Performance evaluation of IBDFEbased strategies for SCFDMA systems
 Adão Silva^{1}Email author,
 José Assunção^{1},
 Rui Dinis^{2} and
 Atílio Gameiro^{1}
https://doi.org/10.1186/168714992013292
© Silva et al.; licensee Springer. 2013
Received: 15 July 2013
Accepted: 9 December 2013
Published: 30 December 2013
Abstract
The aim of this paper is to propose and evaluate multiuser iterative block decision feedback equalization (IBDFE) schemes for the uplink of singlecarrier frequencydivision multiple access (SCFDMA)based systems. It is assumed that a set of single antenna users share the same physical channel to transmit its own information to the base station, which is equipped with an antenna array. Two spacefrequency multiuser IBDFEbased processing are considered: iterative successive interference cancellation and parallel interference cancellation. In the first approach, the equalizer vectors are computed by minimizing the mean square error (MSE) of each individual user, at each subcarrier. In the second one, the equalizer matrices are obtained by minimizing the overall MSE of all users at each subcarrier. For both cases, we propose a simple yet accurate analytical approach for obtaining the performance of the discussed receivers. The proposed schemes allow an efficient user separation, with a performance close to the one given by the matched filter bound for severely timedispersive channels, with only a few iterations.
Keywords
1. Introduction
Singlecarrier frequencydivision multiple access (SCFDMA), a modified form of orthogonal frequencydivision multiple access (OFDMA), is a promising solution technique for high data rate uplink communications in future cellular systems.
When compared with OFDMA, SCFDMA has similar throughput and essentially the same overall complexity. A principal advantage of SCFDMA is the peaktoaverage power ratio (PAPR), which is lower than that of OFDMA[1, 2]. SCFDMA was adopted for the uplink, as a multiple access scheme, of the current longterm evolution (LTE) cellular system[3].
Singlecarrier frequency domain equalization (SCFDE) is widely recognized as an excellent alternative to OFDM, especially for the uplink of broadband wireless systems[4, 5]. As other block transmission techniques, SCFDE is suitable for high data rate transmission over severely timedispersive channels due to the frequency domain implementation of the receivers. Conventional SCFDE schemes employ a linear FDE optimized under the minimum mean square error (MMSE) criterion. However, the residual interference levels might still be too high, leading to performance that is still several decibels from the matched filter bound (MFB). Nonlinear time domain equalizers are known to outperform linear equalizers and DFE are known to have good performancecomplexity tradeoffs[6]. For this reason, there has been significant interest in the design of nonlinear FDE in general and decision feedback FDE in particular, with the IBDFE being the most promising nonlinear FDE[7, 8]. IBDFE was originally proposed in[9] and was extended for a wide range of scenarios in the last 10 years, ranging from diversity scenarios[10, 11], MIMO systems[12], CDMA systems[13, 14], and multiaccess scenarios[15, 16], among many other. Essentially, the IBDFE can be regarded as a low complexity turbo equalizer[17–20] implemented in the frequency domain that do not require the channel decoder output in the feedback loop, although true turbo equalizers based on the IBDFE concept can also be designed[21–23]. An IBDFEbased scheme specially designed for offset constellations (e.g., OQPK and OQAM) was also proposed in[24]. In the context of cooperative systems, an IBDFE approach was derived to separate the quantized received signals from the different base stations (BSs)[25].
Works related to IBDFE specifically designed for SCFDMAbased systems are scarce in the literature. In[26], the authors proposed an IBDFE structure consisting of a frequency domain feedforward filter and a time domain feedback filter for singleuser SCFDMA systems. An iterative frequency domain multiuser detection for spectrally efficient relaying protocols was proposed in[27], and a frequency domain softdecision feedback equalization scheme for single user SISO SCFDMA systems with insufficient cyclic prefix was proposed in[28].
