# Research on Acquisition Performance of FFT Algorithm for Low-Frequency Spread-Spectrum Signals Using Acoustic Sensors

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## Abstract

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## 1. Introduction

## 2. Improved FFT Acquisition Algorithm

#### 2.1. Fuzzy Function and 2D Search Step Size

_{c}is the transmit carrier frequency, f

_{D}is the Doppler frequency, θ is the carrier phase, and n(t) is the noise.

_{D}is the difference between the actual Doppler and the estimated Doppler of the local carrier. In practical acquisition algorithms, if a Doppler and carrier phase cannot be initially estimated, ${\widehat{f}}_{D}$ and $\widehat{\theta}$ are usually set to zero and then related with the locally generated spreading spectrum code:

_{s}is integration time. The fuzzy function is defined as:

_{s}. As can be seen, the longer the integral time, the smaller the Doppler search step size and the higher the accuracy, and hence, the more search time and resources consumed. Noteworthily, the conclusion that the longer the integral time, the higher the frequency search accuracy is drawn from the perspective of the maximum search step size, which does not mean that the highest frequency accuracy can only reach 1/T

_{s}.

_{s}is prone to miss the signal peaks and the failure of correct acquisitions. The smaller the search step size, the more likely the search frequency falls on the signal peak.

#### 2.2. Conventional FFT Acquisition Methods

#### 2.3. FFT-Based Circular Shift Fast Acquisition Algorithm Combining Maximum Correlation and Threshold Discrimination

## 3. Simulation Analysis of Factors Affecting Algorithm Performance

#### 3.1. Effect of Spread-Spectrum Signal Waveform Parameters

_{s}increases by the same factor, and the acquisition performance under a low SNR is significantly boosted. This is because an increasing integral length leads to a higher correlation peak and a larger possibility of successfully extracting useful information from the noise. Furthermore, the acquisition success probability is calculated under the conditions that the code length N = 63 and there are two carriers in each chip (T

_{s}= 126 ms). The calculated integral time is very close to that calculated using N = 31, and the acquisition probabilities in the two cases are not much different either. This suggests that under a constant sampling frequency, only increasing the code length cannot remarkably improve the acquisition performance if the integral time remains unchanged. If the number of carriers in a chip remains unchanged (T

_{c}is constant), lengthening the spreading spectrum code will increase the integral time and thus significantly enhance the acquisition performance. In low-frequency signals, only a limited number of carriers can be used within a chip so as to ensure the bit rate and bandwidth. The larger the number of intra-chip carriers, the wider the chip and the smaller the bandwidth. The simulation results in Figure 4b show that within a certain SNR range, the acquisition probability is significantly raised and the acquisition performance improved as the number of carriers in the chip increases.

#### 3.2. Effect of Sampling Frequency

_{c}= 1000 Hz, and there are four carriers in one chip (chip length T

_{c}= 4 ms). For the received signal, the chip phase Δτ = 30 chips and the Doppler f

_{D}= 1 Hz. The variation in the acquisition probability with the signal-to-noise ratio (SNR) is calculated under four different sampling frequencies. At each sampling frequency, 5000 Monte Carlo simulations are performed. An acquired result with a chip error not exceeding 0.2 chips and a carrier frequency error less than 0.5 Hz is identified as a successful acquisition, and based on this, the corresponding successful acquisition probability P

_{d}is calculated.

## 4. Experimental Verifications

#### 4.1. Accuracy Verification of Improved FFT Acquisition Algorithm

^{−1}. The transmit signal is a spread-spectrum signal containing the bit information, with the spreading spectrum code length of N = 63, the chip width T

_{c}= 4 ms, and the carrier frequency of 1000 Hz. The sampling frequency is set to 16 kHz. Considering that it is only for testing the acquisition performance, the data acquisition time is set to 20 s, which is enough to satisfy the requirement for signal length. Two groups of experiments with high signal-to-noise ratios and low signal-to-noise ratios are adopted. When the correlation peak exceeds the threshold, the acquisition can be achieved. By comparing the time delay difference of the captured reflected signal with the theoretical time delay calculated by the real reflected path, the code phase accuracy of the captured signal can be further judged. As shown in Figure 8, the power of the background noise individually measured is concentrated at 281 Hz, and the peak in the power spectrum is 50.80 dB; the power peak at the center frequency of the measured signal in the background noise environment is 49.86 dB. In the white noise environment, the signal power is much lower than the noise power.

