An Implementation of Nonlinear Multiuser Detection in Rayleigh Fading Channel

A blind nonlinear interference cancellation receiver for code-division multiple-access-(CDMA-) based communication systems operating over Rayleigh ﬂat-fading channels is proposed. The receiver which assumes knowledge of the signature waveforms of all the users is implemented in an asynchronous CDMA environment. Unlike the conventional MMSE receiver, the proposed blind ICA multiuser detector is shown to be robust without training sequences and with only knowledge of the signature waveforms. It has achieved nearly the same performance of the conventional training-based MMSE receiver. Several comparisons and experiments are performed based on examining BER performance in AWGN and Rayleigh fading in order to verify the validity of the proposed blind ICA multiuser detector.


INTRODUCTION
In mobile communication systems, multiuser detection is also known under the names of cochannel interference suppression, multiuser demodulation, or interference cancellation, and requires rather complicated high-precision power control. The design of multiuser detectors has been motivated by the channel environment encountered in many CDMA applications, for channels with fading, multipath, or noncoherent modulation have been considered [1,2]. In particular, recent focus has been on the blind (or non-dataaided) multiuser detectors such as SIC [3], PIC [4], and DFD [5], these require no training data sequence, but only knowledge of the desired user signature sequence and its timing [6]. The main motivation of employing a blind multiuser detector in CDMA is to recover the original sequence from the received signal that is corrupted by noise and MAI, without the help of training sequences and a priori knowledge of the channel. Prior work by Ristaniemi and Joutsensalo [7] leads to the proposal of two types of receivers, RAKE-ICA and MMSE-ICA, in a Rayleigh fading channel using the modified FastICA algorithm [8]. However, the estimation of eigenvectors and eigenvalues becomes an additional burden for the proposed RAKE-ICA. Also, the MMSE-ICA which required training sequences causes an increase in computational load. An adaptive multiuser detector, which converges to the MMSE detector without requiring training sequences, is proposed in [1]. This detector is designed with incomplete knowledge of the received signature waveform of the desired user. In [9], a blind adaptive multiuser detector based on Kalman filtering for both stationary and slowly time-varying environments has been proposed, and it is shown that the steady-state excess output energy of the Kalman filtering algorithm is strictly zero for a statistically stationary environment. A overview of adaptive tentativedecision-based detectors is given by Verdu in [2]. It was mentioned that a linear MMSE detector has the features of a decorrelating detector, except that it requires knowledge of the received amplitudes. On the other hand, the tentativedecision-based multiuser detector is the simplest idea for successive cancellation, but its disadvantage is that it requires extremely accurate estimation of the received amplitudes [2,10]. The work of Verdu has also provided an exceptionally important reference and guide for the implementation of the subsequent work.
The objective of this paper is to introduce a blind multiuser detector that adaptively recovers signals from multiple users. The proposed blind multiuser detector is capable of replacing the conventional MMSE detector which requires training sequences and the original transmitted signals. This 2 EURASIP Journal on Wireless Communications and Networking is achieved based on independent component analysis (ICA) [11,12] used at the outputs of a bank of matched filters.
The rest of this paper is organized as follows. Section 2 describes the CDMA communication model and decision statistics. This is followed by an introduction of the proposed ICA algorithm in Section 3. Performance analysis and simulation results are presented in Section 4 to demonstrate the performance of the new proposed detector and make a comparison with the other conventional detectors. Finally, discussions and conclusions are given in Section 5 and Section 6.

Multiuser communication model
Let us first derive the decision statistic for the proposed blind nonlinear multiuser receiver structure. The main principle of the receiver is that for a given decoding order, the received signal is first passed through a correlator. The soft output of the correlator is then used to detect the first signal. We present expressions for the probability of the bit error for an AWGN channel with constant (although not necessarily equal) received user energies. These analytical results are then compared with simulation results.

