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Final update 2024 for comms notes. Anyone may add PRs to suggest changes
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@ -127,7 +127,7 @@ $$
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\text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\
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\text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\
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\text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\
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\text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\
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\text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}=\frac{1}{T}\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\
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g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\
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\end{align*}
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$$
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@ -157,42 +157,29 @@ $$
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### Shape functions
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| $\text{rect}$ function | $\text{tri}$ function |
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| -------------------------------------------------------------------- | --------------------- |
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|  | TODO: Add graphic. |
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Tri placeholder: For $\text{tri}(t/T)=1-\|t\|/T$, Intersects $x$ axis at $-T$ and $T$ and $y$ axis at $1$.
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### Bessel function
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### Random processes examples
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$$
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\begin{align*}
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\sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\
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J_n(\beta)&=(-1)^nJ_{-n}(\beta)
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&\text{Example: separate RV from expression}\\
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X(t) &= A\cos(2\pi f_c t)\quad A\thicksim \mathcal{N}(\mu=5,\sigma^2=1)\\
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\implies E[X(t)] &= E[A\cos(2\pi f_c t)] = E[A]\cos(2\pi f_c t) = 5\cos(2\pi f_c t)\\
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&\text{Example: random phase}\\
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X(t) &= B\cos(2\pi f_c t+\theta)\quad \theta\thicksim \mathcal{U}(0,2\pi)\\
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\implies E[X(t)] &= E[B\cos(2\pi f_c t+\theta)] = B\int_0^{2\pi}\underbrace{\frac{1}{2\pi}}_{\text{uniform}}\cos(2\pi f_c t+\theta)d\theta=0
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\end{align*}
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$$
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### White noise
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### Wide sense stationary (WSS)
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$$
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\begin{align*}
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R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\
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G_w(f)&=\frac{N_0}{2}\\
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N_0&=kT\\
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G_y(f)&=|H(f)|^2G_w(f)\\
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G_y(f)&=G(f)G_w(f)\\
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\end{align*}
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$$
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Two conditions for WSS:
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### WSS
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$$
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\begin{align*}
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\mu_X(t) &= \mu_X\text{ Constant}\\
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R_{XX}(t_1,t_2)&=R_X(t_1-t_2)=R_X(\tau)\\
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E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)]
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\end{align*}
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$$
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| Constant mean | Autocorrelation only dependent on time difference |
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| ---------------------------------- | ------------------------------------------------- |
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| $\mu_X(t) = \mu_X\text{ Constant}$ | $R_{XX}(t_1,t_2)=R_X(t_1-t_2)=R_X(\tau)$ |
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| $\mu_X(t)=E[X(t)]$ | $E[X(t_1)X(t_2)]=E[X(t)X(t+\tau)]$ |
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### Ergodicity
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@ -205,9 +192,9 @@ $$
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$$
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| Type | Normal | Mean square sense |
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| ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- |
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| ergodic in mean | $$\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0$$ |
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| ergodic in autocorrelation function | $$\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0$$ |
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| ----------------------------------- | ----------------------------------------------------- | --------------------------------------------------------- |
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| ergodic in mean | $\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X$ | $\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0$ |
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| ergodic in autocorrelation function | $\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)$ | $\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0$ |
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Note: **A WSS random process needs to be both ergodic in mean and autocorrelation to be considered an ergodic process**
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@ -289,19 +276,19 @@ $$
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## AM
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### CAM
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### Conventional AM modulation (CAM)
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$$
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\begin{align*}
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x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\
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x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right]\quad\text{CAM signal}\\
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&\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\
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m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
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m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\
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m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\
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P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\
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P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\
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\eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\
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B_T&=2f_m=2B
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P_s &=\frac{1}{4}{m_a}^2{A_c}^2\quad\text{Signal power, \textbf{total} of all 4 sideband power, \textbf{single-tone} case}\\
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\eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_s}{P_s+P_c}=\frac{P_s}{P_x}\quad\text{Power efficiency}\\
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B_T&=2f_m=2B\\
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\end{align*}
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$$
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@ -310,7 +297,7 @@ $B$: Bandwidth of modulating wave
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Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
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### DSB-SC
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### Double sideband suppressed carrier (DSB-SC)
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$$
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\begin{align*}
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@ -334,11 +321,24 @@ $$
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\end{align*}
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$$
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### Bessel function
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$$
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\begin{align*}
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J_n(\beta)&=\begin{cases}
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J_{-n}(\beta) & \text{$n$ is even}\\
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-J_{-n}(\beta) & \text{$n$ is odd}
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\end{cases}\\
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1&=\sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&\text{Conservation of power}\\
