 Research
 Open Access
Bidirectional algorithms for interference suppression in multiuser systems
 Patrick Clarke^{1} and
 Rodrigo C. de Lamare^{1, 2}Email author
https://doi.org/10.1186/s1363801504400
© Clarke and de Lamare. 2015
 Received: 2 March 2015
 Accepted: 31 August 2015
 Published: 19 October 2015
Abstract
This paper presents adaptive bidirectional minimum meansquare error parameter estimation algorithms for fastfading channels. The time correlation between successive channel gains is exploited to improve the estimation and tracking capabilities of adaptive algorithms and provide robustness against timevarying channels. Bidirectional normalized least meansquare and conjugate gradient algorithms are devised along with adaptive mixing parameters that adjust to the timevarying channel correlation properties. An analysis of the proposed algorithms is provided along with a discussion of their performance advantages. Simulations for an application to interference suppression in multiuser DSCDMA systems show the advantages of the proposed algorithms.
Keywords
 Multiuser systems
 Interference suppression
 Adaptive algorithms
 Mobile channels
1 Introduction
Lowcomplexity reception and interference suppression are essential in multiuser mobile systems if battery power is to be conserved, datarates improved and quality of service enhanced. Conventional adaptive schemes fulfill many of these requirements and have been a significant focus of the research literature [1–8]. However, in timevarying fading channels commonly associated with mobile systems, these adaptive techniques encounter tracking and convergence problems. Optimum closedform solutions can address these problems but their computational complexity is high and CSI is required. Lowcomplexity adaptive channel estimation can provide CSI but in highly dynamic channels tracking problems exist due to their finite adaptation rate [9]. An alternative statistical approach is to obtain the correlation structures required for optimal minimum meansquare error (MMSE) or leastsquares (LS) filtering [10, 11]. Although this relieves the tracking demands placed on the filtering process, in a Rayleigh fading channel, a zero correlator is the result due to the expectation of a Rayleigh fading coefficient, and therefore the crosscorrelation vector, equating to zero, i.e., E[h _{1}[ n]]=0 and \(E\left [b_{1}^{\ast }[\!n]\mathbf {r}[\!n]\!\right ]=0\). In slowly fading channels, this problem may be overcome by using a time averaged approach where the averaging period is equal to or less than the coherence time of the channel. However, in fast fading channels, an averaging period equal to the coherence time of the channel is insufficient to overcome the effects of additive noise and characterize the multiuser interference (MUI) [1].
Furthermore, the use of optimized convergence parameters such as step sizes and forgetting factors into conventional adaptive algorithms extend their fading range and lead to improved convergence and tracking performance [8, 12–18]. However, the stability of adaptive stepsizes and forgetting factors can be a concern unless they are constrained to lie within a predefined region [19]. Other alternative schemes include those based on processing the received data in subblocks [20–22] and subspace algorithms [23–28]. In addition, the fundamental problem of obtaining the unfaded symbols whilst suppressing MUI remains. Consequently, the application of such algorithms is restricted to low and moderate fading rates. The limitations of conventional estimation approaches led to the development of methods that attempt to track the faded symbol, such as the channelcompensated MMSE solution [29, 30]. This removes the burden of fading coefficient estimation from the receive filter. However, a secondary process is required to perform explicit estimation of the fading coefficients in order to perform symbol estimation [31].
Approaches that avoid tracking and estimation of the fading coefficients were proposed in [31–33]. Although a channel might be highly time variant, two adjacent fading coefficient will be similar and have a significant level of correlation as studied in [31–33]. These properties can then be exploited to obtain a sequence of faded symbols where the primary purpose of the filter is to suppress multiuser interference and track the ratio between successive fading coefficients; thus, not burdening it with estimation of the fading coefficients themselves. However, this scheme has a number of limitations stemming from the use of only one correlation time instant and a single class of adaptive algorithms.
