A new hybrid blind equalization algorithm with steady-state performance analysis

Digital Signal Processing 22 (2012) 233–237

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Digital Signal Processing www.elsevier.com/locate/dsp

A new hybrid blind equalization algorithm with steady-state performance analysis ✩ Ning Xie a,b , Hengyun Hu a , Hui Wang a,∗ a b

College of Information Engineering, Shenzhen University, Shenzhen, Guangdong 518060, China State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Available online 13 December 2011

The dithered signed-error constant modulus algorithm (DSE-CMA) is an approach less computationally than CMA but exhibits ill-convergence. On the contrary, the concurrent CMA with soft decisiondirected scheme (CMA + SDD) improves the convergence rate of CMA by adding SDD but brings lots of computational complexity. As a tradeoff, in this paper we combine the virtues of DSE-CMA and SDD, then propose a new algorithm, which compensates the phase shift and provides convergence faster than DSE-CMA with lower complexity than CMA + SDD. More importantly, an approximation of its steadystate mean square error performance is derived. Simulation results show the superiority of the proposed algorithm. Crown Copyright © 2011 Published by Elsevier Inc. All rights reserved.

Keywords: Constant modulus algorithm (CMA) Soft decision-directed (SDD) Mean square error (MSE) Phase shift

1. Introduction As an important research subject of adaptive signal processing, the blind equalization is widely used in the wireless communication [1–5]. Recently, the constant modulus algorithm (CMA) becomes one of the most popular algorithms used in blind channel equalization. However, its computational complexity, convergence rate and steady-state mean square error (MSE) are not desirable enough to obtain adequate performance. Moreover, it suffers from the problem of phase shift [6,7]. Many schemes were proposed to improve the CMA. For example, the dithered signed-error constant modulus algorithm (DSE-CMA) [8–10], which is based on the judicious use of dither, aimed to reduce the complexity of CMA by transforming the bulk of its update multiplications into sign operations but failed to provide good convergence; the concurrent CMA with soft decision-directed scheme (CMA + SDD) [7], compared to the CMA, provided better convergence and steady-state mean square error (MSE) behavior, but with higher complexity; the fuzzy-logic tuned constant modulus algorithm and soft decisiondirected scheme (FL-CMA + SDD) [11], achieved significantly faster convergence with the same excellent steady-state performance, in

comparison with the CMA + SDD that employs a constant step-size for the CMA, but imposed extra computational complexity. As a tradeoff, we propose a new algorithm by combining the virtues of the DSE-CMA and SDD. The proposed algorithm is considered to compensate the phase shift, just like the conventional CMA + SDD, and to provide better convergence and steady-state behavior than the conventional DSE-CMA meanwhile keep a low complexity compared to the conventional CMA + SDD. Using the energy preservation approach [12], we derive the first-order excess MSE (EMSE) approximation closed-form solutions of the proposed algorithm for complex-valued cases, which is greater than the CMA + SDD but less than both the CMA and DSE-CMA. Our theoretical analysis is confirmed by simulations. 2. Signal model Considering a baseband model of a digital communication channel in which the received signal at sample n is given by

r (n) = ✩

This work was supported by three National Natural Science Foundations of China (Nos. 61001182 and 60773203), the Natural Science Foundation of Guangdong, China (No. 10451806001004788), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM09122), Fundamental Research Program of Shenzhen City (No. JC201005280556A) and the open research fund of State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications (No. SKLNST-2009-1-06). Corresponding author. E-mail address: [email protected] (H. Wang).

*

1051-2004/$ – see front matter Crown Copyright doi:10.1016/j.dsp.2011.11.007

©

N −1 

h i s(n − i ) + v (n)

(1)

i =0

where N is the length of the channel impulse response (CIR) h and v (n) = v R (n) + jv I (n) is a complex-valued Gaussian white noise. The complex-valued transmitting symbol sequence s(n) = s R (n) + js I (n) is assumed to be independently identically distributed (i.i.d.) and takes the value from the M-QAM symbol set defined by√S = {sil = (2i − Q − 1) + j (2l − Q − 1), 1  i, l  Q } where Q = M.

2011 Published by Elsevier Inc. All rights reserved.

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N. Xie et al. / Digital Signal Processing 22 (2012) 233–237

Fig. 1. System model of the proposed algorithm.

