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.
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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
LTE operates on a timefrequency grid as shown in Fig. 1. The number of subcarriers is always a multiple of 12; 12 adjacent subcarriers over 7(or 6—in case of extended CP) successive OFDM symbols are called resource block (RB). Each RB thus consists of 12×7 (12×6) resource elements (REs), corresponding to the different timefrequency bins. A detailed description of LTE up and downlink is available, e.g. in [27].
We focus on those details necessary to describe our system model at time n ^{1}. LTE employs OFDM(A)^{2} as physical layer modulation and multiple access scheme in the downlink and SCFDM(A), i.e. DFTspreaded OFDM, in the uplink. In a SCFDM model, OFDM can be considered a special case. The major difference is an additional spreading and despreading stage at the transmitter and receiver, highlighted via dashed boxes in Fig. 2. The common parts of the system model will be described from left to right.
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.
Depending on the level of abstraction, our system model can be described via different channel matrices. The physical baseband time domain channel is described by a blockwise Töplitz matrix \(\boldsymbol {H} \in \mathbb {C}^{(N_{\text {FFT}}+N_{\text {CP}})N_{\mathrm {R}} \times (N_{\text {FFT}}+N_{\text {CP}})N_{\mathrm {T}}}\), with N _{T} transmit and N _{R} receive antennas, which turns blockwise circulant (H _{cir}) after addition (P _{addCP}) and removal (P _{remCP}) of an appropriately chosen CP of length N _{CP}. Finally, it turns diagonal after the IFFT and FFT on the transmitter and receiver, respectively. An example of the Töplitz and diagonal structured channel is demonstrated in Fig. 3 a, b, respectively.
The last step of the OFDM demodulator is the reversal of the localized subcarrier mapping M ^{H}. The effective MIMO channel from L transmit layers to N _{R} receive antennas, incorporating the precoder, the OFDM modulator, the timedomain MIMO channel H and the OFDM demodulator, is abstracted to one block matrix H _{eff}. This greatly facilitates the readability of all formulas later on.
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
The special structure of Eq. (1), due to the frequency domain onetap equalizer and the DFT spreading, yields a blockwise circulant inputoutput matrix, cf. Fig. 3 c,
This blockwise circulant structure produces a constant post equalization and post spreading SINR over all subcarriers within one layer [30]. The detailed derivation is provided in the Appendix.
where
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
In contrast to SCFDM, no spreading takes place for OFDM. The dashed boxes in Fig. 2 are replaced by identity matrices; they are simply omitted. Thus, different subcarriers k are orthogonal/independent and the equalizer treats the corresponding subcarrier channel H _{ k } only. We use the subscript k to denote the relevant part of the full channel matrix H _{eff} for the kth subcarrier. The corresponding indices within the diagonal matrix H _{diag} are , with the canonical base vectors e _{ k }. Using this notation, the effective subcarrier channel \(\boldsymbol {H}_{k} \in \mathbb {C}^{N_{R}\times L}\) is
and F _{ k } is its linear onetap equalizer. The SINR formula is quite similar to the SCFDM case, except that the SINR shows subcarrier dependency now. The SINR vector at layer l reads
with the selection vector
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.
The Vienna LTEA uplink simulator calculates the discretetime baseband PAPR with the default oversampling factor J=4 [32]. The discrete time signal on transmit antenna \(t \in \lbrace 1,\dots,N_{\mathrm {T}}\rbrace \) is therefore calculated as
where \(\boldsymbol {x}_{\text {pre}}^{(t)}\) is the transmit vector right after precoding and before the IFFT at transmit antenna t. The PAPR of the stacked vector \(\boldsymbol {s}_{\text {tx}}=\left (\left (\boldsymbol {s}_{\text {tx}}^{(1)}\right)^{T}, \dots, \left (\boldsymbol {s}_{\text {tx}}^{(N_{\mathrm {T}})}\right)^{T} \right)^{T}\) is calculated as
where the Euclidean norm in the denominator serves as an estimate for the ensemble average.
Figure 4 depicts the PAPR of OFDM and SCFDM obtained for different system bandwidths. Already for a small bandwidth (1.4 MHz), there is a significant reduction for SCFDM over OFDM. With increasing bandwidth, OFDM’s PAPR grows and the gains obtained by SCFDM become more and more pronounced. The PAPR also depends on the modulation alphabet; the smaller the alphabet, the smaller the PAPR. This effect is illustrated in dotted lines in Fig. 4, where we have shown the PAPR of 4QAM, exemplarily.