In this work, a bidirectional MMSEbased interference suppression scheme for highly dynamic fading channels is presented. The nonzero correlation between multiple time instants is exploited to improve the robustness, tracking, and convergence performance of existing MMSE schemes. Unlike existing adaptive solutions [8, 31–33], which do not fully exploit the fading correlation between multiple successive time instants, the proposed bidirectional approach exploits the correlation and adaptively weighs the output of the receive filter in order to optimize the estimation performance. Normalized leastmean square (NLMS) and conjugate gradient (CG)type algorithms are presented that overcome a number of problems associated with applying the recursive leastsquares (RLS) algorithm to bidirectional problems. Novel mixing strategies that weigh the contribution of the considered time instants and improve the convergence and steadystate performance, increasing the robustness against the channel discontinuities, are also presented. An analysis of the proposed schemes is developed and establishes the mechanisms and factors behind their behavior and expected performance. The proposed schemes are applied to conventional multiuser DSCDMA [2] and cooperative DSCDMA systems [3, 4] to assess their MUI suppression and tracking capabilities. The application of the proposed scheme and algorithms to multipleantenna and multicarrier systems are also possible. Simulations show that the algorithms improve upon existing schemes with minimal increase in complexity.

Bidirectional MMSEbased interference suppression scheme for highly dynamic fading channels.

Bidirectional adaptive parameter estimation algorithms based on NLMS and CG techniques.

An analysis of the convergence and the computational complexity of the proposed algorithms.

A study of the proposed and existing algorithms in DSCDMA and cooperative DSCDMA multiuser systems.
This paper is organized as follows. Section 2 briefly details the signal models of a conventional DSCDMA system and a cooperative DSCDMA system. Section 3 presents the proposed scheme and its corresponding optimization problems and the motivation behind their development. Switching and mixing strategies that optimize performance are proposed and assessed in Section 4, followed by the derivation of the proposed algorithms in Section 5. An analysis of the proposed algorithms is given in Section 6, whereas performance evaluation results are presented in Section 7. Conclusions are drawn in Section 8.
2 Signal models
In this section, we describe the signal models of a DSCDMA system operating in the uplink and a cooperative DSCDMA system in the uplink equipped with relays and the amplifyandforward (AF) cooperation protocol. These systems are employed for testing the proposed algorithms even though that extensions to multipleantenna and multicarrier can also be considered with appropriate modifications of the algorithms.
2.1 DSCDMA signal model
where w[ i] is an Mdimensional vector that corresponds to the receive filter.
2.2 Cooperative DSCDMA signal model
where \(h_{\text {sr}_{n}}[\!i]\phantom {\dot {i}\!}\) and \(h_{\mathrm {r}_{n}\mathrm {d}}[\!i]\phantom {\dot {i}\!}\) are the channel fading channel coefficients between the source and the nth relay, and the nth relay and the destination, respectively, and \(\mathbf {n}_{\mathrm {r}_{n}}[\!i]\phantom {\dot {i}\!}\) and n _{d}[ i] are additive white Gaussian noise vectors at the relays and the destination, respectively.
where w[ i] is an Mdimensional vector that corresponds to the receive filter for the cooperative system.
3 Proposed bidirectional scheme
where h _{1}[ i] is the channel coefficient of the desired user. The interference suppression of the resulting receive filter is improved in fast fading environments compared to conventional adaptive receivers but only the ratio of adjacent fading samples is obtained. Consequently, differential MMSE schemes are suited to differential modulation where the ratio between adjacent symbols is the data carrying mechanism.
Although the existing differential scheme operates over 2 correlated samples, the proposed scheme is able to exploit the additional correlation present between multiple adjacent samples. Moreover, it is also possible to obtain further gain by weighting the correlation between multiple adjacent samples. However, the benefit of using multiple time instant is dependent on the fading rate of the channel and the related correlation of the channel coefficients. We have investigated the use of multiple time instants and it turns out that a scheme which exploits 3 adjacent samples captures most of the performance benefits. In particular, we have tested the proposed bidirectional scheme and algorithms with various values of adjacent time instants (between 4 and 8) and verified that exploiting extra time instants above 3 does not yield significant gains. In fact, the number of time instants is a parameter to be chosen by the designer.
In what follows, we describe switching and weighting strategies to optimize the proposed scheme and obtain further performance gain.
4 Switching and weighting strategies
where 0≤ρ _{ n }≤3 for n=1,2,3 are the weighting factors.
