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Interferenceaware receiver structure for multiuser MIMO and LTE
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 40 (2011)
Abstract
In this paper, we propose a novel lowcomplexity interferenceaware receiver structure for multiuser MIMO that is based on the exploitation of the structure of residual interference. We show that multiuser MIMO can deliver its promised gains in modern wireless systems in spite of the limited channel state information at the transmitter (CSIT) only if users resort to intelligent interferenceaware detection rather than the conventional singleuser detection. As an example, we focus on the long term evolution (LTE) system and look at the two important characteristics of the LTE precoders, i.e., their low resolution and their applying equal gain transmission (EGT). We show that EGT is characterized by full diversity in the singleuser MIMO transmission but it loses diversity in the case of multiuser MIMO transmission. Reflecting on these results, we propose a LTE codebook design based on two additional feedback bits of CSIT and show that this new codebook significantly outperforms the currently standardized LTE codebooks for multiuser MIMO transmission.
1. Introduction
The spatial dimension surfacing from the usage of multiple antennas promises improved reliability, higher spectral efficiency [1], and the spatial separation of users [2]. This spatial dimension (MIMO) is particularly beneficial for precoding in the downlink of multiuser cellular systems (broadcast channel), where these spatial degrees of freedom at the transmitter can be used to transmit data to multiple users simultaneously. This is achieved by creating independent parallel channels to the users (canceling multiuser interference) and the users subsequently employ simplified singleuser receiver structures. However, the transformation of crosscoupled channels into parallel noninteracting channels necessitates perfect channel state information at the transmitter (CSIT) whose acquisition in a practical system, in particular frequency division duplex (FDD) system, is far from realizable. This leads to the precoding strategies based on the partial or quantized CSIT [3], which limit the gains of multiuser MIMO.
Ongoing standardizations of modern cellular systems are investigating different precoding strategies based on lowlevel quantized CSIT to transmit spatial streams to multiple users sharing the same timefrequency resources. In thirdgeneration partnership project longterm evolution (3GPP LTE) system [4], the CSIT acquisition is based on the precoder codebook approach. These LTE precoders are characterized by low resolution and are further based on the principle of equal gain transmission (EGT). These precoders when employed for the multiuser MIMO mode of transmission are unable to cancel the multiuser interference thereby increasing the suboptimality of conventional singleuser detection. This has led to the common perception that multiuser MIMO mode is not workable in LTE [[5], p. 244].
Considering multiuser detection, we propose in this paper a lowcomplexity interferenceaware receiver [6] for the multiuser MIMO in LTE. Though multiuser detection has been extensively investigated in the literature for the uplink (multiple access channel), its related complexity has so far prohibited its employment in the downlink (broadcast channel). For the multiple access channel, several multiuser detection techniques exist in the literature starting from the optimal multiuser receivers [7] to their nearoptimal reduced complexity counterparts (sphere decoders [8]). The complexity associated with these techniques led to the investigation of lowcomplexity solutions as suboptimal linear multiuser receivers [9], iterative multiuser receivers [10, 11], and decisionfeedback receivers [12, 13]. Since in practice, most wireless systems employ error control coding combined with the interleaving, recent work in this area has addressed multiuser detection for coded systems based on soft decisions [14, 15].
Our proposed lowcomplexity interferenceaware receiver structure not only reduces one complex dimension of the system but is also characterized by exploiting the interference structure in the detection process. Considering this receiver structure, we investigate the effectiveness of the lowresolution LTE precoders for the multiuser MIMO mode and show that multiuser MIMO can bring significant gains in future wireless systems if the users resort to intelligent interferenceaware detection as compared to the suboptimal singleuser detection. We further look at the second characteristic of the LTE precoders, i.e., EGT both for the singleuser and multiuser MIMO modes. We show that the EGT has full diversity in the singleuser MIMO mode (a result earlier derived for equal gain combining for BPSK in [16] and for EGT in MIMO systems in [17]); however, it suffers from a loss of diversity in multiuser MIMO mode [18]. Based on this analysis, we propose a design criteria for the precoder codebooks and show that the additional feedback of two bits for CSIT can lead to significant improvement in the performance of the multiuser MIMO.
Regarding notations, we will use lowercase or uppercase letters for scalars, lowercase boldface letters for vectors and uppercase boldface letters for matrices. The matrix I_{ n } is the n × n identity matrix. . and . indicate norm of scalar and vector while (.) ^{T} , (.)*, and (.)^{†} indicate transpose, conjugate, and conjugate transpose, respectively. (.) _{ R } indicates the real part and (.) _{ I } indicates the imaginary part of a complex number. The notation E (.) denotes the mathematical expectation while denotes the Gaussian Qfunction. All logarithms are to the base 2.
