See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224399988 Legendre-FLANN-based nonlinear channel equalization in wireless communication system Conference Paper · November 2008 DOI: 10.1109/ICSMC.2008.4811554 · Source: IEEE Xplore CITATIONS READS 63 500 4 authors, including: P.K. Meher Goutam Chakraborty Sandhaan Labs Private Limited Iwate Prefectural University 270 PUBLICATIONS 4,856 CITATIONS 199 PUBLICATIONS 1,580 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Study of high performance pairing computation: Ate pairing implementation on FPGA View project Digital System Design View project All content following this page was uploaded by Goutam Chakraborty on 29 March 2018. The user has requested enhancement of the downloaded file. Legendre-FLANN-based Nonlinear Channel Equalization in Wireless Communication System J. C. Patra, W. C. Chin, P. K. Meher G. Chakraborty School of Computer Engineering Nanyang Technological University, Singapore Email: {aspatra, y050045 and aspkmeher}@ntu.edu.sg Faculty of Software and Information Science Iwate Prefectural University, Japan Email: goutam@soft.iwate-pu.ac.jp Abstract— In this paper, we present the result of our study on the application of artificial neural networks (ANNs) for adaptive channel equalization in a digital communication system using 4quadrature amplitude modulation (QAM) signal constellation. We propose a novel single-layer Legendre functional-link ANN (L-FLANN) by using Legendre polynomials to expand the input space into a higher dimension. A performance comparison was carried out with extensive computer simulations between different ANN-based equalizers, such as, radial basis function (RBF), Chebyshev neural network (ChNN) and the proposed LFLANN along with a linear least mean square (LMS) finite impulse response (FIR) adaptive filter-based equalizer. The performance indicators include the mean square error (MSE), bit error rate (BER), and computational complexities of the different architectures as well as the eye patterns of the various equalizers. It is shown that the L-FLANN exhibited excellent results in terms of the MSE, BER and the computational complexity of the networks. Keywords—Legendre functional link network, nonlinear channel equalization. artificial neural I. INTRODUCTION In wireless communication systems, transmission bandwidth is one of the most precious resources. To make efficient utilization this resource signals are usually transmitted through band-limited channels. Some of the inherent properties of the frequency-selective channels are that they are nonlinear and dispersive, and possess long delay spread. Due to these properties, the channel introduces inter-symbol interference (ISI) which reduces the data transmission rate. If the duration of the transmitted pulse is much smaller than the multipath delay spread, then each of the multipath components cannot be resolved in time at the receiver. Therefore, the currently transmitted pulse interferes with the previously and subsequently transmitted pulses, resulting in undesirable ISI and irreducible error floors at the receiver end of digital communication systems [1]. To mitigate the adverse effects of nonlinear channels, usually channel equalization is carried out in the digital system. Equalization refers to signal processing technique used at the front-end of the receiver to combat ISI in dispersive channels in the presence of additive noise. Traditionally, linear adaptive filters (AFs) are used to implement the equalizer. However, the performance of AF equalizers severely deteriorates when the 1-4244-2384-2/08/$20.00 ©2008 IEEE channel is nonlinear and highly dispersing [2]. Therefore, in order to improve the performance of equalizers in nonlinear channels, new equalizer structures are needed. Artificial neural networks (ANNs) can perform complex mapping between its input and output space and are capable of forming complex decision regions with nonlinear decision boundaries [3]. Further, because of nonlinear characteristics of the ANNs, these networks of different architectures have found successful application in channel equalization problem. Siu et al. [4] proposed a multilayer