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remote sensing
Article
Multipath Mitigation for BOC Signals Based on
Prompt-Assisted-Offset Correlator
Zhenyu Tian 1,2 , Xiaowei Cui 1,2, * and Mingquan Lu 1,2
1 Department of Electrical Engineering, Tsinghua University, Beijing 100084, China
2 Beijing National Research Center for Information Science and Technology, Beijing 100084, China
* Correspondence: cxw2005@tsinghua.edu.cn
Abstract: Tracking ambiguity and multipath are two serious threats in processing binary offset carrier
(BOC) signals that are widely used in global navigation satellite systems (GNSS). Three-loop tracking
methods, such as dual binary phase-shift keying tracking (DBT), use a delay lock loop (DLL), a phase
lock loop (PLL), and a subcarrier PLL (SPLL) to track the code, carrier, and subcarrier, respectively,
thus solving the tracking ambiguity problem. However, the multipath remains an important factor
in the ranging performance deterioration of these tracking methods. Using the offset correlator
(OC) technique in the SPLL can effectively mitigate subcarrier multipath errors, but it substantially
raises thermal noise errors. To solve this contradiction, this paper proposes a prompt-assisted-offset
correlator (PAOC) technique that combines the prompt and offset correlators in a complementary
way to mitigate multipath for BOC signals. Compared with the original OC technique, the PAOC
technique has less thermal noise performance loss. Moreover, this paper discovers and quantifies the
interaction between carrier and subcarrier multipath errors by analyzing the coupling effect between
SPLL and PLL. Most multipath mitigation methods for BOC signals ignore the carrier multipath in
the PLL, so their subcarrier multipath performance is not optimal. Thus, this paper further proposes
applying the PAOC technique in both PLL and SPLL to mitigate carrier and subcarrier multipath
errors simultaneously. Theoretical analysis and experimental results demonstrate that the proposed
method can significantly improve the multipath performance with small noise performance loss.
Keywords: GNSS receiver; BOC signal; multipath mitigation; dual BPSK tracking (DBT); offset
correlator (OC); prompt-assisted-offset correlator (PAOC)
Citation: Tian, Z.; Cui, X.; Lu, M.
Multipath Mitigation for BOC Signals
Based on Prompt-Assisted-Offset
Correlator. Remote Sens. 2023, 15, 937.
https://doi.org/10.3390/rs15040937
1. Introduction
Global navigation satellite systems (GNSS) are widely used in high-precision posi-
Academic Editors: Mieczysław
tioning applications, such as surveying, remote sensing, and unmanned ground vehicles
Bakuła and Krzysztof Naus
(UGV) [1]. Differential techniques can effectively eliminate the errors of ionosphere, tropo-
Received: 6 January 2023 sphere, and satellite ephemeris by exploiting the spatial correlation of these GNSS error
Revised: 4 February 2023 sources [2]. However, differential techniques can hardly eliminate multipath errors because
Accepted: 6 February 2023 multipath signals change with the environment. Therefore, multipath has become the
Published: 8 February 2023 dominant error source in high-precision GNSS positioning.
Multipath is the phenomenon where the direct signal is reflected or diffracted by
obstacles. The modulation method, code rate, and carrier frequency of GNSS signals have
a direct impact on multipath effects, so the multipath performance is an important factor to
Copyright: © 2023 by the authors.
be considered in the design of GNSS signals. Modernized GNSS signals generally adopt
Licensee MDPI, Basel, Switzerland.
binary offset carrier (BOC) modulation [3,4]. BOC modulation introduces a subcarrier in
This article is an open access article
the binary phase-shift keying (BPSK) modulation, which moves the main energy from
distributed under the terms and
conditions of the Creative Commons
the center to the edge of the frequency band [5,6]. As a result, the BOC signal has more
Attribution (CC BY) license (https://
high-frequency components and larger root-mean-square bandwidth than the BPSK signal
creativecommons.org/licenses/by/ with the same code rate, indicating that the BOC signal has better ranging capability and
4.0/). multipath resistance in theory.
Remote Sens. 2023, 15, 937. https://doi.org/10.3390/rs15040937 https://www.mdpi.com/journal/remotesensingRemote Sens. 2023, 15, 937 2 of 25
How to make full use of the ranging potential of BOC signals is a topic of great
interest. Tracking ambiguity is a major threat in processing the BOC signal because its auto-
correlation function (ACF) has multiple peaks. The typical delay lock loop (DLL) probably
locks on side peaks of the BOC signal, resulting in large ranging errors. Therefore, many
three-loop tracking methods have been proposed to solve the tracking ambiguity problem,
such as the Dual Estimate Technique (DET) [7], the Double Phase Estimator (DPE) [8], and
Dual BPSK Tracking (DBT) [9,10]. In addition to the DLL and carrier phase lock loop (PLL),
three-loop tracking methods use a subcarrier lock loop (SLL) to track the subcarrier delay.
In the DPE and DBT methods, the SLL is replaced with a subcarrier phase lock loop (SPLL).
With limited signal bandwidth, the SPLL has lower computational complexity and similar
thermal noise performance compared to the SLL [8].
Although three-loop tracking methods address the threat of tracking ambiguity, their
ranging performance deteriorates severely in the presence of multipath. Therefore, GNSS
receivers need to be equipped with multipath mitigation techniques to reduce ranging
errors in practice. There are various GNSS multipath mitigation techniques, among which
the receiver signal processing techniques using advanced correlator architectures are of
great concern. These correlator-based techniques can be divided into two categories:
parametric and non-parametric [11]. Parametric correlator-based techniques, such as the
multipath estimation delay lock loop (MEDLL) [12], require a large amount of computation
and may not converge when there are many multipath signals. By contrast, non-parametric
correlator-based techniques require a small amount of computation, so they are more widely
used in GNSS receivers. There are many non-parametric correlator-based techniques that
can effectively mitigate multipath, such as the narrow correlation (NC) [13] and double delta
(DD) correlator variants (high resolution correlator (HRC) [14] and strobe correlators [15]).
These multipath mitigation techniques were originally designed for BPSK signals and
gradually applied to BOC signals [16–19].
Specifically, the NC and DD techniques are usually used in the DLL to mitigate code
multipath errors for BOC signals [20–22]. Since the code ranging measurement is used to fix
the integer ambiguity of subcarrier ranging measurement, the code multipath error needs to
be suppressed to be less than half of the subcarrier wavelength. The pseudo-range used for
positioning is the subcarrier ranging measurement after fixing integer ambiguity. Multipath
signals can introduce errors to the subcarrier ranging measurement, thus resulting in
pseudo-range bias and positioning errors. Therefore, subcarrier multipath mitigation
techniques are very important for BOC signals to improve positioning accuracy [21–23].