3. The proposed algorithm In order to improve the convergence rate of DSE-CMA, we add the SDD part which operates concurrently with DSE-CMA, just shown in Fig. 1. Note that w = wc + wd , where wc is the weight vector of the DSE-CMA part and wd is the weight vector of the SDD part. The DSE-CMA is a computationally efficient algorithm which is based on the judicious incorporation of controlled noise (sometimes referred to as “dither”) [8]. The update weight vector of the complex-valued DSE-CMA [9] is defined as

wc (n + 1)

 2     = wc (n) + μc α csgn y (n) R −  y (n) + αdn r ∗ (n)  

∂ J LMAP (w(n), y (n)) ∂ wd

(3)

where

ed (n) =

p =2i −1

2i

q=2l−1 exp(−

p =2i −1

2l

 wc (n + 1) = wc (n) + μc e (n) + ε (n) r ∗ (n)

(4)

wd (n + 1) = wd (n) + μd ed (n)r ∗ (n)

(5)

(c) Combining the equalizer stages:

where wc (n) = [ w 0 (n), w 1 (n), . . . , w M (n)] T , M is the number of taps, r(n) = [r (n), r (n − 1), . . . , r (n − M + 1)] T is the received sig(r ) (i ) nal vector and csgn(x) = sgn(Re x) + j sgn(Im x). dn = dn + jdn , (r ) (i ) where both {dn } and {dn } are real-valued independent random processes uniformly distributed on (−1, 1]. According to the theory of dithered quantizers, the sign operator can be subsumed by adopting a quantization noise model of the DSE-CMA error function e ( yn , dn ), which can be written as e ( yn , dn ) = e (n) + ε (n) in terms of the quantization noise ε (n), where e (n) = y (n)( R − | y (n)|2 ) is CMA error function and R = E [|s(n)|4 ]/ E [|s(n)|2 ]. μc is a step-size which controls the convergence speed of the DSE-CMA. Dither amplitude α is a positive constant which is generally selected between αZF in zero-forcing (ZF) equalizers and αOE in open-eye equalizers [9]. The SDD equalizer is designed to maximize the log of the local a posteriori probability density (p.d.f.) criterion ¯J LMAP (w) = E [ J LMAP (w, y (n))] by adjusting wd , where J LMAP (w, y (n)) = ρ log( p (w, y (n))), ρ is the noise covariance and p (w, y (n)) is the a posteriori p.d.f. of y (n). In order to simplify the computation, we divide the complex plane into M /4 regular regions, each region contains four symbol points [7]: S i ,l = {s pq , p = 2i − 1, 2i, q = 2l − 1, 2l, 1  i, l  Q /2}. If the equalizer output y (n) is within the region S i ,l , the SDD equalizer adapts wd according to

∂ J LMAP (w, y (n)) = ed (n)r∗ (n) ∂ wd

2i

2l

(a) Updating the DSE-CMA equalizer (written in terms of the quantization noise) [9]:

(b) Updating the SDD equalizer concurrently with (a):

(2)

e ( yn ,dn )

wd (n + 1) = wd (n) + μd

is the SDD error function. μd is a step-size which controls the convergence speed of the SDD. The choice of ρ should ensure a proper separation of the four clusters in S i ,l . As the minimum distance between the two neighboring symbol points is 2, typically ρ is chosen to be < 1 [7]. The new scheme can be summarized as:

w(n + 1) = wc (n + 1) + wd (n + 1)

(6)

(d) Output: y (n + 1) = r(n + 1) T w(n + 1). The proposed algorithm is similar to the CMA + SDD, but has lower complexity than CMA + SDD by transforming the bulk of CMA’s update multiplications into sign operations [6,9]. 4. Steady-state EMSE analysis EMSE is defined as the steady-state MSE above the level attained by the zero-forcing solution wZF which gives r(n) T wZF = s(n − D )e j θ , where θ denotes the constant phase shift and D denotes the constant time shift [12]. The a priori estimation error can be defined as

ea (n) = r(n) w(n) = s(n − D )e j θ − y (n)

(7)

(n) = wZF − w(n). The steady-state EMSE of the proposed where w algorithm becomes ζproposed = limn→∞ E |ea (n)|2 . By applying (4) and (5) to (6), we can get





w(n + 1) = w(n) + μc r∗ (n) e (n) + ε (n) + β ed (n)

(8)

μ

where β = μd . By subtracting both sides of (8) from wZF we get c the weight error equation





