Metric entropy of causal, discrete-time LTI systems

In [1] it is shown that recurrent neural networks (RNNs) can learn - in a metric entropy optimal manner - discrete time, linear time-invariant (LTI) systems. This is effected by comparing the number of bits needed to encode the approximating RNN to the metric entropy of the class of LTI systems under consideration [2, 3]. The purpose of this note is to provide an elementary self-contained proof of the metric entropy results in [2, 3], in the process of which minor mathematical issues appearing in [2, 3] are cleaned up. These corrections also lead to the correction of a constant in a result in [1] (see Remark 2.5).Comment: [1] arXiv:2105.0255

Intra- and Intersystem Interference in GNSS: Performance Models and Signal Design

The European Galileo, the American Global Positioning System (GPS), and other global navigation satellite systems (GNSSs) transmit direct-sequence spread spectrum (DSSS) signals from space, allowing receivers on Earth to compute their position, velocity, and time (PVT) based on the principle of pseudorange trilateration. However, as multiple satellites and systems transmit signals simultaneously within shared frequency bands, multiple access interference (MAI) in the form of intra- and intersystem interference can affect signal processing at the receiver. To compute a pseudorange, the receiver must estimate synchronization parameters of the respective signal with high resolution. This synchronization is performed in a two-step approach, consisting of signal acquisition (detection) and fine parameter estimation. Most GNSSs rely on asynchronous direct-sequence code-division multiple access (DS-CDMA), assigning different pseudorandom noise (PRN) code to each satellite. This multiple access scheme involves a controlled level of MAI degrading acquisition and parameter estimation performance, which needs to be carefully modeled before launching new signals or raising transmit power levels. The International Telecommunications Union (ITU) regulates that radio frequency compatibility (RFC) of systems, satellites and signals within the radionavigation frequency bands must be ensured, meaning that receiver performance must not be harmed significantly. Conventional receiver performance models are based on the spectral separation coefficient (SSC) between desired and interfering signal, and mostly rely on the idealization that GNSS signals are wide-sense stationary (WSS), circularly-symmetric Gaussian (CSG) random processes. In this work, we propose refined models for performance of coarse and fine estimation of synchronization parameters, taking into account the signals’ wide-sense cyclostationary (WSCS) property and their non-circularity. This is of particular interest in light of the recent signal design trend towards novel coarse/acquisition (C/A) signals with short PRN codes, which are especially vulnerable to MAI but very attractive for the group of mass-market GNSS-enabled electronic devices. Ultimately, our performance model enables the C/A signal designer to minimize the PRN code length while ensuring a given acquisition performance constraint. Moreover, with regard to RFC of an increasing number of navigation systems, satellites, and signals, our detailed models for interference effects on user equipment will allow to make more efficient use of the available radio frequency spectrum