Depending on the loop structure, subcarrier multipath mitigation techniques for BOC
signals can be divided into two categories: multi-correlator techniques and the offset
correlator (OC) technique. Multi-correlator techniques combine multiple correlator outputs
to synthesize a narrower correlation function to reduce the multipath sensitive area, such
as the DD and strobe correlator techniques applied in the SLL/SPLL [21,23]. The OC
technique uses forward offset correlators instead of prompt correlators for subcarrier
discrimination [21,22]. Since multipath signals have longer path delays than the direct
signal, forward offset correlators are less susceptible to multipath than prompt correlators,
so the OC technique can reduce multipath errors. Furthermore, the multipath mitigation
performance of OC technique improves with the increase of forward offset. It has been
demonstrated that the OC technique with a large forward offset has better subcarrier
multipath mitigation performance than that of multi-correlator techniques [22]. However,
the signal-to-noise ratio (SNR) loss of the OC technique increases with the forward offset,
which may cause the tracking loop to lose lock. As a consequence, the OC technique has to
compromise between the thermal noise performance and multipath mitigation performance.
To ensure the tracking stability, the forward offset of the OC technique cannot be very large,
which limits its multipath mitigation performance in practice.
In addition to the code and subcarrier ranging measurements, the BOC signal also
provides carrier phase measurements for centimeter-level positioning applications. The
carrier multipath error can reach up to several centimeters and is a major error source forRemote Sens. 2023, 15, 937 3 of 25
high-precision positioning [24,25]. However, existing multipath mitigation methods for
BOC signals generally focus on the code and subcarrier multipath errors, while neglecting
the carrier multipath error. Moreover, in DPE and DBT methods, the SPLL is coupled with
the PLL, which implies that the carrier and subcarrier multipath errors interact with each
other. When using the DPE or DBT method to track BOC signals, the subcarrier multipath
error cannot be completely eliminated without eliminating the carrier multipath error.
This paper proposes a prompt-assisted-offset correlator (PAOC) technique based on
the DBT loop to mitigate multipath for BOC signals. The PAOC technique uses the prompt
and offset correlators to significantly improve multipath performance with small noise
performance loss. Considering the interaction between carrier and subcarrier errors, this
paper further proposes using the PAOC technique in both the PLL and the SPLL of the DBT
loop. In addition, the code multipath error is mitigated by the NC technique.
The main contributions of this paper are summarized as follows:
1. A new multipath mitigation technique named PAOC is proposed for BOC signals. The
PAOC technique is an improved OC technique with less thermal noise performance
loss. Therefore, the PAOC technique can be configured with a larger forward offset to
obtain better multipath performance in practice.
2. It is proposed that carrier and subcarrier multipath errors affect each other when
using the SPLL to track the subcarrier. Therefore, this paper proposes to apply the
PAOC technique in both the PLL and the SPLL to mitigate carrier and subcarrier
multipath errors simultaneously. Compared with existing methods that ignore the
carrier multipath, the proposed method better mitigates the subcarrier multipath error
and additionally mitigates the carrier multipath error.
The remainder of this paper is organized as follows. Section 2 introduces the multipath
model for BOC signals and analyzes the multipath effects in detail. Section 3 elaborates on
the principle and implementation architecture of the PAOC technique. Section 4 analyzes
the multipath performance and thermal noise performance of the PAOC technique and
provides several simulation results. The results and implications are discussed in Section 5.
Finally, conclusions are given in Section 6.
2. Problem Formulation
This section first describes the BOC signal reception model without multipath and
briefly reviews the DBT method. Then, a double-sideband multipath model of the BOC
signal is introduced. Finally, the multipath effect on the DBT loop is analyzed.
2.1. Double-Sideband Reception Model for BOC Signals
In the time domain, the BOC signal can be viewed as the product of a BPSK signal and
a subcarrier in the form of a square wave, which is written as follows:
s(t) = c(t)sign sin(2π f sc0 t + φ0 ) cos(2π f c0 t + θ0 ) (1)
where c(t) represents the spreading code, f sc0 is the nominal subcarrier frequency, φ0 is
the initial subcarrier phase, f c0 is the nominal carrier frequency, and θ0 is the initial carrier
phase. A BOC-modulated GNSS signal is usually abbreviated as BOC(m, n), indicating that
its subcarrier frequency is m × 1.023 MHz and its code rate is n × 1.023 MHz. The order
of the BOC signal is defined as k = 2m/n, which is generally an integer. Figure 1 shows
the normalized ACF of BPSK(n), BOC(2n, n), and BOC(6n, n) signals. It can be seen from
Figure 1 that the ACF of the BOC signal with a higher order has more peaks.Remote Sens. 2023, 15, 937 4 of 25
BPSK(n)
BOC(2n,n)
Normalized correlation value
BOC(6n, n)
Code delay (chip)
Figure 1. Normalized auto-correlation functions of BPSK(n), BOC(2n,n), and BOC(6n,n) signals.
In practice, the signal transmission bandwidth of GNSS satellites is limited. For BOC
signals with a high subcarrier frequency, the actual subcarrier emitted by GNSS satellites is
not a square wave but approximates a sine wave. For example, the transmission bandwidth
for Galileo E1A signal with the BOC(15, 2.5) modulation is 40.96 MHz [26], so the subcarrier
harmonics are filtered out at the transmitting end. Moreover, the front-end bandwidth
of GNSS receivers is also limited and usually smaller than the GNSS signal transmission
bandwidth. The subcarrier of the received BOC signal can be reasonably considered as a
sine wave. Therefore, the received BOC signal can be written as:
rd (t) = Ac(t − τ ) cos(2π f sc t + φ) cos(2π f c t + θ ) (2)
where A is the amplitude of the received BOC signal, τ is the propagation delay, f sc is the
subcarrier frequency with Doppler, φ is the subcarrier phase, f c is the carrier frequency
with Doppler, and θ is the carrier phase. For the sake of simplicity, the noise is omitted in
Equation (2), which does not affect the following discussions.
Using the trigonometric transformation, Equation (2) can be rewritten as follows:
r d ( t ) =r u ( t ) + r l ( t )
A
ru (t) = c(t − τ ) cos 2π ( f c + f sc )t + (θ + φ) (3)
2
r (t) = A c(t − τ ) cos 2π ( f c − f sc )t + (θ − φ)
l
2
where ru (t) is the upper sideband (USB) of the BOC signal, and rl (t) is the lower sideband
(LSB) of the BOC signal. It can be seen from Equation (3) that both ru (t) and rl (t) are
BPSK-modulated signals.
Equation (3) shows a double-sideband reception model for BOC signals, where the
received BOC signal is regarded as the sum of two BPSK signals. Based on this double-
sideband signal model, the DBT method jointly processes the two BPSK signals in the USB
and LSB to track the BOC signal. Since the ACFs of the two BPSK signals have only one
peak, the DBT method solves the tracking ambiguity problem. The DBT loop employs two
sets of correlators for the USB and LSB of the BOC signal, so it is compatible with classical
GNSS receiver architectures designed for BPSK signals.
