﻿ 复合调制雷达信号时差估计算法
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 雷达学报  2015, Vol. 4 Issue (4): 460–466  DOI: 10.12000/JR15072 0

### 引用本文 [复制中英文]

[复制中文]
Xiong Peng, Liu Zheng, Jiang Wen-li. TDOA Estimation Algorithm of Hybrid Modulation Radar Signals[J]. Journal of Radars, 2015, 4(4): 460–466. DOI: 10.12000/JR15072.
[复制英文]

### 文章历史

(国防科技大学电子科学与工程学院 长沙 410073)

TDOA Estimation Algorithm of Hybrid Modulation Radar Signals

(College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China)
Foundation Item: The National Natural Science Foundation of China (61302141)
Abstract: In the modern radar signal environment, hybrid modulation signals are increasingly used in many systems. This study focuses on the Time Difference Of Arrival (TDOA) estimation problem of FSK-BPSK hybrid modulation signals. The method combines the characteristic of modulated signal and utilizes the sum of simple correlation functions of sub-pulses to elicit the complex correlation function of the entire pulse signal. Finally, a correlation function fitting algorithm is used to estimate the exact TDOA. Experimental results indicate an obvious improvement in the accuracy and noise immunity of the method, and the method is appropriate for the TDOA estimation of low-bandwidth hybrid modulation signals.
Key words: Hybrid modulation signal     Time Difference Of Arrival (TDOA) estimation     Correlation function fitting algorithm
1 引言

2 信号模型

FSK-BPSK信号的解析形式为：

 \begin{aligned} s\left( t \right) = & \sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {A\exp \left\{ {{\rm{j}}2{π} {f \!_m}\left[{t - \left( {m - 1} \right){T_m}} \right.} \right.} } \\ & \left. {\left. { - \left( {n - 1} \right){T_n}} \right] + {\varphi _{mn}}} \right\} \end{aligned} (1)

1/Bw

3 时差估计算法 3.1  信号相关特性

 图 1 时差估计的远场模型 Fig.1 The far-field model of TDOA estimation

 $\left. {\begin{array}{*{20}{l}} {x(t) = s(t) + {n_1}(t)}\\ {y(t) = s(t - D) + {n_2}(t)} \end{array}} \right\}$ (2)

 \begin{aligned} {R_{xy}}(\tau ) = & E[x(t){y^*}(t + \tau )]\\ \quad \quad = & E[s(t){s^*}(t - D + \tau )] + E[s(t){n_2^*}(t + \tau )] \\ & + E[{n_1}(t){s^*}(t - D + \tau )] + E[{n_1}(t){n_2^*}(t + \tau )]\\ \quad \quad = & {R_{ss}}(\tau \! - \! D) \! + \! {R_{s{n_2}}}(\tau ) \! + \! {R_{{n_1}s}}(\tau - D) \! + \! {R_{{n_1}{n_2}}}(\tau ) \end{aligned} (3)

 ${R_{xy}}(\tau ) = {R_{ss}}(\tau - D) = \int_0^T {x(t){y^*}\left( {t + \tau } \right)} {\rm{d}}t$ (4)

 \begin{aligned}\phi (\hat D) = & \mathop {\arg \max }\limits_\tau \phi (\tau ) = \mathop {\arg \max }\limits_\tau \int_0^T {x(t){y^*}\left( {t + \tau } \right)} {\rm{d}}t \\ = & \mathop {\arg \max }\limits_\tau {R_{ss}}(\tau - D)\end{aligned} (5)

 $△T = {1 \mathord{\left/ {\vphantom {1 {{f \!_{s}}}}} \right. } {{f\!_s}}}$ (6)

3.2  FSK-BPSK复合调制信号相关函数

FSK-BPSK复合调制信号是将发射的宽脉冲分为若干个调制参数一致、载频对称分布的子脉冲。两个接收通道接收的FSK-BPSK信号子脉冲q(t)为常规BPSK信号，子脉冲互相关函数Q()可表示为：

 \begin{aligned} Q(△){\rm{ = }} & \int {\left[{q(t) + {n_1}(t)} \right]} \left[{{q^*}(t + d -△ ) } \right.\\ & + \left. {{n_2^*}(t +△)} \right]{\rm{d}}t = {R_{qq}}(d -△) \end{aligned} (7)

 \begin{aligned} Q(△) = & \sum\limits_{n = 1}^N {A\exp \{ {\rm{j}}2{π} {f \!_1}[t - (n - 1){T_n}] + {\varphi _{mn}}\} } \\ & \cdot \sum\limits_{n = 1}^N \! {A \! \exp \{ - {\rm{j}}2{π} {f \!_1}[\left( {t \! +\! d \! - \! △} \right) \! - \! (n \! - \! 1){T_n}] \! \! - \! \! {\varphi _{mn}} \} } \\ \quad \quad = & \frac{{{T_n} - \left| {d -△} \right|}}{{{T_n}}} \cdot \exp \left[{{\rm{j}}2{\rm {\pi}} {f \!_1}(d - △)} \right] \end{aligned} (8)

 \begin{aligned} F(△\left| d \right.) = & \sum\limits_{i = 1}^M {Q(△)} = \sum\limits_{i = 1}^M {{R_{qq}}(d - △)} \,\\ = & \frac{{{T_n} - \left| {(d - △)} \right|}}{{{T_n}}} \cdot \sum\limits_{i = 1}^M {\exp \left[{\rm j}2{π} {{f \!_i}(d - △)} \right]} \end{aligned} (9)

 \begin{aligned} F(△\left| d \right.) = & \frac{{{T_n} - \left| {(d - △)} \right|}}{{{T_n}}} \cdot \exp \left[{{\rm{j}}2{π} \bar f(d - △)} \right] \\ & \cdot \sum\limits_{i = 1}^M {\exp \left[{{\rm{j}}2{π} ({f \!_i} - \bar f)(d - △)} \right]} \end{aligned} (10)

