A Supervised Constant Modulus Algorithm for Blind Equalization

Wireless Pers Commun (2012) 62:151–166DOI 10.1007/s11277-010-0045-1

A Supervised Constant Modulus Algorithm for BlindEqualization

A. Özen · I. Kaya · B. Soysal

Published online: 4 June 2010© Springer Science+Business Media, LLC. 2010

Abstract Blind equalization is a technique for adaptive equalization of a communicationchannel without the aid of the usual training sequence. Although the Constant ModulusAlgorithm (CMA) is one of the most popular adaptive blind equalization algorithms, it suf-fers from slow convergence rate. A novel enhanced blind equalization technique based on asupervised CMA (S-CMA) is proposed in this paper. The technique is employed to initializethe coefficients of a linear transversal equalizer (LTE) filter in order to provide a fast start-up for blind training. It also presents a computational study and simulation results of thisnewly proposed algorithm compared to other CMA techniques such as conventional CMA,Normalized CMA (N-CMA) and Modified CMA (M-CMA). The simulation results havedemonstrated that the proposed algorithm has considerably better performance than others.

Keywords Blind channel equalization · Constant modulus algorithm ·Supervised-CMA · Adaptive blind training

1 Introduction

Blind identification and equalization of wireless communication channels have attracted con-siderable interest during the past few years because they avoid training and thus make efficientuse of the available bandwidth. While many blind training algorithms have been proposedin the last years, due to its simplicity, good performance, and robustness, the CMA [1,2]is widely used in practice. Perhaps the greatest drawback of the CMA is its relatively slow

A. Özen (B) · I. KayaDepartment of Electrical and Electronics Engineering, Karadeniz Technical University DSP LAB,Trabzon, Turkeye-mail: [email protected]

B. SoysalDepartment of Electrical and Electronics Engineering, Atatürk University Faculty of Engineering,Erzurum, Turkey

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convergence, which becomes ever more significant as wireless applications involving rapidchanges in channel characteristics become more prominent [3].

Blind adaptive filtering using the CMA [1,2] utilizes a constant step size to change itscoefficients in response to the changing environment. Using a large step size will cause a fastinitial convergence, but result in larger fluctuation in the steady state; the results are oppositewhen a small step size is used. Therefore, the choice of the step size reflects a trade–offbetween misadjustment and speed of convergence. There are many methods available toimprove performance of the CMA during the different stages of adaptation. Among themthe most commonly used is the automatic switching scheme [4] that utilizes a large step sizeduring the transient state and switches to a smaller step size during the steady state. Othermethods include:

(i) Douglas L. Jones [5] controls the step size, for efficient implementation, by usingthe signal vector energy, ‖xn‖2, is computed recursively as in the normalized LMSalgorithm.

(ii) Chahed et al. [6] adjusts the step size by using a time varying step size parameterdue to the squared Euclidian norm of the channel output vector and on the equalizeroutput.

(iii) De Castro et al. [7] presented a concurrent CMA and Decision Directed (DD) blindequalization scheme. Specifically, they let w = wc + wd , where wc is the weightvector of the CMA equalizer which minimizes the cost function and wd is the weightvector of the DD equalizer which minimizes the decision based mean square error(MSE).

(iv) Chen et al. [8] proposed that operates a CMA equalizer and a Soft Decision Directed(SDD) equalizer concurrently. The CMA part is identical to that of the concurrentCMA and DD scheme. The SDD equalizer is designed to maximize log of the locala posteriori probability density function (PDF) criterion by adjusting wd using a sto-chastic gradient algorithm.

(v) Tong et al. [9,10] proposed an alternative method called channel-surfingre-initialization technique that employs an equalizer input covariance matrix.

However, most of these approaches involve significant increases in complexity or com-putational cost. The aim of this study is to present a low complexity and high performanceblind equalization technique. This paper proposes a novel initialization technique that alladaptive blind training algorithms start in a certain degree of convergence. An initial con-vergence obtained by the proposed technique, including initializing preselected coefficientsof the equalizer filter in accordance with a calculated channel output autocorrelation (COA)model, helps to sustain the convergence profiles of the adaptive blind training algorithms.The essential features of the initialization are Inter Symbol Interference (ISI) componentcoefficient calculations of the COA and employment of the adaptive blind training algorithmfor fine-tuning and, noise and ISI cancellation. Therefore, the initialization is very significantthat after the initialization the performance of the conventional CMA becomes comparableto other adaptive blind training algorithms. In addition, the proposed technique has improvedthe performance of the other adaptive blind training algorithms such as N-CMA [5] andM-CMA [6]. Simulation results show that the proposed algorithm performs better than otherblind training algorithms found in the literature.

