Method, apparatus and computer program for controlling a receiver
View Patent ↗Linear symbol level equalisation is used to estimate a transmitted symbol vector x n (0) in an nth observation window y n of signals received from multiple users each using a different spreading code. An iterative conjugate gradient algorithm resolves a column vector z that satisfies Az=b, A being a square matrix and b being a column vector. The matrix vector product Ap k used in each respective kth iteration of the algorithm is computed using at least one fast transform. The receiver or detector estimates a posterior mean {circumflex over (x)} LMMSE,n (0) from the computed column vector z, then utilises the estimated posterior mean to determine the transmitted symbol vector x n (0) . The posterior mean {circumflex over (x)} LMMSE,n (0) may be estimated from matrix A and vector b. Different examples have the square matrix A with an inner or an outer product form; and have the fast transform as a fast Walsh-Hadamard transform and/or a fast (inverse) Fourier transform.
1. A method for using linear symbol level equalisation to control a receiver to estimate a transmitted symbol vector x n (0) in an nth observation window y n of signals received from multiple users each using a different spreading code, the method comprising:
calculating using an iterative conjugate gradient algorithm to resolve a column vector z that satisfies Az=b, where A is a square matrix and b is a column vector, in which a matrix vector product Ap k used in each respective kth iteration of the iterative conjugate gradient algorithm is computed using at least one fast transform operation, in which p k is a conjugate vector for the kth iteration;
estimating a posterior mean {circumflex over (x)} LMMSE,n (0) , wherein LMMSE is a linear minimum mean squared error, from the column vector z; and
estimating the transmitted symbol vector x n (0) using the estimated posterior mean.
2. The method according to claim 1 , in which the posterior mean {circumflex over (x)} LMMSE,n (0) is estimated from the square matrix A and the column vector b, where A has an inner product form which includes computation of Σ n −1 .
3. The method according to claim 2 , in which the matrix vector product Ap k is computed by:
a) multiplying the vector p k with a spreading code matrix containing all active spreading codes;
b) multiplying the result of part a) by a diagonal scrambling code matrix;
c) multiplying the result of part b) by a block-banded block-Toeplitz channel matrix H (0) ;
d) multiplying the result of part c) by the block-banded block-Toeplitz channel matrix (H (0) ) H ;
e) multiplying the result of part d) by a diagonal descrambling code matrix;
f) multiplying the result of part e) with a despreading code matrix containing the active spreading codes; and
g) computing
∑
x
n
(
0
)
-
1
p
k
by scalar multiplication with
diag
(
σ
x
n
(
0
)
-
2
)
and adding the result to the scaled result of part f), where
∑
x
n
(
0
)
-
1
is an inverse covariance matrix;
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part a) and/or in part f).
4. The method according to claim 3 , in which part c) and/or part d) utilize a fast Fourier transform operation and/or fast inverse Fourier transform operation.
5. The method according to claim 3 , in which part c) and/or part d) utilize a block fast Fourier transform operation and/or block fast inverse Fourier transform operation.
6. The method according to claim 2 , in which the matrix vector product Ap k is computed by:
a) multiplying the vector p k with a spreading code matrix containing all active spreading codes;
b) multiplying the result of part a) by a diagonal scrambling code matrix;
c) perform a fast Fourier transform operation on the result of part b);
d) multiplying the transformed result from part c) by a block-diagonal matrix (D (0) ) H D (0) ;
e) performing an inverse fast Fourier transform operation on the result of part d);
f) multiplying the result of part e) by a diagonal descrambling code matrix;
g) multiplying the result of part f) with a despreading code matrix containing the active spreading codes; and
h) computing
∑
x
n
(
0
)
-
1
p
k
by scalar multiplication with
diag
(
σ
x
n
(
0
)
-
2
)
and adding the result to the scaled result of part g), where
∑
x
n
(
0
)
-
1
is an inverse covariance matrix;
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part a) and/or in part g), and/or the fast Fourier transform operation utilized in part c) and/or the fast inverse Fourier transform operation utilized in part e).
