舰船科学技术  2019, Vol. 41 Issue (4): 124-128 PDF

A waveform design method for reducing sidelobe level in a particular area of ambiguous function
ZHOU Fei, CHENG Jin-sheng
Shanghai Marine Electronic Equipment Research Institute, Shanghai 201108, China
Abstract: When detecting low velocity moving targets, Polyphase pulse has high peak sidelobe that will cause the loss of weak target detection.To solve this Problem, a waveform design method for reducing sidelobe level in a particular area of ambiguous function(AF) was proposed.This method take advantage of the low velocity moving targets with very small doppler frequency shift, simplified the ambiguous function.we put forward a conceptual for Discrete Ambiguous Function(D-AF).The foregoing discussion implies that by minimizing the correlations of AF, we equivalently minimize the D-AF sidelobes. The minimization problem can be solved by the cyclic algorithm described in this paper. Numerical simulation showed that the proposed algorithm sidelobe suppression for designing sequences with a thumbtack-shaped discrete-AF near the origin of the delay-Doppler plane effectively, meanwhile, it improved the detection ability of weak target detection.
Key words: sidelobe suppression     polyphase pulse     waveform design     discrete ambiguous function
0 引　言

1 离散模糊度函数 1.1 相位编码脉冲

 $x\left( n \right) = {e^{j{\phi _n}}}{\text{，}}$ (1)

 $\sum\limits_{n = 1}^N {{{\left| {x\left( n \right)} \right|}^2}} = N{\text{。}}$ (2)

 $s\left( t \right) = \sum\limits_{n = 1}^N {x\left( n \right){p_n}} \left( t \right){\text{，}}$ (3)

 ${\int_0^{{T_p}} {\left| {p\left( t \right)} \right|} ^2}{\rm d}t = 1{\text{，}}$ (4)

 $s\left( t \right) = \sum\limits_{n = 0}^{N - 1} {x\left( n \right)} p\left( {t - n{T_r}} \right){\text{，}}$ (5)

 ${v_0}\left( t \right) = {a_t}s\left( t \right)\exp \left[ {j\left( {2{\text{π}} {f_0}t + \phi } \right)} \right] {\text{。}}$ (6)

1.2 离散模糊度函数

 $\begin{split} & \chi \left( {\tau ,f} \right) = \int_0^T {\left( {\sum\limits_{n = 1}^N {x\left( n \right){p_n}\left( t \right)} } \right)} \left( {\sum\limits_{m = 1}^N {{x^*}\left( m \right){p_m}\left( {t - \tau } \right)} } \right)\times\\ & {e^{j2{\text{π}} f\left( {t - \tau } \right)}}{\rm d}t = \\ &\sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {{x^*}\left( m \right)\left( {\int_0^T {{p_n}\left( t \right){p_m}\left( {t - \tau } \right){e^{j2{\text{π}} f\left( {t - \tau } \right)}}{\rm d}t} } \right)} } x\left( n \right){\text{，}} \end{split}$ (7)

 $\begin{split} \!\! \chi \left( {k{t_p},f} \right) = &\sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {{x^*}\left( m \right)\left( {\int_{\left( {n - 1} \right){t_p}}^{n{t_p}} {{{\left| {{p_n}\left( t \right)} \right|}^2}{e^{j2{\text{π}} f\left( {t - k{t_p}} \right)}}{\rm d}t} } \right)}\times }\!\!\!\!\!\!\!\!\!\!\! \\ & {\delta _{m + k,n}}x\left( n \right) = {\frac{{{e^{j{\text{π}} f{t_p}}}\sin \left( {{\text{π}} f{t_p}} \right)}}{{{\text{π}} f{t_p}}}} \times \\ &\sum\limits_{n = 1}^N {x\left( n \right)} {x^*}\left( {n - k} \right){e^{ - j2{\text{π}} f{t_p}\left( {n - k} \right)}}{\text{，}} \end{split}$ (8)

 $\chi \left( {k{t_p},\frac{p}{{N{t_p}}}} \right) = {e^{j{\text{π}} \frac{p}{N}}}\sin c\left( {{\text{π}}\frac{p}{N}} \right)\bar r\left( {k,p} \right){\text{，}}$ (9)

 $\bar r\left( {k,p} \right) = \sum\limits_{n = 1}^N {x\left( n \right)} {x^*}\left( {n - k} \right){e^{ - j2{\text{π}} f{t_p}\left( {n - k} \right)}}{\text{，}}$ (10)

