Partial interference cancellation with maximum likelihood sequence detection in FBMC spatial multiplexing system
- Rostom Zakaria^{1}Email author and
- Didier Le Ruyet^{1}
https://doi.org/10.1186/s13634-016-0339-x
© Zakaria and Le Ruyet. 2016
Received: 8 October 2015
Accepted: 22 March 2016
Published: 18 April 2016
Abstract
Intrinsic interference in filter bank multicarrier (FBMC) modulation prevents the maximum likelihood (ML) detection in spatial multiplexing (SM) system. This intrinsic interference is caused by the transmultiplexer impulse response in time-frequency domain. Solutions based on interference cancellation are not always effective because they may introduce error propagation. In this paper, we propose to study some receivers based on reducing the interference concerned by the cancellation. These solutions can be seen as a trade-off between the whole Viterbi (ML) detection and the whole interference cancellation. The principle of the proposed receivers is to partially estimate and cancel the interference, and the rest of the interference is processed by a low complexity Viterbi detector. We show in this work that the receiver performance depends on the transmultiplexer impulse response and on the choice of the partial Viterbi detector.
Keywords
1 Introduction
Nowadays, the most well-known multicarrier modulation is the orthogonal frequency division multiplex (OFDM) modulation. Thanks to the cyclic prefix (CP) insertion, OFDM technique provides a high robustness against the multipath fading channels. However, since OFDM uses CP and also rectangular pulse shape filtering that causes some disadvantages concerning the spectrum control and efficiency, to tackle this issue, filter bank multicarrier systems with offset quadrature amplitude modulation (FBMC/OQAM) was proposed as an alternative to CP-OFDM. Many works on the comparison between FBMC/OQAM and OFDM can be found in the literature such as [1]. One of the features of FBMC/OQAM is the possibility of shaping subcarrier signals with well-localized prototype filter in time and frequency axis [1]. That is, sidelobes steeply decay and allow a better spectrum control and offer an important resistance against time and frequency misalignment. Thus, FBMC/OQAM enables asynchronous multiple access and reduces intercarrier interference since each subcarrier signal is spectrally confined in a band and has negligible interference to other bands. In cognitive radios, the filtering capability of FBMC systems makes them the perfect choice for filling in the spectrum holes [1]. In the other hand, thanks to its spectral confinements, FBMC-based systems can reduce their guard band at the frequency boundaries, thereby increasing the spectral efficiency. Moreover, the absence of CP and any guard interval in time domain also contributes to increase the spectral efficiency. Many research works have been carried out in the issues related to doubly spread channels, and it has been shown that FBMC modulations are far better choices when compared to OFDM [2–4]. For instance, the author in [1] clearly shows the outperformance of the FBMC over OFDM in terms of the resilience to the doubly dispersive channels under the same assumptions of channel mobility and symbol density.
In FBMC/OQAM, each subcarrier is modulated with an OQAM transmitting pulse-amplitude modulation (PAM) symbols at each half a period (T/2). The orthogonality condition is considered only in the real field [5]. Indeed, the data at the receiver side is carried only by the real components of the signal, and the imaginary parts appear as interference terms. This interference is caused by the data symbols transmitted in the neighborhood area in the time-frequency domain. Since the interference is orthogonal to the useful data symbols, data detection is easily performed when the channel is flat fading or lowly selective [6]. However, in some situations like when the channel is highly selective, the intrinsic interference may cause performance degradation. Several works based on sub-channel equalization were carried out to tackle to this issue as in [7, 8] where multi-tap equalization is used. In [9], the authors have performed a minimum mean square error (MMSE) per subcarrier equalization. Whereas in [10, 11] the authors propose to implement a per-subcarrier decision-feedback equalizer (DFE). The extension of the latter to the multiple-input multiple output (MIMO) spatial multiplexing (SM) system was addressed in [12]. The per-subchannel equalization in FBMC with MIMO systems was also studied in [13, 14]. Some other works [15–20] address the design of MIMO precoding and decoding techniques for the FBMC modulation. For low MIMO selective channels, linear equalizations such as zero forcing (ZF) or MMSE can be applied to FBMC as shown in [21].