7. The method according to claim 6 , in which the fast Fourier transform operation in part c) is a block fast Fourier transform operation and/or the inverse fast Fourier transform operation in part e) is a block inverse fast Fourier transform operation.
8. The method according to claim 1 , in which the posterior mean {circumflex over (x)} LMMSE,n (0) is estimated from the square matrix A and the column vector b, where A has an outer product form which includes computation Σ n .
9. The method according to claim 8 , in which the matrix-vector product is computed as a result vector a k =Ap k by:
a) initialising the result vector a k to zero;
b) for each cell indexed as c=0 to N cells −1:
i. multiplying the vector p k by a block-banded block-Toeplitz channel matrix (H (c) ) H ;
ii. multiplying the result of part i by a diagonal descrambling code matrix;
iii. multiplying the result of part ii with a despreading code matrix containing all active spreading codes;
iv. multiplying the result of part iii by a diagonal covariance matrix
∑
x
n
(
c
)
;
v. multiplying the result of part iv with a spreading code matrix containing all active spreading codes;
vi. multiplying the result of part v by a diagonal scrambling code matrix;
vii. multiplying the result of part vi by a block-banded block-Toeplitz channel matrix H (c) ;
viii. adding the result from part vii to the result vector a k ; and
c) adding variance σ 2 to all elements of the result vector a k .
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part iii and/or in part v.
10. The method according to claim 9 , in which part i and/or part vii utilize a fast Fourier transform operation and/or fast inverse Fourier transform operation.
11. The method according to claim 9 , in which part i and/or part vii utilize a block fast Fourier transform operation and/or block fast inverse Fourier transform operation.
12. The method according to claim 9 , comprising preconditioning the iterative conjugate gradient algorithm with a square full-rank matrix P according to
P
=
(
∑
c
=
0
N
cells
-
1
H
(
c
)
(
H
(
c
)
)
H
+
σ
2
I
)
-
1
;
wherein N cells is a total number of cells considered,
σ 2 is variance of additive white Gaussian noise AWGN
H (c) is a block-circulant matrix, and
I is an identity matrix.
13. A method according to claim 8 , in which the matrix-vector product is computed as a result vector a k =Ap k by:
a) initialising the result vector a k to zero;
b) for each cell indexed as c=0 to N cells −1:
i. multiplying the vector p k by a block-diagonal matrix (D (c) ) H ;
ii. performing an inverse fast Fourier transform operation on the result of part i;
iii. multiplying the result of part ii by the diagonal descrambling code matrix;
iv. multiplying the result of part iii with a despreading code matrix containing all active spreading codes;
v. multiplying the result of part iv by a diagonal covariance matrix
∑
x
n
(
c
)
;
vi. multiplying the result of part v with a spreading code matrix containing all active spreading codes;
vii. multiplying the result of part vi by a diagonal scrambling code matrix;
viii. perform a fast Fourier transform operation on the result of part vii;
ix. multiplying the result of part viii by a block-diagonal matrix D (c)
x. adding the result from part ix to the result vector a k ; and
c) adding variance σ 2 to all elements of the result vector a k ;
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part iv and/or in part vi, and/or the fast Fourier transform operation utilized in part ii and/or the fast inverse Fourier transform operation utilized in part viii.
14. The method according to claim 13 , in which the inverse fast Fourier transform operation in part ii is a block inverse fast Fourier transform operation and/or the fast Fourier transform operation in part viii is a block fast Fourier transform operation.
15. The method according to claim 13 , comprising preconditioning the iterative conjugate gradient algorithm with a square full-rank matrix P according to
P
=
(
∑
c
=
0
N
cells
-
1
D
(
c
)
(
D
(
c
)
)
H
+
σ
2
I
)
-
1
wherein N cells is a total number of cells considered,
D (c) is a block-diagonal matrix for cell c,
σ 2 is variance of additive white Gaussian noise AWGN, and
I is an identity matrix.