The rest of the paper is organized as follows: Sect. 2 introduces the adaptive blind chan-nel equalization. Section 3 explains the proposed supervised constant modulus algorithm.Section 4 provides computational complexity comparisons of adaptive blind equalizationtechniques. Section 5 derives convergence and steady-state error analysis of the CMA for

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A Supervised Constant Modulus Algorithm for Blind Equalization 153

Blind Adaptive

Algorithm

Channel Equalizer

(LTE)

AWGN

+

)(kv

)(kx

+

)(ˆ kx )(~ kx

)(kη

Fig. 1 Block diagram of a blind equalization system

the best training trajectory. Section 6 presents the computer simulation results to verify thefeasibility and robustness of the proposed algorithms and finally the paper is concluded inSect. 7.

2 Blind Channel Equalization

The general structure of an adaptive blind equalization system is illustrated in Fig. 1.The baseband model of a digital communication channel can be characterized by a sym-

bol-spaced Finite Impulse Response (FIR) filter and additive white Gaussian noise (AWGN)source. Let us consider that a wireless communication system with quadrature phase shiftkeying (QPSK) modulation. The modulated signal passes through a linear time-invariantchannel to provide a received signal, v(k)

v(k) =M∑

i=0

h(i)x(k − i) + η(k) (1)

where, x(k)is the transmit data sequence, assumed to be independent identically distributed(iid), h(i) is the ith tap coefficients of the tapped-delay-line (TDL) filter model of a channel,M + 1 is the tap number of the channel, η(k) is a zero-mean iid AWGN component and k isthe time index.

The objective of blind equalizers is to remove the ISI at its output without using anytraining sequence. A linear transversal equalizer (LTE) or a soft decision data directed deci-sion feedback equalizer (DFE) is used for blind equalizations. Because of required feedbackdata for a DFE, an LTE is more suitable for blind equalizations. It is also well known fromliteratures that the performance differences between DFE and LTE disappears after errorcorrecting coding. Therefore, an LTE is used in this study and is designed as an FIR filterwith N+1 coefficients, C , then its output is given by

x̂(k) =N∑

i=0

c(i)v(k − i) (2)

The most popular blind equalization algorithms are the family of Godard algorithms [1],which are stochastic gradient descent (SGD) methods for minimizing the cost functions

JGodard(C) = E{(∣∣x̂(k)∣∣p − �p)

2}, p = 1, 2, . . . (3)

where C is the equalizer coefficient vector as C = [c0, c1, . . . , cN ]T ([.]T indicates thetranspose of the matrix [.]), �p = E{|x(k)|2p}/E{|x(k)|p} and E {.} denotes mathematicalexpectation. For the particular case p = 2, Eq. 3 is the cost function of the conventionalCMA, which was independently developed using the idea of penalizing the output samples

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154 A. Özen et al.

that do not have the constant modulus property [2]. The error function to verify CMA criterionis

ε̂(k) = x̂(k)(�2 − ∣∣x̂(k)∣∣2

) (4)

Using an SGD algorithm to define the update equation, the coefficient vector is adapted by[1,2]

c(i + 1) = c(i) + με̂(k)v∗(k − i), i = 0, 1, . . . , N (5)

where, μ is the step size parameter of CMA, ε̂(k)is the kth estimate of error function usingCMA criterion and v∗(k − i) is the complex conjugate of v(k − i).

Due to its simplicity the CMA is one of the most widely used blind equalization tech-niques, although typically it requires a large number of samples to converge satisfactorily.An important problem for the conventional CMA, however, is that it is phase-blind, becausethe cost function can only deal with the modulus of the equalizer output. As a result, theequalizer output signal constellation suffers from an arbitrary phase rotation at a rate equalto the carrier frequency offset rate (see [8,11]). Therefore, the phase of estimated symbol isnot processed by a hard detector directly. The defined problem has to be solved by furtheroperations using schemes either a differential modulation or a phase compensating codingtechniques.