(n + 1) = w

(n) − μc r∗ (n) e (n) + ε (n) + β ed (n) w

(9) T

To define the a posteriori estimation error e p (n) = r(n) w(n + 1), multiplying (9) by r(n) from the left, we can obtain



| y (n)−s pq |2 2ρ

q=2l−1 exp(−

)(s pq − y (n))

| y (n)−s pq |2 2ρ

)

2 



e p (n) = ea (n) − μc r(n) e (n) + ε (n) + β ed (n)

(10)

Therefore e (n) + ε (n) + β ed (n) = [ea (n) − e p (n)]/[μc r(n)2 ] and rewrite (9) as

N. Xie et al. / Digital Signal Processing 22 (2012) 233–237

r∗ (n)  ea (n) − e p (n) 2 r(n)

(n + 1) = w

(n) − w

(11)

Table 1 Values of K α ,s and β for

Rearranging (11) leads to

(n + 1) + w

r∗ (n)

(n) + e (n) = w 2 a

r(n)

r∗ (n)

e (n) 2 p

(12)

r(n)

By squaring (12) and then taking expectations of both sides, we obtain:



2

(n + 1) + E E w  2

(n) + E = E w

1

r(n)2

E

+μ 



r(n)2

2

(13)

A



2

r |e + ε + β ed | 

(14)

This implies that the terms A should be equal to B. Using the following conditions [9,12,13]:





∗

∗





E |ε |2 |e i , er = 2α 2 − |e |2 ;



∗



2

E ea e + e ea = E ea y R − | y |



∗



E (ed ) = E (ea ) 2

+ ea y R − | y |   ≈ 2E 2|s| − R · E |ea |2     2 E r2 |e |2 = E r2 | y |2 R − | y |2     ≈ E r2 · E R 2 |s|2 − 2R |s|4 + |s|6 

2

μα 2 E r2

(17)

E (2|s|2 − R )







    ≈ 2μc E 2|s|2 − R · E |ea |2       + 2μc β E |ea |2 = 2μc E 2|s|2 − R + β · E |ea |2    B ≈ μc2 E r2 |e |2 + |ε |2    = μc2 E r2 · E |e |2 + 2α 2 − |e |2 = 2α 2 μc2 E r2 Solving A = B gives the steady-state EMSE of the complex-valued proposed algorithm as

α 2 μc E r2 E (2|s|2

2E (2|s|2 − R )

2E (2|s|2 − R + β)

E r2

(18)

For the same small step-size μ in the four methods and the same β in CMA + SDD and the proposed algorithm, the EMSE of CMA ζ -CMA = and DSE-CMA differ by the multiplicative factor K α ,s = DSE ζ 2α 2 , E ( R 2 |s|2 −2R |s|4 +|s|6 )

CMA

the relationship between CMA + SDD and the ζ

proposed algorithm is the same, i.e., ζ proposed = K α ,s .1 We just conCMA+SDD sider a relative optimal value of α (i.e. α = αZF ), and then K α ,s is a constant larger than 1 which is determined by transmitting symbol [9], so we can get ζDSE-CMA > ζCMA and ζproposed > ζCMA+SDD . Note that the EMSE of CMA and the proposed algorithm differ by the multiplicative factor

χ=

ζproposed ζCMA

= K α ,s ·

E (2|s|2 − R ) . E (2|s|2 − R +β)

Assume

χ = 1 as the critical value for comparison, Table 1 presents values of K α ,s and β . β actually reflects the SDD part’s proportion of the entire weight and usually takes a large value, for example, in [7] β is chosen to be 100 for 16-QAM and 400 for 64-QAM, in [13] β is chosen to be 120 for 16-QAM and 600 for 64-QAM. Therefore, it can be easily satisfied to choose β larger than the corresponding value in Table 1 which will cause χ < 1, i.e., ζproposed < ζCMA . So we have ζDSE-CMA > ζCMA > ζ > ζCMA+SDD .

