Computationally-efficient MIMO equalization algorithm for high-speed, real-time, adaptive polarization multiplexed (POLMUX) OFDM transmission with direct detection
View Patent ↗A polarization-multiplexed (POLMUX) optical orthogonal frequency division multiplexing (OFDM) system with direct detection includes an adaptive dual POLMUX carrier OFDM transmitter; and a block symmetric (B-S) MIMO equalizer coupled to the adaptive dual POLMUX carrier OFDM transmitter through a standard single-mode-fiber (SSMF) feedback path.
1. A polarization-multiplexed (POLMUX) optical orthogonal frequency division multiplexing (OFDM) system with direct detection, comprising:
an adaptive dual POLMUX carrier OFDM transmitter; and
a block symmetric (B-S) MIMO equalizer coupled to the adaptive dual POLMUX carrier OFDM transmitter through a standard single-mode-fiber (SSMF) feedback path,
wherein the B-S MIMO equalizer comprises a B-S channel estimator,
wherein the B-S channel estimator first uses training sequences in determining an estimation matrix such that the received symbols on subcarriers i and N−i+1 in the polarizations X′ and Y′ can be modeled as:
[
c
XX
′
a
XX
′
(
i
)
c
YX
′
a
YX
′
(
i
)
c
YX
′
a
XX
′
(
N
-
i
+
1
)
c
XX
′
a
YX
′
(
N
-
i
+
1
)
c
XY
′
a
XY
′
(
i
)
c
YY
′
a
YY
′
(
i
)
c
YY
′
a
XY
′
(
N
-
i
+
1
)
c
XY
′
a
YY
′
(
N
-
i
+
1
)
c
YX
′
a
XX
′
(
i
)
c
XX
′
a
YX
′
(
i
)
c
XX
′
a
XX
′
(
N
-
i
+
1
)
c
YX
′
a
YX
′
(
N
-
i
+
1
)
c
YY
′
a
XY
′
(
i
)
c
XY
′
a
YY
′
(
i
)
c
XY
′
a
XY
′
(
N
-
i
+
1
)
c
YY
′
a
YY
′
(
N
-
i
+
1
)
]
︸
Channel
Estimation
Matrix
×
[
S
X
,
i
S
Y
,
i
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
︸
Tx
Signals
+
v
=
[
S
X
′
,
i
S
Y
′
,
i
S
X
′
,
N
-
i
+
1
S
Y
′
,
N
-
i
+
1
]
︸
Re
Signals
=
y
where v and y denote a noise vector and a vector of received symbols, respectively and i=1, 2, . . . N, denotes the OFDM subcarrier index, and N denotes the FFT size,
wherein the B-S channel estimator approximates the channel estimation matrix by a matrix of the form
[
A
B
B
A
]
or a matrix of the form
[
A
0
0
C
]
where A and B and C are 2×2 matrices, if the error in such approximation is below an acceptable threshold.
2. The system of claim 1 , comprising a polarization beam splitter (PBS) coupled to the standard single mode fiber (SSMF), a pair of photodetectors (PD) coupled to the PBS, and an Orthogonal Frequency Division Multiplexing (OFDM) receiver coupled to each PD, wherein the OFDM receiver outputs frequency-domain OFDM signals containing cross-polarization interference to the block-symmetry multiple input multiple output (B-S MIMO) equalizer.
3. The system of claim 1 , wherein the B-S MIMO equalizer comprises a B-S zero forcing module.
4. The system of claim 3 , wherein the zero forcing module determines
H
=
[
P
Q
Q
P
]
=
[
A
B
B
A
]
-
1
=
[
(
A
-
BA
-
1
B
)
-
1
-
(
A
-
BA
-
1
B
)
-
1
BA
-
1
-
(
A
-
BA
-
1
B
)
-
1
BA
-
1
(
A
-
BA
-
1
B
)
-
1
]
where A and B are 2×2 matrices.
5. The system of claim 3 , wherein the zero forcing module determines
H
=
[
P
Q
Q
T
]
=
[
A
-
1
0
0
C
-
1
]
where A, C are 2×2 matrices and 0 is a 2×2 matrix with all zero entries.
