R E S E A R C H
Open Access
Blind ICA detection based on second-order cone
programming for MC-CDMA systems
Chih-Wei Jen
*and Shyh-Jye Jou
Abstract
The multicarrier code division multiple access (MC-CDMA) technique has received considerable interest for its potential application to future wireless communication systems due to its high data rate. A common problem regarding the blind multiuser detectors used in MC-CDMA systems is that they are extremely sensitive to the complex channel environment. Besides, the perturbation of colored noise may negatively affect the performance of the system. In this paper, a new coherent detection method will be proposed, which utilizes the modified fast independent component analysis (FastICA) algorithm, based on approximate negentropy maximization that is subject to the second-order cone programming (SOCP) constraint. The aim of the proposed coherent detection is to provide robustness against small-to-medium channel estimation mismatch (CEM) that may arise from channel frequency response estimation error in the MC-CDMA system, which is modulated by downlink binary phase-shift keying (BPSK) under colored noise. Noncoherent demodulation schemes are preferable to coherent demodulation schemes, as the latter are difficult to implement over time-varying fading channels. Differential phase-shift keying (DPSK) is therefore the natural choice for an alternative modulation scheme. Furthermore, the new blind differential SOCP-based ICA (SOCP-ICA) detection without channel estimation and compensation will be proposed to combat Doppler spread caused by time-varying fading channels in the DPSK-modulated MC-CDMA system under colored noise. In this paper, numerical simulations are used to illustrate the robustness of the proposed blind coherent SOCP-ICA detector against small-to-medium CEM and to emphasize the advantage of the blind differential SOCP-ICA detector in overcoming Doppler spread.
Keywords: Multicarrier code division multiple access (MC-CDMA); Second-order cone programming (SOCP); Independent component analysis (ICA); Negentropy
1 Introduction
The use of multicarrier code division multiple access (MC-CDMA) has recently been proposed for future wire-less communication systems, and cellular networks in particular [1], due to its high data rate or spectrum effi-ciency over frequency-selective fading channels [2,3]. MC-CDMA combines orthogonal frequency-division mul-tiplexing (OFDM) with CDMA to spread the data symbol in the frequency domain. The main advantage of OFDM is that the narrowband of the subcarriers may be signifi-cantly increased in order to combat frequency-selective fading. As opposed to the spreading of data in the time domain, as in the direct sequence CDMA technique, the spreading of data associated with the subcarriers in
MC-CDMA is carried out in the frequency domain. In order to ensure orthogonality among users in the frequency do-main, Walsh-Hadamard sequences are often implemented to achieve the spreading [4]. However, a disadvantage of the MC-CDMA system is that it is easily affected by imperfect channel frequency response estimation mis-match [5]. Besides, the system performance may suffer from Doppler spread, which is induced by time-varying fading channels [6].
In general, coherent receivers that utilize channel esti-mation and compensation achieve a gain of approximately 3 to 4 dB in signal-to-noise ratio (SNR) compared to non-coherent receivers [7] under additive white Gaussian noise (AWGN). Therefore, the design of coherent receivers is based on the assumption that the channel estimate is error-free. However, coherent demodulation is difficult to achieve over time-varying fading channels, as this requires * Correspondence:charlesjen10@gmail.com
Department of Electronics Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan
© 2014 Jen and Jou; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
frequency and phase synchronization of all carriers, and therefore, it is logical to consider differential phase-shift keying (DPSK) as an alternative modulation scheme [8].
It should also be mentioned that the performance of the system at the receiver end will be degraded by the perturbation of colored noise, which can be defined as the variation in power spectral density in the frequency domain, and may be induced by narrowband interference (NBI) [9,10], intercarrier interference (ICI) in wideband channels [10,11], oversampling [12], channel shortening [13], analog bandpass filter [14], etc. in multicarrier sys-tems. In most practical circumstances, white noise is sim-ply a particular instance of colored noise. The assumption of the presence of colored noise is therefore often much more realistic than assuming white noise, which may ren-der difficulties in implementation.
A large number of blind multiuser detection methods, making use of only the spreading sequence and timing of the desired user, have been proposed. The main rea-soning behind employing a blind detector is to avoid the requirements of a training sequence and the reduced loss of channel throughput. Suboptimal approaches to the use of blind multiuser detectors, such as [15,16], have also been proposed to mitigate multiple access interference (MAI), which results from multipath fading destroying the orthogonality of the user's spreading sequences. The per-formance of each of these methods is heavily dependent on accurate knowledge of the channel impulse response or the signature waveform of the user of interest at the re-ceiver end. Unfortunately, in most real-world, practical cir-cumstances, a perfect estimation of the information within channel parameters cannot be conveyed to the receiver, resulting in the system's performance degradation accord-ingly [17-20]. In [15], Honig et al. propose a blind version of the linear minimum mean square error (LMMSE) de-tector, using constrained minimum output energy (CMOE) as the cost function for MAI suppression. However, in this method, the receiver's performance is considerably de-graded in the presence of channel and data correlation matrix estimation errors [21,22].
There has been substantial research into the use of subspace-based blind receivers [18,23] in the MC-CDMA system. The second-order statistic (SOS)-based technique utilizes eigenvalue decomposition (EVD) on the covari-ance matrix of the received signal. However, in this tech-nique, the orthogonality between the signal and noise subspaces is subject to serious destruction due to corre-lated noise subspace [24,25]. Another effective diagonal loading technique [26] has been the widely used ap-proach of enhancing robustness against mismatch er-rors and correlated noise. The main drawback of this method is that there is uncertainty as to how to pre-cisely choose the optimal value of the diagonal loading factor [27]. In [28], Wang et al. utilize a blind
variable-diagonal loading (VDL) multiuser detector, based on the concept of optimizing the worst-case performance over properly chosen bounded uncertainty sets that are de-termined by a bounding probability, to provide further robustness against small-to-medium prior signal model-ing errors in MC-CDMA systems.
Another extensively studied technique is the blind sep-aration of sources (BSS) based on higher-order statistics (HOS), which has a potential for application in a variety of areas, such as biomedical engineering, sonar, radar, speech enhancement, and telecommunications [29-32]. SOS-based techniques usually require the power spectra of the sources to be different [24], while in general, HOS-based techniques depend on the non-Gaussian sources in the probability distribution, even if the sources have the same power spectra. The HOS-based independent compo-nent analysis (ICA) algorithm has been implemented to deal with BSS of unknown sources from a mixed-up re-ceived signal. This algorithm is based on the assumption of the independence and non-Gaussianity of the sources be-fore maximizing the approximate negentropies of those ones [30,33]. The approximate negentropy can be described by choosing a proper nonlinear cost function. Thereafter, the classical ICA algorithm performs the pre-processing steps, which include centering and whitening [33,34] based on eigenvector projection, to decorrelate the mixed-up received signal, following which it separates the mixture by using a ‘Gram-Schmidt-like’ technique [30,33] after every iteration step under unit variance constraint. However, when using this technique, the weight order of the source separation will be unpredictable, a phenomenon which is referred to as weight order ambiguity [30,35].
