Characterization of the effects of multitap filtering on FBMC/OQAM systems
 Marius Caus^{1}Email author,
 Ana I PerezNeira^{1, 2} and
 Adrian Kliks^{3}
https://doi.org/10.1186/16876180201484
© Caus et al.; licensee Springer. 2014
Received: 2 December 2013
Accepted: 19 May 2014
Published: 5 June 2014
Abstract
The filter bank multicarrier modulation based on offset quadrature amplitude modulation (FBMC/OQAM) is being considered as an eligible technology for future wireless communications. However, the orthogonality may be destroyed in the presence of multipath fading, and thus, the demodulated data may be affected by intersymbol and intercarrier interference. To restore the orthogonality, it is deemed necessary to either precode the symbols to be transmitted or equalize the demodulated data. Under severe propagation conditions, both the precoders and equalizers should perform multitap filtering. In this paper, we try to determine whether it is the best choice to combat the channel at the transmit, or at the receive side. The answer to this question is not trivial, since with data precoding, the transmit power may increase, while equalization at the receive side may enhance the noise power. To cast light on this issue, this paper characterizes the average transmit and noise power to determine the influence of multitap filtering on the transmit signal power and on the noise variance. The analysis conducted in this paper reveals that if multitap precoding yields a power increase, then the noise variance would increase with the same magnitude if the same linear filter was moved from the transmit to the receive side. Therefore, if symbols are properly scaled when the transmit power increases, there is no degradation due to placing the complexity burden at the reception rather than at the transmission. To scale the symbols, it is mandatory to know the additional transmit power. In this sense, a lowcomplexity method to estimate the power increase is proposed. Simulationbased results confirm that the estimation is reasonably accurate.
Keywords
1 Introduction
Networks are evolving in such a way that different systems with specific characteristics coexist in the same area. Cognitive radio networks, where secondary users share the same spectrum as the primary licensed users, are a good example. In these systems, secondary users have to detect the spectrum conditions to transmit on those bands where the primary user is inactive [1]. In this regard, there are initiatives to use spectral slots in the professional mobile radio (PMR) band and white spaces freed up by the current digital television system. In these scenarios, the devices that transmit in the unoccupied bands have to guarantee that no interference will be induced in legacy primary networks. Therefore, future wireless communications should be able to transmit in a fragmented spectrum where different spectral components are unlikely to be tightly synchronized. This observation highlights that it is deemed necessary to utilize spectrally agile waveforms, and thus, further research beyond the established orthogonal frequency division multiplexing (OFDM) technique is required [2]. It is worth mentioning that due to the aforementioned features, other modulation formats are also considered for the next generation systems [3,4].
The filter bank multicarrier modulation based on offset quadrature amplitude modulation (FBMC/OQAM), also known as OFDM/OQAM, is a potential candidate to satisfy the upcoming needs of future wireless communications [5]. This modulation is gaining momentum since it achieves maximum bandwidth efficiency as no redundancy is transmitted. In addition, the data transmitted on each subcarrier can be shaped with frequency welllocalized waveforms, which allow a flexible use of the spectrum [6]. An efficient implementation of the FBMC/OQAM scheme for noncontiguous spectrum use, which is especially relevant in cognitive radio networks, is presented in [7].