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.
We focus on the two most prominent equalizer concepts and start with the ZF equalizer, for whom the SCFDM signaltonoise ratio (SNR) expression (??) reduces to the harmonic mean
whereas the OFDM expression (8) is subcarrier dependent and becomes proportional to the channel transfer function
The average OFDM SNR
yields an upper bound on the SCFDMA SNR due to the harmonic mean—arithmetic mean inequality [33].
Equality in Eq. (15) holds if and only if the channel is frequency flat. The difference between the harmonic mean and the arithmetic mean gets increasingly pronounced, the more selective the channel becomes. We therefore expect the (uncoded) BER of SCFDM and ZF equalization to perform worse than OFDM, which is also validated by simulations. The BER simulations were carried out with CQI=4 on a PedB channel [34]. This modulation and coding scheme (MCS) employs 4QAM and has an effective coderate of 0.3008. As expected, the BER performance of SCFDM is worse than OFDM, both shown in Fig. 5 a in solid lines. Due to the spreading, SCFDM already expends all channel diversity and coding does not increase the SNR slope of the BER curve. This manifests in an almost parallel shift of the BER curve for SCFDM, as visual in Fig. 5 a in red dashed lines. None exploited diversity allows coded OFDM to increase the BER slope considerably, cf. Fig. 5 a blue dashed line.
The MMSE SINR expression is less intuitive and for the purpose of comparison, similar mathematical transformations as in [35] and [23] are required to arrive at
The detailed derivation is shown in the Appendix. The denominator of Eq. (16) is regularized and less sensitive to spectral notches.
An upper bound on the SINR can be obtained via the maximum of the transfer function H _{ k }
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
As demonstrated in the previous section, SCFDM provides a significant advantage in terms of PAPR over OFDM, thus relaxing linearity requirements of radio frequency power amplifiers for user equipments. Yet, this comes at the cost of coded BER degradation since channel diversity is lost and the performance is mostly dominated by the weakest subcarrier of a user, especially with ZF receivers; c.f. (12). This diversity loss cannot be recovered by forwarderrorcorrection channel coding, since the DFTspreading applied with SCFDM effectively causes an averaging over SINR observed on all scheduled subcarriers according to (??). As a consequence, SCFDM transmission over frequency selective channels achieves worse throughput than OFDM. This is demonstrated in Fig. 6, where we crosscompare the achievable rate, as defined in (18) and (19), and the actual throughput of SCFDM and OFDM transmission as obtained by the Vienna LTEA Uplink Simulator. We consider singleuser transmission over 5 MHz bandwidth assuming N _{T}=N _{R}=2 antennas at the user and the base station and L=2 spatial layers. The precoder is selected as a scaled identity matrix: \(\boldsymbol {W} = 1/\sqrt {L}\, \boldsymbol {I}_{L}\). We consider transmission over independent and identically distributed frequencyselective Rayleigh fading channels, emphasizing the difference between OFDM and SCFDM. The achievable rate in bits per OFDM/SCFDM symbol with Gaussian signalling and equal power allocation over subcarriers and spatial layers is calculated as
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}
In case N _{R}>L, we can easily estimate the achievable rate of SCFDM transmission: The perlayer SNR with ZF receivers is governed by the harmonic mean of the channel responses on the individual subcarriers, similar to (12)
with \(\boldsymbol {H}_{k} \in \mathbb {C}^{N_{\mathrm {R}} \times N_{\mathrm {T}}}\) denoting the OFDM channel matrix on subcarrier k. Assuming constant precoding and semicorrelated Rayleigh fading
with \(\boldsymbol {C}_{T} \in \mathbb {C}^{N_{\mathrm {T}} \times N_{\mathrm {T}}}\) determining the spatial correlation at the user equipment side, the matrix in the denominator of (20) follows a complex inverse Wishart distribution with N _{R} degrees of freedom and scale matrix \(\boldsymbol {C} = \left (\boldsymbol {W}^{\mathrm {H}} \boldsymbol {C}_{T} \boldsymbol {W}\right)^{1}\)
Letting \(N_{\text {SC}} \rightarrow \infty \), we can replace the term in the denominator of (20) with its expected value
This expected value only exists in case N _{R}>L [38]. For N _{R}=L, the diagonal elements of \(\overline {\boldsymbol {H}}\) follow a heavytailed inverted Gamma distribution [39, 40] with nonfinite first moment. Yet, for N _{R}>L, which is a common situation in cellular networks since the base station is mostly equipped with far more antennas than the users, the expected value is
Hence, we can estimate the achievable rate of SCFDMA transmission over semicorrelated Rayleigh fading channels
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.
