Iterative MIMO Turbo Multiuser Detection and Equalization for STTrC-Coded Systems with Unknown Interference

Iterative multiuser detection in a single-carrier broadband multiple-input multiple-output (MIMO) system is studied in this paper. A minimum mean squared error (MMSE) low-complexity multiuser receiver is derived for space-division multiple-access (SDMA) space-time trellis-coded (STTrC) systems in frequency-selective fading channels. The receiver uses MMSE ﬁltering to jointly detect several transmit antennas of the user of interest, while the interference from the undetected transmit antennas, cochannel interference (CCI), and intersymbol interference (ISI) are all cancelled by the soft cancellation. The performances of two extreme receiver cases are evaluated. In the ﬁrst case, only one transmit antenna of the user of interest is detected at a time and the remaining ones are cancelled by soft cancellation. In the second case, all the transmit antennas are detected jointly. The comparison of the two cases shows improvement with the latter one, both in single-user and multiuser communications and in the presence of unknown cochannel interference (UCCI). It is further shown that in the multiuser case, the proposed receivers approach the corresponding single-user bounds. The number of receive antenna elements required to achieve single-user bound is thereby equal to the number of users and not to the total number of transmit


INTRODUCTION
The scarcity of the frequency spectrum resources and the ever-growing demand for new broadband services imposes a need for the bandwidth-efficient transceiver schemes.Signal transmission and reception using multiple-transmit and multiple-receive antennas over a multiple-input multipleoutput (MIMO) channel is one of the most promising approaches for increasing the link capacity and the achievable data rates [1].Two key approaches have been developed to make effective use of the benefits of the MIMO channels.The first one is the Bell-Labs-layered-space-time architecture (BLAST) [2], where independent signals are transmitted from different transmit antennas.Another technique that combines the benefits of transmit diversity and channel coding in an efficient manner is space-time coding either in a form of space-time block coding (STBC) [3] or space-time trellis coding (STTrC) [4].This paper focuses on STTrCcoded systems.
STTrC codes have originally been developed for frequency flat-fading channels.In order to meet the requirements for high-data-rate transmission, their extension to frequency-selective channels and performance in such scenarios are of great interest.The performance of STTrC codes in unequalized frequency-selective channels was studied in [5,6], where it was shown that the dominant factor on their performance is the intersymbol interference (ISI), that causes inevitable error floor in large signal-tonoise ratio (SNR) range.To solve this problem, two research directions have been followed in recent years.One of them is a combination of orthogonal-frequency-divisionmultiplexing (OFDM) and decoding [7], which allows one to perform signal processing for a set of frequency flat-fading channels.The other approach is a combination of equalization and decoding for single-carrier communications.The optimal receiver for the frequency-selective channels is the maximum a posteriori (MAP) equalizer usually implemented by means of the BCJR algorithm [8].Its complexity, however, grows exponentially with the number of multipath components.Moreover, in the channel-coded systems, the optimal receiver is the one that performs joint equalization and decoding and its complexity is exponential in the product of the number of multipath components and the code memory length.Therefore, it is impractical in broadband systems, and low complexity schemes are of a particular interest.The problem of complexity of the optimal joint equalization and decoding can be effectively solved by iterative (turbo) equalization principle that was introduced in [9].Further complexity reductions have been mainly based on simplifications of the equalization part.This paper focuses on the lowcomplexity turbo equalization for single-carrier communications using STTrC codes.
Joint iterative equalization and decoding of STTrC codes is introduced in [10], where the optimal MAP equalizer is used.Its low-complexity extension is proposed in [11], where channel shortening takes place first, and the reduced complexity MAP algorithm is performed afterwards.The technique results in relatively low-performance degradation when compared to the optimal receiver.In the case of decision feedback equalization (DFE) combined with STTrCdecoder in an iterative manner, a method for complexity reduction was studied by [12,13].This method is based on the decoupling between the real and imaginary parts of the received signal, resulting in a reduced total number of equalizer states.An equalizer based on soft interference cancellation and MMSE filtering was proposed in [14] for a convolutionally coded system with diversity signal reception.The receiver can be seen as an extension of the idea introduced for code-division multiple-access (CDMA) in [15].It was further extended to cover higher-order modulations in [16,17], where further complexity reduction methods were proposed as well.In [18], the idea was applied to the multiuser diversity signal detection with convolutional codes.The reduced complexity version of the receiver, based on matched filter approximation, was proposed in [19].An STTrC coded multiuser system in frequency flat MIMO channels was considered in [20].It employs iterative multiuser detection schemes similar to those of [15,18].
