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Blind Equalization Technique for Cross Correlation Constant Modulus
Algorithm (CC-CMA)
Abstract: - Equalization plays an important role for the communication system receiver to correctly
recover the symbol send by the transmitter, where the received signals may contain additive noise and
intersymbol interference (ISI). Blind equalization is a technique of many equalization techniques at which the
transmitted symbols over a communication channel can be recovered without the aid of training sequences,
recently blind equalizers have a wide range of research interest since they do not require training sequence and
extra bandwidth, but the main weaknesses of these approaches are their high computational complexity and
slow adaptation, so different algorithms are presented to avoid this nature.
The conventional Cross Correlation Constant Modulus Algorithm (CC-CMA) suffers from slow
convergence rate corresponds to various transmission delays especially in wireless communication systems,
which require higher speed and lower bandwidth. To overcome that, several adaptive algorithms with rapid
convergence property are proposed based upon the cross-correlation and constant modulus (CC-CM) criterion,
namely the recursive least squares (RLS) version of the CC-CMA (RLS-CC-CMA).
This paper proposes a new blind equalization technique, the Exponential Weighted Step-size Recursive Cross
Correlation CMA (EXP-RCC-CMA), which is based upon the Exponentially Weighted Step-size Recursive
Least Squares (EXP-RLS) and the Recursive Cross Correlation CMA (RCC-CMA) methods, by introducing
several assumptions to obtain higher convergence rate, minimum Mean Squared Error (MSE), and hence better
receiver performance in digital system. Simulations studies show the rate of convergence, the mean square
error (MSE), and the average error versus different signal-to-noise ratios (SNRs) with the other related blind
algorithms.
Key-Words: - Blind Equalization, Constant Modulus Algorithm (CMA), Recursive Least Squared (RLS)
algorithm, Exponentially Weighted Step-size Recursive Least Squares (EXP-RLS) algorithm, Recursive Cross
Correlation Based method for CMA (RCC-CMA) algorithm, Channel Equalization.
1 Introduction
One of the most important advantages of the
digital communication system for voice, data and
video is their higher reliability in noise handling in
comparison with that has the analogue
communication property. In modern digital
communication systems an estimator of the
transmitted symbols represents one of the critical
parts of the receiver, it consists typically of an
equalizer and a decision device as shown in Fig. 1,
and because the equalizer is designed to compensate
the channel distortions, through a process known as
equalization so it plays an important role for the
communication systems. Equalization is a process in
which the symbols sent by the transmitter can be
recovered correctly from the received signal that
suffer from additive noise and the linear channel
Prof. Dr. Amin Mohamed Nassar 1 , Eng. Waleed El Nahal 2
1 Electronics &Communication Dept., Faculty of Engineering, Cairo University, Giza, Egypt.
E-mail: aminnassar@gawab.com
2 Electronics &Communication Dept., Faculty of Engineering, MSA University, 6th October,
Egypt. E-mail: welnahal@gmail.com
WSEAS TRANSACTIONS on SIGNAL PROCESSING
Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
distortion, known as the Inter-symbol interference
(ISI), this means that the transmitted pulses are
spreaded out of their limits and overlapped with the
adjacent pulses, so on the pulses that correspond to
different symbols are not separable and that can
severely corrupt the transmitted signal and make it
difficult for the receiver to directly recover the send
data. Equalization is a process in which the symbols
sent by the transmitter can be recovered correctly
from the received signal that suffer from additive
noise and the linear channel distortion, known as the
Inter-symbol interference (ISI). According to the
transmission media the main causes for ISI are: the
band limited of the cable lines, and the multipath
propagation of the cellular communications.
Fig. 1: Digital transmission system using channel
equalization.
The conventional or trained adaptive algorithms
for equalization use a training signal to update the
tap weights of the equalizer's adaptive filter,
according to that these methods suffer from
consuming large part of the channel bandwidth, i.e.,
when the pilot or the training sequence is
transmitted, no other useful sequences can be
transmitted by the same spectrum, and also
sometimes it is so difficult to characterize the
statistical properties of the training sequence to
estimate the radio environment as an example. To
overcome these limitations, a lot of researches have
been much interest in blind approaches, which can
adapt their tap weight vector by restoring certain
properties of the transmitted signals' structure that
can compensate the transmission of the pilot or the
training sequence. So from the above discussion we
can classify the equalization techniques into two
types which are, blind (non-trained) and non-blind
(trained) equalization.