6. The system of claim 1 , wherein the B-S MIMO equalizer comprises a B-S Minimum Mean Square Error (MMSE) module.
7. The system of claim 6 , wherein the MMSE module determines
H
=
(
[
I
0
0
I
]
+
[
A
*
B
*
B
*
A
*
]
[
A
B
B
A
]
)
-
1
[
A
*
B
*
B
*
A
*
]
=
(
[
I
+
A
*
A
+
B
*
B
︸
C
A
*
B
+
B
*
A
︸
D
A
*
B
+
B
*
A
I
+
A
*
A
+
B
*
B
]
)
-
1
[
A
*
B
*
B
*
A
*
]
=
[
W
F
F
W
]
[
A
*
B
*
B
*
A
*
]
=
[
WA
*
+
FB
*
WB
*
+
FA
*
WB
*
+
FA
*
WA
*
+
FB
*
]
,
W
=
(
C
-
D
C
-
1
D
)
-
1
,
F
=
-
WDC
-
1
where I is the 2×2 identity matrix.
8. The system of claim 6 , wherein the MMSE module determines
H
=
(
[
(
I
+
A
*
A
)
-
1
A
*
0
0
(
I
+
C
*
C
)
-
1
C
*
]
)
.
9. The system of claim 1 , wherein the B-S MIMO equalizer determines:
H
×
[
S
X
′
,
i
S
Y
′
,
i
S
X
′
,
N
-
i
+
1
S
Y
′
,
N
-
i
+
1
]
.
10. The system of claim 1 , wherein the B-S channel estimator approximates the channel estimation matrix by a matrix of the form
[
A
0
0
C
]
and the B-S MIMO equalizer separately equalizes the two received symbols on each sub-carrier corresponding to the two polarizations.
11. The system of claim 1 , wherein the B-S MIMO equalizer comprises a signal-to-interference-and-noise ratio (SINR) determination module.
12. The system of claim 1 , wherein the SINR determination module determines the SINR using MMSE or determines SINR using zero forcing (ZF) as follows:
SINR
MMSE
k
=
1
[
(
[
I
0
0
I
]
+
[
A
*
B
*
B
*
A
*
]
[
A
B
B
A
]
)
-
1
]
k
,
k
-
1
=
1
[
[
W
F
F
W
]
]
k
,
k
-
1
SINR
ZF
k
=
1
[
(
[
A
*
B
*
B
*
A
*
]
[
A
B
B
A
]
)
-
1
]
k
,
k
=
1
[
(
[
A
*
A
+
B
*
B
︸
G
B
*
A
+
A
*
B
︸
D
B
*
A
+
A
*
B
A
*
A
+
B
*
B
]
)
-
1
]
k
,
k
=
1
[
[
M
N
N
M
]
]
k
,
k
M
=
(
G
-
DG
-
1
D
)
-
1
,
N
=
-
MDG
-
1
.
13. The system of claim 11 , wherein the SINR determination module determines the SINR using MMSE or determines SINR using zero forcing (ZF) as follows:
SINR
MMSE
k
=
1
[
[
(
I
+
A
*
A
)
-
1
0
0
(
I
+
C
*
C
)
-
1
]
]
k
,
k
SINR
ZF
k
=
1
[
[
(
A
*
A
)
-
1
0
0
(
C
*
C
)
-
1
]
]
k
,
k
.
14. The system of claim 1 , wherein the B-S MIMO equalizer comprises a group MMSE module.
15. The system of claim 14 , wherein the group MMSE module determines
LL
*
=
G
1
*
R
-
1
G
1
(
I
-
G
1
*
R
-
1
G
1
)
G
1
=
[
A
B
]
,
G
2
=
[
B
A
]
,
R
=
[
I
0
0
I
]
+
[
A
B
B
A
]
[
A
B
B
A
]
*
G
1
*
R
-
1
G
1
=
G
2
*
R
-
1
G
2
=
[
F
1
F
2
]
[
A
*
A
+
B
*
B
B
*
A
+
A
*
B
]
=
F
1
(
A
*
A
+
B
*
B
)
+
F
2
(
B
*
A
+
A
*
B
)
.
16. The system of claim 11 , wherein the B-S MIMO equalizer includes a two symbol Maximum Likelihood Demodulation (MLD) module.