The presence of the inherent interference causes problems when combining FBMC/OQAM with some MIMO techniques such as Alamouti space-time block coding (STBC) scheme and SM systems with maximum likelihood (ML) detection. Regarding the Alamouti scheme, its application in a straightforward manner to the FBMC makes an inherent interference appear that cannot be easily removed [22]. Many works have been carried out on this topic such as [22] where the authors show that Alamouti coding can be performed but only when it is combined with code division multiple access (CDMA). A pseudo-Alamouti scheme was introduced in [23] where it is combined with single-carrier FBMC using the cyclic prefix (CP). Another solution was proposed by Renfors et al. in [24] where the Alamouti coding is performed in a block-wise manner. We have proposed in [25, 26] an iterative Alamouti scheme for FBMC based on intrinsic interference cancellation. On the other side, ML detection in SM system, which is supposed to offer a diversity order equal to the number of the receive antennas [27], cannot be applied straightforwardly with FBMC due to the presence of the interference.
Full ML receivers, in principle, offer the best possible performance but require an impractically high complexity when the impulse response is long. Moreover, the intrinsic interference in FBMC is two-dimensional which further complicates the detection task in maximum likelihood sense. Interference cancellation approaches generally offer the possibility of removing interference with low complexity increase and without enhancing the level of noise already present in the received signal [28]. Interference cancellation schemes are essentially based on using preliminary decisions to estimate and cancel the interference. In [29], we have addressed the possibility to cancel the interference before applying a local ML detection and proposed some receiver schemes. However, the obtained performance was limited and far from the optimum due to the error propagation. We have also proposed in [26, 30] to modify the FBMC/OQAM modulation by transmitting QAM symbols at each period of T instead of transmitting PAM symbols at each T/2. We referred to this modulation as FBMC/QAM. This modulation allows to reduce the intrinsic interference power but at the expense of the orthogonality [26]. Error propagation problem in iterative inter-symbol interference (ISI) cancellation has been addressed in many research fields. In the general case, the authors in [31] have established conditions under which the cancellation scheme is effective. The idea is to consider the interference as the sum of two terms. The first one is cancelled by using tentative decisions, and the second uncancelled one is considered by a maximum likelihood sequence equalizer (MLSE).
In this work, in order to counteract the error propagation in SM-FBMC and make the cancellation scheme effective, we were inspired by [31]. We apply the theory developed in [31] and propose some receiver structures that satisfy the conditions of the interference cancellation effectiveness. The receivers are based on partial interference cancellation followed by a Viterbi detector instead of an ML detector. The tentative detector is first used to only partially cancel the intrinsic interference. We have presented a part of this work in [32] where we have only focused on FBMC/OQAM with one prototype filter. In this paper, both FBMC/OQAM and FBMC/QAM modulations are addressed, and their performances are compared. Furthermore, each of the both modulations are analyzed with two different prototype filters.
The paper is organized as follows. We start in Section 2 by giving an overview on FBMC/OQAM modulation highlighting the issue of FBMC when it is combined with SM-ML detection. Then, in Section 3, we give a background on the use of tentative decisions to cancel the interference, and we present the principle of partial interference cancellation. Sections 4 and 5 are devoted to the analysis and adaptation of the partial interference cancellation to the FBMC/OQAM and FBMC/QAM, respectively. Simulation results of the different proposed receivers are presented in Section 6. Then, we finish by a conclusion in Section 7.
2 The FBMC/OQAM modulation
2.1 System model
where Δ n=n ^{′}−n _{0}, Δ k=k ^{′}−k _{0} and \(\Gamma _{\Delta k, \Delta n}^{(k_{0})}\) is then the transmultiplexer impulse response coefficients. It is worth noticing that the impulse response of the transmultiplexer depends on k _{0}. Indeed, because the multiplicative factor \(\phantom {\dot {i}\!}e^{j\pi k_{0}\Delta n}=(-1)^{k_{0}\Delta n}\), the sign of some impulse response coefficients (with Δ n odd) depends on the parity of k _{0}.