In Fig. 7, we investigate the performance of the rate estimate (25) for N _{T}=L=4 and varying number of receive antennas. We assume \(\boldsymbol {W} = 1/\sqrt {L}\, \boldsymbol {I}_{L}\) and
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.
In principal, link adaptation must be jointly optimized with multiuser scheduling to optimize the performance of the system, since the effective SCFDM SINR (and thus the rate) of a user depends on the assigned RBs according to (??). For reasons of computational complexity, however, we assume that the multiuser schedule is already fixed and determine link adaptation parameters based on this resource allocation. We modify the approach proposed in [37] for LTE downlink transmission to determine the link adaptation parameters in four steps:

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.
In Fig. 8, we evaluate the performance of singleuser MIMO LTE uplink transmission over N _{T}=N _{R}=4 antennas with link adaptation, 1.4 MHz system bandwidth and ZF receiver. We do not consider signalling delays between the base station and the user. We employ the VehA channel model [34] and compare the absolute and relative (to channel capacity) throughput to the performance bounds proposed in [24].^{6} Channel capacity is obtained by applying singular value decomposition (SVD)based transceivers and waterfilling power allocation over subcarriers and spatial streams. Notice that we do not account for guard band and CP overheads when calculating the channel capacity; that is, we only consider subcarriers that are available for data transmission. The achievable channel capacity takes overhead for pilot symbols (DMRS and SRS) into account, corresponding to a loss of 16.7 % in our simulation. The achievable BICM bound additionally accounts for equal power allocation, codebookbased precoding and ZF detection as well as the applied BICM architecture as detailed in [24].
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
In LTE uplink, two types of reference signals are standardized. For CE and coherent detection, DMRS are exploited, while SRS are employed for channel sounding to enable frequency selective scheduling. For the purpose of CE, we will consider DMRS only. The reference symbols are defined in [1] and are explained in more detail in [45, 46]. As shown in Fig. 9, DMRS are multiplexed in the resource grid at OFDM symbol time n=3 in every slot. In a physical uplink shared channel (PUSCH) transmission of the LTEA uplink, a DMRS occupies all scheduled subcarriers. We assume that the user is assigned to all N _{SC} subcarriers starting at 0, i.e., k∈{0,1,…,N _{SC}−1}. We denote the ZadoffChu (ZC) base sequence on N _{SC} subcarriers for one slot by \(\boldsymbol {\bar {r}} \in \mathbb {C}^{N_{\text {SC}} \times 1}\). The base sequences \(\boldsymbol {\bar {r}}\) are complex exponential sequences lying on the unit circle fulfilling
In LTEA, the DMRS of different transmission layers in the same slot are orthogonal in terms of frequency domain code division multiplexing (FDCDM) [45]. This is obtained by cyclically shifting the base sequence. Similar to [47], DMRS on layer l for one slot are given by
with the cyclic shift operator
and the layer dependent cyclic shift α _{ l }. We further conclude from (28) to (30) that (R ^{(l)})^{H}=(R ^{(l)})^{−1} which implies \(\left (\boldsymbol {R}^{(l)}\right)^{H} \boldsymbol {R}^{(l)} = \boldsymbol {I}_{N_{\text {SC}}}\). Exploiting (28), the product of two DMRS from layers l and u with l,u∈{1,…,L} becomes
with Δ α=α _{ u }−α _{ l } being the cyclic phase shift between DMRS of two different spatial layers. The FDCDM orthogonality can therefore be exploited as
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
For channel estimation we exploit the system model only at symbol times, where reference signals are allocated. For a normal CP length, this is the 4th symbol in each slot, i.e. n=3 as shown in Fig. 9. Since we estimate the channel only at this single symbol time per slot, interpolation in time has to be carried out to obtain channel estimates for the whole resource grid. The effects of interpolation will be studied in Section 7. As illustrated in Fig. 2, the DMRS are added after DFT spreading, right before precoding. As the channel estimation takes place after the receiver’s DFT, just before equalization, the system model for CE amounts to an OFDM system. The system model (1) therefore reads as
