Channel Estimation and data detection for the first M

94

4.3.2. Channel Estimation and data detection for the first M

1
OFDM symbols of each block
Since the first M
1
OFDM symbols adopt the comb-type pilot pattern, we can obtain the LS channel estimate at pilot subcarriers in time domain for the i-th OFDM
symbol,
.
,
p ls
h i n and it is given by
1
1 2
. ,
1 1
1
1 ,
, ,
0,1,..., 1,
0,1,..., 1.
P
N j
n mR k N
p ls p ls
P m
P
h i n
H i mR k e
n N
i M
N
π
− +
=
= +
= −
= −
∑
4-5 where
,
, ,
, ,
,
p ls
Y i k H
i k k
X i k β
= ∈
1
{ | ,
0,1,..., 1}
P
k k mR k m N
β = =
+ =
− , , Y i k is the i-th
received OFDM symbol after fast FFT operation, , X i k is the i-th transmitted OFDM
symbol. Next, the MST algorithm [32] has been proposed to get the refined channel estimate in time domain. The MST algorithm deals with each OFDM symbol by reserving
the most significant L′ paths in terms of power and setting the other taps to be zero. The algorithm can reduce the influence of AWGN and other interference significantly,
compared with the LS method. However, the algorithm may choose the wrong paths and omit the right paths because of the AWGN and other interference. Thus, we will improve
the algorithm of [32] by processing several adjacent OFDM symbols jointly. Firstly, we calculate the average power of each tap for the M
1
adjacent OFDM symbols,
LS
P n and it is given by
1
1 2
, 1
1
1 |
, | , 0,1,...,
1.
M LS
p ls P
i
P n h
i n n
N M
− =
= =
−
∑
4-6 Next, we choose the
L′ most significant taps from
LS
P n and reserve the indexes of them
95
into a set α
′
, i.e., { :
0,1,..., 1}
l
l L
α τ
′ ′
= =
− , where
l
τ is the delay of the l-th path in unit of sample point. Finally, the refined channel estimate in time domain,
, p MST
h ,is given by
, ,
1 1
, , ,
, 0,1,...,
1, 0,1,...,
1. 0,
p ls p MST
P
h i n if n
h i n
n N
i M
if k α
α ⎧
′ ∈
⎪ =
= −
= −
⎨ ′
∉ ⎪⎩
4-7 Then, we can obtain the refined channel estimate in frequency domain,
MST
H , and it is
given by
1
2 1
,
, ,
P
N j
nk N
MST p MST
n
H i k
h i n e
π
− −
=
=
∑
, 0,1,...,
1 k
N =
− ,
1
0,1,..., 1.
i M
= − 4-8
So the detected data , X i k is given by
1
, ,
, 0,1,...,
1, 0,1,...,
1. ,
MST
Y i k X i k
i M
k N
H i k
= =
− =
− 4-9
4.3.3. Channel estimation and data detection for the last M
2
OFDM symbols of each block
We use the polynomial of Q order to model the channel [29] shown as follows.
, ,
, , , ,
, , , ,
Q q
l i l q
q l
h i n a n
e i n l h i n
e i n l
τ τ
=
= +
= +
∑
4-10
where
, , i l q
a is the polynomial coefficient and , ,
e i n l is the approximation error,
, ,
, ,
Q q
l i l q
q
h i n a n
τ
=
=
∑
. It is noted that we use Taylor series of , ,
l
h i n τ around 0. Therefore
the approximation error can be expressed by
1 1
1
, , , ,
1
Q Q
Q
d h i n l
n e i n l
Q dn
+ +
+
= +
4-11
96
where [0, ]
n n
∈ . Since the ICI terms which do not significantly affect Yi, k can be
discarded [41], i.e., gi, k, j = 0 if| |
j k c
− , where 2c is the number of dominant ICI terms. We can rewrite the received signal Yi, k as follows.
