From 9636b73b33a0d5b05f74c59e64afe1d0183b06b2 Mon Sep 17 00:00:00 2001 From: Peter Tanner Date: Tue, 29 Oct 2024 16:31:31 +0800 Subject: [PATCH] Replace {{ -> { { to prevent tag errors --- ...ots-guide-to-ELEC4402-Communications-Systems.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md b/_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md index 88cb60c..a19830b 100644 --- a/_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md +++ b/_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md @@ -242,7 +242,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\ x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\ &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\ - P_c &=\frac{{A_c}^2}{2}\quad\text{Carrier power}\\ + P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\ \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\ B_T&=2f_m=2B @@ -288,8 +288,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ ```math \begin{align*} - P_\text{av}&=\frac{{A_c}^2}{2}\\ - P_\text{band\_index}&=\frac{{A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\ + P_\text{av}&=\frac{ {A_c}^2}{2}\\ + P_\text{band\_index}&=\frac{ {A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\ \text{band\_index}&=0\implies f_c+0f_m\\ \text{band\_index}&=1\implies f_c+1f_m,\dots\\ \end{align*} @@ -507,7 +507,7 @@ b_n:\{1,0\}\to a_n:\{1,0\} \begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\ s_1(t)&=0\\ - &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + 0)=\frac{{A_c}^2}{4}T_b$} + &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{ {A_c}^2}{2}\times T_b + 0)=\frac{ {A_c}^2}{4}T_b$} \end{align*} ``` @@ -535,7 +535,7 @@ b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\} \begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{E_b}\varphi_1(t)\\ s_1(t)&=-A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=-\sqrt{E_b}\varphi_2(t)\\ - &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + \frac{{A_c}^2}{2}\times T_b)=\frac{{A_c}^2}{2}T_b$} + &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{ {A_c}^2}{2}\times T_b + \frac{ {A_c}^2}{2}\times T_b)=\frac{ {A_c}^2}{2}T_b$} \end{align*} ``` @@ -594,7 +594,7 @@ tBitstream[bitstream_, Tb_, title_] := Module[{timeSteps, gridLines, plot}, timeSteps = Flatten[Table[{(n - 1) Tb, bitstream[[n]]}, {n, 1, - Length[bitstream]}] /. {t_, v_} :> {{t, v}, {t + Tb, v}}, 1]; + Length[bitstream]}] /. {t_, v_} :> { {t, v}, {t + Tb, v}}, 1]; gridLines = {Join[ Table[{n Tb, Dashed}, {n, 1, 2 Length[bitstream], 2}], Table[{n Tb, Thin}, {n, 0, 2 Length[bitstream], 2}]], None}; @@ -851,7 +851,7 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems | Modulation | $G_x(f)$ | | ---------- | ------------------------------------------------------------------------------------------------- | -| Quadrature | $\color{red}\frac{{A_c}^2}{4}[G_I(f-f_c)+G_I(f+f_c)+G_Q(f-f_c)+G_Q(f+f_c)]$ | +| Quadrature | $\color{red}\frac{ {A_c}^2}{4}[G_I(f-f_c)+G_I(f+f_c)+G_Q(f-f_c)+G_Q(f+f_c)]$ | | Linear | $\color{red}\frac{\|V(f)\|^2}{2}\sum_{l=-\infty}^\infty R(l)\exp(-j2\pi l f T)\quad\text{What??}$ | ### Symbol error probability