The application of ICA has been extended to dealing with complex-valued sources, in addition to real-valued sources [36-40]. In [39], two types of receivers, the RAKE-ICA and MMSE-RAKE-ICA detectors, have been proposed, which are developed by modifying the classical complex-valued fast independent component analysis (FastICA) al-gorithm [36], based on the quasi-Newton iteration as well as the proper starting point initializing iterations. Further-more, a modified stochastic gradient descent ICA detector [40] has been generalized from Amari's natural gradient algorithm [41]. All of these ICA-based detectors first carry out the centering and whitening steps and then select the appropriate starting point initializing iterations for captur-ing the correct weight vector. If colored noise exists, the whitening based on eigenvector projection is unable to decorrelate the mixed-up signals [42,43], which means that the system's performance is degraded accordingly.
Under colored noise environment, the SOS-based tech-nique will fail since some of the correlated noise subspace diverges into the signal eigenvectors causing serious de-struction to this orthogonality especially at low SNR [25]. The HOS-based ICA algorithm depends on the
non-Gaussian sources in the probability distribution for BSS and feature extraction [30,33]. Available information may be used to solve the problem of weight order ambiguity, by either indirectly choosing appropriate close point ini-tializing ICA iterations [44] or directly adding the add-itional constraint [42,45]. Besides, the signal processing technique proposed in [46,47] utilizes the worst-case per-formance optimization based on second-order cone pro-gramming (SOCP) constraint to mitigate estimation error. Therefore, in this paper, a new blind coherent detection will be developed, which utilizes the modified FastICA algorithm [36] and the optimization of the lower bound of the worst-case performance, based on approximate negentropy maximization, as well as the concept of SOCP constraint [46,47]. This method initializes itera-tions using appropriate starting points to improve ro-bustness against small-to-medium channel frequency response estimation mismatch in the downlink binary phase-shift keying (BPSK)-modulated MC-CDMA sys-tem under colored noise.
The application will then be extended to combat large channel estimation mismatch (CEM); that is, the design will be associated with DPSK demodulation without chan-nel estimation and compensation in order to overcome Doppler spread caused by time-varying fading channels in the noncoherent MC-CDMA system under colored noise.
The coherent and differential SOCP-ICA detectors pro-posed in this paper are capable of dealing with the weight order ambiguity present in the classical ICA algorithm [33,36], converging on the correct solution over complex channels with colored noise. Computer simulations indi-cate that the performance of the proposed blind multiuser detectors is superior to those in existing literature.
The remaining of this paper is organized as follows. ‘Section 2’ describes the signal model for the downlink MC-CDMA system over Rayleigh fading channel.‘Section 3’ respectively proposes the blind coherent and differential SOCP-ICA detectors to combat against the small-to-medium CEM and Doppler spread under colored noise. ‘Section 4’ compares both computational complexities with other methods. Simulation results are shown and dis-cussed in‘Section 5,’ and ‘Section 6’ concludes this paper. 2 Signal model
Consider a synchronous downlink MC-CDMA system with M orthogonal subcarriers and K active users. First, the ith input data symbol of the kth user, bk(i)∈ {1, − 1}
(k∈ {1, 2, ⋯, K}), is replicated M times; each of these M parallel copies is spreaded by the corresponding chip of the kth user's orthogonal Walsh-Hadamard code, ckð Þm m¼0; 1; ⋯ ; M−1∈ 1= ffiffiffiffiffi M p ; −1=pffiffiffiffiffiM , in the frequency domain. Then, the spreaded symbol is sent to an M-point inverse discrete Fourier transform (IDFT)
to convert the spread data into the time domain. After the parallel-to-series conversion, a cyclic prefix (CP) - pre-sumed to be longer than the maximum delay spread - is inserted to eliminate intersymbol interference (ISI) caused by multipath fading. For the user k, the downlink sym-bols are finally transmitted over the frequency-selective Rayleigh fading channels and their equivalent effect in discrete time is a finite impulse response tap-delay line filter with channel impulse responseβk(0), βk(1), ⋯, βk
(Lk− 1), where Lkis the channel order of the kth user. It
is assumed that the narrowband signal at each sub-carrier suffers from flat fading.
At the receiver end, after removing the CP, the result-ant is then passed through a series-to-parallel conversion and sent to an M-point DFT to convert the data into the frequency domain. For the desired user 1, the received baseband MC-CDMA signal in the frequency domain can be represented as y ið Þ ¼XK k¼1 Q1 ffiffiffiffiffi Pk p ckbkð Þ þ n ii ð Þ; ð1Þ where ffiffiffiffiffiPk p
is the chip energy of the kth user,ck= [ck(0),
ck(1), ⋯, ck(M− 1)] is the normalized spreading code
vector of the kth user, andQ1= diag{q1, 0, q1, 1,…, q1, M− 1}
denotes the channel matrix of the first user, whose diagonal elements are the channel frequency response given by q1; m¼XLl¼01−1β1ð Þel −j2π ml=M; m ¼ 0; 1; ⋯; M−1 . It is assumed thatn(i) = [ni(0), ni(1),…, ni(M− 1)]Tis a colored
noise vector, each element of which contains an additive channel noise that is a zero-mean random variable with a variance ofσ2
n.
3 Blind ICA detection based on SOCP constraint The HOS-based ICA algorithm is actually a neural net-work learning rule transformed into an efficient fixed-point iteration. It is based on an assumption of the independence and non-Gaussianity of the sources. Due to using the CDMA technique, it is rationally presumed that each user and path is approximately independent of each other, so each received data sequence can be regarded as an inde-pendent source sequence. Afterward, specific features of the received mixed-up signals will be extracted by maxi-mizing their approximate negentropies.