FBMC/OQAM transmultiplexer impulse response (main part) using PHYDYAS filter
n−3 | n−2 | n−1 | n | n+1 | n+2 | n+3 | ||
---|---|---|---|---|---|---|---|---|
k−1 | 0.043j | 0.125j | 0.206j | 0.239j | 0.206j | 0.125j | 0.043j | |
k | −0.067j | 0 | −0.564j | 1 | 0.564j | 0 | 0.067j | |
k+1 | 0.043j | −0.125j | 0.206j | −0.239j | −0.206j | −0.125j | 0.043j |
FBMC/OQAM transmultiplexer impulse response using IOTA filter
n _{0}−3 | n _{0}−2 | n _{0}−1 | n _{0} | n _{0}+1 | n _{0}+2 | n _{0}+3 | ||
---|---|---|---|---|---|---|---|---|
k _{0}−2 | −0.0016j | 0 | −0.0381j | 0 | 0.0381j | 0 | 0.0016j | |
k _{0}−1 | 0.0103j | 0.0381j | 0.228j | 0.4411j | 0.228j | 0.0381j | 0.0103j | |
k _{0} | −0.0182j | 0 | −0.4411j | 1 | 0.4411j | 0 | 0.0182j | |
k _{0}+1 | +0.0103j | −0.0381j | 0.228j | −0.4411j | 0.228j | −0.0381j | 0.0103j | |
k _{0}+2 | −0.0016j | 0 | −0.0381j | 0 | 0.0381j | 0 | 0.0016j |
The last equality is obtained according to Eq. (4).
2.2 Problem statement
where H _{ k,n } is an (N _{ r }×N _{ t }) channel matrix with the element \(h_{k,n}^{(ti)}\) at the tth row and ith column, \(\mathbf {r}_{k,n}=\left [r_{k,n}^{(1)}, r_{k,n}^{(2)},\ldots,r_{k,n}^{(N_{r})}\right ]^{T}\) is the received vector, \(\mathbf {a}_{k,n}=\left [a_{k,n}^{(1)}, a_{k,n}^{(2)},\ldots,a_{k,n}^{(N_{t})}\right ]^{T}\) is the transmitted symbol vector, \(\mathbf {u}_{k,n}=\left [u_{k,n}^{(1)}, u_{k,n}^{(2)},\ldots,u_{k,n}^{(N_{t})}\right ]^{T}\) is the vector of the interference values, and \(\boldsymbol {\gamma }_{k,n}=\left [\gamma _{k,n}^{(1)}, \gamma _{k,n}^{(2)},\ldots,\gamma _{k,n}^{(N_{r})}\right ]^{T}\) is the noise vector.
where G _{ k,n } is the equalization matrix based on the ZF or MMSE criterion. Then, the real part retrieval of \(\tilde {\mathbf {c}}_{k,n}\) yields the real equalized data vector \(\tilde {\mathbf {a}}_{k,n}\) [21].
As for maximum likelihood (ML) detection, the presence of the interference vector term u _{ k,n } in (11) prevents the application of ML separately at each time-frequency position (k,n). This is because the interference terms take their values in a large set and depend on the transmitted data symbols in the neighborhood around the considered position (k,n). The global ML detector is the one that considers all the transmitted data symbols within a frame. Obviously, such a receiver implementation is by far impractical due to its huge complexity.
3 Overview on partial interference cancellation
where \(\mathbf {R}=\mathbb {E}\{\mathbf {ww}^{H}\}/{\sigma _{w}^{2}}\) is the normalized noise autocovariance matrix, with \({\sigma _{w}^{2}}\) is the noise variance and w=[w _{0},w _{1},…,w _{ K−1}]^{ T }. The events whose distance d _{0}(ε) is the minimum are called “first-order” error events. Similarly, events whose distance is the second smallest are called “second-order” error events and so on [31].
- 1.
Errors affecting the main (Viterbi detector) and the tentative detector must be statically independent.
- 2.
The RISI (described by f _{1}) must be small enough to guarantee that the main Viterbi detector can make relatively reliable decisions even when the tentative detector makes a decision error and such that the tentative detector also makes relatively reliable decisions in spite of the ISI.
- 3.
The distance of second-order and higher-order error events that could cause error propagation must be significantly larger than that of first-order error events.
