In this paper, we derive the fundamental limits of low earth orbit (LEO) enabled localization by analyzing the available information in signals from multiple LEOs during different transmission time slots received on a multiple antennas and evaluate the utility of these signals for 9D localization (3D position, 3D orientation, and 3D velocity estimation). We start by deriving the Fisher Information Matrix (FIM) for the channel parameters that are present in the signals received from LEOs in the same or multiple constellations during multiple transmission time slots. To accomplish this, we define a system model that captures i) time offset between LEOs caused by having relatively cheap clocks, ii) frequency offset between LEOs, iii) the unknown Doppler rate caused by high mobility LEOs, and iv) multiple transmission time slots from a particular LEO. We transform the FIM for the channel parameters to the FIM for the location parameters and determine the required conditions for localization. To do this, we start with the 3D localization cases: i) 3D positioning with known velocity and orientation, ii) 3D orientation estimation with known position and velocity, and iii) 3D velocity estimation with known position and orientation. Subsequently, we derive the FIM for the full 9D localization case (3D position, 3D orientation, and 3D velocity estimation) in terms of the FIM for the 3D localization. Using these results, we determine the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas that allow for different levels of location estimation. We then provide insights into the interaction between the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas. One key result is that in the presence of time and frequency offsets and Doppler rate, it is possible to perform 9D localization (3D position, 3D velocity, and 3D orientation estimation) of a receiver by utilizing the signals from three LEO satellites observed during three transmission time slots received through multiple receive antennas.

In this paper, we rigorously derive the information in the signals received from low earth orbit (LEO) satellites, which are unsynchronized in time and frequency, and their utility for 3D position estimation. To enable this derivation, we define a system model that captures i) the time offset between LEOs caused by having cheap clocks, ii) the frequency offset between LEOs, and iii) multiple transmission time slots from a particular LEO. After this definition, we derive the Fisher information matrix (FIM) for the relevant channel parameters and transform the FIM for the channel parameters to the FIM for the 3D position. These derivations show the interactions between the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas. Subsequently, these allow us to determine the minimal number of LEOs, the number of transmission time slots, and the number of receive antennas needed to determine the 3D position. One key result is that when the LEOs are unsynchronized in time and frequency and experience a high Doppler rate, the 3D position can be determined by observing a single LEO for four transmission time slots.

Large antenna arrays and reconfigurable intelligent surfaces (RIS) have been made available due to the use of higher frequency bands, and there is the possibility that these arrays can become disturbed. Hence, their orientation could change after deployment. Since low earth orbits (LEO) are being proposed to provide position, navigation, and timing services, and LEOs from different constellations could be unsynchronized in time and frequency and experience a high Doppler rate. We ask, "can unsynchronized LEOs provide 3D orientation for a ground receiver?" To answer this question, we introduce the Fisher information matrix (FIM) and use the FIM to quantify the available information needed for 3D orientation estimation utilizing signals received from LEOs during multiple transmission time slots across multiple receive antennas. We observe by analyzing the positive definitiveness of the FIM for the 3D orientation that irrespective of the presence or absence of both time and frequency offsets, the 3D orientation of the receiver can be estimated through the multiple TOA measurements received across the receive antennas from two LEO satellites during a single transmission time slot. We also observe by analyzing the positive definitiveness of the FIM for the 3D orientation that irrespective of the presence or absence of both time and frequency offsets, the 3D orientation of the receiver can be estimated through the multiple TOA measurements received across the receive antennas during two transmission time slots from a single LEO satellite.