− R + β)

E r2

The steady-state EMSE of complex-valued DSE-CMA is [9]:

A typical communication channel in [7,13] is used in our simulations. In all cases, an 11-tap equalizer is adopted. Both of weight vectors of the DSE-CMA and SDD use standard center-tap initialization synchronously. The signal to noise ratio (SNR) is 25 dB. And we consider 16-QAM transmitted data symbols with amplitudes ±{1, 3}, E {|s|2 } = 10, E {|s|4 } = 132, E {|s|6 } = 1960 and R = 13.2. For comparison, the corresponding parameters of each algorithm are the same, i.e., α = αZF , μ = μc = 0.000009, μd = 0.001 and ρ = 0.6. From the signal constellations of the equalizer output shown in Fig. 2, we can see that the proposed algorithm compensates the phase shift as well as CMA + SDD. It also can be seen in Fig. 3 that the proposed algorithm provides faster convergence than the DSE-CMA and CMA but slower than CMA + SDD. However, by transforming the bulk of CMA’s update multiplications into sign operations [6,9], the proposed algorithm has lower complexity than CMA + SDD, therefore it can be seen as a tradeoff. In addition, the steady-state MSE shown in Fig. 3 satisfies the theoretical results discussed before thereby proved our analysis. 6. Conclusions

(15) In this paper, a new hybrid algorithm, which is based on operating the computationally efficient DSE-CMA and the fast

The steady-state EMSE of complex-valued CMA is [12]:

μ E ( R 2 |s|2 − 2R |s|4 + |s|6 )

μ E ( R 2 |s|2 − 2R |s|4 + |s|6 )

5. Simulation results



A = μc E ea∗ e + e ∗ ea + ea∗ ε + ε ∗ ea + β ea∗ ed + ed∗ ea

ζCMA =

19 319.8



we obtain

  ζproposed = E |ea |2 =

β

3 .8 13.3

ζCMA+SDD =

B

E (ε ) = 0;

K α ,s

  e p (n)2

1

|ea |2 r2   1 2 | e | =E p r2    1  ea − μc r2 (e + ε + β ed )2 =E r2    1 2 =E | e | − μc E ea∗ (e + ε + β ed ) + (e + ε + β ed )∗ ea a 2 r   2 cE

M-QAM 16-QAM 64-QAM

The steady-state EMSE of complex-valued CMA + SDD is [13]:



1

ζDSE-CMA =

χ = 1, α = αZF .

  ea (n)2

We assume when n → ∞, E  w(n + 1)2 = E  w(n)2 , and write ea  ea (n), e  e (n), ed  ed (n), ε  ε (n), r  r(n), y  y (n) and s  s(n − D ) for short. Then (13) can be simplified to



235

(16)

1 K α ,s is the same as in [9] after some replacement and the values of K α ,s can be seen in Table 2 of [9], where subscripts α and s denote the dither amplitude and transmitted signal, respectively.

236

N. Xie et al. / Digital Signal Processing 22 (2012) 233–237

Fig. 2. Equalizer output signal constellations after convergence.

Fig. 3. Comparison of MSE trajectories.

N. Xie et al. / Digital Signal Processing 22 (2012) 233–237

converging SDD concurrently for blind channel equalization, was presented to compensate the phase shift and provide convergence faster than the DSE-CMA while being less computationally than the CMA + SDD. With the help of the energy preservation approach, we analyzed that the steady-state performance of the proposed algorithm can be better than both the CMA and DSE-CMA. Those theoretical results were confirmed by simulations.

[12] J. Mai, A.H. Sayed, A feedback approach to the steady-state performance of fractionally spaced blind adaptive equalizers, IEEE Trans. Signal Process. 48 (2000) 80–91. [13] B. Lin, R. He, X. Wang, B. Wang, Excess MSE analysis of the concurrent constant modulus algorithm and soft decision-directed scheme for blind equalisation, IET Signal Process. 2 (2008) 147–155.

Ning Xie received his B.E. and Ph.D. degrees in communications and information system from Sun Yat-Sen University, China, in 2002 and 2007, respectively. He joined the College of Information Engineering, Shenzhen University, Guangdong, China, in July 2007. His research interests include adaptive array processing, array pattern synthesis, communications circuits and wireless communications. He is a mem-

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ber of IEEE. Hengyun Hu received her M.S. degree in communications and information system from Shenzhen University, China, in 2011. Her research interests include wireless communication and adaptive signal processing.

Hui Wang received his B.S., M.S. and Ph.D. degrees from Xi’an Jiaotong University, in 1990, 1993, and 1996, respectively. He is now a professor of the College of Information Engineering, Shenzhen University. His research interests include wireless communication, signal processing, and distributed computing systems, in which he is the author or co-author of more than 50 international leading journals, conferences and book chapters. He is a member of IEEE.