermore, to reduce the computational complexity of the LLR-calculation, the same approximation is applied to the LLRs, too.
VII. S PACE -T IME
MAPPING
In Section III-A, we introduced the widely linear ST mapping (2) but did not yet explain the reasons behind the use of such a mapping. The answer is that it provides more degrees of freedom for the design of ST codes compared to a purely linear mapping. Linear dispersion (LD) codes [21] are able to successfully exploit these extra degrees of freedom leading to a superiority over many known ST mappings. For an arbitrary number of transmit and receive antennas, they approach the MIMO channel capacity and further provide transmit diversity. If G1 and G2 describe a LD code, the matrices are designed to maximize the average mutual information h i E 1 x T E log2 det I2nS nR + HR HR (47) C¯S = max G1 ,G2 2nS N0 (in bits/channel use) between the transmitted and received signal. The scaling factor 1/(2 · n S ) considers the nS channel uses as well as the real-valued channel matrix HR of (5). C¯S is upper bounded by the capacity of the MIMO channel. With N chosen equal to min{nT , nR } · nS , these codes achieve the maximum rate. Furthermore, G1 and G2 have to satisfy an energy constraint which is chosen such that it guarantees not only a total transmit energy of s equal to nS · ES but also a dispersion of the data symbols with identical energy in all spatial and temporal dimensions and thus provides transmit diversity [21]. Writing down optimized ST mapping matrices would not provide much insight. However, to give the reader an idea about a possible structure of a certain LD code, we explain the Enhanced Multi-Stratum Space-Time Code (EMSSTC) as an example. The structure is based on MSSTC [30], but enhanced by the use of a quasi-orthogonal Space-Time Block-Code (STBC) [31]. T T To obtain the EMSSTC, the N data symbols x = [x(1) . . . x(nL ) ]T are split up into nL groups of nS symbols x(i) , i.e., N = nL · nS . The transmitted signal is then given by s=
nL X
D(i) [G1,STBC x(i) + G2,STBC x(i) ]. ∗
(48)
i=1
The nT nS × nS matrices G1,STBC and G2,STBC generate the quasi-orthogonal STBC [31] on each of the nL layers and disperse the symbols into space and time. The block diagonal matrix D(i) = diag(di1 InT , . . . , dinS InT ) modifies
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STBC+Hadamard
(+) x1 (+)x2 (+)-x∗2 (+)x∗1
x2 x1 x
STBC+Hadamard
(+) x3 (+)x4 (−)-x∗4 (−)x∗3
x4 x3
Fig. 4.
(−x∗2 + x∗4 )(x1 + x3 )
(x∗1 − x∗3 )(x2 + x4 )
Example of EMSSTC for nT = nR = nS = 2, Ex = Es /4 – Exemple de code EMSSTC pour nT = nR = nS = 2, Ex = Es /4
the signs of the symbols according to the entries dij of an nS × nS Hadamard matrix and nL ≤ nS . To obtain the maximum rate, we choose N = min{nT , nR } · nS . Fig. 4 illustrates an example of EMSSTC for nT = nR = 2. nS = 2 symbols are encoded with the Alamouti scheme [32] on a per layer basis. After multiplication with the entries of the Hadamard matrix, superposition yields the transmitted signal. Note that, if we apply an interference cancelling receiver to eliminate the interference caused for example by the symbols x3 and x4 of the second layer, we are left with the standard Alamouti scheme where we can now exploit the full transmit diversity of two. The iterative receiver has the potential to suppress the interference. This motivates the use of ST mappings that provide transmit diversity. The 10%-outage capacities of different ST mappings can be determined by Monte-Carlo simulations (see Fig. 5). We have chosen the outage instead of the average capacity (47), because it is more meaningful for the quasi-static channel model. Comparisons of EMSSTC with strictly linear ST mappings are given. For nT = nR = 4, both the 15
nT=4,nR=2
n =n =4 R
10
C
10
in bits/channel use
T
5
MIMO channel EMSSTC V−BLAST SWITCH
0
0
5
10
ES/N0 in dB
15
20
25
Fig. 5. 10% outage capacities of different ST mappings and MIMO channels – Capacit´e au taux de perte de 10% pour diff´erents codes spatio-temporels
linear V-BLAST transmitter [22] and EMSSTC achieve the channel capacity. In contrast, for n T = 4, nR = 2, the capacities of all mappings are lower than the channel capacity. However, EMSSTC performs better than the linear switched diversity. For switched diversity, a pair of 2 out of 4 antennas is used per channel use. The antenna pairs
