Blind Equalization of QAM Signals Based On Dual-Mode Multi-modulus Algorithms

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Procedia Engineering

Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 2434 – 2438 www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

Blind Equalization of QAM Signals Based On Dual-Mode Multi-modulus Algorithms Wen SiYuana∗ a

ShanDong Economic University, No.7366, Erhuan East Road, Lixia District, Jinan, Shandong,250014,China

Abstract For blind equalization of high-order quadrature amplitude modulation (QAM) signals, A dual-mode multi-modulus algorithm (DM- MMA) and a stop-and-go dual-mode multi-modulus algorithm (SAG-DMMMA) are proposed. Through Simulation, compared with the recently introduced multi-modulus algorithm, verified the proposed blind equalization algorithms have faster convergence speed and smaller steady-state mean square error.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: multi-modulus algorithm, dual-mode, blind equalization, stop-and-go

1. Introduction The increasing demand for digital communications needs high-speed data transmission over bandlimited channels. Hence, the channels are subject to intersymbol interference (ISI). Channel equalization is one of the techniques to mitigate the effects of ISI. Blind equalization which does not require any known training sequence has been an active area for several decades. The constant modulus algorithm (CMA)[1]has become a favorite of practitioners due to its LMS-like complexity and desirable robustness properties. But the CMA converges independently of carrier recovery, and the output constellation after convergence has a phase rotation. Thus a rotator has to be added at the output of the equalizer, which increases the complexity of implementation of the receiver in steady-state operation. In order to improve the performance of the CMA, a multi-modulus algorithm (MMA) has been proposed [2,3,4]. The MMA provides reliable initial convergence without the need of a rotator in steady-state. The MMA cost function is (1)

(

J MMA = E ⎧⎨ y R2 ( n ) − γ R ⎩

) + (y 2

)⎭

2⎫ 2 ⎬ I (n) − γ I

(1) where E[·] indicates statistical expectation. yR(n) and yI(n) are the real and imaginary parts of the equalizer output y(n), respectively. γR and γI are computed as: ∗

Corresponding author. Tel.:+8613263106676

E-mail address: [email protected]

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.457

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Wen SiYuan SiYuan， / Procedia Engineering 15 (2011) 2434 –000–000 2438 Wen et al/ Procedia Engineering 00 (2011)

2

γR =

( E (a

), )

E a R4 ( n )

γI =

2 R( n )

( (

) )

E a I4 ( n )

E a I2 ( n )

where a(n)=aR(n) + j aI(n) is the transmitted QAM data symbol. The corresponding MMA tap updating algorithm is: W (n + 1) = W (n) + μ ⋅ e(n) ⋅ X ∗ (n)

(2)

(3) Where μ is the step-size parameter and the asterisk denotes complex conjugation. The equalizer complex tap weight-vector and input-vector are respectively defined as W(n)=[w0(n), w1(n),…, wL-1(n)]T and X(n)=[x(n), x(n-1),…, x(n-L+1)]T , where L is the length of the equalizer tap weights and the notation superscript T stands for transpose of vector. The error function e(n) can be expressed as e( n ) = y R ( n) γ R − y R2 ( n ) + j ⋅ y I ( n) γ I − y I2 ( n ) (4)

(

)

(

)

A particular problem of the MMA, however, is that it still suffers from a low convergence when applies to the higher order QAM. The dual-mode blind equalization algorithms are therefore designed to speed up the convergence rate [5,6,7]. These algorithms use the CMA at the first mode and then switch to the second mode, so they suffer from the same pitfall as the CMA, i.e. they can’t correct the phase rotation at the output of the equalizer. In this paper, we propose a new dual-mode MMA to effectively improve the equalizer’s convergence performance and recover the phase rotation simultaneously. Although the dual-mode algorithm switches in two modes, it never stops adjusting the equalizer tap weights even when the adjustment is in the wrong direction. If we can tell whether a particular adjustment is correct or not, we may improve the convergence behavior by making only the right adjustment but bypassing those wrong ones. Such a concept has been applied to blind equalization and is termed “stopand-go” [8,9,10]. In this paper, we develop a stop-and-go dual-mode MMA for blind equalizers. The paper is organized as follows. In Section 2 we derive and analyze the proposed algorithms. Computer simulations are presented in Section 3, and the concluding remarks are contained in Section 4. 2. The Stop-And-Go Dual-Mode Algorithm 2.1. The dual-mode multi-modulus algorithm Suppose ∪Dk (k=1,2,…) denotes the union of the square regions Dk enclosing data points of the QAM data constellation. In these regions, we define a new cost function:

(

~ 1 J = E ⎧⎨ y R ( n ) − a R , p ( n ) 2 ⎩

) + (y ( n ) − a 2

I

Where p = arg min y R ( n) − a R ,i ( n) ,

I ,q ( n )

(i = 1,2,/ )

i

) ⎫⎬⎭ 2

(5)

q = arg min y I ( n ) − a I ,k ( n ) , k

(k = 1,2,/ )