This paper develops fundamental bounds for localization in orthogonal frequency division multiplexing (OFDM) systems aided by reconfigurable intelligent surfaces (RISs). Specifically, we start from the assumption that the position and orientation of a RIS can be viewed as prior information for RIS-aided localization in wireless systems and derive Bayesian bounds for the localization of a user equipment (UE). To do this, we first derive the Bayesian Fisher information matrix (FIM) for channel parameters to derive the Bayesian localization bounds. Then, to focus on the geometric channel parameters, we derive the equivalent Fisher information matrix (EFIM) and show that it has a definite structure. Subsequently, we show through the information loss associated with the EFIM that when the RIS reflection coefficients remain constant across all OFDM symbols, and there is no prior information about the nuisance parameters, the corresponding submatrix in the EFIM related to the RIS angle parameters is a zero matrix. As a result of the EFIM being a zero matrix, estimating the RIS-related angle channel parameters is not possible when the RIS reflection coefficients remain constant across all OFDM symbols. This observation is crucial for the estimation of the RIS-related angle parameters. It dictates that to estimate the RIS-related angle parameters, there must be more than one OFDM transmission with differing RIS reflection coefficients. Furthermore, due to this observation, we note that localization of a single antenna UE through the signals received from reflections from a single RIS to the UE is not feasible in the far-field when the RIS reflection coefficients remain constant across all OFDM symbols. We also show that the FIM for the RIS-related channel parameters can be decomposed into i) information provided by the receiver, ii) information provided by the transmitter, and iii) information provided by the RIS components. We then transform the Bayesian EFIM for geometric channel parameters to the Bayesian FIM for the UE position and orientation parameters and examine its specific structure under a particular class of RIS reflection coefficients.

This paper presents a rigorous examination of the estimation of the misorientation of a reconfigurable intelligent surface (RIS) based on the received signal when the user equipment (UE) is in the near or far fields of the RIS. The Bayesian analysis views the location of the RISs as a priori system-level information. With incorrect a priori information, the position and orientation offsets of the RISs become parameters that need to be estimated and fed back to the Base station (BS) for correction. Two key insights are obtained from our Bayesian analysis. First, the Bayesian equivalent Fisher information matrix (EFIM) for the channel parameters indicates that the RIS orientation offset cannot be estimated when there is an unknown phase offset in the received signal in the far-field propagation regime. Second, the corresponding EFIM for the channel parameters in the received signal observed in the near-field shows that this unknown phase offset does not hinder the estimation of the RIS orientation offset when the UE has more than one receive antenna.

This paper presents a rigorous Bayesian analysis of the information in the signal (consisting of both the line-of-sight (LOS) path and reflections from multiple reconfigurable intelligent surfaces (RISs)) that originate from a single base station (BS) and is received by a user equipment (UE). For a comprehensive Bayesian analysis, both near and far field regimes are considered. The Bayesian analysis views both the location of the RISs and previous information about the UE as a priori information for UE localization. With outdated a priori information, the position and orientation offsets of the RISs become parameters that need to be estimated and fed back to the BS for correction. We first show that when the RIS elements have a half wavelength spacing, this RIS orientation offset is a factor in the pathloss of the RIS paths. Subsequently, we show through the Bayesian equivalent Fisher information matrix (EFIM) for the channel parameters that the RIS orientation offset cannot be corrected when there is an unknown phase offset in the received signal in the far-field regime. However, the corresponding EFIM for the channel parameters in the received signal observed in the near-field shows that this unknown phase offset does not hinder the estimation of the RIS orientation offset when the UE has more than one receive antenna. Furthermore, we use the EFIM for the UE location parameters to present bounds for UE localization in the presence of RIS uncertainty. We rigorously show that regardless of size and propagation regime, the RISs are only helpful for localization when there is a priori information about the location of the RISs. Finally, through numerical analysis of the EFIM and its smallest eigenvalue, we demonstrate the loss in information when the far-field model is incorrectly applied to the signals received at a UE experiencing near-field propagation.

In this paper, we present a differential modulation and detection scheme for use in the uplink of a system with a large number of antennas at the base station, each equipped with low-resolution analog-to-digital converters (ADCs). We derive an expression for the maximum likelihood (ML) detector of a differentially encoded phase information symbol received by a base station operating in the low-resolution ADC regime. We also present an equal performing reduced complexity receiver for detecting the phase information. To increase the supported data rate, we also present a maximum likelihood expression to detect differential amplitude phase shift keying symbols with low-resolution ADCs. We note that the derived detectors are unable to detect the amplitude information. To overcome this limitation, we use the Bussgang Theorem and the Central Limit Theorem (CLT) to develop two detectors capable of detecting the amplitude information. We numerically show that while the first amplitude detector requires multiple quantization bits for acceptable performance, similar performance can be achieved using one-bit ADCs by grouping the receive antennas and employing variable quantization levels (VQL) across distinct antenna groups. We validate the performance of the proposed detectors through simulations and show a comparison with corresponding coherent detectors. Finally, we present a complexity analysis of the proposed low-resolution differential detectors.