17. The system of claim 16 , wherein the inputs to the two symbol MLD module are determined and modeled as
L
-
1
[
z
1
z
2
]
=
L
-
1
G
1
*
R
-
1
G
1
[
S
X
,
i
S
Y
,
i
]
+
v
1
L
-
1
[
z
3
z
4
]
=
L
-
1
G
1
*
R
-
1
G
1
[
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
+
v
2
where L, G 1 , R are defined as in claim 18 , [z 1 , z 2 , z 3 , z 4 ] T =H[y 1 , y 2 , y 3 , y 4 ] denote the equalized received symbols, where y 1 , y 2 , y 3 , y 4 are the received symbols, the matrix H is determined as in claim 10 and v 1 and v 2 denote effective noise vectors each normalized to have identity matrix covariance.
18. The system of claim 16 , wherein the inputs to the two symbol MLD module are modeled as
[
z
1
z
2
]
=
A
[
S
X
,
i
S
Y
,
i
]
+
v
1
[
z
3
z
4
]
=
C
[
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
+
v
2
where A, C are 2×2 matrices, and [z 1 , z 2 , z 3 , z 4 ]=[y 1 , y 2 , y 3 , y 4 ] are the received symbols.
19. A method to provide equalization for a polarization-multiplexed (POLMUX) optical orthogonal frequency division multiplexing (OFDM) system with direct detection, comprising:
in a digital signal processor, generating a channel estimation matrix from a training signal set output by the OFDM receiver and approximating it with a matrix having a block symmetric (B-S) structure;
using the B-S structure to perform a B-S zero-forcing (ZF) operation for B-S equalization;
using the B-S structure to perform a Minimum Mean Square Error (MMSE) operation and a group MMSE operation for B-S equalization;
using the B-S structure to perform a B-S ZF or B-S MMSE equalization;
using the B-S group MMSE operation to perform a Two-Symbol Maximum Likelihood Demodulation (MLD);
using the B-S structure to perform theoretically-optimal Two-Symbol Maximum Likelihood Demodulation (MLD);
using the B-S structure to determine signal to interference noise ratio (SINR), a metric of link quality; and
sending SINR information to the transmitter via a standard single-mode-fiber (SSMF) feedback path to generate an adaptive POLMUX OFDM transmitter.
20. The method of claim 19 , comprising performing modulation and coding format adaptation based on link quality to optimize a transmission rate.
21. The method of claim 19 , comprising using training sequences A X , A Y , B X , B Y in determining the channel estimation matrix:
[
c
XX
′
a
XX
′
(
i
)
c
YX
′
a
YX
′
(
i
)
c
YX
′
a
XX
′
(
N
-
i
+
1
)
c
XX
′
a
YX
′
(
N
-
i
+
1
)
c
XY
′
a
XY
′
(
i
)
c
YY
′
a
YY
′
(
i
)
c
YY
′
a
XY
′
(
N
-
i
+
1
)
c
XY
′
a
YY
′
(
N
-
i
+
1
)
c
YX
′
a
XX
′
(
i
)
c
XX
′
a
YX
′
(
i
)
c
XX
′
a
XX
′
(
N
-
i
+
1
)
c
YX
′
a
YX
′
(
N
-
i
+
1
)
c
YY
′
a
XY
′
(
i
)
c
XY
′
a
YY
′
(
i
)
c
XY
′
a
XY
′
(
N
-
i
+
1
)
c
YY
′
a
YY
′
(
N
-
i
+
1
)
]
︸
Channel
Estimation
Matrix
×
[
S
X
,
i
S
Y
,
i
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
︸
Tx
Signals
+
v
=
[
S
X
′
,
i
S
Y
′
,
i
S
X
′
,
N
-
i
+
1
S
Y
′
,
N
-
i
+
1
]
︸
Re
Signals
=
y
,
where v and y denote a noise vector and a vector of received symbols, respectively, wherein both v and the transmit signal vector are normalized and have identity-matrix covariance.
22. The method of claim 19 , wherein the B-S channel estimator approximates the channel estimation matrix by a matrix of the form
[
A
B
B
A
]
or a matrix of the form
[
A
0
0
C
]
,
where A and B and C are 2×2 matrices, if the error in such approximation is below an acceptable threshold.
23. The method of claim 19 , wherein the zero forcing comprises
H
=
[
P
Q
Q
P
]
=
[
A
B
B
A
]
-
1
=
[
(
A
-
BA
-
1
B
)
-
1
-
(
A
-
BA
-
1
B
)
-
1
BA
-
1
-
(
A
-
BA
-
1
B
)
-
1
BA
-
1
(
A
-
BA
-
1
B
)
-
1
]
where A and B are 2×2 matrices.