perceptron (MLP) structure for channel equalization with decision feedback and have shown that the performance of this network is superior to that of a linear equalizer trained with LMS algorithm. A radial basis function (RBF)-based equalizer structure with satisfactory performance has been reported [5]-[6]. The functional link-ANN (FLANN) is first introduced by Pao [7]. In FLANN, the original input pattern undergoes a pattern enhancement by using some nonlinear functions. Then the enhanced patterns are applied to a single-layer perceptron. Due to the absence of hidden layers, the computational complexity of FLANN is drastically reduced. In order to reduce the computational complexity, efficient functional-link ANN (FLANN)-based equalizer structures have been proposed [8][9]. The functional expansion in these networks was carried out using orthogonal trigonometric functions. Recently, a reduceddecision feedback FLANN channel equalizer is also proposed [10]. Another computational efficient network, i.e., Chebyshev neural network (ChNN) has been proposed for pattern classification [11], functional approximation [12], nonlinear dynamic system identification [13]-[14] and nonlinear channel equalization [15]. In these networks the expansion of input pattern is carried out using Chebyshev polynomials. ChNN provides similar, and in some cases, better performance than an MLP network but with much reduced computational load. Similar to ChNN, the Legendre function-based neural networks, i.e., Legendre functional-link ANN (L-FLANN), provides computational advantage while promising better performance. In this paper, we propose a novel L-FLANN based nonlinear channel equalization technique. By taking several channels and different nonlinearities, with extensive simulations we have shown the effectiveness of the L-FLANNbased equalizer. We have shown that the proposed equalizer performs much better than the RBF-based and a linear FIRbased equalizers. However, its performance is similar to that of SMC 2008 ChNN [15]. We have compared the performance of the four equalizers under different noisy nonlinear channels in terms of computational complexity, bit error rate (BER), mean square error (MSE) and eye patterns, that indicate true performance of an equalizer. II. DIGITAL COMMUNICATION CHANNEL EQUALIZATION Consider the system in Fig. 1 which depicts a commonly used wireless digital communication channel with an equalizer placed right at the front-end of the receiver. The transmitted symbols, represented by t (k ), where k denotes the discrete time index, traverses through the channel, which can be of linear or nonlinear in nature. a(k) t(k) Channel b(k) III. ANN STRUCTURES FOR CHANNEL EQUALIZATION In this section, we present the three different ANN structures, namely: L-FLANN, ChNN and RBF, used in this study. A brief illustration for each of the neural networks is given below. q(k) A. The Legendre-FLANN The L-FLANN structure is shown in Fig. 2. The input pattern is expanded into a nonlinear high dimensional space using Legendre polynomials and a single-layer perceptron network. The enhanced pattern is then used for the channel equalization process. The network is trained using the popular backpropagation (BP) algorithm [3]. The Legendre polynomials are denoted by Pn (x), where n = 0, 1, 2, ..., is the order of the polynomial and - 1 ≤ x ≤ 1 . These are a set of orthogonal polynomials defined as a solution to the differential equation: r(k) Nonlinearity Equalizer _ y(k) + d(k) e(k) Delay Figure 1. A typical wireless digital communication system with equalizer. The channel output at time instant k, which is a convolution between the transmitter filter, the transmission medium and the receiver filter, can be modeled mathematically with a FIR filter, is given by: a(k) = At the receiver, the received signal is first passed into the equalizer to reconstruct the transmitted symbols based on the noisy channel observations r (k ). The output of the adaptive equalizer, y (k ), is then compared with a delayed version of the desired signal d (k ) , to produce an error e(k ). This error is used to update the weights of the network according to some learning