System model
Consider a time-invariant flat-fading asynchronous uplink CDMA channel. The received baseband signal, r(t), in an antipodal K-user BPSK modulated system is given by (1) where 2J + 1 denotes the number of data symbols per user per frame.
(i) g k is the channel gain for the kth user. (ii) A k is the transmitted amplitude of the kth user.
The kth signature waveform s k is determined by the random pseudonoise (PN) spreading sequence c k and pulse-shape waveform p(t), given by where s k (t) is assumed to have unit energy over the symbol interval: T s = N PG T c symbol interval; T c chip interval; N PG processing gain. In this work, s k is generated by Gold code sequences. These signature sequences are independent of the data symbols and have a chip rate much higher than the symbol rate.
(v) The kth signature waveform s k is assumed to have unit energy ( T 0 |s k | 2 dt = 1). τ k ∈ [0, T s ) is the kth user's relative time offset, where T s is the symbol period. (vi) The additive white Gaussian noise n(t) has unit power spectral density. (vii) σ 2 is the variance of the additive noise.
We assume completely asynchronous transmission. In this context, when there are timing errors, each user's code experiences a random delay during the transmission and the received signal is no longer aligned with the locally generated codes [13]. We consider the received signal r(t) over only one symbol period that is asynchronous to the desired user.

Correlator and crosscorrelation matrix
An important multiuser detection issue [14] is the representation via sampling of r(t) as a vector in a continuous finitedimensional linear space: The representation of r(t) could be easier if no narrowband interference were present and the delays were known a priori. However, such an assumption is not realistic and would affect the implementation of adaptive receivers. Therefore, it is customary to apply the r(t) to chip-matched filtering. At the receiver, the matched-filter bank is designated at the first stage in the baseband signal detection. For each user, a correlation is performed between the received signal r(t) and the user spreading waveform. The sampled output of the matched filter for the ith bit of the kth user is where Δτ k denotes timing or synchronization error, which is minimized through the use of, for example, correlationbased time-delay estimation. The (k, j)th element of the K × K normalized signal crosscorrelation matrices β whose entries ρ = β k j are given by Since the modulating signals are zero outside [0, T s ], we define where the NK matrix is Wai Yie Leong et al. Sync 1 Generally, there are no cross-links among the filters. Each branch of the matched-filter bank consists of the correlation operation of the received signal with one particular user's signature sequence as illustrated in Figure 1.

Channel
In multiple-access channels, not only do the received amplitudes vary with time, but so too do the received signature waveforms, due to channel distortion. In this work, we consider Rayleigh distributed signal amplitudes. The Rayleigh distribution model is particularly suitable for non-line-ofsight (NLOS) communication links, to describe the statistical time-varying natures of the received envelope of a flat-fading signal. This arises when the process is zero mean, its phase is uniformly distributed on [0, 2π), and g k has its pdf in the form of where σ is the rms value of the received voltage signal before the envelope detection, σ 2 is the time-average power of the received signal before the envelope detection, and g k is the path amplitude. Referring to (1) and (3) (the first-order statistics of the received amplitude probability density function of |g k (i)|), it can be written as the product of a deterministic component and a random component which is Rayleigh distributed, g k : where R(i) is a stationary ergodic Rayleigh-distributed random process whose first-order probability density function is shown in (8).

Source independence
In the CDMA downlink receiver, both code timing and channel estimation are often prerequisites. Detection of the desired user's symbols in CDMA system is far more complicated than in the previous simpler TDMA and FDMA systems. Our main goal is therefore to estimate and recover the original transmitted symbols. Several techniques are currently available to estimate the desired user's symbols. In general, the matched filter (correlator) is the simplest estimator, but it performs well only if different users' chip sequences are orthogonal and the users' received signals have equal powers [15].
To that case, we propose to apply ICA to design a new blind receiver. The main reason for using ICA in the CDMA receiver is due to its resistance to strong interference [9] and because each user path and user transmitted symbol sequence are approximately independent of one another.

