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 c9a548a..35b3845 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 @@ -78,26 +78,26 @@ along with this program. If not, see . ## Fourier transform identities and properties -| Time domain $x(t)$ | Frequency domain $X(f)$ | -| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | -| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | -| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | -| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ | -| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | -| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | -| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T$ | $T \text{sinc}^2(fT)$ | -| $\delta(t)$ | $1$ | -| $1$ | $\delta(f)$ | -| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | -| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | -| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | -| $\cos(2\pi f_c t+\theta)$ | $\frac{1}{2}[\delta(f - f_c)\exp(j\theta) + \delta(f + f_c)\exp(-j\theta)]\quad\text{Use for coherent recv.}$ | -| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | -| $\sin(2\pi f_c t+\theta)$ | $\frac{1}{2j} [\delta(f - f_c)\exp(j\theta) - \delta(f + f_c)\exp(-j\theta)]$ | -| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | -| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ | -| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | -| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | +| Time domain $x(t)$ | Frequency domain $X(f)$ | +| ---------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | +| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | +| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | +| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ | +| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | +| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | +| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T\quad\text{tri}(t/T)$ | $T \text{sinc}^2(fT)$ | +| $\delta(t)$ | $1$ | +| $1$ | $\delta(f)$ | +| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | +| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | +| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | +| $\cos(2\pi f_c t+\theta)$ | $\frac{1}{2}[\delta(f - f_c)\exp(j\theta) + \delta(f + f_c)\exp(-j\theta)]\quad\text{Use for coherent recv.}$ | +| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | +| $\sin(2\pi f_c t+\theta)$ | $\frac{1}{2j} [\delta(f - f_c)\exp(j\theta) - \delta(f + f_c)\exp(-j\theta)]$ | +| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | +| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ | +| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | +| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | | Time domain $x(t)$ | Frequency domain $X(f)$ | Property | | ---------------------------------------- | ------------------------------------------------ | ----------------------------- | @@ -127,6 +127,7 @@ $$ \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ + \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\ g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ \end{align*} $$ @@ -154,9 +155,13 @@ $$ \end{align*} $$ -### $\text{rect}$ function +### Shape functions -![rect](/assets/img/2024-10-29-Idiots-guide-to-ELEC/rect.drawio.svg) +| $\text{rect}$ function | $\text{tri}$ function | +| -------------------------------------------------------------------- | --------------------- | +| ![rect](/assets/img/2024-10-29-Idiots-guide-to-ELEC/rect.drawio.svg) | TODO: Add graphic. | + +Tri placeholder: For $\text{tri}(t/T)=1-\|t\|/T$, Intersects $x$ axis at $-T$ and $T$ and $y$ axis at $1$. ### Bessel function @@ -288,11 +293,11 @@ $$ $$ \begin{align*} - m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ - m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\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], \\ &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\ + m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ + m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\ + m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\ 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}\\