3 Supervised Constant Modulus Algorithm

The Supervised Constant Modulus Algorithm (S-CMA) is based on the initialization of thelinear transversal equalizer (LTE) filter coefficients using the channel output autocorrela-tion (COA). The received signal of a multipath mobile radio channel can be modeled bya TDL filter. The resulting signal v(k) is made up of the combination of the input symbolsx(k), x(k−1), x(k−2), . . . , x(k−M) multiplied by filter coefficients h(0), h(1), . . . , h(M)

and a noise component η(k), as shown in Eq. 1.The autocorrelation of the received signal from the channel output is calculated as,

y(k) =L∑

i=0

v(k)v∗(k − i) (6)

where, L = N/2. The output ISI profile of the COA is calculated according to the followingequations:

d j =L+ j∑

i=0

v(i)v∗(i + j), j = −L , . . . ,−1 (7)

d j =L∑

i=0

v(i)v∗(i), j = 0 (8)

d j =L− j∑

i=0

v∗(i)v(i − j), j = 1, . . . , L (9)

where, dj’s are ISI components of the COA, the negative and positive indexes represent pre-cursor ISI and post-cursors ISI components, respectively and d0 is the center tap. The profile

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of the COA has the advantage of being symmetrical and exhibiting a large real center tap.Thus the COA based initialization provides the optimum symbol-synchronization point andmultipath diversity at the center tap, d0. The output of the COA can then be fed as an inputto an LTE filter to remove side lobes calculated by Eqs. 7 and 9. Finally consider the initial-ization of the LTE filter coefficients, they would be pre-cursor, post-cursor ISI componentsand center tap of the COA in Eq. 10.

c j = d j , j = −L , −L + 1, . . . , 0, 1, . . . , L (10)

By the initialization of the LTE coefficients using Eq. 10 the starting conditions implies thesymbol synchronization on the center tap of the channels’ ISI profile and if the channel has aunit gain then the blind training may well started. However, most of times the channel doesnot have unity gain and the obtained coefficients by (10) is not compatible to the receivedsignal. Therefore normalization is required using center tap coefficients of the COA.

c jnormali zed = c j

d0, j = −L ,−L + 1, . . . , 0, 1, . . . , L (11)

Since the center tap coefficient, d0, is real value, Eq. 11 is implemented with a real valueddivision and 2L + 1 multiplication’s. When the initialization is in accordance with Eq. 11,the blind training would start with a proper ISI cancellation on the previous and the postsymbols side of the cursor symbol (center tap). Although the features of this initializationtechnique presented here is explained assuming the CMA, the initialization process is validfor all blind training algorithms. However, the autocorrelation based techniques have beenemployed in several blind and non-blind applications for different aims [11–14].

4 Computational Complexity Comparisons

In order to investigate the advantages and disadvantages of the algorithms against each other,their computational complexity needs to be known in each recursion period. The level ofcomputational complexity involved in obtaining the adaptive weights of update equation ofeach algorithm determines the required processing speed, complexity of the hardware, andultimately the cost of the system. In this study, the computational complexity is defined bythe required number of floating point operations running each weight update process.

The greatest advantage of the CMA algorithm is the fact that it requires far less computa-tional complexity as for the other blind algorithms. The complexity incurred by the proposedinitialization technique does not prevent its application. The proposed initialization methodrequires Nlte + 1 complex multiplication (Eqs. 7, 8, 9) and Nlte + 1 real valued division(Eq. 11). The comparison of the computational complexities of the blind equalizers requiredfor per weight update is given in Table 1, where Nlte is the tap number of linear transversalequalizer filter. The computational complexity per weight update of the proposed S-CMA issimilar to whom using the CMA, M-CMA and N-CMA.

Table 1 shows that the additional computational complexity brought by the proposedmethod to the CMA algorithm is Nlte + 1 complex multiplication and Nlte + 1 real valueddivision. Thus, a more robust of version of CMA algorithm is developed with a very smallcomplexity concern.

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Table 1 Comparison of the computational complexities of the blind equalizers

Equalizers Multiplications Additions Divisions Calculation of initializationvalues

CMA [1,2] 12Nlte + 6 2Nlte+2 – –N-CMA [5] 34Nlte + 14 4Nlte+2 2Nlte –M-CMA [6] 22Nlte + 8 3Nlte+2 2Nlte –Proposed S-CMA As in CMA As in CMA – Nlte + 1 complex multip.

Nlte + 1 real valued div.

5 Convergence and Steady State Error Analysis

In this section convergence and steady stare error analysis of CMA is derived based on afundamental energy preserving relation, which was first exploited in [15,16] during studieson the robustness and stability of adaptive filters.