THE PROPOSED ICA ALGORITHM
The proposed ICA learning algorithm (see Figure 2) requires K matched filters to provide multiple mixtures of the K transmitted user signals. We assume the mixtures are linear, so that the relationship between the vector of K received signals X(i) = [x 1 (i), x 2 (i), . . . , x K (i)] T and the vector of K transmitted bit sequences can be expressed as where The proposed algorithm consists of three stages: (i) principle component analysis (PCA), (ii) independent component analysis (ICA), and (iii) denoising. The ICA learning algorithm is generalized from Amari's natural gradient algorithm [16] mainly in terms of applying cost functions to multivariate data. The minimization of the cost function is performed according to stochastic gradient descent and will be discussed later. Wavelet denoising [17] may also be employed to reduce the effects of the Gaussian noise.

Principle component analysis
PCA-based whitening and sphering (the mean becomes zero and the standard deviation one) of the received data X(i) is a common preprocessing technique in ICA. It is usually performed before the application of ICA as a means to reduce the effect of first-and second-order statistics, and to speed up the convergence process. It helps to reduce the number of unknowns in the mixing matrix, so that the remaining mixture can be modelled by a simpler orthogonal matrix [9]. This method has the additional advantage of decorrelating the sensor signals before separation. It makes the subsequent separation task easier, so that the separating matrix is constrained to be orthogonal. There is no explicit assumption on probability densities [18], as long as the first-and secondorder statistics are known or can be estimated from the mixture. The origin of PCA relies on the following problem. For the multidimensional vector X(i), find a linear transform F such that the obtained components are uncorrelated: That is, 4 EURASIP Journal on Wireless Communications and Networking . .

Denoising
Let Ψ x be the matrix formed from the normalized eigenvectors for the covariance matrix § x , then is the corresponding diagonal eigenvalue matrix. The PCA whitened signals are given by where D 1/2 x Ψ H x = F is the PCA whitening linear transform. In the following, we proceed to derive an algorithm for estimating the linear unmixing K × K matrix W such that the elements of the generated output vector y(i) = Wu(i), y(i) = [y 1 (i), y 2 (i), . . . , y K (i)] T , have minimum mutual information. The goal is to determine the gradient of the mutual information with respect to the elements of W. Essentially, W is an estimate of G −1 , where G is unknown mixing matrix. Once such a gradient is computed, update the elements of W in the gradient-based optimization algorithm [16]: where α denotes the learning rate.
In order to compute the gradient algorithm, we expand the mutual information between output signals as follows: When the mutual information (y) is equal to zero, the output variables are statistically independent. The gradient of (y 1 , . . . , y K ) with respect to W can be expressed as since the first term E{log p(u)} does not involve W. We will analyze the two remaining terms separately. For the first term, we have A family of nonlinear functions g i (·) is adapted in ICA, such that they approximate the probability density function of every y i . Accordingly, Wai Yie Leong et al.

5
where g(y) = [g 1 (y 1 ), . . . , g K (y K )]. Finally, we compute an approximation to the gradient of (y 1 , . . . , y K ) and the update step is multiplied with the W H W: where It can be proved that use of the natural gradient not only preserves the direction of the gradient but also speeds up the convergence process. The minimum mutual information algorithm for ICA will repeatedly perform an update of the matrix W: where α is a "small" update coefficient that controls the convergence speed. A robust formulation of (23) requires that each g i (y i ) is a nonlinear function of any symmetric density. These nonlinear functions are essential for the accuracy of the algorithm. Ideally the nonlinear function g i (y i ) approximates the probability density function of y i . It was suggested to model these functions by a weighted sum of parametric logistic functions [16,19]. Some experimental evidence [11] has indicated that even with a single fixed nonlinearity, the minimum mutual information algorithm converges very close to the optimal ICA solution. The success with using only one function is due to the corresponding independent sources being transforms of the initial independent sources. It is, however, important to indicate that "not all functions are equal." Approaches involving prediction of the nature of the sources along with a switch between sub-Gaussian and super-Gaussian functions have been proposed [3,5,10]. Alternatively, we now apply a nonlinear function g(y) as in [16]: After initializing the weight matrix W 0 as an identity matrix I, and choosing α to be a sufficiently small constant, such as α = 0.0001, the weights are iteratively updated according to the learning rule given by where p is the iteration index, and the estimated output y p (i) = W p u(i).
The outline of the proposed algorithm is given by the following.

