# An Inter-Subband Processing Algorithm for Complex Clutter Suppression in Passive Bistatic Radar

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Signal Model and Problem Analysis

#### 2.1. Signal Model

_{r}is the complex altitude and n

_{r}(t) is the white Gaussian noise.

_{0}is the complex altitude of the direct path signal by the side/back lobe of the surveillance signal. a

_{i}and τ

_{ci}are the complex amplitude and the propagation delay (in respect to direct path signal) of ith stationary scatterer component, respectively. Particularly, the delays τ

_{ci}involve fractional and integer multiple of the sampling period. N

_{c}is the number of scatterer components. b

_{m}, τ

_{m}and f

_{tm}are the complex amplitude, delay and Doppler frequency shift of the mth target, respectively. N

_{t}is the target number. n

_{s}(t) is the white Gaussian noise term in the surveillance signal.

_{cp}is the spread Doppler frequency; c

_{pq}and τ

_{pq}are the complex amplitude and delay of the qth time-varying clutter with doppler shift f

_{cp}, respectively. Similarly, τ

_{pq}involves a fractional and integer multiple of the sampling period. N

_{p}∙N

_{q}is the number of scatterers.

#### 2.2. PBR Target Detection

_{d}. Therefore, the CAF is also called as range-Doppler processing, and it can be given as

## 3. ECA and ECA-B Analysis

#### 3.1. ECA Analysis

_{s}. The nth sample is taken at time n/f

_{s}, and we can write the samples s(t) as

^{T}is the transpose operator.

**Γ**is the incidence matrix ensuring that only the last N rows of the following matrix are considered, which is shown as

**Λ**

_{F}is a diagonal matrix which is applied to add the Doppler shift information.

**r**

_{ref}is zero-Doppler projecting subspace written as

**Γ**is a permutation matrix to generate a one-sample-delayed reference signal matched to a clutter scatterer, defined as

**X**. By resorting to least square (LS) criterion, the filter weight to be estimated is calculated as

^{H}is Hermitian transpose operator. Therefore, the surveillance signal without clutter is achieved as

#### 3.2. ECA-B Analysis

_{B}batches, each batch contains N

_{B}= N/L

_{B}samples. The suppression result of the lth batch can be given as

**r**

_{l}; ${\widehat{\alpha}}_{l}$ is the filter coefficient in each batch. As the filter coefficients are updated in a short duration, the cancellation notch is extended in the frequency domain, which produces two improvements. The first is that the system becomes capable of rejecting time-varying clutter. Then, and secondly, the computational cost is relatively reduced, due to the subspace’s dimensional reduction.

_{B}, where T

_{B}is the batch duration. Figure 2 shows the frequency selectivity (the cancellation notch feature in frequency domain) of the ECA-B with different batch durations. In Figure 2, an FM signal is simulated as the direct path signal, at the sampling rate and CPI are f

_{s}= 200 kHz and T = 1 s, respectively. It can be shown that, to remove the clutter with a Doppler spectrum bandwidth of 1/T

_{B}, the received signal should be divided into smaller batches. However, a smaller batch duration will cause a series of periodic peaks and reduce the target SNR simultaneously; in the case of the slow-moving target, this is close to the transition zone. Figure 3 shows the RD result of a target located in the transition-band in which periodic peaks occur, seriously degrading the detection probability. In Figure 3, the batch duration T

_{B}is 10 ms, and the target is located at −28 Hz and in the 50th range bin.

## 4. Complex Clutter Suppression via ECA-FB and ECA-FBD

#### 4.1. ECA-FB

_{ci}, ${n}_{{\tau}_{ci}}={\tau}_{ci}\cdot {f}_{s}$; ${g}_{{f}_{dm}}$ and ${n}_{{\tau}_{m}}$ are the normalized Doppler and delay unit of the mth target with Doppler f

_{dm}and delay τ

_{m},${n}_{{\tau}_{m}}={\tau}_{m}\cdot {f}_{s}$.

_{L}. Each subband contains N

_{L}samples. The lth subband signal is expressed as

_{L}, L is the number of subbands. Note that the direct path signal and stationary clutter are merged in (18), ${n}_{{\tau}_{c0}}=0$ denotes the delay of the direct-path signal.