5.1 Convergence and Stability Analysis

The aim of convergence and stability analysis is to determine a bound on the step-size param-eter μ so that J (k) (cost function of CMA) is guaranteed to be bounded (i.e. J (k) < ∞ forall k). In this context, analysis of the time evolution of MSE plays a central role in character-izing the necessary conditions on the step-size parameter in order to ensure that J (k) doesnot grow without bound. The key challenge in convergence and stability analysis is to deriveclosed-form expressions relating readily computable quantities to time evolution of MSE.

The stability of stochastic gradient algorithm, also called least mean squares (LMS), ismaintained by choosing the step size parameter, μ, as

0 < μ <2

λmax(12)

where λmax is the biggest eigenvalue of the autocorrelation matrix, Rvv. The components ofRvv are obtained by Rvv(i) = E {v(k)v∗(k − i)} ,. i = 0, 1, 2, L and L ≥ �t/Ts . Here,�t is the excess delay time of channel and Ts is the sampling period of the signal correspond-ing to symbol duration. The convergence rate of the LMS algorithm strictly depend on theeigenvalue spread, χ(Rvv), calculated as

χ(Rvv) = λmax

λmin(13)

where λmin is also the smallest eigenvalue of Rvv. A step size, μ > 0, is enough to providea convergence, however there is no clear definition for an adequate value for μ. Severaldiscussions and analyses on the effect of eigenvalue spread on the convergence speed havebeen made by Haykin in [17].

For the blind equalization case the stochastic gradient algorithm turns into form of CMAand its stability and convergence constraints are also maintained by the eigenvalues of Rvv

as Eq. 14, using the evaluation of [18] for a stable training.

0 < μ <2

γ̄ λmax(14)

Here, γ̄ = E{3x2(k − kof f set ) − �2

}, kof f set is a certain delay indicating the cursor sym-

bol in the equalizer filter, x(k) is the kth transmit symbol, �2 is the modulus value in

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Eq. 3. The value of γ̄ provides an inequality γ̄ ≥ 3 for higher order of quadrature ampli-tude modulation, where modulation order is greater or equal to 4. Equation 14 stands formean convergence constraint of CMA, however the mean-square stability constraint of μ isdifferent, as in [18], as

0 < μ <2γ̄

r2(γ )T r{Rvv} (15)

Here, r2(γ ) = E{γ 2(k)

}, γ (k) = 3x2(k − kof f set ) − �2 and T r{Rvv} = ∑L

i=0 |λ(i)| ≥λmax. Therefore, mean square stability constraint refers smaller or equal values for the max-imum value of μ than what Eq.(14) provides. Thus, the stability region of μ for CMA ismuch narrower than whose using the LMS.

On the other hand, in practice a maximum value much smaller than the constraint providedby Eq. 15 is chosen for μ, without relating the eigenvalue. Therefore, the proposed methodalso employs a maximum value for μ smaller than the constraint value stated by Eq. 15, asgiven by Table 2 and also helps to start with a proper ISI cancellation on the previous andthe post symbols side of the center tap.

5.2 Steady State Error Analysis

In this section, we will use the approach in [15,16] to analyze the steady state error per-formance of CMA algorithm. For the lth equalizer (l > 1), the cost function of the CMAalgorithm is

Jl(k) = E

{(∣∣x̂l(k)∣∣2 − �2

)2}

(16)

where �2 is the dispersion constant in (3). In the minimization of cost function, the CMAalgorithm adopts the stochastic gradient descent method. Denoting μεR+ as the step sizeparameter, the equalizer update equation at the kth iteration is written as

cl(k + 1) = cl(k) + μεl(k)v∗(k) (17)

where the constant that arises from differentiation of Eq. 16 is absorbed within the step sizeμ and εl(k) is the instantaneous error signal for the lth equalizer as

εl(k) = x̂l(k)(�2 − ∣∣x̂l(k)∣∣2

) (18)

Hence, by using the update equation shown in (17), the CMA algorithm converges to theglobal minimum asymptotically when the parameter μ is appropriately chosen. As globalconvergence of the CMA algorithm is achievable [19,20], without loss of generality, weassume that the lth equalizer asymptotically converges to the retrieval of the lth source withdelay kof f set . Define a priori error εl

a(k)for the lth equalizer at the kth iteration as

εla(k) = xl(k − kof f set ) − x̂l(k) = vT (k)c̃l(k) (19)

where c̃l(k) = cz f − cl(k) is the lth equalizer tap weight error vector with cz f representingthe equalizer tap vector corresponding to the zero-forcing solution in the combined channelplus equalizer-l space. The expression of the steady-state mean square error at the equalizeroutput is shown in Eq. 20, and this is the quantity that we wish to determine:

M SE = limk→∞ E

{∣∣∣εla(k)

∣∣∣2}

(20)

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158 A. Özen et al.

