1
Basic of Constant Modulus Algorithm
(CMA)
uingrd@lycos.com
1 SISO Fading Channel
Signal FIR channel model:
( ) ( )
0
K
k
k
x t h s t k
=
= −
∑
(1-1)
Where
(
)
s t
is transmitted signal,
(
)
x t
is received signal and
k
h
is channel impulse
response.
The FIR channel cause ISI in the
(
)
x t
, to decode
(
)
s t
correctly, an equalizer is often
required. In the receiver, the output of an equalizer is given by:
( ) ( )
0
L
l
l
y t w x t l
=
= −
∑
(1-2)
It is desired that
(
)
(
)
0
y t s t t
= −
, where
0
t
is an unknown constant.
The
l
w
can be found by the minimization of following cost function [2]:
( )
( )
{
}
2
2
| |
CM
J Ex y t
γ
= − (1-3)
Where
(
)
{
}
( )
{ }
4
2
| |
| |
Ex s t
Ex s t
γ
=
ensures that equalization solution is a stationary point of the
CM
J
It can be seen that if
(
)
s t
is CM (Constant Modulus) and
l
w
is the solution of a zero
forcing equalizer,
( )
( )
0
0
L
l
l
w x t l s t t
=
− = −
∑
(1-4)
Then the
CM
J
will reach its global minimum, i.e.
0
CM
J
=
(1-5)
The receiver equalization in (1-2) is T-spaced equalization, which is reported to
show some local convergence problem[4], while a fractionally space equalizer (FSE)
methods is shown to be able to achieve global convergence under certain condition [3].
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