Using the DBT loop to track BOC signals can yield three ranging measurements: the
code delay, subcarrier phase, and carrier phase. The subcarrier ranging measurement is
more accurate than the code ranging measurement, but it has integer ambiguity. In theRemote Sens. 2023, 15, 937 5 of 25
absence of multipath, code ranging errors are usually less than half of the subcarrier wave-
length. Therefore, the subcarrier ambiguity can be fixed by the code ranging measurement
to obtain the unambiguous and high-precision pseudo-range:
ρcode − ρsc
ρ = ρsc + round( )λsc (4)
λsc
where ρsc is the subcarrier ranging measurement, ρcode is the code ranging measurement,
and λsc is the subcarrier wavelength.
In the presence of multipath, code ranging errors may exceed half of the subcarrier
wavelength, resulting in an incorrect subcarrier ambiguity solution. Therefore, it is required
to reduce the code multipath error to less than half of the subcarrier wavelength. It has
been proved that applying the NC technique in the DBT loop can effectively mitigate code
multipath errors to meet this requirement [22]. When the subcarrier ambiguity is correctly
fixed, the pseudo-range shown in Equation (4) has the same accuracy as the subcarrier
ranging measurement.
The accuracy of pseudo-range positioning is usually at the meter level, while centimeter-
level positioning applications require high-precision carrier ranging measurements. The
carrier phase has integer ambiguity, which needs to be fixed with the pseudo-range. How-
ever, the pseudo-range error of the BOC signal, i.e., the subcarrier ranging error, may
lead to an incorrect resolution of carrier phase ambiguity. Therefore, it is necessary to
design carrier and subcarrier multipath mitigation techniques for BOC signals to improve
positioning accuracy.
2.2. Multipath Model for BOC Signals
In practice, the BOC signal is probably reflected or diffracted by obstacles around
the GNSS receiver, resulting in multipath signals. Compared with the direct signal, the
multipath signal has a longer propagation path. Moreover, the power and carrier Doppler
of a multipath signal are usually different from those of the direct signal. The compound
signal consisting of the direct signal and multipath signals can be expressed as follows:
M
r c ( t ) =r d ( t ) + ∑ rm (t) (5)
m =1
rm (t) =αm Ac(t − τ − τm ) cos(2π f sc t + φ + φm ) cos 2π ( f c + f m )t + θ + θm (6)
where rd (t) is the direct signal shown in Equation (2), rm (t) is the m-th multipath signal,
M is the number of multipath signals, αm is the multipath relative amplitude, τm is the
multipath relative delay, φm = −2π f sc τm is the multipath relative subcarrier phase, f m is
the Doppler difference between the direct signal and m-th multipath signal, which is called
the fading frequency, and θm is the multipath relative phase.
The multipath fading frequency is related to the geometric distribution of the satellite,
reflecting platform, and GNSS receiver. Fast-varying multipath signals with large fading
frequencies will be suppressed by correlators and loop filters of the GNSS receiver [27].
When the reciprocal of the fading frequency is less than the integration time, the multipath
signal is completely averaged out and has no effect on the tracking loop. By contrast, slow-
varying multipath signals with small fading frequencies severely affect the tracking loop
and cause large ranging errors. In conclusion, the slow-varying multipath poses a greater
threat to GNSS positioning accuracy than the fast-varying multipath. For a stationary
GNSS receiver, the multipath fading frequency is very small, typically less than 1 Hz [28,29].
Most multipath signals received by GNSS receivers used in surveying, monitoring stations,
and other low-dynamic scenarios have small fading frequencies and change slowly [30,31].
Therefore, this paper investigates the slow-varying multipath problem for low-dynamic
GNSS receivers.
According to the double-sideband reception model for BOC signals, the compound
signal shown in Equation (5) can be transformed as follows:Remote Sens. 2023, 15, 937 6 of 25
rc (t) =ruc (t) + rlc (t)
M
ruc (t) = A c(t − τ ) cos(2π f u t + θu ) +
∑ m
c ( t − − ) cos 2π ( f + f ) t + + +
α τ τm u m θ u θ m φm
2 m =1 (7)
M
A
rlc (t) = 2 c(t − τ ) cos(2π f l t + θl ) + ∑ αm c(t − τ − τm ) cos 2π ( f l + f m )t + θl + θm − φm
m =1
where ruc (t) is the USB of the compound signal, rlc (t) is the LSB of the compound signal,
f u = f c + f sc and f l = f c − f sc are the carrier frequencies of the direct USB and LSB signals,
respectively, and θu = θ + φ and θl = θ − φ are the carrier phases of the USB and LSB direct
signals, respectively.
Equation (7) shows that the received BOC signal can be regarded as the sum of USB
and LSB signals in the presence of multipath. The BOC multipath signal can be decomposed
into two BPSK multipath signals located in the USB and LSB, respectively. Therefore, it is
feasible to track the compound BOC signal with the DBT loop. However, multipath signals
lead to tracking errors. It is necessary to analyze the effect of multipath on the DBT loop.
2.3. Multipath Effect Analysis for BOC Signals
In the DBT loop, the received BOC signal is first mixed with the in-phase carrier
and quadrature-phase carrier generated by the carrier numerically controlled oscillator
(NCO), respectively. This process is called the carrier wipe-off. Two zero intermediate-
frequency (IF) signals are obtained after wiping off the carrier, which are named the
in-phase signal and the quadrature-phase signal, respectively. In correlator channels, the
in-phase signal is correlated with local codes generated by the code NCO to calculate
in-phase correlation values, and the quadrature-phase signal is correlated with local codes
to calculate quadrature-phase correlation values.
The complex correlation functions of the USB and LSB signals can be written as:
M
A
R(∆τd ) + ∑ αm R(∆τd − τm ) exp j(θm + φm ) exp j(∆θd + ∆φd )
Ru = (8)
2 m =1
M
A
R(∆τd ) + ∑ αm R(∆τd − τm ) exp j(θm − φm ) exp j(∆θd − ∆φd )
Rl = (9)
2 m =1
where ∆τd , ∆θd , and ∆φd are the code delay difference, carrier phase difference, and
subcarrier phase difference between the local signal and the direct signal, respectively.
R(·) denotes the normalized ACF of the spreading code. When the code delay difference
between the local signal and the received signal is greater than one chip, the correlation
value is very close to zero. The real part of the complex correlation function is the in-phase
correlation function, and the imaginary part of the complex correlation function is the
quadrature-phase correlation function.
In the absence of multipath, the correlation functions of USB and LSB signals have the
same shape. The in-phase correlation function of each single sideband (SSB) signal is an
isosceles triangle, as shown by the blue dashed lines in Figure 2. In the DBT loop, each
correlator channel for the USB and LSB signals has three correlators, which are denoted as
early (E), prompt (P), and late (L), respectively. The spacing between every two adjacent
correlators is identical, and the spacing between E and L correlators is called the early-
minus-late (EML) spacing.
The EML code discriminator is designed based on the symmetry of the correlation
function. However, the symmetry of the correlation function is broken by the multipath as
shown in Figure 2, where the multipath relative amplitude is 0.5, the multipath relative
carrier phase is π/6, and the multipath relative delay is 0.4 chips. Thus, the multipath
leads to bias in the EML discrimination output, and the code multipath error is related toRemote Sens. 2023, 15, 937 7 of 25
the EML spacing. Using narrow EML spacing can effectively reduce the code multipath
error, which is the principle of the NC technique.
