Pulse Interference Mitigation Method for BeiDou Receivers Based on Message Randomization

Abstract

In complex electromagnetic environments, especially those with pulsed interference sources, long-period pulsed interference can repeatedly disrupt the real-time data within navigation messages, preventing receivers from obtaining complete message information and significantly extending the time to first fix (TTFF). To address this problem, the interference mechanism is modeled and analyzed from the perspective of navigation message structure. An anti-interference strategy based on navigation message scrambling is proposed, including two key techniques: random scrambling of subframe order and message interleaving encoding. Simulation and experimental results demonstrate that various pulsed interference patterns, with different periods and duty cycles, can significantly impact TTFF. The subframe scrambling method is effective against interference whose period exceeds the subframe duration but is limited when the period is equal to or shorter than the subframe. In contrast, the interleaving method provides more universal resistance across interference patterns. When both techniques are combined, the overall anti-interference performance is further enhanced. Specifically, for interference patterns with periods longer than the subframe duration, the probability that the receiver fails to achieve positioning across multiple consecutive frames is reduced by at least 50% compared to the case without interference mitigation.

1. Introduction

The Global Navigation Satellite System (GNSS) has become an indispensable infrastructure in modern society. However, navigation signals are vulnerable to various interferences during transmission [1], among which pulse interference is a common and highly destructive form. Pulse interference is usually caused by natural phenomena such as lightning, electronic equipment malfunctions, or malicious attacks [2,3]. It can severely affect the reception and decoding process of satellite navigation signals, leading to increased positioning errors or even making it impossible for the terminal to locate [4,5].

When studying the impact of pulse interference on satellite navigation receivers at the signal level, the carrier-to-noise ratio (C/N₀) is primarily used to evaluate the effectiveness of the interference [6]. By analogy to the calculation method of the anti-interference quality factor for narrowband interference [7], the formula for the anti-interference quality factor of pulse interference can be derived. This allows for an analysis of the impact of pulse interference on the C/N₀. Incorporating factors such as the pulse repetition frequency and integration clearing frequency to recalculate the equivalent C/N₀ results in a more accurate assessment. From a time-frequency domain perspective, the smaller the pulse repetition frequency (i.e., the larger the period), the greater the spectral range of the pulse interference covered by the spread spectrum code’s spectral envelope [8]. Consequently, more interference energy enters the correlator, causing greater disruption to the signal and reducing the equivalent C/N₀ accordingly [9]. The most common method for mitigating pulse interference is temporal blanking in the time domain [10]. Temporal blanking can effectively reduce C/N₀ loss. After employing this method, the C/N₀ loss can be decreased by 4–6 dB [11].

From the perspective of navigation messages, pulse interference can significantly impact receiver performance [12]. Long-cycle pulse interference causes sudden, periodic reductions in the C/N₀ and correlation values during the tracking process of software receivers, leading to abnormal demodulation of message data [13]. Equivalent C/N₀ theory can be used to derive relationships between pulse interference parameters, C/N₀, signal-to-noise ratio (SNR), and bit error rate (BER) [14]. However, analyzing only the BER ignores the variability in message data. In fact, navigation messages contain not only critical sensitive data necessary for initial receiver positioning, such as synchronization headers, message information, and check fields, but also non-essential data like almanac parameters and ionospheric delay correction parameters [15]. By synchronizing interference pulses with the aforementioned key message information through hardware-level time synchronization techniques, continuous decoding errors in the critical data within the message occur. Although the receiver can normally capture and track signals, the frame synchronization and decoding verification stages of satellite navigation terminals are compromised, leaving the receiver unable to obtain time and position information [16]. When the pulse interference period matches the navigation message frame period, several different periods of pulse interference can significantly extend the initial positioning time of the receiver.

A commonly used method for suppressing impulse interference is the time-domain blanking technique [17]. By exploiting the concentration of impulse interference energy in the time domain, a detection threshold is set in the time domain. Data samples exceeding the threshold are recorded, and decision conditions are applied. After iteration, the portions above the threshold are set to zero, and the results are output. When impulse interference is present in the signal, the anti-jamming threshold is closely related to the C/N₀ loss. If the threshold is set too high, significant interference residue will remain; if it is set too low, excessive signal loss will occur. In both cases, the interference suppression performance will be unsatisfactory. Generally, the threshold for impulse interference suppression is set to be 5 dB to 10 dB above the noise floor [18]. The block diagram of the time-domain blanking method is shown in Figure 1:

The above figure illustrates the algorithm block diagram of the time-domain blanking method for pulse interference suppression. In this method, sampling signals exceeding the detection threshold are regarded as pulse interference signals and are set to zero. The method is simple to implement and has been widely applied in both civil and military fields. The core issue of the time-domain blanking method is how to determine the zeroing threshold. While suppressing pulse interference, the method also directly causes the loss of the in-pulse duration of the interference in the time domain. If navigation message information exists within this duration, that part of the message will be lost directly [19]. The bit-level errors caused by the lost message are very likely to trigger frame-level verification failures, rendering the entire frame of information unusable. As shown in Figure 2:

Hence, temporal blanking methods are ineffective in reducing the BER of interfered navigation messages. Even after implementing temporal blanking [20], the receiver still cannot obtain complete navigation message information. Therefore, it is urgent to propose other anti-interference measures to mitigate the impact of pulse interference on receiver positioning time.

