Time domain raised cosine-binary coded symbol modulation for satellite navigation

Aerospace Science and Technology 66 (2017) 59–69

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Time domain raised cosine-binary coded symbol modulation for satellite navigation Yanbo Sun, Rui Xue ∗ , Danfeng Zhao College of Information & Communication Engineering, Harbin Engineering University, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 24 May 2016 Received in revised form 13 September 2016 Accepted 1 March 2017 Available online 7 March 2017 Keywords: GNSS Modulation signal TDRC BCS Performance evaluation

a b s t r a c t The number of satellite navigation signals in space grows dramatically as the number of global and regional navigation satellites constant increases. This phenomenon further aggravates an already crowded radio spectrum in-band and increases out-of-band (OOB) emissions. One feasible solution to the issue of signal compatibility is to design a spectrum-efficient modulation signal that has better navigation performance and backward compatibility with adjacent signals and services in operation. In this study, a time domain raised cosine (TDRC) pulse is introduced as an alternative waveform. A binary coded symbol (BCS) modulation family based on TDRC pulse called TDRC-BCS is also proposed as a candidate for future global navigation satellite system (GNSS). An extensive study on the multipath model analysis approach is also provided. The proposed modulation and existing modulations are then comprehensively evaluated. Theoretical analysis and simulation results show that TDRC-BCS signals offer superior navigation performance compared with existing modulations in terms of code tracking accuracy, multipath mitigation capacity, while maintaining comparable or better anti-jamming performance. These signals can also attain higher spectral efficiency and better backward compatibility with the existing GNSS signals. The proposed modulation scheme provides additional degrees of freedom for GNSS signal design. © 2017 Elsevier Masson SAS. All rights reserved.

1. Introduction The modulation scheme, completely determining the upper bound of the performance of the global navigation satellite system (GNSS), is the core part of the signal structure, and the power spectrum envelopes determined by the modulation waveforms have a very significant effect on the performance of a navigation system, such as code tracking accuracy, multipath mitigation, anti-jamming, and compatibility [1,2]. From the perspective of the modulation development process, binary phase shift keying (BPSK) with rectangular pulse shape has been successfully applied to the first-generation Global Positioning System (GPS). However, BPSK modulation has a limited ranging capability when the available bandwidth of the receiver is not sufficiently large [3]. Also, BPSK is very vulnerable to the noise and multipath propagations. The transmitted power of BPSK signals on satellites must be augmented, and a wider bandwidth receiver is also required to improve the tracking accuracy. These requirements undoubtedly increase the cost and implementation complexity at both the transmitter and receiver sides.

*

Corresponding author. E-mail address: [email protected] (R. Xue).

http://dx.doi.org/10.1016/j.ast.2017.03.002 1270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

Since its introduction, the binary offset carrier (BOC) modulation with a square wave subcarrier [4] has received particular attentions. This scheme has been successfully applied to global and regional navigation satellite systems because of its more desired compatibility with legacy navigation signals, potentially better time-resolution capabilities, and higher robustness against the multipath effect and noise compared to BPSK [3,4]. Subsequent modulations developed from the BOC concept are also currently utilized to modernize GPS, Galileo, and BeiDou system (BDS). For examples, multiplexed BOC (MBOC) modulations using composite BOC and time-multiplexed BOC are recommended for Galileo E1 OS signals and GPS L1C signals [5–8] respectively; Galileo E5 signals and BDS B2 signals have adopted alternative BOC modulation [8]. Moreover, several innovative BOC modulation schemes are emerging, such as BOC with adjustable width modulation [9], double-BOC modulation [10], generalized BOC modulation [11], etc. These modulations are specific subclasses of binary coded symbol (BCS) modulation [12] and are generated by a judicious configuration of BCS sequences. Unfortunately, the above-mentioned modulations, including BOC, have relatively high spectral side lobes and low spectral efficiency. These disadvantages consume large amounts of spectrum resources and are prone to introduce larger

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mutual interference with the existing signals and services of other coexisting navigation systems. At present, GNSS is growing from previous US GPS and Russian GLONASS to additional European Galileo and Chinese BDS, as well as space-based augmentation systems and various regional navigation satellite systems, such as the Japanese QZSS and Indian IRNSS. Moreover, the new satellites are capable of transmitting multiple signals in multiple frequency bands [13–17]. A total of 150 satellites and over 400 GNSS signals are predicted to be present in space by 2030 [18,19]. Such large amounts of signals will further aggravate the situation of signal congestion in space and indirectly degrade the performance of all satellite navigation systems sharing same radio frequency resources. To allow the peaceful coexistence with minimal mutual interference among spaced signals and services in operation, a modulation with high spectral efficiency and improved navigation performance must be designed for future GNSS. In views of this, the spectrum profiles of future desired GNSS signals should have the following properties: (1) The signal power near the region of the carrier frequency must be weakened for better compatibility with narrow band signals. (2) The amplitudes of the side lobes closer to the main lobe must be increased to improve the tracking accuracy and robustness against multipath. (3) The larger power fluctuation of high frequency components far from the main lobes must be cut down to minimize mutual interference with wide band signals and ensure lower out-ofband (OOB) emissions within adjacent services. A radical method achieving spectrally efficient signals is to substitute the chip shape with some continuous and smooth waveforms, given that smooth changes in the complex amplitude of signals tends to result in a quite compact spectrum. In this paper, a novel pulse waveform called time domain raised cosine (TDRC) is designed, which satisfies the above-mentioned requirements well. Based on the BCS technique, a particularly promising modulation family called TDRCBCS is proposed. A comprehensive evaluation for GNSS signals is employed to assess the proposed modulation in terms of tracking accuracy, multipath mitigation, anti-jamming, and compatibility, compared with legacy modulations such as BPSK and BOC. The rest of this paper is organized as follows: Section 2 establishes the mathematical model of TDRC-BCS modulation and provides a general derivation of the theoretical power spectrum density (PSD) for TDRC-BCS modulation, along with its particular cases of TDRC-BOC modulation. Section 3 presents a comprehensive evaluation criterion for GNSS signal design and introduces an extensive study on the multipath model analysis approach. A comparison of the performance of the proposed TDRC-BCS modulation with that of current GNSS modulations are also conducted comprehensively in this section. Finally, Section 4 draws the conclusions. 2. TDRC-BCS modulation