The determination of the receive vector samples that correspond to the scenarios depicted in Fig. 2 is essential if correct optimization of the ρ is to be achieved. The use of CSI to achieve this would be an effective but impractical solution due to the difficulty in obtaining CSI; consequently, other methods must be sought. In this section, we propose the use of two alternative metrics: the signal power differential after interference suppression between the considered time instants, and the error between the considered time instants.
and ν is a positive user defined constant greater than unity that scales the threshold. The threshold ν is set with the help of computer experiments in a similar way as the step size of the NLMS algorithm is tuned. The aim is to scale the threshold such that it will be used to inform the algorithm about the relevant differential power which should be used.
The forgetting factor, 0≤λ _{ e }≤1, is user defined and, along with normalization by the total error, e _{ T }[ i], and \({\sum ^{3}_{n=1}}\rho _{n}[\!0]=1\), ensures \({\sum ^{3}_{n=1}}\rho _{n}[\!i]=1\) and a convex combination at each time instant.
5 Adaptive algorithms
This cost function then forms the basis of the adaptive algorithms derived in this section. However, to reduce the complexity of the derivations, enforcement of the nonzero constraint is not included and instead enforced in a stochastic manner at each time instant after the adaptation step is complete [31].
5.1 Normalized leastmean square algorithm
where λ _{ M } is an exponential forgetting factor [31]. The enforcement of the constraint is performed by the denominator of (27) which ensures that the receive filter w[ i] does not tend towards a zero correlator as the adaptation progresses.
as the receive filter update equation.
5.2 Least squares algorithm
for the autocorrelation matrix; a form which (31) is unable to fit into without assumptions that cause a significant performance degradation. Consequently, an alternative lowcomplexity algorithm to implement the LS solution given by (31)–(35) is required.
5.3 Conjugate gradient algorithm
ensures the R[ i] orthogonality between d _{ j }[ i] and d _{ l }[ i] where j≠l. The iterations (43)–(47) are then repeated until j=j _{max}.
The variable switching and mixing factors can be incorporated into the algorithm to improve performance. This is achieved by operating the CG algorithm over the modified correlation structures given by (36) and (37).
6 Analysis
In this section, we analyze the proposed bidirectional algorithms to gain insight of the expected performance but also to obtain further knowledge into the operation of the proposed and existing algorithms. The unconventional form of the proposed cost functions precludes the application of standard MSE analysis. Consequently, we concentrate on the signaltointerferenceplusnoise ratio (SINR) of the proposed algorithms in order to analyze their interference suppression and tracking performance. We firstly study the NLMS algorithm and the features of its weight error correlation matrix in order to arrive at an analytical SINR expression. Following this, we explore the analogy between the form of the bidirectional expression and convex combinations of adaptive receive filters [39, 40].
6.1 SINR analysis
where R _{ S } and R _{ I } are the signal and interference and noise correlation matrices, into a form amenable to analysis.
where w _{ o } is the instantaneous standard MMSE receiver.
From (52), it is clear that we need to pursue expressions for K[ i] and G[ i] in order to reach an analytical interpretation of the bidirectional NLMS scheme.
From the expression above, it is clear that the underlying factor that governs the SINR performance of the algorithms is the correlation between the considered time instants, f _{1−3}, dataruse and the use of f _{2}. Accordingly, it is the additional correlation factors that the proposed bidirectional algorithms possess that enhances its performance compared to the conventional scheme, confirming the initial motivation behind the proposition of the bidirectional approach. Lastly, the f _{1−3} expressions of (60) are the factors that influence the optimum number of considered time instants.
6.2 Combinations of adaptive receive filters
This is equivalent to a convex combination of adaptive receive filters with varying λ [39, 40], where each of the 3 filters focuses on the correlation between the 2 of the 3 considered time instants. However, the presence of the autocorrelation matrices in the inverses of the expression also indicates that the remaining time instants also influence the structure of each filter. Although the mixing factors are not separable, we can interpret them as a form of weighting that is present in conventional combinations of adaptive filters. This explains in part the additional control and performance they provide.
