Indoor Radio Positioning Using Pseudolites in the UHF Band – Signals and Ranging Precision Survey

Jiří Svatoň; Jan Sýkora; Daniel Skopec; František Vejražka; Jan Roháč; Jan Spáčil; Milan Švanda; Martin Tomis,; Jan Škaloud

doi:10.33012/navi.783

Abstract

This contribution presents a theoretical and practical analysis of the precision of deep-indoor radio positioning using sub-GHz (ultra-high frequency) band ranging pseudolites, penetrating whole buildings and blocks. Working deep indoors, pseudolite systems face a negative signal-to-noise ratio and a challenging dynamic power range. These challenges lead to small nuances and limitations compared with global navigation satellite systems. Therefore, ranging codes, modulation techniques, and corresponding time-delay/range, clock, carrier, and signal-to-noise estimators are analyzed and discussed to demonstrate the limits of deep-indoor ranging. Then, the ranging precision is evaluated based on the Cramer-Rao lower bound of ranging variance: its analytical development is followed by numerical computation and comparison with results from real measurement campaigns and a particular model of losses.

Keywords

1 INTRODUCTION

1.1 Background and Motivation

Indoor positioning with small to medium area coverage (on the scale of a whole building, city block, or campus) using a dedicated fixed infrastructure outside individual rooms (unlike Wi-Fi or ultra-wideband-based local or room-coverage-based systems) is a long-term goal of many academic and commercial efforts. Such efforts have utilized a pseudolite approach (Cobb, 1997; Wang, 2002), such as Locata (Barnes et al., 2003), NextNav (Meiyappan et al., 2013), or similar methods using multilateration (time of arrival [ToA], time difference of arrival [TDoA]), as in the works of Soderholm et al. (2001), Hameed et al. (2024), and Borio and Gioia (2014), as well as two-way ranging (TWR), as reported by Myrick (2018), Yu et al. (2023), Herschfelt (2019), and others (see Figure 1).

FIGURE 1

Different positioning topologies (1: ToA, 2: TDoA, 3: TWR)

A priori knowledge of the positioning error (accuracy) is crucial for the design of these systems. As a simplification, the positioning (root mean square [RMS]) error RMS_Δx in ToA topology is, according to Misra and Enge (2010), determined by the (pseudorange) measurement variance σ² and a geometrical factor (i.e., dilution of precision, DOP):

$R M S_{Δ X} = \sqrt{σ^{2}} D O P$ 1

where $\sqrt{σ^{2}}$ is the standard deviation, corresponding to the precision considered in this work. The DOP represents the area of intersection of concentric circles/hyperbolas with spacing given by $\sqrt{σ^{2}}$ (see Figure 2(a)). Based on the topology, σ² will affect RMS_ΔX once (ToA) or twice (TDoA, TWR). The geometry is often firmly set; thus, an analysis of σ² is crucial for system design.

FIGURE 2

(a) Positioning error and geometry and (b) measurement variance

σ² is defined as the variance of the time-delay/range estimator. An estimator for ranging signals is typically implemented as a search for the maximum of the cross-correlation function between the received signal and its replica (see Figure 2(b)).

The lower bound of σ² is given by the Cramér–Rao lower bound (CRLB) (Rao, 1945). This term is a function of the signal-to-noise ratio (SNR) and the power spectrum density (PSD). Achieved values of σ² in practice for specific laboratory experiments are sometimes reported in the literature. However, experiments focusing on ranging across buildings, such as those by Meiyappan et al. (2013), are rare, and relevant analyses are always missing. A deeper analysis of σ² is essential for determining the positioning error in Equation (1) and for extrapolating these results, e.g., for applications in indoor use.

This contribution’s motivation is the deployment of Locata and NextNav and their adoption as a potential global navigation satellite system (GNSS) backup by authorities. However, the performance of these systems in terms of positioning accuracy has not yet been publicly documented in a manner that allows extrapolation to harsh, deep-indoor environments. These systems are also often used with GNSS-origin signals.

The essential assumption in this contribution is through-wall outdoor-indoor and indoor-indoor signal penetration, with local to medium-sized area coverage and target sub-decameter deep-indoor precision. The goal is a future pseudolite ranging augmentation for autonomous pedestrian navigation for emergency and rescue services (with sensor fusion, e.g., an inertial measurement unit for direction/distance and a barometer for altitude). Proposed pseudolite technology is expected to be theoretically less limited in emergencies, such as when a partial or complete building collapse occurs.

1.2 Overview of Existing Systems and Approaches

The Australian Locata (Barnes et al., 2003) is a well-known pseudolite system. Locata offers high, centimeter-level accuracy (carrier-phase measurement) (Rizos et al., 2010); however, its use is primarily in outdoor open spaces, such as open-pit mines or ports. Owing to its frequency of approximately 2.4 GHz in the industrial, scientific, and medical (ISM) band, Locata is not applicable to indoor signal penetration. Its militarized version, the Ultra High Accuracy Reference System, has similar properties (Craig et al., 2012).

The NextNav system originated in the United States and is the only such system with verified indoor penetration, achieving accuracy at the level of tens of meters (10–100 m) (Meiyappan et al., 2013). Its TeraPoint service utilizes the Metropolitan Beacon System (Yang et al., 2016) operating in the licensed sub-GHz band at 920–928 MHz, providing near-nationwide coverage in the United States (Meiyappan et al., 2013). NextNav has been well described in further applications, such as the passive radar reception reported by Navrátil et al. (2017).

Unfortunately, neither system has a mass-market spread because they both rely on relatively fixed and expensive infrastructure. Both systems were also tested as candidates for a “GNSS backup” for the European Union (see the report by Bonenberg et al. (2023)). Nevertheless, the report did not focus on the deep-indoor positioning user case.