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\end{align*}
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$$
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### Bessel form and magnitude spectrum (single tone)
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$$
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\begin{align*}
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s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t]
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s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\\
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\Longleftrightarrow s(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t]
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\end{align*}
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$$
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@ -346,10 +346,12 @@ $$
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$$
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\begin{align*}
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P_\text{av}&=\frac{ {A_c}^2}{2}\\
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P_\text{band\_index}&=\frac{ {A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\
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\text{band\_index}&=0\implies f_c+0f_m\\
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\text{band\_index}&=1\implies f_c+1f_m,\dots\\
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P_\text{av}&=\frac{ {A_c}^2}{2}&\text{Av. power of full signal}\\
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P_\text{i}&=\frac{ {A_c}^2|{J_\text{i}}(\beta)|^2}{2}&\text{Av. power of band $i$}\\
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i=0&\implies f_c+0f_m&\text{Middle band}\\
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i=1&\implies f_c+1f_m&\text{1st sideband}\\
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i=-1&\implies f_c-1f_m&\text{-1st sideband}\\
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&\dots\\
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\end{align*}
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$$
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$$
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\begin{align*}
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B &= 2Mf_m = 2(\beta + 1)f_m\\
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&= 2(\Delta f_\text{max}+f_m)\\
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&= 2(D+1)W_m\\
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B &= 2(\beta + 1)f_m\\
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B &= 2(\Delta f_\text{max}+f_m)\\
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B &= 2(D+1)W_m\\
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B &= \begin{cases}
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2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\
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2(\Delta\phi_\text{max} + 1)f_m=2(\Delta \phi_\text{max}+1)W_m & \text{PM, sinusoidal message}
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\end{align*}
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$$
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### Complex envelope
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### Complex envelope of a FM signal
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$$
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\begin{align*}
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s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t)) \Leftrightarrow \tilde{s}(t) = A_c\exp(j\beta\sin(2\pi f_m t))\\
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s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t))\\
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\Longleftrightarrow \tilde{s}(t) &= A_c\exp(j\beta\sin(2\pi f_m t))\\
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s(t)&=\text{Re}[\tilde{s}(t)\exp{(j2\pi f_c t)}]\\
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\tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t)
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\end{align*}
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P_x&={\sigma_x}^2=\lim_{t\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\quad\text{For zero mean}\\
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P[A\cos(2\pi f t+\phi)]&=\frac{A^2}{2}\quad\text{Power of sinusoid }\\
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E_x&=\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{-\infty}^{\infty}|X(f)|^2df\quad\text{Parseval's theorem}\\
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R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}
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R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}\\
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P_x &= R_x(0)\quad\text{Average power of WSS process $x(t)$}\\
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\end{align*}
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$$
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##
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### White noise
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$$
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\begin{align*}
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R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\
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G_w(f)&=\frac{N_0}{2}\\
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\end{align*}
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$$
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## Noise performance
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Coherent detection system.
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<!-- Coherent detection system.
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$$
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\begin{align*}
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&= \frac{1}{2}m(t)+\frac{1}{2}\cos(4 f_c t+2\theta)\\
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\implies S_u(f) &= \left(\frac{1}{2}\right)^2S_u(f)\quad\text{After LPF.}
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\end{align*}
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$$
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$$ -->
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Use formualas from previous section, [Power, energy and autocorrelation](#power-energy-and-autocorrelation).\
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Use these formulas in particular:
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Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth.
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```math
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$$
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\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B}
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```
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$$
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<!-- MATH END -->
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### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png`)
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Cannot add directly due to copyright!
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**TODO: Make an open source replacement for this diagram [Send a PR to GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/issues/new).**
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<!--  -->
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<!--  -->
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$$
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\begin{align*}
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\Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\
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\Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\quad\text{Quantizer step size $\Delta$}\\
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\end{align*}
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$$
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\begin{align*}
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e &:= y-x\quad\text{Quantization error}\\
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\mu_E &= E[E] = 0\quad\text{Zero mean}\\
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{\sigma_E}^2&=E[E^2]-0^2=\int_{-\Delta/2}^{\Delta/2}e^2\times\left(\frac{1}{\Delta}\right) de\quad\text{Where $E\thicksim 1/\Delta$ uniform over $(-\Delta/2,\Delta/2)$}\\
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\text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise}}\\
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\text{SQNR(dB)}&=10\log_{10}(\text{SQNR})
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P_E&={\sigma_E}^2=\frac{\Delta^2}{12}=2^{-2m}V^2/3\quad\text{Uniformly distributed error}\\
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\text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise power}}=\frac{P_x}{P_E}\\
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\text{SQNR(dB)}&=10\log_{10}(\text{SQNR})\\
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m\to m+A\text{ bits}&\implies \text{newSQNR(dB)}=\text{SQNR(dB)}+6A\text{ dB}
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\end{align*}
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$$
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### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `assets/img/2024-10-29-Idiots-guide-to-ELEC/quantizer.png`)
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Cannot add directly due to copyright!
