 Research
 Open Access
Lowcomplexity interference variance estimation methods for coded multicarrier systems: application to SFN
 Marius Caus^{1}Email author,
 Ana I Perez Neira^{1, 2} and
 Markku Renfors^{3}
https://doi.org/10.1186/168761802013163
© Caus et al.; licensee Springer. 2013
 Received: 18 March 2013
 Accepted: 20 September 2013
 Published: 27 October 2013
Abstract
For singlefrequency network (SFN) transmission, the echoes coming from different transmitters are superimposed at the reception, giving rise to a frequency selective channel. Although multicarrier modulations lower the dispersion, the demodulated signal is sensitive to be degraded by intersymbol interference (ISI) and intercarrier interference (ICI). In view of this, we use channel coding in conjunction either with filter bank multicarrier (FBMC) modulation or with orthogonal frequency division multiplexing (OFDM). To deal with the loss of orthogonality, we have devised an interferenceaware receiver that carries out a soft detection under the assumption that the residual interference plus noise (IN) term is Gaussiandistributed. To keep the complexity low, we propose to estimate the variance of the IN term by resorting to dataaided algorithms. Experimental results show that regardless of the method, FBMC provides a slightly better performance in terms of coded bit error rate than OFDM, while the spectral efficiency is increased when FBMC is considered.
Keywords
 Orthogonal Frequency Division Multiplex
 User Equipment
 Cyclic Prefix
 Orthogonal Frequency Division Multiplex Symbol
 Symbol Error Rate
1 Introduction
With the aim of making an efficient use of the spectrum, 3GPP has introduced the multimedia broadcast and multicast service (MBMS) for delivering multimedia content to mobile users[1]. Among the possible transmission schemes, we focus on singlefrequency network (SFN), which has also been widely studied in the DVBT digital TV context. In a SFN, the frequency reuse factor is one, and thus, the user equipment (UE) receives several delayed versions of the same signal, giving rise to an artificial multipath channel. In this regard, the use of the orthogonal frequency division multiplexing (OFDM) technique facilitates the implementation of the SFN concept. It must be mentioned that depending on the intersite distance and the system parameters such as the sampling frequency and the subcarrier spacing, OFDM may not present a good balance between the resilience against multipath fading and the spectral efficiency. In other words, the minimum cyclic prefix (CP) length that is required to absorb the last echo may lead to a dramatic spectral efficiency reduction. However, if the CP is not long enough to encompass the maximum channel excess delay, the demodulated signal will suffer from intersymbol interference (ISI) and intercarrier interference (ICI). To overcome the OFDM limitations, we can resort to the filter bank multicarrier (FBMC) modulation[2]. The FBMC technique is designed to achieve maximum bandwidth efficiency since no redundancy is transmitted in the form of a CP. Furthermore, the subcarrier signals follow the Nyquist pulse shaping idea, which makes the FBMC modulation more robust against narrowband interferences and synchronization errors than OFDM. However, the channel is dispersive at the subcarrier level and thus the equalization is not a straightforward task; see, e.g.,[3, 4]. For further details about how OFDM and FBMC compare, we address the reader to[5].
To the best of the authors’ knowledge, FBMCbased SFNs have not been considered in the literature earlier. In this work, we assume the worst case scenario in which orthogonality is not restored neither in OFDM nor FBMC cases. To combat the drawbacks of this definitely challenging scenario, OFDM and FBMC transmission are combined with channel coding. Bearing this in mind, we propose mapping the received symbols into soft bits under the assumption that the residual interference plus noise (IN) is Gaussiandistributed. Simulationbased results show that the FBMC system is able to give the same or slightly better performance than OFDM in terms of bit error rate (BER) while the spectral efficiency significantly increases.
It is worth mentioning that previous works have compared coded FBMC and OFDM modulations in the presence of severe multipath fading[6–8]. These studies highlight the importance of characterizing the statistical information of the signal that corrupts the demodulated data. In this sense, we have formulated an analytical model for the additive noise and interference effects, as well as evaluated the complexity that is required to get a closedform expression of its variance. The order of the complexity may render the solution impractical. Therefore, we opt to use lowcomplexity estimation methods for the variance of the IN term. In this work, we have described two dataassisted strategies. The complexity analysis reveals that the proposed methods alleviate the complexity with respect to the cost of implementing the ideal receiver, which perfectly characterizes the variance of the IN term.