The paper is divided into eight sections. In Sec. II, we give a brief overview of LTE and define the system model. In Sec. III, we consider a geometric scheduling strategy for the multiuser MIMO mode in LTE and propose a lowcomplexity interferenceaware receiver structure. In Sec. IV, we look at the information theoretic perspective of the proposed receiver structure. Sec. V is dedicated to the performance analysis of the EGT that is followed by the simulation results. Before concluding the paper, we propose a design criteria for the precoder codebooks of the forthcoming standardizations of LTE. The proof details in the paper have been relegated to appendices to keep the subject material simple and clear.
2. LTE system model
A. LTEA brief overview
In 3GPP LTE, a 2 × 2 configuration for MIMO is assumed as the baseline configuration; however, configurations with four transmit or receive antennas are also foreseen and reflected in the specifications [19]. LTE restricts the transmission of maximum of two codewords in the downlink that can be mapped onto different layers where one codeword represents an output from the channel encoder. Number of layers available for the transmission is equal to the rank of the channel matrix (maximum 4). In this paper, we restrict ourselves to the baseline configuration with the eNodeB (LTE notation for the base station) equipped with two antennas while we consider single and dualantenna user equipments (UEs). Physical layer technology employed for the downlink in LTE is OFDMA combined with bit interleaved coded modulation (BICM) [20]. Several different transmission bandwidths are possible, ranging from 1.08 to 19.8 MHz with the constraint of being a multiple of 180 kHz. Resource blocks (RBs) are defined as groups of 12 consecutive resource elements (REs  LTE notation for the subcarriers) with a bandwidth of 180 kHz thereby leading to the constant RE spacing of 15 kHz. Approximately, 4 RBs form a subband and the feedback is generally done on subband basis. Seven operation modes are specified in the downlink of LTE; however, we shall focus on the following four modes:

Transmission mode 2. Fallback transmit diversity. Transmission rank is 1, i.e., one codeword is transmitted by the eNodeB. Employs Alamouti spacetime or spacefrequency codes [21].

Transmission mode 4. Closedloop spatial multiplexing. Transmission rank is 2, i.e., two codewords are transmitted by the eNodeB to the UE in the singleuser MIMO mode. UEs need to have minimum of two antennas.

Transmission mode 5. Multiuser MIMO mode. Supports only rank1 transmission, i.e., one codeword for each UE.

Transmission mode 6. Closedloop precoding for rank1 transmission, i.e., one codeword for the UE in the singleuser MIMO mode.
In the case of transmit diversity and closedloop precoding, one codeword (data stream) is transmitted to each UE using Alamouti code in the former case and LTE precoders in the latter case. Timefrequency resources are orthogonal to the different UEs in these modes thereby avoiding interference in the system. However, in the multiuser MIMO mode, parallel codewords are transmitted simultaneously, one for each UE, sharing the same timefrequency resources. Note that LTE restricts the transmission of one codeword to each UE in the multiuser MIMO mode.
For closedloop transmission modes (mode 4, 5 and 6), precoding mechanisms are employed at the transmit side with the objective of maximizing throughput. The precoding is selected and applied by the eNodeB to the data transmission to a target UE based on the channel feedback received from that UE. This feedback includes a precoding matrix indicator (PMI), a channel rank indicator (RI), and a channel quality indicator (CQI). PMI is an index in the codebook for the preferred precoder to be used by the eNodeB. The granularity for the computation and signaling of the precoding index can range from a couple of RBs to the full bandwidth. For transmission mode 5, the eNodeB selects the precoding matrix to induce high orthogonality between the codewords so that the interference between UEs is minimized. In transmission modes 4 and 6, the eNodeB selects the precoding vector/matrix such that codewords are transmitted to the corresponding UEs with maximum throughput.
In order to avoid excessive downlink signaling, transmission mode for each UE is configured semistatically via higher layer signaling, i.e., it is not allowed for a UE to be scheduled in one subframe in the multiuser MIMO mode and in the next subframe in the singleuser MIMO mode. For the case of eNodeB with two antennas, LTE proposes the use of following four precoders for transmission modes 5 and 6:
The number of precoders increases to sixteen in the case of four transmit antennas; however, in this paper, we restrict to the case of two transmit antennas. For transmission mode 4, LTE proposes the use of following two precoder matrices on subband basis.
Note that there is a possibility of swapping the columns in P but the swap must occur over the entire band.