First, the differential entropy H of a continuous ran-dom vector z = [z1, z2, ⋯, zK]T with density f(z) is
de-fined as Η(z) = − ∫f(z)log f(z)dz [48]. The kth random variable zk is assumed to be of zero mean and unit
variance for k = 0, 1, ⋯, K. Negentropy can be simply obtained from differential entropy and defined [30,33] as
where zGauss is a Gaussian random vector of the same
covariance matrix asz. Mutual information is the theor-etical measure of the dependence between random vari-ables. Then, exploiting negentropy, mutual information can be expressed [30,33] as
I zð 1; z2; ⋯; zKÞ ¼ J zð Þ−
XK
k¼1J zð Þ;k ð3Þ
where J(z) is a constant for a given covariance matrix. Maximizing the sum of the negentropy of the components subject to a variance constraint equals minimizing mutual information while estimating independence. Therefore, it is clearly shown from (3) that minimizing the mutual in-formation is roughly equivalent to finding the direction in which the negentropy is maximized. Negentropy is a measure that is zero for a Gaussian variable and always nonnegative. An approximation of negentropy for the kth signal can be given [33] by
J zð Þ≈ξ E G zk ½ f ð Þk g−E G υf ð Þg2; ð4Þ where E{‧}, G, and ξ denote, respectively, the expect-ation, any nonquadratic function, and an irrelevant con-stant andυ is a Gaussian variable of zero mean and unit variance. The random variable zkis assumed to be of zero
mean and unit variance [30,33]. The approximation of neg-entropy given in (4) readily provides a cost function for carrying out the ICA. It can measure the non-Gaussianity of data sequences by maximizing their approximate neg-entropies. Generally speaking, super-Gaussian cases, in-cluding binary sources, are extensively used and result in E{G(zk)} > E{G(ν)} [49] with respect to a proper G function.
Therefore, the maximum value of (4) is obtained at a cer-tain optimum of E{G(zk)} [33] that is reduced to
maximiz-ing E{G(zk)} [36,49]. Maximizing the negentropy could be
interpreted as an estimation of a single independent com-ponent, which means that if J(z1) is the largest negentropy
among all J(zk), z1has the largest non-Gaussianity.
For complex-valued sources, find a certain signal tap and make zk=wHy, where w is a certain signal tap-weight
vector and the superscript H denotes the complex conju-gate transposition. This inference is extended to complex-valued sources in [36-38,49], making the cost function
J zð Þ ¼ EfGð wk Hy 2
Þg: ð5Þ
Then, maximizing (5) subject to unit variance constraint E{|wHy|2} = 1, the weight vectorw of the classical complex-valued FastICA can obtain the optimization problem [36]
maximize w EfGð w Hy 2 Þg; subject to Ef w Hy2 g ¼ 1: ð6Þ
It has been demonstrated through stability analysis [33] that almost any nonquadratic, even function can
be regarded as a cost function for non-Gaussianity maximization. For super-Gaussian signals, a suitable choice of the G function is a smooth, even function with fast computation, such as G = log(0.1 + |wHy|2
). Independent components with super-Gaussian distri-bution tend to be found first [33,36].
To achieve distinct convergent points, a Gram-Schmidt-like orthogonalization technique [33] is generally ap-plied after every ICA iteration step to separate the signals of each user. However, the weight order of the source separation is unpredictable [30]. Even though the desired weight will be captured using centering and pre-whitening [33,34], based on eigenvector projection and an appropriate starting point initializing ICA itera-tions [39], it is still sensitive and subject to the complex channel model.
Available information may be used to solve the problem of weight order ambiguity, either by indirectly choosing appropriate close point initializing ICA iterations [44] or by directly adding the additional constraint [42,45]. ICA is a higher-order, statistical signal processing technique for BSS and features extraction by maximizing the approxi-mate negentropy of input signals, which is significantly less sensitive to the colored noise model than the SOS-based EVD [24]. Therefore, the proposed idea - which im-poses the additional SOCP constraint in the classical, complex-valued FastICA algorithm and then initializes it-erations using the desired spreading code - could attain an adaptive, interference suppression technique for coherent and differential detection over complex channels with col-ored noise.
3.1 Robust coherent SOCP-ICA detection
In the coherent BPSK-modulated MC-CDMA system, the received data vector ^y ið Þ at the desired receiver end has been implemented by channel estimation and com-pensation and can be represented as
^ y ið Þ ¼ ^Q−1 1 y ið Þ ¼ ffiffiffiffiffiP1 p ^c1b1ð Þ þ si intþ ^Q−11 n ið Þ; ð7Þ
where ^Q1 is the estimated channel frequency response matrix of the desired user, ^c1¼ ^Q−11 Q1c1 is defined as
the presumed (i.e., compensated) spreading signature, andsint¼ XK k¼2 ^ Q−1 1 Q1 ffiffiffiffiffi Pk p
ckbkð Þ is the interference vec-i
tor with regard to user 1. It is usually assumed that the presumed spreading signature^c1has perfect knowledge of
its spreading code vector,^c1¼ c1, but this may not be true
in real systems since the presumed spreading signature (^c1
equalingc1) may include CEM and is therefore nominal.
actual desired spreading code vectorc1and its presumed
value^c1,
^c1¼ c1þ Δe; ð8Þ
whereΔe denotes an unknown, complex mismatch vector of the desired user. Since the channel frequency response estimation error is known to be complex, circularly sym-metric, and Gaussian-distributed [5], it is reasonably in-ferred that the mismatch termΔe is modeled as a vector of complex, circularly symmetric, Gaussian random vari-ables with zero mean and a variance ofσ2
e.
Now, it is assumed that the norm of mismatch vector Δe would be bounded by a small-to-medium value δ, i.e., ‖Δe‖ ≤ δ, where ‖ ⋅ ‖ denotes the Frobenius norm of a vec-tor. The presumed spreading signature^c1 is expressed as a
vector in the set
C1ð Þ ¼δ f j^c1 ^c1¼ c1þ Δe; Δek k≤ δg: ð9Þ For constraint satisfaction, the weight vector w must satisfy wH^c
1
j j≥1 for all vectors ^c1∈C1ð Þ , where w isδ
the desired signal tap-weight vector of the multiuser detector. Such a constraint guarantees that the data symbols of user 1 can be taken from the mixture, re-gardless of how its channel estimation is mismatched, while the error is bounded by δ. Now suppose that the weight constraint is enforced; the main goal will be to find a vector w that maximizes the cost func-tion EfGð wj H^yj2Þg. Then, the problem of the worst-case
design of the ICA detector based on approximate neg-entropy maximization can be formulated as
maximize w EfGð w H^y 2 Þg; subject to wH^c 1 ≥1 for k k≤δ:Δe ð10Þ
The problem in (10) can be rewritten as
maximize w EfGð w H^y 2 Þg; subject to min Δe k k≤δjw H^c 1j≥1: ð11Þ
The ICA formulation introduces the additional SOCP condition [46], which has infinitely many nonconvex quadratic constraints on w. According to the derivation of [50], it can be easily shown that in the case of a suffi-ciently small uncertainty (i.e., where the uncertainty ra-diusδ ≤ |wHc1|/‖w‖), the left-hand side of the constraint
in (11) can be rewritten as min Δe k k≤δ jw H c 1þ Δe ð Þ ¼j jwHc 1j−δkw :k ð12Þ
Using (12) and taking into account that the cost func-tion in (10) is not affected by an arbitrary phase
rotation of w, the problem of (11) can be described in this form: maximize w EfGð w H^y 2 Þg; subject to wHc 1−δ wk k≥1 ð13Þ
If the cost function in (13) is maximized, the inequality constraint in (13) is satisfied, which means that, while maximizing the cost function in (13), the equality con-straint wHc1− δ‖w‖ = κ is satisfied, where κ ≥ 1. Then,
an extra constraint Im {wHc1} can be omitted because
the equality constraint wHc1− δ‖w‖ = κ guarantees that
the value ofwHc1is real-valued and positive [50]. Using
the equality constraint, the weight vectorw is obtained by rewriting the convex optimization problem of (13) as
maximize w EfGð w H^y 2 Þgsubject to 1 δ2 w Hc 1−κ 2 ¼ wHw: ð14Þ
This solves the optimization problem of worst-case per-formance by maximizing the approximate negentropy sub-ject to the SOCP constraint. The constrained optimization problem in (14) will be derived by the quasi-Newton it-eration. Newton's method is based on the Lagrangian function Jð Þ ¼ EfGð ww H^y2Þg−μ0 1 δ2 w Hc 1−κ 2 −wHw ; ð15Þ
whereμ0is a Lagrange multiplier. In order to avoid the
influence of correlated noise subspace when decorrelat-ing mixed-up signals, the pre-whitendecorrelat-ing step is omitted. Finally, the desired weight vector with unit variance con-straint in the proposed method is as follows (details are presented in Appendix 1):
Step 1. After centering ^y , take a small initial vector w(0) = 0.01c1. Let iteration number p = 0.