Denote a posteriori error εlp(k) at the kth iteration as

εlp(k) = vT (k)c̃l(k + 1) (21)

It can be observed that the instantaneous error εl(k)is related to the a priori and a posteriorierrors εl

a(k) and εlp(k) via the following equation [15]:

εl(k) = εla(k) − εl

p(k)

μ ‖v(k)‖2 (22)

In [15], the following equation concerning the energy preservation during an equalizerupdate is derived:

E{‖c̃l(k + 1)‖2

}+ E

{α(k)

∣∣∣εla(k)

∣∣∣2}

= E{‖c̃l(k)‖2

}+ E

{α(k)

∣∣∣εlp(k)

∣∣∣2}

(23)

where the positive scalar α(k) is defined as α(k) = 1/ ‖v(k)‖2. As in the steady state

limk→∞ E{‖c̃l(k + 1)‖2

}= E

{‖c̃l(k)‖2

}, (23) is simplified to

E

{α(k)

∣∣∣εla(k)

∣∣∣2}

= E

{α(k)

∣∣∣εlp(k)

∣∣∣2}

(24)

Using the expression of εlp(k) in (22), the above equation is written as

E

{α(k)

∣∣∣εla(k)

∣∣∣2}

= E

{α(k)

∣∣∣∣εla(k) − μεl(k)

α(k)

∣∣∣∣2}

(25)

After expansion and simplification, (25) can be simplified to

μ2 E

{‖v(k)‖2

∣∣∣εla(k)

∣∣∣2}

= μE{εl

a(k)∗εl(k) + εla(k)εl(k)∗

}(26)

As in [15,16], we utilize the following assumption.

Assumption 1 The energy of the input vector μ2 ‖v(k)‖2 is independent of the equalizeroutput |εl(k)|2.

This assumption becomes realistic for longer filter lengths and sufficiently small step size,as the value of μ2 ‖v(k)‖2 would then approach a constant and therefore be independent of|εl(k)|2. As the result of this assumption, (26) becomes

μ2 E{‖v(k)‖2} E

{∣∣∣εla(k)

∣∣∣2}

= μE{εl


}(27)

We first consider the term μ2 E{∣∣εl

a(k)∣∣2

}on the left hand side of (27). Substituting the

expression for εl(k) shown in (18), we obtain as following

μ2 E

{∣∣∣εla(k)

∣∣∣2}

= μ2 E

{∣∣∣(�2 − ∣∣x̂l(k)

∣∣2)

x̂l(k)

∣∣∣2}

(28)

where μ2 E

{∣∣∣(�2 − ∣∣x̂l(k)

∣∣2)

x̂l(k)

∣∣∣2}

is called as term A. For term A, its approximation is

derived in [15], as following

term A = μ2 E{|xl |2 �2

2 − 2�2 |xl |4 + |xl |6}

(29)

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With the expression of term A in hand, the left hand side of (27) is written as

μ2 E{‖v(k)‖2} E

{∣∣∣εla(k)

∣∣∣2}

= μ2 E{‖v(k)‖2} [

E{|xl |2 �2

2 − 2�2 |xl |4 + |xl |6}]

(30)

Refer to the right hand side of (27). By substituting the expression of εl(k) in (18), it can beexpanded as

μE{εl


}

= μE{εl

a(k)∗(�2 − ∣∣x̂l(k)

∣∣2)

x̂l(k) + εla(k)

(�2 − ∣∣x̂l(k)

∣∣2)

x̂∗l (k)

}(31)

By employing the approximation given in [15], we obtain

μE{εl


}= 2μE

{ρ |xl(k)|2

∣∣∣εla(k)

∣∣∣2 − �2

∣∣∣εla(k)

∣∣∣2}

(32)

Here ρ is a two-valued scalar whose value depends on whether the system is real or complex[21]. ρ is equal to 3 for a real system and 2 for complex system.