B. System model
We first consider the system model for transmission mode 5, i.e., the multiuser MIMO mode in which the eNodeB transmits one codeword each to two singleantenna UEs using the same timefrequency resources. Transmitter block diagram is shown in Figure 1. During the transmission for UE1, the code sequence c_{1} is interleaved by π_{1} and is then mapped onto the signal sequence x_{1}. x_{1} is the symbol of x_{1} over a signal set with a Graylabeling map where χ_{1} = M_{1} and x_{2} is the symbol of x_{2} over signal set χ_{2} where χ_{2} = M_{2}. The bit interleaver for UE1 can be modeled as π_{1}: k' → (k, i) where k' denotes the original ordering of the coded bits c_{ k' }, k denotes the RE of the symbol x_{1},_{k}, and i indicates the position of the bit c_{ k' }in the symbol x_{1},_{k}. Note that each RE corresponds to a symbol from a constellation map χ_{1} for UE1 and χ_{2} for UE2. Selection of the normal or extended cyclic prefix (CP) for each OFDM symbol converts the downlink frequencyselective channel into parallel flat fading channels.
Cascading IFFT at the eNodeB and FFT at the UE with the cyclic prefix extension, the transmission at the kth RE for UE1 in transmission mode 5 can be expressed as
where y_{1,k}is the received symbol at UE1 and z_{1,k}is zero mean circularly symmetric complex white Gaussian noise of variance N_{0}. x_{1,k}is the complex symbol for UE1 with the variance and x_{2,k}is the complex symbol for UE2 with the variance . symbolizes the spatially uncorrelated flat Rayleigh fading MISO channel from eNodeB to the nth UE (n = 1, 2) at the kth RE. Its elements can therefore be modeled as independent and identically distributed (iid) zero mean circularly symmetric complex Gaussian random variables with a variance of 0.5 per dimension. Note that ℂ^{1 × 2} denotes a 2dimensional complex space. p_{n,k}denotes the precoding vector for the nth UE at the kth RE and is given by (1). For the dualantenna UEs, the system equation for transmission mode 5 is modified as
where y_{1,k}, z_{1,k}∈ ℂ^{2 × 1} are the vectors of the received symbols and circularly symmetric complex white Gaussian noise of doublesided power spectral density N_{0}/2 at the 2 receive antennas of UE1, respectively. H_{1,k}∈ℂ^{2 × 2} is the channel matrix from eNodeB to UE1.
In transmission mode 6, only one UE will be served in one timefrequency resource. Therefore, the system equation for singleantenna UEs at the kth RE is given as
where p_{ k } is given by (1). For the dualantenna UEs, the system equation for mode 6 is modified as
3. Multiuser MIMO mode
We now look at the effectiveness of the lowresolution LTE precoders for the multiuser MIMO mode. We first consider a geometric scheduling strategy [22] based on the selection of UEs with orthogonal precoders.
A. Scheduling strategy
As the processing at the UE is performed on a RE basis for each received OFDM symbol, the dependency on RE index can be ignored for notational convenience. The system equation for the case of singleantenna UEs for the multiuser mode is
The scheduling strategy is based on the principle of maximizing the desired signal strength while minimizing the interference strength. As the decision to schedule a UE in the singleuser MIMO, multiuser MIMO or transmit diversity mode will be made by the eNodeB, each UE would feedback the precoder that maximizes its received signal strength. So this selected precoder by the UE would be the one closest to its matched filter (MF) precoder in terms of the Euclidean distance.
For the multiuser MIMO mode, the eNodeB needs to ensure good channel separation between the coscheduled UEs. Therefore, the eNodeB schedules two UEs on the same RBs that have requested opposite (orthogonal) precoders, i.e., the eNodeB selects as the second UE to be served in each group of allocatable RBs, one of the UEs whose requested precoder p_{2} is 180° out of phase from the precoder p_{1} of the first UE to be served on the same RBs. So if UE1 has requested , q ∈ {±1, ±j}, then eNodeB selects the second UE that has requested . This transmission strategy also remains valid also for the case of dualantenna UEs where the UEs feedback the indices of the precoding vectors that maximize the strength of their desired signals, i.e., Hp^{2}. For the multiuser MIMO mode, the eNodeB schedules two UEs on the same RE, which have requested 180° out of phase precoders. The details of this geometric scheduling strategy can be found in [22].
Though this precoding and scheduling strategy would ensure minimization of the interference under the constraint of lowresolution LTE precoders, the residual interference would still be significant. Singleuser detection, i.e., Gaussian assumption of the residual interference and its subsequent absorption in noise, would lead to significant degradation in the performance. On the other hand, this residual interference is actually discrete belonging to a finite alphabet and its structure can be exploited in the detection process. However, intelligent detection based on its exploitation comes at the cost of enhanced complexity. Here, we propose a lowcomplexity interferenceaware receiver structure that on one hand reduces one complex dimension of the system while on the other hand, it exploits the interference structure in the detection process.