Step 2. Update ^w ¼ R−1 ^y E ^y wð Hð Þp^yÞg jwHð Þp^yj 2 n o − E gjwHð Þp^yj2 þ wj Hð Þp^yj2 g′ wj Hð Þp^yj2 w pð Þ þμ⋅R−1 ^y c1; w p þ 1ð Þ← ^w= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^wHR ^y^w p ; ð16Þ
where R^y¼ Ef^y^yHg is the data correlation matrix and g
and g′ denote the derivatives of G and g, respectively. Step 3. Check the convergence of w(p). If the error measure isXMm¼1jwmðpþ 1Þ−wmð Þp j> ε, where ε is the
terminating error value, let p = p + 1 and go back to step 2. Otherwise, output the vectorw(p).
According to classical FastICA [36], the step that projects ^w through w p þ 1ð Þ← ^w. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip^wHR^y^w could enforce the
unit variance constraint Ef wj H^yj2g ¼ 1 after each step.
For coherent detection, a new modification of the classical FastICA has been proposed that omits the complex pre-whitening step and directly incorporates the SOCP constraint onto the correct solution. Of course, the desired spreading code is still necessary as a starting point. In the coherent MC-CDMA system, the proposed coherent SOCP-ICA detection is developed to combat small-to-medium, norm-bounded CEM under colored noise. It is worth noting that the proposed co-herent detector can be implemented by choosing a proper value for μ without estimating δ. Due to the algorithmic parameter μ in (16) (including to the Lagrange multiplier μ0, the parameter κ, and the
un-known CEMδ), the proposed SOCP-ICA possesses the capability of self-adjusting computation against CEM. Self-adjusting computation refers to a model of com-puting where computations can automatically respond to changes in varying CEMδ, which means that param-eterμ can be kept a proper constant under varying cir-cumstances by using self-adjusted μ0 and κ for ICA
iterations. The proposed algorithm is nonsensitive to small-to-medium CEM.
3.2 Differential SOCP-ICA detection
For a noncoherent receiver, DPSK is usually used to avoid channel estimation and tracking over a time-varying fading scenario [8]. When using the DPSK for-mat, information is encoded in the phase difference between two successive symbol transmissions and the sequence of the transmitted data symbols for user k is given by bk(i) = ak(i)bk(i− 1), where ak(i) is the
unmodu-lated data. In order to decorrelate observed mixtures, after centeringy, the observed data are first projected on the signal eigensubspace for the pre-processing step. To accomplish this, the transformation is given by
˜y ¼ EsEHsy; ð17Þ
where Es is the signal subspace and corresponds to the
principle eigenvectors of the data correlation matrix Ry= E{yyH}.
In contrast to the small-to-medium, norm-bounded CEM in (11), the formulation will be extended to a large-scale CEM over time-varying channels, which means that a differential SOCP-ICA criterion will not re-quire the channel-state information of the desired user. The proposed differential detection maximizing the approximate negentropy that is subject to worst-case performance is still reasonably applied to the largely
norm-bounded mismatch Δe and can be described in the following form:
maximize w EfGð w H~y 2 Þg; subject to min Δe k k≤δjw H^c 1j≥1: ð18Þ
Note that the bounded region δ of the differential scheme in (18) is different from the coherent in (11). The formulation will then be developed to overcome Doppler spread in the noncoherent MC-CDMA system under col-ored noise. Without channel estimation, the norm of Δe would be bounded by a large valueδ. The solution to (18) can then be found by maximizing the function
Jð Þ ¼ EfGð ww H~y2Þg−μ0 1 δ2 w Hc 1−κ 2 −wHw : ð19Þ
Next, the differential SOCP-ICA is performed as follows (details are similar to what is shown in Appendix 1):
Step 1. After centering y and projecting on the signal subspace, take a small initial vectorw(0) = 0.01c1. Let
it-eration number p = 0. Step 2. Update ~w ¼ R−1 ~yE ~y wð Hð Þ~yp Þg jwHð Þ~yp j2 n o − E gjwHð Þ~yp j2 þ wj Hð Þ~yp j2 g′ wj Hð Þ~yp j2 w pð Þ þμ⋅R−1 ~yc1; w p þ 1ð Þ← ~w=pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~wHR~y~w; ð20Þ
whereR~y¼ Ef~y~yHg is the data correlation matrix.
Step 3. Check the convergence of w(p). If the error measure isXMm¼1jwmðpþ 1Þ−wmð Þp j> ε, where ε is the
terminating error value, let p = p + 1 and go back to step 2. Otherwise, output the vectorw(p).
Finally, the decoding rule without training sequence for the desired user is given as ^a1ð Þ ¼ sign Re wi ð fð Hy ið ÞÞ
wHy i−1ð Þ
ð ÞgÞ:
4 Comparison of computational complexity The computational complexities of both the proposed coherent detector in (16) (including projection) and the differential SOCP-ICA detector in (20) (including pre-processing steps) are about O(M2B) in total (details are presented in Appendix 2).
The computational complexity of the proposed detectors will then be compared with the CMOE detector [15], the subspace-based MMSE detector [23], the robust VDL de-tector based on bounding probability [28], and the MMSE-ICA detector [39]. The evaluation of the CMOE detector including autocorrelation matrix R^y and its inverse
O(M2B). In the subspace-based MMSE detector, the EVD of the autocorrelation matrix has the complexity of O(M2B) + O(M2K), and the complexity of the projec-tion of the desired signature waveform onto signal subspace is O(M2). Thus, the final complexity of the subspace-based MMSE is of order O(M2B). The compu-tation of the VDL detector has the complexity of O(M2B). Besides, in the MMSE-ICA, the computational complex-ities of autocorrelation matrix, prior subspace estimation, pre-whitening of the received data, and each unit-gain-based ICA iteration are O(M2B), O(M2K), O(MK), and O(KB), respectively. Thus, the final complexity of the MMSE-ICA is of order O(M2B).