In some situations, the unknown cochannel interference (UCCI) can be present in the channel apart from the users that are to be detected.Those users can originate from the undetected users in the same cell, from the other-cell interference, or from other communication systems.In [21], an iterative UCCI suppression method has been studied, which is based on the covariance-matrix-estimation technique.Subspace estimation methods for the UCCI suppression were considered in [20,22] in CDMA and SDMA systems, respectively.A noniterative receiver for detection of STTrC codes in the presence of UCCI was introduced in [23] in frequency-flat-fading channels.The method of [23] is based on joint detection of all the transmit antennas' signals using MMSE receiver presented in [24].A similar solution was proposed in [25] for the orthogonal transmission using STBC codes in flat-fading channels.In [23,25], however, the MMSE filters are different so that the outputs of the latter one are the combined signals from all the receive antennas, while the outputs of the former are the separated outputs for each receive antenna.
In this paper, new low-complexity turbo equalization schemes for the multiuser MIMO-STTrC-coded system are derived.The first part of the studied receivers is soft cancellation of both ISI and cochannel interference (CCI).The second part is a linear MMSE filtering that is used to cope with the residual interference after soft cancellation and UCCI if the latter is present.The degrees-of-freedom (DoF) of the MMSE receiver are thereby decreased depending on the significance of the residual interference and UCCI.The receiver can be seen as a combination of those considered in [20,25], and their extension to frequency-selective channels.Assuming that each transmitter has N T transmit antennas, the receiver is derived for the general case of jointly detecting signals in the N T /n 0 sets containing n 0 transmit antennas of one particular user, where N T is an integer multiple of n 0 .The derived receiver's performance in special cases corresponding to n 0 = 1 and n 0 = N T is studied through simulations.The case n 0 = 1 can be seen as an extension of the receiver proposed in [20] to the frequency selective channels.The cases of 1 ≤ n 0 ≤ N T are a further receiver extensions where several transmit antennas are detected jointly.The aim of joint detection of several transmit antennas' signals is to preserve the DoFs of the receiver.In case of n 0 = N T , N T − 1, the DoFs are preserved.
The UCCI mitigation capability of the proposed receiver is attained by using the iterative covariance estimation technique shown in [21].It should be noted that the receiver proposed in [20] requires the knowledge of the UCCI channel matrix.Unlike [20], the method of [21] requires only a covariance matrix estimate of the UCCI-plus-noise and it is therefore less complex.
The rest of the paper is organized as follows.Section 2 describes system model.Section 3 presents the proposed receiver and its special cases for which either one antenna or all antennas are detected simultaneously.Section 4 describes the maximum-likelihood (ML) receiver with perfect feedback, whose performance curve is used as a lower bound on the proposed receivers' performance.Section 5 presents numerical results.The paper is concluded in Section 6.
Notation.In the sequel, the following notations were adopted.
is a matrix and the mth element of (•) if (•) is a vector, respectively.However, it is straightforward to extend the receiver derivations to quadrature-amplitude-modulation (QAM) schemes.The coded sequence is then interleaved so that the positions within blocks of length N T remain unchanged, but the positions of the blocks themselves are permuted within a frame according to the user-specific interleaver pattern.Thereby the rank properties of the STTrC codes are preserved [26].The interleaved sequences are then headed by userspecific training sequences consisting of TN T symbols.The entire frame is serial-to-parallel converted, resulting in the sequences b (n) k (i), n = 1, . . .,N T , i = 1, . . ., B + T, and transmitted with N T transmit antennas through the frequencyselective channel.After coherent demodulation in the receiver, the signals from each of N R receive antennas are sampled in time domain to capture the multipath components.Observing the signals from different transmit antennas of different users as the virtual users and arranging them in the vector form similarly as in [18,20], we form the space-time representation of the received signal at time instant i given by

SYSTEM AND RECEIVED SIGNAL MODEL
where y(i) ∈ C LNR×1 is space-time sampled received signal vector, given by where L is the number of paths of the frequency-selective channel, and r m (i) denotes the signal sample obtained after matched filtering at the mth receive antenna.Channel matrix H has the form of 1) , where h (n) k,m (l) denotes the lth path complex gain between kth user's nth transmit antenna and mth receive antenna.Similarly, H I is defined as . .
. . .with The vectors u(i) and u I (i) denote the desired and unknown users' sequences, respectively, which are defined as with Vector n(i) ∈ C LNR×1 contains the spatially and temporally white additive Gaussian noise (AWGN) samples with covariance E{n(i)n H (i)} = σ 2 I.