The signals of these two systems are designed as copies of the original GNSS signals with minimal changes. Thus, binary phase shift keying (BPSK) modulation and relatively short Gold codes are still widely used (Syam et al., 2022; Barnes et al., 2003; Meiyappan et al., 2013).

Further academic works dedicated to this problem are relatively plentiful. A study by Syam et al. (2022) at the boundary between academic and commercial research is closely aligned with our objectives. Navigation in outdoor–indoor and indoor–indoor environments for emergency and rescue services was considered, within a wider very high frequency (VHF)/ultra-high frequency (UHF) band (113–400 MHz, 430 MHz, and 500 MHz). The presented results are promising; however, only signal penetration was considered, not other parameters relevant to ranging performance (precision, SNR).

Other efforts continue with TWR applications, enabled by the expansion of software-defined radio (SDR) platforms, such as the commercial ENSCO Timing, Communications, and Ranging Device (TCR-D) (Myrick, 2018; Yu et al., 2023), or academic efforts such as those of Herschfelt et al. (2019). Yet, neither of these efforts described the penetration and performance of ranging across building structures. The ENSCO TCR-D also focuses on military applications (Appleget et al., 2023).

1.3 Contribution Objectives

The objective of this contribution is to provide an overall analysis of σ² for ranging using pseudolites in the UHF band for deep-indoor positioning. This paper is organized as follows.

After an overview of existing pseudolite systems and approaches (Section 1.2), the ranging and ranging signal model is presented in Section 2. Selected ranging signals with different codes (binary and non-binary sequences), modulations (modulation pulses, orthogonal frequency division multiplexing [OFDM]), and ranging code chip rates of kchip/s to Mchip/s are briefly presented for further analysis.
The CRLB is compared with numerical simulation results in Section 3. Differences significant for ranging are highlighted and described based on a limited code gain and SNR.
A supplemental model for ranging, which compares the CRLB and simulation results with experimental measurements, is described in Section 4, along with corresponding SNR, clock, and carrier estimators.
Experimental measurements obtained using SDR and real-world signals based on the considered signals (Section 2) and estimators (Section 4) are described in detail in Section 5. This discussion includes the methodology (Section 5.1) and an auxiliary building model for SNR calculation (Section 5.2). The agreement among theory, simulation, measurement, and modeling is presented in Section 5.3.
Section 6 presents a brief comparison and extrapolation to existing systems (NextNav).

The parameter selection is intentionally similar to that of NextNav and the work of Syam et al. (2022). A reasonable bandwidth of up to the order of MHz is used. The choice of carrier frequency focuses on the 400-MHz (UHF) band for the following reasons.

The primary reason is that these frequencies penetrate building interiors more effectively than frequencies on the order of a few GHz. Even multipath propagation is limited and is close to line-of-sight propagation (Fischer et al., 2013; Bertoni et al., 1995). This advantage is extensively used and experimentally verified here.
The second reason is that this portion of the frequency spectrum is used for communication, specifically the ISM (unlicensed) band or amateur radio (HAM), which has relatively narrow channels (×10 kHz). We also aim to demonstrate the usability and limits of practical ranging using these narrowband channels. The reasonable dimensions of the antennas typically used for these applications is also beneficial.

2 UHF CHANNEL MODEL AND RANGING SIGNALS

Indoor UHF signal propagation is not geometric-optic, unlike that at a few GHz, and is affected by everything within the first Fresnel zone. The propagation is dominated by diffraction rather than strong reflections. However, owing to the wavelength (70 cm), the effect of furniture and interior equipment is not dominant (as attenuation and scatter sources) (Fischer et al., 2013; Bertoni et. al., 1995).

Therefore, the impulse response, as well as the correlation function, is characterized by a main correlation peak followed by a continuous-diffuse trailing tail, forming a smeared sidelobe structure caused by multipath propagation and a long delay spread (Bertoni et al., 1995). The continuous form of an additive white Gaussian noise (AWGN) model with noise w(t) and I multipath propagation paths with amplitude α is used for the received signal r(t):

$r (t) = \sum_{i = 0}^{I} α_{i} s (t + τ_{i}) + w (t) \approx \int_{- \infty}^{\infty} α (τ) s (t + τ) d τ + w (t)$ 2

A severe problem that arises indoors is direct-path signal blockage (often referred to as a type of multipath). In this case, the direct-path signal is significantly attenuated compared with the diffracted or reflected signal. Nevertheless, the signals are relatively well separated in time. In the resulting correlation function, this situation is reflected by a weaker peak of the correlation function, which precedes the dominant peak.

Multipath propagation can be estimated by conventional estimators, such as those used by Irsigler et al. (2004) or the multipath-estimating delay-locked loop (Van Nee, 1994). Then, we can assume that the error given by the multipath error envelope is smaller than the CRLB error given by the AWGN channel considered below.

The expected multipath effects discussed above are taken into account in this work. It is evident that for a final indoor positioning, any of the above estimators or a more sophisticated method, such as ray tracing or machine learning, as reported by de Sousa & Thomä (2018), must be used. Nevertheless, for the UHF band, multipath is handled in a simplified and semi-manual manner for the measuring purpose of this contribution. The maximal likelihood (ML) time-delay τ estimator $\hat{τ}$ is as follows:

$\hat{τ} = min \arg max_{\tilde{τ}} {| R (τ) |}$ 3

Here, only the first, direct geometric path is taken into consideration, and multipath propagation does not significantly affect measurements in the UHF band. This approach is supported by empirical results, as demonstrated by the verification experiments in Section 5. Moreover, multipath is not the primary focus of this contribution.

The cross-correlation R(τ) of r(t) and its replica s(t) (originally transmitted signal) is defined as follows:

$R (τ) = \int_{- \infty}^{+ \infty} r (t + τ) s^{*} (t) dt$ 4

This cross-correlation is a well-known matched filter or correlative receiver (Proakis & Salehi, 2008).