7 Simulations
In this section, the proposed bidirectional adaptive algorithms are applied to conventional multiuser and cooperative DSCDMA systems using the signal models described in Section 2. The application of the proposed algorithms to multipleantenna and multicarrier systems is straightforward and requires a change in the signal models. The individual Rayleigh fading channel coefficients, h[ i], are generated using Clarke’s model [43] where 20 scatterers are assumed. In all simulations, the number of packets is denoted by N _{ p } and the fading rate is given by the dimensionless normalized fading parameter, T _{ s } f _{ d }, where T _{ s } is the symbol period and f _{ d } is the Doppler frequency shift. The convergence parameters of the algorithms have been optimized resulting in stepsizes forgetting factors of 0.1 and 0.99, respectively, λ _{e}=0.95, λ _{ M }=0.99, and the number of CG iterations, j _{max}=5.
As detailed in Section 6, the proposed algorithms do not minimize the same MSE as a conventional MMSE receiver; therefore, the MSE is not an adequate performance metric. As a result, BER and SINRbased metrics are chosen for the purpose of comparison between existing algorithms and the optimum MMSE solution. Due to the rapidly fading channel, the instantaneous SNR, SNR_{i}, is highly variable and so the SINR alone is also not a satisfactory metric. To overcome this, it is normalized by the instantaneous SNR to give \(\frac {SINR}{SNR_{\mathrm {i}}}\). This value is negative in all simulations and directly reflects the MUI interference suppression and tracking capabilities of the proposed algorithms [31, 32].
7.1 Conventional DSCDMA
Here we apply the adaptive algorithms of Section 5 to interference suppression in the uplink of a multiuser DSCDMA system described in Section 2. Each simulation is averaged over N _{ p } packets and detailed parameters are specified in each plot.
7.1.1 Analytical results
The correlation matrices are calculated via ensemble averages prior to the start of the algorithm and G[0]=K[ 0]=I. In Fig. 3, one can see the convergence of the simulated schemes to the analytical and MMSE plots, validating the presented analysis. Due to the highly dynamic nature of the channel, using the expected values of the correlation matrix alone cannot capture the true transient performance of the algorithms. However, the convergence period of the analytical plots within the first 200 iterations can be considered to be within the coherence time and therefore give an indication of the transient performance relative to other analytical plots. Using this justification and the aforementioned analysis, advantages should be present in the transient phase due to the additional correlation information supplied by F _{2} and F _{3}. This conclusion is supported by Fig. 3 and the similar forms of the analytical and simulated schemes relative to each other and their subsequent convergence.
7.1.2 SINR performance
7.2 Cooperative DSCDMA
To further demonstrate the performance of the proposed schemes in cooperative relaying systems [5], we apply them to an AF cooperative DSCDMA system detailed in Section 2.
8 Conclusions
In this paper, we have presented a bidirectional MMSE framework that exploits the correlation characteristics of rapidly varying fading channels to overcome the problems associated with conventional adaptive interference suppression techniques in such channels.
An analysis of the proposed schemes has been performed and the reasons behind the performance improvements shown to be the additional correlation information, data reuse, and optimized correlation factor weighting. The conditions under which the differential and bidirectional schemes are equivalent have also been established and the steadystate implications of this detailed. Finally, the proposed algorithms have been assessed in standard and cooperative multiuser DSCDMA systems and shown to outperform both differential and conventional schemes.
Declarations
Acknowledgements
Part of this manuscript was presented at the International Symposium on Wireless Communications Systems (ISWCS) in 2013. The work of R. C. de Lamare is partly funded by CNPq and FAPERJ.