#### 4.2. Verification of the Effects of Spread-Spectrum Signal Waveform Parameters and Sampling Frequency on Acquisition Performance

_{c}), and the sampling frequency F

_{s}are changed separately to receive signals and acquire the FFT. The effects of the parameters on the acquisition performance are measured by whether or not an accurate acquisition is achieved and the relative peak value of the normalized correlation curve. The normalized correlation value represents the ratio of the correlation value at the sampling point to the average correlation value in the time period of integration. To a certain extent, a higher normalized correlation peak value suggests a better anti-noise performance and easier achievement of the acquisition.

_{c}vary, but the duration T

_{s}of their spreading spectrum codes is almost the same. Since T

_{s}is dependent on N and T

_{c}, these three cases can be used to study the effect of T

_{s}.

#### 4.2.1. Spread-Spectrum Signal Waveform Parameters

#### 4.2.2. Sampling Frequency

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Effects of different spreading spectrum code parameters on acquisition success probability: (

**a**) Different spreading spectrum code lengths N; (

**b**) Different numbers of carriers in a single chip.

**Figure 9.**Acquired results: (

**a**) Acquired result with background noise; (

**b**) Acquired result with background noise + white noise.

**Figure 11.**Acquired results for spread-spectrum signals with different code lengths N: (

**a**) Case 1, N = 31; (

**b**) Case 2, N = 63; (

**c**) Case 3, N = 127.

**Figure 12.**Acquisition correlation curves at different numbers of intra-chip carriers: (

**a**) Case 4, 2 carriers per chip; (

**b**) Case 5, 8 carriers per chip; (

**c**) Case 6, 16 carriers per chip; (

**d**) Case 7, 16 carriers per chip (N = 31).

**Figure 13.**Acquisition correlation curves at different sampling frequencies: (

**a**) Case 8; (

**b**) Case 9; (

**c**) Case 10; (

**d**) Case 11.

Case | Code Length N | No. of Intra-Chip Carriers | T_{c} | T_{s} | Sampling Frequency F_{s} |
---|---|---|---|---|---|

1 | 31 | 4 | 4 ms | 124 ms | 16 kHz |

2 | 63 | 4 | 4 ms | 252 ms | 16 kHz |

3 | 127 | 4 | 4 ms | 508 ms | 16 kHz |

4 | 63 | 2 | 2 ms | 126 ms | 16 kHz |

5 | 63 | 8 | 8 ms | 504 ms | 16 kHz |

6 | 63 | 16 | 16 ms | 1008 ms | 16 kHz |

7 | 31 | 16 | 16 ms | 496 ms | 16 kHz |

8 | 63 | 4 | 4 ms | 252 ms | 2 kHz |

9 | 63 | 4 | 4 ms | 252 ms | 4 kHz |

10 | 63 | 4 | 4 ms | 252 ms | 8 kHz |

11 | 63 | 4 | 4 ms | 252 ms | 32 kHz |

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**MDPI and ACS Style**

Tang, Y.; Zhou, Q.; Xie, Z.; Liu, W.; Lü, X.
Research on Acquisition Performance of FFT Algorithm for Low-Frequency Spread-Spectrum Signals Using Acoustic Sensors. *Sustainability* **2023**, *15*, 6405.
https://doi.org/10.3390/su15086405

**AMA Style**

Tang Y, Zhou Q, Xie Z, Liu W, Lü X.
Research on Acquisition Performance of FFT Algorithm for Low-Frequency Spread-Spectrum Signals Using Acoustic Sensors. *Sustainability*. 2023; 15(8):6405.
https://doi.org/10.3390/su15086405

**Chicago/Turabian Style**

Tang, Yongzhuang, Qidou Zhou, Zhiyong Xie, Wenxi Liu, and Xiaojun Lü.
2023. "Research on Acquisition Performance of FFT Algorithm for Low-Frequency Spread-Spectrum Signals Using Acoustic Sensors" *Sustainability* 15, no. 8: 6405.
https://doi.org/10.3390/su15086405