The signal s(t) is a digital linear modulation (Proakis & Salehi, 2008). Thus, s(t) is a convolution of code chips d[k] with modulation pulse h(t) per chip period T_chip (1/T_chip is equal to the chip rate f_chip):

$s (t) = \sum_{k} d [k] h (t - k T_{chip})$ 5

The CRLB depends on the PSD |S(f)|². Because the PSD of (pseudorandom) ranging code is close to white noise (i.e., flat), the PSD of the modulated signal s(t) is asymptotically close to the PSD of the modulation pulse and can be calculated via a Fourier transform (FT):

$| S (f) |^{2} = | FT [d [k]] |^{2} | FT [h (t)] |^{2} \approx | FT [h (t)] |^{2}$ 6

2.1 Ranging Codes Considered

Auto- and cross-correlation properties are the main requirements on the ranging code. These requirements include a high ratio between the main peak and side peaks of the auto-correlation and low peaks of the cross-correlation (with other codes from the same class) (see Figures 3(a) and (b)). These properties are given by the Welch bound.

FIGURE 3

(a) Auto-correlation function of MLS, Gold, and FZC (CAZAC) codes (N_code = 1023), (b) side-peaks, and (c) scatter plot of binary and non-binary code

The Welch bound (Welch, 1974) determines the maximal amplitude of side peaks of the auto-correlation function as $\sqrt{} N_{code}$ , where N_code is the code length. This term is the upper bound of the main-to-side peak suppression ratio. In practice, codes reach the bound at different levels. Typically, this property is balanced at the expense of other advantageous properties of the code class.

These advantageous properties include an arbitrary code length and a large subset of codes with desirable cross-correlation properties. Binary and non-binary code types are considered below.

2.1.1 Binary Codes (Maximum-Length Sequence, Gold, Etc.)

The best correlation properties for binary codes are achieved for maximum-length sequences (MLSs). However, the number of MLS codes of a given length is limited (Dixon, 1994). Gold codes have slightly worse correlation properties (higher side peaks) (see Figure 3(b)), but a broader subset of codes of the same length.

Both codes are generated with a fixed length, which is given by one (MLS) or more (Gold) linear feedback shift registers (LFSRs) (N_code = 2ⁿ–1, where n is the number of LFSR bits). The LFSR is often described by a polynomial. For example, an MLS code with 511 chips (identical to GLONASS L1OF) uses an LFSR with nine stages (Dixon, 1994; Ipatov, 2005), which is defined by the following polynomial:

$a_{k} = X^{9} + X^{5} + 1$ 7

Further binary codes have been developed for communications, radars, and GNSSs, including, for example, Kasami, bent, and Jet Propulsion Laboratory (JPL) sequences (Dixon, 1994; Ipatov, 2005), Barker codes (Richards, 2005), and new GNSS codes (Soualle, 2005), such as Legendre (Rushanan, 2007) and memory-numerically optimized codes for Galileo E1. Nevertheless, in principle, these codes all have worse correlation properties than MLSs.

2.1.2 Non–Binary Codes (Frank–Zadoff-Chu)

Polyphase codes (Ipatov, 2005) are good representatives of non-binary codes (see Figures 3(b) and (c)). The basic principles of these codes are known from radar systems, such as Frank codes (Richards, 2005).

The subclass of Frank-Zadoff-Chu (FZC) codes (Frank et al., 1962; Chu, 1972) has desirable properties. This subclass is generated using the following equation:

$a_{k} = \exp (j \frac{M πk (k + (N mod 2))}{N})$ 8

where the choice of length N is arbitrary. M is indivisible and must be smaller than N.

FZC codes enable the generation of relatively broad subsets of codes with good cross-correlation properties, similar to those of MLS codes (Cobb, 1997) (see Figure 3(b)). The discrete Fourier transform of the FZC code is also an FZC code.

These codes are called constant amplitude–zero auto-correlation (CAZAC) codes. Because of their promising properties, FZC codes are utilized for OFDM synchronization signals in modern 4G and 5G communication systems.

Despite the potentially superior properties (cross-correlation, broader subsets, arbitrary length) of FZC codes, as mentioned by Cobb (1997), binary codes persist in current systems such as Locata (Barnes et al., 2003), NextNav (Meiyappan, 2013), GNSSs, and most relevant studies, such as those by Yu et al. (2023), Syam et al. (2022), and others.

2.2 Modulation Pulses Considered

The modulated signal spectrum varies depending on the modulation pulse used. A rectangular (REC) modulation pulse, binary offset carrier (BOC) modulation pulses of various order m, and root raised cosine (RRC) modulation pulses (Figure 4 and Figure 5) are considered here, with various values of the roll-off factor β (Proakis & Salehi, 2008). |H(f)|² is the PSD of h(t).

FIGURE 4

(a) REC and BOC modulation pulses and (b) corresponding PSDs

FIGURE 5

(a) RRC modulation pulse for different roll-off factors and (b) corresponding PSDs

The REC modulation pulse (Figure 4(a)) is typical for GNSSs. Its non-band-limited spectrum is shown in Figure 4(b). The BOC modulation also originates from GNSSs (Galileo signals) and can be expressed as a BPSK signal. However, a non-return-to-zero BOC modulation pulse is used instead of the REC (Figure 4(a)); its spectrum is shown in Figure 4(b). The RRC pulse (Figure 5(a)) originates from general wireless communications and is utilized because of its favorable (band-limited) spectral properties (Figure 5(b)) in bandwidth-limited channels.

Modulation pulse techniques are popular in overused bands, particularly in the sub-GHz (VHF and UHF) frequency range. Appropriate modulation pulses can shape the spectrum according to frequency allocation and coexistence with other communication services in this overcrowded frequency allocation. Specific signs of spectral shaping with similar indications have been described by Meiyappan et al. (2013) for the NextNav signal. The corresponding effect on the CRLB is discussed below.