$\bar f$为各段载频均值，对$F(△\left| d \right.)$取绝对值即可将难以运算操作的复数项$\exp \left[{{\rm{j}}2{π} \bar f(d - △)} \right]$简化为1，利用=欧拉公式，将式(10)中剩余复数项$\sum\nolimits_{i = 1}^M {\exp \left[{{\rm{j}}2{π} ({f \!_i} - \bar f)} \right.}$ $\left. \cdot {(d - △)} \right]$展开得到：

 \begin{aligned} & \sum\limits_{i = 1}^M {\exp \left[{{\rm{j}}2{π} ({f\!_i} - \bar f)(d - △)} \right]} \\ & \quad\quad = \sum\limits_{i = 1}^M {\cos \left[{2{π} ({f\!_i} - \bar f)(d - △)} \right]} \\ & \quad\quad\quad + \sum\limits_{i = 1}^M {{\rm{j}}\sin \left[{2{π} ({f \!_i} - \bar f)(d - △)} \right]} \end{aligned} (11)

 \begin{aligned} & \sum\limits_{i = 1}^M {\exp \left[{{\rm{j}}2{π} ({f \!_i} - \bar f)(d - △)} \right]} \\ & \quad\quad = \sum\limits_{i = 1}^M {\cos \left[{2{π} ({f \!_i} - \bar f)(d - △)} \right]} \end{aligned} (12)

 \begin{aligned} & \sum\limits_{i = 1}^M {\exp \left[{{\rm{j}}2{π} ({f \!_i} - \bar f)(d - △)} \right]} \\ & \quad\quad = \sum\limits_{i = 1}^{(M - 1)/2} {\cos \left[{2{π} ({f \!_i} - \bar f)(d - △)} \right]} + 1 \\ & \quad\quad\quad \ + \sum\limits_{i = (M + 1)/2}^M {\cos \left[{2{π} ({f \!_i} - \bar f)(d - △)} \right]} \end{aligned} (13)

 \begin{aligned} \left| {F(△\left| d \right.)} \right| = & \left| {\frac{{{T_n} - \left| {d - △} \right|}}{{{T_n}}}} \right| \cdot 1 \\ & \cdot \left| {\sum\limits_{i = 1}^M {\cos \left[{2{π} ({f \!_i} - \bar f)(d - △)} \right]} } \right| \end{aligned} (14)
3.3  相关函数拟合算法

 \begin{aligned} {F} = & \left| {F(△\left| {{{D}^\prime}} \right.)} \right| = [F(△\left| {{d_1}} \right.)\,\ F(△\left| {{d_2}} \right.)\ \,\cdots \\ & \quad F(△\left| {{d_{i - 1}}} \right.)\,\ F(△\left| {{d_i}} \right.)] \end{aligned} (15)

 $J(\theta ,△) = {(\theta \cdot {{S}_2} - {{S}_1})^{\rm{H}}}(\theta \cdot {{S}_2} - {{S}_1})$ (16)

 $\theta = \mathop {\arg \min }\limits_\theta [\,{(\theta \cdot {{S}_2} - {{S}_1})^{\rm{H}}}(\theta \cdot {{S}_2} - {{S}_1})]$ (17)

$J(\theta ,△)$其梯度为：

 $\frac{{\partial J(\theta ,△)}}{{\partial \theta }} = 2 \cdot \theta \cdot {{S}_2^{\rm{H}}} \cdot {{S}_2} - \,2 \cdot {{S}_1^{\rm{H}}} \cdot {{S}_2}$ (18)

 $\hat \theta = {({{S}_2^{\rm{H}}} \cdot {{S}_2})^{ - 1}}({{S}_1^{\rm{H}}} \cdot {{S}_2})$ (19)

 ${J_{\min }} = {(\hat \theta \cdot {{S}_2} - {{S}_1})^{\rm{H}}} \cdot (\hat \theta \cdot {{S}_2} - {{S}_1})$ (20)
4 仿真实验

4.1  算法结果

 图 2 FSK-BPSK复合调制信号瞬时频率图 Fig.2 The IF of FSK-BPSK hybrid modulation signal
4.1.1  信号相关特性曲线

 图 3 互相关函数 Fig.3 Cross correlation function
4.1.2  插值算法

 图 4 插值函数 Fig.4 Interpolation function
4.1.3   相关函数拟合算法

 图 5 拟合误差结果图 Fig.5 The result of fitting error
4.2  算法比较

4.2.1   对信噪比的适应性

 图 6 两种算法对信噪比的适应性 Fig.6 The adaptability of SNR

4.2.2  对带宽的适应性

 图 7 两种算法对带宽的适应性 Fig.7 The adaptability of bandwidth

5 结论

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