_{L}exceeds the maximum clutter distance, all the stationary clutter will occupy only one degree of freedom (DOF) in each subband. As a consequence, the influence of stationary clutter (including fractional-order clutter) can be regarded as exactly the same for all the frequency component at the same subband, and thus the lth subband surveillance signal can be written as

**W**

_{l}of each subband is independent of its delay to the direct path signal. Accordingly, the contributions of series of clutter for each frequency component g at the same lth subband can be combined to a complex amplitude correlated with

**W**

_{l}. In particular, we pay more attention to the subband surveillance signal; since the Doppler shift is derived from the target movement, the target signal spectrum ${\mathit{W}}_{{}_{l}}^{{f}_{dm}}$ can be regarded as uncorrelated with

**W**

_{l}. Therefore, applying the ECA to reject stationary clutter in each subband, the corresponding outputs are shown as

**S**

_{l}. Since the DOF of clutter is limited to 1, the subspace ${\tilde{X}}_{l}$ can be regarded as the subband reference signal

**R**

_{l}, formed by just one vector with the dimension N

_{L}× 1, as follows

**S**

_{sur}= [

**S**

_{sur}

_{-0},

**S**

_{sur}

_{-1}, ⋯,

**S**

_{sur}

_{-L-1}], and therefore coherent integration is performed to achieve target information via (5).

#### 4.2. ECA-FBD

_{cp}and delay τ

_{pq};

**N**

_{t}is the combination of noise and residual clutter.

_{cp}. Meanwhile, from the previous section, we know that the division of subbands can limit the DOF of clutter in the range dimension. Therefore, the time-varying projecting subspace can be formed by adding a continuous Doppler shift to the stationary clutter subspace ${\tilde{\mathbf{X}}}_{l}$. Specifically, in ECA-FBD method, the surveillance signal ${\tilde{\mathit{S}}}_{sur}$ in (25) and reference signal

**R**in (17) are redivided into L

_{D}subbands to limit the DOF of the time-varying clutter in the range dimension. The projecting subspace in the lth (l = 1,2,⋯, L

_{D}) subband is given as follows

_{D}can be set flexibly to adapt different time-varying environments. Usually, L

_{D}< L, due to the Doppler-shifted clutter being distributed over short ranges, as shown in Figure 4; otherwise far-distant targets with the same Doppler shift will also be suppressed. Therefore, the ECA-FBD is considered a cascade method of ECA-FB in the presence of a time-varying clutter environment. Additionally, the summarized framework of the proposed method is given in Algorithm 1 while the flowchart is shown in Figure 6.

Algorithm 1. The main processes of the proposed methods | |

1 | Input: Original surveillance signal s(t) and reference signal r(t), subbands number L, L_{D} and Doppler extension number F. |

2 | Discrete Fourier transform: Apply DFT on s(t) and r(t) to obtain S(g) and R(g), respectively. |

3 | Subband division: Divide the signal S[g] and R[g] into L fragments with bandwidth B_{L}. |

4 | Stationary clutter suppression: Go through each l in [1, L] to conduct ECA operation. |

5 | For l = 1, ⋯ , L do |

6 | Construct the one-dimensional clutter subspace ${\tilde{\mathbf{X}}}_{l}={[{R}_{l}[0],{R}_{l}[1],\cdots ,{R}_{l}[{N}_{L}-1]]}^{T}$ via (24). |

7 | Estimate the clutter coefficient ${\tilde{\alpha}}_{l}$. |

8 | Subtract the stationary clutter component to obtain the clutter suppressed signal S_{sur-l}. |

9 | end |

10 | Subband synthesis: Recombine the subband signal to achieve S_{sur}. |

11 | Subband redivision: Redivide the signal S_{sur} and R into L_{D} fragments. |

12 | Time-varying clutter suppression: Go through each l in [1, L_{D}] to conduct ECA operation. |

14 | For l = 1, ⋯ , L_{D} do |

15 | Construct the clutter subspace ${\tilde{\mathbf{X}}}_{D-l}=[{\tilde{\mathsf{\eta}}}_{-}{\mathit{R}}_{l},{\tilde{\mathsf{\eta}}}_{-}^{2}{\mathit{R}}_{l},\cdots ,{\tilde{\mathsf{\eta}}}_{-}^{F}{\mathit{R}}_{l},{\tilde{\mathsf{\eta}}}_{+}{\mathit{R}}_{l},{\tilde{\mathsf{\eta}}}_{+}^{2}{\mathit{R}}_{l},\cdots ,{\tilde{\mathsf{\eta}}}_{+}^{F}{\mathit{R}}_{l}]$ via (26)–(28). |

16 | Subtract the time-varying clutter component. |

17 | end |

18 | Output: Recombine the subband signal after removing time-varying clutter and then output it for coherent integration. |

