Systems and methods for generating sequences that are nearest to a set of sequences with minimum average cross-correlation

1. A method for generating sequences that are nearest to a set of sequences with minimum average cross-correlation, comprising:

projecting, by a communications device, each element of the set of sequences to a nearest constellation point;

converting, by the communications device, the set of sequences into a time domain representation;

producing, by the communications device, a number of orthogonal cyclic shifts equal to a sequence length for each element in the set of sequences;

performing, by the communications device, an inverse discrete Fourier Transform (IDFT) on the set of sequences;

evaluating, by the communications device, a cubic metric of each root sequence of the set of sequences, each root sequence being identical to an original sequence;

removing from the set of sequences, by the communications device, each sequence associated with each of the root sequence having the cubic metric that exceeds a threshold; and

obtaining, by the communications device, a minimum, maximum cross-correlation for the set of sequences.

2. The method of claim 1 , wherein the constellation point is a quadrature phase-shift keying (QPSK) constellation point.

3. The method of claim 1 , wherein projecting each element of each sequence comprises obtaining a point in the set

{

±

2

2

±

j

⁢

2

2

}

that is nearest to the element.

4. The method of claim 3 , further comprising obtaining the point in the set

{

±

2

2

±

j

⁢

2

2

}

that is nearest, in Euclidean metric, to the element.

5. The method of claim 1 , wherein the set of sequences is obtained by:

determining an input multiple input multiple output signal;

obtaining a nearest tight frame to one or more given structured vectors;

obtaining one or more structured vectors from the nearest tight frame;

projecting the one or more structured vectors onto a space of circulant matrices; and

outputting one or more classes of matrices associated with reference signals.

6. The method of claim 1 , wherein the set of sequences is a set of quadrature phase-shift keying (QPSK) sequences.

7. The method of claim 1 , further comprising converting the set of sequences to a time domain representation.

8. The method of claim 1 , wherein the set of sequences is obtained by:

providing a first matrix on a unit hyper-sphere with non-zero components;

computing a second matrix that is a nearest tight frame to the first matrix;

computing a third matrix that is a closest matrix with minimum peak-to-average-power ratio to the second matrix;

computing a fourth matrix from an expansion of the third matrix;

computing a fifth matrix that is a nearest circulant matrix to the forth matrix;

setting the first matrix to the fifth matrix; and

outputting the fourth matrix and the fifth matrix and a maximum of the fourth matrix and the fifth matrix.

9. The method of claim 1 , wherein the set of sequences is obtained by:

providing a matrix Z 0 C dxN on a unit hyper-sphere with non-zero components;

computing

∝

(

ZZ

H

)

1

2

⁢

Z

→

Y

;

computing

(

∝

(

W

m

H

⁢

W

m

)

1

2

⁢

W

m

H

)

H

→

V

m

;

computing a Q matrix that is a nearest circulant matrix to the V matrix;

computing a W matrix that is a closest matrix with minimum peak-to-average-power ratio to the Y matrix;

setting the Z matrix as the Q matrix; and

outputting the V matrix and the Q matrix and a max k≠n <v k , v n > and a max k≠n <q k , q n >;

where Z represents a matrix, C represents a circulant matrix, N represents a number of column vectors, and d represents Euclidean d-space.

10. A communications device that is configured to generate sequences that are nearest to a set of sequences with minimum average cross-correlation, the communications device comprising:

a processor;

memory in electronic communication with the processor;

instructions stored in the memory, the instructions being executable to:

project each element of a set of sequences to a nearest constellation point;

convert the set of sequences into a time domain representation;

produce a number of orthogonal cyclic shifts equal to a sequence length for each element in the set of sequences;

perform an inverse discrete Fourier Transform (IDFT) on the set of sequences;

evaluate a cubic metric of each of the root sequence of the set of sequences, each root sequence being identical to an original sequence;

remove from the set of sequences each sequence associated with the root sequence having the cubic metric that exceeds a threshold; and

obtain a minimum, maximum cross-correlation for the set of sequences.

11. The communications device of claim 10 , wherein the constellation point is a quadrature phase-shift keying (QPSK) constellation point.

12. The communications device of claim 10 , wherein projecting each element of each sequence comprises obtaining a point in the set

{

±

2

2

±

j

⁢

2

2

}

that is nearest to the element.

13. The communications device of claim 12 , wherein the instructions are further executable to obtain the point in the set

{

±

2

2

±

j

⁢

2

2

}

that is nearest, in Euclidean metric, to the element.

14. The communications device of claim 10 , wherein the set of sequences is obtained by the instructions being further executable to:

determine an input multiple input multiple output signal;

obtain a nearest tight frame to one or more given structured vectors;

obtain one or more structured vectors from the nearest tight frame;

project the one or more structured vectors onto a space of circulant matrices; and

output one or more classes of matrices associated with reference signals.

15. The communications device of claim 10 , wherein the set of sequences is a set of quadrature phase-shift keying (QPSK) sequences.

16. The communications device of claim 10 , wherein the instructions are further executable to convert the set of sequences to a time domain representation.

17. A non-transitory computer-readable medium comprising executable instructions for:

projecting each element of a set of sequences to a nearest constellation point;

converting the set of sequences into a time domain representation;

producing a number of orthogonal cyclic shifts equal to a sequence length for each element in the set of sequences;

performing an inverse discrete Fourier Transform (IDFT) on the set of sequences;

evaluating a cubic metric of each of the root sequence of the set of sequences, each of the root sequence being identical to an original sequence;

removing from the set of sequences each sequence associated with the root sequence having the cubic metric that exceeds a threshold; and

obtaining a minimum, maximum cross-correlation for the set of sequences.

18. The non-transitory computer-readable medium of claim 17 , wherein the constellation point is a quadrature phase-shift keying (QPSK) constellation point.