# A Time-Frequency Domain Underdetermined Blind Source Separation Algorithm for MIMO Radar Signals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Assumption**

**1.**

## 3. Mixing Matrix Estimation

#### 3.1. Noise Preprocessing

#### 3.2. Mixing Matrix Estimation Based on Single Source Points

## 4. Sources Recovery Algorithm

#### 4.1. The Introduction of the Smoothed ${l}_{0}$ Norm

#### 4.2. Time-Frequency-Smoothed ${l}_{0}$ Norm Algorithm

- (1)
- Applying STFT on both sides of Equation (3), then we get $\mathbf{X}(t,f)=\mathbf{A}\mathbf{S}(t,f)+\mathbf{N}(t,f)$.
- (2)
- For the observed signal $\mathbf{X}(t,f)$, the SL0 algorithm is utilized to preliminarily reconstruct the TF information of sources $\mathbf{S}(t,f)=[{({S}_{1}(t,f),{S}_{2}(t,f),\cdots ,{S}_{N}(t,f)]}^{T}$ in the TF domain.
- (3)
- For the reconstructed n-$th$ source ${S}_{n}(t,f)$, the TF ridge ${r}_{n}\left(t\right)$ is obtained by extracting the maximum frequency elements at each time point in the TF domain. In essence, ${r}_{n}\left(t\right)$ is the TF coordinate values of the peak sequences, where time is the abscissa axis and peak frequency is the ordinate axis.
- (4)
- The median filter is selected to smooth processing for TF ridge and then get $\tilde{{r}_{n}}\left(t\right)$. In theory, each step of the TF ridge is flat. However, the interference of Gaussian noise in the environment leads to the impulse-noise while searching the optimal solution. Median filter has a good filtering effect on the impulse-noise, especially when the noise is filtered out, and it can protect the edges of the ridge instead of being blurred.
- (5)
- Taking the difference of the smoothed ${\tilde{r}}_{n}\left(t\right)$, we then get ${\tilde{r}}_{n}^{{}^{\prime}}\left(t\right)$. Set the threshold $\rho $, when the ${\tilde{r}}_{n}^{{}^{\prime}}\left(t\right)$ is numerically larger than $\rho $, and its corresponding times are regarded as the jumping-times. The period of coding $\widehat{T}$ can be obtained by using statistically averaging of the differences of the jumping-times.
- (6)
- From Equation (2), we can know that the frequency of each step is the product of coding frequency and $\Delta f$. $\Delta f=1/\widehat{T}$. Assuming that ${\widehat{F}}_{n}\left(t\right)$ is the estimated coding frequency, so ${\widehat{F}}_{n}\left(t\right)$ = ${\tilde{r}}_{n}\left(t\right)/\Delta f$. Meanwhile, it can be seen that the coding frequency ${\widehat{F}}_{n}\left(t\right)$ is an integer sequence. Therefore, the integer part of the ${\tilde{r}}_{n}\left(t\right)/\Delta f$ is kept to make approximations.
- (7)
- Estimate coding frequency in each coding period. In theory, ${\widehat{F}}_{n}\left(t\right)$ remains constant during a coding period; however, due to the effect of noise and errors caused by the approximations of process (6), ${\widehat{F}}_{n}\left(t\right)$ may appear to fluctuate. Therefore, take out the coded values with the most time steps within each coding period as the estimated coding frequency.
- (8)

## 5. Simulation Results and Analysis

#### 5.1. Algorithm Performance Evaluation Criteria

#### 5.2. Parameter Setting

#### 5.3. Experiment 1 and Analysis

#### 5.4. Experiment 2 and Analysis

#### 5.5. Experiment 3 and Analysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The scatter plots of the observed signals before single source point (SSP) detection. (

**a**) without noise pretreatment; (

**b**) after noise pretreatment.

**Figure 2.**The scatter plots of the observed signals after SSP detection. (

**a**) without noise pretreatment; (

**b**) after noise pretreatment.

**Figure 3.**Parameter setting. (

**a**) the proportion of remnant SSPs in all time-frequency (TF) points; (

**b**) the sum of the frequency differences corresponding to the jumping-times of four TF ridges.

**Figure 4.**The TF and time domains diagrams of the four sources. (

**a**) in TF domain; (

**b**) in time domain.

**Figure 5.**The TF and time domains diagrams of the observed signals. (

**a**) in TF domain; (

**b**) in time domain.

**Figure 6.**The TF and time domains diagrams of the recovery signals. (

**a**) in TF domain; (

**b**) in time domain.

**Figure 10.**Performances of the proposed and other algorithms to estimate mixing matrix in the noisy case. NMSE: Normalized mean square error.

**Figure 11.**Performances of the proposed and other algorithms to reconstruct signals in the noisy case. ASNR: Average recovered signal-to-noise ratio; SL0: Smoothed ${l}_{0}$ norm; OMP: Orthogonal matching pursuit; TF-SL0: Time-frequency smoothed ${l}_{0}$ norm.

**Table 1.**The parameter values by the setting method in Experiment 1 under different signal-to-noise ratios (SNRs).

SNR | 5 dB | 10 dB | 15 dB | 20 dB | 25 dB | 30 dB |
---|---|---|---|---|---|---|

$\eta $ | 0.02 | 0.035 | 0.045 | 0.05 | 0.052 | 0.055 |

$\rho $ ($\times {10}^{6}$) | 6.5 | 6.2 | 5.8 | 5.0 | 5.0 | 5.0 |

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**MDPI and ACS Style**

Guo, Q.; Ruan, G.; Liao, Y.
A Time-Frequency Domain Underdetermined Blind Source Separation Algorithm for MIMO Radar Signals. *Symmetry* **2017**, *9*, 104.
https://doi.org/10.3390/sym9070104

**AMA Style**

Guo Q, Ruan G, Liao Y.
A Time-Frequency Domain Underdetermined Blind Source Separation Algorithm for MIMO Radar Signals. *Symmetry*. 2017; 9(7):104.
https://doi.org/10.3390/sym9070104

**Chicago/Turabian Style**

Guo, Qiang, Guoqing Ruan, and Yanping Liao.
2017. "A Time-Frequency Domain Underdetermined Blind Source Separation Algorithm for MIMO Radar Signals" *Symmetry* 9, no. 7: 104.
https://doi.org/10.3390/sym9070104