# Sub-Nyquist SAR Based on Pseudo-Random Time-Space Modulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}relaxation methods [26,27,28,29], and Bayesian-based methods [30,31,32] can be used under the assumption of satisfying the RIP. This paper mainly adopts l

_{1}relaxation methods with small recovery error compared with the other two algorithms [33].

## 2. SAR Image Reconstruction Based on CS

_{2}-norm of vector [37], ${D}_{\mathsf{\Lambda}}$ denotes the sub-matrix formed from the columns of $D$ indexed by $\mathsf{\Lambda}$, and ${\mathsf{\sigma}}_{\mathsf{\Lambda}}$ denotes the sub-vector formed from the rows of $\mathsf{\sigma}$ indexed by $\mathsf{\Lambda}$. The RIP essentially explains that the distance between two signals $D\mathsf{\sigma}$ and $D{\mathsf{\sigma}}^{\prime}$ is proportional to the distance between the two signals $\mathsf{\sigma}$ and ${\mathsf{\sigma}}^{\prime}$, it can guarantee the exact reconstruction. The sparsity $S$, the non-zero number of the unrecovered signal ${\mathsf{\sigma}}_{M\times 1}$, is introduced to express the RIP definition. The fewer the number of non-zero elements, the sparser the signal is and the more easily the RIP is satisfied. Additionally, the improvement of RIP condition contributes to enlarging the sparsity so that the reconstructed scene is not only limited to the sparse type.

_{1}relaxation methods, e.g., the Dantzig selector [26] and basis pursuit denoising (BPDN) [27,28,29]; and the third is Bayesian-based methods, which include maximum a posteriori (MAP) estimation and Hierarchical Bayesian framework [30,31,32]. The greedy method is faster than the other two methods in term of the recovery time, l

_{1}relaxation method performs better in term of small recovery error, Bayesian method balances between small recovery error and short recovery time [33]. This paper mainly adopts l

_{1}relaxation methods with small recovery error. The optimization equation of Equation (3) without matrix subscript is

## 3. Information Channel Model of Microwave Imaging Radar Based on Information Theory

## 4. Pseudo-Random Space-Time Modulation

#### 4.1. Pseudo-Random Space-Time Modulation

#### 4.2. Choice of Pseudo-Random Space-Time Modulation

_{1}and information sink Y is to improve this modulation. In this modulation, the phase of complex exponent is a stochastic variable. The distribution of stochastic variable has many distributions, e.g., uniform distribution or Gaussian distribution, and different distributions lead to different reconstruction performance. While the ratio of the maximum to minimum non-zero singular values is small, i.e., β

_{max}/β

_{min}→1, it can transfer more scene information, where β

_{max}and β

_{min}are the maximum and minimum non-zero singular values of random modulation matrix, respectively. It can be explained by the restricted isometry constant δs. The RIP can be simplified as (1 − δs) ≤ β

_{min}

^{2}≤ β

_{max}

^{2}≤ (1 + δs), which implies that the smaller restricted isometry constant δs leads to the more exact reconstruction, and so it also verifies that the smaller value of β

_{max}/β

_{min}has the better reconstructed performance from the perspective of restricted isometry constant δs. Additionally, singular value decomposition explains that singular value corresponds to the information implied in the matrix and the importance of information is positively correlated with the magnitude of the singular value. Therefore, the maximum non-zero singular value β

_{max}should be improved to maintain the scene details. During the quantitative analysis of β

_{max}/β

_{min}and β

_{max}, the mutual coherence coefficient u is a widely used variable to be introduced [42], this coefficient explains the maximum similarity between any two columns in ${D}_{N\times M}$ and is quantitatively denoted as:

_{max}/β

_{min}and β

_{max}are derived to (see Appendix A)

_{max}/β

_{min}is and the exacter the whole reconstruction is. Equation (9) denotes that the smaller the mutual coherence u is, the smaller β

_{max}is and the less the scene details are maintained. Therefore, it is undesirable for that the distribution of random phase to choose u with a large or small value; rather, it should try to maintain the scene details under guaranteeing the whole error. Equations (8) and (9) indicate the choice of pseudo-random space-time modulation. The sparsity achieved by different modulations can refer to Reference [43].