EXPERIMENTS AND RESULTS
Experiments were presented to compare the performance of the proposed blind ICA multiuser detector with the decorrelating detector, matched-filter bank, SIC [3], PIC [4], and DFD [5]. Example 1. The environment considered was the uplink of a simplified CDMA system over a slow Rayleigh fading channel. The receiver output SNR is used as the evaluation index. Also, the input SNR is defined as SNR i = P/σ 2 , where we assume equal power MAI for convenience. The channel model adopted for simulations was an unknown Rayleigh flat-fading channel in which the fading gains were independent, identically distributed complex Gaussian random variables with zero mean and unit variance. The path delays τ k were assumed uniform over [0, 5T c ]. The chosen learning rate was α = 0.000001 and number of iterations, p = 200.
All CDMA signals were generated with BPSK data modulation and Gold codes of length N GC = 31 and N GC = 127 were used as the spreading codes. Unless otherwise stated, the following parameters were assumed: processing gain, N PG = 31 and N PG = 127, SNR i = 0 dB, doppler bandwidth, f D = 10 Hz, sampling frequency, f s = 2 kHz, and number of users, K = 10.
The bit error rate (BER) performance of the proposed blind ICA receiver in the slow Rayleigh fading channel is presented in Figures 3 and 4. The results show that when the processing gain N PG = 127, the standard matched filter receiver was significantly outperformed by each of the other training (linear MMSE) and/or blind multiuser detectors and that each of these alternative detectors gave nearly equivalent performance.
Greater performance discrimination between the detectors was observed when the processing gain N PG = 31. Again, each of the alternative detectors significantly outperformed the standard matched-filter receiver. However, in this case, at increasing SNR the proposed blind ICA receiver significantly outperformed the other blind multiuser detectors (successive interference cancellation detector (SIC), parallel interference     Hz. The proposed blind ICA detector exhibits nearly equivalent performance to that of the MMSE and decorrelating detectors and at high SNRs outperforms the alternative multiuser detectors PIC, DFD, and SIC. The PIC detector showed the best performance at high SNRs. This is because of the frequent use of signature waveforms to deal with changing levels of MAI. Again, the conventional matched-filter receiver showed relatively poor ability to deal with multiple-access interference.

DISCUSSION
A blind multiuser detector based on blind source separation has been proposed for CDMA systems over Rayleigh flatfading channels. This detector applies an iterative decisionaided procedure to reconstruct the unmixing matrix and the distorted signals from the input data. Simulation studies have shown that the proposed blind ICA multiuser receiver outperforms other more conventional blind multiuser detectors, and achieves nearly the same performance of the ideal training-based MMSE receivers under severe environmental conditions. The main advantage of the proposed receiver over the investigated alternatives is that it does not require the training sequence.    follows.
(i) The conventional CDMA detection and estimation methods do not exploit the powerful but realistic independence assumption [15] (ii) ICA offers an additional interference suppression capability, since the independence of the source signals is utilized [7]. (iii) ICA is worth considering as an additional element, attached to the existing matched filter-based CDMA receiver structure.

CONCLUSION
We have proposed and analyzed a blind multiuser detector based on ICA algorithm. The proposed blind multiuser detector has the potential to replace the conventional MMSE detector which requires training sequences and knowledge of the original transmitted signals. This way, the proposed blind ICA multiuser detector is suitable for the next-generation wireless CDMA communication system. Several simulation results show that the blind multiuser detector provides significant performance improvements over other multiuser detectors. Signal-to-noise ratio TDMA Time-division multiple access Danilo P. Mandic is a Reader in Signal Processing at Imperial College London, UK. He has been working in the area of nonlinear, blind, and adaptive signal processing and nonlinear dynamics. His publication record includes a research monograph on recurrent neural networks and more than 100 publications on signal and image processing. He has been a Member of the IEEE Technical Committee on Machine Learning for Signal Processing, and an Associate Editor for the IEEE Transactions on Circuits and Systems II, and International Journal of Mathematical Modelling and Algorithms. He has produced award winning papers and products resulting from his collaboration with Industry. He is a Senior Member of the IEEE and Member of the London Mathematical Society.