Summarizing the above results, (27) is rewritten as

μE{‖v(k)‖2} [

E{|xl(k)|2 �2

2 − 2�2 |xl(k)|4 + |xl(k)|6}]

≈ 2E{ρ |xl(k)|2 − �2

}E

{∣∣∣εla(k)

∣∣∣2}

(33)

The mean square error in the steady state, i.e. E{∣∣εl

a(k)∣∣2

}when k → ∞, is given by

M SE ≈ μE{‖v(k)‖2} [

E{|xl(k)|2 �2

2 − 2�2 |xl(k)|4 + |xl(k)|6}]

2[E

{ρ |xl(k)|2 − �2

}] (34)

It can be observed that the steady state MSE of the CMA algorithm is affected by the two normof the channel output E

{‖v(k)‖2} and the step size μ. Furthermore, its value is affected bythe second, fourth and the sixth moments of the source. As a result, to achieve small steady-state MSE, small values of μ is preferred in the CMA algorithm and CMA equalizer isstarted with the proposed initialization method. Thus performance of the conventional CMAbecomes comparable to other blind adaptive training algorithms.

6 Computer Simulation Results

The performance of the proposed S-CMA blind equalizer is compared with that of the N-CMA [5] and M-CMA [6] in computer simulations using the conventional CMA blind equal-izer as a benchmark. The simulations are performed over 500 channels by a Monte Carlosimulation using the Quadrature Phase Shift Keying (QPSK). The complex channel impulseresponse (CIR)’s, having 5 symbol storages, given by {-0.2+j0.3, -0.5+j0.4, 0.7-j0.6, 0.4+j0.3,0.2+j0.1} [8] are used in simulations. For the case of the constant channel, two performancecriteria were used to assess the convergence rate of blind equalizers. The first criterion was

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160 A. Özen et al.

Table 2 Algorithm parameter settings in simulation

CMA M-CMA N-CMA Proposed S-CMA

μ µM αM αN σN μS−CMA0.0005 0.0055 0.05 0.038 0.0005 0.0005

a decision-based estimated MSE at each adaptation sample based on a block of NM SE sym-bol-spaced data samples as in [8].

M SE = 1

NM SE

NM SE∑

k=1

∣∣Q(x̂(k)) − x̂(k)∣∣2 (35)

Where, Q(x̂(k)) denotes the quantized equalizer output defined by

Q(x̂(k)) = arg minx(k)

∣∣x̂(k) − x(k)∣∣2 (36)

The second criterion was the Inter Symbol Interference (ISI) measure as in [7] defined by

I S I =∑nc−1

i=0 | f (i)| − | f (i)max|| f (i)max| (37)

Where, { f (i)}nc−1i=0 was the combined impulse response of the channel and equalizer, nc =

N + M − 1 was the length of the combined impulse response, and

f (i)max = max { f (i), 0 ≤ i ≤ nc − 1} (38)

In all the simulations, a 13-tap LTE is used. The CMA, M-CMA and N-CMA equalizer coef-ficients are initialized to zero value, except the central tap which is set to 1. Table 2 lists thealgorithm parameters used in all the simulations for four blind equalizers (μ for the CMA,μM and αM for the M-CMA, αN and σN for the N-CMA, and μS−CMA, for the proposedS-CMA). All the MSE versus iteration number simulations are obtained in the value of Signalto Noise Ratio (SNR) of 23 dB for the QPSK systems whose data sequence length is 4000.

Figure 2 shows the plots of steady-state MSE curves of the conventional CMA and theproposed S-CMA as a function of the step size for the value of SNR of 23 dB. The complexchannel impulse response (CIR)’s, having five symbol storages, given by [8] for fixed channeland a five tap channel profile with average coefficient amplitudes given by (0.227, 0.460,0.688, 0.460, 0.227), which is defined by Proakis [22] for Rayleigh fading channels are usedin this simulation.

As can be seen in Fig. 8 that the proposed S-CMA algorithm performs better than theconventional CMA for all the values of step size in aforementioned both channels. However,performance of blind equalizers for Rayleigh fading channel is worse than fixed channel.

The comparisons of MSE performances related to the blind equalizer trainings are givenin Fig. 3.

The learning curves of the four blind equalizers are compared in Fig. 3, where MSE per-formance of the N-CMA is a little worse than the performance of the CMA. The M-CMAconverges to the lower MSE floor by speeding up the CMA. However, the proposed S-CMAexhibits a clear performance advantages over other blind training techniques simulated forcomparisons in both areas of convergence speeds and MSE floor.

