From 4ebf80ee3a25ae020bc9dc536614b815f47a162f Mon Sep 17 00:00:00 2001
From: Josh Gao <josh_gao@foxmail.com>
Date: Thu, 20 Apr 2023 00:50:40 +0800
Subject: [PATCH] new post, josh-oap-1-2

---
 source/_posts/2023-04-18-josh-oap-part-1-1.md |  2 +
 source/_posts/2023-04-20-josh-opa-part-1-2.md | 66 +++++++++++++++++++
 2 files changed, 68 insertions(+)
 create mode 100644 source/_posts/2023-04-20-josh-opa-part-1-2.md

diff --git a/source/_posts/2023-04-18-josh-oap-part-1-1.md b/source/_posts/2023-04-18-josh-oap-part-1-1.md
index 208ea62..dc9ad3a 100644
--- a/source/_posts/2023-04-18-josh-oap-part-1-1.md
+++ b/source/_posts/2023-04-18-josh-oap-part-1-1.md
@@ -217,6 +217,8 @@ $$\begin{equation}
 
 定义
 
+<a id="DefinitionOfArrayManifoldVector"></a>
+
 $$\begin{equation} \label{DefinitionOfArrayManifoldVector}
   \vec{v}_{\vec{k}}\left(\vec{k}\right) = \left[ \begin{array}{c}
     e^{-j \vec{k}^\mathrm{T}\vec{p}_0} \\
diff --git a/source/_posts/2023-04-20-josh-opa-part-1-2.md b/source/_posts/2023-04-20-josh-opa-part-1-2.md
new file mode 100644
index 0000000..344f5d6
--- /dev/null
+++ b/source/_posts/2023-04-20-josh-opa-part-1-2.md
@@ -0,0 +1,66 @@
+---
+title: 'Josh''s Notes: 最优阵列处理（Part 1.2 —— 均匀线阵模型）'
+mathjax: true
+comments: true
+copyright: true
+toc:
+  enable: false
+  number: true
+  wrap: true
+  expand_all: true
+  max_depth: 4
+tags:
+  - 阵列信号处理
+  - 均匀线阵
+  - ULA
+categories:
+  - - Josh 的学习笔记
+    - 最优阵列处理
+    - 阵列和空域滤波器
+abbrlink: de20fd09
+date: 2023-04-20 00:14:31
+---
+
+我们感兴趣的线阵如[图 1-2-1](#fig.1-2-1) 所示，其中有 $N$ 个位于 $z$ 轴上的阵元，阵元间距均匀，记为 $d$。我们把这种阵列称为**{% label primary @均匀线性阵列（uniform linear array, ULA） %}**。我们已经把阵列的中心放在了坐标系的原点上，这种阵列中心放置方法将在计算量上具有优点。
+
+<a id="fig.1-2-1"></a>
+
+![图 1-2-1 放置在 $z$ 轴上的线阵](https://josh-blog-1257563604.cos.ap-beijing.myqcloud.com/img/2023-04-20-josh-opa-part-1-2/2023-04-20-josh-oap-part-1-2-010-LinearArrayAlongZAaxis.png!sign){width=600px}
+
+<!-- more -->
+
+&emsp;&emsp;阵元的位置为
+
+$$\begin{equation} \label{ElementLocation}
+\begin{cases}
+  p_{x_n} = 0 \\
+  p_{y_n} = 0 \\
+  p_{z_n} = \left( n - \displaystyle\frac{N-1}{2} \right) d
+\end{cases}
+  , \, n = 0,1,\cdots,N-1
+\end{equation}$$
+
+为了确定阵列流形矢量 $\vec{v}_{\vec{k}}\left(\vec{k}\right)$, 把式 $\eqref{ElementLocation}$ 代入到[阵列流形矢量的定义][], 得到
+
+$$\begin{equation}
+  \vec{v}_{\vec{k}}\left(k_z\right) = \left[\begin{array}{c}
+    e^{j \left( \frac{N-1}{2} \right) k_z d} \\
+    e^{j \left( \frac{N-1}{2} - 1\right) k_z d} \\
+    \vdots \\
+    e^{-j \left( \frac{N-1}{2} \right) k_z d} \\
+  \end{array}\right]
+\end{equation}$$
+
+这里
+
+$$\begin{equation}
+  k_z = -\frac{2\pi}{\lambda}\cos\theta = -k_0 \cos\theta
+\end{equation}$$
+
+其中波数的幅度
+
+$$\begin{equation}
+  k_0 \triangleq \left| \vec{k} \right| = \frac{2\pi}{\lambda}
+\end{equation}$$
+
+[阵列流形矢量的定义]: https://josh-gao.top/posts/8b61f5a7.html#DefinitionOfArrayManifoldVector