$k = - N + 1, \cdots ,0, \cdots N - 1$ $p = - N/2, \cdots ,N/2 - 1$

 $\left| {\chi \left( {k{t_p},\frac{p}{{N{t_p}}}} \right)} \right| \approx \bar r\left( {k,p} \right){\text{，}}$ (11)

2 抑制离散模糊度函数原点附近旁瓣级的相位编码设计

 $\mathop {\min }\limits_{\left\{ {x\left( n \right)} \right\}} {C_1} = {\sum\limits_{k \in \kappa } {\sum\limits_{p \in \rho } {\left| {\bar r\left( {k,p} \right)} \right|} } ^2}{\text{，}}$ (12)

 $\begin{split} &\left\{ {{x_1}\left( n \right) = x\left( n \right)} \right\}_{n = 1}^N{\text{，}} \\ &\left\{ {{x}_{2}}\left( n \right)=x\left( n \right){{e}^{j2{\text{π}} \frac{n}{N}}} \right\}_{n=1}^{N}{\text{，}} \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots \\ &\left\{ {{x_P}\left( n \right) = x\left( n \right){e^{j2{\text{π}}\frac{{n\left( {P - 1} \right)}}{N}}}} \right\}_{n = 1}^N{\text{。}} \end{split}$ (13)

 $\begin{split} &{r_{ml}}\left( k \right) = \sum\limits_{n = 1}^N {{x_m}} \left( n \right)x_l^*\left( {n - k} \right) = \\ &{e^{j2{\text{π}} \frac{{\left( {m - 1} \right)k}}{N}}}\sum\limits_{n = 1}^N {x\left( n \right){x^*}} \left( {n - k} \right){e^{ - j2{\text{π}} \frac{{\left( {n - k} \right)\left( {l - m} \right)}}{N}}}{\text{。}} \end{split}$ (14)

 ${{X}} = {\left[ {{{{X}}_1} \cdots {{{X}}_P}} \right]_{\left( {N + K - 1} \right) \times K{P}}}{\text{，}}$ (15)

 ${{{X}}_m} = \left[ {\begin{array}{*{20}{c}} {{x_m}\left( 1 \right)}&{}&0 \\ \vdots & \ddots &{} \\ \vdots &{}&{{x_m}\left( 1 \right)} \\ {{x_m}\left( N \right)}&{}& \vdots \\ {}& \ddots & \vdots \\ 0&{}&{{x_m}\left( N \right)} \end{array}} \right]{\text{，}}$ (16)

$\left\{ {{x_m}\left( n \right)} \right\}$ 如式（12）所定义，可得

 ${{X}}_m^H{{{X}}_l} = {\left[ {\begin{array}{*{20}{c}} {{r_{ml}}\left( 0 \right)}&{r_{ml}^*\left( 1 \right){\text{ }}}& \cdots &{{\text{ }}r_{ml}^*\left( {N - 1} \right)} \\ {{r_{ml}}\left( 1 \right){\text{ }}}&{{r_{ml}}\left( 0 \right)}& \ddots & \vdots \\ {{\text{ }} \vdots {\text{ }}}& \ddots & \ddots &{r_{ml}^*\left( 1 \right)} \\ {{r_{ml}}\left( {N - 1} \right)}& \cdots &{{r_{ml}}\left( 1 \right)}&{{r_{ml}}\left( 0 \right)} \end{array}} \right]_{N \times N}}{\text{，}}$ (17)

 ${\hat C_1} = {\left\| {{{{X}}^H}{{X}} - N{{{I}}_{KP}}} \right\|^2}{\text{。}}$ (18)

 $\mathop {\min }\limits_{{{X}},{{U}}} {\left\| {{{X}} - \sqrt N {{U}}} \right\|^2}{\text{，}}$
 $s.t.\left| {x\left( n \right)} \right| = 1,n = 1, \cdots ,N{\text{，}}$
 ${x_m}\left( n \right) = x\left( n \right){e^{j2{\text{π}} \dfrac{{n\left( {m - 1} \right)}}{N}}}{\text{。}}$ (19)

1）随机的选取现有的序列来初始化序列 $\left\{ {x\left( n \right)} \right\}$

2）固定 ${{X}}$ ，通过式（20）来计算极小值

 ${{U}} = {{{U}}_1}{{U}}_1^H{\text{，}}$ (20)