The OFDM modulation allows for the most considerable freedom in spectral shaping. This principle is widespread in the synchronization of many general telecommunication OFDM systems (4G/5G, digital video broadcasting (DVB)). The ranging code bits are mapped in a straightforward manner to the spectrum coefficients d[n] as follows:

$s [k] = {DFT}^{- 1} [d [n]] = \frac{1}{N} \sum_{n = 0}^{N - 1} d [n] \exp (j k 2 π \frac{n}{N})$ 9

The spectrum can then be shaped in various ways (Figure 6). This approach offers the construction of bespoke ranging signals. For example, some subcarriers can be omitted or zero-padded to control the slope of spectrum edges or controlled in the sense of cognitive radio, etc. OFDM has been discussed in many new-ranging signal studies over the last two decades, but has not yet been applied in practice.

FIGURE 6

PSD of an OFDM-modulated 511-bit-long (MSL) binary sequence with a rate of 4 Mchip/s

2.3 Selected Ranging Signals

The modulations and chip rates listed in Table 1 were chosen for experimental investigation. A 511-chip-long MLS code is used, based on the polynomial in Equation (7). The code and signals correspond to the experimental setup used in Section 5. Multiples of 12.5-kchip/s chip rates were selected to fill the most commonly used communication bandwidths. Lower kchip/s chip rates were also selected for comparison and demonstration only, although they are apparently unsuitable for practical ranging.

View this table:

TABLE 1 Parameters and f_chip Values (in Mchip/s) of Selected Experimental Ranging Signals

3 RANGING PRECISION: COMPARISON OF THEORY AND SIMULATION

3.1 Theory (CRLB)

The lower bound of the estimator variance is given by the CRLB (Rao, 1945; Cramér, 1946). This bound can be transformed from a variance of time delay to a variance of range r in meters (m) as follows (Kay, 1993):

$σ_{\hat{r}}^{2} = \frac{c^{2}}{8 \frac{S}{N} β_{R M S}^{2}}$ 10

where the constant c is the speed of light. This form is de facto a function of two parameters. The parameter β_RMS is the so-called RMS (effective) bandwidth (Richards, 2005). This bandwidth is computed from the PSD |S(f)|² of the signal s(t) and is given in Hz as follows:

$β_{R M S} = \sqrt{\frac{\int_{0}^{T_{0}} {(\frac{d s (t)}{dt})}^{2}}{\int_{0}^{T_{0}} s^{2} (t) dt}} = \sqrt{\frac{\int_{- \infty}^{+ \infty} (2 π f)^{2} | S (f) |^{2} df}{\int_{- \infty}^{+ \infty} | S (f) |^{2} df}}$ 11

The second parameter $\frac{S}{N}$ (signal-to-noise [power] ratio) has the same meaning here as the SNR. The SNR can be calculated from the S/N or the energy per chip (bit, symbol) E_B and the noise PSD N₀ ratio (E_B/N₀), using the sample frequency f_sample (on a logarithmic scale [dB]) as follows:

$\begin{array}{l} S N R & = & 10 \log_{10} (\frac{S}{N}) = 10 \log_{10} (\frac{\frac{1}{K} \sum_{0}^{K - 1} s [k] s^{*} [k]}{\frac{1}{K} \sum_{0}^{K - 1} n [k] n^{*} [k]}) \\ = & 10 \log_{10} (\frac{E_{B}}{N_{0}}) + 10 \log_{10} (\frac{f_{chip}}{f_{sample}}) \end{array}$ 12

Equation (10) is the general form for a ranging signal for an idealistic channel in Equation (2) without a multipath component (I=1). Recalculating the CRLB for the multipath channel in Equation (2) is feasible, as reported by Botteron et al. (2004). However, its form is much less transparent than the original Equation (10). Based on the assumption of de facto multipath-free UHF propagation, the CRLB will be used here.

The CRLB is not the only bound that could be employed. Alternative bounds exist, such as those proposed by Barankin (1949) or Ziv and Zakai (1969). However, as will be shown later, the CRLB fits the estimator and experimental results well.

3.2 Simulation, Code Gain, and SNR

Equation (10) is the general form for a theoretically infinite-length ranging signal. The result differs when the actual finite-length ranging code signal of length N_code is used due to the code gain G_code, which is expressed as follows:

$G_{code} = 10 \log_{10} (N_{code})$ 13

The code gain in Equation (13) is simplified with regard to the upper bound. This expression applies only to long codes reaching the Welch bound (MLS, FZC) and is lower for other codes.

The code gain is a principal limitation for indoor measurements. The dynamic range in topology must not exceed the code gain; otherwise, the near-far effect will occur. (Its practical solution [time, frequency division] is known (Cobb, 1997; Wang, 2002) and used in both NextNav and Locata.)

An illustrative Monte Carlo simulation result is shown in Figure 7(a) (solid line) compared with the CRLB (dashed line). Here, a signal with a 511-chip-long (binary) MLS code with an f_chip of 4 Mchip/s, BPSK modulation, and a REC modulation impulse is used. The two curves are equal until the negative SNR exceeds G_code = 27 dB. The good agreement between the two curves in a higher dynamic range (SNR) can be observed on a logarithmic scale, as shown in Figure 7(b). Both figures present the standard deviation instead of the variance owing to the more natural expression of the standard deviation in meters instead of meters squared.

FIGURE 7

Illustrative comparison of the ranging standard deviation obtained using the CRLB and simulation

The differences are clarified in Figure 8. $\sqrt{σ^{2}}$ increases with decreasing SNR, following the analytical CRLB from Equation (10) in the first region, until the code gain is overcome at the line labeled “2.” Fundamental differences arise in the third and fourth regions. The analytical CRLB results continue to grow according to Equation (10) in the third and fourth regions, while the simulation results deviate and remain almost constant.