16. An apparatus for controlling a receiver to use linear symbol level equalisation for estimating a transmitted symbol vector x n (0) in an nth observation window y n of signals received from multiple users each using a different spreading code, the apparatus comprising:
a processing system comprising at least one processor and at least one memory storing a computer program, where the processing system is configured to cause the apparatus to
utilize an iterative conjugate gradient algorithm to resolve a column vector z that satisfies Az=b, where A is a square matrix and b is a column vector, in which a matrix vector product Ap k used in each respective kth iteration of the iterative conjugate gradient algorithm is computed using at least one fast transform operation, in which p k is a conjugate vector for the kth iteration;
estimate a posterior mean {circumflex over (x)} LMMSE,n (0) ; wherein LMMSE is a linear minimum mean squared error, from the column vector z; and
estimate the transmitted symbol vector x n (0) using the estimated posterior mean.
17. The apparatus according to claim 16 , in which the posterior mean {circumflex over (x)} LMMSE,n (0) is estimated from the square matrix A and the column vector b, where A has an inner product form which includes computation of Σ n −1 .
18. The apparatus according to claim 17 , in which the matrix vector product Ap k is computed by:
a) multiplying the vector p k with a spreading code matrix containing all active spreading codes;
b) multiplying the result of part a) by a diagonal scrambling code matrix;
c) multiplying the result of part b) by a block-banded block-Toeplitz channel matrix H (0) ;
d) multiplying the result of part c) by the block-banded block-Toeplitz channel matrix (H (0) ) H ;
e) multiplying the result of part d) by a diagonal descrambling code matrix;
f) multiplying the result of part e) with a despreading code matrix containing the active spreading codes; and
g) computing
∑
x
n
(
0
)
-
1
p
k
by scalar multiplication with
diag
(
σ
x
n
(
0
)
-
2
)
and adding the result to the scaled result of part f), where
∑
x
n
(
0
)
-
1
is an inverse covariance matrix;
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part a) and/or in part f).
19. The apparatus according to claim 17 , in which the matrix vector product Ap k is computed by:
a) multiplying the vector p k with a spreading code matrix containing all active spreading codes;
b) multiplying the result of part a) by a diagonal scrambling code matrix;
c) perform a fast Fourier transform operation on the result of part b);
d) multiplying the transformed result from part c) by a block-diagonal matrix (D (0) ) H D (0) ,
e) performing an inverse fast Fourier transform operation on the result of part d);
f) multiplying the result of part e) by a diagonal descrambling code matrix;
g) multiplying the result of part f) with a despreading code matrix containing the active spreading codes; and
h) computing
∑
x
n
(
0
)
-
1
p
k
by scalar multiplication with
diag
(
σ
x
n
(
0
)
-
2
)
and adding the result to the scaled result of part g), where
∑
x
n
(
0
)
-
1
is an inverse covariance matrix;
in which the at least one fast transform operation is a fast Walsh-Hadamard transform operation utilized in part a) and/or in part g), and/or the fast Fourier transform operation utilized in part c) and/or the fast inverse Fourier transform operation in part e).
20. A non-transitory computer readable memory storing a set of executable instructions which, when executed on a receiver which utilizes linear symbol level equalisation for estimating a transmitted symbol vector x n (0) in an nth observation window y n of signals received from multiple users each using a different spreading code, causes the receiver to perform steps comprising:
calculating using an iterative conjugate gradient algorithm to resolve a column vector z that satisfies Az=b, where A is a square matrix and b is a column vector, in which a matrix vector product Ap k used in each respective kth iteration of the iterative conjugate gradient algorithm is computed using at least one fast transform, in which p k is a conjugate vector for the kth iteration;
estimating a posterior mean {circumflex over (x)} LMMSE,n (0) , wherein LMMSE is a linear minimum mean squared error, from the column vector z; and
estimating the transmitted symbol vector x n (0) using the estimated posterior mean.