# Joint Design of Complementary Sequence and Receiving Filter with High Doppler Tolerance for Simultaneously Polarimetric Radar

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## Abstract

**:**

## 1. Introduction

- (1)
- Based on the AF, the joint design of unimodular orthogonal CCS (UOCCS) and receiving filter is proposed for SPR waveforms. Specifically, the complementary integrated sidelobe level (CISL) of Auto-AFs, the complementary integrated isolation level (CIIL) of Cross-AFs, and the mismatch constraint with controllable SNR loss are all considered simultaneously in the objective function formulated for optimization. By setting the predefined SNR loss, a trade-off between the suppression of CISL/CIIL and actual SNR loss can be achieved. In other words, the work in [41,42] was extended, i.e., the proposed scheme not only considers the low sidelobe of the pulse compression of CCS but also takes into account the orthogonality for all time delays within appropriate DFR.
- (2)
- The joint design problem is decomposed into subproblems of waveform design and receiving filter design via theoretical derivation, which is solved via an alternatively iterative approach. Concretely, the two subproblems are transformed into nonconvex quadratic terms containing the Hermitian matrix. The MM method is then applied to transforming these two nonconvex quadratic terms into linear programming problems with closed solutions. By utilizing the characteristics of Toeplitz matrix-vector multiplication, the main computation step can be completed via FFT. For further improvement, the convergence speed of the algorithm, an acceleration scheme of the squared iterative method (SQUAREM) is introduced. Compared with the representative and latest MM-CCS method [34] and the L-BFGS algorithm [35], better performance is achieved by the proposed algorithm, benefiting from both the joint design and the application of the MM framework.

## 2. Problem Statement

## 3. Joint Design of UOCCS and Receiving Filters via MM Method

#### 3.1. Reformulation of the Problem

#### 3.2. Joint Optimization via the MM Method

**Lemma**

**1**

**Lemma**

**2**

Algorithm 1. Joint Design CCSs and Receiving Filters with Expected AFs Shape Based on the MM Method via Alternately Iteration |

Initialize: $l=0$, pulse number $K$, sequence and filter length $N$, a predefine SNR loss $\mu $, the DFR as $\left[{f}_{1},{f}_{2}\right]$.1: Compute the predefined ${a}_{max}$ 2: Initialize ${\mathbf{u}}^{\left(0\right)}$, ${\mathbf{v}}^{\left(0\right)}$ of length $2L$ as (11) and (12) 3: compute ${\gamma}_{max}\left({\mathit{Q}}_{m\stackrel{~}{m}q}\right)$ by (28) 4: repeat5: Compute ${\mathbf{y}}_{\stackrel{~}{m}}$ by (34) with the designated ${\mathbf{u}}^{\left(l\right)}$ 6: Update the receiving filters ${{\mathbf{v}}_{1}}^{\left(l+1\right)}$ and ${{\mathbf{v}}_{2}}^{\left(l+1\right)}$ by (35) 7: Compute ${\mathbf{z}}_{m}$ by (40) with the designated ${\mathbf{v}}^{\left(l+1\right)}$ 8: Update the CCSs ${{\mathbf{u}}_{1}}^{\left(l+1\right)}$ and ${{\mathbf{u}}_{2}}^{\left(l+1\right)}$ by (41) 9: $l=l+1$; 10: $\mathbf{until}\mathbf{convergence}$ Output: CCS $\mathbf{u}$ and receiving filter $\mathbf{v}$ with expected AFs. |

#### 3.3. Acceleration Scheme Using SQUAREM

Algorithm 2. The Acceleration Scheme for Algorithm 1 using SQUAREM. |

1: Initialize: $l=0$, ${\mathit{x}}^{\left(0\right)}$2: repeat3: ${\mathbf{x}}_{1}={\mathcal{F}}_{MM}\left({\mathbf{x}}^{\left(l\right)}\right)$ 4: ${\mathbf{x}}_{2}={\mathcal{F}}_{MM}\left({\mathbf{x}}_{1}\right)$ 5: $\mathit{y}={\mathbf{x}}_{1}-{\mathbf{x}}^{\left(l\right)}$ 6: $\mathit{z}={{\mathbf{x}}_{2}-\mathbf{x}}_{1}-\mathit{y}$ 7: Compute the step-length $\alpha =-\Vert \mathit{y}\Vert /\Vert \mathit{z}\Vert $ 8: $\mathbf{x}={\mathcal{P}}_{x}\left({\mathbf{x}}^{\left(l\right)}-2\alpha \mathit{y}+{\alpha}^{2}\mathbf{z}\right)$ 9: while $f\left(\mathbf{x}\right)>f\left({\mathbf{x}}^{\left(l\right)}\right)$ do10: $\alpha \leftarrow \left(\alpha -1\right)/2$ 11: $\mathbf{x}={\mathcal{P}}_{x}\left({\mathbf{x}}^{\left(l\right)}-2\alpha \mathit{y}+{\alpha}^{2}\mathbf{z}\right)$ 12: $\mathbf{e}\mathbf{n}\mathbf{d}\mathbf{w}\mathbf{h}\mathbf{i}\mathbf{l}\mathbf{e}$ 13: ${\mathbf{x}}^{\left(l+1\right)}=\mathbf{x}$ 14: $l\leftarrow l+1$ 15: until convergence. |