In view of the above discussion, the contributions of this paper are summarized as follows:

We evaluate the complexity and memory requirements of each variance estimator. In addition, the coded BER is assessed when each method is applied. The complexity costs and the system performance of the ideal receiver, which perfectly characterizes the variance of the IN term, are also provided. By confronting the ideal receiver with the receiver that relies on the variance estimation, we are able to provide insight into performance degradation when the complexity is reduced.

We carry out a comparison between OFDM and FBMC in the context of SFN transmission. Regardless of which estimator is implemented, the numerical results reveal that FBMC slightly outperforms OFDM in terms of coded BER. In this sense, multitap equalization plays a key role in FBMC systems to improve link reliability especially in highly frequencyselective channels. This allows us to conclude that the FBMC modulation scheme is a potential candidate to be used in a SFN. To the best of our knowledge, FBMC has not been considered earlier in the literature for SFN.
The remainder of the paper is organized as follows. In Section 2, we describe the system model in a SFN. The loss of orthogonality in the FBMC context is studied in Section 3. Based on the analysis in Section 3, we design in Section 4 a receiver that is interference aware. To alleviate the complexity, two dataassisted methods are investigated to estimate the variance of the IN term. Section 5 analyzes the complexity and the memory that is required to estimate the variance by each method. The coded BER is evaluated in Section 6 when the interferenceaware receiver is applied in OFDM and FBMC systems. Finally, Section 7 draws the conclusions.
2 System model
Here s[n] is the signal transmitted by all the BSs and w[n] is the additive white Gaussian noise. The term τ _{ i } stands for the delay of the i th transmitter with respect to the BS of reference, which can be identified without loss of generality with any index. The propagation conditions between the i th transmitter and the UE are modeled by the channel impulse response (CIR) h _{ i }[n] and by the combined effect of the path loss and the shadowing, which is expressed as${L}_{i}(\text{dB})={\stackrel{\u0304}{L}}_{i}(\text{dB})+{X}_{i}(\text{dB})$. The variable X _{ i }(dB) accounts for the shadowing, and it follows a Gaussian distribution with zero mean and standard deviation σ _{ x }. By contrast,${\stackrel{\u0304}{L}}_{i}(\text{dB})$ is a distancedependant term given by${\stackrel{\u0304}{L}}_{i}(\text{dB})=128.1+37.6{\text{log}}_{10}({d}_{i})$, where d _{ i } denotes the distance to the i th transmitter in kilometers[9]. With the aim of studying the most general case, we consider that the UE receives several delayed versions of the signal broadcasted by a given transmitter. Therefore, h _{ i }[n] is modeled as a tapped delay line, which indicates that the channel between the receiver and any transmitter is frequency selective.
3 SFN based on FBMC transmission
where S _{ m } = {mod_{ M }(m  1),m,mod_{ M }(m + 1)} contains three indexes. Note that mod_{ M }(x) accounts for the modulus M of x.
Now the vector y[k] contains complex QAM symbols. The closedform expression of$\mathbf{H}[0],\mathbf{H}[1]\in {\mathbb{C}}^{M\times M}$ can be found in[11]. The matrix formulation written in (12) highlights that only the previous block induces ISI. The reason lies in the fact that there is no time overlapping when OFDM is considered. Conversely, the prototype pulse used in the FBMC case exhibits better frequency localization properties than the sinclike shape. As a consequence, the matrices H[0],H[1] are not so sparse as {G[t]}.
where b _{ q } = [b _{ q }[L _{ b }] ⋯ b _{ q }[L _{ b }]]^{ T }, g _{ qm }[t] = [g _{ qm }[t + L _{ b }] ⋯ g _{ qm }[tL _{ b }]]^{ T }, and the noise vector is w _{ q }[k] = [w _{ q }[k + L _{ b }] ⋯ w _{ q }[kL _{ b }]]^{ T }. Since the OFDM modulation has been widely studied, we refrain from formulating the IN term in the OFDM context. Its expression can be computed as[11] details.