TURBO MIMO EQUALIZERS
The receiver first associates the signals from transmit antennas of the kth user to the N T /n 0 sets of size n 0 , so that antennas indexed by n = 1, . . ., n 0 belong to the first set, those indexed by n = n 0 + 1, . . ., 2n 0 belong to the second set and so forth.Thereby the number of transmit antennas N T is assumed to be an integer multiple of n 0 .However, the receiver derivation for the more general cases of users having different numbers of transmit antennas and/or different sets of transmit antennas having different sizes is straightforward.Without loss of generality, the receiver derivation is presented for the first set of transmit antennas of the kth user in Section 3.1.The derivation is exactly the same for the rest of transmit antenna groups and the rest of users, with a difference only in indexing.The SISO channel decoding part and the extrinsic probabilities calculation are presented in Section 3.2.The special cases of n 0 = 1 and n 0 = N T are considered in more detail in Section 3.3.For n 0 = 1, only one antenna is detected at a time and the receiver can be viewed as an extension of the receiver presented in [20] to the frequency-selective channel.For n 0 = N T , all the transmit antennas are detected jointly, resulting in the preserved DoFs of the receiver.

SC/MMSE equalizer derivation
Figure 3 shows the receiver block diagram.First, an estimate H of the channel matrix H is obtained based on the training sequence u(i), i = 1, . . ., T. Then, the covariance matrix R of the UCCI-plus-noise is estimated.In the first iteration, only the training sequence is used for this purpose, resulting in Starting from the second iteration, we make use of the soft feedback from the SISO decoding when estimating the covariance matrix.We denote the soft feedback vector as u(i).
Its elements are obtained by replacing the corresponding elements of u(i) by their soft estimates, defined as where P APP SISO denotes a posteriori information obtained after SISO decoding (to be defined in (27)).The covariance matrix estimate is now obtained by The estimate R is, therefore, dependent on the iteration index.However, for the simplicity of notation we omit this dependence, since the receiver derivation is identical for all iterations.Moreover, only in the first iteration ( 9) is used for the estimation of R, while in all subsequent iterations (11) is used.Let the kth user be the user of interest.We further denote ] T , (13) and and γ = 1, . . ., N T /n 0 denote elementwise vector product and antenna set index, respectively.The vectors ũ(i) are obtained by replacing the elements of u(i) by their soft estimates, that is, an element is where P ext SISO denotes the extrinsic information obtained after SISO decoding (to be defined in (28)).The signals b (n)  k (i), n = 1, . . ., n 0 , are jointly detected by filtering the signal using a linear MMSE filter whose weighting matrix W 1 k (i) satisfies the following criterion: subject to the constraint [A] j, j = 1, j = 1, . . ., n 0 , to avoid the trivial solution is defined by It is shown in the Appendix that the matrix W 1 k (i) ∈ C LNR×n0 can be derived as where and h (n)  k is the [(L−1)KN T +kN T +n]th column of the matrix H.The matrix Λ 1 k (i) is defined as Note that (20) holds only for the M-PSK case, although it is straightforward to extend the receiver derivation to the more general signal constellations.Detailed derivation of the optimal solution for a pair of matrices [W 1 k (i), A 1 k (i)] is given in the Appendix.Assuming that the MMSE filter output z 1 k (i) ∈ C n0×1 can be viewed as the output of the equivalent Gaussian channel [27], we can write where matrix Ω 1 k (i) ∈ C n0×n0 contains the gains of the equivalent channel, defined as is the equivalent additive Gaussian noise with covariance matrix, The outputs of the equivalent channels z γ k (i) and their parameters Ω γ k (i) and Θ γ k (i) for γ = 1, . . ., N T /n 0 are passed to the APP block that calculates the extrinsic probabilities needed for SISO decoding, as described in Section 3.2.

