Time-Frequency Localization Measures for Packets of Orthogonally Multiplexed Signals

We consider measures of time-frequency localization (TFL) for stochastic signals. The approach is complementary to the use of TFL in prototype filter design; here, TFL is instead applied to multiplexed waveform packets, with the objective to evaluate multi-user interference in a multiple access scenario rather than combat channel dispersion. We show that a generalization of the Heisenberg parameter to N-dimensional stochastic signals directly characterizes the localization of the inter-user interference in the time-frequency phase space. A tight bound is provided that shows the fundamental trade-off between the TFL of a packet and orthogonality among the multiplexed waveforms inside the packet. Hermite-Gauss waveforms are optimally localized with regard to this measure. We also derive expressions for the TFL of a Gabor system consisting of Nt time-and Nf frequency-shifts of a prototype, on conventional and staggered lattices. In the limit of large N, the particular properties of the prototype yield diminishing returns to the overall localization. Lastly, we compare the performance of waveforms in a connectionless and asynchronous random access scenario. At lower access intensities, where the out-of-band emissions are the significant limiting factor, the outage probability for smaller access packets is shown to vary significantly between modulations. This variability diminishes when N is increased, consistent with the presented theory.