From 34ee0358bb9bb36152c9aee93eeb67427a305df6 Mon Sep 17 00:00:00 2001
From: Josh Gao <josh_gao@foxmail.com>
Date: Thu, 20 Apr 2023 10:12:07 +0800
Subject: [PATCH] update post, josh-oap-1-2

---
 source/_posts/2023-04-18-josh-oap-part-1-1.md |  4 +++
 source/_posts/2023-04-20-josh-opa-part-1-2.md | 35 +++++++++++++++++--
 2 files changed, 36 insertions(+), 3 deletions(-)

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 dc9ad3a..ef86828 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
@@ -394,12 +394,16 @@ $$\begin{equation}
 
 &emsp;&emsp;定义一个复数权矢量为
 
+<a id="ComplexWeightVector"></a>
+
 $$\begin{equation} \label{ComplexWeightVector}
   \vec{w}^\mathrm{H} = [w_0^\ast, w_1^\ast, \cdots, w_{N-1}^\ast]
 \end{equation}$$
 
 则式 $\eqref{ArrayResponseToUnitPlaneWaveInTimeDomain}$ 成为
 
+<a id="FrequencyWavenumberResponseWithComplexWeigh"></a>
+
 $$\begin{equation}
   y\left( t,\vec{k} \right) = \vec{w}^\mathrm{H} \vec{v}_{\vec{k}}\left( \vec{k} \right) e^{j\omega t}
 \end{equation}$$
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
index 344f5d6..b9194bb 100644
--- a/source/_posts/2023-04-20-josh-opa-part-1-2.md
+++ b/source/_posts/2023-04-20-josh-opa-part-1-2.md
@@ -35,14 +35,14 @@ $$\begin{equation} \label{ElementLocation}
 \begin{cases}
   p_{x_n} = 0 \\
   p_{y_n} = 0 \\
-  p_{z_n} = \left( n - \displaystyle\frac{N-1}{2} \right) d
+  p_{z_n} = \left( n - \frac{N-1}{2} \right) d
 \end{cases}
   , \, n = 0,1,\cdots,N-1
 \end{equation}$$
 
-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
+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
 
-$$\begin{equation}
+$$\begin{equation} \label{ManifoldVectorOfULA}
   \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} \\
@@ -63,4 +63,33 @@ $$\begin{equation}
   k_0 \triangleq \left| \vec{k} \right| = \frac{2\pi}{\lambda}
 \end{equation}$$
 
+意线阵在 0 方向是没有分辨能力的。将[复权矢量][] 和式 $\eqref{ManifoldVectorOfULA}$ 代入[用复权矢量表示的波数-频率响应函数][]，得到 ULA 在角度域的频率-波数响应函数
+
+$$\begin{equation} \label{WFRFOfULAInAngleDomain}
+  \begin{aligned}
+    \vec{\varUpsilon}(\omega,k_z) &= \vec{w}^\mathrm{H}\vec{v}_{\vec{k}}\left(k_z\right) \\
+    &= \sum_{n=0}^{N-1} w_n^\ast e^{-j\left( n - \frac{N-1}{2} k_z d \right)}
+  \end{aligned}
+\end{equation}$$
+
+定义下面的变量也是很有用的（波数在 $z$ 轴上的投影，乘以阵元间距，也即扫过一个阵元间距时，波数的变化量在 $z$ 轴上的体现）：
+
+$$\begin{equation} \eqref{DefinitionOfWavenumberDiffOfTwoElements}
+  \psi = -k_z d = \frac{2\pi}{\lambda}\cos\theta\cdot d = \frac{2\pi}{\lambda}u_z d
+\end{equation}$$
+
+其中 $u_z$ 是针对 $z$ 轴的方向余弦，
+
+$$\begin{equation}
+  u_z = \cos\theta
+\end{equation}$$
+
+将式 $\eqref{DefinitionOfWavenumberDiffOfTwoElements}$ 代入式 $\eqref{WFRFOfULAInAngleDomain}$，得到
+
+$$\begin{equation}
+  \vec{\varUpsilon}_\psi(\psi) = e^{-j \frac{N-1}{2} \psi } \sum_{n=0}^{N-1} w_n^\ast e^{j n\psi}
+\end{equation}$$
+
 [阵列流形矢量的定义]: https://josh-gao.top/posts/8b61f5a7.html#DefinitionOfArrayManifoldVector
+[复权矢量]: https://josh-gao.top/posts/8b61f5a7.html#ComplexWeightVector
+[用复权矢量表示的波数-频率响应函数]: https://josh-gao.top/posts/8b61f5a7.html#FrequencyWavenumberResponseWithComplexWeigh