## 4. Simulations and Performance Analysis

#### 4.1. Performance of the Proposed Method

**Example 1**: In this example, we show the convergence performances of the MM-CSRF algorithm and that of the L-BFGS algorithm. Suppose $N=64,$ $K=1$6, and the normalized DFR is $f\in [-0.2,0.2]$, which are the same as those in [35]. For the MM-CSRF algorithm, the weight factor is $\lambda =10$, and the predefined SNR loss is $\mu =0.5\mathrm{dB}$. For both algorithms, the iteration stops when the NCIAL in (47) is less than −35 dB or after 10,000 s of processing.

**Example 2:**In this example, the CCSs and receiving filters for zero-Doppler shift are designed by the MM-CSRF algorithm with N = 64 and without the SNR loss, i.e., $\mu =0\mathrm{d}\mathrm{B}$, and the weight factor is $\lambda =10$. The processing stops after 10,000 iterations. A group of zero-Doppler cuts of the AF of CCS sets with K = 2, 4, 5, 6 is given in Figure 3.

**Example 3**: In this example, the influence of the weighting factor $\lambda $ on the NCIAL of the designed CCS is investigated with the predefined SNR loss $\mu =0.5\mathrm{d}\mathrm{B}$. Same as before, N = 64, K = 5, and the normalized DFR $f\in \left[-0.2,0.2\right]$ are used. The waveform is optimized via 10,000 iterations. Here, $\lambda =[0.10.20.5125102050100200500100020005000]$.

#### 4.2. Doppler Effect Analysis

**Example 4**: In this subsection, we compare the performance of UOCCSs designed by different algorithms on Doppler resilience. Figure 5 shows the AFs of CCS designed by the MM-CCS algorithm [34], the L-BFGS algorithm [35], and our algorithm. The subfigures in Figure 5a–f shows the zero-Doppler cuts of the AFs. For all algorithms, the waveforms are optimized by 10,000 iterations or 600 s. Here, N = 64, K = 16, the normalized DFR $f\in \left[-0.2,0.2\right]$, $\mu =0.5$, and $\lambda =10$ are used.

## 5. Experimental Validation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**(

**a**) Transmitting and receiving radar signal by simultaneously polarimetric radar. (

**b**) Signal-processing procedure [28].

**Figure 4.**Influences of the weight $\lambda $ and the SNR loss $\mu $ on the NCIAL suppression. (

**a**) Influence of the weighting factor $\lambda $; (

**b**) Influence of the SNR loss $\mu $; (

**c**) NCIAL evolution curves with respect to time.

**Figure 8.**Sampled waveforms by A/D convertor from our waveforms. (

**a**) In-phase components in the time domain; (

**b**) Spectrums of the sampled signals.

Notation | Meaning |
---|---|

${\left(\xb7\right)}^{T}$ | the transposition of a vector/matrix |

${\left(\xb7\right)}^{*}$ | the conjugate of a complex number/vector/matrix |

${\left(\xb7\right)}^{\u2020}$ | the conjugate transpose of a vector/matrix |

$\left|\xb7\right|$ | the modulus of a complex number |

$\Vert \xb7\Vert $ | the ${l}_{2}$-norm of a vector |

$\mathrm{R}e\left(\xb7\right)$ | the real part of a complex number |

$\mathrm{T}\mathrm{r}\left(\xb7\right)$ | the trace of a matrix |

$\mathrm{v}\mathrm{e}\mathrm{c}\left(\xb7\right)$ | the stacking vectorization of a matrix |

$\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathit{x}\right)$ | a diagonalized matrix of $\mathit{x}$ |

⨀ | Hadamard product |

${\mathbf{0}}_{\mathit{N}}$ | an all-zero column vector with dimension N |

${\mathbf{1}}_{\mathit{N}}$ | an all-one column vector with dimension N |

${\mathbf{0}}_{\mathrm{L}\times \mathrm{L}}$ | an all-zero matrix with dimension $L\times L$ |