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Fig. 2 Steady-state MSEcurves of the conventionalCMA and the proposedS-CMA as a functionof the step size

0.0005 0.0006 0.0007 0.0008 0.0009 0.001-12

-10

-8

-6

-4

-2

0

2

Step Size

MSE

(dB

)

ConventionalCMA

ProposedS-CMA

Simulated channel profile, Proakis [22], {0.227, 0.46, 0.688, 0.46, 0.227}

Simulated channel profile, Chen [8]

0 1000 2000 3000 4000-12

-10

-8

-6

-4

-2

0

Iteration Number

MSE

(dB

)

Proposed S-CMA

N-CMA Jones-1996CMA

M-CMA Chahed-2004

Fig. 3 Comparison of the MSE performances of the blind equalizers

The obtained MSE performances of aforementioned algorithms for non-stationary envi-ronment with 50 kHz CFO during the 750th–3000th iteration period are given by Fig. 4 inthe value of SNR 23 dB.

Figure 4 illustrates that the proposed S-CMA algorithm outperforms the other CMA algo-rithms in all instances (i.e. during the transient stage, steady state and non-stationary stage)in terms of its speed of convergence, excess mean square error and tracking error. As can beseen from the Fig. 4 that the conventional CMA is prevented converging to unstable regionby the proposed technique in non-stationary environment.

Figure 5 shows the MSE versus SNR curves for all considered algorithms and the MSEperformance samples are obtained after 4000 iterations of blind trainings. For an ordinary,training based equalizations bit-error-rate versus SNR curves are more common for per-formance comparisons, however considered blind trainings has no information about theincoming phase of transmit data and there is no way to detect the phase ambiguity errorwithout any aid from data [8,11]. So, the produced MSE and ISI versus SNR curves exhibit

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0 1000 2000 3000 4000-10

-8

-6

-4

-2

0

2

4

Iteration Number

MSE

(dB

)

Proposed S-CMA

CMA

N-CMA Jones-1996

M-CMA Chahed-2004

Fig. 4 Comparison of the MSE performances of the blind equalizers for a non-stationary environment with50 kHz CFO

0 10 20 30 40-10

-8

-6

-4

-2

0

SNR in dB

MSE

(dB

)

Proposed S-CMA

CMA

ISI limited regionNoise limited region

M-CMA Chahed-2004

N-CMA Jones-1996

Fig. 5 Comparison of the MSE versus SNR performances of the blind equalizers

a comparative error performance profile between algorithms like BER performance compar-ison curves except it is impossible to clear the MSE floor even for great values of SNR.

From Fig. 5, the proposed S-CMA algorithm produces better steady state MSE perfor-mances in both SNR regions, noise-limited (SNR < 20 dB) and ISI-limited (SNR > 20 dB).

The ISI performances of blind equalization techniques are compared in Fig. 6.The N-CMA has accelerated performance of the CMA at the beginning, but has converged

nearly to the CMA floor after 500 training iterations as shown in Fig. 6. The M-CMA exceedsthe performance of the CMA and N-CMA after 1000 training iterations and converges to thelower ISI floor. However, the proposed S-CMA has improved a lot the performance of theother CMA techniques and has converged to the lowest ISI floor.

When the proposed technique is applied to the M-CMA and N-CMA, the obtained ISIperformance results are shown in Fig. 7.

The advantage of the proposed technique is quite clear and provides an instant convergenceat the beginning. The proposed technique has improved the performance of the M-CMA and

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0 1000 2000 3000 4000-10

-8

-6

-4

-2

0

2

Iteration Number

ISI

(dB

)

CMA

Proposed S-CMA

M-CMA Chahed-2004

N-CMA Jones-1996

Fig. 6 Comparison of the ISI performances of the blind equalizers

0 1000 2000 3000 4000-15

-10

-5

0

Iteration Number

ISI

(dB

)

S-M-CMA

CMA

S-N-CMA

Proposed S-CMA

M-CMA Chahed-2004

N-CMA Jones-1996

Fig. 7 The obtained ISI performances of the blind equalizers with the proposed technique

N-CMA blind equalizers as shown in Fig. 7. The convergence rate of the supervised M-CMA(S-M-CMA) and S-N-CMA by the proposed technique has a noticeable improvement. TheS-N-CMA catches nearly the performance of the proposed S-CMA. However, the S-M-CMAperforms better than the other techniques subjected in this study and produces the lowest ISIfloor.