In summary, FBMC/OQAM exhibits a low outofband emission while the spectral efficiency is not degraded, which is a desirable feature to protect legacy users in cognitive radio networks or when perfect synchronization between nodes cannot be attained. However, the FBMC/OQAM transmit signal induces interference, which is known as modulationinduced interference or intrinsic interference. Under ideal propagation conditions, the receiver can recover the data perfectly if the subband pulses fulfill the perfect reconstruction (PR) property [5]; thus, interference is eliminated. It must be mentioned that multipath fading destroys this property, and as a consequence, intersymbol interference (ISI) and intercarrier interference (ICI) is present at detection. This highlights that the channel has to be counteracted to restore the orthogonality. It is well known that in general, multiband processing is required to better cope with the interference coming from adjacent subcarriers. On the negative side, the complexity is substantially increased when subcarriers are jointly processed. The complexity order is $\mathcal{O}\left({L}_{s}{L}_{t}\right)$ when each subchannel is equalized after combining the information received in L_{ s } adjacent subcarriers through a multiband processor that uses L_{ t } taps per subband. As L_{ s } increases, the number of arithmetic operations required to counteract the channel increases as well. Thanks to the good spectral confinement exhibited by the subcarrier signals, the performance enhancement brought by the joint processing of L_{ s }=3 subcarriers is marginal with respect to the case that L_{ s }=1 [8]. The improvement could be more significant by increasing L_{ s } beyond 3, yet the complexity would substantially augment. For this reason, we discard the implementation of multiband equalizers, and we focus on filters that equalize the channel on a persubcarrier basis. Some examples can be found in the literature when the channel is compensated at reception [9,10,11,12,13,14] or at the transmit side [15].
When the frequency selectivity of the channel becomes appreciable at the subcarrier level, both the equalizers and the precoders have to perform multitap filtering to compensate the channel. Otherwise, orthogonality is destroyed and residual interference terms may significantly degrade the performance. Among all the possible designs, we favor the subband processing based on the frequency sampling approach described in [9] because it offers a good tradeoff between performance and implementation complexity. As the authors highlight in [16], the complexity needed to design the optimum minimum mean square error (MMSE) equalizer, which is designed in the time domain [10], is higher while the bit error rate (BER) is only lower than that achieved by the frequency sampling method at a very high energybittonoise ratio in highly frequencyselective channels [8]. It is important to mention that the equalizer in [9] can be applied at the transmitter as it is proposed in [15].
This paper studies the impact of multitap filtering in FBMC/OQAM systems to determine whether it is the best choice to combat the channel at the transmit or at the receive side. A similar comparison has been made in narrowband multipleinputmultipleoutput (MIMO) systems by examining different types of linear processing [17]. Therein, the authors have concluded that the receive filters outperform the transmit filters for low signaltonoise ratio (SNR), while the transmit filters provide better results for high SNR. The conclusions drawn in [17] may not necessarily be the same in this paper because of the characteristics of FBMC/OQAM together with the fact that we propose to apply the same processing either at transmission or at reception. Regarding the FBMC/OQAM specificities, it is important to remark that the FBMC/OQAM signal structure may be responsible for boosting the power when multitap precoders are used. By analogy, multitap equalization may result in noise enhancement. These two features highlight the necessity of conducting a new analysis, taking into account the FBMC/OQAM characteristics. Based on that, the contributions of the paper may be summarized as follows:

The average transmit power has been characterized when the channel is preequalized at transmission.

The average noise power has been characterized when the channel is equalized at reception.

A lowcomplexity method to estimate the additional transmit power is devised when precoders are designed according to the frequency sampling approach.
From the closedform expressions derived in this paper, we may conclude that if the transmit power increases/ decreases, then the noise variance increases/decreases as well with the same magnitude, as long as the same filters are used as precoders or equalizers. This reveals that if the transmitted symbols are properly scaled when the transmit processing boosts the power, then there is no degradation due to equalizing the demodulated data instead of precoding the symbols to be transmitted. In view of this discussion, it is of paramount importance to determine if the power is boosted due to transmit processing operations and to state if some normalization is required. The analysis that has been conducted allows us to approximately formulate the power as the function of the statistical expectation of precoders, which depends on the statistical channel information. Thus, a priori knowledge of channel statistics is required. It is worth mentioning that many algorithms, such as the MMSE channel estimation, also make use of channel statistics information [18]. Alternatively, we may compute the instantaneous transmit power given the precoders. However, the power has to be recalculated if precoders are updated to adapt to the new channel conditions. Simulationbased results confirm that the method based on the statistical channel information characterizes the transmit power with reasonable accuracy. Hence, the proposed method may be used to determine if the transmitted symbols should be scaled or not without computing the instantaneous power, thus reducing the complexity.
