A ranging code based on the improved Logistic map for future GNSS signals: code design and performance evaluation
 Dun Wang^{1},
 Rui Xue^{2}Email author and
 Yanbo Sun^{2}
DOI: 10.1186/s1363801708404
© The Author(s) 2017
Received: 1 October 2016
Accepted: 13 March 2017
Published: 27 March 2017
Abstract
Ranging code is the core component of the signal transmission scheme in any global navigation satellite system (GNSS); its performance directly influences on the technical indexes of positioning accuracy, compatibility, interoperability, antiinterference, security, synchronization realization, and so on. Therefore, research on ranging codes could provide theoretical support for the improvement of the performance of ranging codes and extension of their design methods to future satellite navigation signal structures. In order to improve the balance in classical chaotic sequences, a novel ranging code is proposed in this paper and constructed by a series of the improved Logisticmap chaotic sequences with different initial values through weighted optimization, summation, and quantization. Then a comprehensive performance evaluation method based on the Welch bound including three main indexes has been introduced, namely the performance of acquisition, tracking, and robustness against interfering narrowband signals. Finally, the three indexes are combined in a cost function by weighting to evaluate the proposed code, coarse/acquisition (C/A), Gold, Weil, and Random as well as the conventional chaotic codes, and the corresponding weighted coefficients can be adjusted flexibly according to the user groups or application types. Theoretical analysis and simulation results over an additive white Gaussian noise (AWGN) channel show that the proposed ranging code cannot only demonstrate excellent performance in acquisition and antinarrowband interference while maintaining high quality in tracking performance as the C/A code but also significantly improve balance performance and strengthen reliability and security.
Keywords
Satellite navigation Ranging codes Chaotic sequences Weighed processing1 Introduction
The navigation signal is an important part of satellite navigation systems as the coordination work link to satellites, ground control centers, and users [1]. It is directly related to the basic functions of location, time service, and velocity measurement as well as to the key performance indexes of positioning accuracy, compatibility, interoperability, security, antiinterference ability, and so on. The ranging code is the core part of modern satellite navigation signal transmission schemes and has been widely used in Global Navigation Satellite Systems (GNSSs), such as the US Global Positioning System (GPS), Russian Global Navigation Satellite System (GLONASS), European Galileo, and Chinese Compass. The properties of ranging code have a great influence on the performance of acquisition, tracking, demodulation, antijamming, secrecy, and so on [2]. Therefore, study on properties of ranging codes could provide theoretical support for pursuing ranging codes with better performance and expand design methods of ranging codes in future satellite navigation signal structures.
With the development of GNSSs and regional navigation systems, such as Japanese QuasiZenith Satellite System (QZSS) and Indian Regional Navigation Satellite System (IRNSS), the number of navigation signals in space is anticipated over 400 by 2030 [3], which will further aggravate an already crowded radio spectrum in Lband (1164 ∼ 1610 MHz) and negatively impact the new signal scheme design due to excessive spectrum overlapping [4]. Meanwhile, compatibility and interoperability among different navigation systems are becoming a hot research topic around the world in recent years [5–7]. The isolation of mutual operation signals in various GNSSs completely depends on ranging code, and it plays a decisive role in multiaccess interference degree and compatibility among navigation systems [8].
At present, M code and Gold code generated by linear feedback shift register (LFSR) sequences are the most widely used ranging codes in existing satellite navigation systems, but both of codes have some common disadvantages such as limited number of available code groups and poor antidecryption ability. With the development of decryption technology to pseudonoise sequence codes, the above codes are facing the danger of being cracked [9]. In recent 10 years, chaotic sequences have made it possible for ranging as a result of the gradual maturity in theory and application of chaos, which have the following advantages: high sensitivity to initial values, large number of code groups, high linear complexity, and excellent confidentiality. The application of chaotic sequences opens up a new perspective for spread spectrum communication [10], but chaotic sequences also have their shortcomings especially for poor balance. For example, the balance coefficient is larger than 0.02 when sequence length is relatively short, which cannot satisfy the requirement of highprecision positioning systems [11, 12].