where sl is the lth coded symbol of the BCS sequence and takes value ±1, T c is the spreading code period and recτ (t ) denotes the rectangular pulses of duration τ , given by



recτ (t ) =

1, 0,



p τ (t ) =

2 [1 − cos( 2τπ t )], 3

0,

k −1  i =0

 sl rec T c /k t − l

Tc k

 (1)

0≤t ≤τ

(3)

others

The normalized PSD (i.e., unit power over infinite bandwidth) of TDRC is derived as

P TDRC ( f )

 ∞ 2  − j2π f t  p τ (t )e dt  =   τ 1 

−∞

= =

    2  recτ ( f ) − 1 recτ f − 1 − 1 recτ f + 1   3τ 2 τ 2 τ  2 

2 sin2 (π f t ) 3π

2

(4)

τ f 2 ( f 2 τ 2 − 1)2

∞

with rec τ ( f ) = −∞ rec τ (t )e − j2π f t dt = e − j π f τ sin(π f τ )/π f . Fig. 1 compares the TDRC with a conventional rectangular chip in the time and frequency domains. As shown in this figure, TDRC can provide a high spectral efficiency compared to that of the rectangular because of its waveform smoothness and continuity in the time domain. The TDRC also obtains a smaller peak value near the center frequency that is more prone to be compatible with narrow band signals using same carrier frequency. TDRC also imposes stronger spectrum roll-off and less OOB emission in the side lobes, beneficial to minimize mutual interference with wide band signals. Besides, TDRC has a main lobe twice wider than that of the rectangular, offering an opportunity to maximize the ranging potentials and multipath mitigation capability due to the large amounts of high frequency components concentrated at the edge of available receiver bandwidth of great interests. The proposed pulse waveform perfectly fulfills the previously mentioned design requirements of new signals for the future GNSS. Based on the good spectrum property of TDRC, an interesting modulation family with spreading code rate f c = m × 1.023 MHz, denoted as TDRC-BCS([s0 , s1 , . . . , sk−1 ], m), can be obtained by substituting the rectangular shape of BCS modulation with TDRC. The equivalent baseband signal is expressed as

sTDRC -BCS (t ) =

q(t ) =

(2)

Here, a notation of BCS([s0 , s1 , . . . , sk−1 ], m) is used to describe the BCS modulation with the BCS sequence [s0 , s1 , . . . , sk−1 ] for each chip and a spreading code rate of f c = m × 1.023 MHz. The PSD characteristics entirely depend on the signal pulse waveform. As for above-mentioned specific requirements for PSD, we introduce a novel pulse waveform called TDRC:

2.1. Mathematical model BCS modulation is considered to be a generalization of BPSK and BOC. Each code chip can be divided into rectangular pulses, each with an equal length k. The particular generation scheme makes BCS modulation a promising modulation given that the well-selected configuration of the BCS sequence offers clear performance advantages and the possibility of organizing spectrum properties in a more efficient way than the current BPSK and BOC modulations. The chip waveform is defined as

0≤t ≤τ others



mi

i

×



ai , j rec T c (t − jT c − iT m )

j k −1  l =0

 sl p T c /k t − l

Tc k

 − jT c − iT m ,

t>0

(5)

where mi is the ith message data symbol (i.e., the value is always 1 for the pilot channel), ai , j is the jth spreading code chip of the ith message data symbol, T c = 1/ f c denotes the spreading code period, and T m is the symbol period. An example of the time-domain waveforms for BOC and TDRCBCS is shown in Fig. 2. In particular, if the BCS sequence takes k sign representation of a sine waveform with k = 2 f s / f c = 2n/m or

Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

61

Fig. 1. (a) Time domain waveforms of TDRC and rectangular pulses; (b) Normalized PSD of TDRC and rectangular pulses.

Fig. 2. Examples of time-domain waveforms for (a) BOC and (b) TDRC-BCS.

a cosine waveform with k = 4 f s / f c = 4n/m, then a special subclass of TDRC-BCS called sine-phased or cosine-phased TDRC-BOC can be attained and labeled by TDRC-BOCs(n, m) or TDRC-BOCc(n, m), where f c = m × 1.023 MHz is the spreading code chip rate and f s = n × 1.023 MHz is the frequency of the sine-phased or cosinephased carrier. 2.2. PSD

1

 S TDRC-MCS([s

0 ,s1 ,...,sk−1 ],m)

2 ( f )

(6)

where S TDRC-BCS([s0 ,s1 ,...,sk−1 ],m) ( f ) is the Fourier transformation of the TDRC-BCS chip waveform, computed as

S TDRC -BCS([s0 ,s1 ,...,sk−1 ],m) ( f )