3）固定 ${{U}}$ , 式（19）可写为下式：

 $\begin{split} & {\left\| {{{X}} - \sqrt N {{U}}} \right\|^2}{\text{ = }}\sum\limits_{n = 1}^N {\sum\limits_{l = 1}^{KP} {{{\left| {{\mu _{nl}}x\left( n \right) - {f_{nl}}} \right|}^2}} } = \\ & const - 2\sum\limits_{n = 1}^N {\operatorname{Re} \left[ {\left( {\sum\limits_{l = 1}^{KP} {{{\left| {\mu _{nl}^*{f_{nl}}} \right|}^2}} } \right){x^*}\left( n \right)} \right]} {\text{。}} \end{split}$ (21)

 $\begin{split} &\left[ {{\mu _{n1}} \cdots {\mu _{n,KP}}} \right] = \\ &{\left[ {\underbrace {1 \cdots 1}_K\underbrace {{e^{j2{\text{π}} \frac{n}{N}}} \cdots {e^{j2\pi \frac{n}{N}}}}_K \cdots \underbrace {{e^{j2{\text{π}}\frac{{n\left( {p - 1} \right)}}{N}}} \cdots {e^{j2{\text{π}} \frac{{n\left( {p - 1} \right)}}{N}}}}_K} \right]_{1 \times KP}}{\text{，}}\!\!\!\!\!\! \end{split}$ (22)

$\left\{ {{f_{nl}}} \right\}$ 是矩阵 $\sqrt N {{U}}$ 的元素，它的位置和 $\left\{ {{\mu _{nl}}} \right\}$ 在矩阵 ${{X}}$ 的元素位置相同，极小值 $x\left( n \right)$ 所对应的相位为：

 ${\phi _n} = \arg \left( {\sum\limits_{l = 1}^{KP} {\mu _l^*{f_{nl}}} } \right),\begin{array}{*{20}{c}} {}&{n = 1, \cdots ,N}{\text{。}} \end{array}$ (23)

4）重复第2步和第3步直到收敛。

3 性能分析

 $x\left( n \right) = {e^{j\phi \left( n \right)}}{\text{，}}$ (24)

 图 1 随机相位序列的模糊度函数图 Fig. 1 The AF of a length-50 random-phase sequence, 3D plot of the positive Doppler plane

 图 2 随机相位序列的模糊度函数等高线图 Fig. 2 The AF of a length-50 random-phase sequence, 2D plot of the positive Doppler plane

 $\begin{split} & \chi \left( {0,f} \right) = \int_{ - \infty }^{ + \infty } {u\left( t \right)} u{\left( t \right)^*}{e^{ - i2{\text{π}} ft}}{\rm d}t = \int_0^T {{e^{ - i2{\text{π}}ft}}} {\rm d}t = \\ & \frac{{1 - {e^{ - j2{\text{π}} fT}}}}{{j2{\text{π}} f}} = {e^{ - j{\text{π}} fT}}T\sin c\left( {{\text{π}} fT} \right){\text{，}} \end{split}$ (25)

 图 3 设计编码的离散模糊度函数等高线图（用相同长度的随机编码作为算法的初始序列） Fig. 3 A synthesized discrete-AF: $\left| {\bar r\left( {k,p} \right)} \right|$ , 2D plot of the positive Doppler plane K=10, P=3

 图 4 K=10，P=15时设计编码的模糊度函数等高线图 Fig. 4 A synthesized discrete-AF: $\left| {\bar r\left( {k,p} \right)} \right|$ , 2D plot of the positive Doppler plane K=10，P=15

 图 5 K=10，P=50时设计编码的模糊度函数等高线图 Fig. 5 A synthesized discrete-AF: $\left| {\bar r\left( {k,p} \right)} \right|$ , 2D plot of the positive Doppler plane K=10，P=50

 图 6 K=30，P=1 时设计编码的模糊度函数等高线图 Fig. 6 A synthesized discrete-AF: $\left| {\bar r\left( {k,p} \right)} \right|$ , 2D plot of the positive Doppler plane K=30，P=1

 图 7 K=50，P=1时，设计编码的模糊度函数等高线图 Fig. 7 A synthesized discrete-AF: $\left| {\bar r\left( {k,p} \right)} \right|$ , 2D plot of the positive Doppler plane K=50，P=1

4 结　语

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