Exploring the physical layer frontiers of cellular uplink
 Erich Zöchmann^{1, 2}Email author,
 Stefan Schwarz^{1, 2},
 Stefan Pratschner^{1},
 Lukas Nagel^{1},
 Martin Lerch^{1} and
 Markus Rupp^{1}
https://doi.org/10.1186/s1363801606091
© Zöchmann et al. 2016
Received: 8 September 2015
Accepted: 6 April 2016
Published: 27 April 2016
Abstract
Communication systems in practice are subject to many technical/technological constraints and restrictions. Multiple input, multiple output (MIMO) processing in current wireless communications, as an example, mostly employs codebookbased precoding to save computational complexity at the transmitters and receivers. In such cases, closed form expressions for capacity or biterror probability are often unattainable; effects of realistic signal processing algorithms on the performance of practical communication systems rather have to be studied in simulation environments. The Vienna LTEA Uplink Simulator is a 3GPP LTEA standard compliant MATLABbased link level simulator that is publicly available under an academic use license, facilitating reproducible evaluations of signal processing algorithms and transceiver designs in wireless communications. This paper reviews research results that have been obtained by means of the Vienna LTEA Uplink Simulator, highlights the effects of singlecarrier frequencydivision multiplexing (as the distinguishing feature to LTEA downlink), extends known link adaptation concepts to uplink transmission, shows the implications of the uplink pilot pattern for gathering channel state information at the receiver and completes with possible future research directions.
Keywords
1 Introduction
Current cellular wireless communications employs Universal Mobile Telecommunications System (UMTS) Long Term Evolution (LTE) as the high data rate standard [1]. The increasing demand of high data traffic in up and downlink forces engineers to push the limits of LTE [2], e.g. through enhanced multiuser multiple input, multiple output (MIMO) support [3, 4], coordinated multipoint (CoMP) transmission/reception [5, 6] as well as improved channel state information (CSI) feedback algorithms [7]. The authors of [8] predict further evolution of existing LTE/LTEAdvanced (LTEA) systems in parallel to the development of new radioaccess technologies operating at millimetre wave frequencies even beyond the expected rollout of 5G technologies by 2020. Fair comparison of novel signal processing algorithms and transceiver designs has to assure equal testing and evaluation conditions to enable reproducibility of results by independent groups of researchers and engineers [9]. For performing systemlevel simulations, [10, 11] or [12] are freely accessible options. For link level, multiple commercial products are available that facilitate reproducible research, such as, iswireless LTE PHY LAB [13] or Mathworks LTE System Toolbox [14] and some noncommercial projects which were introduced in [15] and [16]. To the best of the authors’ knowledge, however, the Vienna LTE simulators are the only MATLABbased suite of simulation tools including LTE system and link level, publicly available under an academic use licence, thus, free of charge for academic researchers all over the world. The software suite consists of three simulators. The downlink link and system level simulators are comprehensively studied in [2, 9, 17]. In this paper, we introduce the latest member of the family of Vienna LTE Simulators, that is, the Vienna LTEA Uplink Link Level Simulator, downloadable at [18], and highlight our research conducted by means of this simulator.
1.1 Outline and contributions
We start with a brief recapitulation of the LTEA specifics and introduce the modulation and multiple access scheme and the employed MIMO signal processing of LTEA uplink in Section 2. We then develop a matrix model describing the inputoutput relationship of the LTEA uplink and present signaltointerferenceandnoise ratio (SINR) expressions for singlecarrier frequencydivision multiplexing (SCFDM) as well as orthogonal frequencydivision multiplexing (OFDM). The OFDM SINR expression and the performance of OFDM will serve as reference to study the effects of discrete Fourier transform (DFT) spreading imposed by SCFDM.
In Section 3, we investigate the physical layer performance of SCFDM and OFDM, comparing bit error ratio (BER) and peaktoaverage power ratio (PAPR). The BER for LTE single input single output (SISO) transmissions was already analysed in linklevel simulations by [19–21] and semianalytically by [22, 23]. By means of our simulator, we reproduce these results and provide bounds to predict the performance of SCFDM with respect to OFDM. The insights gathered by the BER simulations allow us to interpret the difference in throughput obtained by OFDM and SCFDM, as discussed in Section 4.