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are selected in a circular fashion during four channel uses. EMSSTC benefits in terms of transmit diversity by simultaneously using all antennas. VIII. S IMULATION R ESULTS In all simulations a data frame consists of 4000 information bits which are first encoded with a convolutional code (G(D) = [1 + D 3 + D4 , 1 + D + D 3 + D4 ]) and then randomly interleaved. The coded bits are modulated to complex symbols using Gray mapping. As in the literature we assumed the 1-dimensional Gaussian pdf for the soft-demapping when using the linear filter. In combination with the WL filter the two-dimensional pdf was applied according to (30). For all simulations we used the quasi-static MIMO channel model as introduced in Section III-A.2. Fig. 6 and Fig. 7 depict the frame-error rate (FER) performance over Es /N0 of the iterative receiver of iterations 0, 1, 2 and 5 for V-BLAST and of iterations 0, 1, 2 and 8 for EMSSTC. The curves are compared with the possible performance of interference-free transmission obtained by providing the detector with perfect a priori information. The V-BLAST curves converge after approximately 5 iterations and EMSSTC after 8 iterations. As simulation example we have chosen a MIMO channel with nT = nR = 4. Thus, V-BLAST transmits N = 4 symbols during nS = 1 channel use whereas for EMSSTC N = 16 and nS = 4. In both cases 16-QAM is used yielding a total rate of R = 8 bits/channel use. 0
10
It. 1
It. 2 −1
FER
10
It. 5
WL filter linear filter avrg. linear filter int.−free transmission
−2
10
−3
10
8
9
10
11
ES/N0 in dB
12
13
14
Fig. 6. V-BLAST: Performance comparison of the iterative receiver with different filters, 16-QAM, n T = nR = 4 – V-BLAST: Comparaison des performances du r´ecepteur it´eratif pour des filtres diff´erents, 16-QAM, nT = nR = 4
Fig. 6 shows how the WL detector’s performance compares to that of the standard linear detector for V-BLAST. According to Section V-B, the result of both detectors is identical for zero or perfect a priori information, i.e., in iteration 0 and in case of interference-free transmission. The gain of exploiting the non-circularity of the data with the WL detector is small in iterations 1 and 2 and afterwards hardly visible. Applying averaged covariance matrices reduces the number of necessary filter calculations by a factor of L ≈ 500 and results in a negligible performance degradation. Fig. 7 shows results for EMSSTC where we compare the full WL filter, the averaged WL filter (see Section VI-B) and the approximated WL filter (see Section VI-A). Both simplified WL filters use averaged (pseudo-)covariances. The approximated filter is optimal in iteration 0, since the data as well as the observation are circular. For the subsequent iterations a performance degradation is visible corresponding to the degree of simplifications used in
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0
10
It. 1
It. 2
−1
FER
10
It. 8
−2
10
WL filter WL filter + averaging approx. WL filter + averaging int.−free transmission
−3
10
8
9
10
11
12
E /N in dB S
13
14
15
0
Fig. 7. EMSSTC: Performance comparison of the iterative receiver with different detectors, 16-QAM, n T = nR = 4 – EMSSTC: Comparaison des performances du r´ecepteur it´eratif pour des d´etecteurs diff´erents, 16-QAM, nT = nR = 4
the filters. However, the reduction of necessary filter computations by a factor of L ≈ 125 as well as the reduction due to the approximations justify the performance loss. A comparison with the linear detector is not meaningful for EMSSTC as it would ignore the information transmitted in x∗ resulting in a FER of 1 over the whole SNR range. Our results enable us to make a performance comparison between the widely linear and the linear ST mappings which we are interested in because the former requires the more complex WL filter. Recall from Fig. 5 that both V-BLAST and EMSSTC achieve the channel capacity for nT = nR = 4. Nevertheless, the previously discussed plots show that EMSSTC is superior in terms of FER. For instance a FER of 10−2 is achieved at an SNR of approximately 13.1 dB with EMSSTC using WL filtering and at 13.7 dB with V-BLAST using linear filtering. Fig. 8 depicts iterations 0 and 8 of EMSSTC and switched diversity (see Section VII) for n T = 4, nR = 2, N = 8 and nS = 4. With 8-PSK, the total rate is R = 3 bits/channel use. Aiming at a complexity reduction we used averaged (pseudo-)covariances for both the WL filter (EMSSTC) and the linear filter (switched