Recent systems (such as GSM system) use well
known methods based on training sequences, where
a part of signal is known and repeated, and the
equalizer is based on matching its output to the
reference signal, by adapting its parameters to
minimize some criterion (typically MSE).
Unfortunately the training sequence consumes a
considerable part of the overall message (approx.
25% in GSM) [1]. For this reason, recently much
research effort has been devoted to blind
equalization algorithms.
Blind equalization or self-recovering algorithms
have no training sequence; so they do not require an
extra bandwidth, also the bandwidth efficiency
potential is increased, and hence the bit rate can be
improved [1], but the main weaknesses of these
approaches are their high computational complexity
and slow adaptation [2][3].
Blind equalization is one of the most important
applications of the telecommunication systems, in
which the unknown input sequence is recovered
from the unknown channel distortion based on the
probability roles and statistical properties of the
input sequence to the adaptive equalizer, and its
performance depends on the characteristics of both
the channel and the transmitted sequence [4].
Among all blind equalization algorithms, the
constant modulus algorithm (CMA) [5], is a
popular, low complexity blind algorithm used for
channel equalization and inters symbol interference
(ISI) suppression for constant modulus signals [6].
One limitation of the conventional CMA
algorithm is that, it is in capable of distinguishing
one user data from another in multiuser detection
applications; and so on it fails to lock on the desired
user signal. By adding a cross-correlation term to
the CMA cost function to get the cross correlation
CMA (CC-CMA) algorithm presented in to solve
such problem in a static multipath channel [7] [8].
However, the classical CC-CMA is found to be
ineffective for some applications where a fast
convergence is needed (such as radio-mobile system
or wireless communication systems) and because
the Recursive Least Square (RLS) algorithm is well-
known that it has a very fast convergence rate [6],
so our proposition is to apply a similar method in
order to solve this phenomenon of the CMA.
The rest of the paper is organized as follows. In
section 2, the Constant Modulus Algorithm (CMA)
Channel
Transmitter
Receiver
Adaptive
Equalize
r
Decision
Device
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Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
is presented. In section 3, the Recursive Cross
Correlation CMA (RCC-CMA) method is shown. In
section 4, we formulate the principles of the
proposed Exponential Weighted Step-size Recursive
Cross Correlation CMA (EXP-RCC-CMA)
technique and present its parameters that improve
the rate of convergence. In section 5, simulation
performance results are generated and compared
with the conventional CMA, RCC-CMA, and EXP-
RCC-CMA algorithms. Finally, we conclude the
paper results and simulations in section 6.
2 Channel Equalization
A typical communication system design involves
passing the transmitted signal through a
communication channel by the transmitter, and at
the receiver, the received signal is passed through
the receiver components to recover the original
signal. However, the channel will affect the
transmitted signal because of the channel noise and
dispersion which are leading to the Intersymbol
Interference (ISI) phenomenon, so it is necessary to
pass the received signal at the receiver through an
equalizer as shown in Fig. 1, to minimize the
channel effect [9][10]. The adaptive equalizer and
the decision device at the receiver compensate the
Intersymbol Interference (ISI) created by a time
dispersive channel as mentioned in the introduction
section.
Fig. 2: Digital transmission system using channel
equalization.
3 Blind Equalization
Blind equalization is one of the most important
applications of the field of telecommunication
systems, where the transmitted signal must be
isolated from the multipath and interference effects.