#### 4.3. Carrier of Pseudo-Random Space-Time Modulation

## 5. Sub-Nyquist SAR Based on Pseudo-Random Space-Time Modulation

#### 5.1. Choice of Sub-Nyquist Sampling Method

#### 5.2. Echo Signal Model after Pseudo-Random Space-Time Modulation

_{i},y

_{i}). The different scene position (x

_{i},y

_{i}) corresponding to different beam pointing has different time-varying random phase ${\phi}_{i}\left(\eta \right)$.

#### 5.3. Reconstructed Method after Pseudo-Random Space-Time Modulation

#### 5.4. Reconstructed Performance

## 6. Validation and Analysis

_{u}, u

_{g}, β

_{max_u}, β

_{max_g}, β

_{min_u}, and β

_{min_g}, respectively. As illustrated in Figure 9, u

_{u}is larger than u

_{g}in Figure 9c, β

_{max_u}and β

_{max_u}/β

_{min_u}are larger than β

_{max_g}and β

_{max_g}/β

_{min_g}in Figure 9d, respectively, and more road details are maintained in the red ellipsoid of Figure 9a compared with Figure 9b. It is consistent with Equations (8) and (9) that random phase distribution with large mutual coherence leads to many scene details under guaranteeing the whole error. Pseudo-random space-time can improve the imaging performance, and Equations (8) and (9) indicate the choice of the modulation.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

- (1)
- the calculation of the data information matrices ${J}_{\mathrm{D}}$:$$\begin{array}{l}{J}_{\mathrm{D}}=-E\left[\frac{{\partial}^{2}\mathrm{ln}{p}_{p/\mathsf{\sigma}}\left(p/\mathsf{\sigma}\right)}{\partial {\mathsf{\sigma}}_{m1}\partial {\mathsf{\sigma}}_{m2}}\right]=E\left[\frac{1}{2{\sigma}_{n}{}^{2}}\xb7\frac{\partial {\Vert p-\mathsf{\phi}\mathsf{\sigma}\Vert}_{2}^{2}}{\partial {\mathsf{\sigma}}_{m1}\partial {\mathsf{\sigma}}_{m2}}\right]\\ =E\left\{\frac{1}{2{\sigma}_{n}{}^{2}}\xb7\frac{\partial}{\partial {\mathsf{\sigma}}_{m1}\partial {\mathsf{\sigma}}_{m2}}\left[{p}^{H}p-{p}^{H}\mathsf{\phi}\mathsf{\sigma}-{\mathsf{\sigma}}^{H}{\mathsf{\phi}}^{H}\mathsf{\sigma}+{\mathsf{\sigma}}^{H}{\mathsf{\phi}}^{H}\mathsf{\phi}\mathsf{\sigma}\right]\right\}\\ =E\left\{\frac{1}{2{\sigma}_{n}{}^{2}}\xb7\frac{\partial}{\partial {\mathsf{\sigma}}_{m2}}\left[2{\mathsf{\phi}}^{H}\mathsf{\phi}\mathsf{\sigma}-2{\mathsf{\phi}}^{H}p\right]\right\}\\ =\frac{{\mathsf{\phi}}^{H}\mathsf{\phi}}{{\sigma}_{n}{}^{2}}\end{array}$$
- (2)
- the calculation of the prior information matrix ${J}_{\mathrm{P}}$:$$\begin{array}{l}{J}_{\mathrm{P}}=-E\left[\frac{{\partial}^{2}\mathrm{ln}{p}_{\mathsf{\sigma}}\left(\mathsf{\sigma}\right)}{\partial {\mathsf{\sigma}}_{m1}\partial {\mathsf{\sigma}}_{m2}}\right]=E\left[\frac{1}{2{\sigma}_{x}{}^{2}}\xb7\frac{\partial {\Vert \mathsf{\sigma}\Vert}_{2}^{2}}{\partial {\mathsf{\sigma}}_{m1}\partial {\mathsf{\sigma}}_{m2}}\right]\\ =\frac{I}{{\sigma}_{x}{}^{2}}\end{array}$$$$E\left\{{\left(\widehat{\mathsf{\sigma}}-\mathsf{\sigma}\right)}^{2}\right\}\ge trace\left[{\left(\frac{{\mathsf{\phi}}^{H}\mathsf{\phi}}{{\sigma}_{n}^{2}}+\frac{I}{{\sigma}_{x}^{2}}\right)}^{-1}\right],$$

