A new adaptive step size mcma blind equalizer algorithm based on absoulate erroe

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
INTERNATIONAL JOURNAL OF ELECTRONICS AND
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 2, March – April (2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 2, March – April, 2013, pp. 01-14
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A NEW ADAPTIVE STEP SIZE MCMA BLIND EQUALIZER
ALGORITHM BASED ON ABSOULATE ERROE AND ITERATION
NUMBER

Thamer M. Jamel 1, Mohammed Abed Shabeeb 2
1
Department of Electrical Engineering, University of Technology, Baghdad , Iraq.
2
Foundation of Technical Education, Technical Institute-Nejif
Al-Nejif Al-Ashraf , Iraq

ABSTRACT

Blind equalization is a technique for adaptive equalization of a communication
channel without the aid of the usual training sequence. The Modified Constant Modulus
Algorithm (MCMA) is one of adaptive blind equalization algorithms. The drawbacks of the
fixed step size of (MCMA) are slow convergence speed and high misadjustment. In order to
overcome the tradeoff between fast convergence rate and low level of misadjustment of
MCMA algorithm, we propose an enhanced technique based on an absolute difference error
and iteration number to adjust a step size. The new proposed algorithm is called Combined
Iteration and Absolute Error MCMA (CIAE-MCMA). Then we applied it for 16 QAM and
64 QAM adaptive blind equalizer systems. Simulation of adaptive blind equalizer with a
typical telephone channel is evaluated to compare the performance of the proposed algorithm
with MCMA and other two variable step size MCMA (VSS- MCMA) algorithms. It is
observed from the simulation results, that the proposed algorithm has better performance
compared with other algorithms in terms of fast convergence rate, low level of misadjustment
and small BER.

Keywords: Adaptive blind equalizer, MCMA, variable step size.

1. INTRODUCTION

For bandwidth-limited channels, it is usually found that inter-symbol interference is
the main determining factor in the design of high inter-symbol interference appears in all
QAM systems. QAM signals are sensitive to inter-symbol interference (ISI) caused by multi-

1


path propagation and the fading of the channel, so it is necessary to use equalization
equ
technique to mitigate the effect of ISI [1]. For bandwidth efficient communication systems,
bandwidth-efficient
operating in high inter-symbol interference (ISI) environments adaptive equalizers have
symbol
become a necessary component of the receiver architecture. The basic data communications
basic
process can be explained with the simplified baseband equalizer block diagram of Fig. 1[2].
A typical QAM data transmission system consists of a transmitter, a channel, and a receiver,
where the unknown channel represents all the interconnections between the transmitter and
interconnections
the receiver (Fig.1). The transmitter generates a zero mean, independent input data sequence
s , each element of which comes from a finite alphabet A of the QAM symbols (or
constellation). The data sequence is sent through the channel that its output x is the
receiver input. The received symbol x(n) is corrupted by intersymbol interference (ISI) and
Gaussian white noise.

Fig. 1 Baseband model of the adaptive digital communication system [2]

In order to counter inter-symbol interference effect, the observed signal may first be
-symbol
passed through a filter called the equalizer that its characteristics are the inverse of the
channel characteristics. Filters, with adjustable parameters, are usually called adaptive filters,
ad
especially when they include algorithms that allow the filter coefficients to adapt to the
changes in the signal statistics, the equalizers; thereby using adaptive filters are called
adaptive equalizers. For adaptive filtering, the FIR filter is the most practical and widely
filter
used. The reason is the FIR filter has only adjustable zeros (the stability of FIR) [2].
The equalizer removes the distortion caused by the channel by estimating the channel
inverse. The equalizer output y(n) is sent to a decision device which results in the received
symbol estimate ˆs(n). Blind equalizer as opposed to data trained equalizer, is able to
compensate amplitude and delay distortion of a communication channel using only channel
output sample and knowledge of basic statistical properties of the data symbol [2]. The main
advantage of blind equalization is that a training sequence is not needed. Hence no bandwidth
is wasted by its transmission. Although various blind equalization techniques exist, the best
known algorithms are the constant modulus algorithm (CMA) , the generalized Sato
n
algorithm (GSA) , modified constant modulus algorithm MCMA, Stop and Go SGA and …
etc.