diversity). The results justify the approximation since both systems almost achieve their possible performance of interference-free transmission. Thus, applying a symbol-wise calculated WL filter could only lead to a benefit of a portion of a dB in both cases. Identical to the capacities of Fig. 5, EMSSTC outperforms switched diversity. For a FER of 10 −2 the gain of EMSSTC is approximately 1.8 dB in iteration 8 over the linear ST mapping. This shows that the transmit diversity provided by EMSSTC is nicely exploited with the iterative receiver. IX. C ONCLUSION We have discussed how a widely linear (WL) detector can be applied to iterative detection of MIMO signals. For linear space-time (ST) mappings the WL filter differs from the linear filter since it additionally exploits the non-circularity of the data signal arising inherently within an iterative receiver. However, simulations with VBLAST have shown only small deviations in performance between both approaches. In contrast, for widely linear ST mappings we have explained that WL filtering is necessary to consider the transmitted complex conjugates of the data. For cases where using the full WL filter is too computationally demanding, we have presented approximations in order to reduce the computational load. For EMSSTC these approximations have shown only little degradation
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0
10
It. 0
It. 8
−1
FER
10
It. 8
−2
10
EMSSTC, avrg. WL filter EMSSTC, int.−free transmission SWITCH, avrg. linear filter SWITCH, int.−free transmission
−3
10
5
6
7
8
9
10
E /N in dB S
11
12
13
14
0
Fig. 8. Performance comparison of different ST mappings, 8-PSK, n T = 4, nR = 2 – Comparaison des performances de codeurs spatio-temporels diff´erents, 8-PSK, nT = 4, nR = 2
in performance with significant reduction of computational complexity. Furthermore, comparative results of different ST mappings have shown remarkable gains in favor of the widely linear mappings compared to the linear mappings. This justifies the larger complexity of the required WL approach. R EFERENCES [1] A. Stefanov and T. M. Duman, “Turbo-coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches, and practical considerations,” IEEE J. Select. Areas Commun., vol. 19, no. 5, pp. 958–968, May 2001. [2] B. Hochwald and S. ten Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389–399, Mar. 2003. [3] A. Reial and S. G. Wilson, “Concatenated space-time coding,” in Proc. 34th Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA, USA, vol. 1. IEEE, Oct. 2000, pp. 710–715. [4] S. B¨aro, J. Hagenauer, and M. Witzke, “Iterative detection of MIMO transmission using a list-sequential (LISS) detector,” in Proc. ICC, vol. 4. Anchorage (AK), USA: IEEE, May 2003, pp. 2653–2657. [5] E. Biglieri, A. Nordio, and G. Taricco, “Suboptimum receiver interfaces and space-time codes,” IEEE Trans. Signal Processing, vol. 51, no. 11, pp. 2720–2728, Nov. 2003. [6] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, July 1999. [7] H. E. Gamal and A. R. Hammons, “A new approach to layered space-time coding and signal processing,” IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2321–2334, Sept. 2001. [8] M. Witzke, S. B¨aro, F. Schreckenbach, and J. Hagenauer, “Iterative detection of MIMO signals with linear detectors,” in Proc. 36th Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA, USA, vol. 1. IEEE, Nov. 2002, pp. 289–293. [9] B. Picinbono and P. Chevalier, “Widely linear estimation with complex data,” IEEE Trans. Signal Processing, vol. 43, no. 8, pp. 2030–2033, Aug. 1995. [10] W. M. Brown and R. B. Crane, “Conjugate linear filtering,” IEEE Trans. Inform. Theory, vol. 15, no. 4, pp. 462–465, July 1969. [11] Y. C. Yoon and H. Leib, “Maximizing SNR in improper complex noise and applications to CDMA,” IEEE Commun. Lett., vol. 1, no. 1, pp. 5–8, Jan. 1997. [12] S. Buzzi, M. Lops, and A. M. Tulino, “A new family of MMSE multiuser receivers for interference suppression in DS/CDMA systems employing BPSK modulation,” IEEE Trans. Commun., vol. 49, no. 1, pp. 154–167, Jan. 2001. [13] A. M. Tulino and S. Verd´u, “Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading,” IEEE J. Select. Areas Commun., vol. 19, no. 8, pp. 1544–1555, Aug. 2001. [14] R. Schober, W. H. Gerstacker, and L. H.-J. Lampe, “A widely linear DF-MMSE receiver for DS-CDMA with BPSK modulation,” in Proc. GLOBECOM, San Francisco, CA, USA, vol. 3. IEEE, Dec. 2003, pp. 1548–1552.
