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Quaternary and Component-Binary Spreading Codes with Low Correlation for Navigation Systems

P. Vijay Kumar, Sugandh Mishra, Dileep Dharmappa

TL;DR

The paper develops a low-correlation quaternary spreading-code framework (MFD$_2$) with a period of $2046$ and its interleaved binary counterpart (IZ4$_2$), then extends to lunar PNT applications via the IZ4$_{2S}$ and IZ4$_{10}$ code families, achieving favorable even/odd correlation profiles and zero-shift orthogonality between different lengths. It provides closed-form cross-correlation expressions, bounds, and construction techniques, including a defect-controlled Subfamily ${\cal D}_{\text{MOD}}$ to limit anti-correlation, and a shift-register implementation strategy that yields practical hardware realizations. The lunar PNT comparison demonstrates IZ4$_2$’s superior correlation performance over BDS B1 at length $2046$ and IZ4$_{10}$’s competitive performance against GPS L1C at length $10230$, with notable tail advantages in CDF analyses and ease of implementation via simple shift-register architectures. Overall, the work contributes novel low-correlation code families, rigorous analytic expressions, and actionable guidance for lunar navigation systems, enabling robust ranging with orthogonal pairings across different code lengths and chip rates.

Abstract

In the first part of this two-part paper, we construct a family MFD$_2$ of low-correlation quaternary spreading codes having period $2046$. By quaternary, we mean that the spreading code symbols are drawn from $Z_4$ and are designed to be used in conjunction with QPSK modulation. Apart from low auto and crosscorrelation properties, we also require in addition, to our knowledge for the first time, that the spreading code family IZ4$_2$ obtained by taking the union of the component in-phase and quadrature-phase binary spreading codes associated to each quaternary spreading code in MFD$_2$, also have desirable low-correlation properties. We also investigate the balance of the quaternary and binary spreading codes. The second part is motivated by an application to the design of spreading code, (in this application termed as ranging codes), having parameters suitable for use in a lunar PNT system. Two lengths that are of particular current interest for a planned lunar PNT satellite system are $2046$ and $10230$. We study the applicability of a subset IZ4$_{2S}$ of IZ4$_2$ containing balanced binary spreading codes having length $2046$ to such a lunar PNT system. We show that the spreading codes belonging to IZ4$_{2S}$ compare favorably with the spreading codes of length $2046$ appearing in a recent issue of Inside GNSS. We also show that the IZ4$_{10}$ spreading code family in which the spreading codes have length $10230$, compares well in comparison with spreading codes of length $10230$ described in this article. In addition, the IZ4$_{10}$ and IZ4$_2$ spreading codes have been paired so as to be orthogonal at zero shift despite their different lengths and chipping rates.

Quaternary and Component-Binary Spreading Codes with Low Correlation for Navigation Systems

TL;DR

The paper develops a low-correlation quaternary spreading-code framework (MFD) with a period of and its interleaved binary counterpart (IZ4), then extends to lunar PNT applications via the IZ4 and IZ4 code families, achieving favorable even/odd correlation profiles and zero-shift orthogonality between different lengths. It provides closed-form cross-correlation expressions, bounds, and construction techniques, including a defect-controlled Subfamily to limit anti-correlation, and a shift-register implementation strategy that yields practical hardware realizations. The lunar PNT comparison demonstrates IZ4’s superior correlation performance over BDS B1 at length and IZ4’s competitive performance against GPS L1C at length , with notable tail advantages in CDF analyses and ease of implementation via simple shift-register architectures. Overall, the work contributes novel low-correlation code families, rigorous analytic expressions, and actionable guidance for lunar navigation systems, enabling robust ranging with orthogonal pairings across different code lengths and chip rates.