=

T c  k −1 0

=

 sl p T c /k t − l

l =0

k −1  l =0

πf

sin( kf )

2

c

3π

f2 f( 2 2 k fc

− 1)

e

πf − j kf

k −1 

c

sl e

π fl − j 2kf

(7)

c

l =0

Once a general expression of S TDRC-BCS([s0 ,s1 ,...,sk−1 ],m) ( f ) is obtained, the resulting normalized PSD of TDRC-BCS modulation can be derived by equation (6) as

(l+1) T c

k

Tc



k

 p T c /k t − l

sl lT c k



=

e − j2π f t dt

Tc k



e − j2π f t dt

 1  2 

Tc 

πf

sin( kf ) c

3π f(

f2 k2 f c2

e

− 1)

πf − j kf c

k −1 

sl e

2   

π fl  − j 2kf c

l =0

πf

=

G TDRC -BCS([s0 ,s1 ,...,sk−1 ],m) ( f ) Tc

=

G TDRC -BCS([s0 ,s1 ,...,sk−1 ],m) ( f )

Suppose that all the spreading codes are statistically independent, equally likely, and identically distributed. The normalized PSD of the TDRC-BCS modulation family in an easy and generic method can be calculated by

=



2 f c sin( kf )2 c

3π 2 f 2 (

×

k −1  l =0

f2 k2 f c2

sl2

− 1)2

+2

k −1 k −2  

 sl s j cos ( j − l)

2π f

l=0 j =l+1

kf c

 (8)

As for the particular case of the TDRC-BCS family namely sinephased or cosine-phased TDRC-BOC modulations, their corresponding PSDs can be calculated by substituting the sign of a sine or cosine waveform into vector [s0 , s1 , . . . , sk−1 ] of equation (8), as shown below. When [s0 , s1 , . . . , sk−1 ] takes the sign of a sine waveform and k = 2 f s / f c = 2n/m, the normalized PSD of the sine-phased TDRCBOC modulation is expressed by

G TDRC -BOCs(n,m) ( f ) =

πf

πf

s

fc

2 f c tan2 ( 2 f ) sin2 ( 3π

2 2 f 2( f 4 fs

− 1)2

)

(9)

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Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

Fig. 3. (a) Theoretical and simulated normalized PSDs of TDRC-BOCs(2, 1), i.e. TDRC-BCS([1, −1, 1, −1], 1); (b) An example comparison of theoretical PSDs for BOCs(2, 1) and TDRC-BOCs(2, 1).

for an even 2 f s / f c and

G TDRC -BOCs(n,m) ( f ) =

πf

πf

s

fc

2 f c tan2 ( 2 f ) cos2 ( 3π

2 2 f 2( f 4 fs

)

(10)

− 1)2

for an odd 2 f s / f c . Similarly, if [s0 , s1 , . . . , sk−1 ] adopts the sign of a cosine waveform and k = 4 f s / f c = 4n/m, then the normalized PSD of the cosine-phased TDRC-BOC modulation can be written as

G TDRC -BOCc(n,m) ( f ) =

πf

πf

s

fc

8 f c sin4 ( 4 f ) sin2 ( 3π

2

f2 f 2 ( 16 f s

)

− 1)2 cos2 ( 2π ffs )

(11)

for an even 2 f s / f c , and

G TDRC -BOCc(n,m) ( f ) =

πf

πf

s

fc

8 f c sin4 ( 4 f ) cos2 ( f2

) πf

3π 2 f 2 ( 16 f − 1)2 cos2 ( 2 f ) s

(12)

s

for an odd 2 f s / f c . Theoretical and simulated normalized PSDs for an instance signal of TDRC-BOCs(2, 1) is shown in Fig. 3(a). A perfect agreement can be clearly recognized that the theoretical PSD performs equivalent spectrum envelope with that using signal simulation, demonstrating the validity of theoretical analysis on TDRC-BCS PSD. It clearly appears from Fig. 3(b) that TDRC-BCS signal has most energy concentrated near the main lobes, while the magnitude of side lobes decreases rapidly compared to BOC. Therefore, these properties make the TDRC-BCS helpful to weaken the interference with other adjacent spaced signal sources and services, and enhance signal performance for positioning. 3. Evaluation criterion and signal performance To investigate the potentials of TDRC-BCS modulation for navigation, this section presents the analytical methods and evaluation criteria applied to quantify and assess the GNSS signal performance in terms of code tracking performance, multipath mitigation, anti-jamming performance, and compatibility. An extensive study on multipath model analysis approach is also provided. The performance analysis and comparison of TDRC-BCS with legacy modulations are completed in this section. 3.1. Spectrum property and auto-correlation function The BOCs(5, 2) is a baseline signal transmitted by IRNSS [20, 21], and it is also ever suggested for GPS as an additional civil

Fig. 4. Normalized PSD of some example signals based on BPSK, BOC, and TDRC-BCS modulations.