This paper first briefly analyzes the impact of long-period pulse interference on the navigation message of BeiDou navigation signals and, based on the structure of navigation messages and receiver theory, investigates the mechanism by which pulse interference affects the TTFF of BeiDou receivers. Since the conventional time-domain clipping method cannot effectively suppress the impact of pulse interference on the receiver, an anti-pulse interference method based on navigation message scrambling is proposed. This method enables the receiver to more readily acquire the essential message data for positioning under pulse interference conditions, thereby mitigating the impact of such interference on TTFF. Modeling and simulation analyses are conducted from two perspectives—navigation message subframe order scrambling and message interleaving. The results show that the subframe-order randomization method can effectively counter pulse interference patterns whose interference period is longer than the subframe period, but it is ineffective for patterns with periods less than or equal to the subframe period, indicating certain limitations. The interleaving method, in contrast, exhibits broader applicability. When the two methods are applied jointly, the interference mitigation performance is further improved. Specifically, for three pulse interference patterns with interference periods longer than the subframe period, the probability that the receiver fails to obtain a position fix over multiple consecutive frames is reduced by at least 50% compared with the pre-mitigation scenario.

The remainder of this paper is organized as follows: Section 2 introduces the impact mechanism of pulse interference on the initial positioning time of receivers and provides a detailed description of the proposed anti-pulsed interference method based on navigation message scrambling. Section 3 verifies the effectiveness of the proposed method from an experimental measurement perspective. Section 4 presents a discussion of the main content and related conclusions of the study. Finally, the paper concludes in Section 5.

2. Materials and Methods

This section begins by introducing the impact of pulse interference on navigation messages, followed by an analysis of the mechanism through which pulse interference affects the initial positioning time of receivers. Finally, a pulse-jamming mitigation approach based on navigation message scrambling is proposed.

2.1. Modeling and Analysis of the Impact of Pulse Interference on Navigation Messages

The time-domain expression of pulse interference is

$$P\left(t\right)=\left\{\begin{array}{cc}Af\left(t\right)& n{T}_P\le t\le \left(n+D\right)T_P\\ 0& \left(n+D\right)T_P\le t\le \left(n+1\right)T_P\end{array}\right.$$

(1)

where T_P is the period of the pulse interference, D is the duty cycle, n is an integer, f(t) represents the specific time-domain form of the interference within one period, with its average power normalized, it can be a single-tone interference, narrowband interference, or broadband interference, A denotes the root mean square value of the interference power during the pulse duration, while represents P(t) the baseband signal, which must be modulated onto the navigation signal frequency band. The period of the pulse interference is typically in the order of seconds to tens of seconds, whereas the duration of f(t) is in the order of tens to hundreds of milliseconds. Given that the intervals during which the interference is absent can last for seconds or even tens of seconds, the receiver is able to resume normal tracking and demodulation during these periods. Taking f(t) as a narrowband interference example, the specific impact of the proposed pulse interference on the navigation message is analyzed. The initial C/N₀ is set to 45 dB·Hz, and the jamming-to-signal ratio (JSR) during the pulse duration is 50 dB. Pulse interference period T_P = 6 s, duty cycle D = 0.02. A simulation is conducted on the BeiDou Navigation Satellite System (BDS) B1I signal’s D1 navigation message over a 35 s interval. The simulation results are shown in Figure 3.

The left figure simulates the effect of pulse interference on the output of the phase-locked loop (PLL) loop filter. When a pulse occurs, the interference is so strong that the PLL cannot maintain lock and temporarily loses its locked state. Once the pulse disappears, the PLL quickly tracks and relocks to the signal, restoring the locked state. In the right figure, when the pulse interference is zero, the receiver remains in a normal tracking and demodulation state, allowing the navigation message to be correctly demodulated. However, when the pulse interference is non-zero, the demodulation of the navigation data is severely disrupted, resulting in bit errors. The equivalent C/N₀ theory [21,22] is used to analyze the variation in JSR and bit error rate (BER) when the pulse interference is non-zero. When both interference and white noise are present in the receiver, the equivalent C/N₀ can be expressed as

$${(C/N_0)}_{eff}={\displaystyle \frac{1}{\frac{1}{(C/N_0)}+\frac{JSR}{Q{R}_c}}}$$

(2)

where (C/N₀)_eff is the equivalent C/N₀, R_c denotes the spreading code rate, with units of chips per second (chip/s), Q represents the anti-jamming quality factor. When f(t) is considered as narrowband interference and the signal is modulated using Binary Phase Shift Keying (BPSK), then Q = 1, Based on the C/N₀ and the coherent integration time T_coh, the following expression can be obtained:

$$BER={\displaystyle \frac{1}{2}}\mathrm{erfc}(\sqrt{2{(C/N_0)}_{eff}\cdot T_{coh}})$$

(3)

where BER is the bit error rate, erfc(x) denotes the complementary error function [23]. By solving the above equations simultaneously, the relationship between BER and JSR can be obtained. To validate this relationship, a software receiver [24] simulation is conducted. A 300-s BDS B1I navigation signal is simulated, and an intermediate frequency (IF) navigation signal is generated through pseudocode spreading and IF modulation. A narrowband interference signal is then added, and the signal undergoes tracking and demodulation. The BER of the demodulated navigation message is calculated, and the resulting theoretical and simulated BER curves are shown in Figure 4:

As shown in the figure above, when the JSR is below 35 dB, the theoretical and simulated BER curves are closely aligned. However, when the JSR is between 35 dB and 45 dB, a significant deviation arises between the theoretical and simulated BER. This discrepancy is primarily due to the assumption in the theoretical BER formula that the carrier tracking loop can accurately track the carrier phase. Under low C/N₀ conditions, the loop fails to track the phase, resulting in the inability to demodulate the data. When the JSR exceeds 45 dB, both the simulated and theoretical BER approach their maximum value of 0.5 [25]. Beyond this point, further increases in interference power do not lead to a higher BER. In the subsequent analysis, the BER is assumed to remain at this maximum value of 0.5.

Taking the BeiDou-2 D1 navigation message as an example, the message is structured in a frame–subframe–word–data bit format, as illustrated in Figure 5.