The rest of the paper is organized as follows. Section 2 mainly expounds basic principle of the proposed chaotic sequence and other classical chaotic sequences, and analyses their characteristics of balance and correlation. Section 3 provides a weighted performance evaluation method based on the Welch bound. Section 4 introduces three typical interference. Simulation results are discussed in Section 5. Finally, we conclude the paper in Section 6.
2 Ranging codes
2.1 Classical chaotic sequences
Generally, the onedimensional discrete chaotic maps can be applied to generate pseudorandom sequences, that is to say a qualified initial value is selected firstly and then a decimal chaotic sequence of length N is formed after N iterations according to chaotic map, for instance, the Chebyshev map, Tent map, Bernoulli map, Logistic map, and improved Logistic map. The details of these maps are shown in Ref. [13–17]. At last, the binary chaotic sequence is obtained through binary quantization. The above mentioned maps are described as follows.
(1) The Chebyshev map
where initial value x _{0}∈[−1,1]. If the positive integer q is equal to any integer powers of 2, the generated sequence is in the state of chaos. In this case, the binary quantization threshold is 0.
when constant a∈(0,1) and initial value x _{0}∈[0,1], the generated sequence is in the state of chaos. In this case, the binary quantization threshold is 0.5.
when constant b∈(0,1) and initial value x _{0}∈[0,1], the generated sequence is in the state of chaos. In this case, the binary quantization threshold is 0.5.
where x _{ n }∈(0,1)(n=0,1,⋯), and 0 < c ≤ 4 is called the bifurcation parameter, for c∈(3.5699⋯,4] the sequence generated from Eq. (4) is chaotic. In this case, the binary quantization threshold is 0.5.
(5) The improved Logistic map
when z _{ n }∈(−1,1), the generated sequence is in the state of chaos, and the binary quantization threshold is 0. In addition, they derived the mathematical expression of statistical properties of chaotic sequences generated by the improved Logistic map and found that their statistical properties are identical with those of white noise. Thus, the improved Logistic map is more appropriate to be used for spread spectrum (SS) communications than the traditional Logistic map [20].
2.2 Analysis of balance
Viewed from information transmission perspective, satellite navigation is a special case of SS communications. In majority of SS communications, one of the most basic requirements for the SS sequence should have excellent balance. There is a close correlation between balance of SS sequence and carrier rejection, and an unbalanced sequence will make carrier leakage much bigger, which will easily lead to bit error or information loss [21]. Specifically, on the transmitter side, an unbalance signal will result in leaking some important information of SS sequence, which will make SS signals lost its superiority in concealment and waste emission power simultaneously. For the receiving part, the unbalanced signal as a narrowband interference into the receiver will increase the internal interference of the system and make the improper locking appear at unsubdued carriers, which influences normal work of receiver seriously. Evidently, the study on balance in SS sequences has important significance to authorized users of GNSSs.
2.3 A novel chaotic sequence based on the improved Logistic map with weighted processing
where k=1,2,⋯,N, s _{ k,L } is the k ^{th} element in the L ^{ t h } sequence, and p _{1},p _{2},⋯,p _{ L } are weighted parameters and p _{1}+p _{2}+⋯+p _{ L }=1. d) The proposed ranging code is obtained from the sequence G through binary quantization.
From construction process of the proposed code, we note that the weighted parameters are the key factor which has serious effect on the balance performance, and balance coefficient is equal to or close to zero can be regarded as criterion for weighted parameter selection. Based on this criterion, a selection method for weighted parameters is provided by this paper, then we set an example to illustrate how the method is carried out under computer simulation.
From the above Eq. (7), we can notice that the proposed chaotic sequence based on weighted processing is a special case of the improved Logistic map chaotic sequences. Meanwhile, an excellent balance performance could be gained through adjusting weighted parameters flexibly.
2.4 Analysis of correlation
It is well known that conventional chaotic sequences based on the above maps have superiority in correlation characteristics including autocorrelation (AC) and crosscorrelation (CC) because of high sensitivity to initial values. In chaotic SS communication systems, the capabilities of suppressing multipath and multiuser interference are characterized by the properties of AC and CC, which are usually evaluated using AC function (ACF), CC function (CCF), AC sidelobe, and CC rootmeansquare (RMS) value. However, it is uncertain whether the proposed sequence can provide the similar correlation performance as classical chaotic sequences or not. Next, the above four technical indexes of the proposed sequence will be tested.
Sidelobe peak value of autocorrelation and crosscorrelation for six chaotic codes (dB)
Ranging codes  MaxACFe  MaxACFo  MaxCCFe  MaxCCFo 