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
References
 S Verdu, Multiuser Detection (Cambridge University Press, NY, 1998).MATHGoogle Scholar
 U Madhow, M Honig, MMSE interference suppression for directsequence spread spectrum CDMA. IEEE Trans. Commun. 42(12), 3178–3188 (1994).View ArticleGoogle Scholar
 W Huang, Y Wong, C Kuo, in IEEE Global Communications Conference. Decodeandforward cooperative relay with multiuser detection in uplink CDMA networks (Phoenix, USA, 2007).Google Scholar
 RC de Lamare, Joint iterative power allocation and linear interference suppression algorithms for cooperative dscdma networks. IET Commun. 6(13), 1930–1942 (2012).View ArticleGoogle Scholar
 P Clarke, RC de Lamare, Transmit diversity and relay selection algorithms for multirelay cooperative mimo systems. IEEE Trans. Veh. Technol. 61(3), 1084–1098 (2012).View ArticleGoogle Scholar
 T Wang, RC de Lamare, PD Mitchell, Lowcomplexity setmembership channel estimation for cooperative wireless sensor networks. IEEE Trans. Veh. Technol. 60(6), 2594–2607 (2011).View ArticleGoogle Scholar
 RC de Lamare, R SampaioNeto, Minimum mean squared error iterative successive parallel arbitrated decision feedback detectors for DSCDMA, systems. IEEE Trans. Commun. 5(56), 778–789 (2008).View ArticleGoogle Scholar
 R de Lamare, P Diniz, Setmembership adaptive algorithms based on timevarying error bounds for cdma interference suppression. IEEE Trans. Veh. Technol. 58(2), 644–654 (2009).View ArticleGoogle Scholar
 S Haykin, Adaptive Filter Theory, 4th edn. (Prentice Hall, NJ, 2002).Google Scholar
 D Sadler, A Manikas, MMSE multiuser detection for array multicarrier DSCDMA in fading channels. IEEE Trans. Signal Process. 53(7), 2348–2358 (2005).MathSciNetView ArticleGoogle Scholar
 R Schoder, W Gerstacker, A Lampe, Noncoherent MMSE interference suppression for DSCDMA. IEEE Trans. Commun. 50(4), 577–587 (2002).View ArticleGoogle Scholar
 S Haykin, A Sayed, J Zeidler, P Yee, P Wei, Adaptive tracking of linear time variant systems by extended RLS algorithms. IEEE Trans. Signal Process. 45(5), 1118–1128 (1997).View ArticleGoogle Scholar
 J Wang, Fast tracking RLS algorithm using novel variable forgetting factor with unity zone. IEEE Electron. Lett. 27(23), 2550–2551 (1991).Google Scholar
 R Kwong, E Johnston, A variable step size LMS algorithm. IEEE Trans. Signal Process. 40(7), 1633–1642 (1992).MATHView ArticleGoogle Scholar
 B Toplis, S Pasupathy, Tracking improvements in fast RLS algorithm using a variable forgetting factor. IEEE Trans. Acoust., Speech, Signal Process. 36(2), 206–227 (1988).MATHView ArticleGoogle Scholar
 SH Leung, C So, Gradientbased variable forgetting factor RLS algorithm in timevarying environments. IEEE Trans. Signal Process. 53(8), 3141–3150 (2005).MathSciNetView ArticleGoogle Scholar
 S HyunChool, A Sayed, S WooJin, Variable stepsize NLMS and affine projection algorithms. IEEE Signal Process. Lett. 11(2), 132–135 (2004).View ArticleGoogle Scholar
 Y Zhang, N Li, J Chambers, Y Hao, New gradientbased variable step size LMS algorithms. EURASIP J. Adv. Signal Process, 1–9 (2008).Google Scholar
 S Gelfand, Y Wei, J Krogmeier, The stability of variable stepsize LMS algorithms. IEEE Trans. Signal Process. 47(12), 3277–3288 (1999).MATHView ArticleGoogle Scholar
 P Baracca, S Tomasin, L Vangelista, N Benvenuto, A Morello, Per subblock equalization of very long ofdm blocks in mobile communications. IEEE Trans. Commun. 59(2), 363–368 (2011).View ArticleGoogle Scholar
 S Yerramalli, M Stojanovic, U Mitra, Partial fft demodulation: A detection method for highly doppler distorted ofdm systems. IEEE Trans. Signal Process. 60(11), 5906–5918 (2012).MathSciNetView ArticleGoogle Scholar