Before broadcasting data bits, satellites typically perform channel coding on the data bits to improve the reliability of information transmission [27]. When errors occur in the data bits and exceed the error correction capability of the encoding method used, correct decoding becomes impossible. Taking the D1 message as an example, BCH (15, 11) channel encoding is applied on a word-by-word basis. When the bit errors caused by pulse interference exceed the error-correcting capability of a word, the navigation message for that word cannot be correctly recovered. At this point, the word error rate (WER) is

$$WER=1-{(1-BER)}^j-C\left(1,j\right)\cdot BER\cdot {(1-BER)}^{j-1}-\cdot \cdot \cdot C\left(k,j\right)\cdot BE{R}^k\cdot {(1-BER)}^{j-k}$$

(4)

where WER is the word error rate, k represents the error correction capability of a single word in the navigation message, j represents the number of message bits covered by the interference,

$$j=T_P\cdot D/T_{bit}$$

(5)

where T_bit represents the navigation message data code period. The duty cycle is mainly set based on the interference period and the navigation message bit period. The period of the pulse interference is aligned with the message period, typically on the order of seconds or tens of seconds. The duty cycle is configured so that the pulse duration within one period can cover several message bits, causing the covered bits to be decoded incorrectly and thereby achieving the purpose of interference.

Therefore, pulse interference can cause individual words in the BeiDou-2 navigation message to be incorrectly decoded, resulting in the loss of the information carried by those words. For BeiDou-3 navigation messages, pulse interference may lead to bit errors within an entire message frame. When the frame fails to pass the CRC parity check, the receiver will discard the data contained in that frame [28].

2.2. Analysis of the Mechanism of Pulse Interference on Time to First Fix

For receivers operating in cold-start or warm-start modes, the data in the navigation message can be categorized into two types. The first type is real-time data [16], which includes clock offset parameters (such as satellite time, system time, and their differences) and ephemeris parameters (such as the satellite’s position, velocity, and acceleration). The second type is non-real-time data, including almanac parameters for the entire satellite constellation and the health status of the satellites. The complete and correct acquisition of real-time data from the navigation message is essential for the receiver’s positioning calculations. These real-time data can be regarded as the key information within the message, The distribution of D1 real-time and non-real-time data is shown in Figure 6:

When the pulse interference period aligns with the message frame period, the receiver may fail to acquire complete and correct real-time data over multiple consecutive frames. This results in a significantly longer TTFF for receivers operating in cold-start modes, compared to the TTFF under interference-free conditions, as illustrated in Figure 7:

At this time, the interference period T_P satisfies the following:

$$T_P={\displaystyle \frac{T_{Frame}}{N}},T_P\le t_{realdata}$$

(6)

where T_Frame denotes the duration of one message frame, and t_realdata represents the duration of the real-time data.

Let t denote TTFF for a receiver in cold-start or warm-start mode under interference-free conditions, and t′ denote the TTFF under interference conditions. It can be considered that

$$P_{t\prime >mt}=P_m$$

(7)

where P_t′>mt is the probability that t′ > mt, P_m is the probability that the receiver fails to successfully decode the real-time data in the navigation message over m consecutive frames. Depending on the receiver’s strategy for handling erroneous message frames, when errors occur in the real-time data within a message frame, the receiver discards the entire frame. Accordingly, the following expression can be derived:

$$P_m=FE{R}^m$$

(8)

where FER denotes the Frame Error Rate. When errors occur in the real-time data of a message frame and the receiver discards only the erroneous words rather than the entire frame, the following can be derived:

$$P_m=1-{(1-WE{R}^m)}^i$$

(9)

where i denotes the number of interference pulses within the real-time data portion of a single navigation message frame.

Parameters related to pulse interference include interference power, duty cycle, and period. Among them, interference power (JSR) affects the effectiveness of pulse interference by influencing the BER of the message under interference. When the interference power causes the BER to reach its maximum value of 0.5, further increases in power no longer lead to higher BER. Therefore, the analysis focuses on the impact of pulse interference period and duty cycle on interference effectiveness.

Taking the B1I signal with D1 navigation message as an example, m frames of navigation data are generated in each simulation run. The initial C/N₀ is set to 45 dB, and pulse interference signals with varying periods and duty cycles are introduced, with the JSR set to 50 dB. A total of 10,000 simulations are performed. Under different interference period and duty cycle conditions, the probability P_m that the receiver fails to collect complete real-time data over m consecutive frames is recorded. Based on this, the relationship between P_t’>mt and the period and duty cycle are illustrated in Figure 8:

In the left figure, with the interference period held constant, a larger duty cycle results in more bits being affected by interference, leading to greater disruption of the navigation message and a higher value of P_t′>mt. In the right figure, with the duty cycle kept constant, a longer interference period results in a stronger interference effect. The navigation message employs a block coding scheme, in which the message bits are divided into fixed-length groups, and each group is encoded independently to produce codewords containing redundant check bits for error detection or correction in the channel. When the duty cycle remains constant, a longer interference period causes the pulses to be more concentrated in specific regions within the same time span, thereby exploiting the error-correction capability of only a few groups and enhancing the interference effect. Conversely, when the interference period is shorter, the pulse occurrences are more dispersed, affecting more groups, and allowing a greater number of groups to fully utilize their error-correction capability, thus weakening the interference effect. Meanwhile, as the number of message frames m increases, the probability that the receiver successfully obtains the complete message information rises, further reducing the interference effect.

2.3. Analysis of Pulse-Interference Mitigation Methods Based on Message Scrambling

Pulsed interference can be regarded as a form of periodic burst jamming. As demonstrated in the previous chapter, such interference repeatedly corrupts the real-time data within a navigation-message frame, preventing the receiver from acquiring complete navigation messages for prolonged intervals. To disrupt the periodicity and burst characteristics of this interference—and thereby suppressing its effects—we propose a message-scrambling approach. Specifically, for the frame–subframe structure of the navigation message, two techniques are employed: randomization of subframe order to break the alignment between the interference period and the message period, and interleaving of message bits [30] to fully leverage the error-correction capability of the navigation message and mitigate the impact of pulsed interference.