FIGURE 8

Ranging standard deviation for the CRLB and simulation

A correlation peak is not detected owing to a weak SNR below the code gain in the fourth region. The estimated range is then expected to be uniformly distributed between zero and the (maximum unambiguous) range, equal to the signal period T_code. At that point, the standard deviation is equal to the following:

$\sqrt{σ_{region_4}^{2}} = \frac{{(c T_{code})}^{2}}{12}$ 14

The SNR in each figure is related to the input signal for a better comparison. The difference between the SNR at the input and at the output of the correlator/matched filter is the processing gain. This processing gain equals G_coh plus the coherent integration gain G_coh over I signal periods (see Figure 9). Therefore, the SNR related to the input results in negative values, corresponding to low-power indoor-received signals (as in similar wideband spread-spectrum systems, such as GNSSs).

FIGURE 9

Illustration of the processing gain

3.3 Theory and Simulation Comparison for Chosen Signals

The resulting $\sqrt{σ^{2}}$ of the ranging for selected individual ranging signals (see Table 1) is presented in terms of its calculated theoretical limit (CRLB) and numerical simulation. Results are presented only for SNRs in the range down to –27 dB. This negative SNR value and code gain of the ranging code used (N_code = 511 ~ G_code = 27 dB) result in a positive (detectable) SNR on the correlation output (see Section 3.2), corresponding to the first region in Figure 8. Application outside this region is otherwise unsuitable for ranging.

Numerical simulations use 500 iterations per individual signal and parameter. The multipath-free model (I = 0) is used for simplicity. The simulation signal bandwidth (single-sided) is limited to 125 kHz for a signal chip rate of 12.5 kchip/s, 1.25 MHz for 125 kchip/s, and 20 MHz for 1.25 Mchip/s and 4 Mchip/s. This bandwidth limitation is the source of potential minor differences between the theoretical (CRLB) and simulated curves.

Figure 10 presents results for individual chip rates from 12.5 kchip/s to 4 Mchip/s. The 511-chip-long MLS code with BPSK modulation and an RRC modulation pulse (β = 0.5) was used.

FIGURE 10

Theoretical versus simulated results for chip rates of 12.5 kchip/s to 4 Mchip/s chip, for an MLS code with an RRC modulation pulse

It is evident that narrowband signals with chip rates on the order of 1–100 kchip/s are not sufficiently precise ( $\sqrt{σ^{2}} ≮ 10 m$ ). Nevertheless, these results are presented for demonstration and experimental verification of the theory and measurement.

Different modulations for a fixed rate of 4 Mchip/s are compared in Figure 11. Firstly, the REC signal performs slightly better (lower $\sqrt{σ^{2}}$ ) than the RRC signal. This result is unsurprising when the PSDs of the REC (Figure 4) and RRC (Figure 5) signals are compared. The REC PSD is not strictly band-limited like the RRC PSD. The first decrease in the REC PSD occurs at f_chip but the first side peaks are attenuated by only 13 dB. The RRC PSD is relatively well band-limited, with no side peaks and almost 40-dB attenuation between 0.5 and 1.0 of f_chip, depending on β. Therefore, the effective bandwidth β_RMS of the REC signal is slightly higher, and the resulting $\sqrt{σ^{2}}$ is lower.

FIGURE 11

Comparison of selected ranging signals (f_chip = 4 Mchip/s)

FIGURE 10 Theoretical versus simulated results for chip rates of 12.5 kchip/s to 4 Mchip/s chip, for an MLS code with an RRC modulation pulse

Good agreement between the curves, independent of the choice of binary (MLS) or non-binary (FZC) code, is evident. Therefore, the precision of ranging signals using codes close to the Welch bound and N_code > ×100 chips depends solely on the modulation pulse PSD (the PSD of such a code is flat) according to Equation (6).

The lowest $\sqrt{σ^{2}}$ is achieved for the BOC modulation owing to its greater bandwidth. BOC modulation is seemingly the most straightforward way to increase the effective bandwidth (if an increase is allowed by the frequency allocation; see Figure 4(b)). BOC is very popular in GNSSs (Galileo E1 signal) because the resulting correlation peak is narrower. In contrast, the correlation function has side peaks that increase with the BOC modulation order m. Therefore, an order higher than one is usually not used independently.

Figure 11 presents the OFDM-modulated signal. The performance is practically the same as that of the RRC because their PSDs are similar (see Figure 6). However, OFDM offers much more agile spectrum utilization.

Both Figure 10 and Figure 11 demonstrate good agreement between the calculated CRLB and the simulation. Marginal differences are due to the limited signal bandwidth used in the simulation. For example, there is slightly worse agreement between the theoretical and simulated curves for MLS REC in Figure 11. The same trend is shown in Figure 10 for a chip rate of 12.5 kchip/s.

4 CHANNEL MODEL FOR MEASUREMENTS AND ESTIMATORS

The model in Equation (2) cannot verify the theory and simulation results above with real-world SDR received signals. The parameters of the carrier, including the carrier frequency (f_carr) offset f_offset, and the initial phase φ_ini, are added owing to the local oscillator (clock) imperfection in the following equation:

$r (t) = \sum_{i = 0}^{I} α_{i} s (t + τ_{i}) exp (j 2 π f_{offset} t + φ_{ini}) + w (t)$ 15

The effect of the Doppler shift, caused by a non-zero mutual transmitter–receiver or scatter source velocity, is omitted. The static case is considered here.

To compare previous theoretical and simulated results with those from the measurement campaign, the carrier parameters must be estimated and compensated.