Based on the SINR expressions developed in Section 2, we present a limited feedback strategy for link adaptation in Section 4 and contrast the performance of LTE uplink with channel capacity and other performance upper bounds that account for practical design restrictions [24]. Until Section 5, we assume perfect CSI at the receiver. The remaining sections will describe methods to obtain CSI at the receiver.
In Section 5, we highlight and describe the demodulation reference signal (DMRS) structure employed in LTEA uplink to facilitate channel estimation of the timefrequency selective wireless channel.
Based on the obtained insights, we elaborate on the basic concept of DFTbased time domain channel estimation in Section 6 and review alternative code/frequency domain methods that can outperform DFTbased schemes [25].
Due to the increasing number of mobile users that stay connected while travelling in cars or (high speed) trains, we then shift our focus to high velocity scenarios. Such scenarios entail high temporal selectivity of the wireless channel, rendering accurate channel interpolation very important to sustain reasonable quality of service. We introduce and investigate basic concepts of channel interpolation in Section 7.
We briefly discuss open questions for future research in Section 8 and conclude in Section 9. Details to the handling of the simulator are provided in [26].
1.2 Notation
Matrices are denoted by bold uppercase letters such as H and vectors by bold lowercase letters such as h. The entries of vectors and matrices are accessed by brackets and subscripts, e.g. [h]_{ k } and [H]_{ k,n }. Spatial layers or receive antennas are denoted by superscripts in braces, e.g. x ^{(l)}. The superscripts (·)^{ T } and (·)^{ H } express transposition and conjugate transposition. ∥·∥_{2}, \(\ \cdot \_{\infty }\) and ∥·∥_{ F } symbolize the Euclidean norm, the Maximum norm and the Frobenius norm, respectively. The entrywise (Hadamard) product is denoted by ⊙ and the Kronecker product by ⊗. The all ones vector/matrix is denoted by . The operator X=Diag(x) places the vector x on the main diagonal of X, and conversely, the operator x=diag(X) returns the vector x from the main diagonal of X. A blockwise Toeplitz (circulant, diagonal) matrix is a block matrix with each matrix of Toeplitz (circulant, diagonal) shape. The size of matrices is expressed via their subscripts, whenever necessary.
2 LTEspecific system model and SINR
Right after the DFT spreading, the DMRS is inserted. The DMRS will be considered later for the purpose of channel estimation (CE). Next, MIMO precoding is carried out, exploiting a set of semiunitary precoding matrices W, pooled in the precoder codebook \(\mathcal {W}\), as defined in [1]. For LTEA uplink transmission, the precoding matrix applied for a given user is equal for all RBs assigned to this user. In case of spatial multiplexing, each spatial layer is transmitted with equal power.
Each antenna is equipped with its own OFDM modulator, consisting of subcarrier mapping, inverse fast Fourier transform (IFFT) and a CP addition. To cope with the channel dispersion and to avoid Intersymbol Interference (ISI), LTE employs a CP. As a result of multipath propagation, a previous symbol may overlap with the present symbol, introducing ISI and impairing the orthogonality between subcarriers, i.e. causing Intercarrier Interference (ICI) [28]. Normal and extended CP lengths, with a respective duration of 4.7 and 16.7 μs, are standardized, enabling a simple tradeoff between ISI immunity and CP overhead.
At the transmitter, the processing occurs in a reversed order. First, the OFDM demodulation/FFT takes place to get back into the frequency domain. The immunity to multipath propagation (stemming from the CP) allows to employ onetap frequency domain equalizers F without performance loss. At last, despreading delivers the data estimates.
All this previously informally described processing is linear, and we are able to formulate a matrixvector inputoutput relationship between a (stacked) datavector x and its estimate \(\boldsymbol {\hat {x}}\). For simplicity, we assume that the channel stays constant during one OFDM symbol. A detailed system description based on [29] can be found in [30].
