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Mixed-Norm Constant Modulus Algorithm for Channel Equalization of
Wireless Communication Systems
JUSAK JUSAK and ZAHIR M. HUSSAIN
Centre For Advanced Technologies in Telecommunications (CATT)
School of Electrical and Computer Engineering, RMIT University
Melbourne, Victoria, AUSTRALIA, 3001
Abstract: - Constant Modulus algorithm (CMA) has been widely used for blind channel equalization. The slow
convergence speed of CMA has motivated us to propose a new method of updating filter coefficients based upon
a mixing of Constant Modulus 2-2 (CM 2-2) and Constant Modulus 1-1 (CM 1-1) cost functions. It is shown that
the proposed algorithm satisfies the Benveniste-Goursat-Ruget Theorem. Hence, under perfect blind equalization
conditions, it guarantees that searching by the stochastic gradient descent method will lead to its minimum point.
The performance of the algorithm is demonstrated in channel equalization of wireless communication systems.
Key-Words: - Mixed-Norm, equalization, adaptive algorithm.
1 Introduction
High-speed data communication over a bandlimited
channel is subject to Inter-Symbol Interference (ISI)
as a result of transmitter receiver filtering and multi-
path propagation. Mitigation of such kind of
distortion calls for the use of equalization filter. In the
last few years, blind equalization techniques have
gained an increasing interest. The most popular and
implemented blind adaptation algorithm is the
constant modulus algorithm (CMA).
The need of achieving low complexity
computation of CMA leads to the sign-error
implementation of the algorithm, which is proposed
in [1]. Despite the advantage of computational
reduction, it has been shown that Signed-Error
Constant Modulus Algorithm (SE-CMA) is lack
robustness properties of CMA. Recent studies
associated with the fractionally-spaced equalization
constant modulus algorithm have developed the
dithered signed-error constant modulus algorithm [2],
where the input signal to the equalizer was dithered
by a non-subtractive random process with i.i.d
samples uniformly distributed over (-1, 1] before a
sign operation. The drawback of DSE-CMA as
compared to original form of CMA is the increment
of excess mean-squared error (EMSE). Our studies in
[3] proved that convergence rate of the DSE-CMA
can be increased significantly by using a variable
step-size instead of using a fixed step size.
In this paper, we shall investigate and show that
performance of the filters can also be enhanced by
combining the cost function of CMA and SE-CMA
algorithms. The motivation is that for identical initial
value of the equalizer coefficients and small step-size,
CMA generally provides more accurate global
minima points with small miss-adjustment, but it is
slow in its convergence rate; whereas SE-CMA
converges very fast, but is less accurate with higher
miss-adjustment. We examined the systems based
upon M-PSK modulation schemes only.
2 System Model
For space limitations we focus on the case of
equalizing communication systems with an FIR
channel model and fractionally-spaced filters with
sampling interval
2/T
(
T
being the symbol period).
The source symbol (
n
x
) is drawn from a finite
alphabet; it is a random variable with zero mean,
independent and identically distributed (i.i.d) with
variance
2
2nx xE
, baud-spaced at sample index
n
, while the fractionally-spaced sample is denoted
using the sample index
k
. The vector
t
Nc
cc c 10
is an
1
c
N
vector representing the fractionally-
spaced channel impulse response and
t
Nf
ff
f10
is a fractionally-spaced equalizer coefficient vector of
size
1
f
N
. Transpose operator is represented as
t
.
.
The number of channel and equalizer coefficients is
denoted as
c
N
and
f
N
, respectively. The
fractionally-spaced received signal is denoted as
k
r
.
t
Nnnn x
xxxn x 11 ,,,
is the original source
symbol, which is a vector of length
2/1 fcx NNN
with baud-spaced source
symbols.
The output of the multirate systems may be
expressed as:
Cfxny t
n
. (1)
Defining
C
as the time decimated channel
convolution matrix of
fx NN
, it can be denoted as:
.C
21
21
0121
01
01
cc
cc
cc
NN
NN
NN
cc
cc
cccc
cc
cc
(2)
Under a perfect blind equalization (PBE)
condition, equalizers minimizing the CM cost
function can perfectly recover the original source
symbols for some system delay
(i.e.,
nn xy
).
Requirements for perfect symbol recovery conditions
are stated as follows: In the
2/T
-spaced fractionally-
spaced and a channel with even-length, a necessary
condition for channel invertibility is that
2 cf NN
. No additive channel noise. For the
even-length
2/T
-spaced channel impulse response,
1
even
zc
and
1
odd
zc
have no common roots. Sub-
Gaussian source. I.i.d. zero mean source (white
source).
3 Mixed-Norm CMA Algorithm
The cost function that is minimized by the stochastic
gradient descent method is a linear mixture of CM 2-
2 and CM 1-1 in the form of the following equation:
,1
4
12
2
nn yEyEJ
(3)
where
is a mixing parameter, which satisfies
1,0
. It can be seen in eq. (3), when
1
, eq. (3)
becomes a CM 2-2 for the CMA algorithm, whereas
0
, eq. (3) refers to CM 1-1 for the signed-error
CMA algorithm.