Many researches of the blind equalization has
been existence for a little over twenty years, during
this time the concentration is on developing new
algorithms and formulating a theoretical
justification for these algorithms. Blind equalization
is also known as a self-recovering equalization
[4][13][14], and the objective of blind equalization
is to recover the unknown input sequence from the
unknown channel based on the probability roles and
statistical properties of the input sequence to the
adaptive equalizer, x(n) as shown in Fig. 2. The
receiver can synchronize to the received signal x(n)
and to adjust the equalizer filter weights w(n)
without the training sequence. The term blind is
used in this equalizer because it performs the
equalization on the data without a reference signal
d(n). Instead, the blind equalizer relies on
knowledge of a signal structure and its statistical
properties to perform the equalization, the error
signal e(n), between the desired and the equalizer
output signals, can be used by the adaptive
algorithm to update the equalizer filter weights from
w(n) to w(n+1) by different approaches until reach
the minimum error. The performance of the blind
equalization depends on the characteristics of the
input signal to the channel and the characteristics of
that channel [15].
Many communication signals, such as FM, PM,
BPSK, and some QAM signals, have the constant
modulus (CM) property, which means the amplitude
of the signal is constant after these modulation
schemes are applied, the blind equalization for these
techniques has been widely used [9][13][14].
However, this CM property is changed after the
transmitted signals are corrupted by the channel
effects, noise and interference. The CM property
can be restored by applying some adaptive
algorithms; the source signal can be detected from
the channel effects which do not have CM property
even without the help of training sequences, these
algorithms are generally called Constant Modulus
Algorithms (CMA), which is introduced by Godard
[1]. So the CMA is used when the transmitted signal
possesses the constant envelope property, and the
distortion developed to its envelope is used as the
measure of a criterion to be minimized.
However, the classical CMA is found to be
ineffective for some applications where a fast
convergence is needed (such as radio-mobile
system) and because the Recursive Least Square
(RLS) algorithm is well-known that it has a very
fast convergence rate [15], so our proposition is to
Channel
Transmitter
Receiver
Adaptive
Equalizer
Decision
Device
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Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
apply a similar method in order to solve this
phenomenon of the CMA.
4 Constant Modulus Algorithm
(CMA)
The Constant Modulus Algorithm (CMA) is a
special case of Godard algorithm which is a steepest
descent algorithm with no training period is present
[1]. We can summarize the Godard algorithm
equations as the following.
The cost function CM
Jfor the equalization
process can be calculated by:
]
2
)
2
)
([()( P
kyEk
CM
J
α
−= (1)
where E(.) is the expectation operator, y(k) is the
filter output and P
α
is called a dispersion constant
which is a positive real constant.
The P
α
can be obtained from:
))((
)
2
)
((
p
kIE
p
kIE
P=
α
(2)
where I(k) is the decision output from the decision
device. This cost function’s optimization results in
the filter coefficients update which equalize only the
symbol amplitude, without depending on the carrier
phase, and also it is differentiated to the derivative
at which an LMS type algorithm is obtained. The
resulting update equation for the equalizer
coefficients is as below
)(
2
)
()()1( p
p
k
y
p
k
y
k
y
ikxk
i
wk
i
w
αµ
−
−
−+=+ (3)
where µ is a suitable step-size, wi(k) is the ith tap of
the filter at time k, x(k-i) is the input at time (k-i), p
is a positive integer.
The algorithm must stop adaptation when perfect
equalization is achieved, so the constant P
α
’s value
results in the gradient of the cost function to be
equal to zero, when y(k)=I(k).
The error term for the algorithm is:
)(
2
)
(p
p
k
y
p
k
y
k
yke
α
−
−
= (4)
with this error term the CMA algorithm uses the
LMS algorithm to update the coefficients, so the
resulting update equation for the equalizer
coefficients is as below
)()()()1( keikxk
i
wk
i
w−+=+
µ
(5)
where µ is a suitable step-size, wi(k) is the ith tap of
the filter at time k, x(k-i) is the input at time (k-i).
In the case which p=2, the algorithm is called
constant modulus algorithm (CMA).
5 Recursive Cross Correlation Based
Method for CMA (RCC-CMA)
The Recursive Cross Correlation Constant
Modulus Algorithm (RCC-CMA) relies on the proof
work in [16], which the CMA equalizer can be a
version of a Minimum Mean Squared Error
(MMSE) equalizer. By using the link between
CMA and MMSE equalizers, the different
transmission delays can be approximated by the
CMA equalizer output, and then be found by the
RLS algorithm [17].