In order to adapt the data transmission to the current channel state, LTEA applies limited feedback; a comprehensive specification follows in Section 4. Limited feedback is depicted via the feedback arrow in Fig. 2. The data vector \(\boldsymbol {x}^{(l)} \in \mathbb {C}^{N_{\text {SC}}\times 1}\) of layer l∈{1,…,L} contains modulated symbols for each of the N _{SC} subcarriers. The number of transmit layers depends on the LTEA specific rank indicator (RI) feedback. The data symbols are coded with a punctured turbo code whose rate is determined by the channel quality indicator (CQI). Subsequently, the codewords are mapped onto a quadrature amplitude modulation (QAM) alphabet (4/16/64 QAM), where the size of the alphabet depends on the CQI as well. All x ^{(l)} are stacked into one vector \(\boldsymbol {x}\in \mathbb {C}^{N_{\text {SC}}L\times 1}\) on which layerwise spreading and joint precoding—according to the precoding matrix indicator (PMI)—of all subcarriers take place. The subsequent OFDM modulator consists of the localized subcarrier mapping M, mapping N _{SC} subcarriers to the centre of an N _{FFT} point IFFT and the addition of the CP.
The additive noise is assumed independent across antennas and is distributed zero mean, white Gaussian \(\boldsymbol {n}^{(i)} \sim \mathcal {CN}\lbrace \boldsymbol {0}, {\sigma _{n}^{2}} \boldsymbol {I} \rbrace, \;\; i \in \lbrace 1,\dots, N_{\mathrm {R}}\rbrace \). The stacked noise vector \(\boldsymbol {n}=\left (\left (\boldsymbol {n}^{(1)}\right)^{T}, \dots, \left (\boldsymbol {n}^{(N_{\mathrm {R}})}\right)^{T} \right)^{T}\) is thus zero mean, white Gaussian as well.
The frequency domain onetap equalizer^{3} F is chosen conforming to different criteria, either the zero forcing (ZF) criterion, which removes all channel distortions at risk of noise enhancement, or the minimum mean squared error (MMSE) criterion, which tries to minimize the effects of noise enhancement and channel distortion.
After the despreading operation, the data estimates \(\boldsymbol {\hat {x}}\) of the noisy, received signal are given in Eq. (1), with the beforementioned convenient abbreviation (3), and \(\boldsymbol {D}_{N_{\text {FFT}}}\) is the DFT matrix of size N _{FFT}.
2.1 SCFDM SINR
selects that part of F H _{eff} effecting the lth layer. The second moment (power) of the zero mean transmit symbols is depicted by \({\sigma _{x}^{2}}\).
2.2 OFDM SINR
with appropriate number of zeros and a one at the lth position.
3 SCFDM features
We first discuss the main reason to apply SCFDM at uplink transmissions, namely PAPR. Then, we look at the expenses of employing it. We will see a worse performance of the coded transmission.
3.1 Peaktoaveragepower ratio
SCFDM is employed as the physical layer modulation scheme for LTE uplink transmission, due to its lower PAPR compared to OFDM [31]. Lower PAPR, or similarly lower crest factor, leads to reduced linearity requirements for the power amplifiers and to relaxed resolution specifications for the digitaltoanalogue converters at the user equipments, entailing higher power efficiency.
where the Euclidean norm in the denominator serves as an estimate for the ensemble average.
3.2 BER comparison over frequency selective channels
The additional spreading of SCFDM leads to an SINR expression that is constant on all subcarriers as for singlecarrier transmission, legitimating its name. The aim of this subsection is to analyse the SINR expression more carefully for the SISO case^{4} and draw conclusions on BER performance.
The detailed derivation is shown in the Appendix. The denominator of Eq. (16) is regularized and less sensitive to spectral notches.
In the low SNR regime \(\frac {{\sigma _{n}^{2}}}{{\sigma _{x}^{2}}} \gg \boldsymbol {H}_{k}^{2}\), this bound becomes tight. The higher the inverse SNR \(\frac {{\sigma _{n}^{2}}}{{\sigma _{x}^{2}}}\) in relation to the maximum of the transfer function, the tighter the bound becomes. The average OFDM SNR can never be larger to its maximum entry and is only equal for frequency flat channels. At low SNR, a lower BER is thus expected. Again, this presumption is validated by our simulation, showing that the uncoded BER is lower for SCFDM as for ODFM, cf. Fig. 5 b in solid lines. Although the uncoded BER shows superior performance, the coded BER is lower for OFDM due to the coding gains stemming from channel diversity, cf. Fig. 5 b dashed lines.