SUBMITTED TO ANNALS OF TELECOMMUNICATIONS
16
[15] G. Gelli, L. Paura, and A. R. P. Ragozini, “Blind widely linear multiuser detection,” IEEE Commun. Lett., vol. 4, no. 6, pp. 187–189, June 2000. [16] A. Lampe, R. Schober, W. Gerstacker, and J. Huber, “A novel iterative multiuser detector for complex modulation schemes,” IEEE J. Select. Areas Commun., vol. 20, no. 2, pp. 339–350, Feb. 2002. [17] F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inform. Theory, vol. 39, no. 4, pp. 1293–1302, July 1993. [18] B. Picinbono, “Second-order complex random vectors and normal distributions,” IEEE Trans. Signal Processing, vol. 44, no. 10, pp. 2637–2640, Oct. 1996. [19] B. Picinbono and P. Bondon, “Second-order statistics of complex signals,” IEEE Trans. Signal Processing, vol. 45, no. 2, pp. 411–420, Feb. 1997. [20] P. J. Schreier and L. L. Scharf, “Second-order analysis of improper complex random vectors and processes,” IEEE Trans. Signal Processing, vol. 51, no. 3, pp. 714–725, Mar. 2003. [21] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inform. Theory, vol. 48, no. 7, pp. 1804–1824, July 2002. [22] G. D. Golden, C. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electronics Letters, vol. 35, no. 1, pp. 14–16, Jan. 1999. [23] K. R. Narayanan, “Turbo decoding of concatenated space-time codes,” in Proc. 37th Annu. Allerton Conf. on Communication, Control, and Computing, Monticello, IL, USA, Sept. 1999. [24] Y. Liu and M. P. Fitz, “Space-time turbo codes,” in Proc. 37th Annu. Allerton Conf. on Communication, Control, and Computing, Monticello, IL, USA, Sept. 1999. [25] A. Stefanov and T. M. Duman, “Turbo coded modulation for wireless communications with antenna diversity,” in Proc. Veh. Techn. Conference (VTC1999-Fall), Amsterdam, The Netherlands, vol. 3. IEEE, Sept. 1999, pp. 1565–1569. [26] X. Lin and R. S. Blum, “Improved space-time codes using serial concatenation,” IEEE Commun. Lett., vol. 4, no. 7, pp. 221–223, July 2000. [27] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inform. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003. [28] P. Robertson, E. Villebrun, and P. Ho¨ her, “A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain,” in Proc. ICC, Seattle, USA, vol. 2. IEEE, June 1995, pp. 1009–1013. [29] M. T¨uchler, A. Singer, and R. K¨otter, “Minimum mean squared error equalization using a-priori information,” IEEE Trans. Signal Processing, vol. 50, no. 3, pp. 673–683, Mar. 2002. [30] U. Wachsmann, J. Thielecke, and H. Schotten, “Exploiting the data-rate potential of MIMO channels: Multi-stratum space-time coding,” in Proc. Veh. Techn. Conference (VTC2001-Spring), Rhodos, Greece, vol. 1. IEEE, May 2001, pp. 199–203. [31] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, pp. 1–4, Jan. 2001. [32] S. Alamouti, “A simple transmitter diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451–1458, 1998.