Chebyshev  21.4636  21.6613  21.7129  20.7860 
Tent  21.0266  20.8169  21.1114  20.8866 
Bernoulli  21.7543  21.4409  21.0964  20.7382 
Logistic  21.4893  21.5406  21.2028  20.8531 
Improved Logistic  21.5507  21.3960  21.5682  21.2776 
The proposed code  21.3509  21.2705  21.6414  21.2772 
2.5 Weil sequence
Weil sequences [22] have excellent properties of pseudorandom and autocorrelation as well as relatively high linear complexity. They are based on Legendre sequences, which in turn are constructed from quadratic residues. The quadratic residue is defined as follows: if the greatest common divisor of α and β is 1, which is denoted as (α,β)=1; if β divided by (x ^{2}−α) has a solution, namely x ^{2}≡α(mod β), and α is called the quadratic residue of module β.
where t is the shift value of the Legendre sequence.
2.6 Random sequence
3 A weighted performance evaluation method based on the Welch bound
The continuous improvements of the Compass signal structure have lead to the need of designing new spreading codes for the different open signals. In order to compare different candidates of code families, selection criteria have been set up and constantly strengthened over the last decade. These criteria model the impact of the spreading code characteristics and in particular of their auto and crosscorrelation functions onto the receiver performance. For this purpose, a comprehensive assessment method based on the Welch bound [24] including three main indexes has been proposed in [25], namely the performance in the acquisition mode, the tracking mode, and the robustness against interfering narrowband signals. Furthermore, these three indexes can be combined in a final figure or cost function in order to evaluate different candidate sets of codes. The corresponding weighted coefficients can be decided by the user groups, since the code impact varies for every type of application. The three performance indexes and cost function will be expounded in the following sections.
3.1 The acquisition performance
Any correlation value exceeding ϕ _{min} can be considered as a degradation risk for the acquisition performance. The higher the distance between these correlation values and the Welch bound is the higher the degradations have to be expected. The criterion built on the Welch bound is called Excess Welch Square Distance (EWSD). Two subcriteria based on the EWSD have been distinguished in [25]. One (EWSD^{MP}) quantifies the effects of multipath onto the acquisition performance and mainly uses the ACF. The other (EWSD^{CT}) considers the effects of the nondesired signals onto the performances.
where A C ^{ e }(l,f _{ offs }) and A C ^{ o }(l,f _{ offs }) are respectively the even and odd ACF calculated for a relative delay ltimes T _{ c } and for a Doppler frequency offset f _{ offs }, and the sample number of Doppler frequency offset is \(n_{f_{offs}}\). In other words, the MEWSD^{MP} represents the average value (over all possible delays and Doppler offsets) of the squared distances between the Welch bound and all even or odd ACF components exceeding this bound.
where \({CC}_{i,j}^{e}(l,{f_{offs}})\) and \({CC}_{i,j}^{o}(l,{f_{offs}})\) denote the even and odd CCF respectively for the sequences i and j when a relative delay is ltimes T _{ c }, and Doppler frequency offset is f _{ offs }. Note that the value for \(n_{f_{offs}}\) is identical to this used for the computation of the MESWD^{MP}.
3.2 The tracking performance
For the acquisition mode, only the highest correlation peaks had to be considered to avoid any performance degradation. Now, in the tracking mode, all correlation components have to be taken into account, since they gather in one aggregate perturbation called average interference parameter. The corresponding criterion is called Merit Factor (MF). Similarly to the MEWSD, two cases, one for the multipath and another for the crosstalk, have been distinguished.
3.3 The robustness against interfering narrowband signals
When the spreading code period becomes smaller, the power spectral density (PSD) for the corresponding navigation signals does not match the exact envelop of the pulse shape, as it would be the case for pure random codes (with infinite period), but rather shows peaks exceeding this envelop by several dBs (e.g., 8 dB for the GPS coarse/acquisition (C/A) codes). This property leads to an increase of the sensitivity against continuous wave signals or narrowband signals which are sent on these particular frequencies. Hence, a code showing good robustness against narrowband interfering signals must have as less peaks which exceed the ideal PSD for the pulse shape envelop as possible. Even for small data rates, these peaks do not represent any more points of vulnerability. Consequently, the Excess Line Weight (ELW) aims to reduce these peak exceedings.
3.4 Sequence cost function
where i is the index for different code sets, j is the index for different criteria, ρ _{ j } is the weighted coefficient of criteria j and ρ _{1}+ρ _{2}+⋯+ρ _{5}=1, c v _{ i,j } is the criterion value of criteria j and code set i, \(\overline {{cv}_{j}}\) is the mean value of criteria j over all different code sets, and R _{ i } is the result of performance in percentage or cost function. In this paper, we mainly consider five performance indexes, namely AMEWSD^{MP}, AMEWSD^{CT}, AMF^{MP}, AMF^{CT}, and AELW. The comprehensive performance including capture, tracking, and antijamming of the proposed sequences is evaluated in the following section compared to the traditional chaotic sequences, C/A, Gold, Weil, and Random sequences.
4 The traditional types of interference
There are various kinds of jamming modes. Active jamming can be divided into blanketing jamming and deceptive jamming, where the blanketing jamming can be divided into several modulation modes [27]. Three of the jamming modes, namely noise amplitude modulation (AM) jamming, noise frequency modulation (FM) jamming, and noise phase modulation (PM) jamming are discussed in this paper as follows.
4.1 Noise AM jamming
4.2 Noise FM jamming
where n _{ J2}(t) is the signal of noise FM jamming, U _{ j } is the amplitude of noise FM jamming, K _{FM} is frequency modulating factor, which indicates the frequency change caused by unit noise signal, u _{ n }(t) is the bandlimited AWGN whose mean and variance are 0 and σ ^{2} respectively, ω _{ j } is angular frequency of carrier, and initial phase ψ is uniform distribution in [0,2π] and uncorrelated with u _{ n }(t).
4.3 Noise PM jamming
where n _{ J3}(t) is the signal of noise PM jamming, U _{ j } is the amplitude of noise PM jamming, K _{ PM } is phase modulating factor which indicates the phase change caused by unit noise signal, u _{ n }(t) is the bandlimited AWGN whose mean and variance are 0 and σ ^{2} respectively, ω _{ j } is angular frequency of carrier, and initial phase ψ is uniform distribution in [0,2π] and uncorrelated with u _{ n }(t).
5 Simulation results and analysis
The comparison of performance indexes for various ranging codes
Ranging codes  AMEWSD^{MP}  AMEWSD^{CT}  AMF^{MP}  AMF^{CT}  AELW 