A bound for the maximum likelihood (ML) detection performance was derived in [36]. As the bandwidth increases, the slope of the BER curve achieved with MMSE receivers tends to the slope of ML detection, demonstrating the full exploitation of channel diversity by the MMSE equalizer, cf. Fig. 5 b black line.
4 Link adaptation
In the previous section, we investigated BER performance of OFDM and SCFDM transmission with different channel models and receivers. We observed significant BER degradation of SCFDM as compared to OFDM when ZF detection is employed, whereas coded BER is very similar when MMSE detection is used. In this section, we evaluate how such BER differences impact the actual throughput performance of LTEA uplink when transmission rate adaptation is employed. We first consider ideal rate adaptation and compare SCFDM transmission to OFDM with ZF and MMSE receivers. Then, we extend our singleuser MIMO CSI feedback algorithms proposed for LTE downlink in [37] to LTE uplink and evaluate their performance comparing to the throughput bounds developed in [24]. We also highlight some important basic differences between link adaptation in LTE up and downlink transmissions.
4.1 Performance with ideal rate adaptation
with the receiverspecific postdespreading (postequalization) SINRs from (??) and (8), respectively.
We observe a significant loss of achievable rate of SCFDM transmission compared to OFDM in Fig. 6, which is especially pronounced with ZF receivers due to noise enhancement. In Fig. 6, we also show the actual rate achieved by LTE uplink SCFDM transmission with ideal rate adaptation and compare to the performance obtained by OFDM transmission; the corresponding curves are denoted by LTE rate. We determine the performance of ideal rate adaptation by simulating all possible transmission rates, corresponding to CQI1 to CQI15, and selecting at each subframe the largest rate that achieves error free transmission. The figure also shows the throughput of the individual CQIs. We observe a gap between the LTE throughput with OFDM and SCFDM that is similar to the gap in terms of achievable rate. Notice that the performance loss with MMSE receivers is significantly smaller than with ZF detection, since MMSE avoids excessive noise enhancement.
We also observe in Fig. 6 a that the gain achieved by instantaneous rate adaptation, as compared to rate adaptation based on the longterm average SNR, is much larger for ZF SCFDM than for ZF OFDM; this is evident from the distance between the curves with rate adaptation (LTE rate) and the curves with fixed CQI. The reason for this behaviour is that the SNR of ZF SCFDM shows strong variability around its means, since it is dominated by the worstcase persubcarrier SNR according to (12); the average SNR over subcarriers of ZF OFDM, however, approximately coincides with its mean value. This implies that the optimal CQI of ZF SCFDM can vary significantly inbetween subframes, as reflected by the large average SNR variation required to increase the rate with fixed CQI from zero to its respective maximum. Yet, for ZF OFDM, the throughput of the individual CQIs follows almost a step function; hence, rate adaptation can be based on the longterm average SNR without substantial performance degradation.^{5}
Here, (26) resembles the high SNR approximation of the achievable rate of OFDM transmission with ZF detection as proposed in ([41] Eq. (14)); even more, for fixed L and letting N _{R} grow to infinity, (26) and ([41] Eq. (14)) tend to the same limit, due to channel hardening on each subcarrier with growing number of receive antennas.
and consider the smallest LTE bandwidth of N _{SC}=72 subcarriers. We observe that the proposed estimate performs very well even at this small bandwidth; notice, though, that a more realistic channel model with correlation over subcarriers may require larger bandwidth to validate the proposed estimate. Figure 7 also confirms the observation that singleuser MIMO OFDM and SCFDM with ZF detectors tend to the same limiting performance with increasing number of receive antennas.
This statement, however, will not hold true if the total number of layers grows proportionally with the number of receive antennas. For example, multiuser MIMO transmission with ZF equalization and singleantenna users achieves only a diversity order of N _{R}−L+1 [42], with L denoting the total number of layers being equal to the number of spatially multiplexed users. Hence, if L scales proportionally with N _{R}, channel hardening on each subcarrier will not occur and thus the performance of OFDM and SCFDM will not coincide.
4.2 Performance with realistic link adaptation
Instantaneous rate adaptation is an important tool for exploiting diversity of the wireless channel in LTE, by adjusting the transmission rate according to the current channel quality experienced by a user. LTE specifies a set of 15 different MCSs; the selected MCS is signalled by the CQI.