5 Simulation results
Consider a synchronous multiuser MC-CDMA system under colored noise environment, using the following fixed parameters: the central frequency fc= 5.2 GHz,
num-ber of subcarrier M = 64, sampling rate 20 MHz, Doppler spread equaling 20 Hz. The length of the adopted Walsh-Hadamard code is equal to 64 chips. Thus, the MC-CDMA system can support the maximum number of ac-tive users K = 64. At the desired receiver end, an unknown Rayleigh multipath fading channel model is adopted where the channel consists of L1= 4 multipath with gains β1(l)
being independent, identically distributed complex Gauss-ian random variables with zero mean, identical varGauss-iance 1/ L1, and delay uniform over [0, 3Tc]. Now, it is
appropri-ately assumed that the small-to-medium CEM Δee N 0; σ2
e
happens at the receiver end. An additional 15Tc
guard interval duration is used to provide protection from ISI due to channel multipath delay spread. The additive
colored noise can be modeled by a moving average (MA) process [51] and generated by applying a white Gaussian noise sequence to a corresponding finite impulse response (FIR) filter with transfer function H(z) = 1− αz− 1 [52], where α is the correlation coefficient. In order to analyze the performance of coherent and differential detectors under the scenario of highly correlated noise, the correlated coefficientα is equal to 0.8. The power of each interfering user is 3 dB more than that of the desired user at each trial. For the proposed coherent SOCP-ICA in (16) and the dif-ferential SOCP-ICA in (20), the parameter values μ are equal to 0.91 and 10, respectively. The terminating errors of both the iterative processes are set to 10−5at each trial. For comparison purposes, the performance is also compared with the CMOE [15], the subspace-based MMSE [23], the robust VDL [28], and the MMSE-ICA [39] methods. The number of data symbols taken in the simulations is 1,000 data symbols at each trial. The number of Monte Carlo simulations is averaged over 5,000 independent trials.
First, Figure 1 shows the desired symbol error rate (SER) performance versus the received symbol size for each de-tector while the number of active users K = 15, the desired user's SNR =15 dB, and CEMδ = 0.1. It can be found that when sample size is 600 data symbols, the proposed blind differential SOCP-ICA detector can achieve steady per-formance, and all the other coherent and differential de-tectors can achieve steady performance within 1,000 data symbols. The results show the speed of convergence of all the schemes and confirm that the 1,000-symbol is enough implementing for all detectors.
Then, consider the influence of the correlated coeffi-cient α on each blind coherent detector with perfect channel estimation and blind DPSK detector. Figure 2
1 200 400 600 800 1000 1200 1400 1600 1800 2000 10-4 10-3 10-2 10-1 100 SER
Number of Received Symbols
CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
depicts the sensitivity in terms of the SER versusα while K= 15, SNR =15 dB, andδ = 0. In general, the perform-ance of the coherent scheme is better than the differen-tial under AWGN noise. The subspace-based MMSE and MMSE-ICA coherent detectors are much more sen-sitive to the correlated coefficient than the others since parts of the variation of the noise subspace diverge into the signal eigenvectors causing serious destruction to this orthogonality. The blind differential CMOE cannot capture the desired symbols and actually diverges after
decoding. The proposed blind coherent SOCP-ICA de-tector in (16) omitting the complex pre-whitening step based on SOS eigenvector projection has the best per-formance; the proposed blind differential SOCP-ICA de-tector degrades with increasing the correlated coefficient α but is still better than the other blind differential ones. Clearly, HOS-based separation technique is less sensitive to colored noise compared to SOS-based EVD.
Figure 3 represents the sensitivity about the SER ver-sus the CEMδ for K = 15 and SNR =15 dB. For coherent
0 0.2 0.4 0.6 0.8 1 10-5 10-4 10-3 10-2 10-1 100 SER CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 2 SER versus the correlated coefficientα. (—: coherent, - - -: differential).
0 0.05 0.1 0.15 0.2 10-5 10-4 10-3 10-2 10-1 100 SER CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
schemes, the coherent CMOE is sensitive to the estima-tion mismatch which can lead to serious errors. The VDL detector based on the bounded condition provides a design trade-off between sensitivity and interference suppression. The proposed coherent SOCP-ICA detector which is a HOS-based source separation technique subject to SOCP constraint can provide robustness in the pres-ence of small-to-medium channel mismatch and heavily correlated coefficients and substantially improves the sys-tem performance. The differential modulation schemes do not need channel estimation and compensation and are independent withδ. Once δ > 0.11, the proposed blind dif-ferential SOCP-ICA detector process is much more robust than the coherent SOCP-ICA.
Figure 4 shows the SER of the coherent schemes for K= 15 and K = 30 as a function of the desired user's input SNR while CEMδ = 0.1. It is noticed that the performance of the coherent CMOE detector could not get better while SNR is increasing. The CMOE detector based on the opti-mal solution enforces the response of the interest signal to be of unity and minimizes the total variance of the de-tector output. While CEMδ exists in the constraint of the CMOE detector, the response of the interest signal is no longer constrained to be of unity. A part of the power of the interest signal will donate towards the interference and thus is seriously degraded by the CMOE method while minimizing the total variance of the detector output. This phenomenon is called signal cancellation or self-nulling. About the subspace-based MMSE and MMSE-ICA coher-ent detectors, it is seen that the performance for K = 30 becomes better than that for K = 15 at a larger SNR region. This performance improvement can be explained by the
fact that the subspace-based methods are less sensitive to the heavily correlated coefficient by reason of the influence of noise subspace on higher-power signal eigenvectors be-coming weaker while increasing more users. The perform-ance of the proposed coherent SOCP-ICA detector is more robust than that of the other coherent ones. Figure 5 shows the SER of the differential schemes for K = 15 and K= 30 as a function of the desired user's input SNR. From Figures 4 and 5, it can be seen that the proposed SOCP-ICA detectors have better performance compared with the others. Figure 6 displays a comparative analysis of the dif-ferential schemes and the coherent schemes with perfect channel-state information for K = 15. It is shown that there is an approximately 3- to 4-dB gap between the differential and the coherent SOCP-ICA with higher SNR.
The near-far effect is exhibited in Figure 7 that the power of each interfering user is more than the de-sired user's input power for K = 15, SNR =15 dB, and CEM δ = 0.1. The near-far resistance of the proposed SOCP-ICA detectors have better performance than the other ones, and the differential SOCP-ICA is especially the best while each interference-to-noise ratio (INR) is greater than 8 dB.
