Method and system for selecting spreading sequences with variable spreading factors

A method for selecting an N length ternary sequence with elements {1, 0, −1} from a spreading sequence of length N with elements {1, 0}, the method comprising:

• generating all combinations of binary sequences with at least one of alphabets {+1, −1} for said spreading sequence with said variable spreading factor of length N;
• generating combinations of ternary sequences of length N with at least one of alphabets {+1, 0, −1} for said spreading sequence of length N, wherein said combinations of ternary sequences are generated by substituting positions of ones in said selected spreading sequence of length N by all combinations of at least one of elements {1, −1};
• selecting a set of sequences with minimum sum of squares of cyclic correlations of all said combinations of said ternary sequences;
• computing a probability of correct detection over a range of Signal-to-noise ratio (SNR) values for said set of sequences with minimum sum of squares of said cyclic correlations;
• and
• selecting a ternary sequence for which said probability of correct detection is maximum as ternary synchronization preamble.

A method for selecting an N length ternary sequence with elements {1, 0, −1} from a spreading sequence of length N with elements {1, 0}, the method comprising:

• generating a ternary sequence with elements {1, 0, −1} of length N from said spreading sequence with elements {1, 0} of length N;
• generating an PN sequence with elements {1,−1} of length N/2−1, wherein said PN sequence is appended with a ‘−1’ to obtain balanced binary sequences with N/4+1s and N/4−1s;
• creating a benchmark ternary sequence by substituting said balanced binary sequence at positions of ones in said spreading sequence with elements {1, 0}, wherein said balanced binary sequence is multiplied with various masking functions and computing all cyclic auto-correlation values of said balanced binary sequence at all lags;
• creating unbalanced binary sequences from balanced binary sequences using at least one masking function, wherein said at least one masking function is multiplied with said balanced binary sequences such that number of ones exceed number of zeros;
• creating unbalanced ternary sequences from said unbalanced binary sequences by substituting said unbalanced binary sequences with alphabets {+1, −1s} in positions of ones in said selected N length spreading sequence with elements {1, 0} and computing all cyclic auto-correlation values of each sequence of said unbalanced binary sequences at all lags;
• selecting all ternary sequences from said unbalanced ternary sequences whose maximum out of phase cyclic auto-correlations are less than maximum out of phase cyclic autocorrelations of said benchmark ternary sequence as candidate ternary spreading sequences;
• and
• selecting, from among the selected all ternary sequences, at least one said ternary sequence with a probability of correct detection performance satisfying constraints as said candidate ternary spreading sequences for synchronization preamble sequences.

A system for selecting spreading sequences with variable spreading factors, the system is configured to:

• generate circular shift equivalent sets of spreading sequences by circularly shifting a base sequence with elements {1, 0} and having at least one said variable spreading factor;
• determine whether each said spreading sequence in said circular shift equivalent set meets a pre-defined spreading sequence criteria, wherein said spreading sequence criteria comprises a balanced criteria, a non-repetition criteria, a non-circular criteria, and a conjugate criteria;
• and
• select at least one spreading sequence from expansions of at least one said spreading sequence of said circular shift equivalent sets in response to determining that said spreading sequences in said circular shift equivalent sets meets said pre-defined spreading sequence criteria.

A system for selecting an N length ternary sequence with elements {1, 0, −1} from a spreading sequence of length N with elements {1,0}, the system is configured to:

• generate all combinations of binary sequences with at least one of alphabets {+1, −1} for said spreading sequence with said variable spreading factor of length N;
• generate combinations of ternary sequences of length N with at least one of alphabets {+1, 0, −1} for said spreading sequence of length N, wherein said combinations of ternary sequences are generated by substituting positions of ones in said selected spreading sequence of length N by all combinations of at least one of elements {1, −1};
• select a set of sequences with minimum sum of squares of cyclic correlations of all said combinations of said ternary sequences;
• compute a probability of correct detection over a range of SNR values for said set of sequences with minimum sum of squares of said cyclic correlations;
• and
• select a ternary sequence for which said probability of correct detection is maximum as ternary synchronization preamble.

A system for selecting an N length ternary sequence with elements {1, 0, −1} from a spreading sequence of length N with elements {1, 0}, the system is configured to:

• generate a ternary sequence with elements {1, 0, −1} of length N from said spreading sequence with elements {1, 0} of length N;
• generate an PN sequence with elements {1,−1} of length N/2−1, wherein said PN sequence is appended with a ‘−1’ to obtain balanced binary sequences with N/4+1s and N/4−1s;
• create a benchmark ternary sequence by substituting said balanced binary sequence at positions of ones in said spreading sequence with elements {1, 0}, wherein said balanced binary sequence is multiplied with various masking functions and computing all cyclic auto-correlation values of said balanced binary sequence at all lags;
• create unbalanced binary sequences from balanced binary sequences using at least one masking function, wherein said at least one masking function is multiplied with said balanced binary sequences such that number of ones exceed number of zeros;
• create unbalanced ternary sequences from said unbalanced binary sequences by substituting said unbalanced binary sequences with alphabets {+1, −1s} in positions of ones in said selected N length spreading sequence with elements {1, 0} and computing all cyclic auto-correlation values of each sequence of said unbalanced binary sequences at all lags;
• select all ternary sequences from said unbalanced ternary sequences whose maximum out of phase cyclic auto-correlations are less than maximum out of phase cyclic autocorrelations of said benchmark ternary sequence as candidate ternary spreading sequences;
• and
• select, from the selected all ternary sequences, at least one said ternary sequence with a probability of correct detection performance as said candidate ternary spreading sequences for synchronization preamble sequences.