algorithm [3]. N f −1 ∑h(i) t(k − i) , (1) i=0 where h(i ) represents the channel tap values and N f is the length of the FIR channel. The “Nonlinearity” block represents the type of nonlinearity present in the channel that may cause distortion of the transmitted symbols and thus can be expressed as: b( k ) = Ψ{(t (k ), t ( k − 1), t (k − 2),..., t (k − N f + 1); h(0), h(1), h(2),..., h( N f − 1)}, (2) where Ψ (.) represents the nonlinearity function present in the existing channel block. Since practically, noise prevails in all channels, the channel output after the passing through the nonlinearity block, b(k ) , is be corrupted with noise. Assuming the noise to be additive white Gaussian noise (AWGN), represented by q (k ) , with variance σ 2 , the channel output with the noise is given by: r ( k ) = b( k ) + q( k ) , which is the corrupted signal received by the receiver. d d [(1 − x 2 ) Pn ( x)] + n(n + 1) Pn ( x) = 0 dx dx The first few Legendre polynomials are as follows: P0 ( x) = 1 P1 ( x) = x 1 (3 x 2 − 1) 2 1 P3 ( x) = (5 x 3 − 3 x) 2 1 P4 ( x) = (35 x 4 − 30 x 2 + 3). 8 P2 ( x) = (5) The higher order Legendre polynomials can be derived from the following recursion formula: Pn +1 ( x) = 1 [(2n + 1) xPn ( x) − nPn −1 ( x)] n +1 (6) Since the input pattern undergoes a nonlinear enhancement process by the use of the Legendre polynomials, there is no need of any hidden layer as needed in an MLP. Note that due to pattern enhancement, the dimension of original pattern is increased to a higher dimension. It is expected that the patterns that are nonlinearly separable in the original pattern space will be separable linearly in the high-dimension pattern space. Consider an input pattern given by: X = [ x1 , x 2 ] (3) (4) (7) Using the Legendre polynomials, this pattern can be enhanced as: SMC 2008 X ' = [ 1 P1 ( x1 ) P2 ( x1 ) ... P1 ( x2 ) P2 ( x2 ) ... ] (8) This enhanced pattern is applied to a single-layer perceptron as shown in Fig. 2. x x1 φ1(x) w1 x2 φ2(x) w2 x3 Legendre Expansion w1 w2 φ3(x) xM ∑ y(k) ρ(.) w3 wN φN(x) … x2 w0 … x1 φ1(x1) φ2(x1) φ3(x1) … x +1 output ∑ Figure 3. Structure of an RBF network. φ1(x2) φ2(x2) φ3(x2) - Adaptive Algorithm e(k) The basis function involves a Gaussian kernel that uses the Euclidean distance between its own reference, called the centre, and the network input, and is given by: ∑ … + d(k) φ ( x) = exp − (|| x − c ||) 2 / 2σ 2 Figure 2. Structure of an L-FLANN. B. Chebyshev Neural Network - ChNN The ChNN consists of Chebyshev polynomials which are a set of orthogonal polynomials defined on the interval (-1, 1) as the solution to the Chebyshev differential equations and is denoted by Tn (x) . The network structure is similar to that of the L-FLANN except that it uses Chebyshev polynomials in the functional expansion block. The Chebyshev polynomials for this defined range of values can be found with the recursive formula which is given by: where x is the input vector, c is the center of the node and σ is the spread parameter. Training of the RBF network involves the selection of suitable centers of the nodes and finding the weights of the output layer. Usually, the selection of the centers is carried out with the K-Means algorithm. Thereafter, the training patterns and their corresponding target patterns are applied to the network using some algorithm (e.g., least squares (LS) or orthogonal least squares (OLS)), the weights are determined. It is shown that the RBF-based equalizers are quite effective in channel equalization and provides near optimal solutions [5]-[6]. IV. Tn +1 ( x) = 2 xTn ( x) − Tn−1 ( x) (9) C. The Radial Basis Function - RBF Fig. 3 shows the structure of an RBF network. Let f(x) be the sum of the basis functions {φ(.)