Channel estimation for frequency selective multipath channels with long delay spreads based on an assumed physical channel
A channel impulse response is determined for a channel by receiving a signal from the channel, and by determining a least squares estimate of the channel impulse response. The received signal contains a training sequence and unknown data. The least squares estimate of the channel impulse response is determined by multiplying the received signal by a stored quantity. The stored quantity is based on (i) a stored replica of the training sequence, and (ii) an assumed covariance matrix that is based on a noise variance and an initial channel impulse response that assumes a unit physical channel.
1. A method for determining a channel impulse response for a channel comprising:
receiving a signal from the channel, wherein the received signal contains a training sequence and unknown data; and,
determining a least squares estimate of the channel impulse response by multiplying the received signal by a stored quantity, wherein the stored quantity is based on (i) a stored replica of the training sequence, and (ii) an assumed covariance matrix that is based on a noise variance and an initial channel impulse response that assumes a unit physical channel.
2. The method of claim 1 wherein the stored quantity is given by the following expression:
( A H [{tilde over (K)} v ( {tilde over (H)} )] −1 A ) −1 A H [{tilde over (K)} v ( {tilde over (H)} )] −1
wherein A is based on the stored replica of the training sequence, wherein {tilde over (K)} v is the covariance matrix, and wherein {tilde over (H)} is based on the initial channel impulse response that assumes a unit physical channel.
3. The method of claim 2 wherein the covariance matrix {tilde over (K)} v is given by the following equation:
K
~
v
(
H
~
)
=
1
2
ɛ
d
H
~
H
~
H
+
σ
~
η
2
Q
Q
H
wherein {tilde over (σ)} η 2 is the noise variance and is given by the following equation:
σ
~
η
2
=
ɛ
d
q
2
C
and wherein ε d , Q, and q are based on a matched filter, wherein C is a constant given by the following equation:
C
=
10
SNR
[
db
]
10
.
4. The method of claim 3 wherein is given by the following equation:
{tilde over (H)}= S T
wherein is given by the following equation:
{tilde over ( h =J{tilde over (h)}
wherein {tilde over (h)} is based on the initial channel impulse response that assumes a unit physical channel, wherein J is a matrix given generally by the following equation:
J
=
[
0
…
0
1
0
…
1
0
⋮
⋮
⋮
1
0
…
0
]
wherein S is a matrix given generally by the following equation:
S
=
[
I
0
0
0
0
I
]
wherein I is the identity matrix.
5. A method for determining a channel impulse response for a channel comprising:
receiving a signal from the channel, wherein the received signal contains a training sequence and unknown data; and,
determining a least squares estimate of the channel impulse response in accordance with only two factors, wherein the two factors consist of the received signal and a constant.
6. The method of claim 5 wherein the constant is given by the following expression:
( A H [{tilde over (K)} v ( {tilde over (H)} )] −1 A ) −1 A H [{tilde over (K)} v ( {tilde over (H)} )] −1
wherein A is based on the stored replica of the training sequence, wherein {tilde over (K)} v is the covariance matrix, and wherein {tilde over (H)} is based on the initial channel impulse response that assumes a unit physical channel.
7. The method of claim 6 wherein the covariance matrix {tilde over (K)} v is given by the following equation:
K
~
v
(
H
~
)
=
1
2
ɛ
d
H
~
H
~
H
+
σ
~
η
2
Q
Q
H
wherein {tilde over (σ)} η 2 is the noise variance and is given by the following equation:
σ
~
η
2
=
ɛ
d
q
2
C
and wherein ε d , Q, and q are based on a matched filter, wherein C is a constant given by the following equation:
C
=
10
SNR
[
db
]
10
.
8. The method of claim 7 wherein {tilde over (H)} is given by the following equation:
{tilde over (H)}= S T
wherein is given by the following equation:
{tilde over ( h =J{tilde over (h)}
wherein {tilde over (h)} is based on the initial channel impulse response that assumes a unit physical channel, wherein J is a matrix given generally by the following equation:
J
=
[
0
…
0
1
0
…
1
0
⋮
⋮
⋮
1
0
…
0
]
wherein S is a matrix given generally by the following equation:
S
=
[
I
0
0
0
0
I
]
wherein I is the identity matrix.
9. The method of claim 5 wherein the determining of a least squares estimate of the channel impulse response comprises multiplying the received signal by the constant.
10. The method of claim 5 wherein the constant is based on (i) the training sequence and (ii) an assumed covariance matrix that combines an assumed noise variance and an assumed channel.
11. The method of claim 10 wherein the determining of a least squares estimate of the channel impulse response comprises multiplying the received signal by the constant.
12. A method for determining a channel impulse response for a channel comprising:
receiving a signal from the channel, wherein the received signal contains a training sequence and unknown data;
determining a least squares estimate of the channel impulse response in accordance with only two factors, wherein the two factors consist of the received signal and a constant; and,
determining tap weights of an equalizer directly from the least squares estimate of the channel impulse response.
13. The method of claim 12 wherein the constant is given by the following expression:
( A H [{tilde over (K)} v ( {tilde over (H)} )] −1 A ) −1 A H [{tilde over (K)} v ( {tilde over (H)} )] −1
wherein A is based on the stored replica of the training sequence, wherein {tilde over (K)} v is the covariance matrix, and wherein {tilde over (H)} is based on the initial channel impulse response that assumes a unit physical channel.
14. The method of claim 13 wherein the covariance matrix {tilde over (K)} v is given by the following equation:
K
~
v
(
H
~
)
=
1
2
ɛ
d
H
~
H
~
H
+
σ
~
η
2
Q
Q
H
wherein {tilde over (σ)} η 2 is the noise variance and is given by the following equation:
σ
~
η
2
=
ɛ
d
q
2
C
and wherein ε d , Q, and q are based on a matched filter, wherein C is a constant given by the following equation:
C
=
10
SNR
[
db
]
10
.
15. The method of claim 14 wherein {tilde over (H)} is given by the following equation:
{tilde over (H)}= S T
wherein is given by the following equation:
{tilde over ( h =J{tilde over (h)}
wherein {tilde over (h)} is based on the initial channel impulse response that assumes a unit physical channel, wherein J is a matrix given generally by the following equation:
J
=
[
0
…
0
1
0
…
1
0
⋮
⋮
⋮
1
0
…
0
]
wherein S is a matrix given generally by the following equation:
S
=
[
I
0
0
0
0
I
]
wherein I is the identity matrix.
16. The method of claim 12 wherein the determining of a least squares estimate of the channel impulse response comprises multiplying the received signal by the constant.
17. The method of claim 12 wherein the constant is based on (i) the training sequence and (ii) an assumed covariance matrix that combines an assumed noise variance and an assumed channel.
18. The method of claim 17 wherein the determining of a least squares estimate of the channel impulse response comprises multiplying the received signal by the constant.