signal centered on carrier frequency of 1575.42 MHz because of its excellent spectrum isolation [22]. An optimal receiver bandwidth of 24 MHz is recommended to BOCs(5, 2) [20,22]. This section takes the BOCs(5, 2) case to investigate the potentials of TDRC-BCS signals for GNSS. For a fair analysis, we also propose a few TDRC-BCS signals and BPSK signal with similar spectral occupancy and comparable envelope construction as the BOCs(5, 2). These competitors include TDRC-BCS([1, −1, 1, −1, −1], 2) (TDRCBCS1), TDRC-BCS([1, −1, 1, −1, 1, 1], 2) (TDRC-BCS2), BPSK(10), and TDRC-BOCs(5, 2). The normalized PSDs of these signals and BOCs(5, 2) are shown in Fig. 4. As recognized from Fig. 4, TDRC-BCS signals have two peak lobes and still maintain the characteristic of spectrum splitting similar to BOC signals. Moreover, the smoothness and continuity of pulse waveforms make the PSD of TDRC-BCS signals more compact and impose stronger spectrum roll-off in side lobes than legacy BOC and BPSK modulations. For example, at a distance from the center frequency of approximately 25 MHz, the TDRC-BCS1 signal has spectral lobe amplitudes at least 10 and 20 dB lower than those of BPSK(10) and BOCs(5, 2) signals, respectively. This result indirectly implies that a large amount of the power of TDRC-BCS signals can be concentrated near the main lobes. Fig. 5 depicts the percentage of the signal power contained within a certain bandwidth for several investigated modulation signals. The required bandwidths containing 90% of the power of all the signals are also reported in this figure. Notably, for a power containment factor of 90%, the TDRC-BOCs(5, 2) consumes minimal frequency re-

Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

63

Fig. 6. Normalized auto-correlation functions for all considered modulations. Fig. 5. Power containment for investigated modulation signals.

sources, closely followed by BPSK(10), TDRC-BCS1, TDRC-BCS2, and BOCs(5, 2). Furthermore, TDRC-BOCs(5, 2), TDRC-BCS1, and TDRCBCS2 have 90% of their power concentrated within 12.61, 17.45, and 19.69 MHz. These values are at least 12 MHz smaller than BOCs(5, 2). For power containment factor of 95%, all these TDRCBCS signals also occupy less available bandwidths compared with existing modulations. This advantage ensures that TDRC-BCS signals are more desirable to allow the peaceful coexistence of wide band signals with minimal mutual interference. The normalized auto-correlation function (ACF) is one of the most significant characteristics in the analysis of the signal structure. The sharper ACF behaves, the more precise the ranging is with this signal waveform. Thus, a promising modulation scheme for navigation with a good normalized ACF is necessary for precise range measurement. To explore the auto-correlation property of TDRC-BCS modulation, according to the Wiener–Khintchine theorem, the normalized ACF can be derived from the inverse Fourier transformation of the corresponding PSD:.

B /2 (t ) =

πf

2 f c sin( kf )2 c

− B /2

×

3π

k −1  l =0

2

f2 f 2( 2 2 k fc

sl2

+2

− 1)2

k −2  k −1  l=0 j =l+1

  2π f sl s j cos ( j − l) e j2π f t df kf c

(13) where B is always recognized as the receiver pre-filtering bandwidth.

The normalized ACFs of previous modulation examples are illustrated in Fig. 6. A 24 MHz receiver pre-filtering bandwidth is adopted. The TDRC-BCS signals all show a relatively narrow main peak contrary to the BOC and BPSK, because of more high frequency components concentrated within the receiver bandwidth. Moreover, the TDRC-BCS signals also behave similar or smaller absolute side peaks in comparison with legacy modulations. These ACF characteristics make TDRC-BCS advantageous to attain better ranging potentials for positioning, as we are going to see in Fig. 7. 3.2. Interference on the BDS RDSS service The Radio Determination Satellite Service (RDSS) and Radio Navigation Satellite Service (RNSS) are two kinds of radio satellite services that can be used for satellite navigation. The International Telecommunication Union Radio Regulations define RNSS as a special subclass of RDSS. The Chinese BDS offers both satellite services, and the BDS satellites broadcast B1 signals that are simultaneously accompanied by the user signal reception of the RDSS band (1610–1626.5 MHz) [23–25]. The interference imposed on RDSS unlink signals from other transmitted signals is likely to seriously degrade RDSS signal quality if no strong output filters are present at satellite transmitters. From the perspective of signal design, the driving factor to be addressed is the design of a spectrum-efficient modulation with quite small OOB emission. Such a scheme is very necessary given that a signal with less OOB emission can significantly mitigate the signal distortion induced by non-ideal filter characteristics. In addition, a spectrum-efficient modulation scheme also can significantly decrease the design complexity and cost of filters on boards to realize a high spectrum separation for minimizing the interference on the RDSS service.

Fig. 7. (a) Gabor bandwidths and (b) Cramér–Rao lower bounds of all the modulation candidates.

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Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

Table 1 OOB emission power in the BDS RDSS band and OOB loss for all the modulation candidates. Signal

BOCs(5, 2)

BPSK(10)

TDRC-BCS1

TDRC-BCS2

TDRC-BOCs(5, 2)

OOB emission power in the RDSS Band (dBc) OOB loss within the 24 MHz bandwidth (dBW)

−19.357

−22.6771

−45.205

−44.550

−43.430

0.815

0.432

0.171

0.291

0.1877

The spectrum isolation degree of all the investigated candidate signals with RDSS service can be clearly observed in Fig. 4. These TDRC-BCS signals appear to introduce less interference into the RDSS band as a result of its higher spectral efficiency. Table 1 computes and compares the OOB loss within the 24 MHz bandwidth and the OOB emission of all the modulation candidates within the RDSS service. The OOB loss is defined as

  B /2      λloss = 10 × log10 G s ( f )df   

(14)

