Linear equalization method and device for OTFS systems
Disclosed is a low-complexity linear equalization method for an Orthogonal Time Frequency Space (OTFS) system. The method may include: receiving a time domain signal passing through a linear time-varying (LTV) channel; sampling the time domain signal to obtain a sampled signal; demodulating the sampled signal to obtain a demodulated signal; performing a Symplectic Finite Fourier Transform (SFFT) on the demodulated signal to obtain a sampled delay-Doppler domain signal; determining an effective channel matrix in a delay-Doppler domain under a restriction of a rectangular window according to a time domain channel matrix; determining a linear equalization matrix according to the effective channel matrix; and equalizing the sampled delay-Doppler domain signal in a low-complexity way according to the linear equalization matrix. The disclosure also discloses a linear equalization device of an OTFS system for realizing the linear equalization method and a computer-readable storage medium.
1. A linear equalization method for an Orthogonal Time Frequency Space (OTFS) system implemented by an OTFS receiver, comprising:
receiving a time domain signal passing through a linear time-varying (LTV) channel;
sampling the time domain signal to obtain a sampled signal;
demodulating the sampled signal to obtain a demodulated signal;
performing a Symplectic Finite Fourier Transform (SFFT) on the demodulated signal to obtain a sampled delay-Doppler domain signal;
determining an effective channel matrix in a delay-Doppler domain under a restriction of a rectangular window according to a time domain channel matrix;
determining a linear equalization matrix according to the effective channel matrix; and
equalizing the sampled delay-Doppler domain signal according to the linear equalization matrix.
2. The linear equalization method of claim 1 , wherein, the effective channel matrix H eff is a block circulant matrix composed of N first circulant matrices A n .
3. The linear equalization method of claim 2 , wherein, determining an effective channel matrix in a delay-Doppler domain under a restriction of a rectangular window according to a time domain channel matrix comprises:
obtaining a time domain channel sub-matrix {tilde over (H)} p from the time domain channel matrix; wherein, {tilde over (H)} p represents the time domain channel sub-matrix of the time domain channel matrix {tilde over (H)}=diag[ , . . . , ] experienced by the p th symbol; {tilde over (H)}=diag[ , . . . , ] is a NM×NM block diagonal matrix, which represents the time domain channel matrix when a length of a cyclic prefix is greater than a maximum delay of the LTV channel;
determining each first circulant matrices A n based on the time domain channel sub-matrix {tilde over (H)} p ; and
determining the effective channel matrix H eff based on the first circulant matrices A n .
4. The linear equalization method of claim 3 , wherein, determining each first circulant matrices A n based on the time domain channel sub-matrix {tilde over (H)} p comprises:
determining the circulant matrices A n according to the following equation:
A n =FFT Mtx ( {tilde over (H)} p )
wherein, FFT Mtx ( ) represents a Fast Fourier Transform operation on a set of matrices.
5. The linear equalization method of claim 3 , wherein, determining the effective channel matrix H eff based on the circulant matrices A n comprises:
determining the effective channel matrix H eff according to the following equation:
H
eff
=
(
A
1
A
2
…
A
N
1
A
N
A
N
A
1
A
N
-
2
A
N
-
1
⋮
⋱
⋮
A
3
A
4
…
A
1
A
2
A
2
A
3
A
N
A
1
)
.
6. The linear equalization method of claim 2 , wherein, the linear equalization matrix is a zero-forcing equalization matrix W ZF ; wherein,
determining the linear equalization matrix according to the effective channel matrix comprises: determining the zero-forcing equalization matrix according to the following expression:
W ZF =H eff −1 .
7. The linear equalization method of claim 2 , wherein, the linear equalization matrix is a zero-forcing equalization matrix W ZF ; wherein,
determining the linear equalization matrix according to the effective channel matrix comprises:
for each first circulant matrices A n , performing an Inverse Fast Fourier Transform (IFFT) on the first circulant matrices A n to obtain a first intermediate matrix S t ; inverting the first intermediate matrix S t to obtain a first inversed intermediate matrix S t −1 ; performing a Fast Fourier Transform (FFT) on the first inversed intermediate matrix S t −1 to obtain a second circulant matrix B q ; and
determining the zero-forcing equalization matrix W ZF according to the second circulant matrices B q .