The rest of the paper is organized as follows. Section 2 describes a FBMC/OQAMbased communication system where channel impairments are combated either at the transmission or at the reception side. Next, Section 3 addresses the design of equalizers and precoders, which may have multiple taps. The effects of performing multitap filtering are studied in Section 4. To that end, we provide analytical expressions to determine if multitap precoding boosts the average transmit power and if multitap equalization enhances the average noise power. In order to validate the closedform expressions derived in Section 4, some numerical results are presented in Section 5. Finally, Section 6 draws the conclusions.
2 System model
Finally, the realvalued PAM symbols are estimated after compensating the phase term and extracting the real part, i.e., ${\widehat{d}}_{q}\left[\phantom{\rule{0.3em}{0ex}}k\right]=\Re \left({\theta}_{q}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}k\right]{\widehat{x}}_{q}\left[\phantom{\rule{0.3em}{0ex}}k\right]\right)$. After that, ${\widehat{d}}_{q}\left[\phantom{\rule{0.3em}{0ex}}k\right]$ can be sent to the symbol demapper.
which is obtained by fixing b_{ m }[ k]=1 for all m.
3 Subband processing
In notation terms, let H(w) be the channel frequency response evaluated on the radial frequency w. The target points in (7) and (8), indicated explicitly by the righthand side of these formulas, can be selected according to the zero forcing (ZF) or the mean square error (MSE) criteria. In this regard, we set η=0 when the ZF approach is followed. By contrast, the target points used in the MSE criterion are such that η=N_{0}. Finally, for both criteria, the vectors are scaled as follows: ${\alpha}_{m}=\frac{{p}_{m}}{{\u2225{\mathbf{u}}_{m}\u2225}_{2}^{2}}$. The power distribution can be designed to satisfy different criteria without violating the constraint $\sum _{m=0}^{M1}{p}_{m}\le {P}_{S}$, where P_{ S } denotes the maximum power that can be assigned to one multicarrier symbol.
When it comes to designing equalizers, the system of Equations in (7) and (8) can also be used to determine the value of each tap. Therefore, we obtain ${a}_{m}\left[\phantom{\rule{0.3em}{0ex}}k\right]=\sqrt{{\alpha}_{m}}{u}_{m}\left[\phantom{\rule{0.3em}{0ex}}k\right]$. Now, the scaling factor is independent of the power distribution and is formulated as ${\alpha}_{m}=\frac{1}{{\u2225{\mathbf{u}}_{m}\u2225}_{2}^{2}}$.
4 Detrimental effects of multitap filtering
The perfect reconstruction property derived in [5] indicates that in the FBMC/OQAM context, the transmitted data can be perfectly recovered at the receive side under ideal propagation conditions. However, orthogonality properties are only satisfied in the real field. As a consequence, multipath propagation will certainly induce ISI and ICI. This justifies the need to counteract the channel in order to restore the orthogonality. As Section 2 proposes, signal processing techniques aimed at combating the channel can be performed either at the transmit or at the receive side. Previous works on this topic have concluded that multitap filtering becomes mandatory if the channel frequency selectivity is appreciable at the subcarrier level [9,10,12,13,14,15]. Nevertheless, the use of multiple taps may involve some detrimental effects. When broadband processing is applied at the transmitter, the power emitted might be boosted as if there was no precoding, because consecutive multicarrier symbols are correlated. By contrast, the consequence of performing multitap equalization is a possible noise enhancement. It is worth mentioning that in the FBMC/OQAM context, the power from the input of the modulator to the channel input is preserved if the transmit processing is based on singletap precoding. That is because the symbols transmitted in different subcarriers are independent along with the fact that $\sum _{\forall n}{\left\phantom{\rule{0.3em}{0ex}}{f}_{m}\left[\phantom{\rule{0.3em}{0ex}}n\right]\right}^{2}=1$ for all m, since the energy of the prototype pulse is equal to one. In the following, we will study the effects of multitap precoding and multitap equalization on the transmit and the noise power, respectively.