B. Lowcomplexity interferenceaware receiver
First, we consider the case of singleantenna UEs. Soft decision of the bit c_{ k' } of x_{1}, also known as loglikelihood ratio (LLR), is given as
We introduce the notation for the bit metric that is developed on the lines similar to the (7) and 9 in [20], i.e.,
where denotes the subset of the signal set x_{1} ∈ χ_{1} whose labels have the value c_{ k' } ∈ {0, 1} in the position i. Here, we have used the logsum approximation, i.e., and this bit metric is therefore termed as maxlog MAP bit metric. As LLR is the difference of two bit metrics and these will be decoded using a conventional softdecision Viterbi algorithm, (a common scaling factor to all LLRs) can be ignored thereby leading to
where indicates the crosscorrelation between the two effective channels. Here, we have used the relation a  b^{2} = a^{2} + b^{2}  2 (a*b) _{ R } where the subscript (.) _{ R } indicates the real part. Note that the complexity of the calculation of bit metric (10) is .
In (10), we now introduce two terms as the outputs of MF, i.e., and . Ignoring y_{1}^{2} (independent of the minimization operation), the bit metric is written as
where
Note that the subscript (.) _{ I } indicates the imaginary part.
For x_{1} and x_{2} belonging to equal energy alphabets, and can be ignored as they are independent of the minimization operation. The values of x_{2,R}and x_{2,I}that minimize Eq. (11) need to be in the opposite directions of ψ_{ A } and ψ_{ B } , respectively, thereby avoiding search on the alphabets of x_{2} and reducing one complex dimension in the detection, i.e.,
As an example, we consider the case of QPSK for which the values of x_{2,R}and x_{2,I}are , so the bit metric is written as
For x_{1} and x_{2} belonging to nonequal energy alphabets, the bit metric is same as (13) but and can no longer be ignored thereby leading to
Note that the minimization is independent of χ_{2} though x_{2} appears in the bit metric. The reason of this independence is as follows. The decision regarding the signs of x_{2,R}and x_{2,I}in (14) will be taken in the same manner as for the case of equal energy alphabets. For finding their magnitudes that minimize the bit metric (14), it is the minimization problem of a quadratic function, i.e., differentiating (14) w.r.t x_{2,R} and x_{2,I} to find the global minima that are given as
where → indicates the discretization process in which among the finite available points of x_{2,R}and x_{2,I}, the point closest to the calculated continuous value is selected. So if x_{2} belongs to QAM256, then instead of searching 256 constellation points for the minimization of (14), the metric (15) reduces it to merely two operations thereby trimming down one complex dimension in the detection, i.e., the detection complexity is independent of χ_{2} and reduces to .
As a particular example of the discretization of continuous values in (15), we consider the case of x_{2} belonging to QAM16. The values of x_{2,R}and x_{2,I}for the case of QAM16 are so their magnitudes in (14) are given as
and I (.) is the indicator function defined as
Now we look at the receiver structure for the case of dualantenna UEs. The system equation for UE1 (ignoring the RE index) is
The receiver structure would remain same with being replaced by H_{1}, i.e., the channel from eNodeB to the two antennas of UE1. Subsequently and are the MF outputs while ρ_{12} = (H_{1}p_{1})^{†}H_{1}p_{2} is the crosscorrelation between two effective channels.
For comparison purposes, we also consider the case of singleuser receiver, for which the bit metric is given as
Table 1 compares the complexities of different receivers in terms of the number of realvalued multiplications and additions for getting all LLR values per RE/subcarrier. Note that n_{ r } denotes the number of receive antennas. This complexity analysis is independent of the number of transmit antennas as the operation of finding effective channels bears same complexity in all receiver structures. Moreover UEs can also directly estimate their effective channels if the pilot signals are also precoded. The comparison shows that the complexity of the interferenceaware receiver is of the same order as of singleuser receiver while it is far less than the complexity of the maxlog MAP receiver. Figure 2 further shows the performancecomplexity trade off of different receivers for multiuser MIMO mode in LTE. The performance of the receivers is measured in terms of the SNR at the frame error rate (FER) of 10^{2} whereas the complexity is determined from Table 1. It shows that the performance of the singleuser receiver is severely degraded as compared to that of the interferenceaware receiver. In most cases, the singleuser receiver fails to achieve the requisite FER in the considered SNR range. On the other hand, interferenceaware receiver achieves same performance as maxlog MAP receiver but with much reduced complexity.