2


However, since the CMA and MCMA methods yield slow convergence speed and
high MSE, several blind algorithms are available to improve performance of the CMA and
MCMA during the different stages of adaptation. One of these is a variation of step size.
There are many methods available to adjust step size of different algorithms during
adaptation [ 3-12].
One of these methods is proposed by Jones.D. L. [3] who controls the step size by
using the channel output signal vector energy. Chahed et al. [4] adjusts the step size by using
a time varying step size parameter depending upon squared Euclidian norms of the channel
output vector and on the equalizer output. Xiong et al. [5] employed the lag error
autocorrelation function between the current and previously output error of the blind system.
Zhao, B. [6], proposed that the variable step size of CMA algorithm is controlled by
difference between current and previous MSE .Kevin Banovi´c., [7] proposed adjustment
process to step size based on the length of the equalizer output radius. An alternative scheme
that considers a nonlinear function of instantaneous error for adjusting the step-size parameter
is proposed by Liyi et al. [8]. Meng Zhang [9], proposed a fine projection blind equalization
CMA algorithm based on quantization estimation errors and variable step-size. Variable step
size MCMA is proposed by Wei Xue, [10] in which the step size is adjusted according to the
region where the received signal lies in the constellation plane.
In this paper, a new variable step size method is used for MCMA algorithm in order
to enhance the performance of the traditional MCMA algorithm and to overcome its
drawbacks which are the slow convergence rate, large steady state mean square error (MSE)
and phase-blind nature [13]. The proposed algorithm is called Combined Iteration number
with Absolute Error MCMA (CIAE-MCMA). In this proposed algorithm, the step size is
controlled by two parameters; the first parameter is the absolute difference between current
and previous error, while the second one is the iteration number. As it will be shown later in
the simulations, the proposed algorithm has better performance as compared with MCMA
and other variable step size (VSSMCMA) algorithms which were proposed in [6 and 8].
This paper is organized as follows: basic concept of the Modified constant modulus
algorithm (MCMA) is given in section 2 .The proposed algorithm and its analysis are
described in section 3. In section 4, a simulation system which includes system model,
channel model, other VSS-MCMA algorithms and simulation results are presented. Finally,
section 5 provides main conclusions.

2. MODIFIED CONSTANT MODULUS ALGORITHM (MCMA)

Modified constant modulus algorithm (MCMA) shows the improved performance of
the convergence behavior and can correct the phase error and frequency offset at the same
time [14]. MCMA algorithm modifies the cost function of CMA in the form of real and
imaginary parts, the modified cost function can be written as [14]:

‫ܬ‬ሺ݊ሻ ൌ ‫ܬ‬ோ ሺ݊ሻ ൅ ‫ܬ‬ூ ሺ݊ሻ (1)

Where ‫ܬ‬ோ ሺ݊ሻ and ‫ܬ‬ூ ሺ݊ሻ are the cost function of real and imaginary parts of the
equalizer output ‫ݕ‬ሺ݊ሻ ൌ ‫ݕ‬ோ ሺ݊ሻ ൅ ‫ݕ‬ூ ሺ݊ሻ respectively and they are defined as:
ଶ
‫ܬ‬ຯ ሺ݊ሻ ൌ ‫ ܧ‬ቀቂ൫|‫ݕ‬ோ ሺ݊ሻ|ଶ െ ܴ௠ೃ ൯ ቃቁ (2)

3


ଶ
‫ܬ‬ூ ሺ݊ሻ ൌ ‫ ܧ‬ቀቂ൫|‫ݕ‬ூ ሺ݊ሻ|ଶ െ ܴ௠಺ ൯ ቃቁ (3)

Where ܴ௠ೃ and ܴ௠಺ are the real constants determined by the real and imaginary parts
of the transmitted data sequence respectively and is determined by the mean distance between
the symbols and the real axis as follows [14] :

