Efficient square-root free 2 symbol max-log receiver
View Patent ↗The present invention employs a look up table based implementation for the metric computations which eliminate redundancy and substantially reduce the number of multiplications. Moreover, inventive method exploits the fact that the un-normalized constellation symbols are complex integers so that the product of a real-number and an un-normalized constellation symbol can be implemented by only additions. The inventive method also enables a greater efficiency for whitening colored noise prior to demodulation, one of which involves no square-root operation. The invention results in less complexity, faster operation, lower power consumption, without sacrificing performance.
1. A method for a square-root free 2 symbol max-log receiver comprising:
i) obtaining linear transformations of a received two stream signal and a channel matrix without implementing square-root operations;
ii) listing out all possibilities for a first symbol of the received two stream signal;
iii) building look-up tables needed for computing first metrics associated with all possibilities for a first symbol of the two stream signal;
iv) determining a second symbol of the two stream signal for each said first symbol listed out;
v) evaluating said first metrics for each said first symbol and second symbol pair using said look-up tables;
vi) listing out all possibilities for said second symbol;
vii) building look-up tables needed for computing second metrics associated with all possibilities for a second symbol of said received two stream signal;
viii) determining a first symbol for each choice of said second symbol listed out;
ix) evaluating said second metrics for each said second symbol and first symbol pair using the look-up tables;
x) determining an exact max-log log likelihood ratio for each coded bit using said second metrics; and
xi) decoding at least one codeword in the two stream signal using the determined exact max-log log likelihood ratios for all bits.
2. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
obtaining
Q
~
=
Q
[
1
/
r
0
,
0
0
0
1
/
r
1
,
1
]
,
r
⋓
0
,
1
=
r
0
,
1
/
r
0
,
0
and
θ
=
r
1
,
1
2
/
r
0
,
0
2
.
where Q is an N r ×2 semi-unitary matrix,
N r is the number of receive antennas,
R is a 2×2 upper triangular matrix that is expanded as
R
=
[
r
o
,
o
r
o
,
1
O
r
1
,
1
]
,
{tilde over (H)}=[{tilde over (h)} 0 ,{tilde over (h)} 1 ] is a channel matrix, and
{tilde over (H)}=QR.
3. The method of claim 2 , wherein said step i) obtaining linear transformations of a received signal comprises:
multiplying a received vector z with {tilde over (Q)} H giving
{tilde over (z)}={tilde over (Q)} H z={tilde over (Q)} H ( QRx+n )= {tilde over (R)}x+ñ,
where
z
~
=
[
z
~
(
0
)
z
~
(
1
)
]
=
[
1
r
0
,
1
/
r
0
,
0
0
1
]
[
x
(
0
)
x
(
1
)
]
+
n
~
=
[
x
(
0
)
+
r
⋓
0
,
1
x
(
1
)
x
(
1
)
]
+
[
n
~
(
0
)
n
~
(
1
)
]
,
and
Q H Q=I with ñ˜N c (0,diag{1 /r 0,0 2 ,1 /r 1,1 2 }).
4. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
obtaining {tilde over (Q)}, {hacek over (r)} 0,1 , r 1,1 2 and θ via a modified QR decomposition for any N r ≧2;
using the channel matrix {tilde over (H)};
considering {tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ]; and
obtaining α={tilde over (h)} 0 H {tilde over (h)} 1 .
5. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
{tilde over (Q)}, {hacek over (r)} 0,1 , r 1,1 2 and θ via a modified QR decomposition for any N r ≧2;
using the channel matrix {tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ]; and
obtaining δ=∥{tilde over (h)} 0 ∥ 2 and a first column of {tilde over (Q)}=[{tilde over (q)} 0 , {tilde over (q)} 1 ] as {tilde over (q)} 0 ={tilde over (h)} 0 /δ.
6. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
obtaining {tilde over (Q)}, {hacek over (r)} 0,1 , r 1,1 2 and θ via a modified QR decomposition for any N r ≧2;
using the channel matrix {tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ]; and
obtaining a vector {tilde over (v)}={tilde over (h)} 1 −α{tilde over (q)} 0 , where α={tilde over (h)} 0 H {tilde over (h)} 1 , r 1,1 2 =∥{tilde over (v)}∥ 2 , and a second column of {tilde over (Q)}=[{tilde over (q)} 0 ,{tilde over (q)} 1 ] as {tilde over (q)} 1 ={tilde over (v)}/r 1,1 2 .
7. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
obtaining {tilde over (Q)}, {hacek over (r)} 0,1 , r 1,1 2 and θ via a modified QR decomposition for any N r ≧2;
using the channel matrix {tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ]; and
obtaining {hacek over (r)} 0,1 as {hacek over (r)} 0,1 =α/δ, where α={tilde over (h)} 0 H {tilde over (h)} 1 and δ=∥{tilde over (h)} 0 ∥ 2 .
8. The method of claim 1 , wherein said step i) obtaining linear transformations of a received signal comprises:
obtaining {tilde over (Q)}, {hacek over (r)} 0,1 , r 1,1 2 and θ via a modified QR decomposition for any N r ≧2;
using the channel matrix {tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ]; and
obtaining θ as θ=r 1,1 2 /δ, where δ=∥{tilde over (h)} 0 ∥ 2 .
9. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
generating a soft bit k:k=0, 1, . . . B (v) −1, corresponding to transmitted symbol x (v) :v=0,1 using max-log approximation of a likelihood ratio (LLR) as follows:
L
(
b
k
,
v
)
≈
min
c
j
∈
S
k
,
v
,
0
(
z
~
-
R
~
c
j
)
H
[
r
0
,
0
2
0
0
r
1
,
1
2
]
(
z
~
-
R
~
c
j
)
-
min
c
j
∈
S
k
,
v
,
1
(
z
~
-
R
~
c
j
)
H
[
r
0
,
0
2
0
0
r
1
,
1
2
]
(
z
~
-
R
~
c
j
)
,
wherein a vector c j =[c j (0) , c j (1) ] T with each element c j (v) is a complex symbol in a modulation constellation applied to layer v,
S k,v,1 and S k,v,0 denote a set containing all possible vector c j such that bit k of element c j (v) is equal to 1 and 0 respectively,
Q is an N r ×2 semi-unitary matrix,
N r is the number of receive antennas,
R is a 2×2 upper triangular matrix that is expanded as
R
=
[
r
o
,
o
r
o
,
1
O
r
1
,
1
]
,
{tilde over (H)}=[{tilde over (h)} 0 , {tilde over (h)} 1 ] is a channel matrix,
{tilde over (H)}=QR,
{tilde over (z)}={tilde over (Q)} H z={tilde over (Q)} H (QRx+n)={tilde over (R)}x+ñ, and
z
~
=
[
z
~
(
0
)
z
~
(
1
)
]
=
[
1
r
0
,
1
/
r
0
,
0
0
1
]
[
x
(
0
)
x
(
1
)
]
+
n
~
=
[
x
(
0
)
+
r
⋓
0
,
1
x
(
1
)
x
(
1
)
]
+
[
n
~
(
0
)
n
~
(
1
)
]
.
10. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
generating a soft bit k:k=0, 1, . . . B (v) −1, corresponding to transmitted symbol x (v) :v=0,1 using max-log approximation of a likelihood ratio (LLR),
wherein a vector c j =[c j (0) , c j (1) ] T with each element c j (v) is a complex symbol in a modulation constellation applied to layer v, and
S k,v,1 and S k,v,0 denote a set containing all possible vector c j such that bit k of element c j (v) is equal to 1 and 0 respectively.
11. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
considering soft bit calculation for a second layer where v=1;
performing for each c j (1) εS (1) :j=0, 1, . . . , K (1) −1; and
computing d j (1) =θ|{tilde over (z)} (1) −c j (1) | 2 .
12. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
considering soft bit calculation for a second layer where v=1;
performing for each c j (1) εS (1) :j=0, 1, . . . , K (1) −1; and
computing soft estimate {tilde over (z)} (0) −{hacek over (r)} 0,1 c j (1) , and
selecting a symbol ĉ j (0) εS (0) that is closest to {tilde over (z)} (0) −{hacek over (r)} 0,1 c j (1) .
13. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
considering soft bit calculation for a second layer where v=1;
performing for each c j (1) εS (1) :j=0, 1, . . . , K (1) −1; and computing d j (0) =|{tilde over (z)} (0) −ĉ j (0) −{hacek over (r)} 0,1 c j (1) | 2 .
14. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
considering soft bit calculation for a second layer where v=1;
performing for each c j (1) εS (1) :j=0, 1, . . . , K (1) −1; and computing total distance d j TOTAL =d j (0) +d j (1) .
15. The method of claim 1 , wherein said step iv), v) or viii), ix) comprises:
a) determining
d
^
(
1
)
=
min
c
j
(
1
)
∈
S
(
1
)
{
d
j
TOTAL
}
,
c
^
j
(
1
)
=
arg
min
c
j
(
1
)
∈
S
(
1
)
{
d
j
TOTAL
}
,
where ĉ j (1) εS (1) is a symbol of layer−1 in a maximum likelihood (ML) decision and {circumflex over (b)} k,1 :k=0, 1 . . . , B (1) −1 is a bit-label of ĉ j (1) εS (1) ; and
b) computing each soft bit L(b k,1 ):k=0, 1, . . . , B (1) −1 as
L
(
b
k
,
1
)
=
r
0
,
0
2
(
d
^
(
1
)
-
min
d
m
TOTAL
∈
D
k
1
d
m
TOTAL
)
if
b
^
k
,
1
=
0
or
L
(
b
k
,
1
)
=
r
0
,
0
2
(
min
d
m
TOTAL
∈
D
k
0
d
m
TOTAL
-
d
^
(
1
)
)
if
b
^
k
,
1
=
1.
c) where D k 1 and D k 0 denote a set containing all distances d j TOTAL correspond to c j (1) that has that bit k equal to 1 and 0, respectively.
16. The method of claim 1 , wherein said step iii) or vii) comprises:
looking at a max-log likelihood ratio (LLR) expression; and
after expanding d j (0) , d j (1) , dropping the terms that do not influence the LLR.
17. The method of claim 1 , wherein said step iii) or vii) comprises:
pulling out normalizing factors so that modulation symbols are from un-normalized quadrature amplitude modulation (QAM) constellations which contain integers.
18. The method of claim 1 , wherein said step iii) or vii) comprises:
forming LUT: containing common terms that depend on a channel coefficient,
wherein entries of the look-up tables (LUTs) are repeatedly accessed while computing all d j (0) , d j (1) , and multiplications are avoided.
19. The method of claim 1 , wherein said step iii) or vii) comprises:
computing d j (1) =θ(c j,R (1) ) 2 +θ(c j,I (1) ) 2 −2{tilde over (z)} R (1) c j,R (1) θ−2{tilde over (z)} I (1) c j,I (1) θ.
20. The method of claim 1 , wherein said step iii) or vii) comprises:
computing d j (1) =θ|{tilde over (z)} (1) −c j (1) | 2 ,
where all {d 0 (1) } can be computed with only 20 real multiplications.