The interferenceaware receiver is therefore not only characterized by low complexity but also resorts to intelligent detection by exploiting the structure of residual interference. Moreover, this receiver structure being based on the MF outputs and devoid of any division operation can be easily implemented in the existing hardware. However, the proposed receiver needs both the channel knowledge and the constellation of interference (coscheduled UE). As the UE already knows its own channel from the eNodeB and the requested precoder, it can determine the effective channel of the interference based on the geometric scheduling algorithm, i.e., the precoder of the coscheduled UE is 180° out of phase of its own precoder. Consequently there is no additional complexity in utilizing this receiver structure as compared to using singleuser receivers except that the UE needs to know the constellation of interference.
4. Information theoretic perspective
Sum rate of the downlink channel is given as
where P = [p_{1}p_{2}] is the precoder matrix, is the mutual information of UE1 once it sees interference from UE2 and is the mutual information of UE2 once it sees interference from UE1. Y_{1} is the received symbol at UE1 while X_{1} is the symbol transmitted by the eNodeB to UE1. Note that interference is present in the statistics of Y_{1} and Y_{2}. No sophisticated power allocation is employed to the two streams as the downlink control information (DCI) in the multiuser mode in LTE includes only 1bit power offset information, indicating whether a 3 dB transmit power reduction should be assumed or not. We therefore consider equalpower distribution between the two streams. For the calculation of mutual information, we deviate from the unrealistic Gaussian assumption for the alphabets and consider them from discrete constellations. The derivations of the mutual information expressions for the case of finite alphabets have been relegated to Appendix A for simplicity and lucidity.
We focus on the LTE precoders but to analyze the degradation caused by the lowlevel quantization and the characteristic of EGT of these precoders, we also consider some other transmission strategies. Firstly, we consider unquantized MF precoder [23] that is given as
For EGT, the unquantized MF precoder is given as
To be fair in comparison with the geometric scheduling algorithm for multiuser MIMO in LTE, we introduce a geometric scheduling algorithm for unquantized precoders. We divide the spatial space into four quadrants according to the spatial angle between and , which is given as
The geometric scheduling algorithm ensures that the eNodeB chooses the second UE to be served on the same RE as the first UE such that their channels and lie in the opposite quadrants.
Figure 3 shows the sum rates of a broadcast channel with the dualantenna eNodeB and two singleantenna UEs for QAM64 alphabets. SNR is the transmit SNR, i.e., whereas the two UEs have equalpower distribution, i.e., , MF and MF EGT precoders are the unquantized precoders given in (20) and (21), respectively, while LTE precoders are the quantized precoders given in (1). The sum rates of unquantized precoders along with those of LTE quantized precoders are shown for the case of singleuser receivers and for the case of lowcomplexity interferenceaware receivers. The results show that under the proposed transmission strategy, the sum rate can be significantly improved (unbounded in SNR) if the lowcomplexity interferenceaware receivers are used as compared to the case when the UEs resort to suboptimal singleuser detection where rates are bounded (in SNR). The behavior of singleuser detection is attributed to the fact that this detection strategy considers interference as noise so the SINR is low once no geometric scheduling has been employed by the eNodeB while the SINR improves due to the reduction of interference once geometric scheduling is employed. However, the rates remain bounded in the SNR if the UEs resort to the singleuser detection that is due to the fact that increasing the SNR (transmit SNR) also increases the interference strength thereby bounding the SINR at high values of the transmit SNR. On the other hand, there is significant improvement in the sum rate once UEs resort to intelligent detection by employing the lowcomplexity interferenceaware receivers. In this case, the sum rate is unbounded if the rate (constellation size) of each UE is adapted with the SNR. Note that the quantized CSIT (LTE precoders) appears to have no effect at high SNR once UEs resort to intelligent interferenceaware detection. This behavior is because the rate is not adapted with the SNR in these simulations, i.e., the constellation size is fixed to QAM64 and is not increased with the increase in the SNR. At high SNR, the rate of each UE gets saturated to its constellation size (six bits for QAM64) if the UE resorts to intelligent interferenceaware detection. However, the approach to this saturation point (slope of the rate curve) depends on the quantization of channel information.
Another interesting result is the effect of the two characteristics of LTE precoders, i.e., low resolution and EGT. There is a slight improvement in the sum rate at medium SNR when the restriction of low resolution (LTE quantized precoders) is relaxed, i.e., eNodeB employs MF EGT precoders; however, there is a significant improvement in the sum rate when the restriction of EGT is eliminated, i.e the eNodeB employs MF precoders. This shows that the loss in spectral efficiency due to the employment of LTE precoders is mainly attributed to the EGT rather than their low resolution (quantization).