BeiDou navigation messages can be classified into two categories based on their framing structure. The first category employs a frame–subframe–word–bit hierarchy, i.e., N_subframe > 1. For example, the D1 navigation message on the B1I signal is organized in this way. Although the receiver must capture and track at least one full frame of message data to derive its position, it actually decodes the positioning parameters on a subframe basis. If one frame does not yield a complete set of parameters, the receiver can simply search for the same subframe in the next frame and continue decoding—there is no need to reprocess the entire frame. Consequently, scrambling the order of subframes at the transmitter does not disrupt receiver synchronization or decoding. The second category consists of frame–bit-structured messages, for which N_subframe = 1. In this case, the receiver interprets the entire frame as a single decoding unit; if the current frame contains errors, all data in that frame are discarded. It should be noted that, although the B-CNAV1 message format used on the B1C signal also superficially employs a frame–subframe structure, its three subframes are of unequal length: subframe 1 is dedicated to achieving frame synchronization, while subframes 2 and 3 employ data interleaving (as illustrated in the figure below). Therefore, the relative ordering of these three subframes cannot be arbitrarily permuted, and the B-CNAV1 message [31] must effectively be treated as having N_subframe = 1. Our message-scrambling-based pulse-interference mitigation methods are analyzed separately for these two cases (N_subframe > 1 and N_subframe = 1).

• The number of subframes is greater than 1 (N_subframe > 1):

Because each subframe carries its own sequence-number identifier, the receiver can determine which message data are contained in a given subframe by reading that identifier. Therefore, scrambling the order of subframes at the transmitter does not disrupt receiver synchronization or decoding. Consequently, the transmitter may randomly permute the subframes within a frame without impairing the receiver’s ability to correctly decode the navigation message, as illustrated in Figure 9.

Let a message frame consist of N_subframe subframes, indexed 1, 2, …, N_subframe. Due to pulsed interference, exactly N_jamming (N_jamming ≤ N_subframe) of these subframes are affected in each frame, and their positions within the frame remain fixed. The error probability of an interfered subframe is denoted by WER. To obtain a valid position solution, the receiver must successfully collect the complete set of subframes numbered 1, 2, …, L. Denote this collection of L subframes by the set H:

$$H=\left\{1,2,3\dots \dots L\right\},L\le N_{subframe}$$

(10)

When the transmitter randomizes the subframe transmission order, the probability that—after m consecutive message frames—the receiver still fails to collect all subframes in H is equivalent to the probability that, after m frames, there exists at least one l ∈ H that has not been successfully received. For each required subframe l ∈ H, we therefore define the event E_l as follows:

$$E_l=\left\{subframe l is not correctly received in any of the m consecutive message frames\right\}$$

(11)

Hence, within a single navigation-message frame, the probability that subframe l is not correctly received is

$${\mathrm{p}}_l={\displaystyle \frac{N_{\mathit{jamming}}}{N_{subframe}}}WER$$

(12)

Since the frames are independent, the probability that the subframe is not received over m frames is

$$P\left(E_{\mathrm{j}}\right)={\mathrm{p}}_{l^m}={({\displaystyle \frac{N_{jamming}}{N_{subframe}}}WER)}^m$$

(13)

For any subset S ⊂ H containing z elements, define the event E_S as

$$E_S=\underset{x\in s}{\cap }{E}_x$$

(14)

That is, all x ∈ S are not correctly received over m frames. The probability of simultaneously losing these z subframes in one frame is

$$p_z={\displaystyle \frac{{\left(N_{jamming}\right)}_z}{{\left(N_{subframe}\right)}_z}}WE{R}^z$$

(15)

in the above formula, where

$${\displaystyle \frac{{\left(N_{jamming}\right)}_z}{{\left(N_{subframe}\right)}_z}}={\displaystyle \frac{N_{jamming}\left(N_{jamming}-1\right)\cdot \cdot \cdot \left(N_{jamming}-z+1\right)}{N_{subframe}\left(N_{subframe}-1\right)\cdot \cdot \cdot \left(N_{subframe}-z+1\right)}}$$

(16)

The probability that, after m consecutive frames, all x ∈ S are not correctly received in any of the frames is

$$P\left(E_S\right)=p_z^m={\left({\displaystyle \frac{{\left(N_{jamming}\right)}_z}{{\left(N_{subframe}\right)}_z}}WE{R}^z\right)}^m$$

(17)

The probability that the receiver still fails to fully collect all subframes in set H after m consecutive frames is equivalent to the probability of the event ‘at least one subframe is missing’. The event ‘at least one subframe is missing’ can be defined as

$$\underset{l=1}{\overset{L}{\cup }}{E}_l$$

(18)

According to the inclusion–exclusion principle in probability, we obtain the following:

$$P_m=P\left(\underset{l=1}{\overset{L}{\cup }}{E}_l\right)={\displaystyle \sum _{z=1}^L{(-1)}^{z-1}}{\displaystyle \sum _{\begin{array}{l}S\subseteq H\\ \left|S\right|=z\end{array}}P\left(E_S\right)}$$

(19)

Since, for any z, P(E_S) depends only on the cardinality of S, and the number of subsets of H is given by the binomial coefficient, with P(E_S) = 0 for z > N_jamming, the expression can be rewritten as

$$P\left(\underset{l=1}{\overset{L}{\cup }}{E}_l\right)=={\displaystyle \sum _{z=1}^{\min (L,N_{jamming})}{(-1)}^{z-1}}\left(\begin{array}{c}L\\ z\end{array}\right){\left({\displaystyle \frac{{\left(N_{jamming}\right)}_z}{{\left(N_{subframe}\right)}_z}}WE{R}^z\right)}^m$$

(20)

The z-th term in the expression can be written as

$${(-1)}^{z-1}\left(\begin{array}{c}L\\ z\end{array}\right){\left({\displaystyle \frac{{\left(N_{\mathrm{j}amming}\right)}_z}{{\left(N_{subframe}\right)}_z}}WE{R}^z\right)}^m$$

(21)

with the summation upper bound equal to min (L, N_jamming).