with (preequalization) noise
and the stacked vector r consisting of DMRS \(\boldsymbol {r}^{(l)} \in \mathbb {C}^{N_{\text {SC}} \times 1}\) from all active spatial layers \(l\in \lbrace 1,\dots,L \rbrace \), i.e. \(\boldsymbol {r} = \left (\left (\boldsymbol {r}^{(1)}\right)^{T},\dots,\left (\boldsymbol {r}^{(L)}\right)^{T}\right)^{T}\). To consider the received signal separately for each receive antenna i, we can select the according part from y by left multiplying with the selector matrix S ^{(i)} from (6). The received signal y ^{(i)}=S ^{(i)} y on antenna i is given by
with the preequalization noise n ^{′} ^{(i)}=S ^{(i)} n ^{′} on receive antenna i and \(\boldsymbol {H}^{(i,l)}_{\text {eff}} = \boldsymbol {S}^{(i)} \boldsymbol {H}_{\text {eff}} \left (\boldsymbol {S}^{(l)}\right)^{T}\) being the (i,l)^{th} block of H _{eff}. Since \(\boldsymbol {H}^{(i,l)}_{\text {eff}}\) is diagonal, we exploit the relations \(\boldsymbol {R}^{(l)} = \text {Diag} \left (\boldsymbol {r}^{(l)} \right)\) and \(\boldsymbol {h}_{\text {eff}}^{(i,l)} = \text {diag} \left (\boldsymbol {H}_{\text {eff}}^{(i,l)} \right)\) to estimate a channel vector rather than a matrix and rearrange terms in (35) leading to
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
First, we present a MMSE estimator where we exploit (36) and estimate the stacked vector h ^{(i)} consisting of effective channels from all L active layers to receive antenna i. The MMSE CE for receive antenna i is given by
which leads to the wellknown solution [48]
with \(\boldsymbol {C}_{\boldsymbol {h}^{(i)}} = \mathbb {E} \lbrace \boldsymbol {h}^{(i)} \boldsymbol {h}^{(i)H}\rbrace \).
6.2 Correlationbased estimation
As a low complexity approach, we correlate (matched filter) the received signal with the reference symbol of layer l to obtain a channel estimate for the effective channel h ^{(i,l)} from layer l to receive antenna i
In fact, this correlation approach is optimum in a least squares (LS) sense
Inserting our system model (36) and exploiting (31), we obtain
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 physically meaningful channels, neighbouring subcarriers will be correlated within the coherence bandwidth [50]. We utilize this property and exploit the DMRS structure to perform frequency domain CE. As explained in [25], applying a sliding averaging on the initial estimate \(\boldsymbol {\tilde {h}}^{(i,l)}\) from (41) over \(\bar {\gamma }\) adjacent subcarriers (\(\bar {\gamma }\) equals 1,2,4,4 for L equals 1, 2, 3, 4, respectively) cancels the interlayer interference, assuming the channel to be frequency flat on these \(\bar {\gamma }\) consecutive subcarriers. The sliding average is given by
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
Another method exploiting channel correlations to estimate the channel in frequency domain is quadratic smoothing (QS). This scheme cannot remove the interlayer interference entirely, which manifests in a higher error floor, but shows improved performance at lower SNR in return. As explained in [25], this estimation method, exploiting the smoothing matrix Q and a smoothing factor γ, is given by
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
We assume a single user 2×2 MIMO transmission with N _{SC}=72 subarriers, a fixed number of layers L=2 and a typical urban (TU) channel model [34] at zero speed. We perform a simulation with onepoint extrapolation, cf. Section 7, and show the MSE curves of the proposed estimators in Fig. 10 a. The DFTbased CE (DbCE) shows the highest error flow of all estimators at high SNR while the MMSE estimator of course shows best performance over the whole SNR range. Compared to these two methods, the SlidingAveraging estimator (42), denoted by SAV, encounters an 8dB SNR penalty when compared to MMSE but comes closest to MMSE performance at high SNR. The quadratic smoothing estimation is denoted by QS and shows a significant improvement for low SNRs because it smooths over several observed channel coefficients. Quadratic smoothing performs uniformly better than DbCE over the whole SNR range and comes close to 4 dB to MMSE at low SNR. The high error floor shows that QS is not able to cancel all the interlayer interference.
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.