4.1 Range, Frequency Offset, and Phase Estimators

For the channel in Equation (15), the original ML estimator in Equation (3) for the channel in Equation (2) leads to a search for the argument of the maximum of the cross-ambiguity function (CAF) peak:

$(\hat{τ}, \hat{f_{offset}}) = min \arg \underset{\tilde{τ}, f_{offset}}{max_{\sim}} {| \int_{- \infty}^{+ \infty} r (t + \tilde{τ}) s^{*} (t) exp (- j 2 π \tilde{f_{offset} t}) dt |}$ 16

The resulting CAF for binary code-ranging signals is shown in Figure 12. However, the CAF appears differently when the class of non-binary codes is employed. For example, the result in Figure 13 is typical for polyphase FZC code ranging signals. (A rate of 4 Mchip/s is used in both figures.) Then, the utilization of Equation (16) becomes complex.

FIGURE 12

CAF of binary ranging code

FIGURE 13

CAF of polyphase (FZC-CAZAC) ranging code

Alternatively, f_offset can be estimated by using a conventional atan2 discriminator (Proakis & Salehi, 2008; Misra & Enge, 2010). The discriminator is expressed in Hz as follows:

$\hat{f_{offset}} = \frac{φ_{2} - φ_{1}}{t_{2} - t_{1}}$ 17

This discriminator is based on the CAF peak phase, which can be computed in radians (rad):

$φ = atan 2 (Re {max {R (τ)}}, Im {max {R (τ)}})$ 18

The third alternative is the estimator described in Section 4.2 below.

4.2 Clock Offset and Range Drift Estimators

The nominal f_carr (carrier), f_sample (sample), and f_chip (chip rate) are derived by a frequency synthesizer from the same receiver clock with nominal frequency f_clk. Clock imperfections are modeled by a clock offset Δf_clk. Then, f_offset and Δf_clk can be calculated in Hz as follows:

$f_{offset} = Δ f_{clk} \frac{f_{carr}}{f_{clk}} = Δ f_{chip} \frac{f_{carr}}{f_{chip}} = \dot{r} (\frac{f_{chip}}{N_{code} f_{samp}}) f_{carr}$ 19

$Δ f_{clk} = \dot{r} (\frac{f_{chip}}{N_{code} f_{samp}}) f_{clk}$ 20

The chip rate offset Δf_chip causes a linear drift Δr of the estimated range $\hat{r} [k]$ among k consecutive periods by rate $\dot{r}$ (see Figure 14). The clock offset drift $Δ {\dot{f}}_{c l k}$ is omitted. The least-squares method is used for $\dot{r}$ estimation (in samples/period):

FIGURE 14

Correlation peak drift as an effect of clock offset

$r [k] = r [0] + \dot{r} k$ 21

$where \dot{r} = \frac{Δ r}{Δ k}$

Up to three expressions (Equations (16), (17), and (19)) can estimate the frequency and clock offset. Offset estimation is then used to compensate for the offsets in the experimental verification. The frequency offset does not significantly affect TDoA measurements. Outdoor-situated pseudolites are usually well time/frequency-synchronized via a GNSS. However, the clock offset is critical in TWR between two transmitters/receivers with free-running clocks (Herschfelt, 2019; Myrick, 2018).

4.3 SNR Estimators

For a real-world received signal, the SNR must be estimated to properly compare theoretical (CRLB) and simulated results with measured results. Principally, SNR estimation (Misra & Enge, 2010) is not straightforward (Falletti et al., 2011). Two SNR estimators are discussed here:

$SNR = S - (N + N F) = 20 {log}_{10} (max {R (τ)}) - (N + N F)$ 22

$SNR = 20 \log_{10} (max {R (τ)}) - 20 \log_{10} (mean {R (τ)})$ 23

These estimators are expressed in decibels (dB) and differ slightly, considering the high dynamic range of signals received indoors.

The estimation of useful signal power S is the same for both estimators. This term is equal to the peak value of the cross-correlation function R(τ) of r(t) and s(t) in Equation (4) (see Figure 15(a)).

FIGURE 15

(a) Cross-correlation function of the ranging signal and its replica and (b) SNR estimators

The estimators differ only in their estimation of noise power N. The first SNR estimator, Equation (22), uses a knowledge of the N lower bound in decibels (dB) as a product of (thermodynamic) temperature T (K), Boltzmann constant k (J ⋅ K^–1), and bandwidth B (equal to f_sample) (Hz) with the receiver’s noise figure NF in decibels (dB):

$N = 10 \log_{10} (k T B) = 10 \log_{10} (N_{0} B) = - 173, 9 + 10 \log_{10} (f_{samp.})$ 24

If NF is not known a priori, the formula (N + NF) can be replaced by the measured noise power. The noise power is measured and computed from a received signal r(t) in the absence of a useful signal as follows:

$N = {10 \log_{10} (r (t) r (t))^{*} |}_{No useful signal}$ 25

The second SNR estimator, Equation (23), is more suitable in practice. Noise is estimated from the mean level of R(τ) based on its main peak (see Figure 15(a)). This approach is helpful when NF is unknown and the useful signal cannot be turned off.

As discussed in Section 3.2, the SNR is considered for the input. The first SNR estimator, Equation (22), corresponds directly to this SNR. However, the second estimator, Equation (23), corresponds to the SNR of the output. Therefore, the code gain from Equation (13) must be subtracted from Equation (23) to obtain the same values.

Both estimators work well even for very low SNRs, down to exceeding the code gain (the area of interest for this contribution; see Figure 15(b)). However, the second estimator, Equation (23), works well only for long codes (N_code > ×100 chips). In that case, the R(τ) side peaks are lower than the noise level (suppression ratio > 20 dB), and they do not affect the noise estimation. For the same reason, this estimator fails for high positive SNRs, because the non-zero value of the side peaks alone limits the noise estimation in Equation (23).

5 RESULTS

The achievable standard deviations of ranging for the chosen signals (see Table 1) are briefly compared here, including the theoretical (CRLB), simulated, and measured values.