Taking the D1 navigation message used on the BeiDou B1I signal as an example—where N_subframe = 5 and L = 3—and assuming that, under pulsed-interference conditions, exactly two subframes (N_jamming = 2) in each message frame are always corrupted by interference (corresponding to an interference period T_P = 15 s), the probability that the receiver fails to fully receive the first L subframes over m consecutive frames is

$$\begin{array}{l}{P}_m=\mathrm{Pr}\left[At least one of the subframes 1, 2, and 3 has never been received correctly\right]\\ =\mathrm{Pr}\left[1 never received\cup 2 never received\cup 3 never received\right]\\ ={\displaystyle \sum _{x=1}^3\mathrm{Pr}\left[x never received in m frames\right]}-{\displaystyle \sum _{x<y}\mathrm{Pr}\left[x y never received\right]}\\ +\mathrm{Pr}\left[ all never received in m frames\right]\end{array}$$

(22)

Let m = 4 and D = 0.02, Under randomization of subframe order, the probability P_m that the receiver still fails to fully collect the navigation-message data over m consecutive frames, as a function of the pulse-interference period, is shown in Figure 10:

As shown in the figure above, with T_P = T_subframe (T_subframe denotes the subframe period) as the threshold, when T_P > T_subframe, subframe-order randomization can effectively reduce the impact of pulse interference on the receiver’s ability to correctly and completely receive the navigation message. However, when T_P ≤ T_subframe, this method fails to mitigate the interference effect. The underlying reason can be explained using a combination of numerical and graphical analysis. When T_P > T_subframe, the relative positions of the message subframes and interference pulses in the time domain—before and after interference mitigation—are illustrated in Figure 11:

Under the effect of interference pulses, the probability that a subframe is corrupted—denoted as WER—approaches 1. Due to the periodic nature of the interference, the same subframes are consistently targeted. Without any interference mitigation, P_m approaches 1. However, after applying subframe-order randomization, the interference pulses can no longer consistently attack the same subframes, causing P_m to decrease rapidly with the number of message frames. When T_P ≤ T_subframe, the relative positions of the message subframes and interference pulses in the time domain—before and after interference mitigation—are illustrated in Figure 12:

When T_P ≤ T_subframe, all five subframes are affected by interference pulses both before and after applying the mitigation technique, rendering the subframe-order randomization method ineffective. Moreover, the longer the pulse interference period, the more significant the mitigation effect. For a fixed duration, a longer interference period results in fewer interference pulses, thereby increasing the probability of successfully acquiring complete subframe information through random subframe reordering.

When the impulse interference period is equal to the subframe period, all subframes will be affected by interference, and random rearrangement of subframes will not effectively mitigate the impulse interference. Considering that each subframe consists of several words, and each word uses BCH coding capable of correcting one erroneous data bit, the subframe structure is shown in Figure 13:

Therefore, by performing data interleaving on the message within subframes at the transmitter and deinterleaving at the receiver, the error-correcting capability of the channel coding is maximized. Since the first word of each subframe is the Telemetry Word (TLW), which contains an 8-bit synchronization code used for subframe synchronization [32], the telemetry word is excluded from interleaving. The remaining nine words participate in the interleaving process. The specific message interleaving method is illustrated in Figure 14:

The interleaving method is implemented using a two-dimensional array. During the write-in process, each row corresponds to one word from a subframe, so M = 9 and N = 30. After all the data is written into the array, it is then read out column by column. Starting from the first column, the data is read from top to bottom, then the next column is read, and so on until the last column is complete. At this point, the interleaving process is finished. Through the above interleaving process, since each word has the ability to correct one erroneous bit, if the number of consecutive erroneous bits is fewer than 9, the deinterleaved errors will be dispersed across different words. Channel decoding can correct all erroneous bits. Compared to non-interleaved navigation messages, the frame error rate can be significantly reduced. Due to the random occurrence of interference pulses within a navigation message frame, these pulses may appear in either the interleaved or non-interleaved regions of the message. Let p₁ be the probability that an interference pulse falls within the interleaved region, and p₂ the probability that it falls within the non-interleaved region (for the interleaving scheme described above, p₁ = 0.9, p₂ = 0.1). Then, after interleaving, the probability that the receiver fails to fully decode the real-time data in a given satellite’s navigation message over m consecutive frames under pulse interference can be expressed as

$$P_m=p_1{P}_{\mathit{interleaved}}+p_2{P}_{\mathit{non}\text{-}\mathit{interleaved}}$$

(23)

where P_interleaved represents the probability that the receiver fails to fully decode the message information over m consecutive frames when the interference pulse falls within the interleaved region, while P_non-_interleaved represents the corresponding probability when the interference pulse falls within the non-interleaved region. The calculation of P_non-_interleaved follows the same method as that used to compute P_m for non-interleaved navigation messages. For P_interleaved, which represents the probability that the receiver fails to fully decode the real-time data of a given satellite’s navigation message over m consecutive frames when the interference pulse falls within the interleaved region, it is considered that, after interleaving, the message can correct up to k′ erroneous bits. When the number of bits affected by the interference pulse is less than the message’s error-correction capability k′,

$$T_P\cdot D/T_{bit}\le k^{\prime }$$

(24)

In this case, P_interleaved approaches zero. However, when the number of bits affected by the interference pulse exceeds the message’s error-correction capability k′,

$$T_P\cdot D/T_{bit}>k^{\prime }$$

(25)

The probability of an error occurring within a given interleaving block is

$$P_{error}=1-{(1-BER)}^{T_P\cdot D-k}$$

(26)

Thus, P_interleaved can be calculated as

$$P_{\mathit{interleaved}}=1-{(1-{P_{error}}^m)}^i$$

(27)

The message interleaving method leverages the full error-correction potential of the navigation message. This approach offers greater generality and is capable of resisting pulse interference patterns with varying periods and duty cycles.