C/A  0.0757  0.0746  6.3700×10^{−5}  1.0017  18.0553 
Gold  0.0715  0.0706  1.7903×10^{−4}  1.0001  17.5296 
Logistic  0.0640  0.0649  1.7443×10^{−4}  1.0014  17.2405 
Improved Logistic  0.0712  0.0718  2.2310×10^{−4}  1.0028  17.2680 
Chebyshev  0.0701  0.0719  1.1625×10^{−4}  1.0026  17.2296 
Bernoulli  0.0729  0.0728  8.4607×10^{−5}  1.0044  17.2903 
Tent  0.0758  0.0740  1.7949×10^{−4}  1.0185  17.4540 
Weil  0.0703  0.0710  1.1586×10^{−4}  1.0053  17.4074 
Random  0.0937  0.0927  1.9021×10^{−4}  1.0047  18.3198 
The proposed code  0.0640  0.0641  7.5566×10^{−5}  0.9992  17.2249 
The comparison of performance indexes for various ranging codes through weighting (%)
Ranging codes  AMEWSD^{MP}  AMEWSD^{CT}  AMF^{MP}  AMF^{CT}  AELW 

C/A  0.7625  0.4833  10.9145  0.0472  0.6323 
Gold  0.3895  0.6150  5.5349  0.0791  0.0316 
Logistic  2.4465  2.1801  4.8788  0.0532  0.2988 
Improved Logistic  0.4717  0.2856  11.8205  0.0253  0.2673 
Chebyshev  0.7735  0.2581  3.4194  0.0293  0.3112 
Bernoulli  0.0055  0.0110  7.9326  0.0066  0.2418 
Tent  0.7899  0.3185  5.6005  0.2874  0.0548 
Weil  0.7186  0.5052  3.4750  0.0245  0.1080 
Random  5.6994  5.4530  6.1295  0.0125  0.9346 
The proposed code  2.4465  2.3998  9.2221  0.0970  0.3166 
From both of Tables 2 and 3, we can note that the proposed sequence has the minimum value the same as the Logistic sequence in AMEWSD^{MP} and has the minimum value in AMEWSD^{CT}. Comprehensively, the proposed sequence has better acquisition performance than other sequences. In terms of AMF^{MP}, the proposed sequence is smaller than other sequences except C/A sequence, which indicates the proposed sequence has better tracking performance against multipath effect than other sequences, inferior to C/A. In addition, the proposed sequence has the minimum value in AMF^{CT} and AELW, which means the proposed sequence has the best tracking performance against other satellites’ signal jamming and optimal performance of antinarrowband interference.
6 Conclusions
In view of the disadvantage of conventional chaotic sequences in terms of balance, a novel ranging code is presented, which is constructed by some improved Logistic map chaotic sequences using different initial values through weighted optimization, summation, and quantization. From construction process, we can see that the proposed codes are not only related to initial values but also related to weighted parameters, and it means that the ability of antidecipher is enhanced. A weighted performance assessment method based on the Welch bound is introduced to evaluate the proposed ranging codes, C/A, Gold, Weil, Random, and the classical chaotic codes. A lot of simulation results in AWGN channel show that the proposed ranging code cannot only have the best performance in acquisition and antinarrowband interference among the above ranging codes and keep excellent tracking performance as C/A code but also greatly improve the balance performance and strengthen reliability and security.
Abbreviations
 AC:

Autocorrelation
 ACF:

Autocorrelation function
 AELW:

Average excess line weight
 AM:

Amplitude modulation
 AMEWSD:

Average mean excess Welch square distance
 AMF:

Average merit factor
 AWGN:

Additive white gaussian noise
 BER:

Bit error rate
 BPSK:

Binary phase shift keying
 C/A:

Coarse/Acquisition
 CC:

Crosscorrelation
 CCF:

Crosscorrelation function
 ELW:

Excess line weight
 EWSD:

Excess Welch square distance
 FM:

Frequency modulation
 GLONASS:

Russian global navigation satellite system
 GNSS:

Global navigation satellite system
 GPS:

Global positioning system
 IRNSS:

Indian regional navigation satellite system
 JSR:

Jammingtosignal ratio
 LFSR:

Linear feedback shift register
 MEWSD:

Mean excess Welch squared distance
 MF:

Merit factor
 PM:

Phase modulation
 PSD:

Power spectral density
 QZSS:

QuasiZenith satellite system
 RMS:

Rootmeansquare
 SNR:

Signaltonoise ratio
 SS:

Spread spectrum
Declarations
Acknowledgements
This paper is supported by the Open Research Fund of State Key Laboratory of SpaceGround Integrated Information Technology (Grant No. 2015_SGIIT_KFJJ_DH_03), the Open Research Fund of State Key Laboratory of Tianjin Key Laboratory of Intelligent Information Processing in Remote Sensing (Grant No. 2016ZWKFJJ01),the National Natural Science Foundation of China (Grant No. 61403093), the Science Foundation of Heilongjiang Province of China for Returned Scholars (Grant No. LC2013C22), the Assisted Project by Heilongjiang Province of China Postdoctoral Funds for Scientific Research Initiation (Grant No. LBHQ14048), and the Fundamental Research Funds for the Central universities (Grant No. HEUCF0817).
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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