SSB of the direct signal SSB of the direct signal
USB of the compound signal USB of the compound signal
Normalized correlation value
Normalized correlation value
LSB of the compound signal LSB of the compound signal
P
L
E L
P
E
Code delay (chip) Code delay (chip)
(a) (b)
Figure 2. Normalized correlation functions for each single sideband of the BOC(15, 2.5) signal, where
E, P, and L denote early, prompt, and late correlators, respectively: (a) in-phase correlation functions;
(b) quadrature-phase correlation functions.
The P correlation values are used for subcarrier and carrier discrimination in the
DBT loop. In the absence of multipath, the quadrature-phase P correlation values of the
USB and LSB signals are 0. However, in the presence of multipath, the quadrature-phase
P correlation values of the USB and LSB signals are not 0. As a result, the outputs of the
carrier and subcarrier discriminators are affected by the multipath.
As shown in Equations (8) and (9), the carrier phase and subcarrier phase are coupled
to each other in the USB and LSB correlation values, which implies that the carrier and
subcarrier multipath errors interact with each other. Thus, the DBT method subtly combines
the USB and LSB correlation values to separate the carrier phase from the subcarrier
phase. Specifically, the synthesized correlation function for subcarrier discrimination is
calculated as:
M
Rsc = Ru + R∗l = A R(∆τd ) cos(∆θd ) + ∑ αm R(∆τd − τm ) cos(∆θd + θm ) exp( jφm ) exp( j∆φd ) (10)
m =1
Equation (10) demonstrates that multipath signals lead to a phase rotation of the
complex correlation function for subcarrier discrimination. The phase rotation of Rsc is the
subcarrier multipath error. When there are many multipath signals, it is difficult to derive
an analytical formula for the subcarrier multipath error. Assuming that there is only one
multipath signal, the subcarrier multipath error of the DBT method can be calculated as:
αm R(∆τd − τm ) cos(∆θd + θm ) sin(φm )
∆φ ME = arctan (11)
R(∆τd ) cos(∆θd ) + αm R(∆τd − τm ) cos(∆θd + θm ) cos(φm )
The synthesized correlation function for carrier discrimination is calculated as:
M
Rc = Ru + Rl = A R(∆τd ) cos(∆φd ) + ∑ αm R(∆τd − τm ) cos(∆φd + φm ) exp( jθm ) exp( j∆θd ) (12)
m =1
Equation (12) demonstrates that multipath signals lead to a phase rotation of the
complex correlation function for carrier discrimination. The phase rotation of Rc is theRemote Sens. 2023, 15, 937 8 of 25
carrier multipath error. Assuming that there is only one multipath signal, the carrier
multipath error of the DBT method can be calculated as:
αm R(∆τd − τm ) cos(∆φd + φm ) sin(θm )
∆θ ME = arctan (13)
R(∆τd ) cos(∆φd ) + αm R(∆τd − τm ) cos(∆φd + φm ) cos(θm )
It is worth noting that the estimated carrier phase ∆θd and subcarrier phase ∆φd in
Equations (11) and (13) contain multipath errors, indicating that the carrier and subcarrier
multipath errors affect each other. If multipath mitigation techniques are only used in
the SPLL, the subcarrier discrimination output still has deviations due to the indirect
effect of carrier multipath errors. Similarly, if multipath mitigation techniques are only
used in the PLL, the carrier discrimination output still has deviations due to the indirect
effect of subcarrier multipath errors. Therefore, it is required that the multipath mitigation
method for BOC signals based on the DBT loop can mitigate both the carrier and subcarrier
multipath errors.
3. Materials and Methods
This section proposes a prompt-assisted-offset correlator (PAOC) technique to miti-
gate carrier and subcarrier multipath errors for BOC signals. Firstly, the principle of the
PAOC technique is introduced. Then, the PAOC discriminators and complementary filters
are described in detail. Finally, an implementation architecture for applying the PAOC
technique in the DBT loop is provided.
3.1. Principle of the PAOC Technique
The multipath signal has a longer propagation delay than the direct signal, implying
that the correlation peak of the multipath signal lags behind that of the direct signal.
Therefore, the forward offset correlator is less susceptible to the multipath than the prompt
correlator. Figure 3 illustrates the effect of a multipath signal with a relative delay of
0.4 chips on different correlators. The quadrature-phase correlation value between the local
signal and the direct signal is 0, so the quadrature-phase correlation value of the compound
signal is equal to that of the multipath signal. As a result, in Figure 3b, the red and green
lines overlap, and the purple and yellow lines overlap. Figure 3 clearly shows that the
prompt (P) correlator is affected by the multipath signal, while the offset correlator (OC)
with a forward offset of 0.8 chips is not affected by the multipath signal.
SSB of the direct signal SSB of the direct signal
USB of the multipath signal USB of the multipath signal
Normalized correlation value
Normalized correlation value
USB of the compound signal USB of the compound signal
LSB of the multipath signal LSB of the multipath signal
LSB of the compound signal LSB of the compound signal
OC P OC P
Code delay (chip) Code delay (chip)
(a) (b)
Figure 3. Schematic diagram of the multipath effect on the offset correlator (OC) and the prompt (P)
correlator: (a) in-phase correlation functions; (b) quadrature-phase correlation functions.
In fact, the prompt correlator is affected by multipath signals with relative delays
less than 1 chip. The offset correlator with a forward offset of τoc chips is only affectedRemote Sens. 2023, 15, 937 9 of 25
by multipath signals with relative delays less than 1 − τoc chips. Therefore, using the
forward offset correlator instead of the prompt correlator for discrimination can effectively
reduce the multipath sensitive area, thus mitigating carrier and subcarrier multipath errors.
Furthermore, the multipath performance of the offset correlator improves with the increase
of forward offset.
However, as shown in Figure 3, the magnitude of the offset correlator output is smaller
than that of the prompt correlator output. In the presence of noise, the SNR of the offset
correlator output is poor. Moreover, the thermal noise performance of the offset correlator
degrades with the increase of forward offset. As a consequence, using the offset correlator
for discrimination requires a compromise between the thermal noise performance and
multipath mitigation performance.
It is observed that the prompt correlator output has a high SNR but is susceptible to the
multipath. In contrast, the offset correlator output has a low SNR but is less susceptible to
the multipath. The prompt correlator and the offset correlator are complementary in terms
of both multipath performance and thermal noise performance. Based on this insight, this
paper proposes a PAOC technique using the prompt correlator to assist the offset correlator
for carrier and subcarrier discrimination. The PAOC technique combines the advantages
of the prompt and offset correlators to substantially improve multipath performance with
small noise performance loss.
3.2. PAOC Technique for Subcarrier Multipath Mitigation
There are two specially designed subcarrier discriminators in the PAOC technique.
One subcarrier discriminator uses the offset correlator and is called the S-OC discrimina-
tor. The other subcarrier discriminator uses the prompt correlator and is called the S-P
discriminator. The forward offset of the offset correlator relative to the prompt correlator is
assumed to be τoc chips.