LTE additionally supports spatial link adaptation by means of codebookbased precoding with variable transmission rank. With this method, the precoding matrix \(\boldsymbol {W} \in \mathbb {C}^{N_{\mathrm {T}} \times L}\) satisfying \(\boldsymbol {W}^{\mathrm {H}} \boldsymbol {W} = 1/L\, \boldsymbol {I}_{L}\) is selected from a standarddefined codebook \(\mathcal {W}_{L}\) of scaled semiunitary matrices; furthermore, the number of spatial layers L can be adjusted to achieve a favourable tradeoff between beamforming and spatial multiplexing. The selected precoder and transmission rank are signalled, employing the PMI and the RI. In singleuser MIMO LTE uplink transmission, the same precoder is applied on all RBs that are assigned to a specific user, whereas frequencyselective precoding is supported in LTE downlink.
There is a basic difference between the utilization of CQI, PMI and RI in up and downlink directions of frequency division duplex (FDD) systems. In the downlink, the base station is reliant on CSI feedback from the users for link adaptation and multiuser scheduling [43], since channel reciprocity cannot be exploited in FDD. CQI, PMI and RI can be employed to convey such CSI from the users to the base station via dedicated feedback channels [44]. In the uplink, on the other hand, the base station can by itself determine CSI exploiting the sounding reference signals (SRSs) transmitted by the users. In this case, CQI, PMI and RI are employed by the base station to convey to the users its decision on link adaptation that has to be applied by the users during uplink transmission.
 1.Determine the optimal precoder for each transmission rank \(L \leq \min \left (N_{\mathrm {T}},N_{\mathrm {R}}\right)\) by maximizing transmission rate$$ \hat{\boldsymbol{W}}(L) = {\underset{\boldsymbol{W} \in \mathcal{W}_{L}}{\text{arg\,max}}}\, \sum_{l = 1}^{L} f\left({\operatorname{{SINR}}}^{\text{SCFDM},\;(l)} \left(\boldsymbol{W} \right)\right). $$(27)
Here, function f(·) maps SINR to rate; this could be either an analytical mapping, such as (19), or a mapping table representing the actual performance of LTE. In our simulations, we employ the bit interleaved codedmodulation (BICM) capacity as proposed in [37], since LTE is based on a BICM architecture.
 2.
Determine the optimal LTE transmission rates per layer for each L and \(\hat {\boldsymbol {W}}(L)\). We employ a target block error ratio (BLER) mapping in our simulations to determine the highest rate that achieves BLER≤0.1.
 3.
Select the transmission rank \(\hat {L}\) that maximizes the sum rate over spatial layers, utilizing the LTE transmission rates determined above.
 4.
Set the RI and PMI according to \(\hat {L}\) and \(\hat {\boldsymbol {W}}(L)\), respectively, and set the pCQI conforming to the corresponding LTE transmission rates.
The performance of LTE uplink transmission with full link adaptation (PMI and rank adaptive) is similar to the achievable BICM bound but shifted by approximately 3 dB. Notice that the saturation value is not the same because the highest CQI of LTE achieves 5.55 bit/channel use, whereas the BICM bound saturates at 6 bit/channel use. We also show the performance of LTE uplink when restricted to fixed precoding (rank adaptive) and fixed rank transmission (ranks 1, 2, 3, 4). We observe that rank adaptive transmission even outperforms the envelope of the fixed rank transmission curves, since instantaneous rank adaptation selects the optimal rank in each subframe independently. In terms of relative throughput, we see that LTE uplink with ZF receivers achieves around 40–50 % of channel capacity; remember, though, that this does not include CP and guard band overheads.
5 Reference symbols
After transmission over a frequency selective channel, this orthogonality has to be exploited to separate all effective MIMO channels at the receiver.
6 Channel estimation
with the stacked vector \(\boldsymbol {h}_{\text {eff}}^{(i)} = \left (\left (\boldsymbol {h}_{\text {eff}}^{(i,1)}\right)^{T},\dots,\left (\boldsymbol {h}_{\text {eff}}^{(i,L)}\right)^{T}\right)^{T}\) of all effective channels from L active layers to receive antenna i for which we will drop the subscript in the following.