5. Performance analysis
We now focus on the EGT characteristic of the LTE precoders and carry out the performance analysis of the EGT in singleuser and multiuser MIMO systems. We restrict to the case of singleantenna UEs while the eNodeB has two antennas. For singleuser case, the received signal at the kth RE is given by
For EGT, the precoder vector is given by . So the received signal after normalization by is given by
where . The PEP has been derived in Appendix B and is given as
where is the normalized minimum distance of the constellation χ_{1}, d_{free} is the free distance (minimum Hamming distance) of the code. Note that c_{1} and are the correct and error codewords, respectively. Eq. 25 clearly shows full diversity of the EGT for singleuser MIMO. Note that this result was earlier derived in [16] but was restricted to the case of BPSK. The same result was derived in [17] for EGT in MIMO systems using the approach of metrics of diversity order. Here, we have generalized this result and have adopted the natural approach of pairwise error probability to show the diversity order. Analysis of the EGT for multiuser MIMO system seemingly does not have closed form solution so we shall resort to the simulations for its analysis in Sec. 6.
6. Simulation results
Simulations are divided into three parts. In the first part, we look at the performance of the proposed interferenceaware receiver structure for the multiuser MIMO mode in LTE while second part is dedicated to the sensitivity analysis of this receiver structure to the knowledge of the constellation of interference. This sensitivity analysis is motivated by the fact that the DCI formats in the transmission mode 5 (multiuser MIMO) do not include the information of the constellation of the coscheduled UE. Third part looks at the diversity order of the EGT in both singleuser and multiuser MIMO modes in LTE.
For the first part (Figures 4 and 5), we consider the downlink of 3GPP LTE that is based on BICM OFDM transmission from the eNodeB equipped with two antennas using rate1/3 LTE turbo code [24] with rate matching to rate 1/2 and 1/4. We deliberate on both the cases of single and dualantenna UEs. We consider an ideal OFDM system (no ISI) and analyze it in the frequency domain where the channel has iid Gaussian matrix entries with unit variance and is independently generated for each channel use. We assume no power control in the multiuser MIMO mode so two UEs have equalpower distribution. Furthermore, all mappings of the coded bits to QAM symbols use Gray encoding. We focus on the FER while the frame length is fixed to 1,056 information bits. As a reference, we consider the fallback transmit diversity scheme (LTE mode 2Alamouti code) and compare it with the singleuser and multiuser MIMO modes employing singleuser receivers and lowcomplexity interferenceaware receivers. To analyze the degradation caused by the low resolution and EGT of LTE precoders, we also look at the system performance employing the unquantized MF and unquantized MF EGT precoders. To be fair in the comparison of the LTE multiuser MIMO mode (mode 5) employing the geometric scheduling algorithm with the multiuser MIMO mode employing unquantized MF and MF EGT precoders, we consider the geometric scheduling algorithm (Sec. 4) based on the spatial angle between the two channels (22). Perfect CSIT is assumed for the case of MF and MF EGT precoding while error free feedback of two bits (PMI) to the eNodeB is assumed for LTE precoders. It is assumed that the UE has knowledge of the constellation of coscheduled UE in the multiuser MIMO mode. It is further assumed that the UE knows its own channel from the eNodeB. So in multiuser MIMO mode, the UE can find the effective interference channel based on the fact that the eNodeB schedules the second UE on the same RE whose precoder is 180° out of phase of the precoder of the first UE. Figure 4 shows the results for the case of singleantenna UEs. It shows enhanced performance of the multiuser MIMO mode once the UEs resort to intelligent detection by employing the lowcomplexity interferenceaware receivers. The performance is severely degraded once the UEs resort to singleuser detection. An interesting result is almost the equivalent performance of the unquantized MF EGT and lowresolution LTE precoders, which shows that the loss with respect to the unquantized CSIT is attributed to the EGT rather than the low resolution of LTE precoders. Performance degradation is observed for LTE multiuser MIMO mode for higher spectral efficiencies. Figure 5 shows the results for the case of dualantenna UEs and focuses on different LTE modes employing LTE precoders. It shows that singleuser detection performs close to interferenceaware detection at low spectral efficiencies once UE has two antennas; however, its performance degrades at higher spectral efficiencies. This behavior is attributed to the fact that the rate with singleuser detection gets saturated at high SNR due to the increased interference strength as was shown in Sec. 4. So the performance of singleuser detection degrades for high spectral efficiencies as these spectral efficiencies are higher than the rate or mutual information of the singleuser detection. For singleuser MIMO (Mode 6), there is no saturation of the rate at high SNR as there is no interference. So mode 6 performs better than mode 5 at high SNR for higher spectral efficiencies once UEs employ singleuser detection. However, if UEs resort to the intelligent interferenceaware detection, the multiuser MIMO mode shows enhanced performance over other transmission modes in LTE. No degradation of LTE multiuser MIMO mode is observed at higher spectral efficiencies once UEs have receive diversity (dual antennas).