The S-OC discriminator is designed as:
QRuoc − QRloc
Disc(∆φoc ) = arctan (14)
IRuoc + IRloc
where IR denotes the in-phase correlator output, QR denotes the quadrature-phase cor-
relator output, the subscript oc denotes the offset correlator, and the superscripts u and l
denote the USB and LSB, respectively.
The S-P discriminator is designed as:
QRup − QRlp
Disc(∆φ p ) = arctan (15)
IRup + IRlp
where the subscript p denotes the prompt correlator.
When the multipath relative delay is greater than 1 − τoc chips, the S-OC discriminator
output contains no multipath error and can be modeled as:
∆φoc = ∆φd + eoc (16)
where ∆φd is the subcarrier phase difference between the local signal and the direct signal,
and eoc is the S-OC noise error.
The S-P discriminator output contains multipath errors and can be modeled as:
∆φ p = ∆φd + ∆φ ME + e p (17)
where ∆φ ME is the subcarrier multipath error, and e p is the S-P noise error. Because the
prompt correlator output has a higher SNR than the offset correlator, the variance of e p is
smaller than that of eoc .Remote Sens. 2023, 15, 937 10 of 25
Combining the discriminator outputs shown in Equations (16) and (17), a coarse
estimate of the subcarrier multipath error can be obtained as:
∆φ̂ ME = ∆φ p − ∆φoc = ∆φ ME + e p − eoc (18)
The subcarrier multipath error estimate ∆φ̂ ME shown in Equation (18) is noisy. To
obtain a fine estimate of the subcarrier multipath error, a first-order low-pass filter is used
to reduce the noise error of ∆φ̂ ME . The low-pass filtering process can be expressed as:
N−1 1
∆φ̂ MEF [k] = ∆φ̂ MEF [k − 1] + ∆φ̂ ME [k] (19)
N N
where N is the smoothing constant of the low-pass filter, and k is the current epoch. The
initial value ∆φ̂ MEF [0] can be set as 0.
There is an implicit prerequisite for using the low-pass filter to reduce noise errors,
which is that the subcarrier multipath error changes slowly. For a low-dynamic GNSS
receiver, the fading frequencies of received multipath signals are usually very small. There-
fore, the multipath errors of a static or low-dynamic GNSS receiver change slowly, which
allows the low-pass filter to be used.
The smoothing constant N determines the noise error of the filtered estimate ∆φ̂ MEF .
As the smoothing constant increases, the noise error of ∆φ̂ MEF reduces, while the dynamic
performance of the GNSS receiver degrades. In practice, the smoothing constant N can be
set flexibly as needed. Considering that the fading frequency of multipath signals received
by a static or low-dynamic receiver is generally less than 1 Hz, the smoothing time of
the low-pass filter is preferably set to be no longer than 1 s. More discussions about the
low-pass filter will be given in Section 4.
After low-pass filtering, an accurate estimate of the subcarrier multipath error can be
obtained as:
∆φ̂ MEF = ∆φ ME + eres (20)
where eres is the residual noise error with a small variance.
Subtracting the subcarrier multipath error estimate ∆φ̂ MEF from the S-P discriminator
output ∆φ p , an unbiased subcarrier estimate with small noise errors can be obtained as:
∆φPAOC = ∆φ p − ∆φ̂ MEF = ∆φd + e p − eres (21)
Equation (21) shows that the PAOC subcarrier estimate ∆φPAOC contains no multipath
error. Moreover, the thermal noise error of ∆φPAOC is smaller than that of ∆φoc but lager
than that of ∆φ p .
The processing described in Equations (18)–(21) can be modeled as a PAOC comple-
mentary filter, which is shown in Figure 4. When the smoothing constant is 1, ∆φ̂ MEF [k ] is
equal to ∆φ̂ ME [k ], and the PAOC subcarrier estimate ∆φPAOC is equal to the S-OC discrimi-
nator output ∆φoc . That is to say, the OC technique is a special case of the PAOC technique
with a smoothing constant of 1.
oc _ ˆME First-order ˆMEF _ PAOC
S-OC discriminator
low-pass filter
+ +
p
S-P discriminator
Figure 4. Schematic diagram of the PAOC technique for subcarrier multipath mitigation.Remote Sens. 2023, 15, 937 11 of 25
3.3. PAOC Technique for Carrier Multipath Mitigation
In addition to subcarrier multipath errors, the PAOC technique can also be used
to mitigate carrier multipath errors. Similar to the subcarrier discrimination, there are
two carrier discriminators in the PAOC technique. One carrier discriminator uses the offset
correlator and is called the C-OC discriminator. The other carrier discriminator uses the
prompt correlator and is called the C-P discriminator.
The C-OC discriminator is designed as:
QRuoc + QRloc
Disc(∆θoc ) = arctan (22)
IRuoc + IRloc
The C-P discriminator is designed as:
QRup + QRlp
Disc(∆θ p ) = arctan (23)
IRup + IRlp
The outputs of C-OC and C-P discriminators are combined using the PAOC comple-
mentary filter shown in Figure 5 to obtain a high-precision carrier estimate ∆θ PAOC without
multipath errors. The PAOC complementary filter for the carrier has the same architecture
as that for the subcarrier. In practice, the PAOC complementary filters for the carrier and
subcarrier can be configured with different parameters if necessary.
oc _ ˆME First-order ˆMEF _ PAOC
C-OC discriminator
low-pass filter
+ +
p
C-P discriminator
Figure 5. Schematic diagram of the PAOC technique for carrier multipath mitigation.
3.4. Implementation of the PAOC Technique
Figure 6 shows the implementation architecture of the PAOC technique in the DBT
loop, which consists of four main parts: correlator channels, a PAOC-PLL for the carrier,
a PAOC-SPLL for the subcarrier, and a DLL for the code. The S-OC discriminator, S-P
discriminator, and PAOC complementary filter are implemented in the PAOC-SPLL. The
C-OC discriminator, C-P discriminator, and PAOC complementary filter are implemented
in the PAOC-PLL.
The correlator channels are divided into correlator channels for the USB signals and
correlator channels for the LSB signals. Every correlator channel has four complex correla-
tors, denoted as the offset correlator (OC), early (E), prompt (P), and late (L), respectively.
The received BOC signal is correlated with local replicas in correlator channels to obtain
correlation values.
The outputs of the OC and P correlators are fed into the PAOC-PLL to estimate the
carrier frequency and carrier phase of the received BOC signal. Moreover, the outputs
of the OC and P correlators are also fed into the PAOC-SPLL to estimate the subcarrier
frequency and subcarrier phase of the received BOC signal. The outputs of the E and L
correlators are fed into the DLL to estimate the code rate and code delay of the received
BOC signal. It is worth noting that the spacing between the E and L correlators is usually
smaller than 0.1 chips, making the code multipath error less than half of the subcarrier
wavelength. According to the parameter estimation results, the frequency words of carrier
NCOs and code NCOs are adjusted in real-time to generate local replicas, thus forming a
closed tracking loop.Remote Sens. 2023, 15, 937 12 of 25
Correlator channels for the upper sideband signal
OC P E L
Upper sideband Upper sideband
carrier NCO code NCO
BOC PAOC-PLL PAOC-SPLL DLL
signal
Lower sideband Lower sideband
carrier NCO code NCO
OC P E L
Correlator channels for the lower sideband signal
Figure 6. Implementation architecture of the PAOC technique in the DBT loop for BOC signals.