5.1 Measurement Campaign Scenario (Polygon)

The previous results are verified through a measurement campaign conducted in a large, massive concrete public building with a 235 × 195 m floor plan. The building comprises several wings with up to nine above-ground floors with offices and laboratories, two underground floors, and a wing of heavy machinery laboratories. Long corridors and staircases connect all of the floors (see Figure 16(a)).

FIGURE 16

(a) Photograph and (b) floor plan of the measurement campaign polygon (signal coverage highlighted by a semi-transparent circle sector); Global Positioning System coordinates: 50.1024983° N, 14.3927758° E

Two locations were selected for the transmitter of the ranging signal. The first location is in an office on the top floor of the central tower. The second location is on the ground floor in the main building entrance. Several locations for a receiver were selected throughout the building, including different floors (including underground), long corridors, rear fire staircases, and a heavy machinery laboratory. A drawing of the measurement polygon is provided in Figure 16(b).

An MLS ranging code with N_code = 511 chips was used (see Equation (7)) owing to its simplicity as well as hardware limitations in the testing generator (with sufficient power output). Then, the code was BPSK-modulated with an RRC modulation pulse and transmitted on a 448.490-MHz (amateur radio) carrier frequency. Only the chip rate was changed; chip rates of 12.5 kchip/s and 4 Mchip/s were used. Both signals were initially transmitted with an output power (P_Tx) of 25 dBm and then 0 dBm into a λ/4 monopole antenna.

The signal was not strong enough to be received at all locations. These locations are distinguished by empty markers in Figure 16. The circle and triangle markers refer to two different transmitter locations, e.g., the circle locations focus on signal coverage from a ground-level transmitter. The signal transmitted from a single location covers approximately one half of the building, as highlighted in the floor plan (Figure 16(b)) by a semi-transparent circular sector. Triangle locations were primarily used for $\sqrt{σ^{2}}$ measurement, as discussed in Section 5.3.

The scenarios do not intersect everywhere, and an exhaustive measurement was not conducted. Instead, this study aimed to test certain critical cross-sections of the building.

5.2 Loss Model

The following model was used to approximate the SNR inside the building. The goal was to achieve situational awareness of the signal power across the building and to roughly distinguish between direct (geometric) and multipath propagation paths.

The SNR presumed for individual locations, as derived from Equation (22) and using N from Equation (24), is given in decibels (dB) as follows:

$S N R = 10 \log_{10} (\frac{S}{N}) = P_{R x} - N - N F$ 26

The NF of the SDR used (ADALM Pluto, f_sample = 48 MHz) is calculated to be 2.9 dB, based on a comparison of Equation (24) and Equation (25).

The presumed received power P_Rx is calculated in dBm as follows:

$P_{R x} = P_{T x} + G_{T x R x} - F S L - L_{a d d}$ 27

The transmitting power in dBm is P_Tx. The combined gain of both antennas, G_TxRx, is assumed to be 2 × 2.14 dBi. The free-space loss (FSL) is calculated in decibels (dB) from the following equation:

$F S L = 20 \log_{10} (r) + 20 \log_{10} (f_{carr}) - 147.55$ 28

where r is the true geometric distance. The FSL model is a simplification; losses for greater distances (outdoor) should be modeled via a more complex model for propagation above terrain (Kaplan & Hegarty, 2017).

Additional losses are modeled using Loss_add, a simplified loss model for losses caused by indoor obstacles, walls, and windows along the direct line between the transmitter and receiver. A model, given in decibels (dB), has been reported by Kaplan and Hegarty (2017) as a loss of man-made structures and by Bertoldo et al. (2019) as follows:

${Loss}_{add} = F S L + 5 N o_{walls} + 15 N o_{windows}$ 29

The attenuation of 5 dB per wall and 15 dB per window is based on rare data presented by Zhekov et al. (2018). These constants are multiplied by the number of walls and ceilings (No_walls) and the number of windows (No_windows), respectively. A direct analytical solution does not exist, and only partial empirical methods are available, as provided by the International Telecommunication Union Radiocommunication Sector (2015). Nevertheless, the Loss_add obtained using Equation (29) shows a surprisingly good match, as presented in Section 5.3.

5.3 Measurement Results

Detailed results for one set of measurements are presented in Table 2. The point numbers in Table 2 correspond to the numbers in Figure 16. The subsequent columns contain the true geometric distance r, corresponding FSL, and Loss_add. A comparison of the pre-calculated SNR obtained using Equation (26) with the SNR_est estimated by Equation (22) and Equation (23) is provided in the last columns of the table, together with the measured $\sqrt{σ^{2}}$ of the range. The SNR estimator from Equation (22) was used for SNRs higher than 10 dB, and the estimator in Equation (23) was used for lower SNRs.

View this table:

TABLE 2 Detailed Results of the Measurement Campaign

Values from the entire measurement campaign are presented in Figure 17, based on the comparison of the CRLB and simulation results in Figure 10. A relatively good agreement among the CRLB, simulated, and measured data is apparent.

FIGURE 17

Final comparison with results from the measurement campaign

The measured data points in Figure 17 correspond to the data acquired at various measuring points throughout the measurement campaign polygon (Figure 16). Therefore, the number of these points is limited. The number of measuring points for the secondary, lower chip-rate in Figure 17(a) was limited to even fewer points.

Data were acquired stepwise using up to three SDRs. The SDRs were of the same type. Nevertheless, the noise figure and other parameters not included in the model in Equation (15) (such as local oscillator phase noise and drift) typically vary with time, temperature, coherence time, and the specific SDR used. Therefore, the data points should appear somewhat sparse, as shown in Figure 17. Hence, the data points are not fitted to any curve or trend.

The remaining residual multipath error may have an effect on the results. However, in the context of this contribution’s objective, i.e., verifying the achiev-ability of sub-decameter precision, individual propagation paths were well time-separated based on post-processing feed-forward estimation applied to an entire signal snapshot.