} for N neurons. Therefore, the equation at the output can be represented by: N y = ∑ wiφi (x) (10) i =1 where x is the input vector and wi are the weight values. The RBF networks have become increasingly popular due its simple structure and efficient learning. The RBF network is a universal approximator. The network structure consists of two layers, each performing a specific task. The input contains the source symbols. In the hidden layer, the input space is expanded into a high dimensional space with a set of basis functions by applying non-linear transformations. At the output layer, it is a linear combination of the output generated by the basis functions. (11) COMPUTATIONAL COMPLEXITY In this section, we compare the computational complexity among the different ANN structures of the RBF, ChNN and LFLANN. Let us consider an RBF structure with M-inputs and N-hidden units. As for the ChNN and L-FLANN networks, the number of nodes in the input and output layers are represented by D and K, respectively. The ChNN and L-FLANN architectures are trained using the BP algorithm. In the case of the RBF, it will require an additional division operation since the squared distance between the centre and the network input vector is divided by a width parameter [5], [15]. The respective computational complexities are shown in Table I. TABLE I CALCULATION OF COMPUTATIONAL COMPLEXITY Operation RBF ChNN L-FLANN Addition(+) 2MN+M+N+1 3DK+3K 3DK+3K Multi(*) NM+2N+M+1 6DK+6K 6DK+6K Division(/) M+N - - Exp(.) N - - tanh(.) - K K SMC 2008 Since both the ChNN and L-FLANN networks do not have any hidden layer as opposed to RBF, the computational complexity for the these networks are relatively lower compared to the RBF neural network. V. SIMULATION STUDIES In order to study the channel equalization problem as depicted in Fig. 1, in-depth investigations with extensive simulations were initiated using a linear FIR equalizer that is trained with the LMS algorithm as well as the three ANN architectures as discussed in the previous sections. For the purpose of our study, we have used the following channel impulse response [2] and is given by: ⎧1 ⎧ ⎡ 2π ⎤⎫ (i − 2)⎥ ⎬, i = 1, 2,3 ⎪ ⎨1 + cos ⎢ h(i ) = ⎨ 2 ⎩ Λ ⎣ ⎦⎭ ⎪ 0 , otherwise . ⎩ Together with a linear equalizer of order eight trained with the LMS algorithm, all the four network architectures were simulated. The learning rate and momentum rate for the linearbased equalizer were both set to 0.01, whereas for the ChNN and L-FLANN networks, the parameters were set to 0.7 and 0.5, respectively. In our study, we have used four different channels with the normalized impulse response as follows: CH = 1 : 0.209 + 0.995 z −1 + 0.209 z −2 CH = 2 : 0.260 + 0.930 z −1 + 0.260 z − 2 (13) CH = 3 : 0.304 + 0.903z −1 + 0.304 z − 2 CH = 4 : 0.341 + 0.876 z −1 + 0.341z − 2 (12) Since our input symbols are of the 4-QAM signal constellation, therefore the transmitted message will be in the form of {±1±j1}, consisting of both the real and imaginary components of the signal. In addition, AWGN is added to the channel output to simulate the actual conditions of the channel. The variable Λ was varied between 2.9 to 3.5 in increments of 0.2 in order to investigate the performance of each of the equalizers under different eigen value ratio (EVR) conditions of the channel. The variations of Λ produce EVR values of 6.08, 11.12, 21.71 and 46.82, respectively [2]. Extensive simulations were carried out with different test cases on the various parameters of the types of equalizers to be experimented. These include the spread of the RBF, learning rate, the momentum rate, number of expansion levels to be used in the functional block of the orthogonal networks etc. The architectures of the three networks are shown in Table II. These four channels will correspond to the parameter Λ of values of 2.9, 3.1, 3.3 and 3.5 for channels 1 to 4, respectively. To further analyze the effect of the degree of nonlinearity on the performance of the equalizers, three different nonlinear channel models with the following types of nonlinearities are chosen: NL = 0 : b(k ) = a(k ) NL = 1 : b(k ) = tanh(a (k )) NL = 2 : b(k ) = a(k ) + 0.2a 2 (k ) − 0.1a 3 (k ) (14) NL = 3 : b(k ) = a (k ) + 0.2a (k ) 2 − 0.1a 3 (k ) + 0.5 cos(πa (k )). Note that NL=0 represents a purely linear channel model, since the output after the onset of the nonlinearity is the same as its input (refer Fig. 1). NL=1 may occur to systems impaired by such distortion caused by the saturation of amplifiers in the transceivers. The other NL channel models are arbitrarily selected. TABLE II NEURAL NETWORK ARCHITECTURES ANN