In this paper, we consider a broadband wireless transmission over severely timedispersive channels, and we design and evaluate multiuser receiver structures for the uplink singleinput multipleoutput (SIMO) SCFDMA systems that are based on the IBDFE principle. It is assumed that a set of single antenna user equipment (UE) share the same physical channel to transmit its own information to the base station, which is equipped with an antenna array. Two multiuser IBDFEbased processing schemes are considered, both with the feedforward and feedback filters designed in space frequency domain: iterative successive interference cancellation (SIC) and parallel interference cancellation (PIC). In the first approach, the equalizer vectors are computed by minimizing the mean square error (MSE) of each individual user at each subcarrier. In the second one, the equalizer matrices are obtained by minimizing the overall MSE of all users at each subcarrier. For both cases, we propose a quite accurate analytical approach for obtaining the performance of the proposed receivers.
The remainder of the paper is organized as follows: Section 2 presents the multiuser SIMO SCFDMA system model. Section 3 presents in detail the considered multiuser IBDFEbased receiver structures. The feedforward and feedback filters are derived for both cases and analytical approach for obtaining the performance is discussed. Section 4 presents the main performance results, both numerical and analytical. The conclusions will be drawn in Section 5.
Notation: Throughout this paper, we will use the following notations. Lowercase letters, uppercase letters, are used for scalars in time and frequency, respectively. Boldface uppercase letters are used for both vectors and matrices in frequency domain. The index (n) is used in time while the index (l) is for frequency. (.)^{ H }, (.)^{ T }, and (.)^{ * } represent the complex conjugate transpose, transpose, and complex conjugate operators, respectively, [.] represents the expectation operator, I_{ N } is the identity matrix of size N × N, CN(.,.) denotes a circular symmetric complex Gaussian vector, tr(A) is the trace of matrix A, and e_{ k } is an appropriate column vector with 0 in all positions except the k th position that is 1.
2. System model
assuming that the cyclic prefix is long enough to account for channel impulse responses between the UEs and the BS. In (1),${H}_{k,l}^{\left(m\right)}={\alpha}_{k}{H}_{k,l}^{\mathit{\text{cfr}}\left(m\right)}$ represents the channel between user k and the m th antenna of the BS on subcarrier l, where${H}_{k,l}^{\mathit{\text{cfr}}\left(m\right)}$ denotes the normalized channel frequency response, i.e.,$\mathbb{E}\left[{\left{H}_{k,l}^{\mathit{\text{cfr}}\left(m\right)}\right}^{2}\right]=1$, while the coefficient α_{ k } is a weighting factor that accounts for the combined effects of power control and propagation losses. The average received power associated to the k th UE is therefore α_{ k }^{2} and${N}_{l}^{\left(m\right)}\sim \mathcal{\text{CN}}\left(0,{\sigma}_{N}^{2}\right)$ is the noise.
The channel vector of the k th user is defined as${H}_{k,l}=\left[{H}_{k,l}^{\left(1\right)}\dots {H}_{k,l}^{\left(M\right)}\right]$.
3. Multiuser IBDFE receiver strategies
In this section, we present in detail the multiuser iterative frequency domain receiver design strategies based on the IBDFE concept[6]. Two iterative approaches are considered: SIC and PIC.
3.1 IBDFE SIC approach
with${\mathbf{R}}_{s}={\sigma}_{S}^{2}{\mathbf{I}}_{K}$ and${\mathbf{R}}_{N}={\sigma}_{N}^{2}{\mathbf{I}}_{M}$, being the correlation matrices of data symbols and noise on each carrier.
The Lagrangian multiplier is selected, at each iteration i, to ensure the constraint$\frac{1}{\mathit{L}}\sum _{l=0}^{L1}{\mathbf{F}}_{k,l}^{\left(i\right)T}{\mathbf{H}}_{k,l}^{T}=1$. It should be emphasizes that for the first iteration (i = 1), and for the first UE to be detected, P^{(0)} is a null matrix and${\overline{\mathbf{S}}}_{k,l}^{\left(0\right)},k=1$ is a null vector.