By computing the gradient of (5) with respect to wR(n) and wI(n) separately, and replacing the value of statistical expectation by the instantaneous value, we obtain: ~ ∂J = y R ( n ) − a R , p ( n ) ⋅ sgn ( y R ( n )) ⋅ X R ( n ) + ∂W R

(

)

(y ( n ) − a I

I ,q ( n )

)⋅ sgn(y ( n ))⋅ X I

I(n)

~ ∂J = y R ( n ) − a R , p ( n ) ⋅ sgn ( y R ( n )) ⋅ X I ( n ) − ∂W I

(

)

(y ( n ) − a I

I ,q ( n )

)⋅ sgn(y ( n ))⋅ X I

R( n )

Then the corresponding tap updating algorithm is: W (n + 1) = W(n) + μ ⋅ ~ e ( n) ⋅ X ∗ ( n) The error signal in (8) is expressed as:

(6)

(7) (8)

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Wen SiYuanProcedia / Procedia Engineering (2011) 2434 – 2438 Wen SiYuan/ Engineering 00 15 (2011) 000–000

e~ (n) = e~R (n) + j ⋅ e~I (n),

y ( n) ∈ D k

(9)

Where ~ eR ( n ) = y R ( n ) − a R , p ( n ) ⋅ sgn ( y R ( n ))

( ~ e ( n ) = (y ( n ) − a I

I

)

I ,q ( n )

3

(10)

)⋅ sgn(y ( n )) I

(11) We refer to the above algorithm as the constellation region based MMA (CR-MMA). From (3), (4), (8), (9) and the preceding discussion, we may express the dual-mode MMA (DM-MMA) as: W (n + 1) = W (n) + μ ⋅ e(n) ⋅ X ∗ (n), y ( n) ∉ ∪ D k (12) ∗ ~ W (n + 1) = W (n) + μ ⋅ e (n) ⋅ X (n), y ( n) ∈ D k (13) The DM-MMA operates as follows. Because of the distortion introduced by the channel during the transient stages, the output of the equalizer will be scattered in a very large area around the transmitted data point. Therefore, most of the equalizer outputs will not be in ∪Dk and the MMA (i.e. (12)) will be used to adjust the tap weights most of the time. This provides a MMA-like initial convergence behavior for the DM-MMA. On the other hand, in the steady state, since the output of the equalizer will be very close to the transmitted data point, the CR-MMA (i.e. (13)) will be used to adjust the tap weights most of the time. When yR(n)≈aR(n) and yI(n)≈aI(n), the error signal in (9) will have a very small value, which provides a good steady state behavior for the DM-MMA. 2.2. The stop-and-go dual-mode multi-modulus algorithm Although the DM-MMA correctly decide whether the equalizer is in a transient state or a steady state, it does not tell whether a particular adjustment is correct or not. The “stop-and-go” method can solve this problem by using a simple flag. The flag suggests “go” if the error signal is sufficiently reliable for adaptation, and suggests “stop” otherwise [8, 9]. In the following, we present a stop-and-go DM-MMA (SAG-DMMMA) for blind equalization. The SAG-DMMMA can be described by the following equations: W (n + 1) = W (n) + μ ⋅ [ f R (n)e R (n) + j ⋅ f I (n)e I (n)]⋅ X ∗ (n), y (n) ∉ 8D k , k = 1,2, / W (n + 1) = W (n) + μ ⋅ [ f R (n)e~R (n) + j ⋅ f I (n)e~I (n)]⋅ X ∗ (n), y (n) ∈ D k , k = 1,2, /

where

⎧⎪1, if sgn(e R ( n )) = sgn(e~R ( n )) f R ( n) = ⎨ ⎪⎩0, if sgn(e R ( n )) ≠ sgn(e~R ( n ))

⎧⎪1, f I ( n) = ⎨ ⎪⎩0,

(14) (15)

if sgn(e I ( n)) = sgn(e~I ( n)) if sgn(e I ( n)) ≠ sgn(e~I ( n))

3. Simulations We have demonstrated the performance of the DM-MMA and SAG-DMMMA using computer simulations. The data sequences for simulations were generated according to the model of Fig. 1. The channel used in simulations was taken from [8]. A decision-feedback equalizer with a 5-tap T/2spaced feedforward filter and a 3-tap feedback filter was used. The equalizer was initialized so that the center taps were set to one and the other taps were set to zero. The signal to noise ratio (SNR) was fixed at 20 dB in all simulations. From figure 2, it can be seen that the performance of the algorithm deteriorates when d is increased beyond 0.85, and for d=1 the algorithm fails to converge. Note that when d=1, the square regions touch each other, and this case is very similar to the CR-MMA equalization scheme.