The MMSE equalizer weights can be
approximated by
i
w
i
xs
R
xx
R
m
i
w≈
−
=1 (6)
The autocorrelation matrix of the equalizer input
vector [4][17]
))()(( k
T
XkXE
xx
R= (7)
The cross correlation vector between the equalizer
input vector and the transmitted signal at delay i
[4][17].
))()(( ikskXE
i
xs
R−= (8)
The estimated autocorrelation and cross
correlation matrices can be written as [4].
WSEAS TRANSACTIONS on SIGNAL PROCESSING
Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
Wi
m and Wi respectively denote the MMSE and
CMA equalizers at delay i. Assuming that the CMA
equalizer retrieves the transmission signal with
delay i, we form another equalizer such that its
output does not have any contribution from s(k-i).
Based upon the above relation in (6), we suggest the
RLS algorithm for finding the equalizer
corresponding to a different delay. The estimated
autocorrelation and cross correlation matrices can be
written as [4]
∑
=
−
=k
mm
T
xmx
ik
k
xx
R1)()()(
λ
(9)
∑
=−
−
=k
mimsmx
ik
k
i
xs
R1)()()(
λ
(10)
where λ is a forgetting factor, and isolating the
term corresponding to n=k from the rest of the
summation on the right hand side of (9) and (10),
)()()1()( k
T
XkXk
xx
Rk
xx
R+−=
λ
(11)
)
()()1()( ikskXk
i
xs
R
k
i
xs
R−+
−=
λ
(12)
Let P(k)=RXX(k)-1 and using the matrix inversion
lemma with (11) and (12), we can write
)1()()(
1
)1
(
1
)( −
−
−−
−
=k
Pk
T
xkKkPkP
λλ
(13)
where
)()1()(
1
-
1
)()1(
1-
)(
kxkPk
T
x
kxkP
kK
−+
−
=
λ
λ
(14)
The update weight vector )(kw equation can be
)()()
1
()( kekKkwkw +−= (15)
6 Exponentially Weighted Step-size
RCC-CMA (EXP-RCC-CMA)
In this section, we propose the Exponentially
Weighted version of the RCC-CMA, (EXP-RCC-
CMA) algorithm, which depends on the estimation
method of the inverse exponentially weighted
correlation matrix.
The proposed technique in this paper relies on the
work done in [18][19][20] by Y. Chen and et. al. for
the RCC-CMA algorithm, [17] K.Skowratananont
and D.Ratanapanich, [21][22][23][24] by S. Makino
and Y. Kaneda for the EXP-RLS algorithm, and by
using the link between CMA and MMSE equalizers
proved in Zeng’s work [29], we can develop the
proposed algorithm (EXP-RCC-CMA), where the
different transmission delays that can be
approximated by the CMA equalizer output, can be
treated by the MMSE equalizer that uses the (EXP-
RLS) algorithm.
We can summarize the algorithm as the
following, as seen in the previous section the
autocorrelation matrix of the equalizer input vector
))()(( k
T
XkXE
xx
R= (16)
The cross correlation vector between the equalizer
input vector and the transmitted signal at delay i
))()(( ikskXE
i
xs
R−= (17)
The estimated autocorrelation and cross correlation
matrices can be written as [1].
)()()1
()( k
T
XkXk
xx
Rk
xx
R+−=
λ
(18)
)()()1()( ikskXk
i
xs
R
k
i
xs
R−+
−=
λ
(19)
where λ is a forgetting factor, and isolating the term
corresponding to n=k from the rest of the
summation on the right hand side of equation (18)
and equation (19), and using the matrix inversion
lemma so P(k) can be written as
WSEAS TRANSACTIONS on SIGNAL PROCESSING
Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
)1()()(
1
)
1(
1
)( −
−
−−
−
=k
Pk
T
xkKkPkP
λλ
(20)
To get faster convergence speed and minimum
Mean Square Error (MSE), the EXP-RCC-CMA
algorithm uses the inverse of the exponentially
weighted correlation matrix estimation seen in
[19][20][26].