ாൣ|௔ሺ௡ሻ|ర ൧
ܴ௠ೃ ൌ ாሾ|௔ሺ௡ሻ|మ ሿ
(4)

Where a(n) is the signal to be transmitted. The mean distance between the symbols
and the imaginary axis is identical:

ாൣ|௔ሺ௡ሻ|ర ൧
ܴ௠಺ ൌ (5)
ாሾ|௔ሺ௡ሻ|మ ሿ

And the error signal is given by:

݁ோ ሺ݊ሻ ൌ ‫ݕ‬ோ ሺ݊ሻ൫|‫ݕ‬ோ ሺ݊ሻ|ଶ െ ܴ௠ೃ ൯ (6)

݁ூ ሺ݊ሻ ൌ ‫ݕ‬ூ ሺ݊ሻ൫|‫ݕ‬ூ ሺ݊ሻ|ଶ െ ܴ௠಺ ൯ (7)

݁ெ஼ெ஺ ሺ݊ሻ ൌ ݁ோ ሺ݊ሻ ൅ ݆ ‫݁ כ‬ூ ሺ݊ሻ (8)

In contrast, the cost function of MCMA separates the output of equalizer to the real
and the imaginary parts and estimates the error signal for real and imaginary parts
respectively.

3. A NEW PROPOSED ADAPTIVE STEP SIZE MCMA ALGORITHM

As explained previously, this paper proposes a new algorithm in order to enhance the
performance of the traditional MCMA algorithm and to overcome its drawbacks.

3.1 Algorithm formulation
Most of variable step size algorithms used the error signal ݁ሺ݊ሻ to directly control the
step size. The error signal is calculated as:

݁ሺ݊ሻ ൌ ܽ ොሺ݊ሻ െ ܹ ் ሺ݊ሻܺሺ݊ሻ
ොሺ݊ሻ െ ‫ݕ‬ሺ݊ሻ ൌ ܽ (9)

Where ܹሺ݊ሻ ൌ ሾܹ଴ ሺ݊ሻ ܹଵ ሺ݊ሻ ܹଶ ሺ݊ሻ … … ܹ௅ିଵ ሺ݊ሻ ሿ் is filter coefficients, L
being the order of the filter, ‫ݕ‬ሺ݊ሻ = ܺ ் ሺ݊ሻܹሺ݊ሻ is adaptive filter output. e(n) is an error
signal , ܺ ் ሺ݊ሻ ൌ ሾܺሺ݊ሻ ܺሺ݊ െ 1ሻ ܺሺ݊ െ 2ሻ … … . . ܺሺ݊ െ ‫ ܮ‬൅ 1ሻሿ is input data and ܽሺ݊ሻ is ො
the desired signal. The proposed algorithm calculates the difference value between current
and previous remaining errors, and then it will use to adjust the step size as:

ߤሺ݊ ൅ 1ሻ ൌ ߤሺ݊ሻ ‫ߩ כ‬ (10)

4


Where ߤሺ݊ሻ is time varying step size and the term (ߩ) is:

ߩ ൌ ߚ ‫ି݌ݔ݁ כ‬ሺ௙ା௚ሻ
௡ൗ ൯ቁ
ߤ ሺ݊ ൅ 1ሻ ൌ ߤሺ݊ሻ‫ି݌ݔ݁ כ‬ቀሺ|௘ሺ௡ሻି ௘ሺ௡ିଵሻ|ሻା൫ ఊ (11)