## 5. Performance Analysis and Some Remarks

#### 5.1. Computational Complexity

_{2}(N) Mcs. Then, the signal in the frequency domain is divided into L subbands with the length of N

_{L}. In each subband, ECA is performed, i.e., ${\mathit{S}}_{sur-l}={\mathit{S}}_{l}-{\tilde{\mathbf{X}}}_{l}{({\tilde{\mathbf{X}}}_{l}^{H}{\tilde{\mathbf{X}}}_{l})}^{-1}{\tilde{\mathbf{X}}}_{l}^{H}{\mathit{S}}_{l}$ is repeated in each subband. For simplicity, we assume that ${x}_{l}={({\tilde{\mathbf{X}}}_{l}^{H}{\tilde{\mathbf{X}}}_{l})}^{-1}$ and ${y}_{l}={\tilde{\mathbf{X}}}_{l}^{H}{S}_{l}$. Note that both the calculation of x

_{l}and y

_{l}include N

_{L}Mcs, since the matrix ${\tilde{\mathbf{X}}}_{l}$ is a N

_{L}× 1 vector, and therefore 1 Mc is needed for achieving the filter coefficient, α

_{l}= x

_{l}∙y

_{l}. After that, the estimation of stationary clutter ${\tilde{\mathbf{X}}}_{l}{\alpha}_{l}$ requires N

_{L}Mcs. Last, the clutter is removed via subtracting ${\tilde{\mathbf{X}}}_{l}{\alpha}_{l}$ from S

_{l}. The overall computational cost of ECA-FB is Nlog

_{2}(N) + (3N

_{L}+ 1)L Mcs. As for ECA method, which has been appropriately optimized in [26], and 3N(1 + log

_{2}(N)) + K

^{2}(1 + log

_{2}(K)) Mcs are required. Due to the subspace dimension of ECA-FB being unrelated to the clutter order, its computational load is significantly reduced compared with ECA, especially for large clutter orders and long CPIs. Additionally, it is worth noting that the ECA-FB is more suitable in parallel processing since the matrix multiplications in each subband are much smaller and independent of each other.

_{D}subbands with the length of N

_{D}, and the time-varying projection matrix ${\tilde{\mathbf{X}}}_{D-l}$ is generated with dimension N

_{D}× 2F. Therefore, the computational complexities of ECA-FBD are L

_{D}[3N

_{D}(1 + log

_{2}(N

_{D}))+ F

^{2}((1 + log

_{2}(F)))] Mcs. Besides, the ECA-B needs L

_{B}[3 N

_{B}(1 + log

_{2}(N

_{B})) + K

^{2}((1 + log

_{2}(K)))] Mcs, in which the received signal is divided into L

_{B}batches, where the length of each batch is N

_{B}. In ECA-B, the projecting subspace dimension is related to the stationary clutter order K, which is usually much larger than 2F. Therefore, the computational complexity of ECA-FBD is lower than that of ECA-B.

_{s}= 200 kHz and the other relevant parameters of the clutter suppression are given as: K = 300, F = 20, L = 1000, L

_{D}= 100, L

_{B}= 40. The relationship between computational complexity and CPI for different suppression methods are shown in Figure 7. Note that the ECA-FBD is a cascade method of ECA-FB in the presence of time-varying clutter. On the contrary, the ECA-B can reject the stationary and time-varying clutter simultaneously. Therefore, in the time-varying clutter scenario, the sum of the computational complexity of the ECA-FB and ECA-FBD is used to compare with ECA-B method. It can be seen that the proposed methods have a relatively lower computational cost than that of the ECA and ECA-B, respectively.

#### 5.2. Some Remarks

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 6. Simulation Results

#### 6.1. Stationary Clutter Scenario

#### 6.2. Time-Varying Clutter Scenario

_{D}was set as 50. As is apparent, the time-varying clutter was rejected completely, and target A formed a unique peak without ambiguous Doppler peaks. Additionally, it can be seen from Figure 10f that the SNR of targets A and B are consistent with the desired system energy gain. In conclusion, considering the time-varying clutter environment, the proposed ECA-FBD can obtain not only better suppression performance, but also superior detection performance in the presence of slow-moving targets, as compared with ECA-B.

