Time-frequency space constructions of families of signals
A computer-implemented method and system for generating large families of sequences with desirable properties for many applications, including communications and radar applications, applies constraints to a sequence in the Zak space, modulates the constrained sequence in the Zak space, and determines permutations of the modulated sequence in the Zak space. The constraints are associated with predetermined properties, including predetermined autocorrelation and cross-correlation properties. Other embodiments of the computer-implemented method and system transform an input sequence into a transformed sequence using the finite Zak transform and determine at least one other different sequence based on the transformed sequence. The at least one other different sequence can be determined by collecting a plurality of sequences that are finitely supported on an algebraic line in the Zak space and modulating and/or determining permutations of some or all of the sequences.
1. A computerized-method of generating a sequence set in a computer, comprising:
determining through the computer L−1 sequences for the set by:
a) selecting through the computer one L×L complex-value array for each value of ā, where 1≦ā<L and where each L×L complex-valued array is in the finite-Zak-transform space, each complex-valued array comprising a time index k and a frequency index j, and
(b) setting through the computer the magnitude of each L×L complex-valued array equal to (1) L if āk+j≡0, modulo L, where 1≦ā<L, and (2) zero, otherwise;
generating (L−2)! through the computer additional sequence sets by permuting non-zero entries in the L−1 sequences; and
storing the (L−2)! in a memory of the computer the additional sequence sets such that the additional sequence sets can be retrieved or transmitted during signal processing.
2. A computerized-method of generating a sequence set in a computer, comprising:
determining through the computer L−1 sequences for the set by:
a) selecting through the computer one L×L complex-value array for each value of ā, where 1≦ā<L and where each L×L complex-valued array is in the finite-Zak-transform space, each complex-valued array comprising a time index k and a frequency index j, and
(b) setting through the computer the magnitude of each L×L complex-valued array equal to (1) L if āk+j≡0, modulo L, where 1≦ā<L, and (2) zero, otherwise;
generating through the computer L to the L-th power additional sequence sets by:
multiplying through the computer non-zero entries in the finite-Zak Transform space of the sequence set by a unique set of values from a set {1, e_L(1), . . . e_L(L−1)}; and
storing in a memory of the computer the L to the L-th power additional sequence sets such that the additional sequence sets can be retrieved or transmitted during signal processing.