8. The linear equalization method of claim 7 , wherein, performing a Fast Fourier Transform (FFT) on the first inversed intermediate matrix S t −1 to obtain a second circulant matrix B q comprises:
determining the second circulant matrix B q according to the following equation:
B
q
=
1
N
F
F
T
Mtx
(
S
t
-
1
)
wherein, FFT Mtx ( ) represents a Fast Fourier Transform operation on a set of matrices.
9. The linear equalization method of claim 7 , wherein, inverting the first intermediate matrix S t to obtain a first inversed intermediate matrix S t −1 comprises:
initializing an L matrix as L=I M ;
initializing a U matrix as U=0 M ;
initializing an intermediate variable matrix as Y=0 M ;
initializing an auxiliary vector as d=[d chan ,M−D P ]; wherein, d chan =[D 1 , D 2 , D 3 , . . . , D P ] is a vector representing delays of the LTV channel; P represents a number of paths of the LTV channel; D 1 , D 2 , D 3 , . . . , D P represents a coordinate value of a delay of each path on a delay axis in the delay-Doppler domain;
determining a first M−D P columns of the L matrix according to L(i+D P ,i)=S t (i+D P ,i)/S t (i,i); where, L(i+D P , i) represents an element at a position of row i+D P and column i in the L matrix; and i is a first cyclic variable, where i=1: M−D P ;
determining a first M−D P rows of the U matrix according to U(ξ,:)=S t (ξ,:)−Σ k=2 n Σ j=1 M L(ξ,ξ−D k )U(ξ−D k ,j); where, U(ξ,:) represents row ξ of the U matrix; S t (ξ,:)) represents row ξ of the first intermediate matrix S t ; the vector ξ increases from D p +1 to D p+1 one by one; and p is a second cyclic variable, where p=1: P+1;
determining a last D P columns of the L matrix through an LU decomposition method;
determining a last D P rows of the U matrix through the LU decomposition method;
obtaining the first inversed intermediate matrix S t −1 according to the L matrix and the U matrix.
10. The linear equalization method of claim 9 , wherein, obtaining the first inversed intermediate matrix S t −1 according to the L matrix and the U matrix comprises:
setting a third cyclic variable n increasing from 1 to P one by one;
in loop n, setting a vector k increasing from d(n)+1 to d(n+1) one by one;
wherein, d(n) represents a n th element in the vector d(n); d(n+1) represents a n+1 th element in the vector d(n);
calculating the row k of the intermediate variable matrix Y according to the following expression Y(k,:)=I(k,:)−Σ i=1 n L(k,k−d(i)Y(k−d(i),:);
setting a fourth cyclic variable k increasing from D P +1 to M one by one;
in loop k, calculating row k of the intermediate variable matrix Y according to Y(k,:)=I(k,:)−L(k,1:k−1)Y(1:k−1,:);
setting a fifth cyclic variable t decreasing from N to 1 one by one;
in loop t, setting a variable f represents a bigger one between M−D P and k+1;
obtain row t of the inversed matrix S t −1 through
S
t
-
1
(
t
,
;
)
=
Y
(
t
,
:
)
-
U
(
t
,
f
:
M
)
X
(
f
:
M
,
:
)
U
(
t
,
t
)
.
11. The linear equalization method of claim 2 , wherein, the linear equalization matrix is a Minimum Mean Square Error (MMSE) equalization matrix W MMSE ; wherein,
determining the linear equalization matrix according to the effective channel matrix comprises:
determining a second intermediate matrix R according to the effective channel matrix and a transpose matrix of the effective channel matrix and a variance of channel noises;
inverting the second intermediate matrix R to obtain a second inverted intermediate matrix R −1 ; and
multiplying the second inverted intermediate matrix R −1 by the transpose matrix of the effective channel matrix to obtain the equalization matrix of the MMSE equalization W MMSE .
12. The linear equalization method of claim 11 , wherein, determining a second intermediate matrix R according to the effective channel matrix and a transpose matrix of the effective channel matrix and a variance of channel noises comprises:
determining the second intermediate matrix R according to the following expression:
R=H eff H H eff +σ 2 I
wherein, H eff H represents the conjugate transpose matrix of the effective channel matrix H eff ; σ 2 represents the variance of channel noises; and I represents an identity matrix with a same dimension with the effective channel matrix H eff .