4. Performance Evaluation and Results
This section first analyzes the theoretical multipath performance and thermal noise
performance of the PAOC technique. Then, some simulation results are presented to
compare the PAOC technique with existing multipath mitigation techniques based on the
DBT loop.
4.1. Multipath Performance Analysis
The outputs of offset correlators for the USB and LSB signals can be expressed as:
M
A
R(∆τd − τoc ) + ∑ αm R(∆τd − τoc − τm ) exp j(θm + φm ) exp j(∆θd + ∆φd )
Ru−oc = (24)
2 m =1
M
A
R(∆τd − τoc ) + ∑ αm R(∆τd − τoc − τm ) exp j(θm − φm ) exp j(∆θd − ∆φd )
Rl −oc = (25)
2 m =1
The synthesized correlation function used by the S-OC discriminator is calculated as:
Rs−oc = Ru−oc + R∗l −oc
M
(26)
= A R(∆τd − τoc ) cos(∆θd ) + ∑ αm R(∆τd − τoc − τm ) cos(∆θd + θm ) exp( jφm ) exp( j∆φd )
m =1Remote Sens. 2023, 15, 937 13 of 25
Multipath signals cause a phase rotation of Rs−oc , which is the subcarrier multipath
error. Assuming that there is only one multipath signal, the subcarrier multipath error of
the S-OC discriminator can be calculated as:
αm R(∆τd − τoc − τm ) cos(∆θd + θm ) sin(φm )
∆φ ME = arctan (27)
R(∆τd − τoc ) cos(∆θd ) + αm R(∆τd − τoc − τm ) cos(∆θd + θm ) cos(φm )
When |∆τd − τoc − τm | ≥ 1, R(∆τd − τoc − τm ) = 0. This demonstrates that the S-OC
discriminator can eliminate subcarrier multipath errors caused by the multipath signal
with a relative delay greater than 1 − τoc chips. The subcarrier multipath error of the S-P
discriminator is equal to that of the DBT method, which is given in Equation (11). The
PAOC technique subtly combines the outputs of the S-OC and S-P discriminators. Thus,
the subcarrier multipath performance of the PAOC technique is the same as that of the
S-OC discriminator shown in Equation (27).
The synthesized correlation function used by the C-OC discriminator is calculated as:
Rc−oc = Ru−oc + Rl −oc
M
(28)
= A R(∆τd − τoc ) cos(∆φd ) + ∑ αm R(∆τd − τoc − τm ) cos(∆φd + φm ) exp( jθm ) exp( j∆θd )
m =1
Assuming that there is only one multipath signal, the carrier multipath error of the
C-OC discriminator can be calculated as:
αm R(∆τd − τoc − τm ) cos(∆φd + φm ) sin(θm )
∆θ ME = arctan (29)
R(∆τd − τoc ) cos(∆φd ) + αm R(∆τd − τoc − τm ) cos(∆φd + φm ) cos(θm )
When |∆τd − τoc − τm | ≥ 1, R(∆τd − τoc − τm ) = 0. This demonstrates that the C-
OC discriminator can eliminate carrier multipath errors caused by the multipath signal
with a relative delay greater than 1 − τoc chips. The carrier multipath error of the C-P
discriminator is equal to that of the DBT method, which is given in Equation (13). The
PAOC technique subtly combines the outputs of the C-OC and C-P discriminators. Thus,
the carrier multipath performance of the PAOC technique is the same as that of the C-OC
discriminator shown in Equation (29).
The multipath error envelope (MEE) is a commonly used criterion to evaluate the
multipath performance, which shows the maximum error under different multipath relative
delays [32]. Equation (27) indicates that the subcarrier multipath error reaches a maximum
value when θm = 0 and π. Equation (29) indicates that the carrier multipath error reaches
a maximum value when θm = ±π/2. Figure 7 shows the carrier and subcarrier MEEs of
three techniques for the BOC(15, 2.5) signal, where the multipath relative amplitude is
0.5. Two forward offsets are tested for the OC and PAOC techniques, which are 0.5 and
0.8 chips, respectively.
Figure 7 demonstrates that the PAOC technique effectively mitigates carrier and
subcarrier multipath errors when the multipath relative delay is greater than 1 − τoc chips.
The multipath mitigation performance of the PAOC technique improves with the forward
offset. Moreover, the PAOC technique has the same multipath performance as the OC
technique with the same forward offset.Remote Sens. 2023, 15, 937 14 of 25
DBT DBT
OC ( oc=0.5) OC ( oc=0.5)
Subcarrier multipath error (m)
Carrier multipath error (°)
PAOC ( oc=0.5) PAOC ( oc=0.5)
OC ( oc=0.8) OC ( oc=0.8)
PAOC ( oc=0.8) PAOC ( oc=0.8)
Multipath relative delay (chip) Multipath relative delay (chip)
(a) (b)
Figure 7. MEEs of the DBT, OC, and PAOC techniques for the BOC(15, 2.5) signal with a bandwidth
of 40.96 MHz: (a) subcarrier MEE; (b) carrier MEE.
4.2. Thermal Noise Performance Analysis
The PAOC technique employs two subcarrier discriminators in the PAOC-SPLL, i.e.,
the S-OC and S-P discriminators. The S-P discriminator is the same as the subcarrier
discriminator used in the original DBT loop. Thus, the thermal noise error of the S-P
discriminator output can be expressed as [10]:
BSPLL (1 − 0.5BSPLL T )
σs2− p = R + β /2 (30)
C/N0 − βrr/2 Gs ( f )d f
where BSPLL is the noise bandwidth of the PAOC-SPLL, T is the integration time, C/N0
is the carrier-to-noise ratio, Gs ( f ) is the normalized power spectral density (PSD) of each
single sideband of the BOC signal, and β r is the received bandwidth.
The S-OC discriminator uses the offset correlator instead of the prompt correlator.
According to the relationship between the offset and prompt correlators, the thermal noise
error of the S-OC discriminator output can be calculated as:
R + β /2
BSPLL (1 − 0.5BSPLL T ) − βrr/2 Gs ( f )d f
σs2−oc = R + β /2 2 (31)
C/N0 − βrr/2 Gs ( f ) cos(2π f τoc )d f
The derivation of Equation (31) is given in Appendix A.
The outputs of the S-P and S-OC discriminators are combined in the subcarrier PAOC
complementary filter to obtain an accurate subcarrier estimate ∆φPAOC . Therefore, the
noise variance σs2− PAOC of the PAOC subcarrier estimate ∆φPAOC is between σs2− p and σs2−oc .
