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Peter 2024-10-29 16:31:31 +08:00
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14
_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md
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@ -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 $m(t)$)}\\ 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], \\ 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}\\ &\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\\ 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}\\ \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\
B_T&=2f_m=2B B_T&=2f_m=2B
@ -288,8 +288,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
```math ```math
\begin{align*} \begin{align*}
P_\text{av}&=\frac{{A_c}^2}{2}\\ 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{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}&=0\implies f_c+0f_m\\
\text{band\_index}&=1\implies f_c+1f_m,\dots\\ \text{band\_index}&=1\implies f_c+1f_m,\dots\\
\end{align*} \end{align*}
@ -507,7 +507,7 @@ b_n:\{1,0\}\to a_n:\{1,0\}
\begin{align*} \begin{align*}
s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\ s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\
s_1(t)&=0\\ 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*} \end{align*}
``` ```
@ -535,7 +535,7 @@ b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\}
\begin{align*} \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_1(t)\\
s_1(t)&=-A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=-\sqrt{E_b}\varphi_2(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*} \end{align*}
``` ```
@ -594,7 +594,7 @@ tBitstream[bitstream_, Tb_, title_] :=
Module[{timeSteps, gridLines, plot}, Module[{timeSteps, gridLines, plot},
timeSteps = timeSteps =
Flatten[Table[{(n - 1) Tb, bitstream[[n]]}, {n, 1, 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[ gridLines = {Join[
Table[{n Tb, Dashed}, {n, 1, 2 Length[bitstream], 2}], Table[{n Tb, Dashed}, {n, 1, 2 Length[bitstream], 2}],
Table[{n Tb, Thin}, {n, 0, 2 Length[bitstream], 2}]], None}; 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 | $G_x(f)$ | | 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 | $\color{red}\frac{\|V(f)\|^2}{2}\sum_{l=-\infty}^\infty R(l)\exp(-j2\pi l f T)\quad\text{What??}$ | | 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 ### Symbol error probability