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here, αm and β m represents the multiplicative distortion at the
desired subcarrier and the ICI, respectively. If the channel keeps
invariant during a block period, β m will be zero, which means no
ICI.
In the general case where the multipath cannot be regarded as
time-invariant during a block period, (7) can be expressed in
vector form as:
Y = H X + W
(9)
where Y=[Y 0,…,Y N-1] T , X=[X 0,…,X N-1] T , W=[W 0,…,W N-1] T , and
⎡a0,0 a0,1La0, N −1
⎤
⎢
⎥
a1,0 a1,1 a1,
N −1
H
⎢ L
=
⎥
⎢M O M ⎥
⎢ ⎥
⎢a a a ⎥
⎣ N−1,0 N−1,1 N−1, N−1⎦
here, a m,k in (10) is defined as
( m−k) − j2 π k( L−1)/ N
+ HL1 e ,0 ≤(
m, k) ≤ N −1.
* * * *
= + (14)
⎤
⎦
(10)
( m−k) ( m−k) − j2 π k/ N
amk , = H0 + H1 e + K (11)
−
In this paper, we assume the multipath fading channel is
slowly-varying, so most energy concentrate in the dominant
adjacent subcarriers, the ICI terms which do not significantly
affect Y m in (10) can be ignored. And then by transforming the
matrix H of order N× N to a block-diagonal matrix H * of order
×
(N-q)(q+1) (N-q)(q+1). We obtain:
where,
⎡Aq/2
0 ⎤
⎢
⎥
A
*
q /2+ 1
H =
⎢ ⎥
⎢ O ⎥
⎢ ⎥
⎢⎣0AN−− 1 q/2⎥⎦
⎛an−q/2, n−q/2 L an−q/2,
n L
0 ⎞
⎜ ⎟
⎜an− q/2+ 1, n−q/2 ⎟
⎜ M M ⎟
⎜ ⎟
⎜ann , −q/2
O
⎟
An
= ⎜ ⎟
⎜ O
an+
1, n+ q/
2 ⎟
⎜ M M ⎟
⎜ ⎟
⎜ an+
q/2− 1, n+ q/2⎟
⎜
⎟
0
an+ q/2, n a ⎟
⎝
L L n+ q/2, n+ q/2
⎠
(12)
(13)
We can locate some of the unused subcarriers at the first q/2 and
the last q/2 IFFT grids to get the estimation of all symbols. Then,
the input-output relationship of the multipath channel is expressed
in a vector form as:
where,
Y H X W
*
T
X ⎡xq/2
xq/2+ 1 L xN−1
−q/2
= ⎣
xn = ⎡
⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦
*
T
Y ⎡yq/2 yq/2+ 1 L y ⎤ N−1 −q/2
= ⎣ ⎦
yn = ⎡
⎣Xn−q/2 Xn− q/2+ 1 L X ⎤ n+ q/2⎦
*
T
W ⎡wq/2 wq/2+ 1 L w ⎤ N−1 −q/2
= ⎣ ⎦
wn = ⎡
⎣Wn−q/2 Wn− q/2+ 1 L W ⎤ n+ q/2⎦
T
T
T
(15)
V. FREQUENCY DOMAIN EQUALIZER
A. Zero-Forcing Equalizer
Zero-Forcing equalization compensates both multiplicative
distortion and ICI by multiplying the inverse of , which is the
*
estimated value of H , to (14). The resulting signal can be
expressed as:
* 1 * − % % (16)
X = H Y
where X % is the estimated signal, and
H%
* −1
−1
⎡ A%
0 0 ⎤
⎢ ⎥
−1
⎢ A%
1 ⎥
=
⎢ O ⎥
⎢ ⎥
−1
⎢0 A% ⎣
⎥ N−− 1 q⎦
Also, (16) can be rewritten as
1
X% −
= A% y , q/2≤ n≤ N −1
−q/
2.
(18)
n n n
Finally, the transmitted symbols are estimated by selecting
the elements in the middle of X .
n
%
B. MMSE Equalizer
H %
*
(17)
Using the fact that the ICI power is localized to the dominant
adjacent subcarriers, only a few neighborhood subcarriers can be
used for equalization without much performance penalty. We find
the solution for each desired subcarrier individually[4]. The
problem of MMSE is to find the equalizer coefficient vector
g m=[g m,0,…,g m,q-1] which minimizes the mean-squared error
2
ˆ { m − m }
E X X
where ˆ T
X m = gmy and , y
m y ⎡Y , Y ⎤
where Hm=A m.
The MMSE solution is
m ( m− q/2) L m is then
( m+ q/2)
= ⎣ ⎦
ym = Hmx+ w (19)
m
g R R −
= (20)
1
m X ,
mym ym
assuming H is known, x is a zero-mean i.i.d. random vector with
variance
2
σ x , and w is an AWGN vector with variance
{ } 2
H H
X mym m m
2
σ w ,then
R = E X y = σ xh (21)
m
where hm is the q/2 th column of the matrix H m, i.e.,
Also, we can get
hm = ⎡
⎣am− q/2, q/2 , L , a ⎤
(22)
m+ q/2, q/2⎦
H 2 H 2
{ } σ σ
R = E y y = H H + wIq (23)
ymm m x m m
After inserting (21) and (23) into (20), the q-tap equalizer vector
g m becomes
T