# Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution

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## Abstract

**:**

## 1. Introduction

## 2. System Model and Frequency Hopping Parameters

#### 2.1. Frequency Hopping Signal Model

#### 2.2. Space-Time Frequency Distribution of Frequency Hopping Signal

#### 2.3. Frequency Hopping Hop Extraction and Space Time Frequency Matrix Construction

## 3. Frequency Hopping Signal Parameter Estimation

#### 3.1. Hopping Basic Parameter Estimation

#### 3.2. Joint Diagonalization of Space-Time-Frequency Matrix Based on Minimum Mean Square Error

#### 3.3. FH Signal DOA Estimation

#### 3.4. Frequency Hopping Network Table Sorting

## 4. Simulation Experiment

#### 4.1. Simulation Environment and Parameter Design

_{s}), observation duration (t), the hop period (Th), where the corresponding sampling points are used instead of the Th, the frequency hopping frequency set (${f}_{h,k}$). A total of three sets of frequency hopping signals are set here, Group1, Group2 and Group3, each group contains three frequency hopping network stations. For example, the frequency hopping signals sent by three frequency hopping network stations in Group1 are $F{h}_{11}$, $F{h}_{12},F{h}_{13}$. The F

_{s}is uniformly set to 100 MHz. Each frequency hopping signal in the first group contains 5 complete hops with a hop period of 250. The second set of frequency hopping signals $F{h}_{21}$ contains 3 complete hops, where Th is 250 and the observation time is 12.5 microseconds; $F{h}_{22}$ contains 4 complete hops and $F{h}_{23}$ contains three complete hops. The third group of frequency hopping signals has a duration of five microseconds and a hop period of 100. Each network station transmits a frequency hopping signal containing five complete hops.

#### 4.2. Feasibility Experiment Verification

#### 4.3. Estimated Accuracy Experiment

#### 4.4. Robust Experiment

#### 4.5. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Time-Frequency Model Diagram of Frequency Hopping Signal. This picture shows the law of signal frequency changing with time. (

**a**) There are three frequency hopping signals in the graph. (

**a**) total of 1250 time points are taken in the time domain. Each hop consists of 250 time points. (

**b**) There are 25 time points in the graph, and each hop consists of 5 time points.

**Figure 3.**Space-time frequency distribution diagram of multi-hop frequency signal. (

**a**) When SNR = 0 dB, the space-time-frequency distribution of the signal, (

**b**) the time-frequency distribution of the signal after threshold transformation.

**Figure 4.**Space-time frequency distribution diagram of multi-hop frequency signal. (

**a**) The figure is a space-time-frequency map of time-hopping signal after threshold processing when SNR = 0 dB, which is composed of self-term time-frequency points;(

**b**) the figure is a space-time-frequency distribution map which is composed of only 0,1 elements after threshold processing and removing part of noise.

**Figure 5.**Space-time-frequency distribution with only 0,1 elements. (

**a**) there is three hops surrounded by red frame in the graph, (

**b**) there are four hops in the graph.

**Figure 6.**The rectangular process diagram of hop space-time frequency value. (

**a**–

**c**) shows how to construct a space-time-frequency matrix from the extracted space-time-frequency values of each hop.

**Figure 7.**The corresponding diagram of each hop in the space-time frequency matrix of the frequency hopping signal.

**Figure 8.**Space-time-frequency diagram composed of self-term time-frequency points. (

**a**) SNR = −3 dB, Space-time-frequency-diagram; (

**b**) SNR = −2 dB, Space-time-frequency-diagram; (

**c**) SNR = 5 dB, Space time frequency diagram.

**Figure 9.**Simulation results diagram.the graph shows the relationship between parameter estimation accuracy and signal-to-noise ratio. (

**a**) The diagram shows the curve of hop duration (Th) estimation with SNR; (

**b**) the diagram shows the curve of direction of arrival (DOA) with SNR; (

**c**) the diagram shows the curve of frequency estimation with SNR; (

**d**) the diagram shows the results of the frequency hopping station; (

**e**) the diagram shows the curve of hopping start time with SNR. (

**f**) the diagram shows the curve of hopping end time with SNR

**Figure 10.**Parameter estimation accuracy rate with frequency hopping duration change graph. The figure shows the accuracy of the frequency hopping parameter estimation with the frequency hopping time. (

**a**) Frequency hopping period estimation accuracy varies with frequency hopping duration (

**b**) Estimation accuracy of frequency hopping frequency set varies with frequency hopping duration.

**Table 1.**Multi-hop frequency signal parameter table. The table describes the jump frequency and frequency hopping period information of the selected simulation signal.

Group | ${\mathit{F}}_{\mathit{s}}\text{}\left(\mathbf{MHz}\right)$ | t (μs) | Signal | Hopping Duration (${\mathit{T}}_{\mathit{h}}$) | Normalized Frequency ${\mathit{f}}_{\mathit{h},\mathit{k}}\text{}\left(\mathbf{MHz}\right)$ |
---|---|---|---|---|---|

Group1 | 100 | 12.5 | $F{h}_{11}$ | 250 | [0.50,0.86,0.52,0.30,0.74] |

$F{h}_{12}$ | 250 | [0.76,0.20,0.72,0.42,0.90] | |||

$F{h}_{13}$ | 250 | [0.96,0.36,0.10,0.62,0.52] | |||

Group2 | 100 | 12.5 | $F{h}_{21}$ | 250 | [0.50,0.86,0.52] |

$F{h}_{22}$ | 250 | [0.76,0.20,0.72,0.42] | |||

$F{h}_{23}$ | 250 | [0.96,0.36,0.10,0.62,0.52] | |||

Group3 | 100 | 5 | $F{h}_{31}$ | 100 | [0.50,0.86,0.52,0.30,0.74] |

$F{h}_{32}$ | 100 | [0.76,0.20,0.72,0.42,0.90] | |||

$F{h}_{33}$ | 100 | [0.96,0.36,0.10,0.62,0.52] |

**Table 2.**Hop number estimation result table. The table shows the estimation of the number of hops of frequency hopping signals under different SNRs.

SNR (dB) | Simulation Hop Number (${\mathit{N}}_{\mathit{o}}$) | Estimation Hop Number (${\mathit{N}}_{\mathit{e}}$) | Accuracy (%) |
---|---|---|---|

−10 | 15 | 30 | 30.42% |

−5 | 15 | 20 | 60.28% |

−4 | 15 | 16 | 70.43% |

−3 | 15 | 15 | 95.56% |

0 | 12 | 12 | 99.41% |

5 | 12 | 12 | 100% |

6 | 12 | 12 | 100% |

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**MDPI and ACS Style**

Wan, J.; Zhang, D.; Xu, W.; Guo, Q.
Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution. *Symmetry* **2019**, *11*, 648.
https://doi.org/10.3390/sym11050648

**AMA Style**

Wan J, Zhang D, Xu W, Guo Q.
Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution. *Symmetry*. 2019; 11(5):648.
https://doi.org/10.3390/sym11050648

**Chicago/Turabian Style**

Wan, Jian, Dianfei Zhang, Wei Xu, and Qiang Guo.
2019. "Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution" *Symmetry* 11, no. 5: 648.
https://doi.org/10.3390/sym11050648