APP block and SISO decoding
The SISO channel decoding algorithm used in this paper is a symbol-level MAP algorithm from [28].For the sake of simplicity, we omit the full derivation of the MAP algorithm and we refer to [20,28].It should be noted that the input required by the decoder is the probability P(S i , S i+1 ) associated with the transition between two trellis states, S i and S i+1 , of the STTrC code.The transition probability can be calculated as where d i,i+1 n ∈ Q are the encoder outputs that are associated with the transition (S i , S i+1 ).P ext MMSE (b (n) k (i) = α q ) are extrinsic probabilities obtained by the MMSE detection.For the first set of jointly detected signals (n = 1, . . ., n 0 ), the extrinsic probabilities are calculated in the APP block as for q = 1, . . ., 2 k0 and n = 1, . . ., n 0 , where Based on the transition probabilities P(S i , S i+1 ), the SISO channel decoder calculates the a posteriori probabilities for the symbols b (n) k (i), defined as The decoder extrinsic probability is then calculated as The similar procedure is repeated for all N T /n 0 groups of transmit antennas that are jointly detected, in order to obtain all probabilities P ext SISO (b (n) k (i) = α q ) for n = 1, . . ., N T .The parameter Q ext is an ad-hoc parameter that was introduced in [9,20].It is shown in [20] that if the value of Q ext is appropriately chosen so as to be between 0 and 1, the receiver performance can be significantly improved.This is due to the fact that the extrinsic information in the initial iterations is not accurate enough, especially with relatively small SNR values.By imposing the parameter Q ext , the effect of this inaccuracy is reduced, at the expense of slower receiver convergence.The result of simulations conducted to evaluate the influence of this parameter on the receiver performance is presented in Section 5.
The receiver complexity is dominated by the MMSE part which requires inversion of the matrix M γ k (i) as well as by the APP block which calculates the extrinsic information of the MMSE detector.The overall complexity is therefore O{max(L 3 N 3 R , 2 k0n0 )}.It can be seen that the complexity of the MMSE part does not depend on the number of antennas to be jointly detected.The complexity of the APP part of the receiver, however, increases exponentially with n 0 .

Special cases of n
, transmit antennas detected one-byone.Since the complexity of the receiver depends exponentially on n 0 , this option has the lowest complexity.Signal from only one antenna is detected at a time, while the rest of the antennas are cancelled by the soft feedback and the MMSE filtering, together with CCI and ISI.By doing this, effective DoFs of the receiver are preserved.It should be noted that the number of effective DoFs depends on the reliability of the soft feedback information, which can be seen from ( 19).In the ideal case of perfect feedback (which is not realistic in practice), the number of effective DoFs reaches its maximum value, which is equal to LN R − 1.The fact that feedback is nonperfect in reality will result in a number of effective DoFs that is smaller than LN R − 1, since the matrix HΛ γ k (i)H H − h (1)  k h (1)   k H is in general a nonzero matrix.The preserved DoFs are then used to suppress UCCI, if it is present.If the number of effective DoFs is large enough so that in the asymptotic case of large SNR the matrix M γ k (i) does not have a full rank, then the ISI, CCI, and UCCI can be perfectly suppressed.Otherwise, the receivers' performance saturates to an error floor for large SNR values.
(ii) Receiver 2, n 0 = N T , all transmit antennas detected jointly.In this case, the complexity of the receiver is the largest.However, the N T − 1 effective DoFs of the receiver are now perfectly preserved.This can be seen from ( 19).The signals from N T antennas can be seen as being passed jointly to the receiver output by the third term on the right-hand side of (19).The jointly detected signals are then optimally separated in the APP block.In general, when n 0 out of N T antennas are detected jointly, the n 0 − 1 DoFs are perfectly preserved, while at most N T − n 0 ones are preserved by soft cancellation, depending on the feedback reliability.Also, one should notice that in the theoretical case of the perfect feedback, the proposed receivers' performance will be identical for any n 0 value.

PERFECT FEEDBACK
A lower bound on the receiver performance curve can be obtained in the case of the perfect feedback where all antennas of the single-user signal are detected jointly (n 0 = N T ) using ML detector followed by the MAP-SISO decoder.With the perfect feedback (15) becomes The probability associated with the transition (S i , S i+1 ) between two trellis states S i and S i+1 can be calculated as The receiver described in this section with the assumption of perfect feedback is used in Section 5 to compare with the proposed receivers' performance curves with its lower bound.