3.2 IBDFE PIC approach
where${\mathbf{F}}_{l}^{\left(i\right)}=\left[\begin{array}{ccc}\hfill {\mathbf{F}}_{1,l}^{\left(i\right)}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{F}}_{K,l}^{\left(i\right)}\hfill \end{array}\right]$ is a matrix of size MxK with all UEs' feedforward vector coefficients,${\mathbf{B}}_{l}^{\left(i\right)}={\left[\begin{array}{ccc}\hfill {\mathbf{B}}_{1,l}^{\left(i\right)}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{B}}_{K,l}^{\left(i\right)}\hfill \end{array}\right]}^{T}$ is a matrix of size KxK with all UEs' feedback vector coefficients, and${\tilde{\mathbf{S}}}_{l}^{\left(i\right)}={\left[\begin{array}{ccc}\hfill {\tilde{S}}_{1,l}^{\left(i\right)}\hfill & \hfill \cdots \hfill & \hfill {\tilde{S}}_{K,l}^{\left(i\right)}\hfill \end{array}\right]}^{T}$.
Note that the correlation matrices${\mathbf{R}}_{l}^{\mathbf{Y}}$,${\mathbf{R}}^{\overline{\mathbf{S}}\mathbf{,}\overline{\mathbf{S}}}$, and${\mathbf{R}}_{l}^{\overline{\mathbf{S}}\mathbf{,}\mathbf{Y}}$ were already defined in (14).
In this approach, the Lagrangian multiplier is selected, at each iteration i, to ensure the constraint$\frac{1}{\mathit{L}}\sum _{l=0}^{L1}\text{tr}\left({\mathbf{F}}_{l}^{T}{\mathbf{H}}_{l}^{T}\right)=K$. Since all users are detected in parallel, for the first iteration (i = 1), P^{(0)} is a null matrix and${\overline{\mathbf{S}}}_{l}^{\left(0\right)}$ is a null vector.
The complexity of the SIC approach is slightly higher than the PIC one. For the SIC, we need to invert a matrix of size M xM for each user on each iteration, while for the PIC one, we need to invert a matrix of size M xM for all users on each iteration, i.e., the SIC approach requires K − 1 more matrix inversions per iteration. Since in the receiver SIC structure, each user is detected individually and sequentially, the delay is also higher.
4. Performance results
In this section, we present a set of performance results, analytical and numerical, for the proposed IBDFEbased PIC and SIC receiver schemes. Two different scenarios are considered:

Scenario 1, we assume two UEs (K = 2) and a BS equipped with two antennas (M = 2).

Scenario 2, we assume four UEs (K = 4) and a BS equipped with four antennas (M = 4).
For both scenarios, the main parameters used in the simulations are NFFT size of 1,024; LDFT size set to 128 (this represents the data symbols block associated to each UE); sampling frequency set to 15.36 MHz; useful symbol duration is 66.6 μs, cyclic prefix duration is 5.21 μs; overall OFDM symbol duration is 71.86 μs; subcarrier separation is 15 kHz, and a QPSK constellation under Gray mapping rule, unless otherwise stated. Most of the parameters are based on LTE system[33].
The channel between each UE and the BS is uncorrelated and severely time dispersive, each one with rich multipath propagation and uncorrelated Rayleigh fading for different multipath components. Specifically, we assume a L_{ p } = 32path frequencyselective block Rayleigh fading channel with uniform power delay profile (i.e., each path with average power of 1/L_{ p }). The same conclusions could be drawn for other multipath fading channels, provided that the number of separable multipath components is high. Also, we assume perfect channel state information, synchronization and α_{ k }^{2} = 1, ∀ k. The results are presented in terms of the average bit error rate (BER) as a function of E_{ b }/N_{0}, with E_{ b } denoting the average bit energy and N_{0} denoting the onesided noise power spectral density. In all scenarios, we present the theoretical and simulation average BER performances for both proposed receiver structures: IBDFE PIC and SIC. For the sake of comparisons, we also include the matched filter bound (MFB) performance.
