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 $N$ -D stochastic signals directly characterizes the localization of the inter-user interference in the time–frequency phase space. A tight bound is provided, which shows the fundamental tradeoff between the TFL of a packet and the orthogonality among the multiplexed waveforms inside the packet. The Hermite–Gauss waveforms are optimally localized with regard to this measure. We also derive the expressions for the TFL of a Gabor system consisting of $N_{t}$ time and $N_{f}$ frequency shifts of a prototype, on the conventional and staggered lattices. In the limit of large $N$ , the particular properties of the prototype yield diminishing returns to the overall localization. Finally, 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 the modulations. This variability diminishes when $N$ is increased, which is consistent with the presented theory.