# Performance Bound for Joint Multiple Parameter Target Estimation in Sparse Stepped-Frequency Radar: A Comparison Analysis

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## Abstract

**:**

## 1. Introduction

**Notation**

**1.**

## 2. Signal Model

## 3. CRLB of Basic LSF and SSF Signals—Time Delay And Doppler Stretch

#### 3.1. CRLB of SSF-Chirp Signals

#### 3.2. SSF-Rect Signals

## 4. CRLB of Joint Multiple Parameter Estimation

#### 4.1. Basic Model of CRLB

#### 4.2. Series Expressions of Partial Derivative

#### 4.3. Derivations of the FIM

## 5. CRLB of Sparse Based Estimator—Compressive Sensing

## 6. Experiments and Discussion

#### 6.1. CRLB Comparison with Different Waveforms

#### 6.2. Comparison of Different CRLBs

#### 6.3. CRLBs Using Different Estimators

#### 6.4. RD Comparison between Different Methods

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of CRLBs with different waveform. (

**a**) CRLB of time delay; (

**b**) CRLB of Doppler-stretch.

**Figure 2.**CRLB comparison with different parameters. (

**a**) different frequency step; (

**b**) different initial carrier frequency; (

**c**) different frequency number; (

**d**) different sparse frequency combination.

**Figure 3.**Comparison of CRLBs with changing SNR. (

**a**) Monte Carlo Comparison of CRLBs; (

**b**) Monte Carlo simulation of CRLB-$\mathit{\theta}$ with multiple variables.

**Figure 4.**Comparison of CRLBs using different model. (

**a**) CRLB using time delay and Doppler stretch estimator; (

**b**) CRLB using multiple parameter estimator; (

**c**) CRLB using the sparse based estimator.

**Figure 5.**Target RD estimation with different method. (

**a**) IDFT method; (

**b**) Correlation function method; (

**c**) compressive sensing method; (

**d**) actual location of targets as a reference (The theoretical values of the actual range and velocity location are the discrete points in (

**d**), which only provide references for estimation in (

**a**–

**c**)).

**Figure 6.**Monte Carlo tests of target estimation. (

**a**) MMSE of IDFT and CORR method; (

**b**) CRLB and MMSE of CS method.

Variables | Initial Value | the Value of ${\mathit{i}}^{\mathit{th}}$ Groups | Value Range |
---|---|---|---|

${\sigma}^{2}$ | ${({\sigma}^{2})}^{\left(0\right)}=1$ | ${({\sigma}^{2})}^{\left(i\right)}={10}^{2(i/20)}$ | $i=[1,50]$ |

$\Delta f$ (kHz) | $\Delta {f}^{\left(0\right)}=1$ | $\Delta {f}^{\left(i\right)}=\Delta {f}^{\left(0\right)}+\left(\right)open="("\; close=")">10i-1$ | $i=[1,50]$ |

${f}_{0}$ (MHz) | ${{f}_{0}}^{\left(0\right)}=1$ | ${{f}_{0}}^{\left(i\right)}={{f}_{0}}^{\left(0\right)}+\left(\right)open="("\; close=")">i-1$ | $i=[1,50]$ |

N | ${N}^{\left(0\right)}=40$ | ${N}^{\left(i\right)}={N}^{\left(0\right)}+(i-1)$ | $i=[1,50]$ |

T(ms) | ${T}^{\left(0\right)}=1$ | ${T}^{\left(i\right)}={T}^{\left(0\right)}+(i-1)$ | $i=[1,50]$ |

$\mathbf{C}$ | Random | Random | ${C}_{n}\in \left(\right)open="["\; close="]">1,M$ |

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## Share and Cite

**MDPI and ACS Style**

Chen, Q.; Zhang, X.; Yang, Q.; Ye, L.; Zhao, M.
Performance Bound for Joint Multiple Parameter Target Estimation in Sparse Stepped-Frequency Radar: A Comparison Analysis. *Sensors* **2019**, *19*, 2002.
https://doi.org/10.3390/s19092002

**AMA Style**

Chen Q, Zhang X, Yang Q, Ye L, Zhao M.
Performance Bound for Joint Multiple Parameter Target Estimation in Sparse Stepped-Frequency Radar: A Comparison Analysis. *Sensors*. 2019; 19(9):2002.
https://doi.org/10.3390/s19092002

**Chicago/Turabian Style**

Chen, Qiushi, Xin Zhang, Qiang Yang, Lei Ye, and Mengxiao Zhao.
2019. "Performance Bound for Joint Multiple Parameter Target Estimation in Sparse Stepped-Frequency Radar: A Comparison Analysis" *Sensors* 19, no. 9: 2002.
https://doi.org/10.3390/s19092002