From 8ec098bc5154b0fdca5712d64895d8fa588b223c Mon Sep 17 00:00:00 2001 From: Noah Huetter Date: Tue, 7 Jan 2020 12:25:07 +0100 Subject: [PATCH] erfc table and formulas --- .../communication_systems.tex | 25 ++++++++++-------- Communication Systems/img/erfcq.png | Bin 0 -> 252328 bytes 2 files changed, 14 insertions(+), 11 deletions(-) create mode 100644 Communication Systems/img/erfcq.png diff --git a/Communication Systems/communication_systems.tex b/Communication Systems/communication_systems.tex index 3d8cb3e..0d0c819 100644 --- a/Communication Systems/communication_systems.tex +++ b/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*} diff --git a/Communication Systems/img/erfcq.png b/Communication Systems/img/erfcq.png new file mode 100644 index 0000000000000000000000000000000000000000..f853e21414e1c3d6e63ff64e2cf013e2a6d60b36 GIT binary patch literal 252328 zcmZ^~cU)6lw=E1JMXF+>1`t7!CLp~8Md4AT2uklA>Aj`MgMxGw1gW7%q)G1yNUx#y z-lan*Nl5wP^Pcyd_q*rb{E=U>*Pd&wIrrLQjXmesk-FL%O!OS|6ciLp&;NR=M?rC^ zih_b_l#Ygcf< z7h77vv_jbtV%$#!?uNc+wRxgj&L74vckA_2UG{52IZNMfurt`S%Fx}nf$`?##D!6F zKDc-%(7y|5yx0p`#O{-{2bDMHF6Pq?lql{8Ch2~=B24i~p!?c0n!wT3>#V=)?om7e z1ZC0h8oQQ%x_0f_rK>6IBrzU$e#FPQs&6r;2S}3Pt2NIKSBfmU_xf@+{_(I9SI#)HmNq( za-9`MA0tF-MSY4jpHzD9xnu9j+QVGOZ^CEMsn)Oj^F$1PbVo?Ei4411JXeqDA6%8R zKSBo%dr77D#WST@d#S%ZERK)w7yI(GcYBlmQj;SZ*Ef+VK~zMP)pvhdAv-7B8P z)nDp#ed*OyEW72TUC6gV`lxT# zOAL5by5ph8j6bQhTWI7c@@Ba9D|)2pK@}hRp5Y%^FlU6$Ns46%S(y&z&zj7dP(2KC zgvYlxD(VMZ%UO1Gn^vZf8|T+mbg>nn8VcY25=AT?IOL@`)}fFaaL{~OKU&kXbNS)B zkNDg7sr<5-PC~EzRNeNu?B2pqC`b1`%;^)wElQcCP!ZlsnyoK>GT->y=I71b%j;RJ zToiswbaFH%twNs6W|t;aS+;1;LvJvKC0pG9QLJ9Qyv$4g^urh*clTAv2v4)_jz8+ zJ-GHYiZ$CHH%d4E`-9i~Ke)#~G(>*MvB*}*Eq8j58dsy*m`yb-;Up7$C0IK6dIOhv zg!^Ml)y1bQIU2)@5;tN5JGoY)tZh~#oWJFNd-77KfMztN{^Rd1Mq3rDawCojww?D( z?SfXL<-$MHeuk85O)YrVPRzskHixPtR+tJ4W&wV?Hg zKJU7>LX81|gosy(j0hGIBT!qHmyOlz%JOAqYk6yi?9?Icy2osSG1qTKc0LWTc_fih z7iXmIde`Ev3NM2H^IyStZ;yajVO2@5)fHYeyyUnG78L$G(50`CUz^5bjk?(45^3fv6&YV_6q!OaKoH+he4 z41Da&vmdD$iW_zxMSOo-DEYen2j?5kHy5vgKL%c*^;Un}ER-%hEdsxS7hM!Oe)rJ# zcmw?&I`MGyTUq&QTUh6_SVN!0&AI_=lu)XV>!7lh(dbJlJ)aL7lmk1dwk9(slqOF~ z=1n%MEu2TInyW^u1Ds&PfZ^=zz^y|+|E=`#pvQ*UQd=n}k~nJI_tFCQtkS0eNcY

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