In the second part of simulations, we look at the sensitivity of the proposed receiver structure to the knowledge of the constellation of coscheduled UE for the multiuser MIMO mode in LTE. The simulation settings are same as of the first part except that we consider the case when UE has no knowledge of the constellation of coscheduled UE. The UE assumes this unknown interference constellation to be QPSK, QAM16, or QAM64, and the results for these different assumptions are shown in Figure 6. Results show that there is negligible degradation in the performance of the proposed receiver if the interfering constellation is assumed to be QAM16 or QAM64. However, there is significant degradation if the interference is assumed to be QPSK when it actually comes from QAM64. It indicates that assuming interference to be from a higher order modulation among the possible modulation alphabets leads to the best compromise as this assumption includes the lower modulation orders as special cases (with proper scaling). However, the converse is not true, i.e., assuming interference from lower modulation order cannot include higher order modulations. As LTE and LTEAdvanced restrict the transmission to three modulations (QPSK, QAM16, and QAM64), assuming interference to be QAM64 (or even QAM16) leads to better performance. The proposed receiver structure, therefore, can still exploit the discrete nature of the interference even if it does not know its modulation order. As the complexity of this receiver structure is independent of the constellation of interference, the assumption of higher order modulation does not add to the complexity of detection.
In the third set of simulations, we look at the diversity order of the singleuser MIMO and multiuser MIMO schemes in LTE. The system settings are same as in the first part, but now we consider slow fading environment, i.e., the channel remains constant for the duration of one codeword. Figure 7 shows that the MF precoders have full diversity both in multiuser MIMO and singleuser MIMO modes. However, once the constraint of EGT is imposed on the MF precoders, multiuser MIMO mode loses diversity while singleuser MIMO still exhibits full diversity, which is in conformity with the analytical results of Sec. 5. This fundamental result holds even when the lowlevel quantization of LTE is imposed on these EGT precoders. Earlier conclusion that the performance loss in the multiuser MIMO mode in LTE is attributed to the EGT rather than the low resolution of LTE precoders is further confirmed. These results give a general guideline for the possible employment of the singleuser MIMO and multiuser MIMO in LTE under different environments. Once not enough diversity is available in the channel, singleuser MIMO is the preferred option while multiuser MIMO is the possible choice once the channel is rich in diversity.
7. Design of LTE precoder codebook with additional feedback
It was shown in the information theoretic analysis and was subsequently confirmed in the simulations that the loss in spectral efficiency due to the lowlevel quantized CSIT (LTE precoders) in the multiuser MIMO mode is more attributed to the EGT of the LTE codebook rather than its low resolution. It was also shown that EGT loses diversity in the multiuser MIMO mode. Focusing on these fundamental results, we now look at the design of the precoder codebook for future standardizations of LTE. Feedback of CSIT is expected to increase in these forthcoming wireless systems. However, the complexity associated with the feedback overhead combined with the low rate feedback channels would allow only a limited increase in the feedback. We therefore consider the case of two additional feedback bits for the quantized CSIT (precoder codebook) and look how these additional bits can be efficiently employed.
We consider two options for the employment of these additional feedback bits as illustrated in Figure 8. As the LTE precoder is [1 exp(jθ)] ^{T} , so additional bits can be used to increase the angular resolution of θ, i.e., more points on the unit circle but restricting to EGT as shown in Figure 8a. Another option is to increase the levels of transmission, i.e., the additional feedback bits are used by the UE to indicate an increase of the power level on either of the two antennas as [1 2 exp(jθ)] ^{T} or [2 exp(jθ) 1] ^{T} .
With this new precoding codebook design, the earlier described scheduling strategy remains same, i.e., for a UE to be scheduled in the multiuser MIMO mode, the eNodeB selects the second UE to be served on the same timefrequency resources (coscheduled UE) such that the desired signal strength is maximized while interference strength is minimized for both the UEs. So if UE1 has requested the precoder p_{1}, the eNodeB finds the precoding vector p_{2} in the codebook, which minimizes their crosscorrelation and then schedules the second UE with UE1, which has requested p_{2} as its desired precoding vector. The receiver structure being independent of the codebook design also remains same for these new precoding codebooks.