4 Application to MIMO-SM FBMC/OQAM
Now that the conditions for effective RISI cancellation are summarized, we will attempt to apply them to FBMC. Hence, the problem is, essentially, how to select the functions f _{0} and f _{1} such that the three conditions cited above are fulfilled. The intrinsic interference in FBMC is seen as a two-dimensional intersymbol interference (2D-ISI). An extension of the works of Agazzi and Seshardi [31] to 2D-ISI channels was treated in [38] assuming that the noise samples are uncorrelated (which is not the case in FBMC). Hence, in general, the target response f _{0} may also represent a 2D-ISI channel. Then, a 2D-Viterbi algorithm is required to match with f _{0}. Designing a 2D-Viterbi is quite challenging. Therefore, for simplicity reasons, we opted to set the additional constraint that the target response f _{0} must be one-dimensional and that f _{1} covers the rest of 2D-ISI.
The first configuration \(\left (f_{0}^{(1)}\right)\) corresponds to the whole ISI cancellation which has been studied in [29, 39]. Since, in FBMC, we have \(\sum _{p,q}|\Gamma _{p,q}^{(k)}|^{2} = 2\) [6], it is easy to calculate the power of the RISI (represented by f _{1}) for each configuration.
Regarding the first condition, it is easily satisfied when the tentative detector is different from the main one (Viterbi) [31]. We recall that we consider the case of spatial multiplexing system. Then, we chose the MIMO-MMSE equalizer as the tentative detector.
where the last equality is obtained thanks to Eq. (9).
Spectrum distances and RISI power for FBMC/OQAM with the PHYDYAS filter
First configuration \(\left (f_{0}^{(1)}\right)\) | Second configuration \(\left (f_{0}^{(2)}\right)\) | Third configuration \(\left (f_{0}^{(3)}\right)\) | |
---|---|---|---|
First-order distance | 2 | 1.8857 | 1.9189 |
Second-order distance | \(2\sqrt {2}\) | 2.4936 | 2.6170 |
w(ε _{1}) | 1 | 1 | 1 |
w(ε _{2}) | 2 | 2 | 2 |
Power of the RISI | 1 | 0.6819 | 0.3638 |
Spectrum distances and RISI power for FBMC/OQAM with the IOTA filter
First configuration \(\left (f_{0}^{(1)}\right)\) | Second configuration \(\left (f_{0}^{(2)}\right)\) | Third configuration \(\left (f_{0}^{(3)}\right)\) | |
---|---|---|---|
First-order distance | 2 | 1.8985 | 1.8871 |
Second-order distance | \(2\sqrt {2}\) | 2.5395 | 2.6594 |
w(ε _{1}) | 1 | 1 | 1 |
w(ε _{2}) | 2 | 2 | 2 |
Power of the RISI | 1 | 0.8054 | 0.6109 |
Let us denote by ε _{1} and ε _{2} the first- and second-order error events, respectively. We also define w(ε) as the number of error position in the error event ε. The values of w(ε) are also shown in the table. We remark that the difference between the second-order and the first-order distances is almost the same for all the configurations (0.8±0.03), so we consider (as considered in [31]) that the higher-order distances are sufficiently larger than the minimum distance for each configuration. Hence, condition 3 is fulfilled for the three configurations.
Now, we have only to determine the configuration(s) for which the second condition is satisfied. Unfortunately, the determination of the RISI power for which the cancellation starts to be effective (or equivalently, error propagation ceases) is not trivial and depends also on the noise variance σ ^{2} [31]. We will show by simulations that only the third configuration \(\left (f_{0}^{(3)}\right)\) allows to obtain effective RISI cancellation.