6.1 Minimum mean square error estimation
with \(\boldsymbol {C}_{\boldsymbol {h}^{(i)}} = \mathbb {E} \lbrace \boldsymbol {h}^{(i)} \boldsymbol {h}^{(i)H}\rbrace \).
6.2 Correlationbased estimation
Here, \(\boldsymbol {\tilde {n}}^{(i)}\) has the same distribution as n ^{ ′ } ^{(i)} since (R ^{(l)})^{ H } is unitary and introduces phase changes only, cf. (29). Due to the allocation of DMRS on the same time and frequency resources on different spatial layers, the initial estimate \(\boldsymbol {\tilde {h}}^{(i,l)}\) of one effective MIMO channel actually consists of a superposition of all L effective MIMO channels to receive antenna i. The unintentional contributions in (41), from layers u≠l are interlayer interference, making it unsuited as initial estimate for coherent detection. Different methods to separate the different effective MIMO channels in (41) will be presented in the following.
6.2.1 DFTbased channel estimation
A wellknown approach for CE in LTEA uplink is DFTbased estimation [46], which aims to separate the MIMO channels contributing to (41) in time domain. For this, the individual cyclic shift of each DMRS is exploited. Applying a DFT on the receive signal, the individual phase shifts will translate into shifts in time domain. This makes a separation of channel impulse response (CIR)s from different MIMO channels possible by windowing. In our simulator, we implemented a DFTbased estimator as in [49] or [47].
6.2.2 Averaging
for \(\bar {\gamma } \leq k \leq N_{\text {SC}}\bar {\gamma } +1\). The second sum describes the averaging of \(\bar {\gamma }\) elements while the first sum describes the shift of this averaging window.
6.2.3 Quadratic smoothing
Similar to (42), this can be interpreted as another way to cope with the interlayer interference in (41) by post processing. This method does not use the DMRS structure explicitly but suppresses the interference by smoothing. It is therefore not able to cancel the complete interlayer interference but shows an improved performance at low SNR.
6.3 MSE and BER comparison
In terms of BER performance, at high SNR, naturally the estimation method with the lowest MSE leads to the smallest BER. At low SNR, the difference in CE MSE translates into very small differences in BER, meaning, we cannot gain too much from a good low SNR MSE performance of QS or MMSE estimation. Considering estimation complexity and that MMSE as well as QS require prior channel knowledge, SAV estimation is a good complexity performance tradeoff.
7 Channel interpolation
Under fast fading conditions, additional effects influence the performance of LTE uplink transmissions. Doppler shifts degrade the SINR by introducing velocity dependent ICI [51] whereas the SINR increases with increasing subcarrier spacing. The subcarrier spacing of 15 kHz that is used in LTE makes transmissions quite robust against ICI. The impact of ICI becomes only evident at high velocities and high SNR. Figure 12 b shows the BER for the case of perfect channel knowledge where the performance is only degraded by noise and ICI. At 200 km/h, the BER saturates due to ICI at high SNR whereas ICI mitigation techniques [52] show promising results to reduce this impact of ICI.
For a measurementbased comparison of interpolation techniques using channel estimates form both, the previous and the subsequent subframe, the reader is referred to [53].
8 Future research questions
Until now our research efforts on the Vienna LTEA Uplink Simulator have been concentrated on single links between user and base station, focusing on basic transceiver issues such as link adaptation and channel estimation. Our treatment of the link performance analysis is not considered complete. There are still important parameters to investigate, such as different forms of channel coding, enhanced channel estimation and detection [54, 55] and analysis of SCFDM sensitivity to synchronization mismatch, similar to our downlink investigations [56].
In the future, our scope will shift to multiuser multibase station scenarios, enabling on one hand exploitation of multiuser diversity in space, time and frequency and, on the other hand, consideration of interference inbetween simultaneous transmissions from multiple base stations. Even though, for reasons of computational complexity, simulations will be confined to comparatively small scenarios containing some few base stations, we still expect to extract valuable performance indicators for coordinated multipoint reception schemes [57], accounting for practical constraints, such as, limited backhaul capacity.