Where ߚ is the proportionality factor. It is used to control the value scope of ߤሺ݊ ൅
1ሻ. When 0 ≤ ݁‫ି݌ݔ‬ሺ௙ା௚ሻ ≤ 1 , the value scope of ߤ௜ (Initial step size) satisfies 0 ൑ ߤሺ݊ ൅
1ሻ ൑ ߚ ‫ߤ כ‬௜ . In order to guarantee the algorithm restrain, the step size must satisfy 0 ൑
ߤ ሺ݊ሻ ൑ 2ൗ3trሺ‫܀‬ሻ [15] . Where R is the input signal autocorrelation matrix, tr(R) is the
trace of R . The rule of using the term [e(n)- e(n-1)] is that the difference is large in the
iteration initial period, then it is gradually reducing along with algorithm restraining. When
the algorithm enters the stable state, [e(n) - e(n-1)] achieves the minimum. Therefore; µ (n)
has a corresponding change rule with difference, and the adaptive step-size µ (n) control was
realized. The step-size is large in the algorithm iteration initial period, and the convergence
rate is faster. After algorithm restraining, the step-size was reduced to enhance the restraining
precision. An exponential function is carefully designed in this proposed algorithm in order
to adjust the value of the step size. An additional parameter was used together with absolute
difference error for adjusting the step size; that is the iteration number .The parameter ߛ in
(11) is a constant used to control the converges rate. Then the calculated value of µ(n+1) is
bounded between two values (ߤ௠௔௫ ) and (ߤ௠௜௡ ) as:

ߤሺ݊ ൅ 1ሻ ൌ ߤ௠௔௫ ݂݅ ߤሺ݊ሻ ൐ ߤ௠௔௫
ߤሺ݊ ൅ 1ሻ ൌ ߤ௠௜௡ ݂݅ ߤሺ݊ሻ ൏ ߤ௠௜௡ (12)
‫ ߤ ݁ݏ݅ݓݎ݄݁ݐ݋‬ሺ݊ ൅ 1ሻ ൌ ߤሺ݊ ൅ 1ሻ

ߤሺ݊ ൅ 1ሻ is set to ߤ௠௜௡ or ߤ௠௔௫ when it falls below or above these lower and upper
bounds, respectively. Table I below illustrates the steps required for the proposed algorithm:

3.2 Analysis of Variable Step-size rule

The new algorithm utilized a nonlinear function of remainder error besides the
iteration number to control the step-size. To analyze the equation (11), we start with the first
part: ݂ ൌ ሺെ|݁ሺ݊ሻ െ ݁ሺ݊ െ 1ሻ|ሻ , at the beginning of the adaptation process, the difference
between current and previous error is high so the value of this part ߩ ൌ ݁‫ି݌ݔ‬ሺ௙ሻ starts with a
large value and as iteration number increases, it becomes a small value, as shown in Fig. 2.
The second part (݃ ൌ ൫݊ൗߛ൯ ሻ depends on the number of iterations. At the beginning, the
ቀି൫௡
ൗఊ ൯ቁ
iteration number is small, so the value of (ߩ ൌ ݁‫݌ݔ‬ ) is equal to the initial value of
the step size (ߤ) and as the number of iteration increases, the value of (ߩ) decreases. The
relation between (ߩ) and the iteration number is illustrated in Fig. 3.
By combining both parts as in (11) i.e. (ߩ ൌ ݁‫ି݌ݔ‬ሺ௙ା௚ሻ ), we observe that they cause
large step size at a low number of iterations (dedicate a faster convergence rate) and a small
value at the steady state reduces low MSE and low misadjustment at the steady state as
shown in Fig. 4.

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Table 1. A New Proposed CIAE-MCMA Algorithm