The interleaving process takes place at the satellite transmission end and does not cause any delay in the receiver’s real-time positioning. The deinterleaving process, however, does introduce delays in real-time positioning at the receiver, primarily including buffering latency and deinterleaving processing latency. Before deinterleaving, a complete interleaving block or several frames of data must be received. For block interleaving, the delay is approximately equal to the time required to receive one complete interleaving block:

$$T_{buffer}\approx N_{block}\cdot T_{bit}$$

(28)

where T_buffer represents the buffering latency, N_block is the number of bits in the interleaving block. Deinterleaving requires rearranging the data according to the interleaving order, typically through matrix operations. For modern processors, when dealing with blocks ranging from several hundred to several thousand bits, the processing delay is usually in the order of tens of microseconds to a few milliseconds.

• The number of subframes is equal to 1 (N_subframe = 1):

Navigation messages of the N_subframe = 1 type primarily include the B-CNAV1, B-CNAV2, and B-CNAV3 messages of the BeiDou-3 system. These messages are received on a per-frame basis, making it infeasible to apply subframe-order randomization for pulse-interference mitigation. Instead, intra-frame message interleaving can be employed to enhance resistance to pulsed interference. In fact, for the B1C signal of BeiDou-3, the B-CNAV1 message adopts a 36 × 48 block interleaving scheme for subframes 2 and 3 after LDPC encoding, as illustrated in Figure 15:

Code interleaving artificially transforms a burst-error channel into a statistically independent error channel, which can significantly reduce the bit error rate (BER) and frame error rate (FER) under bursty or continuous error conditions. As a result, it effectively mitigates the impact of pulse interference on the receiver’s ability to correctly decode the navigation message. It should be noted, however, that block interleaving and de-interleaving introduce transmission delay, which can affect the real-time performance of positioning. Therefore, the B-CNAV2 navigation message used on the B2a signal and the B-CNAV3 message on the B2b signal can also employ similar block interleaving schemes after LDPC encoding to reduce the frame error rate caused by burst interference.

3. Results

In this section, the effectiveness of the proposed method and the corresponding conclusions are first validated through simulations, followed by the development of an anti-jamming experimental platform to carry out relevant real-world tests.

3.1. Simulation

First, the impact of message interleaving on the anti-pulse-interference performance of navigation messages with an N_subframe = 1 structure is evaluated through simulation. Taking the B-CNAV2 navigation message used on the B2a signal as an example, JSR is set to 50 dB, and the pulse interference period is equal to the frame duration of the B-CNAV2 message. By varying the duty cycle of the pulse interference, the frame error rates (FER) after decoding are measured both before and after block interleaving. The simulation results are shown in Figure 16:

As shown in Figure 16, block interleaving significantly reduces the frame error rate of navigation message bit demodulation at the receiver when subjected to burst interference.

For navigation messages with an N_subframe > 1 structure, taking the D1 message on the B1I signal as an example, a simulation is conducted over 4 message frames (i.e., m = 4). The initial carrier-to-noise ratio is set to 45 dB, the JSR is 50 dB, and the duty cycle D = 0.02. The pulse interference period is varied, and for each period, 10,000 simulation trials are performed. Under each condition, the probability P_m that the receiver fails to collect the complete real-time data over m consecutive frames is recorded for three scenarios: without interference mitigation, with subframe-order randomization, and with message interleaving. The simulation results are shown in Figure 17:

As shown in Figure 17, it can be observed that the subframe-order randomization method is effective only when T_P > T_subframe, and its anti-interference performance improves as the interference period increases. This is because a longer interference period results in fewer interference pulses within the same duration, thereby increasing the likelihood that the receiver can successfully acquire the first three subframes after randomization. In contrast, the message interleaving method is more universal in nature and significantly reduces the value of P_m after interference mitigation. Combining both methods yields better anti-interference performance than using either method alone.

Since the subframe scrambling method is only effective when the pulse period T_P is greater than the subframe duration T_subframe, pulsed interference patterns with T_P > T_subframe are selected to analyze the impact of duty cycle and the number of navigation frames on anti-interference performance. According to Equation (6), let N = 2 and N = 3, corresponding to pulsed interference patterns with T_P = 15 s and T_P = 10 s, respectively. Taking the D1 message on the B1I signal as an example, a simulation is conducted over 6 message frames (m = 6), with pulse interference of varying duty cycles applied under otherwise identical conditions. The relationship between P_m and the duty cycle under the two interference period conditions is shown in Figure 18:

Keeping the duty cycle constant and varying the number of navigation message frames m, the relationship between P_m and the number of frames m under the two interference period conditions is shown in Figure 19:

As shown in Figure 18, the anti-interference performance of the subframe-order randomization method is relatively insensitive to variations in the duty cycle of the pulse interference. When the interference period and the number of message frames m remain constant, the duty cycle affects the number of navigation message bits covered by interference pulses and the WER. When the duty cycle becomes large enough, WER approaches 1 and no longer increases, causing P_m to stabilize.