NUMERICAL EXAMPLES
Performance of the proposed receivers was evaluated through computer simulations.The channel estimates were assumed to be perfect.All users transmitted with the same power, and fading was constant over each transmitted frame, but changed independently frame-by-frame.The fading was assumed to be frequency selective with the number of paths L = 5, each of which is Rayleigh distributed and uncorrelated.It was assumed that antennas are spatially uncorrelated, and that signals at all receive antennas have the same average powers.E s /N 0 is defined as the SNR per symbol per receive antenna.The exponentially decaying power delay profile with decay exponent −τ was assumed, so that τ = 0 results in the equal-average-power-multipath and τ → ∞ in the flat-fading channels, respectively.The 4-state QPSK code with N T = 2 presented in [4] was used to encode signals of all MIMO users.All the users transmit with the same powers.
The log-MAP space-time trellis decoder shown in [20,28] was used.User-specific random interleavers were assumed.As mentioned in Section 3.1 it is shown in [20] that the appropriate choice of parameter Q ext significantly improves the receiver performance in a flat-fading case for n 0 = 1.The same approach was used in the simulations and the impact of the parameter Q ext in the multipath channel with τ = 0 for the receivers (receiver 1 and receiver 2) was first evaluated.The results are presented in Figure 4 for the case of (K, K I , N R ) = (1, 0, 1).The symbol error rate (SER) versus iteration index and Q ext is presented.It can be seen that Q ext = 0.5 yields the best performance for both receivers.Therefore, this value was used in the further simulations.
In Figure 5, SER and frame error rate (FER) performances of receivers 1 and 2 are presented versus per-antenna E s /N 0 for (K, K I , N R ) = (1, 0, 1) and τ = 0 with the iteration number as a parameter.It is seen that receiver 2 offers performance improvement over receiver 1, due to the joint detection of all transmit antennas.The gain, however, is not very large.This is due to the fact that the effective number of DoFs is increased only by the factor of 1 with the joint detection.Also the feedback is relatively reliable and the DoFs can be preserved almost equally well with soft cancellation as with the joint detection.The performance of the receiver with perfect-feedback and ML detection is also shown for comparison.In Figure 6, the SER and FER performances of receiver 1 are presented versus per-antenna E s /N 0 , for (K, K I , N R ) = (3, 0, 3) and τ = 0 with the iteration number as a parameter.In Figure 7, the same set of curves is given for the receiver 2. It is seen that the single-user bound can be achieved after approximately 8 iterations.It should be noticed that with (K, K I , N R ) = (1, 0, 3), the performances of both receivers after 5 iterations are within only 0.5 dB from the perfect-feedback bound.It is important to mention that the number of receive antennas required to achieve the single-user bound is equal to the number of users and not to the total number of transmit antennas, as it was the case in [20] for the flat fading.It can also be seen from Figures 6 and 7 that receiver 2 slightly outperforms receiver 1.The reason for the relatively small gains is very similar to that given for the situation described in Figure 5.    Figure 8 shows SER and FER performances of both receivers versus per-antenna E s /N 0 for (K, K I , N R ) = (2, 1, 3).It was assumed in this scenario that the UCCI uses only a single antenna.The simulation results for two values of signal-to-UCCI-interference ratio (SIR) are presented.In the cases of SIR equal to 3 dB and 0 dB, the power of the signal transmitted from the UCCI's antenna was assumed to be the same as the power of the signal from the single and two antennas of any of the desired users, respectively.For comparison, the single-user bound described by (K, K I , N R ) = (1, 0, 3) is presented.It can be seen that both receivers are rather robust against the presence of unknown interference in a wide range of E s /N 0 values, for both SIR values.Moreover, in case of SIR = 3 dB, the receivers can perfectly suppress the UCCI if the E s /N 0 value becomes large.This is due to the fact that after convergence the receivers have enough effective DoFs to separate and detect two desired users' signals and suppress one UCCI.It can also be seen from Figure 8 that receivers 1 and 2 show very similar performance.This is due to the fact that the soft feedback is relatively reliable and preserving one additional DoF is of less significance.
In Figure 9, SER and FER performances of both receivers are presented versus per-antenna E s /N 0 for (K, K I , N R ) = (2,1,3).In this scenario, the UCCI uses two antennas in the same way as the desired users.The curves are plotted with SIR, frame length B, and channel decay exponent τ as parameters.For comparison, the single-user bound described by (K, K I , N R ) = (1, 0, 3) is also presented.It can be seen again that both receivers are robust against interference over a wide range of E s /N 0 values.However, due to the lack of effective DoFs, the performance curves tend to saturate an error floor with high E s /N 0 values.This can be solved in a straightforward manner by adding more receive antennas.However, it should be noted from Figure 9 that the error floor can be reduced by increasing the frame length while keeping the ratio T/B constant.This behavior can be explained by two reasons: first, increasing the frame length results in more samples for R estimation; second, the feedback becomes more accurate with the increased frame length.
It can also be seen by comparing the results for τ → 0 and τ → ∞ from Figure 9 that the gain from using receiver 2 is larger if the number of significant multipath components is smaller.This is due to the fact that in the rich multipath environment (τ → 0), the number of effective DoFs is much smaller than the number needed to perfectly suppress the UCCI.Therefore, preserving one DoF with the receiver 2 does not have any significant impact on the receivers' performance.On the other hand, in the flat fading, (τ → ∞) the number of effective DoFs is comparable to the number needed to suppress the UCCI, and preserving one DoF improves performance.The performances of both receivers improve with increased SIR and in the absence of UCCI they are expected to approach the corresponding single-user bounds.