− B /2

where G s ( f ) is the desired signal normalized PSD and symmetrical to the center frequency. As illustrated in Table 1, TDRC-BCS1 shows the smallest OOB loss within the 24 MHz bandwidth among all the signals, followed by TDRC-BOCs(5, 2) and TDRC-BCS2. All these TDRC-BCS signals have a smaller OOB loss than BOC and BPSK at over 0.5 and 0.13 dB levels, respectively. These results are in agreement with previous analysis on spectrum property and also demonstrate that TDRC-BCS is a spectrum-efficient modulation. Furthermore, the OOB emission of TDRC-BCS1, TDRC-BCS2, and TDRC-BOCs(5, 2) within the BDS RDSS band are −45.205, −44.550, and −43.430 dBc, respectively. These signals can even perform at least 20 and 22 dB lower than BPSK and BOC signals. Considering the relatively small OOB emission of TDRC-BCS modulation, the spectrum isolation of future navigation signals with the BDS RDSS service can be improved significantly. 3.3. Tracking accuracy Given that code tracking is essential to maintain chip synchronization in receiver processing, tracking accuracy is one of the basic performances in navigation and position systems. The Cramér–Rao lower bound is the generic theorem to reflect the lower bound of code tracking accuracy. With discriminators using coherent early-late processing (CELP), the code-tracking error variance in white noise can be written as [26]

B /2

2 CELP

σ

=

B L (1 − 0.5B L T ) − B /2 G s ( f ) sin2 (π f d)df

(2π

B /2 )2 NC s0 ( − B /2

f G s ( f ) sin(π

f d)df )2

(15)

where B L is the one-side code-tracking loop bandwidth, T denotes the integration time, d is referred as the early-late spacing, C s is commonly called the signal carrier power, and N 0 is the PSD of white noise such that the signal has a carrier power-to-noise density ratio of C s / N 0 . When the discriminators using CELP take a very small early-late spacing, the code-tracking error variance can approach the Cramér–Rao lower bound. If the reference signal is ideal and B L T is small enough in the limit as d becomes negligibly small, then equation (15) becomes 2 CELP ,d→0

σ

2 ∼ = σlower bound =

BL

(2π

)2 NC s0

B /2

2 − B /2 f G s ( f )df

(16)

− B /2

3.4. Multipath errors The multipath error is one of the dominant sources of the error budget because of its random nature. As a result, multipath mitigation capacity has become another relevant design driver for new GNSS signals. The multipath error is mainly caused by signals reflected from diverse transmission paths and is described as [27–29]

r (t ) = a0 e j ψ0 c (t − τ0 ) +

N −1 

ai e j ψi c (t − τi )

(18)

i =1

where a0 , τ0 , and ψ0 are the received amplitude, propagation delay, and received phase of the direct signal, respectively. Similarly, ai , τi , and ψi denote the received amplitude, propagation delay, and received phase of the ith reflected signal. N − 1 is taken as the total path number of the reflected signals, whereas c (t ) denotes the code modulation signal. The multipath error envelopes and their running averages based on a conventional two-path model with a direct and a single reflected path are common tools of describing the anti-multipath performance of a signal. In a two-path model, the resultant multipath error envelopes based on the CELP discriminator are estimated as [30]

ετ (τ1 ) =

a1 ±

B /2

B /2

− B /2 G s ( f ) sin(2π f τ1 ) sin(π f d)df

2π − B /2 f G s ( f ) sin(π f d)[1 ± a1 cos(2π f τ1 )]df (19)

A quantity called Gabor bandwidth is given by

  B /2    f Gabor =  f 2 G s ( f )df 

The Gabor bandwidth can also be considered another way of interpreting the Cramér–Rao lower bound. Intuitively, signals with larger Gabor bandwidth would result in very small CRLB and offer the potentials for more accurate code tracking. Thus, the key factor for optimizing and designing signals for GNSS is the maximization of the Gabor bandwidth. The Gabor bandwidths and Cramér–Rao lower bounds of all the investigated modulation candidates with 1 Hz loop bandwidth and 24 MHz receiver pre-filtering bandwidth are shown in Fig. 7. As shown in Fig. 7(a), when the receiver bandwidth is less than 10 MHz, TDRC-BCS signals have Gabor bandwidths similar to those of any other modulation. The Gabor bandwidths of TDRC-BCS signals continue to enlarge with an increase in the receiver prefiltering bandwidth and outperform BOCs(5, 2) and BPSK(10) until the receiver pre-filtering bandwidth greater than 10 MHz. This result demonstrates that TDRC-BCS signals have the best tracking accuracy especially for a receiver with pre-filtering bandwidth near main lobes. As revealed in Fig. 7(b), for the 24 MHz receiving bandwidth suggested for BOCs(5, 2), these TDRC-BCS signals all provide the quite small lower bound on code-tracking errors for C s / N 0 between 20 and 50 dB Hz. Therefore, TDRC-BCS signals present prominent advantages in terms of tracking accuracy compared to BOCs(5, 2) and BPSK(10).

(17)

where a1 is the multipath-to-direct signal ratio that satisfies a1 = a1 /a0 and τ1 = τ1 − τ0 is the propagation delay differences of the reflected signal relative to the direct signal. The running average multipath error envelopes are given by

Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

65

Fig. 8. (a) Multipath error envelopes and (b) running average multipath errors of all the modulation candidates.