As shown in Figure 18 and Figure 19, when the duty cycle is low and the number of message frames is small, the interleaving method outperforms subframe-order randomization. However, when the duty cycle is high and the number of message frames increases, subframe-order randomization exhibits better anti-interference performance than interleaving. Since the two methods operate based on different principles, combining them results in enhanced overall interference resistance. The interleaving-based method is primarily designed to fully exploit the error-correction capability of the navigation message, thereby reducing the bit error rate of message bits affected by interference pulses and mitigating the impact of pulse interference. Under the same period condition, when the duty cycle is small, the number of message bits covered by interference pulses is limited, and all erroneous bits can be corrected; when the duty cycle is large, the number of message bits covered increases, making it impossible to correct all erroneous bits. The subframe-order randomization method aims to disrupt the alignment between the pulse interference and the real-time data of the navigation message. Once the subframe transmission order is randomized, the probability that interference pulses can continuously disrupt the same subframe decreases significantly as the number of message frames increases, while the probability that the receiver can obtain interference-free, real-time data from multiple frames increases notably. Furthermore, due to the different principles of the two anti-interference methods described above, their combined use yields an even better interference mitigation effect.

3.2. Experimental

3.2.1. Experimental Measurement of the Impact of Impulse Interference on Receiver TTFF

An interference test platform was built using a navigation signal generator and a Universal Software Radio Peripheral (USRP) device, The equipment used in this study is USRP B210. The manufacturer is Ettus Research, located in Sunnyvale, CA, USA [34]. The receiver is the UM980 positioning module developed in collaboration with Unicore Communications Inc., Beijing. The module supports simultaneous reception of multiple satellite navigation signals, including GPS and BDS signals [35]. A computer is used to monitor the receiver’s experimental results and control the generation and transmission of interference signals. The schematic diagram of the experimental platform is shown in Figure 20 and Figure 21:

Two sets of real-world experiments were conducted using the above interference test platform. In the first experiment, the navigation signal generator produced a B1I signal, and a pulse interference signal targeting the BeiDou B1I signal was injected. The signal was modulated to the navigation signal frequency band using a USRP, C/N₀ = 45 dB·Hz, T_P = 15 s, D = 0.04, JSR = 50 dB, and the u-center (version 21.06, u-blox, Thalwil, Switzerland) software was used to observe the receiver’s performance under both pulse interference and non-interference conditions, as shown in Figure 22:

Experimental findings: Under non-interference conditions, the receiver was able to use satellite signals for positioning 28 s after a cold start. However, under interference conditions, despite all BeiDou satellite signals having a C/N₀ above 35 dB, the receiver still failed to achieve effective positioning even after approximately 128 s (about 6 times longer). In the diagram, the Chinese flag indicates the signal type, and the color represents whether the signal can be used for positioning—green indicates normal reception and usable for positioning, while blue indicates normal reception but not usable for positioning. This experiment demonstrates that the applied pulse interference significantly affects the receiver’s ability to perform accurate positioning.

In the second experiment, the navigation signal generator produced a B1I signal, and pulse interference with varying periods and duty cycles was injected. The time t′, defined as the moment when a satellite signal first becomes usable for positioning (i.e., the signal turns green), was measured. Multiple measurements were taken to calculate the probability P_t′>6t that the time t′ for a single satellite signal to become usable exceeds 6t. The relationship between P_t′>6t and the pulse period and duty cycle is shown in Figure 23:

The above experiment confirms that several types of pulse interference patterns with different periods and duty cycles can significantly affect the receiver’s TTFF. Moreover, the longer the pulse interference period, the greater the impact on the receiver’s TTFF.

3.2.2. Verification of the Effectiveness of the Navigation Message Scrambling Method Against Pulse Interference

The pulse interference test platform designed in Section 3.2 was further used to conduct anti-interference experiments. In real-world testing, it is not feasible to actually change the transmission order of subframes in the navigation message. In practice, randomizing the subframe order within the navigation message can be equivalently modeled as randomizing the position of interference pulses within one cycle. Therefore, the equivalent treatment of subframe order randomization is illustrated in Figure 24:

The pulsed waveforms before and after equivalence, as observed on the oscilloscope, are shown in Figure 25:

In the figure above, the period and duty cycle of the pulse interference remain unchanged, but the position of the pulse within each cycle is randomized. Similarly, under pulse interference, interleaved encoding of the navigation message can be regarded as equivalent to a discrete distribution of interference pulses within one period, with the total duty cycle remaining unchanged, as illustrated in Figure 26:

The pulsed waveforms before and after equivalence, as observed on the oscilloscope, are shown in Figure 27:

As shown in the figure above, when the pulse interference period and total duty cycle remain unchanged, the equivalent processing of message interleaving results in multiple discretely and uniformly distributed interference pulses within one period. In practical measurements, interference pulses similar to those illustrated above are transmitted to simulate the effects of subframe-order randomization and message interleaving.

Taking the D1 message on the B1I signal as an example, real-world measurements are conducted to validate the effectiveness of the proposed methods. First, scenarios where T_P ≤ T_subframe are tested, Based on Equation (6), pulsed interference patterns with N = 5 and N = 10 are selected, which correspond to T_P = 6 s and T_P = 3 s, respectively. The duty cycle is varied, and the time t′ during which the satellite signal can be used for positioning is measured. The probability that the time required for a single satellite signal to become usable for positioning exceeds 6t, denoted as P_t′>6t, is then calculated. The relationship between P_t′>6t and the interference duty cycle under the two period conditions, before and after applying the interference mitigation strategies, is shown in Figure 28:

Keeping the duty cycle constant and varying the number of navigation message frames, the probability that the time required for a single satellite signal to become available for positioning exceeds mt, denoted as P_t′>mt, is calculated. The relationship between P_t′>mt and the number of navigation message frames under two interference period conditions, before and after applying interference mitigation strategies, is shown in Figure 29:

The experimental results shown above lead to the same conclusion: For pulse interference patterns where T_P ≤ T_subframe, the subframe-order randomization method is ineffective in mitigating the impact of pulse interference on the receiver’s time-to-first-fix. In contrast, the message interleaving method proves effective in alleviating the effects of pulse interference.