4.1 Average transmit power
It is worth mentioning that the variable s in (10) cannot be omitted because the equality $\mathbb{E}\left\{\left({d}_{m}\left[\phantom{\rule{0.3em}{0ex}}k\right]{\theta}_{m}\left[\phantom{\rule{0.3em}{0ex}}k\right]\ast \phantom{\rule{0.3em}{0ex}}{b}_{m}\left[\phantom{\rule{0.3em}{0ex}}k\right]\phantom{\rule{0.3em}{0ex}}\right){\left({d}_{m}\left[\phantom{\rule{0.3em}{0ex}}s\right]{\theta}_{m}\left[\phantom{\rule{0.3em}{0ex}}s\right]\ast \phantom{\rule{0.3em}{0ex}}{b}_{m}\left[\phantom{\rule{0.3em}{0ex}}s\right]\phantom{\rule{0.3em}{0ex}}\right)}^{\ast}\right\}=P{m}^{\delta}k,s$is not satisfied. This can be verified by realizing that any two samples of the sequence d_{ m }[ k]θ_{ m }[ k]∗b_{ m }[ k], which are evaluated in consecutive time epochs, are correlated. The term s can only be omitted if precoders have a single tap. The summation zone in the last equality of (11) has been reduced since b_{ m }[ k] is other than zero for −1≤k≤1.
Based on this result, the analysis of the transmit power hinges on the evaluation of the positivity of (21). Hence, if ${P}_{{\mathrm{T}}_{1}}<0$, we will conclude that there is no penalty for using the addressed transmit filters. On the contrary, if ${P}_{{\mathrm{T}}_{1}}>0$, we shall scale the symbols so that (14) is satisfied, which is equivalent to multiplying the symbols by $\beta =\sqrt{\frac{\sum _{m=0}^{M1}{p}_{m}{\mathit{\text{NR}}}_{{f}_{m}}\left[0\right]}{{P}_{{\mathrm{T}}_{1}}+{P}_{{\mathrm{T}}_{2}}}}$. Taking into account (20) and (22), we can lower bound the scaling factor as follows: $\beta \ge {\left(1+\frac{{P}_{{\mathrm{T}}_{1}}}{N\sum _{m=0}^{M1}{p}_{m}{R}_{{f}_{m}}\left[0\right]}\right)}^{1/2}$. Nevertheless, it is difficult to formulate P_{T1} in a closedform expression, since the expectation of a fraction cannot be computed straightforwardly. In this sense, Appendices 1 to 4 give details on how to obtain an approximate value of (21).
One alternative to compensate the possible boost of power due to multitap precoding is to evaluate (21) given the precoders, which is equivalent to dropping the expectation, and then using this value to compute β. While this ensures that the power is not increased, it entails the recalculation of β whenever the precoder is modified. If we are able to characterize the expected value of ${P}_{{\mathrm{T}}_{1}}$, then β does not have to be updated since its value is based on the statistical knowledge of the channel. The reduction of the complexity burden justifies the attempt to derive an analytical expression of (21).
4.2 Average power of the equalized noise
The denominator on the lefthand side corresponds to the transmit power, when precoders have a single tap. The denominator on the righthand side accounts for the summation of the average noise power in all subcarriers, when the length of the equalizers is one. In light of condition (31), we may conclude that if symbols are properly scaled when the transmit power increases, then there is no degradation for counteracting the channel at reception rather than at transmission, as long as the same filters are used as precoders or equalizers.