So referring to the work done in [21][22][27][28],
each element of the impulse response variation
∆w(k) is assumed to be a statistically independent
random variable, so the covariance matrix Q(k) of
the variation ∆w(k) becomes a diagonal matrix,
where the diagonal components are the E[∆wi(k)].
The w0(k), which represents the magnitude of the
variation, is assumed to take time-invariant value
w0, and based on these assumptions, we set Q(k) as
=
L
w
w
w
kQ
......00
0.......
2
0
0......0
1
)(
(21)
where
),......,1(
1
0L
i
i
w
i
w=
−
=
γ
(22)
γ
: exponential attenuation ratio of the impulse
response (0 <
γ
≤ 1). Elements wi are time-
invariant and decrease exponentially from w1 to wL
by the same ratio
γ
as the impulse response w(k).
We will define the matrix )(kP
kwhich is a priori
coefficient error covariance matrix,
])}(
^
)()}{(
^
)([{)( T
kwkwkwkwEk
k
P−−=
(23)
where )(kw is the filter coefficient vector, )(
^kw is
the updated filter coefficient vector, and E[.] is the
statistical expectation. Also we will define the
matrix )(kR which is a power of the ambient noise
n(k),
]
2
)([)( knEkR =
(24)
and the vector K(k) is the Lth order Kalman gain
vector,
)()()()(
)
()(
)(
kxk
k
P
T
kxkR
kxk
k
P
kK
+
=
(25)
assuming that the ambient noise n(k) is stationary
[R(k) ≡ R], we introduce ES
P(k) by multiplying the
a priori coefficient error covariance matrix k
P(k) of
the Kalman filter by (l/R).
)()( k
ES
PRk
k
P×
= (26)
)()()1
( kQk
k
Pk
k
P+
=+ (27)
substituting (27) into (26), using R(k) ≡ R, and by
using (21), (24), and (26) into (27), we get the
following EXP-RLS-CMA algorithm [12][13].
T
k
xkxkQkQ
knkx
T
kwkyke
R
kQ
k
ES
P
T
kxkkk
ES
Pk
ES
P
kxk
ES
P
T
kx
kxk
ES
P
kK
kekKkwkw
)()()()1(
)()()(
^
)()(
)(
)()()()()1(
)()()(1
)()(
)(
)()()(
^
)1(
^
+=+
+−=
+−=+
+
=
+=+
λ
(28)
(29)
(30)
(31)
(32)
where
λ is the forgetting factor
)(kP
ES : L×L matrix
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Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
The equation (31) is obtained from the estimate of
the exponentially weighted correlation matrix Q(k)
by using the matrix inversion lemma. Elements
}
{
L
ww
w.....,,........., 21 of the Q(k) matrix are not
really step-sizes like in the conventional NLMS or
the projection algorithms. However these elements
function as if they were step-sizes, and from the
relationship between the previously proposed ES
algorithm [23] and ESP algorithm [24], we call
matrix Q(k) a step-size matrix.
On the other hand, the step-size is known to be
related to the forgetting factor λ of the RLS
algorithm. In fact, according to (30), when the value
(Q(k)/R) is large compared to ES
P(k), the
proportion of ES
P(k) in ES
P(k+l) becomes small. In
other words, old information is forgotten quickly.
The covariance matrix Q(k) of the impulse
response variation is added in (27) and (30),
according to that the exponentially attenuating bias
is always added in diagonal elements of the matrix
ES
P(k), as a result, the gain vector K(k) attenuates
exponentially in (29), the filter coefficient vector
w(k) is adjusted by the exponentially attenuating
adjustment vector in (28).
7 Results and Simulations
In this section we evaluate the performance of the
proposed EXP-RCC-CMA algorithm, where the
main parameters of concern are the rate of
convergence, the mean square error (MSE), and the
average error in dB with different signal to noise
ratios, compared with the CMA and RCC-CMA
algorithms. In the simulations the transmitted signal
s(n) is a QPSK symbol sequence; the different
samples are coded with binary sequences taking the
two possible values –1 and 1 (s(n) is a sequence of –
1s and 1s). The signal to noise ratio of the
transmitted sequence SNR=20 dB, the channel is
modelled with a FIR filter of third order and the
equalizer is realized as a FIR adaptive filter of third
order. The CMA, RCC-CMA, and EXP-RCC-CMA
algorithms are implemented according to the steps
presented in sections 4, 5, and 6. For CMA
algorithm µ= 0.08, RCC-CMA algorithm λ = 0.99,
and for EXP-RCC-CMA algorithms λ = 0.99,
γ
=
0.95.