When the smoothing constant of the subcarrier PAOC complementary filter is one, the
PAOC technique degenerates to the OC technique, and σs2− PAOC is equal to σs2−oc . As the
smoothing constant increases, σs2− PAOC gradually approaches σs2− p .
The PAOC technique employs two carrier discriminators in the PAOC-PLL, i.e., the
C-OC and C-P discriminators. The C-P discriminator is the same as the carrier discriminator
used in the original DBT loop. Thus, the thermal noise error of the C-P discriminator output
can be expressed as [10]:
B (1 − 0.5B T )
σc2− p = PLL R + β /2 PLL (32)
C/N0 − βrr/2 Gs ( f )d f
where BPLL is the noise bandwidth of the PAOC-PLL.Remote Sens. 2023, 15, 937 15 of 25
The C-OC discriminator uses the offset correlator instead of the prompt correlator.
According to the relationship between the offset and prompt correlators, the thermal noise
error of the C-OC discriminator output can be calculated as:
R + β /2
BPLL (1 − 0.5BPLL T ) − βrr/2 Gs ( f )d f
σc2−oc = R + β /2 2 (33)
C/N0 − βrr/2 Gs ( f ) cos(2π f τoc Tc )d f
The derivation of Equation (33) is given in Appendix A.
The outputs of the C-P and C-OC discriminators are combined in the carrier PAOC
complementary filter to obtain an accurate carrier estimate ∆θ PAOC . Therefore, the noise
variance σc2− PAOC of the PAOC carrier estimate ∆θ PAOC is between σc2− p and σc2−oc . When
the smoothing constant of the carrier PAOC complementary filter is one, σc2− PAOC is equal
to σc2−oc . As the smoothing constant increases, σc2− PAOC gradually approaches σc2− p .
Figure 8 shows the subcarrier and carrier noise errors of three techniques for the
BOC(15, 2.5) signal, where the noise bandwidths of the SPLL and the PLL are 1 Hz and
10 Hz, respectively. With the identical forward offset, the PAOC technique has better noise
performance than the OC technique. As the smoothing constant N increases, thermal noise
errors of the PAOC technique gradually decrease and approach those of the DBT method.
0.35 25
DBT DBT
(m)
0.3 OC ( oc
=0.8) OC ( oc
=0.8)
(°)
20
PAOC ( oc
=0.8, N=10) PAOC ( oc
=0.8, N=10)
0.25
Subcarrier noise error
PAOC ( =0.8, N=20) PAOC ( =0.8, N=20)
Carrier noise error
oc oc
0.2 15
Threshold = 15°
0.15 10
0.1
5
0.05
0 0
36 38 40 42 44 46 48 50 36 38 40 42 44 46 48 50
C/N0 (dB-Hz) C/N0 (dB-Hz)
(a) (b)
Figure 8. Tracking jitters of the DBT, OC, and PAOC techniques for the BOC (15, 2.5) signal with a
bandwidth of 40.96 MHz: (a) subcarrier noise error; (b) carrier noise error.
The tracking threshold of the PLL usually requires the standard deviation of the carrier
phase error to be no more than 15◦ . Figure 8b demonstrates that the application of the OC
technique in the PLL may cause the tracking loop to lose lock. Compared with the OC
technique, the PAOC technique can greatly reduce the risk of losing lock.
4.3. Simulation Results
In the following experiments, four methods are compared in terms of thermal noise
performance and multipath performance, which are shown in Table 1. The first method is
the original DBT, where both the SPLL and the PLL use prompt correlators. The second
method is called the OC-P, where the SPLL uses the OC technique, and the PLL uses the
prompt correlator. The third method is called the OC-OC, where both the PLL and the
SPLL use the OC technique. The fourth method is called the PAOC-PAOC, where both
the PLL and the SPLL use the PAOC technique. The first two methods, DBT [10] and
OC-P [22], are old methods that have been proposed before. The last two methods, OC-OC
and PAOC-PAOC, are new methods proposed in this paper for the first time.Remote Sens. 2023, 15, 937 16 of 25
Table 1. Configurations of the four tracking methods for BOC signals.
Method DLL SPLL PLL
DBT Narrow correlation Prompt correlator Prompt correlator
OC-P Narrow correlation Offset correlator Prompt correlator
OC-OC Narrow correlation Offset correlator Offset correlator
PAOC-PAOC Narrow correlation PAOC PAOC
A GNSS receiver equipped with the DBT loop was developed to evaluate the perfor-
mance of various methods. In all subsequent experiments, the common parameters are
configured as shown in Table 2: the receiver front-end bandwidth is 40.96 MHz, the EML
spacing is 0.1 chips, the integration time is 10 ms, the noise bandwidths of the DLL, SPLL,
and PLL are 1 Hz, 1 Hz, and 10 Hz, respectively, C/N0 of the direct signal is 42 dB-Hz, and
the multipath relative amplitude is 0.5.
Table 2. Common parameters for all subsequent simulations.
Parameter Value
Front-end bandwidth 40.96 MHz
Integration time 10 ms
EML spacing 0.1 chips
DLL bandwidth 1 Hz
SPLL bandwidth 1 Hz
PLL bandwidth 10 Hz
C/N0 of the direct signal 42 dB-Hz
Multipath relative amplitude 0.5
4.3.1. Parameter Configuration of the OC and PAOC Techniques
The forward offset τoc has a great impact on the performance of the OC and PAOC
techniques. Figure 9 shows the carrier and subcarrier noise errors of four methods for
the BOC(15, 2.5) signal without multipath, where the smoothing constant of the PAOC
technique is 20. The OC-P and OC-OC methods have the same subcarrier noise performance
because they use the same offset correlator in the SPLL. The OC-P and DBT methods have
the same carrier noise performance because they use the same prompt correlator in the
PLL. The thermal noise error increases with the forward offset for both the OC and PAOC
techniques. Moreover, the thermal noise error of the PAOC technique is much smaller than
that of the OC technique with the same forward offset.
0.12 9
DBT DBT
8
(m)
OC-P OC-P
(°)
0.1 OC-OC OC-OC
7
PAOC-PAOC PAOC-PAOC
Subcarrier noise error
Carrier noise error
0.08 6
5
0.06 4
3
0.04
2
0.02 1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Forward offset oc (chip) Forward offset oc (chip)
(a) (b)
Figure 9. Tracking jitters of various methods for the BOC(15, 2.5) signal without multipath: (a) sub-
carrier noise error; (b) carrier noise error.
Thermal noise performance affects the tracking stability, which is the primary consid-
eration for GNSS receivers. It is fair to compare the multipath performance of the OC andRemote Sens. 2023, 15, 937 17 of 25
PAOC techniques under the same thermal noise performance. Therefore, in the subsequent
experiments, the forward offset of the OC technique is set as 0.5 chips, and the forward
offset of the PAOC technique is set as 0.8 chips with a smoothing constant of 20. The
integration time is 10 ms, so a smoothing constant of 20 means that the smoothing time
is 0.2 s.
4.3.2. Static Multipath Scenarios
In practice, ranging errors include the noise error and multipath error. Moreover,
multipath signals lead to variations in the SNR, thereby affecting the thermal noise error. It
is difficult to separate the noise error from the multipath error. Thus, the root-mean-square
error (RMSE) is used to evaluate the ranging performance of the four methods.