5 Numerical results
This section provides some numerical results to provide insight into the effects of multitap filtering. In particular, we evaluate the average transmit power and the BER. As for the communication system, we simulate the model pictured in Figure 1. As it is proposed in Section 2, we concentrate on two simpler cases. The first scenario that is studied is based on a transmultiplexer that performs broadband filtering at the transmitter without equalizing the signals at reception. By contrast, the second scenario that is assessed is based on a system that exclusively hinges on multitap equalization to counteract the channel. Concerning the design criteria, we have favored the ZF approach described in Section 3. As Appendix 4 demonstrates, the ZF criterion offers better analytical tractability than the MSE alternative. Besides, the performance difference between ZF and MSE is small, as it is shown in [9]. As for the system parameters, the FBMC/OQAM modulation scheme splits the 10MHz bandwidth into M=1,024 or 512 subcarriers. The frame transmission comprises N=20 multicarrier symbols, and the sampling frequency is set to f_{s}=10 MHz leading to subcarrier spacing equal to 9.76 or 19.53 KHz, depending on the total number of subcarriers. The channel is generated following the ITU Vehicular B (VehB) or the ITU Vehicular A (VehA) guidelines [21]. The transmitted symbols are generated by staggering real and imaginary parts of complexvalued symbols drawn from the 16QAM constellation. It must be mentioned that the study of the impact that the power distribution may have on the results is out the scope of the paper, and thus, the power is equally split among subcarriers, i.e. p_{ m }=1 ∀m.
5.1 Average transmit power
Computation of ${P}_{{\mathrm{T}}_{1}}$
M = 1,024  M = 512  M = 1,024  M = 512  

(VehB)  (VehB)  (VehA)  (VehA)  
SCI(frequency sampling)  −139.29  −222.16  −0.55  −1.09 
CFR(frequency sampling)  −133.21  −229.96  −0.76  −1.49 
CFR(MMSE)  304.00  337.98  −0.15  5.66 
In all the scenarios that have been simulated, it has been observed that multitap filtering has no harmful effects in terms of transmit power, thus supporting its utilization. In accordance with (31), we can also state that the average noise power is not enhanced due to multitap equalization. Although this result cannot be generalized for any channel, we provide the mathematical analysis to draw conclusions in other scenarios.
To determine if other techniques behave similarly to the frequency sampling approach, ${P}_{{\mathrm{T}}_{1}}$ has been computed when the taps of the precoders, i.e., {u_{ m }[ −1],u_{ m }[ 0],u_{ m }[ 1] }, are designed according to the MMSE criterion proposed in [10]. As Table 1 indicates, the MMSE precoder has a harmful effect in the average transmit power for VehB and VehA channels. However, for other types of channels, the conclusions may differ. It is left for future work to determine, in a mathematical or deductive way, which type of channels and precoders yield a positive ${P}_{{\mathrm{T}}_{1}}$.
5.2 BER performance
Our current investigation confirms that there is still some space for improvement, mostly in highly frequencyselective channels, so finding a better tradeoff between the performance and the complexity still remains as an open research problem in the FBMC/OQAM context. However, the goal of the paper is not to propose a new subband processing to combat the channel impairments with affordable complexity but to characterize the average transmit and noise power with the emphasis on systems with tractable complexity.
