Systems and methods for designing a reference signal to be transmitted in a multiplexed cellular system

1. A method for designing a reference signal to be transmitted in a multiplexed cellular system, comprising:

selecting a first set of matrices that comprises a plurality of matrices;

determining, by a processor, an optimal partition of the first set of matrices;

forming a third matrix by a processor, wherein the third matrix represents a cross-correlation of a second matrix and each of the plurality of first matrices;

setting a minimum corresponding to the first set of matrices;

updating the second matrix using the minimum corresponding to the first set of matrices; and

scaling each column of the second matrix by a first value.

2. The method of claim 1 , further comprising:

computing a complex conjugation of the second matrix;

computing a function from each column of the second matrix;

obtaining a complex conjugation of the function with minimum Peak to Average Power Ratio (PAPR);

mapping the complex conjugation of the function to a first vector associated with the second matrix; and

computing a second vector nearest to the first vector with minimum PAPR in the frequency domain.

3. The method of claim 1 , further comprising using clustering algorithms to minimize cross-correlation for matrices comprising sub-matrices of Constant Amplitude Zero Autocorrelation (CAZAC) sequences.

4. The method of claim 1 , further comprising using a matrix metric based on a magnitude of a normalized cross-correlation.

5. The method of claim 1 , further comprising using initial conditions that are based on the Frobenius norm of d×d sub-matrices of the Gram matrix of a set of initial blocks of sequences.

6. The method of claim 5 , further comprising determining an initial representation of a set of sequences which can be sub-divided into subsets.

7. The method of claim 6 , wherein the subsets are CAZAC sequences.

8. The method of claim 5 , further comprising using the initial conditions to design a set of reference signals for transmission of uplink demodulation reference signals.

9. The method of claim 1 , further comprising using initial conditions that are based on choosing a series of random subsets of a training set and choosing the initial set of matrices from the series of random subsets that has minimum average cross-correlation.

10. The method of claim 1 , further comprising optimizing PAPR in time and frequency domains.

11. The method of claim 1 , wherein each column of the second matrix is scaled by its Euclidean norm.

12. A communications device that is configured to design a reference signal to be transmitted in a multiplexed cellular system, the communications device comprising:

a processor;

memory in electronic communication with the processor;

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

select a first set of matrices that comprises a plurality of matrices;

determine an optimal partition of the first set of matrices;

form a third matrix, wherein the third matrix represents a cross-correlation of a second matrix and each of the plurality of first matrices;

set a minimum corresponding to the first set of matrices;

update the second matrix using the minimum corresponding to the first set of matrices; and

scale each column of the second matrix by a first value.

13. The communications device of claim 12 , wherein the instructions are further executable to:

compute a complex conjugation of the second matrix;

compute a function from each column of the second matrix;

obtain a complex conjugation of the function with minimum Peak to Average Power Ratio (PAPR);

map the complex conjugation of the function to a first vector associated with the second matrix; and

compute a second vector nearest to the first vector with minimum PAPR in the frequency domain.

14. The communications device of claim 12 , wherein the instructions are further executable to use clustering algorithms to minimize cross-correlation for matrices comprising sub-matrices of Constant Amplitude Zero Autocorrelation (CAZAC) sequences.

15. The communications device of claim 12 , wherein the instructions are further executable to use a matrix metric based on a magnitude of a normalized cross-correlation.

16. The communications device of claim 12 , wherein the instructions are further executable to use initial conditions that are based on the Frobenius norm of d×d sub-matrices of the Gram matrix of a set of initial blocks of sequences.

17. The communications device of claim 16 , wherein the instructions are further executable to determine an initial representation of a set of sequences which can be sub-divided into subsets.

18. The communications device of claim 17 , wherein the subsets are CAZAC sequences.

19. The communications device of claim 16 , wherein the instructions are further executable to use the initial conditions to design a set of reference signals for transmission of uplink demodulation reference signals.

20. The communications device of claim 12 , wherein the communications device is a mobile cellular device.

21. A computer-readable medium comprising executable instructions to perform the actions of:

selecting a first set of matrices that comprises a plurality of matrices;

determining an optimal partition of the first set of matrices;

forming a third matrix, wherein the third matrix represents a cross-correlation of a second matrix and each of the plurality of first matrices;

setting a minimum corresponding to the first set of matrices;

updating the second matrix using the minimum corresponding to the first set of matrices; and

scaling each column of the second matrix by a first value.