As for the receiver complexity, it strongly depends on that of the Viterbi detector. When we consider a spatial multiplexing system with N _{ t } transmit antennas, the Viterbi detector has to compute \(q^{i\times N_{t}}\phantom {\dot {i}\!}\) branch metrics, where q is the number of all possible symbols a _{ k,n } (constellation size) and i∈{1,2,3} is the number of the taps in \(f_{0}^{(i)}\). In order to reduce the receiver complexity, we can replace the Viterbi detection algorithm by the M-Algorithm [40] which keeps only a fixed number (J) of inner states instead of all the inner states \(\left (q^{(i-1)\times N_{t}}\right)\phantom {\dot {i}\!}\). Hence, the M-algorithm has to compute only the \(\phantom {\dot {i}\!}J\times q^{N_{t}}\) branch metrics. In the rest of the paper, we call the proposed receivers “PaIC/Viterbi” (for Partial Interference Cancellation with Viterbi detector) and can be followed by an index to indicate the considered configuration.
5 Application of PaIC/Viterbi receivers in FBMC/QAM
FBMC/QAM transmultiplexer impulse response using the PHYDYAS filter
n−1 | n | n+1 | |
---|---|---|---|
k−1 | 0.125j | 0.239j | 0.125j |
k | 0 | 1 | 0 |
k+1 | −0.125j | −0.239j | −0.125j |
FBMC/QAM transmultiplexer impulse response using the IOTA filter
n _{0}−1 | n _{0} | n _{0}+1 | |
---|---|---|---|
k _{0}−1 | 0.0381j | 0.4411j | 0.0381j |
k _{0} | 0 | 1 | 0 |
k _{0}+1 | −0.0381j | −0.4411j | −0.0381j |
where \(f_{0}^{(1)}\) obviously corresponds to the IIC-ML receiver [26] where the whole interference cancellation is performed. Hence, the advantage for PaIC/Viterbi receivers in FBMC/QAM lies in the fact that the Viterbi algorithms are performed in the frequency axis direction. Whereas in FBMC/OQAM, they are performed in the time axis direction. Consequently, from an implementation point of view, the PaIC/Viterbi receivers are less complicated for implementation in FBMC/QAM than in the conventional FBMC, because only one Viterbi algorithm has to be performed for each one received multicarrier symbol, whereas in FBMC/OQAM, we need to perform a Viterbi detector simultaneously for each subcarrier.
Spectrum distances and RISI power for FBMC/QAM with the PHYDYAS filter
First configuration \(\left (f_{0}^{(1)}\right)\) | Second configuration \(\left (f_{0}^{(2)}\right)\) | Third configuration \(\left (f_{0}^{(3)}\right)\) | |
---|---|---|---|
First-order distance | 2 | 1.8857 | 1.7185 |
Second-order distance | 2.5407 | 2.6520 | 1.9189 |
w(ε _{1}) | 1 | 1 | 2 |
w(ε _{2}) | 2 | 2 | 1 |
Power of the RISI | 0.1771 | 0.1198 | 0.0626 |
Spectrum distances and RISI power for FBMC/QAM with the IOTA filter
First configuration \(\left (f_{0}^{(1)}\right)\) | Second configuration \(\left (f_{0}^{(2)}\right)\) | Third configuration \(\left (f_{0}^{(3)}\right)\) | |
---|---|---|---|
First-order distance | 2 | 1.8985 | 1.7833 |
Second-order distance | 2.3561 | 2.5395 | 1.7892 |
w(ε _{1}) | 1 | 1 | 3 |
w(ε _{2}) | 2 | 2 | 4 |
Power of the RISI | 0.3956 | 0.2010 | 0.0065 |
It is worth noticing that in the third configuration using the IOTA filter, the values of the two smallest distances are practically the same. Moreover, the first- and second-order error events contain, respectively, 3 and 4 error positions (w(ε _{1})=3 and w(ε _{2})=4), whereas in the other cases, they only contain 1 and 2 errors, respectively. Consequently, even if the residual interference is perfectly removed, the bit error rate (BER) performance would be poor because the most likely error events are those that contain 3 or 4 errors.
Finally, we note that if we assume imperfect channel estimation, the performance of the proposed detectors (for both FBMC/OQAM and FBMC/QAM) will be affected. Indeed, channel estimation error will lead additional noise to the detected signals. However, according to the considered system model given by Eqs. (8) or (10), the channel estimation error is uncorrelated with the intrinsic interference. Hence, since the proposed configurations \(f_{0}^{(i)}\), i∈{1,2,3} are only related to the FBMC intrinsic interference, then they will not be changed in the context of imperfect channel estimation. Nevertheless, the metric used in the Viterbi algorithm might be changed to improve the detection performance under imperfect channel knowledge assumption. It is worthwhile to note that a generalized robust (against channel estimation errors) ML decoder that yields near optimal BER performance was proposed in [41].