Another effect that hampers LTE transmissions at high velocities are temporal channel interpolation errors. While in the LTE downlink, the pattern used to multiplex data and reference symbols is a good tradeoff between a small temporal and spectral spacing accounting for highly frequency selective channels and fastfading channels and a rather small overhead, this is different in the uplink. As shown in Fig. 11 a, uplink DMRSs occupy the whole subband. While there is no need for interpolation over frequency, the temporal spacing is about twice the spacing of the reference symbols in the downlink. Furthermore, if frequency hopping is performed, the number of adjacent pilots transmitted in the same subband is two for intersubframe frequency hopping and only one for intrasubframe frequency hopping where frequency hopping is performed on a perslot basis. Due to this special structure channel, interpolation in the LTE uplink is a challenging problem. Therefore, we investigated various channel interpolation techniques using a single, two or three consecutive pilot symbols. Figure 11 b–e illustrates the channel interpolation techniques considered. The highest channel interpolation errors (Fig. 12 a) are observed for 1 point extrapolation where the channel estimate obtained in a certain slot is used to equalize the symbols within that slot and no interpolation is performed at all. The higher the number of pilots involved in channel interpolation, the lower the MSE gets. The results in terms of BER in Fig. 12 b show a similar behaviour.
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
The signal estimates are described via the inputoutput relationship in Eq. (1). We first slice out that part of K which acts on layer l by multiplying with the selector matrix S ^{(l)} from the left. As indicated in (1), the signal estimate consists of three contributions.

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

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

\(\hat {\boldsymbol {x}}_{\mathrm {n}}=\boldsymbol {S}^{(l)} \tilde {\boldsymbol {n}} \)
As x and \(\tilde {\boldsymbol {n}}\) are zero mean random quantities, their power is described by means of the second moment. To calculate the second moments, we take out the diagonal elements of the respective covariance matrices of each contribution.
Before, we derive the different covariance matrices, we recapitulate a required property of circulant matrices. A circulant matrix \(\boldsymbol {C} \in \mathbb {C}^{N \times N}\) is fully described by its first column c, as its eigenvectors are the DFT basisvectors and its eigenvalues are the DFT of \(\boldsymbol {c} = (c_{0}, c_{1}, \dots, c_{N1}) \).
The main diagonal elements c _{0} of C are given by
11.1.1 \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {s}} \hat {\boldsymbol {x}}_{\mathrm {s}}^{H} \rbrace \):
The inputoutput matrix K is of blockcirculant structure, as illustrated in Fig. 3 c. The eigenvalues of the diagonal blocks are given by \(\text {diag}\left (\boldsymbol {\Lambda }\right)=\boldsymbol {S}^{(l)} \text {diag}\left (\boldsymbol {F} \boldsymbol {H}_{\text {eff}} \right)\) and the diagonal elements of the lth diagonal block are then as asserted by Eq. (47), thus
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 \(\mathbb {E}\lbrace \hat {\boldsymbol {x}}_{\mathrm {i}} \hat {\boldsymbol {x}}_{\mathrm {i}}^{H} \rbrace \):
If C is circulant
is circulant as well and the diagonal elements of \(\tilde {\boldsymbol {C}} \tilde {\boldsymbol {C}}^{H}\) are the sum of the magnitude squares of \(\tilde {\boldsymbol {c}} = (0, c_{1}, \dots, c_{N1}) \). Using Parseval’s theorem, we arrive at
The interlayer interference consists of L−1Ctype blocks, where we simply average the magnitude squares of the eigenvalues, i.e. the corresponding blockpart of F H _{eff}. The intralayer interference is described via a \(\tilde {\boldsymbol {C}}\) block and is given in Eq. (50). Both contributions can be compactly written as
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
For a SISO system and a onetap equalizer, the expression F H _{eff} is of a diagonal shape. [30] has shown, that the MMSE equalizer for SCFDM equals the OFDM expression, i.e. \(\boldsymbol {F}=\left (\frac {{\sigma _{n}^{2}}}{{\sigma _{x}^{2}}} \boldsymbol {I} + \boldsymbol {H}_{\text {eff}}^{H} \boldsymbol {H}_{\text {eff}}\right)^{1} \boldsymbol {H}_{\text {eff}}^{H}\). Thus, the elements on the main diagonal of F H _{eff} are simply given by \(\boldsymbol {H}_{k}^{2}\left (\frac {{\sigma _{n}^{2}}}{{\sigma _{x}^{2}}}+\boldsymbol {H}_{k}^{2}\right)^{1}\), and we rewrite (??) to (55), where we have used the identity
from [23].
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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.
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Zöchmann, E., Schwarz, S., Pratschner, S. et al. Exploring the physical layer frontiers of cellular uplink. J Wireless Com Network 2016, 118 (2016). https://doi.org/10.1186/s1363801606091
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DOI: https://doi.org/10.1186/s1363801606091