The obtained ISI performances of aforementioned algorithms for non-stationary environ-ment with 50 kHz CFO during the 1000th–3000th iteration period are given by Fig. 8 in thevalue of SNR 23 dB.

Figure 8 demonstrates similar performances what it has been obtained for startup. Ascan be seen from the Fig. 8 the proposed S-CMA algorithm outperforms the other CMAalgorithms in all stages in terms of its speed of convergence, excess ISI and tracking error.

Figure 9 shows the ISI versus SNR curves for all considered algorithms and the ISI per-formance samples are obtained after 4000 iterations of blind trainings.

The proposed S-CMA algorithm produces better steady state ISI performances thanthe M-CMA and N-CMA as shown in Fig. 9. The proposed technique has improved the

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0 1000 2000 3000 4000-15

-10

-5

0

Iteration Number

ISI

(dB

)

S-M-CMA

CMA

S-N-CMA

Proposed S-CMA

M-CMA Chahed-2004

N-CMA Jones-1996

Fig. 8 Comparison of the ISI performances of the blind equalizers for a non-stationary environment with50 kHz CFO

0 10 20 30 40-15

-10

-5

0

SNR in dB

ISI

(dB

)

Proposed S-CMA

CMA

S-M-CMA

S-N-CMA

M-CMA Chahed-2004

N-CMA Jones-1996

Fig. 9 Comparison of the ISI versus SNR performances of the blind equalizers

performance of the M-CMA and N-CMA blind equalizers. The S-N-CMA converges nearlythe performance of the proposed S-CMA. However, the S-M-CMA performs better perfor-mance improvement than the other techniques subjected in this study.

7 Conclusions

In this paper, a novel low complexity blind equalization scheme based on initialization usingthe COA has been proposed. It has been shown that a combination of initialization and CMAequalizer provides an effective and robust way for adaptive blind equalization. Comparedwith a state of art low complexity blind training schemes, namely the recently introducedN-CMA [5] and M-CMA [6] blind equalizer, the proposed S-CMA blind equalizer has sim-pler in computational requirements, faster convergence and lower steady state error. Thisnew blind equalizer offers practical alternatives to blind equalization of QPSK channels and

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provides significant convergence improvement over the conventional CMA blind equalizer.Additionally, the proposed technique improves the performance of N-CMA and M-CMA,and is also valid for all adaptive blind training algorithms.

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10. Evans, S., et al. (1997). Adaptive channel surfing re-initialization of the constant modulus algorithm.In Thirty-First Asilomar Conference on Signals, Systems and Computers (Vol. 1, pp. 823–827).

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Author Biographies

A. Özen was born in Adana, Turkey, on September, 1967. Hereceived B.S., M.Sc. and Ph.D. degrees in the Department of Electri-cal and Electronics Engineering from Karadeniz Technical University,in 1994, 1999, and 2005, respectively. He has been a research assistantat the Department of Electrical and Electronics Engineering in Karade-niz Technical University between February 1996 and May 2004. Laterhe qualified as a university lecturer at the same department betweenSeptember 2006 and November 2008. Later, he has been working as alecturer (Assistant Professor) in the same department since 2008. Hiscurrent research interests are wireless communications, fuzzy basedchannel estimation and equalization, signal processing, blind equaliza-tion, wimax radio, cognitive radio and cooperative communications.

I. Kaya He has graduated from Karadeniz Technical University(KTU), and completed Ph.D. in 1998 in The University of Bristol. Hehas been working as a lecturer (Assistant Professor) in The Depart-ment of Electrical and Electronics Engineering Department since 1999.His academic interests are on channel caunter measures, equalization,blind equalization, OFDM receivers and MIMO. He has established aresearch lab at KTU and completed three patent applications on thearea of fast equalization techniques in UoB.

B. Soysal was born in Bayburt, Turkey, on February, 1966. He receivedB.S., M.Sc. and Ph.D. degrees in the department of Electrical andElectronics Engineering from Karadeniz Technical University, in 1989,1998, and 2004, respectively. He has been a research assistant at theDepartment of Electrical and Electronics Engineering in Atatürk Uni-versity between February 1995 and July 2006. Later he qualified as anAssistant Professor at the same department. His major research inter-ests are wireless communications, wireless network, channel equaliza-tion, digital signal processing, coding theory.

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A Supervised Constant Modulus Algorithm for Blind Equalization

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