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**TODO: Make an open source replacement for this diagram [Send a PR to GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/issues/new).**
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<!--  -->
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<!--  -->
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@ -550,6 +565,8 @@ $$
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\end{align*}
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$$
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**TODO: Someone please make plots of the PSD for all line code types in Mathematica or Python! [Send a PR to GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/issues/new).**
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## Modulation and basis functions
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### $Q(x)$ function
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You should use [$\text{erf}$ function table](#function-1) instead in exams using the identity $Q(x)=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)$. Use this for validation.
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| $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ |
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| ------ | ---------- | ------ | ----------------------- | ------ | ------------------------ | ------ | ------------------------ |
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| $0.00$ | $0.5$ | $2.30$ | $0.010724$ | $4.55$ | $2.6823 \times 10^{-6}$ | $6.80$ | $5.231 \times 10^{-12}$ |
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### $\text{erf}(x)$ function
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$$
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Q(x)=\frac{1}{2}-\frac{1}{2}\text{erf}(\frac{x}{\sqrt{2}})
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$$
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| $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ |
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| ------ | --------------- | ------ | --------------- | ------ | --------------- |
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| $0.00$ | $0.00000$ | $0.75$ | $0.71116$ | $1.50$ | $0.96611$ |
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@ -805,6 +828,15 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
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\*\*The value of $\text{erf}(3.30)$ should be $\approx0.999997$ instead, but this value is quoted in the formula table.
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### $Q(x)$ fast reference
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Using identity.
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| $x$ | $Q(x)$ |
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| ----------- | --------- |
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| $\sqrt{2}$ | $0.07865$ |
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| $2\sqrt{2}$ | $0.00234$ |
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<div style="page-break-after: always;"></div>
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### Receiver output shit
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@ -921,6 +953,14 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems
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## Information theory
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### Stats
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$$
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\begin{align*}
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P(A|B) &= \frac{P(B|A)P(A)}{P(B)} = \frac{P(A,B)}{P(B)}\\
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\end{align*}
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$$
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### Entropy for discrete random variables
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$$
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@ -939,6 +979,25 @@ $$
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Entropy is **maximized** when all have an equal probability.
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### Transition probability diagram
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Example for binary erasure channel where $X$ is input and $Y$ is output:
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Equivalent to:
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$$
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\begin{align*}
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P[Y=0|X=0] &= 1-p\\
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P[Y=e|X=0] &= p\\
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P[Y=1|X=1] &= 1-p\\
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P[Y=e|X=1] &= p\\
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P[X=0|Y=0] &= 0\quad\text{Note the direction}\\
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P[Y=0] &= P[Y=0|X=0] P[X=0]
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\end{align*}
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$$
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### Differential entropy for continuous random variables
|
||||
|
||||
TODO: Cut out if not required
|
||||
|
|
@ -964,7 +1023,8 @@ $$
|
|||
### Channel model
|
||||
|
||||
Vertical, $x$: input\
|
||||
Horizontal, $y$: output
|
||||
Horizontal, $y$: output\
|
||||
Remember $\mathbf{P}$ is a matrix where each element is $P(y_j|x_i)$
|
||||
|
||||
$$
|
||||
\mathbf{P}=\left[\begin{matrix}
|
||||
|
|
@ -1005,13 +1065,23 @@ $$
|
|||
\begin{align*}
|
||||
&\text{1. Find }H(x)\\
|
||||
&\text{2. Find }[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] = [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}\\
|
||||
&\text{3. Multiply each row in $\textbf{P}$ by $p_X(a_i)$ since $p_{XY}(x_i,y_i)=P(y_i|x_i)P(x_i)$}\\
|
||||
&\text{3. Multiply each row in $\textbf{P}$ by $p_X(a_i)$ since $p_{XY}(a_i,b_i)=P(b_i|a_i)P(a_i)$}\\
|
||||
&\text{4. Find $H(x,y)$ using each element from (3.)}\\
|
||||
&\text{5. Find }H(x|y)=H(x,y)-H(y)\\
|
||||
&\text{6. Find }I(y;x)=H(x)-H(x|y)\\
|
||||
&\text{6. Find }I(x;y)=H(x)-H(x|y)\\
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Example of step **3**:
|
||||
|
||||
$$
|
||||
\mathbf{P_{XY}}=\left[\begin{matrix}
|
||||
P(y_1|x_1) P(x_1) & P(y_2|x_1) P(x_1) & \dots\\
|
||||
P(y_1|x_2) P(x_2) & P(y_2|x_2) P(x_2) & \dots\\
|
||||
\vdots & \vdots &\ddots
|
||||
\end{matrix}\right]
|
||||
$$
|
||||
|
||||
### Channel types
|
||||
|
||||
| Type | Definition |
|
||||
|
|
@ -1031,6 +1101,8 @@ $$
|
|||
\end{align*}
|
||||
$$
|
||||
|
||||
**Note that the channel capacity is realized when the channel inputs are uniformly distributed** (i.e. $P(x_1)=P(x_2)=\dots=P(x_N)=\frac{1}{N}$)
|
||||
|
||||
#### Channel capacity of an AWGN channel
|
||||
|
||||
$$
|
||||
|
|
@ -1249,4 +1321,11 @@ $$
|
|||
|
||||
- Transfer function in complex envelope form $\tilde{h}(t)$ should be divided by two.