13. The linear equalization method of claim 11 , wherein, determining a second intermediate matrix R according to the effective channel matrix and a transpose matrix of the effective channel matrix and a variance of channel noises comprises:
determining the second intermediate matrix R according to the following expression:
R
=
[
∑
n
=
1
N
A
n
H
A
[
n
]
N
+
σ
2
I
∑
n
=
1
N
A
n
H
A
[
n
+
1
]
N
…
∑
n
=
1
N
A
n
H
A
[
n
+
N
-
1
]
N
∑
n
=
1
N
A
n
H
A
[
n
+
N
-
1
]
N
∑
n
=
1
N
A
n
H
A
[
n
]
N
+
σ
2
I
∑
n
=
1
N
A
n
H
A
[
n
+
N
-
2
]
N
⋮
⋱
⋮
∑
n
=
1
N
A
n
H
A
[
n
+
1
]
N
+
σ
2
I
∑
n
=
1
N
A
n
H
A
[
n
+
2
]
N
+
σ
2
I
…
∑
n
=
1
N
A
n
H
A
[
n
]
N
+
σ
2
I
]
wherein, [⋅] N represent a modulo operation; [N] N =N.
14. The linear equalization method of claim 11 , wherein, inverting the second intermediate matrix R to obtain a second inverted intermediate matrix R −1 comprises:
initializing an L matrix as L=I M ;
initializing a U matrix as U=0 M ;
initializing an intermediate variable matrix as Y=0 M ;
initializing an auxiliary vector as d=[d chan ,M−D P ]; wherein, d chan =[D 1 , D 2 , D 3 , . . . , D P ] is a vector representing delays of the LTV channel; P represents a number of paths of the LTV channel; D 1 , D 2 , D 3 , . . . , D P represents a coordinate value of a delay of each path on a delay axis in the delay-Doppler domain;
determining a first M−D P columns of the L matrix according to L(i+D P ,i)=M(i+D P ,i)/M(i,i); where, L(i+D P , i) represents an element at a position of row i+D P and column i in the L matrix; and i is a first cyclic variable, where i=1: M−D P ;
determining a first M−D P rows of the U matrix according to U(ξ,:)=M(ξ:)−Σ k=2 n τ j=1 M L(ξ,ξ−D k )U(ξ−D k ,j); where, U(ξ,:) represents row of the U matrix; S t (ξ,:) represents row of the first intermediate matrix M; the vector increases from D p +1 to D p+1 one by one; and p is a second cyclic variable, where p=1: P+1;
determining a last D P columns of the L matrix through an LU decomposition method;
determining a last D P rows of the U matrix through the LU decomposition method;
obtaining the second inversed intermediate matrix R −1 according to the L matrix and the U matrix.
15. The linear equalization method of claim 14 , wherein, obtaining the second inversed intermediate matrix R −1 according to the L matrix and the U matrix comprises:
setting a third cyclic variable n increasing from 1 to P one by one;
in loop n, setting a vector k increasing from d(n)+1 to d(n+1) one by one;
wherein, d(n) represents a n th element in the vector d; d(n+1) represents a n+1 th element in the vector d;
calculating the row k of the intermediate variable matrix Y according to the following expression Y(k,:)=I(k,:)−L(k,k−d(i))Y(k−d(i),:);
setting a fourth cyclic variable k increasing from D p +1 to M one by one;
in loop k, calculating row k of the intermediate variable matrix Y according to Y(k,:)=I(k,:)−L(k,1:k−1)Y(1:k−1,:);
setting a fifth cyclic variable t decreasing from N to 1 one by one;
in loop t, setting a variable f represents a bigger one between M−D P and k+1;
obtain row t of the inversed matrix R −1 through
R
-
1
(
t
,
;
)
=
Y
(
t
,
:
)
-
U
(
t
,
f
:
M
)
X
(
f
:
M
,
:
)
U
(
t
,
t
)
.
16. An electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the linear equalization method of claim 1 .
17. A non-transitory computer-readable storage medium, wherein the non-transitory computer-readable storage medium stores computer instructions, and the computer instructions being used to make a computer execute the linear equalization method of claim 1 .