ετ ,av (τav ) =

1

τav

 τav

abs(ετ (τ1 )|0 ) + abs(ετ (τ1 )|π ) 2

errors induced by the τ1 and τ2 , we take respectively the partial derivatives of ετ with respect to ψ1 and ψ2 :

dτ1

(20)

0

with the ετ (τ1 )|0 and ετ (τ1 )|π as multipath errors under the condition that a single multipath signal and a direct signal are in the same and inverse phase. abs(·) is the absolute operator. The performance of various analyzed modulation candidates in terms of multipath mitigation capability using the multipath error envelopes and running average multipath errors are compared in Fig. 8. In this situation, a multipath-to-direct signal ratio of −6 dB, an early-late spacing of 0.1 chip, and a receiver bandwidth of 24 MHz are considered for visual comparison. As observed in Fig. 8(a), the multipath error envelopes of TDRC-BCS1 and TDRC-BCS2 are much smaller for most of variations of the multipath delay and are more likely to diminish at a shorter multipath delay than BOCs(5, 2). Fig. 8(b) also reflects that TDRC-BCS1 and TDRC-BCS2 show the highest robustness against multipath and provide the smallest peak running average multipath errors of 2.6 and 2.5 m among all the other investigated modulation options. These results are expected because TDRC-BCS signals concentrate more high-frequency components at the edge of the receiving bandwidth of great interest, significantly improving the anti-multipath performance. Next, a three-path model analysis approach for the presence of two multipath signals is introduced in this paper. With two multipath signals and a direct signal, the output S-curve of the CELP discriminator, an approximation of the signal ACF derivative, can be expressed as

     d d − R ετ + = a0 cos(εψ ) R ετ − 2

{a0 cos(εψ )[ R (ετ − d2 ) − R (ετ + d2 )] + a1 cos(εψ + ψ1 ) × [ R (ετ − τ1 − d2 ) − R (ετ − τ1 + d2 )] + a2 cos(εψ + ψ2 ) × [ R (ετ − τ2 − d2 ) − R (ετ − τ2 + d2 )]}

≡0

(22)

If εψ is viewed as 0, the cases in which ψ1 is 0 or π and simultaneously ψ2 is 0 or π correspond to the worst ετ , then equation (21) can be rewritten as

D CELP (ετ )

     d d − R ετ + = a0 R ετ − 2

 

± a1 R ετ − τ1 −   ± a2 R ετ − τ2 −

d 2 d 2



2



− R ετ − τ1 + 

 − R ετ − τ2 +

d 2 d

 

2

≡0

(23)

2



ετ2



(24)

with the o(·) as a higher-order and indefinitely small quantity. Through the Wiener–Khintchine theorem, D CELP (0) and D CELP (0) can be attained by substituting 0 into equation (23) and its first order derivative as follows:

2

+ a2 cos(εψ + ψ2 )      d d × R ετ − τ2 − − R ετ − τ2 + ≡0 2

−ai sin(εψ + ψi )[ R (ετ − τi − d2 ) − R (ετ − τi + d2 )]

D CELP (ετ ) = D CELP (0) + D CELP (0)ετ + o

+ a1 cos(εψ + ψ1 )      d d × R ετ − τ1 − − R ετ − τ1 + 2

=

Although the D CELP (ετ ) is nonlinear, we can take a linear approximation by expanding the D CELP (ετ ) with the first-order Taylor’s series about 0:

D CELP (ετ )

2

∂ D CELP /∂ψi ∂ ετ = ∂ψi ∂ D CELP /∂ ετ

B /2

(21)

where ετ and εψ are the propagation delay estimation error and phase estimation error of the direct signal, respectively; R (·) is the signal ACF; and ψi = ψi − ψ0 and τi = τi − τ0 are the phase and propagation delay differences of the ith reflected signal relative to the direct signal. To explore the worst ετ , the multipath

D CELP (0) = ±2a1

G s ( f ) sin(−2π f τ1 ) sin(π f d)df

− B /2

B /2 ± 2a2 − B /2

G s ( f ) sin(−2π f τ2 ) sin(π f d)df

(25)

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Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

Table 2 SSC between the modulation candidates and existing signals in the L1 band [dB]. SSC

Reception 24 MHz receiver bandwidth

Emission

BOCs(5, 2) BPSK(10) TDRC-BCS1 TDRC-BCS2 TDRC-BOCs(5, 2)

40.92 MHz receiver bandwidth

BPSK(1)

BPSK(10)

MBOC

BPSK(2)

BOCs(10, 5)

BOCs(14, 2)

BOCc(15, 2.5)

−77.93 −70.29 −76.58 −75.17 −79.37

−74.12 −71.87 −73.35 −74.63 −73.88

−78.62 −70.86 −73.72 −75.95 −78.16

−80.13 −83.65 −82.37 −83.69 −82.27

−84.88 −80.90 −78.88 −77.62 −82.89

−80.82 −83.42 −82.32 −83.46 −82.49

−79.86 −85.09 −85.22 −86.08 −83.24

D CELP (0)

peak errors with BPSK, but it is obviously more superior to BOCs(5, 2). As a whole, the TDRC-BCS modulation also can present more immunity to multipath than legacy modulations in the presence of two multipath signals.