Structure No. of input dimensions No. of hidden nodes (N)/ nonlinear dimensions (D) No. of output nodes (K) RBF 8 10 (N) 2 ChNN 8 18 (D) 2 L-FLANN 8 18 (D) 2 For the RBF equalizer, we have utilized the newrb function available in MATLAB for the simulation. The spread parameter was set at 50. As for the ChNN and L-FLANN networks, their input vectors were expanded into an 18dimensional space network with the Chebyshev and Legendre polynomials, respectively, and BP algorithm was used for updating of the weights. VI. PERFORMANCE EVALUATION Performance evaluation of the three ANN structures along with a linear FIR LMS-based equalizers using the four channel models with four NL models. A. The Computational Complexity The training times were tabulated for one iteration of the network on an Athlon XP system, 1.86 GHz computer. Hence, the training times, in seconds, recorded were 8.0, 6.5 and 6.4 for the RBF, ChNN and L-FLANN, respectively. B. The Convergence Characteristics Fig. 4 shows the convergence characteristics of the various equalizers defined for Channel 3 at SNR=15dB. Clearly, it can be seen that all the three neural network architectures outperformed the linear-based equalizer. Also, the ChNN and L-FLANN achieved a much faster convergence as compared to the RBF network. SMC 2008 Next, a closer comparison is made between the ChNN and the L-FLANN. At NL=0, the ChNN performed slightly better than the L-FLANN. But as the channel complexity increases, the L-FLANN performance is comparable to that of the ChNN. This shows that L-FLANN is also able to work well under harsh conditions of the channel. pattern for CH=1 at NL=1 with SNR=15dB. It can be observed that the three neural networks showed similar classification of the output. This further justifies the use of the L-FLANN architecture for channel equalization where high precision in classification is needed. Similar observations were also made for the different channels under different nonlinearity settings. Figure 4. Convergence characteristics of the four architectures for CH=3 using different nonlinear model settings at SNR=15dB. (a) NL=0. (b) NL=1. (c) NL=2. (d) NL=3. Figure 5. BER performance of the four architectures for CH=2 with varying SNR values from 10-18. (a) NL=0. (b) NL=1. (c) NL=2. (d) NL=3. C. BER Performance The BER represents the percentage of error bits relative to the total number of bits sent in a given transmission. It serves as an indication of how often the data has to be retransmitted due to an error; hence it determines the true performance of the various equalizers. Computation and comparison of the BER was carried out with the 4-QAM signal constellation of the linear-based equalizer and the three neural network architectures. In order to observe the effect of the different variations of the SNR has on the BER, graphs were plotted with MATLAB for CH=2 across all the linear and nonlinear channel models, as shown in Fig. 5. From the plots, we can see that the neural networks performed far more efficiently than the LMS equalizer with increased SNR. On a closer observation, we also note that the L-FLANN performed slightly better than the other two artificial neural networks. This again, reaffirmed the efficiency of L-FLANN for use in the channel equalization problem. D. The Eye Patterns The eye patterns, or the equalizer output values, determine how well the equalizers can perform the equalization process. In obtaining the eye pattern, each equalizer undergoes 5000 iterations during the training phase and tested with 1000 data samples where the output is captured. Fig. 6 shows the eye Figure 6. Eye pattern of the four architectures for CH=1 at NL=1 with SNR=15dB with 1000 data symbols. (a) LMS equalizer. (b) RBF equalizer. (c) ChNN equalizer. (d) L-FLANN equalizer. VII. CONCLUSIONS Three different artificial neural networks are proposed to solve the adaptive channel equalization problem with 4-QAM SMC 2008 signal constellation. Especially in the case of L-FLANN, it employs a novel functional expansion model to cast the input vectors into a higher dimensional space with a single-layer perceptron network. 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