CIAE-MCMA ALGORITHM
1. Initialization:
- Given the input vector:-
Xሺkሻ ൌ ሾxሺkሻ, xሺk െ 1ሻ, … xሺk െ L ൅ 1ሻሿ୘
- N is iteration number, , L is the order of the taps FIR filter ;
for accuracy: P=N-L,
- Set the parameters ߛ, ߚ constant ߤ, ߤ௜ , ߤ௠௔௫ , ߤ௠௜௡ for
adjusting the step size.
- a (k) is the output of QAM modulation with dimension
[1xN]
- Set Wሺ0ሻ ൌ zerosሾ1xNሿ୘
eሺkሻ ൌ zerosሾ1xPሿ
Rr=mean(abs(real(a)).^4)/mean(abs(real(a)).^2);
Ri=mean(abs(imag(a)).^4)/mean(abs(imag(a)).^2);
2. For k= 1,2,….. Iterations
Compute the following:-
y=W'*X : Equalizer Output
er(k)=real(y)*( Rr -real(y)^2)
ei(k)=imag(y)*( Ri -imag(y)^2)
e(k)=er(k)+j*ei(k)
ߤሺ1ሻ ൌ ߤ௜
௞ൗ ൯
ߤሺ݇ ൅ 1ሻ ൌ ߤሺ݇ሻ‫ି ݁ כ‬൫௘௫௣ሺି|௘ሺ௞ሻି ௘ሺ௞ିଵሻ|ሻ൯ା൫ ఊ
ߤሺ݇ ൅ 1ሻ ൌ ߤ௠௔௫ ݂݅ ߤሺ݇ሻ ൐ ߤ௠௔௫
ߤሺ݇ ൅ 1ሻ ൌ ߤ௠௜௡ ݂݅ ߤሺ݇ሻ ൏ ߤ௠௜௡
‫݁ݏ݅ݓݎ݄݁ݐ݋‬ ߤሺ݇ ൅ 1ሻ ൌ ߤሺ݇ ൅ 1ሻ
w =w+ ߤሺ݇ ൅ 1ሻ*e(k)'*X;
ߤሺ݇ሻ ൌ ߤሺ݇ ൅ 1ሻ

6


Fig. 2 Updating the step size with iteration number using absolute difference error part only.

Fig. 3 Updating the step size with iteration number using iteration number part only.

7


Fig. 4 The complete step size profile curve for CIAE-MCMA with iteration number using
both parts.

4. SIMULATION SYSTEM

4.1 System Model
Figure 1 provides a block diagram of adaptive blind equalizer for the digital
communications system that will be simulated in this section. For the work presented in this
paper, we consider the channel and equalizer are both constrained to be linear as well as time
invariant FIR filters. Noise in the channel is modeled as zero-mean additive white Gaussian
noise (AWGN).

4.2 Telephone Channel Model
The channel used for testing comes from Proakis [16]. This channel is given by the
11-tap impulse response h = [0.04 -0.05 0.07 -0.21 -0.5 0.72 0.36 0.0 0.21 0.03 0.07] and it is
real-valued. It is a typical response of a good quality telephone channel and exhibits a non-
linear phase distortion [16].

4.3 Another VSS MCMA Algorithms
Two variable step size algorithms are explored here to make a comparison with the
proposed algorithm in this section:-

4.3.1 VSS-MCMA1 Algorithm
This variable step size algorithm is proposed by Wei Xue and Xiaoniu Yang in 2010
[8]. The step size is adjusted according to the region where the received signal lies in the
constellation plane.
During the transient stages the output of equalizer will be scattered around a large area of the
transmitted data symbols. However, in the steady state the output of equalizer will lie in a close
8


neighborhood of the transmitted data symbols. In the variable step size scheme, M regions D1, D2,. . .
, DM are chosen in the constellation for M-QAM signals. The region Di(i = 1; 2; : : : ;M) represents a
small circular area around the point of the transmitted data symbol constellation with the radius (d).

ߤሺ݊ ൅ 1ሻ ൌ ߤ଴ ݂݅ ‫ݕ‬ሺ݇ሻ ‫ܦڂ ב‬௜
ߤሺ݊ ൅ 1ሻ ൌ ߤଵ ݂݅ ‫ݕ‬ሺ݇ሻ ‫ܦ א‬௜ (13)

4.3.2 VSS-MCMA2 Algorithm
This variable step size algorithm is proposed by Zhang Liyi and Chen Lei1 2009 [6]. The
new algorithm utilized a nonlinear function of remainder error to control the step-size as follows:
ߤሺ݊ሻ ൌ ߚ ‫ כ‬ൣ1 െ ݁ ିఙ|௘ሺ௡ሻ| ൧
ߤሺ݊ሻ ൌ ߤ௠௔௫ ݂݅ ߤሺ݊ሻ ൐ ߤ௠௔௫
ߤሺ݊ሻ ൌ ߤ௠௜௡ ݂݅ ߤሺ݊ሻ ൏ ߤ௠௜௡ (14)

Where, β is the proportionality factor that is used to control the value scope of ߤሺ݊ሻ.