We will address crosslayer multiuser scheduling, jointly optimizing multiuser resource allocation and peruser link adaptation; this is an intricate issue in LTE, due to the nonlinear relationship between the resources assigned to a user and its corresponding SCFDM SINR (??); we have already addressed this issue for the downlink in [43]. Multiuser scheduling, furthermore, has to find a favourable tradeoff between transmission efficiency and fairness of resource allocation. We will extend existing downlink schedulers, which enable Paretoefficient transmission with arbitrary fairness, to the uplink specifics and compare to other proposals, e.g. [58].
The realization of massive MIMO in LTE compliant systems is another highly important research topic, since it promises an order of magnitude network efficiency gains through spatial multiplexing of users [59–61]. Yet, many issues still need to be better understood and resolved to enable efficient massive MIMO transmission in practice. One important step towards reasonable performance investigation of massive antenna arrays is to employ realistic channel models, such as, the 3GPP threedimensional channel model [62], which we plan to incorporate in future releases of our simulator.
9 Conclusions
For an LTEA uplink transmission model, we derived SINR expressions, both with and without DFT prespreading. We specialized these equations to ZF and MMSE receivers and showed that ZF performance is strongly affected by the worst subcarrier. Comparing the resulting BER we revealed that SCFDM performance is generally inferior to OFDM and that applying MMSE equalization is crucial to get closer to OFDM performance.
Based on the system’s SINR, we analysed the achievable rate. We also introduced a method to estimate the SCFDM rate for N _{R}>L. Further, a possible calculation of LTEA link adaptation parameters was proposed to achieve throughout close to performance bounds.
Lastly, we considered methods to gather CSI at the receiver. We compared the performance of various channel estimation and interpolation techniques. By incorporating the channel estimates of the previous subframe, we showed superior performance in terms of channel interpolation.
10 Endnotes
^{1} Note that we use the symbol n as time index and the vector n for noise, the distinction should be clear from the context.
^{2} Within this paper we focus on a single user’s link performance. Multi user / multi basestation simulations are possible to perform, but come at very long simulation times. For sake of readability we use the nonstandardized OFDM, SCFDM notation in the remainder of this manuscript.
^{3} A multitap equalizer applied on the intralayer interference visual in Fig. 3 c could possibly enhance the link performance.
^{4} The reduction to SISO is done to make our results comparable even to older frequency domain equalization (FDE) works, e.g., [63].
^{5} Notice, however, that instantaneous rate adaptation for ZF OFDM can be advantageous in case of frequencycorrelated channels [44].
^{6} Notice that the simulation setup is the same as employed in [24] for the investigation of LTE downlink transmission, thus, facilitating the comparison of up and downlink performance.
11 Appendix
11.1 General MIMO SCFDMA SINR expression

\(\hat {\boldsymbol {x}}_{\mathrm {s}}= \boldsymbol {S}^{(l)} \left (\boldsymbol {I}\odot \boldsymbol {K}\right) \boldsymbol {x}\)

\(\hat {\boldsymbol {x}}_{\mathrm {i}}= \boldsymbol {S}^{(l)}\left (\boldsymbol {K}\boldsymbol {I}\odot \boldsymbol {K}\right) \boldsymbol {x}\)

\(\hat {\boldsymbol {x}}_{\mathrm {n}}=\boldsymbol {S}^{(l)} \tilde {\boldsymbol {n}} \)
11.1.1 \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {s}} \hat {\boldsymbol {x}}_{\mathrm {s}}^{H} \rbrace \):
Assuming zero mean, white data with variance, \({\sigma _{x}^{2}}\) the diagonal elements of \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {s}} \hat {\boldsymbol {x}}_{\mathrm {s}}^{H} \rbrace \) are given by .
11.1.2 \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {i}} \hat {\boldsymbol {x}}_{\mathrm {i}}^{H} \rbrace \):
11.1.3 \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {n}} \hat {\boldsymbol {x}}_{\mathrm {n}}^{H} \rbrace \):
The noise covariance matrix is circulant as well and the detailed derivations can be found in [30].
11.1.4 SISO MMSE SCFDMA SINR expression
Declarations
Acknowledgements
This work has been funded by the Christian Doppler Laboratory for Dependable Wireless Connectivity for the Society in Motion and the Institute of Telecommunications, TUWien. The financial support by the Austrian Federal Ministry of Science, Research and Economy and the National Foundation for Research, Technology and Development is gratefully acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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