Sequences with minimal time-frequency spreads

Abstract

For a given time or frequency spread, one can always find continuoustime signals, which achieve the Heisenberg uncertainty principle bound. This is known, however, not to be the case for discrete-time sequences; only widely spread sequences asymptotically achieve this bound. We provide a constructive method for designing sequences that are maximally compact in time for a given frequency spread. By formulating the problem as a semidefinite program, we show that maximally compact sequences do not achieve the classic Heisenberg bound. We further provide analytic lower bounds on the time-frequency spread of such signals.

Achievable simultaneous time and frequency domain energy concentration for finite length sequences

“…This phenomenon and related properties are well defined and extensively studied in the literature for continuous signals [20,21]. However, for discrete sequences, the Heisenberg uncertainty bound is not achievable [22,23]. Parhizkar et al [23] have introduced the uncertainty minimisers for discrete sequences and termed them as most compact sequences.…”