|
||||
- Convolutions: do not forget width when using graphical method
|
||||
- todo: add more items to check
|
||||
- $2W$ for rectangle functions
|
||||
- Scale sampled spectrum by $f_s$
|
||||
- $2f_c$ for spectrum after IF mixing.
|
||||
- Square transfer function for PSD $G_y(f)=|H(f)|^\mathbf{2}G_x(f)$
|
||||
- Square besselJ function for FM power $|J_n(\beta)|^\mathbf{2}$
|
||||
- Bandwidth: only consider positive frequencies (so the bandwidth of an AM signal will be the range from the lowest to greatest sideband frequency. For a rectangular function, it will be from 0 to W).
|
||||
- TODO: add more items to check
|
||||
- TODO: add some graphics for these checklist items
|
||||
|
|
|
|||
|
|
@ -0,0 +1,34 @@
|
|||
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<text transform="matrix(1 0 0 1 144 143.0513)" font-family="Times" font-style="normal" font-size="24">1</text>
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<text transform="matrix(1 0 0 1 54 19.9438)"><tspan x="0" y="0" font-family="Times" font-style="normal" font-size="24">1 – </tspan><tspan x="36" y="0" font-family="Times" font-style="italic" font-size="24">p</tspan></text>
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<text transform="matrix(1 0 0 1 54 157.0513)"><tspan x="0" y="0" font-family="Times" font-style="normal" font-size="24">1 – </tspan><tspan x="36" y="0" font-family="Times" font-style="italic" font-size="24">p</tspan></text>
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||||
<text transform="matrix(1 0 0 1 47.0117 67.3862)" font-family="Times" font-style="italic" font-size="24">p</text>
|
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<text transform="matrix(1 0 0 1 144.6738 89.9438)" font-family="Times" font-style="italic" font-size="24">e</text>
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<text transform="matrix(1 0 0 1 47.0107 107.0732)" font-family="Times" font-style="italic" font-size="24">p</text>
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<g>
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<line fill="none" stroke="#000000" x1="15" y1="28.944" x2="131.999" y2="28.944"/>
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<path d="M134.361,27.919c-1.654-0.5-2.729-0.959-4.055-1.492v5.044c0.475-0.226,2.4-0.993,4.055-1.492
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c1.77-0.535,3.373-0.9,4.414-1.031C137.734,28.819,136.131,28.454,134.361,27.919z"/>
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</g>
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<g>
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<line fill="none" stroke="#000000" x1="15" y1="136.944" x2="131.999" y2="136.944"/>
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<path d="M134.361,135.919c-1.654-0.5-2.729-0.959-4.055-1.492v5.044c0.475-0.226,2.4-0.993,4.055-1.492
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c1.77-0.535,3.373-0.9,4.414-1.031C137.734,136.819,136.131,136.454,134.361,135.919z"/>
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</g>
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<g>
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<line fill="none" stroke="#000000" x1="15" y1="37.944" x2="131.999" y2="77.944"/>
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<path d="M134.565,77.738c-1.403-1.008-2.271-1.791-3.354-2.723l-1.632,4.773c0.521-0.06,2.593-0.163,4.318-0.1
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c1.848,0.066,3.484,0.239,4.511,0.453C137.467,79.681,136.067,78.816,134.565,77.738z"/>
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</g>
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<g>
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<line fill="none" stroke="#000000" x1="15" y1="127.944" x2="131.999" y2="86.944"/>
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<path d="M133.889,85.195c-1.726,0.076-2.892-0.003-4.319-0.067l1.669,4.761c0.373-0.37,1.938-1.731,3.332-2.749
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c1.493-1.09,2.887-1.966,3.825-2.433C137.371,84.929,135.735,85.114,133.889,85.195z"/>
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</g>
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</svg>
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|
After Width: | Height: | Size: 2.7 KiB |
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|
Before Width: | Height: | Size: 70 KiB After Width: | Height: | Size: 126 KiB |
Loading…
Reference in New Issue
Block a user