Abstract

In the first part of this two-part paper, we construct a family MFD of low-correlation quaternary spreading codes having period . By quaternary, we mean that the spreading code symbols are drawn from and are designed to be used in conjunction with QPSK modulation. Apart from low auto and crosscorrelation properties, we also require in addition, to our knowledge for the first time, that the spreading code family IZ4 obtained by taking the union of the component in-phase and quadrature-phase binary spreading codes associated to each quaternary spreading code in MFD, also have desirable low-correlation properties. We also investigate the balance of the quaternary and binary spreading codes. The second part is motivated by an application to the design of spreading code, (in this application termed as ranging codes), having parameters suitable for use in a lunar PNT system. Two lengths that are of particular current interest for a planned lunar PNT satellite system are and . We study the applicability of a subset IZ4 of IZ4 containing balanced binary spreading codes having length to such a lunar PNT system. We show that the spreading codes belonging to IZ4 compare favorably with the spreading codes of length appearing in a recent issue of Inside GNSS. We also show that the IZ4 spreading code family in which the spreading codes have length , compares well in comparison with spreading codes of length described in this article. In addition, the IZ4 and IZ4 spreading codes have been paired so as to be orthogonal at zero shift despite their different lengths and chipping rates.

Paper Structure

This paper contains 37 sections, 8 theorems, 94 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Closed-Form Expression for Family ${\cal D}$ Sequence Family
  4. Component Binary Sequence Correlation
  5. Relating Quaternary and Binary Signals
  6. Computing the Anti-Correlation term $\Delta_{12}(\tau)$
  7. Correlation Expressions for $\Delta_{12}(\tau)$ in Summary
  8. Can $|\Delta_{12}(\tau)|$ be Large of $O(L)$ ?
  9. Summary Expressions for Quaternary and Binary Correlation
  10. Quaternary Correlation Summary
  11. Binary Correlation Summary
  12. Shift-Register Generation
  13. Linear Recursions
  14. Application to Lunar PNT
  15. LunaNet Signal-in-Space Document (Aug. 2023 Draft)
  16. ...and 22 more sections

Key Result

Theorem 3.1

We have the following closed-form expression for the correlation of two Family ${\cal D}$ PRN sequences associated to $y_1,y_2 \in H$ in the case of $\tau$ even: When $\tau$ is odd, we have where $y_3=y_1+\theta$.

Figures (7)

  • Figure 1: The IZ4$_2$ spreading code can be generated using $2$ 11-bit binary shift registers. The upper register is a $7$-tap linear feedback shift register. The lower register is updated using nonlinear logic using the contents of both shift registers. The figure also shows where to draw outputs corresponding to generating the in-phase $\{W(t)\}$, quadrature phase sequence $\{V(t)\}$ as well as as composite quaternary spreading code $\{Q(t)\}$.
  • Figure 2: Data and pilot waveforms, chipping rates and spreading code periods as presented in the LunaNet Signal-in-Space standardLunaNet_SIS.
  • Figure 3: The figure plots the Cumulative Distribution Function (CDF) of all correlation values of the IZ4$_2$ and BDS B1 (truncated-Gold) spreading codes respectively. Values of the CDF are expressed as percentages.
  • Figure 4: Zooming in to a segment of the CDF of the correlation values of the IZ4$_2$ and BDS B1 spreading codes. Only a portion of the CDF corresponding to percentiles $90$ and above is plotted here. As can be seen, above the $95$th percentile, the IZ4$_2$ spreading code does better throughout in terms of having a larger number of correlation values below a specified threshold. As an example, corresponding to a correlation value of $80$ (equivalent to -28.16 $\text{dB}$), the percentiles associated to the IZ4$_2$ and BDS B1 spreading codes are $96.58$ % and $93.69$% respectively.
  • Figure 5: The figure plots the Cumulative Distribution Function (CDF) of all correlation values of the IZ4$_{10}$ and Weil spreading codes incorporated into ISRO's NavIC L1 signal and the GPS L1 C pilot signal respectively. Values of the CDF are expressed as percentages. The maximum correlation magnitudes associated with the IZ4$_{10}$ and the Weil spreading codes are given by $484$ and $500$ respectively.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 1: Quaternary Family ${\cal D}$
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 4.1
  • proof
  • Definition 2
  • Example 1
  • Corollary 4.2
  • Theorem 5.1
  • ...and 7 more