For the case where T_P > T_subframe, according to Equation (6), let N = 2, N = 3 and N = 4, corresponding to pulsed interference patterns with T_P = 15 s, T_P = 10 s and T_P = 7.5 s. The duty cycle of the interference is varied, and the time t′ during which the satellite signal becomes usable for positioning is measured. After multiple measurements, the probability that the time required for a single satellite signal to become available for positioning exceeds 6t, denoted as P_t′>6t, is calculated. The relationship between P_t′>6t and the interference duty cycle under the two period conditions, before and after applying interference mitigation strategies, is shown in Figure 30:

Keeping the duty cycle constant and varying the number of navigation message frames, the probability that the time required for a single satellite signal to become available for positioning exceeds mt, denoted as P_t′>mt, is calculated. The relationship between P_t′>mt and the number of navigation message frames under two interference period conditions, before and after applying interference mitigation strategies, is shown in Figure 31:

As shown in Figure 31a, for the pulsed interference pattern with T_P = 15 s, when both anti-interference methods are applied simultaneously, the probability that the receiver fails to achieve positioning drops below 5% after just two navigation frames. From Figure 31b, it can be seen that for the T_P = 10 s interference pattern, this probability decreases to below 10% after approximately four navigation frames. Figure 30 and Figure 31 validate the simulation results from the previous chapter: when the duty cycle is low and the number of navigation frames is small, the message interleaving method outperforms the subframe order scrambling method. Conversely, when the duty cycle is high and more navigation frames are available, the subframe order scrambling method shows better anti-interference performance than the interleaving method. When both methods are applied together, the anti-interference effectiveness improves further. For the two pulsed interference patterns with periods greater than the subframe duration mentioned above, the probability that the receiver fails to achieve positioning over multiple consecutive frames is reduced by at least 50% compared to the case without interference mitigation.

4. Discussion

In this study, the time-domain characteristics of pulse interference were first analyzed, and a corresponding mathematical model was established. Using a software-defined receiver, we simulated the impact of the proposed pulse interference on both the tracking loop PLL and the demodulation of navigation messages. The results demonstrate that the BER of navigation message demodulation is strongly dependent on the interference power. When the interference power is sufficiently high, the BER saturates at the maximum value of 0.5 and remains unchanged. Furthermore, the WER and FER are determined jointly by the channel coding/decoding capability of the navigation message and the duty cycle of the interference.

From the perspective of navigation message structure and receiver theory, we divided the navigation message into immediate data and non-immediate data. By aligning the interference period with the navigation frame period in the time domain, the immediate data can be continuously attacked, preventing the receiver from obtaining complete positioning information and thus significantly extending the TTFF. To quantitatively evaluate the interference effectiveness of pulse interference, two key metrics were employed: the probability P_m that the receiver fails to obtain complete navigation message frames over mmm consecutive frames, and the TTFF. Simulation results revealed that, with a fixed interference period, an increased duty cycle leads to a larger number of affected bits and stronger interference, reflected in larger values of P_m. Conversely, with a fixed duty cycle, longer interference periods also yield stronger interference and larger P_m, as the interference becomes more concentrated and suppresses the error correction capability of the navigation message. In contrast, shorter interference periods disperse the interference, allowing the one-bit error correction mechanism of the navigation message to take effect, thereby weakening the interference.

Based on the identified interference mechanism, an anti-pulse interference strategy was proposed, consisting of two approaches: subframe order scrambling and message interleaving. For frame–subframe structured navigation messages, although the receiver must capture and track at least one full frame for positioning, the positioning parameters are decoded at the subframe level. Thus, if complete parameters cannot be obtained from the current frame, the receiver can continue decoding the corresponding subframe in the subsequent frame without re-decoding the entire frame. Scrambling the subframe order at the transmitter does not affect synchronization or decoding, but effectively disrupts the alignment between the interference period and the navigation message period. This method is effective against pulse interference with periods longer than the subframe period, but has limited effectiveness when the interference period is equal to or shorter than the subframe period. In comparison, message interleaving fully exploits the error correction capability of the navigation message, making it a more universally applicable approach.

To validate the proposed interference mechanism and anti-interference strategies, an experimental platform was developed and multiple real-world experiments were conducted. The results show that pulse interference with different periods and duty cycles can significantly influence the TTFF of the receiver. Subframe order scrambling is effective against interference periods longer than the subframe period, but not against shorter or equal periods. When the duty cycle is low and the number of frames is small, message interleaving provides superior anti-interference performance compared with subframe order scrambling. Conversely, when the duty cycle is high and the number of frames is large, subframe order scrambling outperforms message interleaving. Importantly, when both methods are combined, the anti-interference performance is further enhanced. For interference patterns with periods longer than the subframe period, the probability that the receiver fails to position over multiple consecutive frames is reduced by at least 50% compared with the case without anti-interference measures.

5. Conclusions

This paper addresses the problem that long-period pulsed interference can significantly prolong the TTFF of satellite navigation receivers. From the perspectives of navigation message structure and receiver algorithms, the mechanism by which such interference affects TTFF is briefly analyzed. An anti-pulsed interference strategy based on message scrambling is then proposed, focusing on two scrambling techniques: random subframe order scrambling and message interleaving. Modeling and simulation analyses are conducted for both methods. Simulation and experimental results show that the subframe order scrambling method is effective in resisting pulsed interference with a period longer than the subframe duration but is limited when the interference period is equal to or shorter than the subframe duration. In contrast, the message interleaving method demonstrates greater generality and is applicable to navigation messages of different structural types. When both scrambling techniques are applied together, the anti-interference performance is further enhanced and superior to using either method alone. Specifically, for interference patterns with periods longer than the subframe duration, the probability that the receiver fails to achieve positioning over multiple consecutive frames is reduced by at least 50% compared to the unprotected case. These findings provide important insights for improving the robustness of satellite navigation receivers against long-period pulsed interference.