Abstract

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.

1. Introduction

Constant modulus algorithm (CMA) [1-6] is widely used for blind equalization [2,5,6,7-10] 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, [11-14] 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 [16-18]. 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 non-  linear 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 nonlinear blind equalizers on 16-QAM symbols are reported in Section 5, which are compared to their linear counterparts, followed by conclusion in Section 6.

Notation: represents expectation of a random variable. denotes the modulus of a complex number or the absolute value of a real number. The superscripts and denote the conjugate and derivative, respectively. Both and denote the real component while both and denote the imaginary component.

2. Effect of NTF

2.1. CMA

The cost function of CMA can be expressed as [1]

(1)

where denotes the time index; represents the equalized symbol; assuming that is the complex-valued transmitted symbol, the constant modulus, , is given by. The 3D performance function of CMA, , is shown versus in [1].

2.2. Performance Function without NTF

Considering the case without a NTF in CMA [1],

(2)

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, , (2) can be expressed as

(3)

where is a constant in this case. Using (3) together with (1), the 2D performance function, , is shown versus in Figure 1, similar as

versus in [1]. In Figure 2, two stable points, and, are given by and, respectively, denotes an arbitrary phase;; and the value at is given by.

2.3. Performance Function with NTF

Let us consider the blind equalizer with a NTF, i.e. NCMA. Define its corresponding cost function by, i.e.

Without loss of generality, a NTF with the expression of, where is a nonlinear coefficient and 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

(4)

Since the performance function is symmetrical around the, i.e. the cost function is independent with the phase of or, without loss of generality, let us consider the simplest case, i.e. is real. In this case, (4) can be approximated as

(5)

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) is determined by the item containing. Furthermore, we have

(6)

Based on (6) together with (1), the 2D performance function, , is shown versus in Figure 1, compared to that without NTF. One can see that using this NTF, its performance function becomes sharper than its previous linear counterpart. Once the convergency point is not exactly the minimum value but one of its surrounding points, namely, there is an estimation bias, i.e., its estimation error gets smaller after implanting this NTF. In other words, NCMA is more robust against the convergency error than CMA. It is noticed that, the NTF will provide a good equalization performance under the constraint that, i.e.. Particularly, when, the NTF exhibits the optimal equalization performance.

2.4. Discussion and Extension

As shown in Figure 1, where the cost function, , is plotted versus the adaptive weights, , looks similar with except that it has a sharper decision region than. The resulting sharper decision region will lead to a better equalization performance in the end. On the other hand, denote by and the cost functions for MCMA and NMCMA, respectively. Since MCMA is similar to CMA except that in NMCA, the real part and the imaginary part are separately considered, after adding a NTF into the linear equalizer, the relationship between and is the same with that between and. Therefore, one can know that also looks similar with except that it has a sharper decision region than.

3. Proposed NCMA

In Figure 2, , as mentioned, are the corrupted symbols at the receiver, , as mentioned, are the equalizer taps with the length of, and the variable is an intermediate variable for convenience. For a time index of, the input and its corresponding output can be formulated as

(7)

  and

(8)

where

(9)

  and

(10)

  Furthermore, the real component and the imaginary component of can be obtained as

(11)

  and

(12)

Based on (1), using the statistic gradient descent (SGD) in terms of, we have

(13)

where

(14)

  and

(15)

Assuming is the learning rate, the weights update from the step to the step can be expressed as

(16)

4. 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, , is given by

(17)

where

(18)

  and

(19)

  Similar to the derivation in NCMA, considering, we have

(20)

  Where

(21)

and

(22)

  Organizing (19)-(21), and assuming as the learning rate, the weights update from the step to the step can be expressed as

(23)

where

(24)

where

(25)

and

(26)

5. 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 [18], is used in this simulation, which is given by

(27)

We set signal-to-noise ratio (SNR) to 30dB for all cases. For 16-QAM symbols, the constant modulus, , is equal to 13.2. The real component, , and the imaginary component, , are both equal to 8.2. The length of NCMA, , is set as 15, the same as CMA. The length of NMCMA, , is set as 21, the same as MCMA. The adaptation step-size is set as 0.00001 for all. All initialized weights are set randomly with small values of approximately except for the center weights, which are set as 1.0 + j0. The NTF is used, where the nonlinear coefficient is set as 0.3 and the frequency of the sine function, , is chosen as for NCMA, and or for NMCMA.

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

MSE of approximate –13 dB, performs better than CMA, with the MSE of approximately –7 dB.

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.

5. 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.

6. 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.

7. References

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