Figure 8 shows the error measure versus the average number of iterations p for K = 15, SNR =15 dB, and CEM δ = 0.1. Here, coherent and differential SOCP-ICA detectors with four different numbers of users, K, are considered: 1, 15, 30, and 64. In order to attain the ter-minating error value 10−5, it can be observed that the lar-ger K is, the more the iteration numbers are. In general, the differential SOCP-ICA converges much faster than the coherent SOCP-ICA. 0 3 6 9 12 15 18 10-4 10-3 10-2 10-1 100 SER SNR CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
The section will analyze the influence of time-varying fading channels. Figure 9 illustrates the performance of the coherent and differential schemes with a high Dop-pler spread of 200 Hz for K = 15 and CEM δ = 0.1. Then, Figure 10 illustrates the performance of the differ-ential schemes over fast fading channels with a Doppler spread of 2 MHz. Here, 1,000 data symbols are used throughout the system at each trial. The notable feature of Figures 9 and 10 is that all the coherent schemes fail in 200 Hz and the significant performance of the differential SOCP-ICA detector is robust to rapid channel time varia-tions and outperforms the other differential ones. Figure 11 compares the performance of the proposed differential SOCP-ICA employing symbol sizes 1,000 and 2,000 over
time-varying fading channels with Doppler spreads of 200 and 2 MHz. The results show that there is an obvious dif-ference in performance for different Doppler spread values under the desired user's input SNR >18 dB and that both different symbol sizes have almost the same performance over the same fading scenario.
Figure 12 shows a comparative analysis between two different sets of delay taps [0, Tc, 2Tc, 3Tc] and [0, 4Tc,
8Tc, 12Tc] with equal Doppler spread of 20 Hz at the
four diversity branches for K = 15 and CEMδ = 0.1. Due to slightly time-varying fading channels, the coherent detectors degrade with increasing the delay taps.
In summary, the outcomes of these experiments have il-lustrated that the proposed blind coherent SOCP-ICA
0 3 6 9 12 15 18 10-5 10-4 10-3 10-2 10-1 100 SER SNR CMOE Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 5 SER of the differential schemes versus the desired user’s input SNR. (—: K =15, - - -: K = 30).
0 3 6 9 12 15 18 10-6 10-5 10-4 10-3 10-2 10-1 100 SER SNR CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
detector can provide robustness against small-to-medium CEM and the blind differential SOCP-ICA detector can successfully overcome Doppler spread under colored noise environment.
6 Conclusions
In this paper, the proposed ICA based on approximate negentropy maximization subject to SOCP constraint has been applied to blind multiuser detection for a synchron-ous MC-CDMA system. The proposed blind coherent SOCP-ICA detector can provide robustness against small-to-medium CEM under colored noise to achieve better
performance than the other coherent ones. Besides, the proposed blind differential SOCP-ICA can successfully overcome Doppler spread over heavily time-varying fading channels with colored noise. Simulation results have dem-onstrated that the new blind coherent and differential SOCP-ICA detectors substantially improve the system performance.
Appendix 1
This Appendix provides the derivation of (16). Accord-ing to a new cost function in (15), its Jacobian vector is
0 3 6 9 12 15 10-4 10-3 10-2 10-1 100 SER INR CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 7 SER versus the strong INR of each interfering user. (—: coherent, - - -: differential).
0 50 100 150 200 10-5 10-4 10-3 10-2 10-1 100 101 Error measure Number of iterations p coherent K=1 coherent K=15 coherent K=30 coherent K=64 differential K=1 differential K=15 differential K=30 differential K=64
∇J wð Þ ¼ 2Ef^yðwH^yÞgð w H^y2Þg−μ 0 1 δ2 c1c T 1w−κc1 −w : ð21Þ
Using the Newton iteration technique solving (14), the optimum of the cost function EfGð wj H^yj2Þg subject to
the constraint (1/δ2)|wHc
1− κ|2=wHw is obtained at
points where ∇J(w) = 0. Then, the Jacobian matrix of ∇EfGð wj H^yj2Þg can be approximated as
∇2EfGð wj H^yj2
Þg ¼ 2Ef^y^yHgð wj H^yj2
Þ þ ^y^yHjwH^yj2
g′ð wj H^yj2Þg
≈2E ^y^yf HgEfgð wj H^yj2
Þ þ wj H^yj2
g′ð wj H^yj2
Þg;
ð22Þ
where the approximation is done by separating the expectations. The Jacobian of 1=δ 2 c
1cT1w−κc1
−w is 1=δ2
c1cT1−I, where I is the identity matrix. Therefore,
the total approximate Jacobian of (21) is
∇ ∇J wð ð ÞÞ ¼ 2E^y^yHEfgð w H^y2Þ þ jwH^yj2g′ð w H^y2Þg
−μ0 1=δ 2 c1cT1−I : ð23Þ
To utilize the approximate Newton iteration, the weight-updating algorithm is given by
0 3 6 9 12 15 18 10-5 10-4 10-3 10-2 10-1 100 SER SNR CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 9 SER versus the desired user’s input SNR for Doppler spread of 200 Hz. (—: coherent, - - -: differential).
0 3 6 9 12 15 18 10-5 10-4 10-3 10-2 10-1 100 SER SNR CMOE Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 10 SER of the differential schemes versus the desired user’s input SNR for Doppler spread of 2 MHz.
0 3 6 9 12 15 18 21 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 SER SNR 1000-symbol, 200 Hz 1500-symbol, 200 Hz 1000-symbol, 2 MHz 1500-symbol, 2 MHz
Figure 11 SER versus the desired user’s input SNR for different sample sizes and Doppler spreads.
