# Adaptive Waveform Design for Cognitive Radar in Multiple Targets Situation

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## Abstract

**:**

## 1. Introduction

## 2. System Model

_{s}is the energy of transmitted waveform. Let ${\mathit{g}}_{i,k}$ represent the TIR of the i-th target in the k-th cycle. According to [28,30,31], the extended target behaves as a linear time-invariant filter with random complex impulse response, where the complex amplitude of every cell is a zero-mean complex Gaussian variable. When the target backscattering surface is assumed to be rough compared with the wavelength of the radar carrier, the variables of different range cells are unrelated to each other. We assume ${\mathit{g}}_{i,k}$ is a zero-mean Gaussian random vector with full rank covariance matrix ${\Sigma}_{i,\mathit{g}}$. The extended targets are described by the wide sense stationary uncorrelated scattering (WSSUS) model [28], and the TIR of the i-th target is:

## 3. Optimization of Waveform Design

#### 3.1. Target Impulse Response Tracking

#### 3.2. Maximum Detection Probability-Based Waveform Optimization Design

#### 3.3. Maximum MI-Based Waveform Optimization Design

Algorithm 1: The proposed waveform optimization design algorithm. |

S1: Transmit initial LFM signal ${\mathit{s}}_{0}$, where ${\overline{\mathit{S}}}_{0}=\mathit{F}{\mathit{s}}_{0}$, ${\mathit{S}}_{0}=diag\left({\overline{\mathit{S}}}_{0}\right)$. |

S2: Capture the signal reflected from the target and interference. The received waveform vector is ${\mathit{Y}}_{i,0}=\left({\mathit{S}}_{0}{\hat{\mathit{G}}}_{i,0|0}+{\mathit{S}}_{0}{\mathit{C}}_{0}+{\mathit{N}}_{0}\right){\mathit{H}}_{0}$. |

S3: Estimate the initialize the estimated TSC and MSE matrix of the target as ${\hat{\mathit{G}}}_{i,0|0}$ and ${\mathit{P}}_{i,0|0}$ by the received waveform vector ${\mathit{Y}}_{0}$ and transmit waveform matrix ${\mathit{S}}_{i,0}$. |

S4: Set the iteration index $k=1$, and the maximal number of iteration as ${K}_{\mathrm{max}}$. |

S5: while $k\le {K}_{\mathrm{max}}$ do |

S6: Predict the estimated TSC ${\hat{\mathit{G}}}_{i,k|k-1}$ according to ${\hat{\mathit{G}}}_{i,k-1|k-1}$ by (10). |

S7: Update the MSE matrix ${\mathit{P}}_{i,k|k-1}$ according to ${\mathit{P}}_{i,k-1|k-1}$ by (11). |

S8: Define ${\hat{\mathit{G}}}_{i,k}=dia\mathit{g}\left({\hat{\mathit{G}}}_{i,k|k-1}\right)$, solve for ${\left|{\overline{S}}_{k}\left(m\right)\right|}^{2}$. |

S9: Obtain ${\hat{\mathit{G}}}_{i,k|k}$ and ${\mathit{P}}_{i,k|k}$ with ${\mathit{S}}_{k}$, ${\mathit{Y}}_{i,k}$, ${\hat{\mathit{G}}}_{i,k|k-1}$ and ${\mathit{P}}_{i,k|k-1}$ by Kalman filter. |

S10: Let $k=k+1$. |

S11: end while |

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Power spectra: (

**a**) Power spectra of targets, clutter, and noise; (

**b**) Power spectra of detection probability-based waveforms; (

**c**) Power spectra of MI-based waveforms.

Waveform | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ |

Design1 | 0 | 1 |

Design2 | 0.5 | 0.5 |

Design3 | 1 | 0 |

**Figure 4.**Normalized MSE performance: (

**a**) Normalized MSE performance of detection probability-based waveform of Target1; (

**b**) Normalized MSE performance of detection probability-based waveform of Target2; (

**c**) Normalized MSE performance of MI-based waveform of Target1; (

**d**) Normalized MSE performance of MI-based waveform of Target2.

**Figure 5.**Joint normalized MSE performance: (

**a**) Joint normalized MSE performance of detection probability-based waveform; (

**b**) Joint normalized MSE performance of MI-based waveform.

**Figure 6.**SINR performance: (

**a**) SINR performance of detection probability-based waveform; (

**b**) SINR performance of MI-based waveform.

**Figure 7.**Detection performance: (

**a**) Detection performance of detection probability-based waveform; (

**b**) Detection performance of MI-based waveform.

**Figure 8.**MI performance: (

**a**) MI performance of detection probability-based waveform; (

**b**) MI performance of MI-based waveform.

Waveform | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ |

Design1 | 0 | 1 |

Design2 | 0.5 | 0.5 |

Design3 | 1 | 0 |

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**MDPI and ACS Style**

Zhang, X.; Liu, X.
Adaptive Waveform Design for Cognitive Radar in Multiple Targets Situation. *Entropy* **2018**, *20*, 114.
https://doi.org/10.3390/e20020114

**AMA Style**

Zhang X, Liu X.
Adaptive Waveform Design for Cognitive Radar in Multiple Targets Situation. *Entropy*. 2018; 20(2):114.
https://doi.org/10.3390/e20020114

**Chicago/Turabian Style**

Zhang, Xiaowen, and Xingzhao Liu.
2018. "Adaptive Waveform Design for Cognitive Radar in Multiple Targets Situation" *Entropy* 20, no. 2: 114.
https://doi.org/10.3390/e20020114