## 7. Experimental Results

#### 7.1. Experimental Results for the DTMB-Based PBR

#### 7.2. Experimental Results for the FM-Based PBR

_{D}, in ECA-FBD was set as 20 for covering short-range time-varying clutter. The corresponding results are given in Figure 15g,h, from which we see that the time-varying clutter is effectively reduced, and the SNR of targets at (169, 98.6 Hz) and (198, −20.5 Hz) are greatly improved. Moreover, two additional targets are clearly observed at (20, 57.6 Hz) and (238, 5.8 Hz), which means that the proposed methods were more effective, compared with ECA-B, in terms of detecting slow-moving targets.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Complex clutter (stationary time-varying clutter) spectrum of measured DTMB data: (

**a**) stationary clutter; (

**b**) time-varying clutter.

**Figure 5.**Zero Doppler cut of RD result after clutter cancellation via ECA-FB with different subband numbers.

**Figure 7.**The relationship between the computational complexity and CPI of different methods in stationary and time-varying clutter scenarios: (

**a**) stationary clutter scenario; (

**b**) time-varying clutter scenario.

**Figure 8.**RD results after integer-order clutter cancellation via ECA-FB and ECA: (

**a**) ECA-FB; (

**b**) ECA.

**Figure 9.**RD results after fractional-order clutter cancellation via ECA-FB and ECA: (

**a**) ECA-FB; (

**b**) ECA.

**Figure 10.**RD results before and after clutter cancellation via ECA-B and ECA-FBD: result before clutter cancellation in top view: (

**a**) and Doppler dimension (

**b**); result of ECA-B in top view (

**c**) and Doppler dimension (

**d**); result of ECA-FBD in top view (

**e**) and Doppler dimension (

**f**).

**Figure 12.**RD results before and after ECA cancellation: result before clutter cancellation in range dimension (

**a**) and Doppler dimension (

**b**); result of ECA cancellation in range dimension (

**c**) and Doppler dimension (

**d**).

**Figure 14.**RD results after clutter cancellation via ECA-FB: (

**a**) range dimension; (

**b**) Doppler dimension.

**Figure 15.**RD results after clutter suppression by different methods: result of ECA in the range dimension (

**a**) and the Doppler dimension (

**b**); result of ECA-B with batches duration 100 ms in the range dimension (

**c**) and the Doppler dimension (

**d**); result of ECA-B with batches duration 20 ms in the range dimension (

**e**) and the Doppler dimension (

**f**); results of ECA-FB&ECA_FBD in the range dimension (

**g**) and the Doppler dimension (

**h**).

Methods | Number of Mcs |
---|---|

ECA-FB | Nlog_{2}(N) + (3N_{L} + 1)L |

ECA-FBD | L_{D}[3N_{D}(1 + log_{2}(N_{D})) + F^{2}((1 + log_{2}(F)))] |

ECA | 3N(1 + log_{2}(N)) + K^{2}(1 + log_{2}(K)) |

ECA-B | L_{B}[3N_{B}(1 + log_{2}(N_{B})) + K^{2}((1 + log_{2}(K)))] |

Description | Parameter | Value |
---|---|---|

total subcarriers | - | 3780 |

carrier frequency | f_{c} | 666 MHz |

sample frequency | f_{s} | 8 MHz |

carrier spacing | ∆f | 2 kHz |

bandwidth | B | 7.56 MHz |

frame header mode | - | 1 |

CPI | T | 0.5 s |

Motion Parameters | Target A | Target B | Stationary Clutter | Time-Varying Clutter | |
---|---|---|---|---|---|

Integer Order | Fractional Order | ||||

range bins | 90 | 40 | 0:1:50 | 0.5:1:10.5 | 0:1:10 |

Doppler (Hz) | 100 | −20 | 0 | 0 | −10:2:10 |

CNR/SNR (dB) | −39 | −34 | 40:−1:−10 | 25:−2:5 | 10:−1:0 |

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**MDPI and ACS Style**

Zuo, L.; Wang, J.; Sui, J.; Li, N.
An Inter-Subband Processing Algorithm for Complex Clutter Suppression in Passive Bistatic Radar. *Remote Sens.* **2021**, *13*, 4954.
https://doi.org/10.3390/rs13234954

**AMA Style**

Zuo L, Wang J, Sui J, Li N.
An Inter-Subband Processing Algorithm for Complex Clutter Suppression in Passive Bistatic Radar. *Remote Sensing*. 2021; 13(23):4954.
https://doi.org/10.3390/rs13234954

**Chicago/Turabian Style**

Zuo, Luo, Jun Wang, Jinxin Sui, and Nan Li.
2021. "An Inter-Subband Processing Algorithm for Complex Clutter Suppression in Passive Bistatic Radar" *Remote Sensing* 13, no. 23: 4954.
https://doi.org/10.3390/rs13234954