4 Interferenceaware receiver
Since w[n] is modeled as a complex circularly symmetric Gaussian variable with mean 0 and variance N _{0}, then$\mathbb{E}\left\{\Re \left(w[n]\right)\right\}=\mathbb{E}\left\{\Im \left(w[n]\right)\right\}=\mathbb{E}\left\{\Re \left(w[n]\right)\Im \left(w[n]\right)\right\}=0$. From this definition, it follows that (18) and (19) are zero. With that we conclude the proof that demonstrates that$\mathbb{E}\left\{{i}_{q}[k]\right\}=0$.
where L _{ v } is the maximum channel excess delay of the virtual channel. Even knowing$\left\{{e}^{j\pi \mathit{\text{qk}}}{\alpha}_{\mathit{\text{qm}}}^{k}[t]\right\}$ beforehand, it can be deduced from (21) that the complexity cost in terms of multiplications is 2L _{ v }. Taking into account which values of g _{ qm }[k] are different from zero, the total number of operations is approximately 30L _{ v } M. According to the expressions provided in[11], the complexity in the OFDM case is in the order of M ^{3}. From the perspective of reducing the complexity, we propose to estimate the power using two different methods.
4.1 Direct decision method
Unless the decisions are correct, i.e.,${d}_{q}[k]={s}_{q}^{0}[k]$, the estimator will be biased as (25) shows. This highlights the importance of regenerating the message as accurately as possible.
4.2 Refined direct decision method
Notice that the extrinsic LLRs computed by the first turbo decoder are not directly forwarded to the second turbo decoder to be used as a priori information. That is because the term${\stackrel{\u030c}{\sigma}}_{q,0}^{2}$, which is computed as (24) specifies, may excessively deviate from the real value. If so, errors will propagate on subsequent turbo iterations since the decoding algorithms are sensitive to the variance errors. It is also important to remark that, contrary to[13], the estimated symbols are not used to cancel out the interferences but to get a more accurate estimation of the transmitted symbols when compared to the approach followed in Section 4.1. As (25) indicates, the lower the symbol error rate is, the lower is the bias. The reason why we have discarded canceling out the interferences has to do with the complexity burden that is required to calculate the coefficients of the equivalent channels {g _{ qm }[k] }.
It is worth mentioning that the symbols in FBMC systems are modulated at a rate twice that of the symbols in OFDM. Hence, for a fixed window, the number of symbols that are used to calculate (23), (24), and (26) will be T/2 in the OFDM case.
5 Comparison of different estimation techniques
Complexity order and memory requirements of computing the variance of the IN term in all the subcarriers
Estimation method  Complexity order  Memory 

DDM  2^{ b } TM   
RDDM  TM(2^{ b } + 3b + 4)  TM 
5.1 DDM
The direct decision method relies on performing an exhaustive search over all the elements of the modulation alphabet as (24) highlights. Provided that b bits are used to represent any point of the constellation diagram, then the number of norms that has to be calculated is equal to 2^{ b } TM. On the positive side, the approach followed in Section 4.1 does not need to store any data.