6 Simulation results
Therefore, the performance curves of CP-OFDM exhibit an SNR loss of about 10 log10(17/16)≈0.26 dB.
6.1 FBMC/OQAM
As for the first and second configurations, the performance degradation compared to CP-OFDM/ML begins from about 12 dB. This relatively high degradation is due to the high values of the corresponding RISI powers causing error propagation.
6.2 FBMC/QAM
As for PaIC/Viterbi-2 and PaIC/Viterbi-1 receivers, we can notice that with the IOTA filter the BER performance does not reach the CP-OFDM/ML one. We obtain an SNR loss at BER=10^{−2} with respect to CP-OFDM/ML of about 2.25 dB for PaIC/Viterbi-2, and 6.25 dB for PaIC/Viterbi-1. Hence, both latter configurations suffer from error propagation and the interference cancellation is not effective. This performance limitation is explained by the fact that the residual interference variances are not sufficiently small; according to Table 8, we have \(\sigma _{\text {RISI}}^{2}\approx 0.2\) and \(\sigma _{\text {RISI}}^{2}\approx 0.4\) for PaIC/Viterbi-2 and PaIC/Viterbi-1, respectively.
Because the orthogonality condition is lost in FBMC/QAM, we can observe the effect of the interference by comparing the BER-performance of the MMSE equalizer in both cases where the PHYDYAS or IOTA filter is used. We notice in Fig. 6 that the MMSE equalizer reaches the BER value of 10^{−1} at SNR=15.5 dB, whereas in Fig. 7, we have BER=10^{−1} at SNR=9.5 dB. This is due to the fact that the BER floor in the IOTA case is much higher than the BER floor in the case where the PHYDYAS filter is used.
7 Conclusions
The intrinsic interference in FBMC is a 2D-ISI in the time-frequency plan. In order to avoid a full 2D-Viterbi detector, we have proposed in this paper to study a trade-off between a whole interference cancellation receiver and a full 2D-Viterbi detector. The proposed receiver is composed by a tentative detector giving decisions which serve to partially cancel the interference, followed by a Viterbi detector matching to the non-cancelled interference. Three configurations were treated. The first one is called PaIC/Viterbi-1 and correspond to the whole interference cancellation. The second one is PaIC/Viterbi-2, where the Viterbi detector matches with the two largest coefficients of the transmultiplexer impulse response. The third one is PaIC/Viterbi-3 and the Viterbi detector matches with the three largest coefficients. We have shown that the performance of the PaIC/Viterbi receivers essentially depends on the power of the residual interference (RISI) which is not concerned by the Viterbi detector. When the RISI power is not sufficiently low, the receivers suffer from error propagation.
We have studied the proposed receivers in both FBMC/OQAM and FBMC/QAM. This latter can offer the best performance in some situation since the global intrinsic interference is reduced. For FBMC/OQAM, we have shown by simulations that only PaIC/Viterbi-3 with the PHYDYAS filter can provide satisfactory performance compared to CP-OFDM/ML one. Whereas for FBMC/QAM, all the three PaIC/Viterbi configurations with the PHYDYAS filter exhibit the same performance as CP-OFDM/ML. However, the interference coefficients of the IOTA filter do not allow any PaIC/Viterbi configuration to reach the optimal BER performance. Indeed, in spite of the lowest RISI power for PaIC/Viterbi-3 with the IOTA filter, the BER performance is limited because of the fact that the most likely error events in the Viterbi detector contain more than 3 errors.
Finally, since the Viterbi detection increases the computational complexity, we can replace the Viterbi detector by another one based on M-Algorithm in order to reduce the receiver complexity without performance deterioration.
As a future work, one can consider extending the present work to the context of imperfect channel state information. As we have aforementioned, in order to minimize the BER performance degradation, we should adapt the Viterbi algorithm and investigate an efficient Viterbi metric capable of achieving near optimal performance.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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