0 3 6 9 12 15 18 10-4 10-3 10-2 10-1 100 SER SNR CMOE Robust-VDL Subspace-MMSE MMSE-ICA SOCP-ICA
Figure 12 SER of the coherent schemes versus the desired user’s input SNR for two sets of delay taps. (—: [0, Tc, 2Tc, 3Tc],
w ¼ w−∇ ∇J w pð∇J wðð Þð ÞÞÞ ¼ w− Ef^y wð H^yÞ gð wj H^yj2 Þg− μð0=2Þ 1=δ 2c 1cT1w−κc1 −w Ef^y^yHgE g w j H^yj2 þ wj H^yj2g′ w j H^yj2 − μ 0=2 ð Þ 1=δ2c 1cT1 ð Þ−I : ð24Þ
The algorithm can be further simplified by multiplying both sides of (24) by− 1=2ð ÞR−1^y ∇ ∇J wð ð ÞÞ. While choosing a sufficiently small μ0, it can be easily shown that the
left-hand side of the equality in (24) w multiplying − 1=2
ð ÞR−1
^y ∇ ∇J wð ð ÞÞ can be set as ^w ¼ γ⋅w, where γ
=−Efgð wj H^yj2Þ þ jwH^yj2g′ð wj H^yj2Þg∈R . Then, project-ing ^w through w p þ 1ð Þ← ^w=pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^wHR^y^w could enforce
the unit variance constraint Ef wj H^yj2g ¼ 1 after each
ICA iteration. The scalar factorγ will not affect the con-vergence point. Finally, it can be explicitly simplified as
^w ¼ R−1
^y Ef^y wð Hð Þ^yp Þgð wj Hð Þ^yp j2Þg
− Efgð wj Hð Þp^yj2Þ þ wj Hð Þp^yj2g′ð wj Hð Þp^yj2Þgw pð Þ þμ⋅R−1 ~y c1; w p þ 1ð Þ← ^w= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^wHR ^y^w p ; ð25Þ whereμ = (μ0κ/2δ2). Appendix 2
The desired weight vector of the coherent SOCP-ICA detector can be obtained by calculating (16). It is assumed that the number of symbols B is used for each batching processing to form the data matrix ^Y≡ ^y 1½ ð Þ; ^y 2ð Þ; ⋯; ^y Bð Þ . First, the computational complexity of computing the autocorrelation matrix R^y ¼ Ef^y^yHg ¼ 1=Bð Þ
XB i¼1^y
i ð Þ^y ið ÞH
needs M2Bmultiplications implementing and is thus of order O(M2B). The inverse autocorrelation matrix R−1
^y can be improved from O(M3) to O(M2) [53]. Then,
the iteration step used by the formula in (16) is
^w←R^y−1ðh pð Þ þ μc1Þ−χ pð Þw pð Þ; ð26Þ where h pð Þ ¼ 1=Bð ÞXBi¼1hið Þ , the vector hp i(p) is the
ith vector of { ^Y □hH(p)}, □ is the Khatri-Rao product, andh pð Þ ≡ g w Hð Þ^pY2 ⊙ w Hð Þ^p Y . Here, |• |2
is taken elementwise and⊙ stands for elementwise multiplication. Furthermore, χ pð Þ ¼ 1=Bð ÞXBi¼1χið Þ, the χp i(p) is the ith
data ofχ pð Þ ≡ g w Hð Þ^p Y2 þg′ w Hð Þ^pY2 ⊙ w Hð Þ^pY2.
wH^Y is of order O(MB). h(p) is of order O(MB) + O(B),
where the last term is due to the elementwise multiplica-tions. The computational complexity ofR−1^y multiplying by an M × 1 vector is of order O(M2). Theχ(p) is of order O
(MB) + O(B). Consequently, the computational complexity of the proposed coherent detector in (16) including projection is of order about O(M2B) + O(M2) + O(MB) + O(B) = O(M2B) (since B > M) in total.
The differential SOCP-ICA detector needs to addition-ally utilize the subspace-based pre-processing step in (17). The computational complexity of autocorrelation matrix R~y and its inverse operation is of order O(M2B) + O(M2). The computational complexity of the classical subspace approach is O(M3) for computing the EVD of a M× M dimensional matrix. If only K eigenvectors are used, its computational complexity is reduced to O(M2K) by recursive algorithms [54]. The computational complex-ity of the pre-processing in (17) is of order O(M2) + O (MB) for B symbols. Therefore, the computational com-plexity of the differential SOCP-ICA detector in (20) in-cluding the pre-processing steps O(M2K) + O(M2) + O (MB) is of order about O(M2B) in total.
Competing interests
The authors declare that they have no competing interests. Received: 25 December 2013 Accepted: 19 August 2014 Published: 8 October 2014
References
1. L Hanzo, B-J Choi, Near-instantaneously adaptive HSDPA-style OFDM versus MC–CDMA transceivers for WiFi, WiMAX, and next generation cellular systems. Proc. IEEE 95(12), 2368–2392 (2007)
2. T Hwang, Y(G) Li, Y Yuan-Wu, Energy spreading transform for down-link MC-CDMA. IEEE Trans. Wireless Commun. 7(5), 1522–1526 (2008) 3. BA Al-fuhaidi, HEA Hassan, MM Salah, SS Alagooz, Parallel interference
cancellation with different linear equalisation and rake receiver for the downlink MC-CDMA systems. IET Commun. 6(15), 2351–2360 (2012) 4. C-C Lin, W-C Chen, C-D Chung, Spectral sidelobe decaying property of
Walsh-Hadamard code in MC-CDMA systems. IEEE Trans. Wireless Commun. 10(10), 3151–3156 (2011)
5. P Ting, C-K Wen, J-C Chen, J-T Chen, BER analysis of the optimum multiuser detection with channel mismatch in MC-CDMA systems. IEEE J. Select Area Commun. 24(6), 1221–1235 (2006)
6. JG Proakis, Digital Communications, 4th edn. (Prentice-Hall, Englewood Cliffs, 2001)
7. T Himsoon, W Su, KJR Liu, Single-block differential transmit scheme for broadband wireless MIMO-OFDM systems. IEEE Trans. Signal Process. 54(9), 3305–3314 (2006)
8. SM Elnoubi, E Sourour, A Elshamly, Performance of multicarrier CDMA with DPSK modulation and differential detection in fading multipath channels. IEEE Trans. Veh. Technol. 51(3), 526–536 (2002)
9. M Morelli, M Moretti, Channel estimation in OFDM systems with unknown interference. IEEE Trans. Wireless Commun. 8(10), 5338–5347 (2009) 10. CR Berger, J Gomes, JMF Moura, Sea-trial results for cyclic-prefix OFDM with
long symbol duration, in Proceedings of the MTS/IEEE OCEANS Conference (Santander, Spain, 2011)
11. CR Berger, W Chen, S Zhou, J Huang, A simple and effective noise whitening method for underwater acoustic orthogonal frequency division multiplexing. J. Acoust. Soc. Am. 127(4), 2358–2367 (2010)
12. Z Wang, S Zhou, GB Giannakis, CR Berger, J Huang, Frequency-domain oversampling for zero-padded OFDM in underwater acoustic communications. IEEE J. Ocean. Eng. 37(1), 14–24 (2012)
13. W Chen, J Huang, Z Wang, S Zhou, Blind channel shortening for zero-padded OFDM, in Proceedings of MTS/IEEE OCEANS Conference 2010 (Seattle, 2010), pp. 20–23
14. S Wu, Y Bar-Ness, A phase noise suppression algorithm for OFDM-based WLANs. IEEE Commun. Lett. 6(12), 535–537 (2002)
15. M Honig, U Madhow, S Verdu, Blind adaptive multiuser detection. IEEE Trans. Inform. Theory 41(7), 944–960 (1995)
