Solution to MATLAB and Simulink Challenge project '192' 'Improve the Accuracy of Satellite Navigation Systems'

Improve the Accuracy of Satellite Navigation Systems

Program: https://github.com/mathworks/MATLAB-Simulink-Challenge-Project-Hub

Project Description: https://github.com/mathworks/MATLAB-Simulink-Challenge-Project-Hub/blob/main/projects/Improve%20the%20Accuracy%20of%20Satellite%20Navigation%20Systems/README.md

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BeiDou GF(64) Non-Binary LDPC Communication System

A complete MATLAB implementation of a BeiDou B-CNAV2 compliant Non-Binary LDPC communication system over GF(64). The project implements the entire digital communication chain, from navigation message generation and LDPC encoding to AWGN transmission and iterative decoding using the Extended Min-Sum (EMS) algorithm.

The implementation closely follows the BeiDou B-CNAV2 Interface Control Document (ICD) while emphasizing algorithmic clarity and educational value through explicit Galois Field arithmetic, parity-check matrix operations, and iterative belief propagation.

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Features

• Complete end-to-end BeiDou B-CNAV2 communication chain
• GF(64) arithmetic using precomputed lookup tables
• Matrix-based Non-Binary LDPC encoder
• Extended Min-Sum (EMS) decoder with configurable Top-M truncation
• Tanner graph message passing
• Soft-decision decoding using Log-Likelihood Ratios (LLRs)
• Binary Phase Shift Keying (BPSK) modulation
• AWGN channel simulation
• BER and FER performance evaluation
• Automated SNR sweep for waterfall curve generation

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System Architecture

Navigation Message
        │
        ▼
 GF(64) Symbol Mapping
        │
        ▼
 Non-Binary LDPC Encoder
        │
        ▼
 BPSK Modulator
        │
        ▼
     AWGN Channel
        │
        ▼
 Soft LLR Generation
        │
        ▼
 EMS LDPC Decoder
        │
        ▼
Decoded Navigation Message

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Project Structure

NBLDPC/
│
├── main.m                  # Run a single communication simulation
├── ber_sweep.m             # BER/FER simulation over multiple SNR values
├── BER_results.mat
├── ber_plot.pdf
│
└── src/
    ├── buildGF.m
    ├── buildParityCheckMatrix.m
    ├── generateNavigationMessage.m
    ├── ldpcencodegf64.m
    ├── channel_awgn.m
    ├── llr2gf64.m
    ├── ldpcdecodegf64.m
    ├── ldpcdecodegf64_ems.m
    ├── ems_combine_messages.m
    ├── ems_truncate.m
    ├── runSimulation.m
    └── ...

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How It Works

Transmitter

• Generates a BeiDou navigation message
• Converts binary data into GF(64) symbols
• Encodes the message using the Non-Binary LDPC parity-check matrix
• Converts the encoded symbols back into bits
• Modulates using BPSK

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Channel

The encoded signal is transmitted through an Additive White Gaussian Noise (AWGN) channel.

Noise level is controlled using the desired Eb/N₀/SNR during simulation.

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Receiver

The receiver performs:

1. Soft-decision demodulation
2. LLR computation
3. LLR-to-GF(64) conversion
4. Iterative EMS decoding
5. Syndrome verification
6. Bit reconstruction
7. BER/FER calculation

The EMS decoder reduces computational complexity by retaining only the Top-M most reliable symbol candidates during check-node processing.

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Running the Project

Requirements

• MATLAB R2024b or newer (recommended)
• Communications Toolbox

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Run a Single Simulation

main

The script automatically

• Loads GF(64) lookup tables
• Builds the parity-check matrix
• Generates a navigation frame
• Encodes the message
• Simulates transmission
• Decodes using EMS
• Displays BER, decoder status, and iteration count

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Generate BER Curve

ber_sweep

This performs an automatic SNR sweep and computes

• Bit Error Rate (BER)
• Frame Error Rate (FER)

The generated performance curve is saved as:

ber_plot.pdf

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Example Output

=======================================
 Non-Binary LDPC Demonstration
=======================================

Decoder    : EMS
Top-M      : 16
BER        : 0.000e+00
Success    : 1
Iterations : 7

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Key Algorithms

• GF(64) finite field arithmetic
• Matrix-based LDPC encoding
• Tanner graph belief propagation
• Extended Min-Sum (EMS) decoding
• Check-node message truncation
• Variable-node message updates
• Syndrome-based early stopping

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Performance

The implementation supports configurable

• Decoder iterations
• Top-M parameter
• SNR
• Message length

The BER/FER performance can be evaluated across different noise conditions using the provided simulation scripts.

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References

1. BeiDou Navigation Satellite System Signal In Space Interface Control Document (B1C & B2a Open Service Signals, Test Version, August 2017)

2. Declercq, D., & Fossorier, M. (2007). Decoding Algorithms for Non-Binary LDPC Codes over GF(q). IEEE Transactions on Communications.

3. Davey, M. C., & MacKay, D. J. C. (1998). Low Density Parity Check Codes over GF(q).

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License

This project is released under the BSD 2-Clause License.