Low-complexity estimation of QAM symbols and constellations
View Patent ↗Disclosed are methods and structures for soft symbol and variance estimation for QAM constellations including a big-flipping framework and efficient methods for soft symbol estimation and variance estimation for QAM. Disclosed are efficient Gray mapping which provides a much lower complexity, i.e., log N for N-QAM for both squared and non-squared QAM constellations. Also disclosed is an approximation method that avoids multiplications completely while exhibiting only a slight performance degradation. Finally, a low complexity method for variance estimations, particularly second moment estimations for both squared and non-squared QAM constellations with Gray mapping are disclosed. Advantageously—using the disclosed methods—the complexity of the second moment estimation is reduced to O((log N)^2) for an N-QAM symbol for both squared and non-squared QAM.
1. A computer implemented method of producing a soft symbol estimate of a pulse amplitude modulation (PAM) symbol, the method comprising:
determining from one PAM symbol, s N′ the following relationship:
Pr
(
x
=
s
N
′
)
=
∏
i
P
b
i
(
1
)
=
∏
i
1
ⅇ
λ
i
+
1
,
where λ i denotes a log-likelihood ratio (LLR) for the i-th bit b i ;
determining, for n=N′−1, . . . , 1, (Pr(x=s n )) sequentially from the following relationship:
Pr
(
x
=
s
n
)
=
Pr
(
x
=
s
n
+
1
)
π
i
⋓
n
+
1
,
s
n
+
1
,
i
⋓
n
+
1
,
n
⟶
s
n
,
i
⋓
n
+
1
,
n
,
where {hacek over (i)} n+1,n denotes the index of the only different bit between a labeling bit sequence of s n+1 and s n ; and
determining, after determining Pr(x=s n ) for all n, the soft PAM estimation according to the following:
x
~
=
∑
s
∈
𝒮
PAM
s
Pr
(
x
=
s
)
=
∑
n
=
1
N
′
s
n
∏
i
=
1
Q
P
b
i
(
s
n
,
i
)
,
where Q is a length of a bit sequence, N′=2 Q , s n,i is the ith bit value mapped to the PAM symbol s n and P bi (s n,i ) is determined according to
P
b
i
(
0
)
=
Δ
Pr
(
b
i
=
0
)
=
1
ⅇ
-
λ
i
+
1
,
P
b
i
(
1
)
=
Δ
Pr
(
b
i
=
1
)
=
1
ⅇ
λ
i
+
1
.
2. The method of claim 1 , which further determines PAM estimations for both I and Q components x I and x Q , and then determines an estimation of a quadrature amplitude modulation (QAM) symbol x qam according to the following relationship:
x qam =x I +jx Q .