${\mathbf{I}}_{\mathrm{L}}$ | the identity matrix with dimension $L\times L$ |

Parameter | Quantity |
---|---|

Carrier Frequency ${f}_{c}$ | 13.58 GHz |

Waveform bandwidth B | 40 MHz |

Pulse duration ${T}_{p}$ | $1.6\mathsf{\mu}\mathrm{s}$ |

Pulse PRI ${T}_{r}$ | $2.1\mathsf{\mu}\mathrm{s}$ |

CPI | $10\mathsf{\mu}\mathrm{s}$ |

Code length $N$ | 64 |

Pulse number $K$ | 5 |

A/D sampling rate | 500 MHz |

ADC resolution | 12 bit |

Waveform of [34] | Waveform of [35] | Our Waveform | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

HH | HV | VH | VV | HH | HV | VH | VV | HH | HV | VH | VV | ||

${f}_{q}=0$ | Simulated 1 | −61.23 | −60.19 | −60.19 | −60.65 | −34.10 | −33.56 | −33.56 | −34.20 | −42.91 | −42.29 | −41.56 | −42.87 |

Simulated 2 | −40.81 | −39.85 | −41.02 | −40.96 | −32.17 | −33.20 | −32.28 | −32.98 | −38.11 | −38.46 | −39.29 | −38.68 | |

Measured | −39.41 | −39.32 | −38.41 | −39.17 | −31.81 | −31.71 | −31.44 | −31.96 | −37.54 | −37.13 | −38.17 | −37.76 | |

${f}_{q}=0.07$ | Simulated 1 | −38.83 | −37.74 | −38.06 | −38.31 | −32.66 | −32.70 | −32.78 | −32.73 | −42.04 | −41.72 | −40.10 | −43.17 |

Simulated 2 | −35.80 | −35.36 | −34.02 | −37.27 | −32.01 | −32.45 | −32.03 | −32.66 | −38.41 | −39.45 | −38.84 | −39.62 | |

Measured | −34.17 | −35.55 | −33.78 | −36.34 | −31.01 | −30.88 | −31.76 | −32.05 | −37.90 | −38.13 | −36.88 | −38.73 | |

${f}_{q}=0.14$ | Simulated 1 | −32.89 | −31.70 | −31.10 | −32.60 | −31.18 | −30.93 | −31.32 | −32.45 | −41.65 | −41.34 | −39.16 | −42.55 |

Simulated 2 | −32.19 | −30.56 | −29.72 | −33.04 | −30.90 | −30.47 | −30.36 | −32.29 | −38.61 | −38.73 | −37.59 | −37.53 | |

Measured | −30.04 | −30.64 | −30.39 | −32.47 | −30.64 | −30.35 | −30.02 | −30.19 | −38.29 | −37.82 | −36.53 | −37.16 | |

${f}_{q}=0.21$ | Simulated 1 | −29.58 | −28.37 | −27.80 | −29.25 | −30.00 | −30.11 | −30.2 | −33.25 | −41.27 | −40.39 | −38.28 | −41.13 |

Simulated 2 | −29.18 | −28.06 | −27.26 | −29.51 | −29.13 | −29.74 | −29.18 | −32.73 | −38.42 | −38.97 | −38.42 | −37.96 | |

Measured | −27.90 | −28.0 | −27.19 | −29.47 | −29.54 | −29.33 | −29.42 | −30.43 | −37.36 | −36.73 | −37.33 | −36.81 | |

${f}_{q}=0.28$ | Simulated 1 | −27.05 | −26.19 | −26.85 | −26.96 | −29.93 | −29.83 | −29.13 | −31.06 | −40.62 | −39.51 | −38.96 | −39.87 |

Simulated 2 | −26.84 | −26.30 | −25.56 | −27.12 | −28.98 | −29.56 | −28.65 | −30.91 | −38.35 | −38.54 | −38.59 | −37.49 | |

Measured | −25.70 | −26.00 | −25.57 | −26.25 | −29.28 | −29.87 | −28.19 | −29.22 | −36.88 | −37.08 | −37.89 | −36.30 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Zhang, Y.; Li, D.; Yang, J.
Joint Design of Complementary Sequence and Receiving Filter with High Doppler Tolerance for Simultaneously Polarimetric Radar. *Remote Sens.* **2023**, *15*, 3877.
https://doi.org/10.3390/rs15153877

**AMA Style**

Chen Y, Zhang Y, Li D, Yang J.
Joint Design of Complementary Sequence and Receiving Filter with High Doppler Tolerance for Simultaneously Polarimetric Radar. *Remote Sensing*. 2023; 15(15):3877.
https://doi.org/10.3390/rs15153877

**Chicago/Turabian Style**

Chen, Yun, Yunhua Zhang, Dong Li, and Jiefang Yang.
2023. "Joint Design of Complementary Sequence and Receiving Filter with High Doppler Tolerance for Simultaneously Polarimetric Radar" *Remote Sensing* 15, no. 15: 3877.
https://doi.org/10.3390/rs15153877