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Research ArticleRegular Papers
Open Access

Bicomplex Representation and Processing of GNSS Signals

Daniele Borio
NAVIGATION: Journal of the Institute of Navigation December 2023, 70 (4) navi.621; DOI: https://doi.org/10.33012/navi.621
Daniele Borio
1Joint Research Centre (JRC), European Commission, Ispra (VA), Italy
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  • FIGURE 1
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    FIGURE 1

    Spectral representation of a signal with components on two different frequencies

  • FIGURE 2
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    FIGURE 2

    Recovery of a bicomplex signal using either two synchronous front-ends (a) or a single wideband front-end (b) ADC: analog-to-digital converter.

  • FIGURE 3
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    FIGURE 3

    Schematic representation of the acquisition block for a bicomplex GNSS signal

  • FIGURE 4
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    FIGURE 4

    Schematic representation of the triple-loop tracking architecture obtained using bicomplex signals

  • FIGURE 5
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    FIGURE 5

    View of the NI USRP-2944R front-end used for collection of the AltBOC wideband signal

  • FIGURE 6
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    FIGURE 6

    Bi-dimensional CAF obtained using the acquisition algorithm described in Section 4.2. The pilot components from both E5a and E5b signals are used. The subcarrier Doppler frequency is constrained to the carrier Doppler.

  • FIGURE 7
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    FIGURE 7

    Comparison between normalized correlation functions considering standard sideband processing (“upper sideband”) and bicomplex processing using two pilot components (“joint”)

  • FIGURE 8
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    FIGURE 8

    Normalized code, subcarrier, and carrier Doppler estimates from the DLL, SPLL, and PLL in the triple-loop architecture implemented for processing the AltBOC signal

  • FIGURE 9
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    FIGURE 9

    Comparison between C/N0 values obtained from the joint bicomplex tracking architecture and C/N0 estimates obtained when processing the single sideband components individually for the Galileo AltBOC signal

  • FIGURE 10
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    FIGURE 10

    Normalized code, subcarrier, and carrier Doppler estimates from the DLL, SPLL, and PLL in the triple-loop architecture implemented for processing the meta-signal formed by the B1I/BIC components

  • FIGURE 11
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    FIGURE 11

    Comparison between C/N0 values obtained from the joint bicomplex tracking architecture and C/N0 estimates obtained when processing the single sideband components individually for BeiDou B1I and B1C signals

Tables

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    TABLE 1

    Multiplication Between Imaginary Units

    ijk
    i−1k−j
    jk−1−i
    k−j−i1
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    TABLE 2

    Characteristics of the Digital Baseband Galileo AltBOC and BeiDou B1 Signals Used for Analysis

    ParameterValue
    Galileo E5 AltBOC
    Sampling frequency, fs50 MHz
    Center frequency, f01191.795 MHz
    Residual IF, fIF0 MHz
    Sampling typeComplex I/Q
    No. of bits8
    BeiDou B1
    Sampling frequency, fs40 MHz
    Center frequency, f01575.42 MHz
    Residual IF, fIF−7.161 MHz
    Sampling typeComplex I/Q
    No. of bits8
    • View popup
    TABLE 3

    Parameters of the Triple-Loop Architecture Implemented for Processing the AltBOC Signal

    ParameterValue
    DLL order2
    DLL bandwidth2 Hz
    Early-minus-late spacing0.5 chips
    SPLL order2
    SPLL bandwidth2 Hz
    PLL order3
    PLL bandwidth15 Hz
    Integration time after secondary code synchronization5 ms
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    TABLE 4

    Parameters of the Triple-Loop Architecture Implemented for Processing the BDS B1 Signals

    ParameterValue
    DLL order2
    DLL bandwidth2 Hz
    Early-minus-late spacing BPSK(2)0.5 chips
    Early-minus-late spacing BOC(1, 1)0.333 slots
    BPSK(2) chip/BOC(1, 1) slot duration0.48876 μs
    SPLL order2
    SPLL bandwidth2 Hz
    PLL order3
    PLL bandwidth15 Hz
    Initial integration time1 ms
    Integration time after secondary code synchronization10 ms
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    TABLE A1

    Summary Table with the Basic Properties of Bicomplex Numbers

    NameDefinitionReference
    Bicomplex numbersEmbedded Image(Alpay et al., 2014, p. 10)(A.1)
    Bar-conjugationEmbedded Image
    The terms multiplying i and k are negated
    (Alpay et al., 2014, p. 10)(A.2)
    †-conjugationEmbedded Image
    The terms multiplying j and k are negated
    (Alpay et al., 2014, p. 10)(A.3)
    ConjugationEmbedded Image
    The terms multiplying i and j are negated
    (Alpay et al., 2014, p. 10)(A.4)
    Conjugation propertiesEach conjugation is additive, involutive, and multiplicative(Alpay et al., 2014, p. 11)(A.5)
    i-modulusEmbedded Image(Alpay et al., 2014, p. 11)(A.6)
    j-modulusEmbedded Image(Alpay et al., 2014, p. 12)(A.7)
    k-modulusEmbedded Image(Alpay et al., 2014, p. 12)(A.8)
    Idempotent decompositionEmbedded Image(Alpay et al., 2014, p. 15)(A.9)
    | · |k idempotent decompositionEmbedded Image(Alpay et al., 2014, p. 21)(A.10)

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NAVIGATION: Journal of the Institute of Navigation: 70 (4)
NAVIGATION: Journal of the Institute of Navigation
Vol. 70, Issue 4
Winter 2023
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Bicomplex Representation and Processing of GNSS Signals
Daniele Borio
NAVIGATION: Journal of the Institute of Navigation Dec 2023, 70 (4) navi.621; DOI: 10.33012/navi.621

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Bicomplex Representation and Processing of GNSS Signals
Daniele Borio
NAVIGATION: Journal of the Institute of Navigation Dec 2023, 70 (4) navi.621; DOI: 10.33012/navi.621
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  • Article
    • Abstract
    • 1 INTRODUCTION
    • 2 BICOMPLEX SIGNAL REPRESENTATION
    • 3 BICOMPLEX SIGNAL RECOVERY
    • 4 BASEBAND PROCESSING
    • 5 EXPERIMENTAL SETUP
    • 6 SAMPLE RESULTS
    • 7 CONCLUSIONS
    • HOW TO CITE THIS ARTICLE
    • CONFLICT OF INTEREST
    • A PROPERTIES OF BICOMPLEX NUMBERS
    • B SQUARE MODULUS OF THE SUM OF TWO BICOMPLEX NUMBERS
    • C IDEMPOTENT DECOMPOSITION OF A PURE I-COMPLEX EXPONENTIAL
    • D PHASES OF A BICOMPLEX NUMBER
    • E GAIN OF THE BICOMPLEX NONCOHERENT EARLY-MINUS-LATE ENVELOPE DISCRIMINATOR
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More in this TOC Section

  • Ranging Performance Evaluation for Higher-Order Scalable Interplex
  • Combinatorial Watermarking Under Limited SCER Adversarial Models
  • Wide-Sense CDF Overbounding for GNSS Integrity
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Keywords

  • acquisition
  • bicomplex numbers
  • double-phase estimator (DPE)
  • dual-frequency
  • GNSS meta-signals
  • tracking

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