According to Equations (27) and (29), when the multipath relative phase is 0 and π,
the subcarrier multipath error reaches its maximum value, while the carrier multipath error
is almost 0. Figure 10 shows the RMSE envelop of the four methods for the BOC(15, 2.5)
signal when the multipath relative phase is 0 and π. The subcarrier RMSE varies with
multipath relative delay in Figure 10a due to the multipath error.
2 8
DBT DBT
OC-P OC-P
OC-OC OC-OC
Subcarrier RMSE (m)
1.5 6
Carrier RMSE (°)
PAOC-PAOC PAOC-PAOC
1 4
0.5 2
0 0
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
Multipath relative delay (chip) Multipath relative delay (chip)
(a) (b)
Figure 10. RMSE envelop of various methods for the BOC(15, 2.5) signal, where the multipath relative
phase is 0 and π: (a) subcarrier RMSE; (b) carrier RMSE.
According to Equations (13) and (29), the carrier multipath errors of all four methods
are 0 when the multipath relative phase θm is 0 and π. Therefore, the carrier errors shown
in Figure 10b are noise errors. Since the PAOC technique has thermal noise performance
loss compared with using the prompt correlator, the carrier RMSE of the PAOC-PAOC
method in Figure 10b is larger than that of the OC-P and DBT methods. Moreover, the
carrier RMSE varies with multipath relative delay in Figure 10b because the SNR is affected
by multipath.
According to Equations (27) and (29), when the multipath relative phase is ±π/2, the
subcarrier multipath error reaches its minimum value, while the carrier multipath error
reaches its maximum value. Figure 11 shows the RMSE envelop of four methods for the
BOC(15, 2.5) signal when the multipath relative phase is ±π/2.
When the multipath relative delay is greater than 1 chip, there is only thermal noise
error and no multipath error in the RMSE. When the multipath relative delay is less than
1 chip, the RMSE of the DBT method contains both thermal noise error and multipath
error. Comparing the RMSE in the two cases, it can be found that the multipath error
is significantly larger than the thermal noise error. Moreover, Figures 10 and 11 show
that the PAOC-PAOC method has better multipath mitigation performance than the other
three methods.Remote Sens. 2023, 15, 937 18 of 25
Both OC-P and OC-OC methods use the OC technique with the same forward offset
in the SPLL, which suggests that they have the same subcarrier multipath performance.
However, Figure 11a shows that the subcarrier RMSE of the OC-P method is greater
than that of the OC-OC method. This is because the OC-P and OC-OC methods use
different techniques in the PLL, and the carrier error indirectly affects the subcarrier error.
Figure 11b shows that the carrier RMSE of the OC-P method is greater than that of the
OC-OC method. The OC-OC method using the OC technique in the PLL not only directly
improves the carrier multipath performance, but also indirectly improves the subcarrier
multipath performance. Therefore, it is desirable to use multipath mitigation techniques in
both the PLL and the SPLL to improve multipath performance.
0.4 30
DBT DBT
OC-P 25 OC-P
OC-OC OC-OC
Subcarrier RMSE (m)
0.3
Carrier RMSE (°)
PAOC-PAOC PAOC-PAOC
20
0.2 15
10
0.1
5
0 0
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
Multipath relative delay (chip) Multipath relative delay (chip)
(a) (b)
Figure 11. RMSE envelop of various methods for the BOC(15, 2.5) signal, where the multipath relative
phase is ±π/2: (a) subcarrier RMSE; (b) carrier RMSE.
The area enclosed by the RMSE envelop can reflect the average ranging performance
of the multipath mitigation methods, which is calculated as:
Z τmax
Area(τmax ) = RMSE(τm )dτm (34)
0
where τmax is the maximum multipath relative delay. Considering that multipath signals
with relative delays greater than 1 chip do not cause multipath errors, τmax is set to 1.2 chips.
Table 3 shows the area enclosed by RMSE envelopes of the four methods. When the
multipath relative phase is ±π/2, the PAOC-PAOC method has the smallest subcarrier
and carrier RMSE area. When the multipath relative phase is 0 and π, the PAOC-PAOC
method has the smallest subcarrier RMSE area but has a large carrier RMSE area. This is
because the PAOC technique has thermal noise performance loss compared with using the
prompt correlator. When the multipath relative phase is 0 and π, the carrier multipath error
is almost 0, and the carrier RMSE contains only the thermal noise error. Since the PAOC-
PAOC method greatly improves multipath performance, the thermal noise performance
loss is acceptable.
Table 3. Area enclosed by RMSE envelopes of the four methods for the BOC(15, 2.5) signal.
Subcarrier Subcarrier Carrier Carrier
Method
(θm = 0&π) (θm = ±π/2) (θm = 0&π) (θm = ±π/2)
DBT 0.51 0.09 1.96 8.84
OC-P 0.28 0.08 1.98 8.72
OC-OC 0.28 0.07 3.77 6.47
PAOC-PAOC 0.15 0.05 4.04 4.71
θm denotes the multipath relative phase.Remote Sens. 2023, 15, 937 19 of 25
4.3.3. Dynamic Multipath Scenario
To further verify the ranging performance of the PAOC technique in the dynamic
multi-path scenario, a slow-varying multipath signal is simulated. Figure 12 illustrates the
variation of this multipath signal.
80
Multipath relative delay (m) 70
60
50
40
30
20
10
0 100 200 300 400 500 600
Time (s)
Figure 12. Variations of the multipath relative delay.
Figure 13 shows the ranging errors of the four methods for the BOC(15, 2.5) signal in
the dynamic multipath scenario. The carrier and subcarrier ranging errors of the PAOC-
PAOC method are always smaller than those of the other three methods. Table 4 shows
the ranging errors of the four methods in the dynamic multipath scenario. Both the RMSE
and maximum ranging error of the PAOC-PAOC method are much smaller than those of
the other three methods. In summary, the PAOC-PAOC method can effectively reduce the
carrier and subcarrier ranging errors caused by the low-dynamic multipath.
Table 4. Ranging errors of the four methods for the BOC(15, 2.5) signal in the dynamic multipath scenario.
Subcarrier Subcarrier Carrier Carrier
Method
RMSE Maximum Error RMSE Maximum Error
DBT 0.49 m 1.01 m 9.65◦ 23.06◦
OC-P 0.29 m 0.77 m 9.63◦ 23.98◦
OC-OC 0.28 m 0.76 m 6.70◦ 16.97◦
PAOC-PAOC 0.11 m 0.41 m 3.92◦ 14.47◦
DBT DBT
Subcarrier ranging error (m)
OC-P OC-P
Carrier ranging error (°)
OC-OC OC-OC
PAOC-PAOC PAOC-PAOC
Time (s) Time (s)
(a) (b)
Figure 13. Ranging errors of various methods for the BOC(15, 2.5) signal in the dynamic multipath
scenario: (a) subcarrier ranging error; (b) carrier ranging error.You can also read