The error function of the above cost function is
obtained by the following equation:
.sign1 22 nnnn
yn
yyyy
Jy n
(4)
The equalizer coefficients are updated according
to the following algorithm:
,)()(1 n
ynnn
r ff
(5)
where
is a constant called the step-size (usually
small). The dispersion constant,
, is defined as
follows
][][ 24 nxEnxE
.
In order to prove that the proposed algorithm
satisfies the Benveniste-Goursat-Ruget (BGR)
Theorem in [4], we seek the second derivative of the
error function as follows:
. 6''
02' 2
nn
nnn
yy
yyy
(6)
For
a positive number and
1,0
, it can be
seen that the proposed algorithm will obey the BGR
Theorem according to the following restrictions:
,0for ,0 ''
and ,0 0
nn yy
(7)
where at least one of the two inequalities must be
strict. We conclude that the Mixed-Norm CMA
algorithm will converge to its minimum point under
the PBE conditions.
4 Simulation Results
In the first simulation, we used a well-behaved
channel with the following channel impulse response
2.0,0.1,6.0,3.0 c
. As the channel impulse
response has length
4
c
N
, then, according to the
PBE condition, the minimum length of the
fractionally spaced equalizer should be
2
f
N
. The
fixed step-size has been set up to
3
105
. The
constant
was set to 0.1 and 0.8 for Fig. 1 and Fig.
2, respectively, while 16-PSK (
1
) modulation was
used to modulate the transmitted signal. It can be
noticed in Fig. 1, for mixing parameter
1.0
the
surface contour of the Mixed-Norm CMA resembles
the Sign-Error CMA, where its local minima points
are away from MSE minima points. This deformation
apparently can be remedied by increasing the value of
the mixing parameter as in Fig. 2. In this example we
used
8.0
. It can also be seen in the two figures
the trajectories of the algorithm initialised at point
2.1,8.0
, where mixed-norm trajectory for
1.0
in Fig. 1 exhibits jitter.
The second simulation is implementation of the
Mixed-Norm CMA algorithm to equalize the wireless
channel, modelled using the hyperbolic channel
model proposed in [5]. The impulse response of the
channel has been produced to satisfy the condition of
urban area environments (with angle spread
0
20
)
with
62
c
N
. The relevant parameters of the channel
model were set as follows: distance between a mobile
and base station was set to 1000 m, distance of local
scatterer was 50 m, distance of dominant scatterer
was 600 m, and path loss exponent was set to 3. The
length of the fractionally spaced equalizer was set to
31
f
N
, while the fixed step-size of the adaptation
algorithm was set to
5
101
. The mixing parameter
was set to
9.0
. Signal transmission was achieved
with 32-PSK (
1
) modulation. Two different
noise conditions were considered, SNR= 50 dB and
SNR = 25 dB. It can be seen in Fig. 3 that the mixed-
norm CMA exhibits the best performance of all. For
an equalization of wireless communication system
with SNR=50dB, the mixed-norm CMA converges
faster as compared to CMA and SE-CMA. Moreover,
it gives the lowest MSE (all graphs are averaged over
100 realizations). On the other hand, for a system
with SNR=25 dB, the mixed-norm CMA provides the
same MSE as CMA, but it achieves a distinguishably
faster convergence speed.
5 Conclusion
In this paper we proposed a mixed-norm constant
modulus algorithm. Performance of the algorithm is
affected by careful choice of the mixing parameter
. Simulation results showed that the mixed-norm
CMA exhibits the fastest convergence rate as
compared to the existing CMA and SE-CMA when
M-PSK signal modulation is used.
References:
[1] D. R. Brown, P. B. Schniter and C. R. Johnson,
Jr., Computationally Efficient Blind
equalization, Proc. 35th Allerton Conf. on
Communications, Control, and Computing
(Monticello, IL), Jan. 1997, pp. 54-66.
[2] P. Schniter and C. R. Johnson, Jr., Dithered
signed-error: robust, computationally efficient
blind adaptive equalization, IEEE Trans. Signal
Processing, vol. 47, no. 6, Jun. 1999, pp. 1592-
1603.
[3] J. Jusak, Z. M. Hussain and R. J. Harris,
Performance of Variable Step-Size Dithered
Signed-Error CMAs for Adaptive Blind
Equalization, Proc. IEEE TENCON 2004, Nov.
2004.
[4] A. Benveniste, M. Goursat and G. Ruget, Robust
Identification of a Non-minimum Phase System:
Blind Adjustment of a Linear Equalizer in Data
Communications, IEEE Trans. Automatic
Control, vol. AC-25, no. 3, Jun. 1980, pp. 385-
399.
[5] S. S. Mahmoud and Z. M. Hussain and P. O'Shea,
A space-time model for mobile radio channel
with hyperbolically distributed scatterers, IEEE
Antennas and Wireless Prop. Letters, vol. 1,
Dec. 2002, pp. 211-214.
Fig. 1. Mixed-Norm CMA trajectories for 16-PSK
signal with
1.0
. Global MSE minima marked by
"*", local MSE minima marked by "+", Mixed-Norm
CMA minima marked by "o".
Fig. 2. Mixed-Norm CMA trajectories for 16-PSK
signal with
8.0
. Global MSE minima marked by
"*", local MSE minima marked by "+", Mixed-Norm
CMA minima marked by "o".
Fig. 3. Averaged MSE trajectories for Mixed-Norm
CMA, modulated by 32-PSK signals with
9.0
.