1 0 ,
1 1
1 1
, , , , , ,
1 1
, , , , ,
, , , ,
, , , ,
, , , ,
, , , ,
, ,
N j
j k k c
j k c N
L N
L l
i n l k i n l k
n l
n l
Q N
L q
i l q i n l k
n l
q
Y i k X i k g i k k
g i k j X i j W i k
g i k j X i j W i k
h i n e i n l
W i k a
n W i k
τ φ φ
φ
− =
≠ +
= − −
− −
− =
= =
= −
− =
= =
= +
+ ≈
+ =
+ +
′ =
+
∑ ∑
∑ ∑ ∑ ∑
∑ ∑ ∑
4-12
where
2 2
, , ,
1 ,
l
k c j
q N
j n k q N
i n l k q k c
e e
X i q N
π τ π
φ
+ −
− −
= −
=
∑
4-13
and
1 1
, , ,
, , ,
,
N L
i n l k n
l
W i k e i n l
W i k φ
− −
= =
′ =
+
∑∑
. Equation 4-12 can be written as the vector form, shown as follows.
, 1
1 1
2
, , ,
, 1,...,
1.
i k i i
Y i k W i k i M M
M M
′ =
+ =
+ +
−
φ K a 4-14
where
, ,0,
,1, ,
1, 1
i k i
k i k
i N k
LN −
×
⎡ ⎤
= ⎣ ⎦
φ φ
φ φ
,
, , , ,0,
, ,1, , ,
1, 1
i n k i n
k i n k
i n L k
L
φ φ
φ
− ×
⎡ ⎤
= ⎣ ⎦
φ ,
,0 ,1
, 1
1 T
T T
T i
i i
i N NL Q
L −
× +
⎡ ⎤
= ⎣ ⎦
K K
K K
,
, 1
n i n
n L Q L
× +
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
θ K
0 θ
,
1 1
1
Q n
Q
n n
× +
⎡ ⎤
= ⎣ ⎦
θ ,
,0 ,1
, 1
1 1
T T
T T
i i
i i L
Q L
− +
×
⎡ ⎤
= ⎣ ⎦
a a
a a
,
, , ,0
, ,1 , ,
1 1 T
i l i l
i l i l Q
Q
a a
a
+ ×
⎡ ⎤
= ⎣ ⎦
a
,
T
i denotes transpose operation. Since the number of
the unknown parameters a
i
is Q+1×L, the number of pilot groups, N
group
, should satisfy N
group
≥ Q+1×L, where L is the number of resolvable paths. When the index k in equation
97
4-14 is chosen from the index set γ , where
{ 0, 1,..., 1}
group
P P
P N γ =
− , we can stack
the signal Yi, k k = P0, P4-1,…,PN
group
-1 together and obtain
1 1
1 2
, ,
1,..., 1.
i i i
i i i M M
M M
′ =
+ =
+ +
− Y
φ K a W
4-15 where
1
, 0 , 1
, 1
group
T group
N
i Y i P
Y i P Y i P N
×
⎡ ⎤
= −
⎣ ⎦
Y ,
, 0 , 1
, 1
group group
T T
T T
i i P
i P i P N
N LN
− ×
⎡ ⎤
= ⎣ ⎦
φ φ
φ φ
,
1
, 0 , 1
, 1
group
T group
N
i W i P
W i P W i P N
×
′ ′
′ ′
⎡ ⎤
= −
⎣ ⎦
W .
Therefore, the estimated coefficients ˆ
i
a can be given by
ˆ
i i
i
i
+
=
a φ K
Y 4-16
where
+
i denotes moore-penrose inverse operation. Then, we can obtain the estimated channel impulse response
, ,
l
h i n
τ from 4-10 and the estimated , , g i p q from 4-3.
Next, we perform data detection. Similar to the detection method in [29], we perform the following steps.
Step 1: Initialization of the detected data , X i k ,
1 1
1 2
, 1,...,
1 i M M
M M
= +
+ − .
Let
[0]
, , , , ,
0,1,..., 1
X i k
Y i k g i k k k N
= =
− , and k ζ
∉ ,where ζ is the set of indexes of the pilots in one OFDM symbol and ζ = {P0-c, P0-c+1,…,
P0+c, …, PN
group
-c, PN
group
-c+1, PN
group
+c}. Step 2: At the j-th iteration j = 1,2,…, we update
[ ]
,
j
X i k by computing
[ ] ,
[ 1]
, , ,
, ,
, ,
k c j
q k c q k j
Y i k g i k q X
i q X
i k g i k k
+ = −
≠ +
− =
∑
4-17
98
Step 3: When
1 [
1] [ ]
0, 1
[ 1]
0,
| ,
, | |
, |
N j
j k
k N
j k
k
X i k
X i k
X i k
ζ ζ
ε
− +
= ∉
− +
= ∉
−
∑ ∑
, where ε is a tolerance threshold,
end the detection. Otherwise, go back to step 2 and increase the iteration number j = j+1.

4.3.4. Summary of the proposed channel estimation and data detection