MIMO系统的恒模算法CMA

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1 SISO Fading Channel
Signal FIR channel model: x ( t ) = ∑ hk s ( t − k )
k =0 K
(1-1)
Where s ( t ) is transmitted signal, x ( t ) is received signal and hk is channel impulse response. The FIR channel cause ISI in the x ( t ) , to decode s ( t ) correctly, a equalizer is often required. In the receiver, the output of an equalizer is given by: y ( t ) = ∑ wl x ( t − l ) It is desired that y ( t ) = s ( t − t0 ) , where t0 is an unknown constant. The wl can be found by the minimize of following cost function [2]:
2
(1-3)
J CM It can be seen that if s ( t ) is CM and wl is the solution of a zero forcing equalizer,
∑ wl x ( t − l ) = s ( t − t0 )
l =0
L
(1-4)
Then the J CM will reach its global minimum, i.e. J CM = 0
1
chre 1.
s(t )
h1(z)
x1(t)
w1(z)
h2(z)
x2(t)
w2(z)
Σ
y(t )
hK(z)
xK(t)
wK(z)
CAM
Figure 1. Multi-channel representation of FSE Where hk ( z ) = ∑ hk ,l z
w io
(2-4)
ˆi ( t ) = wT s i x (t )
(2-5)
There are two problems 1 For each w io , one can not make sure which s j ( t ) ( j = 1, 2," , M ) is recovered 2 To recover all si ( t ) for i = 1, 2," , M one have to try all different w . Since J CM is usually minimized by iteration, the selection of the initial value w i0 determines which s j ( t ) it will finally recovered 3 J CM is not convex, so it is possible the selection of w i won’t converge to desired signal
y ( t ) = as1 ( t ) + bs2 ( t )
(2-10)
The modulus of y ( t ) is found to be: | y ( t ) |2 =| as1 ( t ) + bs2 ( t ) |2 =| ae
jθ p
+ be
jφq 2
|
( ) ( ) =| ( a cos θ p + b cos φq ) + j ( a sin θ p + b sin φq ) |2
s1 ( t ) = e
jθ p
, p = 1, 2," , P , q = 1, 2," , Q
(2-7) (2-8)
and
s2 ( t ) = e
jφ p
Let the system is recovered by a constant modulus criterion, i.e. minimize the cost function:
(
)
(
)
(
)
(2-11) Note that if | y ( t ) | kept constant ∀p, q , then | θ p − φq | must keep constant ∀p, q . One example is:
s1 = e ± jπ
(2-12)
and
4
Where | θ p − φq |=
l =0 Lh −l
and wk = ∑ wk ,l z −l . The Lh and Lw are the length of
l =0
Lw
channel and equalizer, respectively. The FSE-CAM cost function is:
J FSE −CAM = Ex | y |2 −γ
T is symbol period and Ts is sample period. There are K samples in each symbol period, xi ( i = 1, 2," , K ), they be regarded as the output of K separated sub-
(2-6)
3
Where si ( t ) is transmitted signal, xi ( t ) is transmitted signal and H is the channel matrix. The signal si ( t ) is constant modulus signal, i.e.
{(
)}
2
(2-3)
Let w io be the optimal combination vector that minimize J CM , i.e.
2⎫ ⎧ 2 ⎪⎛ T ⎪ ⎞ = arg min Ex ⎨⎜ w i x ( t ) − γ ⎟ ⎬ wi ⎠ ⎭ ⎪⎝ ⎪ ⎩ Then the si ( t ) is recovered by:
{(
)}
2
(1-6)
Where
y ( t ) = ∑ hk ( z ) wk ( z ) x ( t )
k =1 K
(1-7)
It is proved in [3] that suppose the system is noiseless, if the length of equalizers are chosen such that ( Lw + 1) ( K − 1) ≥ Lh , then FSE-CAM is globally convergent if the
T
n (t ) = ⎡ ⎣ n1 ( t ) n2 ( t ) " nM ( t ) ⎤ ⎦ is AWGN The receiver tries to recover si ( t ) from the received vector x ( t ) by linear
T
combination, i.e.
J CM = Ex | y ( t ) |2 −γ
{(
)}
2
(2-9)
Where y ( t ) = w1 x1 ( t ) + w2 x2 ( t ) J CM = 0 means that y has constant modulus. We will see whether y is the mixture of s1 and s2 when it has constant modulus. The y ( t ) can be expressed by linear combination of si ( t ) since xi ( t ) is the linear combination of si ( t ) , i.e.
The first problem is common among blind recovery problem. While the second problem can be solved by modified cost function or by subtracting recovered data one by one from x ( t ) . The modified cost function method will be discussed in next section. A simple example of MIMO CMA Consider a simple MIMO model ⎛ x1 ( t ) ⎞ ⎛ s1 ( t ) ⎞ ⎜ ⎟ = H⎜ ⎟ ⎝ x2 ( t ) ⎠ ⎝ s2 ( t ) ⎠
J CM = Ex | y ( t ) |2 −γ
l =0 L
(1-2)
Where γ =
{ } ensures that equalization solution is a stationary point of the Ex {| s ( t ) | }
Ex | s ( t ) |4
2
{(
)}
K sub-channels satisfy the “length and zero” condition, i.e. 1 hk ,0 ≠ 0 for some 1 ≤ k ≤ K
2 3
hk , Lh ≠ 0 for some 1 ≤ k ≤ K
{hk ( z )}1