5.2 RDDM
The complexity required to implement the refined direct decision method is tantamount to computing the complexity of the grey blocks in Figure3. Towards this end, we first analyze${\stackrel{\u030c}{\sigma}}_{q,0}^{2}$. According to (24), the number of norms to be computed is 2^{ b } TM, where b is the number of bits that constitutes the symbols. The next process that contributes to the increase of complexity is the conversion from soft bits to binary data. Considering that a single mapping only takes one operation together with the fact that the code rate is set to r _{code} = 1/3 implies performing$\frac{b}{3}\mathit{\text{TM}}$ operations. Then each bit has to be coded again by concatenating two identical systematic convolutional codes. Then it follows that the turbo code computes$\frac{2b}{3}\mathit{\text{TM}}$ coded bits, and each one is obtained after performing four logical operations. To regenerate the message, the coded bits are mapped into OQAM symbols by performing MT lookup operations. As (26) indicates, the refined estimation requires computing MT norms. In the last step, we multiply$\text{LLR}({c}_{l}{\u010f}_{q}[k])$ by$\frac{{\stackrel{\u030c}{\sigma}}_{q,0}^{2}}{{\stackrel{\u030c}{\sigma}}_{q,1}^{2}}$, which takes TM divisions and TM multiplications. According to the values gathered in Table1, the complexity costs when b = 2 results approximately in 14T M operations. Bearing in mind the complexity analysis conducted in Section 4, the number of operations to get the exact value of$\left\{{\sigma}_{q}^{2}\right\}$ is in the order of M ^{3} and 30L _{ v } M when the OFDM and the FBMC modulation scheme is considered, respectively. This highlights that although the strategy devised in Section 4.2 is the most complex, the method is still interesting because there is a good prospect of L _{ v } and M ^{2} being higher than T. Therefore, the refined direct decision method is likely to be more efficient than the computation of the real variance. Unlike what happens in the DDM, the regenerated message has to be stored so that it can be loaded later on to estimate the variance. As a result, there should be enough available memory to save TM symbols.
A feature that is common to all the algorithms described in Section 4 is that they do not operate in real time. That is, the variance is estimated after receiving T consecutive multicarrier symbols and storing the decision variables$\left\{{\u010f}_{q}[k]\right\}$ for 0 ≤ k ≤ T  1 and 0 ≤ q ≤ M  1. This observation reveals that in addition to the memory requirements that are summarized in Table1, the receiver has to reserve some additional space to save TM equalized symbols.
6 Numerical results
where the noise samples are generated as follows$w[n]\sim \mathcal{C}\mathcal{N}(0,{N}_{0})$ and E _{ s } is the symbol energy. The constant 4 accounts for the number of bits that constitutes the 16QAM symbols. It is worth mentioning that CP = 0 for FBMC systems and$\text{CP}=\frac{M}{4}$ in the OFDM case.
6.1 Benchmark
where S _{ a } contains the indices of those subcarriers that are active. Figure7 confirms that multitap equalization removes the interference more effectively than the singletap counterpart in highfrequency selective channels. Hence, the results of Figure7 are in accordance with the coded BER versus$\frac{{E}_{b}}{{N}_{0}}$ curves.
Regarding bandwidth efficiency, the spectral efficiency reaches 1.20 bits/s/Hz for the FBMC case. The OFDM counterpart results in 0.96 bits/s/Hz.
6.2 Evaluation of the proposed interferenceaware receiver
7 Conclusions
In the worst case scenario, in the context of OFDM and FBMC, orthogonality is destroyed in a SFN. Based on this and taking into account that both schemes are combined with channel coding, we have devised an interferenceaware receiver, which relies on the knowledge of the statistics of the IN term. When the residual interference is perfectly characterized, the OFDM technique and the FBMC modulation based on singletap equalization perform reasonably close regardless of the channel. Therefore, we may opt to use the same singletap equalizer both in OFDM and FBMC systems, resulting in a similar performance in terms of BER. To alleviate the complexity we propose two different dataassisted methods to estimate the variance of the IN term. The complexity analysis reveals that the RDDM is more complex than the DDM. The simulationbased results demonstrate that the additional complexity is well justified because the receiver based on the RDDM achieves the lowest coded BER. When the RDDM is considered under severe multipath fading, it becomes important to use multitap equalization in FBMC systems. Anyway, it is enough to use threetap subcarrier equalizers, which leads to relatively minor increase in the overall computational complexity. With proper multitap equalization, FBMC is able to reach or slightly exceed the error performance of CPOFDM while supporting about 25% higher data rate.
Declarations
Acknowledgements
This work has received funding from the Spanish Ministry of Economy and Competitiveness (Ministerio de Economia y Competitividad) under project TEC201129006C0302 (GRE3NLINKMAC) and from the Catalan Government (2009SGR0891). This work was partially supported by the European Commission through the Emphatic project (ICT318362).
Authors’ Affiliations
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