24. The method of claim 19 , wherein the zero forcing comprises
H
=
[
P
Q
Q
T
]
=
[
A
-
1
0
0
C
-
1
]
where A, C are 2×2 matrices and 0 is a 2×2 matrix with all zero entries.
25. The method of claim 19 , wherein the MMSE module determines
H
=
(
[
I
0
0
I
]
+
[
A
*
B
*
B
*
A
*
]
[
A
B
B
A
]
)
-
1
[
A
*
B
*
B
*
A
*
]
=
(
[
I
+
A
*
A
+
B
*
B
︸
C
A
*
B
+
B
*
A
︸
D
A
*
B
+
B
*
A
I
+
A
*
A
+
B
*
B
]
)
-
1
[
A
*
B
*
B
*
A
*
]
=
[
W
F
F
W
]
[
A
*
B
*
B
*
A
*
]
=
[
WA
*
+
FB
*
WB
*
+
FA
*
WB
*
+
FA
*
WA
*
+
FB
*
]
,
W
=
(
C
-
DC
-
1
D
)
-
1
,
F
=
-
WDC
-
1
.
26. The method of claim 19 , wherein the MMSE module determines
H
=
(
[
(
I
+
A
*
A
)
-
1
A
*
0
0
(
I
+
C
*
C
)
-
1
C
*
]
)
.
27. The method of claim 19 , wherein the B-S group MMSE module determines
LL
*
=
G
1
*
R
-
1
G
1
(
I
-
G
1
*
R
-
1
G
1
)
G
1
=
[
A
B
]
,
G
2
=
[
B
A
]
,
R
=
[
I
0
0
I
]
+
[
A
B
B
A
]
[
A
B
B
A
]
*
G
1
*
R
-
1
G
1
=
G
2
*
R
-
1
G
2
=
[
F
1
F
2
]
[
A
*
A
+
B
*
B
B
*
A
+
A
*
B
]
=
F
1
(
A
*
A
+
B
*
B
)
+
F
2
(
B
*
A
+
A
*
B
)
.
28. The method of claim 19 , wherein the two symbol MLD module determines
L
-
1
[
z
1
z
2
]
=
L
-
1
G
1
*
R
-
1
G
1
[
S
X
,
i
S
Y
,
i
]
+
v
1
L
-
1
[
z
3
z
4
]
=
L
-
1
G
1
*
R
-
1
G
1
[
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
+
v
2
where L, G 1 , R are defined as in claim 18 , z 1 , z 2 , z 3 , z 4 denote received symbols, and v 1 and v 2 denote effective noise vectors each normalized to have identity matrix covariance.
29. The method of claim 19 , wherein the two symbol MLD module determines
[
z
1
z
2
]
=
A
[
S
X
,
i
S
Y
,
i
]
+
v
1
[
z
3
z
4
]
=
C
[
S
X
,
N
-
i
+
1
S
Y
,
N
-
i
+
1
]
+
v
2
where A, C are 2×2 matrices, and no additional filtering is needed at the two-symbol MLD outputs to achieve theoretically optimal performance.
30. A polarization-multiplexed (POLMUX) optical orthogonal frequency division multiplexing (OFDM) system with direct detection, comprising:
an adaptive dual POLMUX carrier OFDM transmitter; and
a block symmetric (B-S) MIMO equalizer coupled to the adaptive dual POLMUX carrier OFDM transmitter through a standard single-mode-fiber (SSMF) feedback path, the B-S MIMO equalizer generating a channel estimation matrix with a block symmetric (B-S) structure; using the B-S structure to perform a B-S zero-forcing (ZF) operation for B-S equalization; using the B-S structure to perform a Minimum Mean Square Error (MMSE) operation and a group MMSE operation to perform B-S equalization; using the B-S group MMSE operation to perform a Two-Symbol Maximum Likelihood Demodulation (MLD); using the B-S structure to perform theoretically-optimal Two-Symbol Maximum Likelihood Demodulation (MLD); using the B-S structure to determine signal to interference and noise ratio (SINR); and sending SINR information to the transmitter via a standard single-mode-fiber (SSMF) feedback path to generate an adaptive POLMUX OFDM transmitter.