CONCLUSIONS
New iterative receiver schemes for the STTrC-coded multiuser system in frequency-selective channels have been derived for single-carrier broadband signalling.It has been shown through computer simulations that the receiver that jointly detects signals from all the transmit antennas of the user of interest performs slightly better than the receiver that   detects only one antenna at a time.In the presence of relatively strong UCCI, the gain from joint detection is larger if the channel has less multipath components.This is due to the fact that preserving DoFs of the receivers with the joint detection has a greater impact on performance if the number of effective DoFs is comparable to the number needed to perfectly suppress UCCI.The complexity of the MMSE part of the receiver is independent of the number of antennas n 0 that are to be jointly detected.However, the complexity of the APP part, that calculates the extrinsic probabilities needed for SISO channel decoding, grows exponentially with n 0 .It has been shown that the performance of both receivers improves by increasing frame length, due to the improved feedback reliability.The performance also improves with higher SIR values.The gain from joint detection, however, is smaller if the feedback is more reliable.
In a multiuser scenario without UCCI, the proposed receivers can achieve corresponding single-user bounds.Thereby, the required number of receive antennas is equal to the number of users and not to the total number of transmit antennas.Furthermore, the receivers' single-user performances are very similar to each other and relatively close to the performance of the ML receiver with perfect feedback.
Future work may include further receiver structure generalization, where the antennas from more than one user are to be detected jointly.This would result in more DoFs that can be used for the UCCI cancellation.It is also of interest to determine the maximum number of users K for which the single-user bound can be achieved with N R = K receive antennas with N T as a parameter.Further study is also required to evaluate the receivers' sensitivity to the spatial correlation.The joint detection is expected to be more robust than the antenna-by-antenna detection [29].

APPENDIX
Without loss of generality, derivation of the optimal pair [W 1 k (i), A 1 k (i)] of matrices is presented only for the first group of n 0 jointly detected antennas.The derivation is similar for the other groups, with difference only in indexing.We denote the nth columns of the matrices W 1 k (i) and A 1 k (i) as w (n) and a (n) , respectively.For simplicity of notation, we omit the dependence of w (n) and a (n) on user index k, antenna group index γ, and time instant i.The cost function in (16) that is to be minimized can be written as where This is equivalent to minimizing each of the component cost functions, defined as To avoid the trivial solution, [a (n) ] n is set to be equal to 1.
Introducing the Lagrange multiplier λ, the equivalent cost function to be minimized becomes Differentiating (A.5) with respect to m (n) gives where From (A.7), the optimal value of m (n) can be found as gg j (n)   .(A.9) Applying the property of the inverse of the block-matrix [30] A B C D to the inverse of matrix R gg , where Q = A − BD −1 C, we obtain .11)from which it follows that . (A.12) After repeating similar procedure for all the n 0 transmit antennas from the first group of the kth user, the optimal pair of matrices [W 1 k (i), A 1 k (i)] is obtained as (1) , . . ., w (n0) , A 1 k (i) = a (1) , . . ., a (n0) .(A. 13)

Figure 2 :
Figure 2: Transmitter block diagram for the first user.