16. H Xiaodong Wang, V Poor, Blind multiuser detection: a subspace approach. IEEE Trans. Inform. Theory 44(2), 677–690 (1998)
17. R Wang, H Li, T Li, Robust Multiuser Detection for Multicarrier CDMA Systems. Select Area Commun. 24(3), 673–683 (2006)
18. H Cheng, SC Chan, Blind linear MMSE receivers for MC-CDMA systems. IEEE Trans. Circuits Syst. 54(2), 367–376 (2007)
19. SA Vorobyov, Robust CDMA multiuser detectors: probability-constrained versus the worst-case-based design. IEEE Signal Process Lett. 15(7), 273–276 (2008)
20. L Hu, X Zhou, L Zhang, Blind multiuser detection based on Tikhonov regularization. IEEE Commun. Lett. 15(5), 482–483 (2011)
21. Z Xu, P Liu, X Wang, Blind multiuser detection: from MOE to subspace methods. IEEE Trans. Signal Process. 52(2), 510–524 (2004)
22. S-J Chern, P-S Chao, New Capon-like blind MIMO-CDMA receiver with LCCM-RLS algorithm for multipath time-varying channels, in Proc. of the 2008 IEEE International Symposium on Intelligent Signal Processing and Communications (Bangkok, 2008), pp. 102–105
23. P Xiao, R Liu, Multi-user detector for multi-carrier CDMA systems. Electronics Lett. 44(23), 1366–1368 (2008)
24. MC Dogan, JM Mendel, Applications of cumulants to array processing. I. Aperture extension and array calibration. IEEE Trans. Signal Process. 43(5), 1200–1216 (1995)
25. Y Ma, KH Li, AC Kot, G Ye, A blind code timing estimator and its implementation for DS-CDMA signals in unknown colored noise. IEEE Trans. Veh. Technol. 51(6), 1600–1607 (2002)
26. Z Tian, KL Bell, HL Van Trees, Robust constrained linear receivers for CDMA wireless systems. IEEE Trans. Signal Process. 49(7), 1510–1522 (2001) 27. A Elnashar, SM Elnoubi, HA El-Mikati, Further study on robust adaptive
beamforming with optimum diagonal loading. IEEE Trans. Antennas Propag. 54(12), 3647–3658 (2006)
28. R Wang, H Li, T Li, Robust multiuser detection for multicarrier CDMA systems. IEEE J. Select Area Commun. 24(3), 673–683 (2006) 29. J-F Cardoso, Blind signal separation: statistical principles. Proc. IEEE
86(10), 2009–2025 (1998)
30. A Hyvärinen, E Oja, Independent component analysis: algorithms and applications. Neural Netw. 13(4–5), 411–430 (2000)
31. Y Washizawa, Y Yamashita, T Tanaka, A Cichocki, Blind extraction of global signal from multi-channel noisy observations. Trans. Neural Netw. 21(9), 1472–1481 (2010)
32. F Nesta, P Svaizer, M Omologo, Convolutive BSS of short mixtures by ICA recursively regularized across frequencies. Trans. Audio Speech Lang. Process 19(3), 624–639 (2011)
33. A Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 10(3), 626–634 (1999) 34. MA Vicente, PO Hoyer, A Hyvärinen, Equivalence of some common linear
feature extraction techniques for appearance-based object recognition tasks. Trans. Pattern Anal Mach. Intell. 29(5), 896–900 (2007)
35. V Zarzoso, P Comon, R Phlypo, A contrast function for independent component analysis without permutation ambiguity. IEEE Trans. Neural Netw. 21(5), 863–868 (2010)
36. E Bingham, A Hyvärinen, A fast fixed-point algorithm for independent component analysis of complex-valued signals. Int. J. Neural Syst. 10(1), 1–8 (2000)
37. T Adah, H Li, M Novey, J-F Cardoso, Complex ICA using nonlinear functions. IEEE Trans. Signal Process. 56(9), 4536–4544 (2008)
38. M Novey, T Adalı, Complex ICA by negentropy maximization. IEEE Trans. Neural Netw. 19(4), 596–609 (2008)
39. T Ristaniemi, J Joutsensalo, Advanced ICA-based receivers for block fading DS-CDMA channels. Signal Process. 82(3), 417–431 (2002)
40. WY Leong, J Homer, Blind multiuser receiver for DS-CDMA wireless system. IEE Proc. Commun. 153(5), 733–739 (2006)
41. S-i Amari, Natural gradient works efficiently in learning. Neural Comput. 10, 251–276 (1997)
42. D Blanco, B Mulgrew, ICA in signals with multiplicative noise. IEEE Trans. Signal Process. 52(8), 2648–2657 (2005)
43. WY Leong, DP Mandic, Noisy component extraction. IEEE Trans. Circuit Syst. 57(3), 664–671 (2010)
44. J Igual, L Vergara, A Camacho, R Miralles, Independent component analysis with prior information about the mixing matrix. Neurocomputing 50, 419–438 (2003)
45. Z-L Sun, K-M Lam, Depth estimation of face images based on the constrained ICA model. Trans. Inform. Foren Sec. 6(2), 360–370 (2011) 46. Z-Q Luo, W Yu, An introduction to convex optimization for communications
and signal processing. IEEE J. Select Area Commun. 24(8), 1426–1438 (2006) 47. S Cui, M Kisialiou, Z-Q Luo, Z Ding, Robust blind multiuser detection against signature waveform mismatch based on second-order cone programming. IEEE Trans. Wireless Commun. 4(4), 1286–1291 (2005)
48. A Papoulis, Probability, Random Variables, and Stochastic Processes, 4th edn. (McGraw-Hill, New York, 1991)
49. PA Regalia, E Kofidis, Monotonic convergence of fixed-point algorithms for ICA. IEEE Trans. Neural Netw. 14(4), 943–949 (2003)
50. SA Vorobyov, AB Gershman, Z-Q Luo, Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem. IEEE Trans. Signal Process. 51(2), 313–323 (2003)
51. C-Y Yang, B-S Chen, Adaptive channel-tracking method and equalization for MC-CDMA systems over rapidly fading channel under colored noise. EURASIP J. Adv. Signal Process, (2010). Article ID 871650
52. AJ Rojas, R Middleton, JS Freudenberg, Infimal feedback capacity for a class of additive coloured Gaussian noise channels, in Proceedings of the 17th IFAC World Congress, 2008
53. S Piotr, Dynamic transitive closure via dynamic matrix inverse: extended abstract, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004, pp. 509–517
54. Y Kai-Bor, Recursive updating the eigenvalue decomposition of a covariance matrix. IEEE Trans. Signal Process. 39(5), 1136–1145 (1991)
doi:10.1186/1687-6180-2014-151
Cite this article as: Jen and Jou: Blind ICA detection based on second-order cone programming for MC-CDMA systems. EURASIP Journal on Advances in Signal Processing 2014 2014:151.
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