erfc table and formulas

Communication Systems/communication_systems.tex

@@ -5,7 +5,7 @@
 % @Author: Noah Huetter
 % @Date:   2019-09-24 17:26:28
 % @Last Modified by:   noah
-% @Last Modified time: 2020-01-07 11:46:45
+% @Last Modified time: 2020-01-07 12:21:57
 % ---------------------------------------------------------------------------
 
 \documentclass[a4paper, fontsize=8pt, landscape, DIV=1]{scrartcl}
@@ -1987,16 +1987,6 @@
   $f^{(n)}$ $n^\text{th}$ derivation \\
   $f^*$ complex conjugate
 
-  \subsection{Probability}
-  \begin{empheq}{align*}
-    &\P(X > x) = \frac{1}{2}\erfc\left(\frac{x}{\sqrt{2}\sigma_X}\right) \\
-    &\erf(x) = \frac{1}{\sqrt{\pi}}\int_{-x}^x \e^{-t^2}\dt = \frac{2}{\sqrt{\pi}}\int_{0}^x \e^{-t^2}\dt \\
-    &\erfc(x) = 1 - \erf{x} = \frac{2}{\sqrt{\pi}}\int_{x}^\infty \e^{-t^2}\dt \\
-    &Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty\exp(-y^2/2)\dy\leq\exp(-x^2/2)
-  \end{empheq}
-
-  \vfill\null
-  \columnbreak
   \subsection{Sums}
   \begin{empheq}{align*}
     & \sum_{k=0}^n q^k k &&=&& \frac{nq^{n+2}-(n+1)q^{n+1}+q}{(q-1)^2} \\
@@ -2006,6 +1996,19 @@
     & \sum_{k=0}^\infty q^k &&=&& \frac{1}{1-q} \\
   \end{empheq}
 
+  \subsection{Probability}
+  \begin{empheq}{align*}
+    &\P(X > x) = \frac{1}{2}\erfc\left(\frac{x}{\sqrt{2}\sigma_X}\right) \\
+    &\erf(x) = \frac{1}{\sqrt{\pi}}\int_{-x}^x \e^{-t^2}\dt = \frac{2}{\sqrt{\pi}}\int_{0}^x \e^{-t^2}\dt \\
+    &\erfc(x) = 1 - \erf{x} = 2Q(\sqrt{2}x) = \frac{2}{\sqrt{\pi}}\int_{x}^\infty \e^{-t^2}\dt \\
+    &Q(x) =  \frac{1}{\sqrt{2\pi}}\int_x^\infty\exp(-y^2/2)\dy\leq\exp(-x^2/2) \\
+    &Q(x) = \frac{1}{2}\erfc\left(\frac{x}{\sqrt{2}}\right)
+  \end{empheq}
+  \cgraphic{0.999}{img/erfcq.png}
+
+  % \vfill\null
+  % \columnbreak
+
 
 \end{multicols*}