Systems and methods for minimum-power leakage windowing for VDSL using least square technique
View Patent ↗Systems and method for minimum power leakage windowing for VDSL using least squares technique are provided. A coefficient vector is defined for the first N samples. The window is then defined in terms of this coefficient vector. The leakage power of the K th tone is defined in terms of the window. A least squares minimization is performed on the leakage power to determine the optimal value of x. The window is implemented using this value of x, thereby reducing the power leakage in the out-of-band and RFI band regions in the frequency domain.
1. A method of shaping an outbound signal in a very high bitrate digital subscriber line (VDSL) transceiver comprising:
defining, using a leakage power module at the VDSL transceiver, a leakage power function of an N th discrete multi-tone (DMT) sub-channel, where N is an integer between 1 and a maximum number of available sub-channels, wherein defining a leakage power function of an N th DMT sub-channel comprises defining a function [Aw] k , where w is the window function, A is the matrix,
A
=
2
x
[
1
1
…
1
cos
(
π
(
L
-
1
)
N
)
cos
(
π
(
L
-
3
)
N
)
…
cos
(
π
N
)
⋮
⋮
⋰
⋮
cos
(
π
(
N
/
2
-
1
)
(
L
-
1
)
N
)
cos
(
π
(
N
/
2
-
1
)
(
L
-
3
)
N
)
…
cos
(
π
(
N
/
2
-
1
)
N
)
]
N
2
×
L
2
a
n
d
k
=
0
:
N
2
-
1
;
minimizing the leakage power function based on the defined leakage power function;
generating a transmit window based on the minimized function; and
reducing power leakage by windowing the outbound signal transmit window.
2. The method according to claim 1 , wherein minimizing the leakage power function comprises performing the minimization problem Min|A 1 x+B| 2 , where
A
1
=
A
(
0
:
N
2
-
1
,
0
:
β
-
1
)
,
B
=
A
2
1
,
a
n
d
A
2
=
A
(
0
:
N
2
-
1
,
β
:
L
-
1
)
,
and where 1 is an all one vector.
3. The method according to claim 2 , wherein minimizing the leakage power function further comprises defining a function P=|A 1 x+B| 2 =x H A 1 H A 1 x+B H B+2x H A 1 H B, wherein the minimum of P is attained at the value of x that makes the gradient of P equal to zero.
4. The method according to claim 3 , wherein gradient of P is given by the expression ∇P=2A 1 H A 1 x+2A H B.
5. The method according to claim 3 , wherein the value of x that makes the gradient of P equal to zero is attained by the solution of linear equations A 1 H A 1 x opt =−A 1 H B and x opt =−(A 1 H A 1 ) −1 A 1 H B.
6. The method according to claim 1 , wherein the maximum number of available sub-channels is 4096.
7. A method of forming a transmit window for a discrete multi-tone (DMT)-based transceiver comprising:
defining, using a leakage power module at the very high bitrate digital subscriber line (VDSL) transceiver, a leakage power function of an N th sub-channel of a communication channel, wherein defining a leakage power function of an N th sub-channel comprises defining a function [Aw] k , where w is the window function, A is the matrix,
A
=
2
x
[
1
1
…
1
cos
(
π
(
L
-
1
)
N
)
cos
(
π
(
L
-
3
)
N
)
…
cos
(
π
N
)
⋮
⋮
⋰
⋮
cos
(
π
(
N
/
2
-
1
)
(
L
-
1
)
N
)
cos
(
π
(
N
/
2
-
1
)
(
L
-
3
)
N
)
…
cos
(
π
(
N
/
2
-
1
)
N
)
]
N
2
×
L
2
a
n
d
k
=
0
:
N
2
-
1
;
performing a least squares minimization on the leakage power function; and reducing power leakage by applying a window function to a transmit signal based on an optimum function as determined by the least squares minimization.
8. The method according to claim 7 , wherein performing a least squares minimization on the leakage power function comprises performing the minimization problem Min|A 1 x+B| 2 , where
A
1
=
A
(
0
:
N
2
-
1
,
0
:
β
-
1
)
,
B
=
A
2
1
,
a
n
d
A
2
=
A
(
0
:
N
2
-
1
,
β
:
L
-
1
)
,
and where 1 is an all one vector.
9. The method according to claim 8 , wherein minimizing the leakage power function further comprises defining a function P=|A 1 x+B| 2 =x H A 1 II A 1 x+B H B+2x H A 1 II B, wherein the minimum of P is attained at the value of x that makes the gradient of P equal to zero.
10. The method according to claim 9 , wherein gradient of P is given by the expression ∇P=2A 1 H A 1 x+2A H B.
11. The method according to claim 9 , wherein the value of x that makes the gradient of P equal to zero is attained by the solution of linear equations A 1 H A 1 x opt =−A 1 H B and x opt =−(A 1 H A 1 ) −1 A 1 H B.
12. The method according to claim 7 , wherein N is an integer ranging from 1 to 4096.
13. A very high bitrate digital subscriber line (VDSL) modem circuit comprising:
a transceiver including a windowing function block comprising:
a leakage power function determination module for determining a leakage power function of an N th sub-channel of a VDSL-based communication channel;
a least squares minimization module for solving minimizing a total leakage power from sub-channel S to sub-channel N/2−1, where N is the maximum number of sub-channels on the communication channel and S<N;
a windowing function determination module that determines an optimum window function for a VDSL communication line based on the leakage power function determination module and least squares minimization module, wherein the leakage power function determination module is adapted to determine a leakage power function of an N th sub-channel comprises defining a function [Aw] k , where w is the window function, A is the matrix,
A
=
2
x
[
1
1
…
1
cos
(
π
(
L
-
1
)
N
)
cos
(
π
(
L
-
3
)
N
)
…
cos
(
π
N
)
⋮
⋮
⋰
⋮
cos
(
π
(
N
/
2
-
1
)
(
L
-
1
)
N
)
cos
(
π
(
N
/
2
-
1
)
(
L
-
3
)
N
)
…
cos
(
π
(
N
/
2
-
1
)
N
)
]
N
2
x
L
2
;
a
n
d
a window function application module for applying the optimum window function to an outbound signal.
14. The modem circuit according to claim 13 , wherein the least squares minimization module is adapted to perform a least squares minimization on the leakage power function comprising the minimization problem Min|A 1 x+B| 2 , where
A
1
=
A
(
0
:
N
2
-
1
,
0
:
β
-
1
)
,
B
=
A
2
1
,
a
n
d
A
2
=
A
(
0
:
N
2
-
1
,
β
:
L
-
1
)
,
and where 1 is an all one vector.
15. The modem circuit according to claim 14 , wherein the least squares minimization module is further adapted to derive aP=|A 1 x+B| 2 =x H A 1 H A 1 x+B H B+2x H A 1 H B, wherein the minimum of P is attained at the value of x that makes the gradient of P equal to zero.
16. The modem circuit according to claim 15 , wherein gradient of P is given by the expression ∇P=2A 1 H A 1 x+2A H B.
17. The modem circuit according to claim 15 , wherein the value of x that makes the gradient of P equal to zero is attained by the solution of linear equations A 1 H A 1 x opt =−A 1 H B and x opt =−(A 1 H A 1 ) −1 A 1 H B.
18. The modem circuit according to claim 13 , wherein N is an integer ranging from 1 to 4096.