B /2 f G s ( f ) sin(π f d)df ± 4π a1

= 4π a 0 − B /2

3.5. Compatibility

B /2 ×

f G s ( f ) cos(−2π f τ1 ) sin(π f d)df

− B /2

B /2 ± 4π a 2

f G s ( f ) cos(−2π f τ2 ) sin(π f d)df

(26)

− B /2

Combine equations (24)–(26) with neglecting the influence of higher-order term, the resultant multipath error envelopes based on the three-path model are estimated as

ετ ,three-path (τ1 , τ2 ) ≈ −

D CELP (0)

B /2 ± − B /2 G s ( f )[ a1 sin(2π f τ1 ) + a2 sin(2π f τ2 )] sin(π f d)df =

B /2 2π − B /2 f G s ( f ) sin(π f d)[1 ± a1 cos(2π f τ1 ) ± a2 cos(2π f τ2 )]df

(27) where ai is the ith multipath-to-direct signal ratio that satisfies ai = ai /a0 . By observing the expression of the resultant multipath error envelopes from the two- and three-path models, a few common characteristics or similarities can be found between the equations (19) and (27). Similarly, if N − 1 reflected signals are considered, then the generic expression of the multipath error envelopes based on the N-path model can be deduced by

=

±

B /2

B /2

 N −1

− B /2 G s ( f ) sin(π f d)[

2π − B /2 f G s ( f ) sin(π f d){

i =1

 N −1 i =1

B /2

χs, J =

D CELP (0)

ετ ,N -path (τ1 , τ2 , . . . , τN −1 )

Ensuring the peaceful coexistence of the existing signals with minimal mutual interference is necessary for future GNSS signals. The spectral separation coefficient (SSC) is widely accepted by the GNSS community as an effective parameter to characterize mutual interference, give that this parameter provides a measure of how the spectral shape of interfering signal affects the desired signal sharing the same frequency band. The SSC is an inner product of the cross PSD of the desired and interfering signals, defined as [31, 32]

ai sin(2π f τi )]df

1 [ N− ± ai cos(2π f τi )]}df 1

(28) Fig. 9 shows this details on the multipath error envelopes using the three-path model for the investigated modulation candidates with a1 = −6 dB and a2 = −8 dB. Their multipath-induced peak errors are also reported in this figure. All the other parameters are the same as those in the previous simulation. As shown in Fig. 9, the multipath error envelopes of each analyzed signal are much higher than those of the two-path model because more reflected signals undoubtedly would result in additional multipath-induced error contributions. Similarly, TDRC-BCS2 has the best multipath mitigation capacity with the smallest multipath-induced peak errors of approximately 13.99 m. Compared with other candidates, the TDRC-BCS2 multipath error envelopes almost become invisible earlier for a multipath delay of τ1 and for τ2 greater than 23 and 70 m. The TDRC-BCS1 signal closely follows TDRC-BCS2. Besides, TDRC-BOCs(5, 2) also exhibits comparable multipath-induced

G s ( f )G J ( f )df

(29)

− B /2

where G s ( f ) and G J ( f ) are the normalized PSD of the desired and interfering signals, respectively. We assume that all the other modulation solutions have the same carrier frequency of 1575.42 MHz as BOCs(5, 2) for civil purposes. Table 2 lists the SSC of all the considered modulation signals with the existing signals in the L1 band. The characteristics of the existing signals in the L1 band for GPS, Galileo, and BDS are summarized in Table 3 [1]. For a fair analysis, the PSDs of all the signals are normalized over an infinite transmission bandwidth in this calculation. The comparison data in the Table 2 demonstrate that the TDRCBCS signals present better or comparable compatibility with existing signals in the L band than legacy modulations. For example, the compatibility of TDRC-BCS1, TDRC-BCS2, and TDRC-BOCs(5, 2) with BOCc(15, 2.5) are 5.36, 6.22, and 3.38 dB lower than that of the BOCs(5, 2). These signals also have compatibility with BOCs(14, 2) that is 1.5, 2.64, and 1.67 dB lower than that of BOCs(5, 2). Similarly, TDRC-BCS2 provides the best compatibility with BPSK(2) and BPSK(10), whereas TDRC-BOCs(5, 2) is slightly smaller than or comparable than BOCs(5, 2) in terms of SSC with MBOC. These results are not surprising because TDRC-BCS signals concentrate minor signal power near the center frequency, which makes them easier to coexist with the narrow signals, such as GPS L1C/A and BDS B1. Besides, spectrum-efficient TDRC-BCS signals have relatively low side lobes that help degrade noise level on wide band signals, such as BDS B1A and Galileo PRS signals. Although TDRCBCS signals show a little worst SSC with BOCs(10, 5), their values are still acceptable. Given the design flexibility of TDRC-BCS signals, the compatibility of TDRC-BCS signals using a suitable configuration of the BCS sequence with existing signals can be improved substantially. 3.6. Anti-jamming performance In satellite navigation systems, the narrowband-jamming and matched-spectrum-jamming are the main threats to the tracking

Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

67

Fig. 9. Multipath error envelopes of all the modulation candidates in the three-path model: (a) TDRC-BCS1; (b) TDRC-BCS2; (c) BOCs(5, 2); (d) TDRC-BOCs(5, 2); (e) BPSK(10).

and demodulation process. Anti-jamming merit factors are employed to assess the anti-jamming performance of the signals The effective signal-to-noise ratio ( E b / N 0 )eff and effective carrier-to-noise ratio (C s / N 0 )eff of a signal are important quantities and determine navigation signal demodulation and tracking performance, respectively. Assuming an interference decomposed into the sum of white noise and a single high power non-white interfering signal, the ( E b / N 0 )eff and (C s / N 0 )eff are approximately expressed by [26,31,33]



Eb N0



Cs N0

 = eff

 = eff

Cs R ( N 0 + J χs, J ) Cs N 0 + J ηs, J

≈

≈

1

Cs

J R χs, J

Cs J ηs, J

∝

1

ηs, J

∝

1 R χs, J

(30)

(31)

where R denotes the navigation data rate and J is the received power of the interfering signal. ηs, J is the code-tracking spectral sensitivity coefficient given by [34]:

B /2

ηs, J =

2

− B /2 G s ( f )G J ( f ) sin (π f d)df

B /2 2 − B /2 G s ( f ) sin (π f d)df

(32)

When d is sufficiently small, equation (32) becomes

 lim

d→0

Cs N0

B /2



2 − B /2 G s ( f ) f df

eff

∝ B /2

(33)

2 − B /2 G s ( f )G J ( f ) f df

Next, a delta function is adopted to represent narrowbandjamming with the G J ( f ) = δ( f − f J ) that satisfies G s ( f J ) is the peak of the desired signal PSD. That is because when the narrowband-jamming is centered on a local maximum in signal spectrum, it causes a local maximum degradation. We also define the interfering signal as G J ( f ) = G s ( f ) for matched-spectrumjamming, and then substitute the two interferences into equations (30) and (33), given below. The anti-narrowband-jamming merit factor for the demodulation process in dB level is



Q DemAJNW = 10 × log10

1 R × max[G s ( f )]



[dB]

(34)

68

Y. Sun et al. / Aerospace Science and Technology 66 (2017) 59–69

Table 3 GPS, Galileo, and BDS signal parameters in the L1 band [dB]. System

Service type

Carrier frequency (MHz)

Chip rate (Mcps)

Modulation

Type service

GPS

L1C/A L1CD L1CP L1P(Y) L1M

1575.42 1575.42 1575.42 1575.42 1575.42

1.023 1.023 1.023 10.023 5.115

BPSK(1) MBOC(6, 1, 1/11)

Open Open

BPSK(10) BOCs(10, 5)

Authorized Authorized

Galileo

E1PRS E1OSD E1OSP

1575.42 1575.42 1575.42

2.5575 1.023 1.023

BOCc(15, 2.5) MBOC(6, 1, 1/11)

PRS Open

BDS

B1I B1Q B1CD B1CP B1A

1561.098 1561.098 1575.42 1575.42 1575.42

2.046 2.046 1.023 1.023 2.046

BPSK(2) MBOC(6, 1, 1/11)

Open Authorized Open

BOCs(14, 2)

Authorized

Table 4 Anti-jamming merit factors of all the considered modulation candidates [dB]. Merit factors

BOCs(5, 2)

BPSK(10)

TDRC-BCS1

TDRC-BCS2

TDRC-BOCs(5, 2)

Q CTAJMS Q CTAJNW Q DemAJMS Q DemAJNW

68.36 65.67 52.37 49.84

73.92 70.24 54.88 53.11

72.26 69.17 54.71 52.57

71.41 68.67 54.61 52.51

68.04 66.62 51.14 49.28

The anti-matched-spectrum-jamming merit factor for the demodulation process in dB level is

 Q DemAJMS = 10 × log10



1

B /2

R × − B /2 G 2s ( f )df

[dB]

(35)

The anti-narrowband-jamming merit factor for the code tracking process in dB level is

Q CTAJNW = 10 × log10

 B /2 f 2 G ( f )df  s − B /2 max[ f 2 G s ( f )]

better anti-jamming performance. Also, the TDRC-BCS signals with optimal parameters have the capacity to substitute BOC signals for the future GNSS, allowing the peaceful coexistence of larges number of signals and adjacent services with minimal mutual interference. In summary, the TDRC-BCS is a promising modulation scheme for satellite navigation, and it is also particularly appropriated for band-limited C band in the future. Conflict of interest statement

[dB]

(36)

The authors declare that there is no conflict of interest.

The anti-matched-spectrum-jamming merit factor for the code tracking process in dB level is

Acknowledgements

 B /2 f 2 G ( f )df  s − B /2 [dB] Q CTAJMS = 10 × log10 B /2 2 2 − B /2 f G s ( f )df

This research work was supported by the National Natural Science Foundation of China (Grant No. 61403093), the Assisted Project by Heilongjiang Province of China Postdoctoral Funds for Scientific Research Initiation (Grant No. LBH-Q14048), the Open Research Fund of State Key Laboratory of Tianjin Key Laboratory of Intelligent Information Processing in Remote Sensing (Grant No. 2016-ZW-KFJJ-01), the Open Research Fund of State Key Laboratory of Space-Ground Integrated Information Technology (Grant No. 2015_SGIIT_KFJJ-DH_03).

(37)

Table 4 gives the numerical results for anti-jamming merit factors of the various analyzed modulation candidates. Notably, TDRCBCS1 and TDRC-BCS2 present at least 3 dB higher than those of the potential BOCs(5, 2) in the aspects of Q CTAJMS and Q CTAJNW . TDRC-BCS1 and TDRC-BCS2 also have better anti-jamming performance for the code-tracking process than BOCs(5, 2) at more than 2 and 2.5 dB levels in terms of Q DemAJMS and Q DemAJNW respectively. Moreover, the difference between the TDRC-BOCs(5, 2) and BOCs(5, 2) in each anti-jamming merit factor is relatively small. Therefore, TDRC-BCS signals have comparable or superior antijamming performance relative to legacy modulations.

Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ast.2017.03.002. References

4. Conclusions In this paper, the TDRC pulse is introduced by BCS modulation to generate a spectrum-efficient modulation scheme for the next generation GNSS. The proposed modulations can provide higher spectral efficiency and less OOB emissions than the legacy modulations because of the smoothness and continuity of the TDRC pulse. The theoretical analysis and simulation results indicate that TDRCBCS is more superior to the legacy modulations in terms of tracking accuracy, multipath mitigation while maintaining similar or

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