 L Li, A Burr, R de Lamare, in IEEE 77th Vehicular Technology Conference (VTC Spring) 2013. Joint iterative receiver design and multisegmental channel estimation for ofdm systems over rapidly timevarying channels (Dresden, Germany, 2013), pp. 1–5.Google Scholar
 ML Honig, JS Goldstein, Adaptive reducedrank interference suppression based on the multistage wiener filter. IEEE Trans. Commun. 6(50), 986–994 (2002).View ArticleGoogle Scholar
 Y Sun, V Tripathi, ML Honig, Adaptive, iterative, reducedrank (turbo) equalization. IEEE Trans. Wirel. Commun. 4(6), 2789–2800 (2005).View ArticleGoogle Scholar
 RC de Lamare, R SampaioNeto, Reduced–rank adaptive filtering based on joint iterative optimization of adaptive filters. IEEE Signal Process. Lett. 14(12), 980–983 (2007).View ArticleGoogle Scholar
 RC de Lamare, R SampaioNeto, Reducedrank space–time adaptive interference suppression with joint iterative least squares algorithms for spreadspectrum systems. IEEE Trans. Veh. Technol. 59(3), 1217–1228 (2010).View ArticleGoogle Scholar
 RC de Lamare, R SampaioNeto, Adaptive reducedrank equalization algorithms based on alternating optimization design techniques for MIMO systems. IEEE Trans. Veh. Technol. 60(6), 2482–2494 (2011).View ArticleGoogle Scholar
 RC de Lamare, R SampaioNeto, Adaptive reducedrank processing based on joint and iterative interpolation, decimation, and filtering. IEEE Trans. Signal Process. 57(7), 2503–2514 (2009).MathSciNetView ArticleGoogle Scholar
 M Honig, M Shensa, S Miller, L Milstein, in IEEE Vehicular Technology Conf. Performance of adaptive linear interference suppression for DSCDMA in the presence of flat rayleigh fading (Phoenix, 1997).Google Scholar
 HV Poor, X Wang, in Annual Allerton Conf. Commun., Control and Computing. Adaptive multiuser detection in fading channels (Monticello, 1996).Google Scholar
 U Madhow, K Bruvold, LJ Zhu, Differential MMSE: A framework for robust adaptive interference suppression for DSCDMA over fading channels. IEEE Trans. Commun. 53(8), 1377–1390 (2005).View ArticleGoogle Scholar
 LJ Zhu, U Madhow, in IEEE Global Telecoms. Conf. Adaptive interference suppression for direct sequence CDMA over severely timevarying channels (Pheonix, 1997).Google Scholar
 M Honig, S Miller, M Shensa, L Milstein, Performance of adaptive linear interference suppression in the presence of dynamic fading. IEEE Trans. Commun. 49(4), 635–645 (2001).MATHView ArticleGoogle Scholar
 PSR Diniz, Adaptive Filtering: Algorithms and Practical Implementation, 3rd edn. (Springer, 2008).Google Scholar
 C Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2001).Google Scholar
 DG Luenburger, Y Ye, Linear and Nonlinear Programming, 3rd edn. (Springer, 2008).Google Scholar
 J Lee, H Yu, Y Sung, Beam tracking for interference alignment in timevarying mimo interference channels: A conjugategradientbased approach. IEEE Trans. Veh. Technol. 63(2), 958–964 (2014).View ArticleGoogle Scholar
 PS Chang, AN Wilson, Analysis of conjugate gradient algorithms for adaptive filtering. IEEE Trans. Signal Process. 48(2), 409–418 (2000).MATHView ArticleGoogle Scholar
 NJ Bershad, JCM Bermudez, JY Yourneret, An affine combination of two LMS adaptive filters  transient mean square analysis. IEEE Trans. Signal Process. 56(8), 1853–1864 (2008).MathSciNetView ArticleGoogle Scholar
 J ArenasGarcia, A FigueirasVidal, A Sayed, Meansquare performance of a convex combination of two adaptive filters. IEEE Trans. Signal Process. 45(3), 1078–1090 (2006).View ArticleGoogle Scholar
 H Kushner, Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic System Theory (MIT Press, 1984).Google Scholar
 A Sayed, Adaptive Filters (John Wiley & Sons, 2008).Google Scholar
 W Jakes, Microwave Mobile Communications (WileyIEE Press, 1994).Google Scholar