Nonlinear Blind Equalizers: Ncma and Nmcma

This paper proposes two nonlinear blind equalizers: the nonlinear constant modulus algorithm (NCMA) and the nonlinear modified constant modulus algorithm (NCMA) by applying a nonlinear transfer function (NTF) into constant modulus algorithm (CMA) and modified constant modulus algorithm (MCMA), respectively. The effect of the NTF on CMA and MCMA is theoretically analyzed, which implies that the NTF can make their decision regions much sharper so that the proposed two nonlinear blind equalizers are more robust against the convergency error compared to their linear counterparts. The embedded single layer in NCMA and NMCMA simultaneously guarantees a comparably speedy convergency. On 16-quadrature amplitude modulation (QAM) symbols, computer simulations show that NCMA achieves an 8dB lower convergency mean square error (MSE) than CMA, and NMCMA achieves a 15dB lower convergency MSE than MCMA.


Introduction
Constant modulus algorithm (CMA) [1][2][3][4][5][6] is widely used for blind equalization [2,5,6,[7][8][9][10] for constant modulus transmissions in communication systems in order to overcome the propagation channel corruption, mitigate the inter-symbol interference (ISI) and recover the transmitted symbols, which usually has a satisfied performance in common situations. However, under a complicated multipath channel, the transmitted symbols suffer from severe distortion and CMA will perform poor for multi-modulus symbols, i.e. for high-order quadrature amplitude modulation (QAM) symbols, mainly due to the inability of CMA on phase error correction [11].
To suppress the convergence error and improve the equalization performance for multi-modulus symbols, [11][12][13][14] proposed a classical modified constant modulus algorithm (MCMA), in which the real component and the imaginary component of the equalizer output are respectively considered to compress the phase error, leading to a better performance. However, its performance is not good enough in some severe cases, since its decision region is comparably smooth, which does not tolerate the convergency error very much.
The method for further improvement is to bring in nonlinearity instead of linearity, which can be realized by utilizing multilayer architecture, nonlinear transfer function (NTF) [15] or neural network [16][17][18]. However, as a tradeoff, the complicated multiple architecture results in a slower convergency. As we all know, the speedy convergency is significant for adaptive blind equalization. Consequently, in this paper, we preferentially consider introducing a NTF into blind equalization to improve the performance. A NTF, ( ) , is proposed in [18] for blind equalizer according to its provided properties. However, there is a remaining unsolved question: in essence, why can this NTF be helpful for equalization performance? Or equivalently, what is the theoretical effect of NTF on equalization performance? This paper will answer this question via theoretical analysis. The following theoretical derivation provides that the NTF can make the decision region much sharper so that the proposed nonlinear blind equalizers are more robust against the convergency error. Based on this discovery, by applying the nonlinear transfer function (NTF) to CMA and MCMA, the nonlinear CMA (NCMA) and nonlinear modified CMA (NMCMA) are thus proposed, and their adaptive learning rules are also theoretically derived in this paper.
The remainder of this paper is organized as follows. Section II theoretically analyzes the effect of the NTF on blind equalizers. Based on the analysis given in Section 2, the nonlinear blind equalizer, NCMA, is proposed in Section III. Moreover, another blind equalizer, NMCMA, is proposed in Section 4. Simulation results of the proposed

Performance Function without NTF
Considering the case without a NTF in CMA [1], where, is the corrupted signal at the receiver, which is also the input signal of the equalizer, and is the adaptive weight vector. Assuming keeping all weights unchanged except where is a constant in this case.
Using ( [1]. In Figure 2, two stable points, and , are given by spectively,  denotes an arbitrary phase; ; and the value at

Performance Function with NTF
Let us consider the blind equalizer with a NTF, i.e. NCMA. Define its corresponding cost function by Without loss of generality, a NTF with the expression of ( ) , where  is a nonlinear coefficient and f denotes the frequency of the sine function, is considered in this paper for performance analysis. With this NTF, based on the proposed architecture as shown in Figure 2, we have Since the performance function is symmetrical around the J axis  , i.e. the cost function is independent with the phase of   y n or p w , without loss of generality, let us consider the simplest case, i.e. p w is real. In this case, (4) can be approximated as where, the approximation is based on the fact that the closer the directions of two vectors with fixed modules, the bigger their summation, and the approximation is true because in (5) is a constant, same as that in (3) and the value of the sine function in (5)  , the NTF exhibits the optimal equalization performance.

Discussion and Extension
As shown in Figure 1, where the cost function, J , is plotted versus the adaptive weights,

Proposed NCMA
In Figure 2

Proposed NMCMA
The cost function of NMCMA is the same as MCMA shown in [11] and [12]. In order to derive the NMCMA using SGD, the expectation operation is removed and the resulting cost function, NMCMA J , is given by

Computer Simulations
The proposed NCMA and NMCMA are demonstrated by using the 16-QAM symbols through a multipath channel. Their performances are compared to those of the preexisting CMA and MCMA. A typical complex-valued 10-path communication propagation channel, labeled ( ) H z [18], is used in this simulation, which is given by   After 12000 training symbols, the following 2000 received symbols are tested for evaluating the equalization performance. The symbols' constellation after CMA equalizer is illustrated in Figure 3, whose estimation error is comparatively large. For comparison, the symbol's constellation after NCMA equalizer is shown in Figure 4, where the equalized symbols more concentrate on their supposed position and their bias are much smaller. To be clear, the MSEs of CMA and MCMA are plotted in Figure 5. One can see that, NCMA, with the   Similarly, the symbols' constellation after MCMA equalizer is illustrated in Figure 6, whose estimation error is comparatively large. For comparison, the symbol's constellation after NMCMA equalizer is shown in Figure 7, where the equalized symbols much more concentrate on their supposed position and their bias are much smaller. To be clear, the MSEs of MCMA and NMCMA are plotted in Figure 8. One can see that, NMCMA, with the MSE of approximately -30 dB, performs better than MCMA, with the MSE of approximately -15 dB.

Conclusions
Two Nonlinear blind equalizers: NCMA and NMCMA, were proposed in this paper by applying the NTF into the existing CMA and MCMA, respectively. The NTF effect   on linear blind equalizers was theoretically analyzed. It is shown that the NTF can make their decision regions sharper so that the proposed NCMA and NMCMA are more robust against the convergency error than CMA and MCMA, respectively. Computer simulations demonstrate that, for 16-QAM symbols, NCMA can reach up to approximately -13 dB MSE compared with -7 dB by CMA, and NMCMA can reach up to approximately -30 dB MSE compared with -15 dB by MCMA.

Acknowledgements
The author would like to thank all the anonymous reviewers of the paper. The critical comments by all the reviewers have helped us to improve the quality of our paper.