From Figure 5, we can also see that the analytical approach proposed for the IBDFE SIC structure is very accurate. The BER performance approaches, with a number of iterations as low as 4, very closely to the limit obtained with the MFB. This means mean that this receiver structure is also able to efficiently separate the UEs, while taking advantage of the spacefrequency diversity inherent to the MIMO SCFDMAbased systems. Comparing the SIC and the PIC approach, it is clear that for the first iteration the SIC approach outperforms the PIC one. It can be observed a penalty of approximately 1 dB of the PIC against the SIC, for a BER = 10^{−3}. This is because the SICbased structure to detect a given user takes into account the previous detected ones, with the exception for the first user. However, when the number of iteration increases, the performance of the PIC approach tends to the one given by the SIC approach. We can observe that the BER performance of both approaches is basically the same for four iterations.
The previous results indicate that IBDFE receivers can have excellent performance, close to the MFB, for MIMO systems with QPSK constellations. One question that arises naturally is if this is still valid for larger constellations such as QAM constellations. In fact, the performance of a DFE for larger constellations can be seriously affected due to error propagation effects. As an example, we present in Figure 8 the performance results for 16QAM constellations in the second scenario, considering IBDFE SIC approach. Clearly, we are still able to approach the MFB, although we need more iterations, the convergence is less smooth and we only approach the MFB for lower BER (and, naturally, larger SNR). Although these good results might be somewhat surprising, we should have in mind that an IBDFE is not a conventional DFE due to the noncausal nature of the feedback. Moreover, the error propagation effects are much lower in IBDFE receivers due to the following issues:

Symbol errors (which are in the time domain) are spread over all frequencies. Due to the frequencydomain nature of the feedback loop input, a symbol error has only a minor effect on all frequencies.

The FDE is designed to take into account the reliability of estimates employed in the feedback loop. When we have a large number of symbol errors, the reliability decreases and the weight of the feedback part decreases.

When we have a decision error, we usually move to one of the closer constellation symbols, i.e., the magnitude of the error is usually the minimum Euclidean distance of the constellation, regardless of the constellation size. This is especially important for larger constellations.
As we pointed out, an IBDFE can be regarded as a complexity turbo equalizer implemented in the frequencydomain which does not employ a channel decoder in the feedback loop. For this reason, it has a turbolike behavior with good performance provided that the BER is low enough. That is why we can only approach the MFB for larger SNR.
5. Conclusions
In this paper, we designed and evaluated multiuser receiver structures based on the IBDFE principle for the uplink SIMO SCFDMA systems. Two multiuser IBDFE PIC and SICbased processing schemes were considered. In the first approach, the equalizer vectors were computed by minimizing the mean square error (MSE) of each individual user at each subcarrier. In the second one, the equalizer matrices were obtained by minimizing the overall MSE of all users at each subcarrier. For both cases, we proposed a quite accurate analytical approach for obtaining the performance of the proposed receivers.
The results have shown that the proposed receiver structures are quite efficient to separate the users, while allowing a closetooptimum spacediversity gain, with performance close to the MFB (severely timedispersive channels) with only a few iterations. The performance of both PIC and SIC receiver structures is basically the same after three or four iterations. However, the main drawback of the SIC approach is the delay in the detection procedure, which is larger than for the PIC, since it detects one user at each time. Thus for practical systems, where the delay is a critical issue, the PIC approach can be the best choice.
To conclude, we can clearly state that these techniques are an excellent choice for the uplink SCFDMAbased systems, already adopted by the LTE standard.
Declarations
Acknowledgements
This study was supported by the Portuguese Fundação para a Ciência e Tecnologia (FCT) COPWIN (PTDC/EEITEL/1417/2012), CROWN (PTDC/EEATEL/115828/2009), and ADIN (PTDC/EEITEL/2990/2012) projects.
Authors’ Affiliations
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