First we will examine the performance of the
three algorithms, according to the average error in
dB, and the simulation results as the following:
Algorithm Average Error in
Db
CMA -20.3964
RCC-CMA -32.1122
EXP-RCC-CMA -39.1067
Table 1. EXP-RCC-CMA versus CMA and RCC-
CMA.
From Table 1, Fig.3 and Fig.4 we can see that
EXP-RCC-CMA has the fastest convergence rate
compared with the other algorithms, and also has
the best performance by achieving the highest
average error in dB for different values of signal to
noise ratio (SNR), and its MSE is much lower than
the CMA algorithm.
0 100 200 300 400 500 600 700 800 900 1000
10
-6
10
-5
10
-4
10
-3
10
-2
CMA,EXP-RCC-CMA,RCC-CMA MSE in dB
Number of Iterations
MSE in dB
CMA
RCC-CMA
EXP-RCC-CMA
Fig.2. EXP-RCC-CMA versus RCC-CMA and
CMA Mean Square Error (MSE).
WSEAS TRANSACTIONS on SIGNAL PROCESSING
Amin Mohamed Nassar, Waleed El Nahal
ISSN: 1790-5052
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Issue 2, Volume 6, April 2010
0 5 10 15 20 25 30 35 40
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
CMA,EXP-RCC-CMA, RCC-CMA error average dB
SNRdB
Av erage E rror in dB
CMA
RCC-CMA
EXP-RCC-CMA
Fig.3. EXP-RCC-CMA versus RCC-CMA and
CMA Average Error with SNR.
Fig. 5 shows the comparison of the three
algorithms with respect to the absolute error, and
from it we can see that the EXP-RCC-CMA
algorithm provides better performance than the
other algorithms. The maximum value of the
absolute error of the EXP-RCC-CMA algorithm not
exceed 0.5 V, while the values of the RCC-CMA
and the CMA algorithms exceed 2 V.
The absolute error of the EXP-RCC-CMA
algorithm is settled after a few samples not exceed
10 samples, but for the RCC-CMA algorithm after
60 samples and the CMA algorithm after 80
samples. Those results lead fastest convergence rate
and minimum absolute error of the EXP-RLS-CMA
algorithm rather than the other algorithms.
8 Conclusion
This paper introduces a new blind equalization
technique, the EXP-RLS-CMA algorithm, which is
simulated and tested for noise minimization in
QPSK signal or 4 QAM symbol sequences, the
obtained results show that the EXP-RLS-CMA
algorithm is very promising, and has an improved
performance by providing a double convergence
speed compared to the conventional CMA, and the
RLS-CMA algorithms.
The main advantages of the proposed algorithm
could be summarized as follows, sufficient
robustness in fixed-point applications, minimum
mean square error (MSE) as shown from Fig.3,
minimum average error with lower signal to noise
ratios (SNR) as shown from Fig.4, and minimum
absolute error and high adaptation rate as shown
from Fig.5. Our previous work showed that the
variation of a channel impulse response becomes
progressively smaller by the same exponential ratio
as the impulse response.
0 20 40 60 80 100 120 140 160 180 200
0
1
2
CMA Absolute Error
Number of Iterations
A b s o l u t e E r ro r
0 20 40 60 80 100 120 140 160 180 200
0
1
2
RCC-CMA Absolute Error
Number of Iterations
A b s o l u t e E r r o r
0 20 40 60 80 100 120 140 160 180 200
0
1
2
EXP-RCC-CMA Absolute Error
Number of Iterations
A b s o l u t e E r r o r
Fig.5. Absolute error simulation results.
References:
[1] D. N. Godard, “Self-Recovering
Equalization and Carrier Tracking in a Two-
Dimensional Data Communication System”,
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