4.4 Simulation mythology
The parameters used for MCMA, VSSMCMA1, VSSMCMA2 and CIAE-MCMA algorithms
were chosen to achieve a better performance in terms of a fast convergence time and a low level
misadjustment in order to make fairly comparable between these algorithms. They are chosen as
follows:
• Number of symbols equal to 3000.
• The signal to noise ratio for all simulations is 30 dB .The noise source used for all simulations
was white Gaussian noise with zero mean and with unity variance.
• The length of the equalizer was taken as (11) taps.
• To carry out the BER performance of different algorithms for 16-QAM and 64-QAM, the
signal to noise ratio at the primary input for all simulations is alternated between 0 and 30 dB with an
increment step equals to 2 dB.
• The remaining factors used for these algorithms for telephone channel are shown in Table 3.

Table 3 Parameters used in telephone channel
Constant MCMA VSSMCMA1 VSSMCMA2 CIAE-
MCMA
ߤ 0.0015 ------- ------- --------
ߤ௠௔௫ ------- ------- 0.08 0.08
ߤ௠௜௡ ------- ------- 0.00008 0.00002
ߤ଴ ------- 0.0003 ------- -------
ߤଵ ------- 0.00006 ------- -------
d ------- 0.6 ------- -------
ߛ ------- ------- ------- 500
ߪ ------- ------- 0.9 -------
ߚ ------- ------- 0.009 0.99
4.5 Simulation results

4.5.1 Case 1
In order to illustrate the performance and activity measurement for our proposed algorithm. it
must be compared with other algorithms which are traditionally MCMA ,the variable step size
modified constant modulus algorithms (VSSMCMA1) and (VSSMCMA2) respectively . The
Equalizer output for four algorithms is shown in Fig. 5 (16-QAM) and Fig.6 (64-QAM) respectively
using telephone channel.
9


It can be observed that a better constellation is gotten for CIAE-MCMA algorithm
compared with other algorithms. Figure 7 shows the learning curves (MSE) curves for all
algorithms. It is clear that the proposed algorithm has fast convergence rate and low level of
misadjustment compared with the other algorithms. The relation between the signal noise
ratio (SNR) and the bit error ratio (BER) for all algorithms is shown in Fig. 8 (16-QAM) and
Fig. 9 (64-QAM). From these figures, we observed that BER performance was improved
using the proposed algorithm.

Fig. 5 Equalizer output using all algorithms for 16-QAM scheme (a) MCMA (b)
VSSMCMA1 (c) VSSMCMA2 (d) proposed algorithm.

Fig. 6 Equalizer output using all algorithms for 64-QAM scheme (a) MCMA (b)
VSSMCMA1 (c) VSSMCMA2 (d) proposed algorithm.

10


(a)

(b)
Fig. 7 Learning curves for all algorithms for (a)16-QAM (b) 64-QAM scheme

11


Fig. 8 BER performance using all algorithms for 16-QAM system.

Fig. 9 BER performance for all algorithms using the 64-QAM system.

4.6 Effect of change the parameter (ࢽሻ
Figure 10 illustrates the effect of changing a constant (ߛሻ that is used in the proposed
algorithm. From the Fig. 10, it is found that the best value of (ߛሻ to achieve fast convergence
and low level of misadjustment is equal to 500 when using a fixed value for the parameter β
which is equal to 0.99.

12


Fig. 10 The effect of changing parameter ( ሻ on the algorithm performance.

5. CONCLUSIONS

In this paper, a variable step size modified constant modulus algorithm (CIAE-
MCMA) is proposed. The step size of the algorithm is adjusted according to the combined
absolute difference error with iteration number. The (CIAE-MCMA) can obtain both fast
convergence rate, and a small steady state MSE compared with traditional MCMA and other
variable step size MCMA algorithms. The simulation results for 16-QAM and 64-QAM
signals demonstrate the effectiveness of the (CIAE-MCMA) in the equalization performance.
Moreover, the optimum parameters for the proposed algorithms are β equals to 0.99 and γ
equals to 500.

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