6 Conclusions
FBMC/OQAM has the key ingredients to deal with the restrictions that will be introduced by future wireless systems, such as the transmission in a fragmented spectrum. It is well known that the channel has to be counteracted to some extent in the FBMC/OQAM context to guarantee a certain quality of service. This translates into precoding the symbols at the transmit side or equalizing the demodulated data at the receive side. Both precoders and equalizers should perform multitap filtering when the channel is highly frequencyselective. The work presented here characterizes the average transmit power and the noise power when multitap precoders and equalizers are used. The closedform expressions reveal that if the same filter is used as a precoder or equalizer, then the transmit power and the noise power increase or decrease with the same magnitude. Therefore, we can conclude that there is no degradation due to combating the channel at reception rather than at transmission, as long as the transmitted symbols are properly scaled if transmit processing boosts the power. To determine whether the symbols should be scaled or not, we have formulated the transmit power as the function of the statistical knowledge of precoders when the criterion of design is based on the frequency sampling approach. The main reason to focus on the frequency sampling method is because it offers a good analytical tractability, which paves the way to get closedform expressions, while it has a performance comparable to the optimum MMSE. The numerical results show that the analytical expressions derived in this paper are reasonably accurate, and thus, they can be used to address the issue related to the power boost. The alternative to using the precoder statistics consists in computing the instantaneous power. However, this solution is very demanding in terms of complexity, since the power has to be recalculated whenever precoders are updated. This highlights that the characterization of the transmit power derived in this paper, which relies on precoder statistics, is useful since it provides reliable information with reduced complexity. Both the numerical results and the closedform expressions allow us to conclude that in the simulated scenarios, multitap precoding and equalization based on the frequency sampling approach do not boost the power. It is left for future work to investigate which precoding designs have a negative impact on the transmit power. Although the design of new equalization techniques is out of the scope of this paper, the numerical results have revealed that there is still space to improve the tradeoff between complexity and performance of the stateoftheart solutions.
Appendices
Appendix 1: expectation of the ratio of two random variables
The coefficient p_{ m } can be taken out of the expectation as long as it is independent of the taps of the filter. From the expressions that are derived in Appendices 2 and 3, it is possible to infer how to compute (38), which enables us to evaluate (21) that is the ultimate goal. It is worth mentioning that the expressions provided in Appendices 2 and 3 are only valid when the subband processing is designed according to the ZF criterion. That is, when η=0 in (7) and (8). The analysis in the MSE case cannot be presented as concisely as in the ZF case. However, we indicate in Appendix 4 how to generalize the mathematical developments so that the MSE criterion is covered.
Appendix 2: computation of $\mathbb{E}\left\{{u}_{m}\left[\phantom{\rule{0.3em}{0ex}}s\right]{u}_{m}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}z\right]\phantom{\rule{0.3em}{0ex}}\right\}$
With (45) and (42), which are independent of m, we can obtain an approximated value of (39) as function of the statistics of the channel.
Appendix 3: computation of $\mathbb{E}\left\{{u}_{m}\left[\phantom{\rule{0.3em}{0ex}}s\right]{u}_{m}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}z\right]\left{u}_{m}\right[\phantom{\rule{0.3em}{0ex}}l]{}^{2}\right\}$
To formulate (48) as function of the channel statistics, it is necessary to compute $\mathbb{E}\left\{h\left[\phantom{\rule{0.3em}{0ex}}i\right]h\left[\phantom{\rule{0.3em}{0ex}}{i}^{\prime}\right]\cdots {h}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}n\right]{h}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}{n}^{\prime}\right]\phantom{\rule{0.3em}{0ex}}\right\}$. In this sense, we can follow the approach presented in [24] to characterize the expectation as function of $\left\{{\sigma}_{n}^{2}\right\},\left\{{\sigma}_{n}^{4}\right\},\left\{{\sigma}_{n}^{6}\right\}$ and $\left\{{\sigma}_{n}^{8}\right\}$, depending on the value of i,i^{′},l,l^{′},k,k^{′},n,n^{′}. Since there are a lot of combinations that result in different expressions and thus a lot of space will be needed, we have not included the closedform expression of (48). However, the steps required to compute (48) using the statistics of the channel are indicated, which allows us to approximately compute (46). Note that again, the result will be constant regardless of the subcarrier index.
Appendix 4: comments on the MSE approach
where r_{1}+r_{2}+…+r_{ m }=n, r_{ i }≥1, and {Z_{ i }} are complexvalued random variables. The effective and generic way to calculate the expectation of the product of m complex random variables is presented in [24], particularly in Definition 1 and the succeeding formula (5). Following that approach, one can calculate the closedform for MSE criteria, and, in general, for any other criteria which could be presented in the form discussed in that paper.
Declarations
Acknowledgments
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 also supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM # (Grant agreement no. 318306).
Authors’ Affiliations
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