WenWen SiYuan / Procedia 15 (2011) 00 2434 – 2438 SiYuan ，et al/ Engineering Procedia Engineering (2011) 000–000

4

Next, to examine the performance of the proposed algorithm, we compared the DM-MMA and SAGDMMMA with the CMA and the MMA for the 32-QAM, 128-QAM and 256-QAM data constellations. The step-size μ is set to 0.000012 and 0.00004 for the CMA and the MMA with the 32-QAM data set, respectively. For the DM-MMA and SAG-DMMMA, d=0.7 and μ=0.002, 0.0001 and 0.00002 for the 32QAM, 128-QAM and 256-QAM constellations, respectively. Fig. 3 shows the constellations after convergence. The phase rotation has been recovered for these algorithms and the DM-MMA and SAGDMMMA can achieve better performance compared with the MMA. In addition, we can see from Fig. 4 that the symbols are more clearly distinguishable at the output of the SAG-DMMMA equalizers than those of the DM-MMA. Note that the CMA and the MMA did not succeed in opening the eye for the 128QAM and 256-QAM signals, so we have not presented their data constellations. Finally, we consider the convergence behavior and steady state MSE of these algorithms. Fig. 4 shows the ensemble-averaged MSE, obtained from 100 Monte Carlo runs. From the results, we can see that the SAG-DMMMA has the fastest convergence rate and the smallest steady state MSE among all the algorithms.

Fig. 1 Baseband communication system model with T/2- spaced receiver MSE of DM-MMA for different d values

0

-5

d=1

d=0

M S E(dB)

-10

d=0.1 -15 d=0.5 -20

d=0.3

d=0.7 d=0.85

-25

0

0.2

0.4

0.6

0.8

1 1.2 Iterations

1.4

1.6

1.8

2 4

x 10

Fig. 2 Learning curves for DM-MMA for a 16-QAM system. For d=0, the corresponding step-size are µ=0.00002; for d=1, µ=0.001, and others µ=0.002

(a)CMA for 32QAM

(b) MMA for 32QAM

(c) DM-MMA for 32QAM

(d) SAG-DMMMA for 32QAM

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Wen / Procedia – 2438 WenSiYuan SiYuan/ ProcediaEngineering Engineering15 00(2011) (2011)2434 000–000

(e) DM-MMA for 128QAM (f) SAG-DMMMA for 128QAM Fig. 3 Output of the equalizers after 4000 iterations.

(g) DM-MMA for 256QAM

(h) SAG-DMMMA for 256QAM

-2 CMA for 32QAM

-4

MMA for 32QAM

-6 -8 MSE(dB)

DM-MMA for 128QAM -10 DM-MMA for 32QAM

-12

DM-MMA for 256QAM -14 SAG-DMMMA for 32QAM

-16

SAG-DMMMA for 128QAM SAG-DMMMA for 256QAM

-18 -20

0

0.2

0.4

0.6

0.8

1 1.2 Iterations

1.4

1.6

1.8

2 4

x 10

Fig.4 Comparison curves of ensemble averaged MSE.

4. CONCLUSIONS In this paper we have introduced two new multi-modulus algorithms for blind equalization and have analyzed their performance. The DM-MMA and the SAG-DMMMA outperform the MMA and offer practical alternatives to blind equalization of higher-order QAM signals. References [1] Godard DN. Self-recovering equalization and carrier tracking in two- dimensional data communication systems. IEEE Transactions on Communications 1980; 28(11):1867-1875. [2] Yang J, Werner JJ, Dumont GA. The multimodulus blind equalization and its generalized algorithms. IEEE Journal on Selected Areas in Communications (ISACEM) 2002; 20(5): 997-1015. [3] Yuan JT, Tsai KD. Analysis of the Multimodulus Blind equalization Algorithm in QAM Communication Systems. IEEE Transactions on Communications 2005; 53(9): 1427-1431. [4] Yuan JT, Chang LW. Carrier Phase Tracking of Multimodulus Blind Equalization Algorithm Using QAM Oblong Constellations. IEEE International Conference on Communications, 2007(ICC 2007), Glasgow, Scotland, 24-28 June 2007;29912998. [5] Weerackody V, Kassam SA. Dual-mode type algorithms for blind equalization. IEEE Transactions on Communications 1994; 42(1): 22- 28. [6] Shahmohammadhi M, Kahaei MH. A new dual-mode approach to blind equalization of QAM signals. Proceedings ISCC, Antalya,Turkey, June 2003;277-281 [7] Banovic K, Abdel-Raheem E, Khalid MAS. Hybrid methods for blind adaptive equalization: new results and comparisons. Proceedings ISCC, June 2005;1341-1346. [8] Picchi G, Prati G. Blind equalization and carrier recovery using a stop-and-go decision-directed algorithm. IEEE Transactions on Communications 1987; 35(9): 877-887. [9] Tseng CH, Lin CB. A stop-and-go dual-mode algorithm for blind equalization. Proceedings of Globecom , 1996;1427- 1431. [10] Abrar S, Amin A, Siddiq F. Stop-and-go square-contour blind equalization algorithms: design and implementation. IEEE International Conference on Emerging Technologies, Islamabad, 2005; 157-162.

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