The presence of multipath was monitored. The cross-correlation functions in Figure 18 are normalized measured data. The multipath effect is not present even across four floors, as is obvious by comparing the 4^th floor (point (3)) function in Figure 18(b) with the multipath-free function (point (1)) in Figure 18(a). This result arises from the lower height of the neighboring buildings (four floors). This scenario is comparable to that of most small buildings (houses). However, reflections from the neighboring lower building are evident in Figures 18(c) and (d), where signals are received closer to the ground level.

FIGURE 18

Typical measured cross-correlation functions: (a) 1-m distance (1), (b) 4^th floor (3), (c) ground floor (4), (d) in front of the building (7), (e) -2^nd underground floor (6), (f) -1^st underground floor (5); (a) – (e) f_chip = 4 Mchip/s, f_sample = 48 Msample/s; (f) f_chip = 20 Mchip/s, f_sample = 60 Msample/s

The path differences of multipath components observed in Figures 18(c)–(e) result from the true geometry of the building and the expected propagation. For example, the distance to the neighboring building results in a path difference of 140 m, which corresponds to Figure 18(c). The distances between the building’s wings result in a 100-m difference in the path shown in Figures 18(d) and (e).

Nevertheless, these components are well time-separable from the main peak. Components with small differences are not actually present. Therefore, these results support the presumptions about the multipath behavior in the UHF band presented in Section 2. The first clear peak is always apparent, followed by continuous-diffuse multipath in Figure 18(d), as well as well isolated reflections from neighboring buildings passing through windows in Figures 18(c) and (e). (Windows have at least twice the attenuation of the wall (Zhekov et al., 2018).)

Measurements for selected points were repeated at a five-fold higher chip-rate (f_chip = 20 Mchip/s) and at the highest available f_sample = 60 Msample/s to increase the temporal resolution and to more precisely identify possible multipath effects. One result for the most critical scenario (-2^nd underground floor) is shown in Figure 18(f). The presence of multipath, which disrupts the dominant peak, is not visible in the measured data here or elsewhere.

Good agreement in this scenario is not the result of advanced signal processing but rather of a selected UHF band. Therefore, the results cannot be generalized for considerably different buildings and higher bands. However, the results presented here can serve as a guide, as the most similar publications (Syam et al., 2022; Meiyappan et al., 2013) focus on a considerably different setup or are not suitable for similar extrapolation.

6 EXTRAPOLATION OF RESULTS TO EXISTING (COMMERCIAL) SYSTEMS

Verifying all of the above assumptions, estimators, and models enables an analysis and evaluation of pseudolite systems under development, as well as comparisons with existing systems and their effective extrapolation to the indoor case. A lack of this consideration was a weakness of all known previous works.

For example, no studies have evaluated the NextNav pseudolite system in a harsh, deep-indoor environment, aside from a brief, outdated study by Meiyappan et al. (2013). Extrapolating the above findings allows us to evaluate and predict the indoor performance of NextNav.

A comparison of the wideband (1.25 Mchip/s) signal, referred to as the reference model, with NextNav is presented in Table 3. The table clearly shows that the two cases are comparable. Thus, the results ( $\sqrt{σ^{2}}$ of ranging) using NextNav for indoor applications should also be comparable to those mentioned above, indicating feasible ranging with $\sqrt{σ^{2}}$ below 10 m.

View this table:

TABLE 3 Comparison of the Reference Case and NextNav

This statement can also be supported by the study of Meiyappan et al. (2013), who utilized the NextNav Metropolitan Beacon System network. The study presented RMS_ΔX values of 18-45 m. This result agrees with the above-mentioned assumptions if a realistic geometric dilution-of-precision factor, between 4 and 1.5, is used in Equation (1).

7 CONCLUSION

This contribution provided an analysis of ranging precision, corresponding to the standard deviation of this ranging, for its application in deep-indoor positioning using pseudolite ToA, TDoA, or other (two-way, round-trip) positioning topology in the UHF band near 400 MHz, promising outdoor-indoor penetration of these premises, with a ranging signal bandwidth from tens of kHz to MHz.

This work used CRLB analysis, a Monte Carlo simulation, and results from a measurement campaign utilizing real SDRs. For a valid comparison, estimators of range, (clock) frequency offset, and SNR were described. The presentation of their behavior in a large dynamic-range environment of the received signal, and the relation to the code gain, is novel.

A comprehensive evaluation was performed from the perspective of the currently most widely used (binary) codes, as well as potentially suitable (non-binary, polyphase) codes, modulation techniques, and modulation pulses. For example, polyphase codes (FZC class) for pseudolites are presented in detail for the first time.

For cross-validation, the entire data set was fitted using the building SNR model. The effect of multipath propagation is negligible or can be controlled owing to the favorable propagation properties of the low-radiofrequency UHF band. Good agreement was demonstrated among the standard deviations of ranging using CRLB, simulation, and indoor measurements. Reference case values were also compared with study results for the existing NextNav system. The results demonstrate that a standard deviation of ranging below 10 m in deep-indoor buildings is achievable.

HOW TO CITE THIS ARTICLE:

Svatoň, J., Sýkora, J., Skopec, D., Vejražka, F., Roháč, J., Spáčil, J., Švanda, M., Tomis, M., & Škaloud, J. (2026). Indoor radio positioning using pseudolites in the UHF band – signals and ranging precision survey. NAVIGATION, 73. https://doi.org/10.33012/navi.783

ACKNOWLEDGMENTS

This contribution arose during work on the project “Monitoring the position of IRS members even during an intervention in large buildings using elements of artificial intelligence” (project number VJ02010037 of the Ministry of the Interior of the Czech Republic). This work was also supported by the Technology Agency of the Czech Republic under grant no. TE01020186.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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