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**Figure 1.**(

**a**) Reconstructed result of ocean containing ships from Nyquist samples based on chirp scaling; (

**b**) Reconstructed result of ocean containing ships from sub-Nyquist samples based on compressive sensing (CS); (

**c**) Reconstructed result of an urban scene from Nyquist samples based on chirp scaling; (

**d**) Reconstructed result of an urban scene from sub-Nyquist samples based on CS.

**Figure 2.**Analogous to the communication system in information theory, the scene, the mapping relation, and the echo correspond to information source, channel, and sink, respectively.

**Figure 3.**Channel capacity against signal-to-noise ratio (SNR) at different sampling frequencies. As the sampling frequency decreases, channel capacity decreases.

**Figure 4.**The information channel after pseudo-random space-time modulation. Information channel is based on Doppler movement under the sub-Nyquist sampling method. After pseudo-random space-time modulation, the mutual information between information source X

_{1}and information sink Y is increased so that the scene with large value of sparsity is reconstructed.

**Figure 5.**The amplitude and phase patterns of the phased array antenna and compressive reflector antenna (CRA), compared with the linear phase of the phased array antenna; the phase of the CRA is random.

**Figure 6.**Illustration of sub-Nyquist sampling method for sampling duration T

_{1}. The dashed line denotes the Nyquist sample. The solid line denotes the sub-Nyquist sample in random and uniform sub-Nyquist sampling method randomly and uniformly selects from dashed line, respectively. (

**a**) random sub-Nyquist sampling: the conflict between transmitted pulse and receiving echo. (

**b**) uniform sub-Nyquist sampling: echo is completely received and has no conflict with the transmitted pulse.

**Figure 7.**(

**a**) Reconstructed result from Nyquist samples; (

**b**) Reconstructed result from sub-Nyquist samples; (

**c**) Reconstructed result from sub-Nyquist samples after the pseudo-random space-time modulation.

**Figure 9.**Simulated results of (

**a**,

**b**) reconstructed scene, (

**c**) mutual coherence coefficient, and (

**d**) singular value.

Parameter | Values of the Nyquist Sampling Method | Values of the Sub-Nyquist Sampling Method |
---|---|---|

Orbital height (km) | 693 | 693 |

Wavelength (m) | 0.0555 | 0.0555 |

Pulse width (µs) | 50 | 50 |

Antenna height (m) | 8.93 | 8.93 |

Signal bandwidth (MHz) | 100 | 100 |

The sampling frequency (MHz) | 110 | 110 |

Incidence angles (°) | 38.73–43.50 | 38.73–43.50 |

(Average) pulse repetition frequency (PRF) (Hz) | 1946 | 278 |

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## Share and Cite

**MDPI and ACS Style**

Chen, W.; Li, C.; Yu, Z.; Xiao, P.
Sub-Nyquist SAR Based on Pseudo-Random Time-Space Modulation. *Sensors* **2018**, *18*, 4343.
https://doi.org/10.3390/s18124343

**AMA Style**

Chen W, Li C, Yu Z, Xiao P.
Sub-Nyquist SAR Based on Pseudo-Random Time-Space Modulation. *Sensors*. 2018; 18(12):4343.
https://doi.org/10.3390/s18124343

**Chicago/Turabian Style**

Chen, Wenjiao, Chunsheng Li, Ze Yu, and Peng Xiao.
2018. "Sub-Nyquist SAR Based on Pseudo-Random Time-Space Modulation" *Sensors* 18, no. 12: 4343.
https://doi.org/10.3390/s18124343