We now look at the effect of two additional bits of feedback for PMI on the performance. We focus on the two options of improved angular resolution and additional levels of transmission. The simulation settings are same as of the previous section. Figure 9 illustrates the performance both in fast and slow fading channels. These results show significant improvement in the performance of the multiuser MIMO mode when the additional feedback bits are employed to increase the levels of transmission as compared to the case of increasing the angular resolution. The performance is within 1.7 dB of the lower bound where lower bound is the performance curve for MF precoder without any interference. In slow fading environment, the change of the slope of FER curve with increased levels of transmission indicates improved diversity as compared to the case of increased angular resolution. On the other hand, little gain is observed in the singleuser mode with additional feedback bits, which is expected as the standard LTE precoders have been optimized for the singleuser transmission. These results indicate that the design of precoders for the forthcoming versions of LTE should consider increasing transmission levels rather than enhancing the angular resolution of the precoders. This proposed design is not merely restricted to the framework of LTE but gives fundamental design guidelines for precoding in modern wireless systems.
8. Conclusions
In this paper, we have looked at the feasibility of the multiuser MIMO for future wireless systems that are characterized by lowlevel quantization of CSIT. We have shown that multiuser MIMO can deliver its promised gains if the UEs resort to intelligent detection rather than the suboptimal singleuser detection. To this end, we have proposed a lowcomplexity interferenceaware receiver structure that is characterized by the exploitation of the structure of residual interference. We have analyzed two important characteristics of the LTE precoders, i.e., low resolution and EGT. We have shown that the performance loss of the LTE precoders in the multiuser MIMO mode is attributed to their characteristic of EGT rather than their low resolution. We have further shown that the EGT is characterized by full diversity in the singleuser MIMO mode but it loses diversity in the multiuser MIMO. Based on these fundamental results, we have proposed a design of the precoder codebook for forthcoming standardizations of LTE incorporating more levels of transmission.
Appendix A
Mutual information for finite alphabets
The mutual information for UE1 for finite size QAM constellation with χ_{1} = M_{1} takes the form as
where is the entropy function. The second term of (26) is given as
where and . Conditioned on the channel and the precoder, there is one source of randomness, i.e., noise. So (27) can be extended as
where M_{2} = χ_{2}, x = [x_{1}x_{2}] ^{T} , and . The mutual information for UE1 can be rewritten as
The above quantities can be easily approximated using sampling (MonteCarlo) methods with N_{ z } realizations of noise and realizations of the channel where the precoding matrix depends on the channel. So we can rewrite (29) as (30).
Similarly the mutual information for UE2 is given as
where .
For the case of singleuser MIMO mode, the mutual information is given by
where the second term is given by
where are the number of channel realizations of the channel . Note that the precoding vector p_{1} is dependent on the channel .
Appendix B
Diversity analysis of EGT in singleuser MIMO
Consider the system equation 24, i.e.,
The maxlog MAP bit metric [20] for the bit c_{ k' }can be written as
The conditional PEP i.e is given as
where indicates the complete channel from the eNodeB to UE1 for the transmission of the codeword c_{1}. Assume for c_{1} and under consideration for the PEP analysis, which is the worst case scenario between any two codewords. Therefore, the inequality on the right hand side of (36) shares the same terms on all but d_{free} summation points and the summations can be simplified to only d_{free} terms for which . Let's denote
As , this leads to PEP being given as
where we have used Chernoff bound . Averaging over channel leads to
Eq. 39 follows from the channel independence at each RE that is the consequence of the interleaving operation. Here we have used the notation with being the normalized minimum distance of the constellation χ_{1}. Using the moment generating function (MGF) of the SNR at the output of two branch EGC as per equations (2) and (23) in [25], PEP at high SNR is upper bounded as
Using the identity , we have
Taylor series expansion [26] of is given as
where x! is the factorial of x while (x) _{ n } is the Pochhammer symbol, i.e., (x) _{ n } = x (x + 1) ⋯ (x + n  1). For x closer to 1, a case that shall be occurring at high SNR in (41), first term will be dominant, i.e.,
Taylor series expansion of at x = 1 is
In the expansion of , first two terms will be dominant at high SNR thereby leading to
So rewriting (40), we get
At high SNR, second term converges to while the third term converges to . So
PEP at high SNR is upper bounded as
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Acknowledgements
Eurecom's research is partially supported by its industrial partners: BMW, Bouygues Telecom, Cisco Systems, France Télécom, Hitachi Europe, SFR, Sharp, ST Microelectronics, Swisscom, Thales. The research work leading to this paper has also been partially supported by the European Commission under SAMURAI and IST FP7 research network of excellence NEWCOM++.
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Ghaffar, R., Knopp, R. Interferenceaware receiver structure for multiuser MIMO and LTE. J Wireless Com Network 2011, 40 (2011). https://doi.org/10.1186/16871499201140
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Keywords
 Long Term Evolution
 Matched Filter
 Frame Error Rate
 Precoding Vector
 3GPP Long Term Evolution