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README.md
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@ -1,9 +1,5 @@
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# Idiot's guide to ELEC4402 communication systems
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## [Download PDF 📄](/README.pdf)
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It is recommended to refer to use [the PDF copy](/README.pdf) instead of whatever GitHub renders.
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## License and information
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Notes are open-source and licensed under the GNU GPL-3.0. **You must include the [full-text of the license](/COPYING.txt) and follow its terms when using these notes or any diagrams in derivative works** (but not when printing as notes)
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@ -29,24 +25,26 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
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</details>
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[Access a web-copy of the notes for printing](/README.html)
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I accept pull requests or suggestions but the content must not be copyrighted under a non-GPL compatible license.
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## Fourier transform identities
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| **Time Function** | **Fourier Transform** |
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| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
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| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ |
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| --------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
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| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \, \text{sinc}(fT)$ |
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| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ |
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| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ |
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| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ |
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| $\exp(-at)u(t), \, a>0$ | $\frac{1}{a + j2\pi f}$ |
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| $\exp(-a\lvert t \rvert), \, a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ |
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| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ |
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| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T$ | $T \text{sinc}^2(fT)$ |
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| $\begin{cases} 1 - \frac{\lvert t \rvert}{T}, & \lvert t \rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}$ | $T \, \text{sinc}^2(fT)$ |
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| $\delta(t)$ | $1$ |
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| $1$ | $\delta(f)$ |
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| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ |
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| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ |
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| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ |
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| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}$ |
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| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift \& scale}$ |
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| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ |
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| $G(t)$ | $g(-f)\quad\text{duality property}$ |
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| $g(t)h(t)$ | $G(f)*H(f)$ |
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@ -55,19 +53,17 @@ I accept pull requests or suggestions but the content must not be copyrighted un
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| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ |
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| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ |
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| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ |
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| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ |
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| $\frac{1}{\pi t}$ | $-j \, \text{sgn}(f)$ |
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| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ |
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| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ |
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|
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```math
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\begin{align*}
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u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\
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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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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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| **Function Name** | **Formula** |
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| -------------------- | ------------------------------------------------------------------------------------------------------- |
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| Unit Step Function | $u(t) = \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}$ |
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| Signum Function | $\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}$ |
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| sinc Function | $\text{sinc}(2Wt) = \frac{\sin(2\pi W t)}{2\pi W t}$ |
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| Rectangular Function | $\text{rect}(t) = \Pi(t) = \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}$ |
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| Convolution | $g(t)*h(t)=(g*h)(t)=\int_\infty^\infty g(\tau)h(t-\tau)d\tau$ |
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### Fourier transform of continuous time periodic signal
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@ -77,21 +73,21 @@ Required for some questions on **sampling**:
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Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$:
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```math
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$$
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X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
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```
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$$
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Calculate $C_n$ coefficient as follows from $x_p(t)$:
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|
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<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->
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```math
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$$
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\begin{align*}
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% C_n&=X_p(nf_s)\\
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C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\
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&=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
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\end{align*}
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```
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$$
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### $\text{rect}$ function
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@ -99,16 +95,16 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$:
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### Bessel function
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```math
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$$
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\begin{align*}
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\sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\
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\sum_{n\in\Z}{J_n}^2(\beta)&=1\\
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J_n(\beta)&=(-1)^nJ_{-n}(\beta)
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\end{align*}
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```
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$$
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### White noise
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```math
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$$
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\begin{align*}
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R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\
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G_w(f)&=\frac{N_0}{2}\\
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@ -116,27 +112,27 @@ N_0&=kT\\
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G_y(f)&=|H(f)|^2G_w(f)\\
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G_y(f)&=G(f)G_w(f)\\
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\end{align*}
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```
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$$
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### WSS
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```math
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$$
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\begin{align*}
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\mu_X(t) &= \mu_X\text{ Constant}\\
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R_{XX}(t_1,t_2)&=R_X(t_1-t_2)=R_X(\tau)\\
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E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)]
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\end{align*}
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```
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$$
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### Ergodicity
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|
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```math
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$$
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\begin{align*}
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\braket{X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t)dt\\
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\braket{X(t+\tau)X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t+\tau)x(t)dt\\
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E[\braket{X(t)}_T]&=\frac{1}{2T}\int_{-T}^{T}x(t)dt=\frac{1}{2T}\int_{-T}^{T}m_Xdt=m_X\\
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\end{align*}
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```
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$$
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|
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| Type | Normal | Mean square sense |
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| ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- |
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|
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@ -147,17 +143,17 @@ G_y(f)&=G(f)G_w(f)\\
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### Other identities
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|
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```math
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$$
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\begin{align*}
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f*(g*h) &=(f*g)*h\quad\text{Convolution associative}\\
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a(f*g) &= (af)*g \quad\text{Convolution associative}\\
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\sum_{x=-\infty}^\infty(f(x a)\delta(\omega-x b))&=f\left(\frac{\omega a}{b}\right)
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\end{align*}
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```
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$$
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### Other trig
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|
||||
```math
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$$
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\begin{align*}
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\cos2\theta=2 \cos^2 \theta-1&\Leftrightarrow\frac{\cos2\theta+1}{2}=\cos^2\theta\\
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e^{-j\alpha}-e^{j\alpha}&=-2j \sin(\alpha)\\
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|
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@ -168,11 +164,11 @@ G_y(f)&=G(f)G_w(f)\\
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\sin(A-\pi/2)&=-\cos(A)\\
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\cos(A-\pi/2)&=\sin(A)\\
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\cos(A+\pi/2)&=-\sin(A)\\
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\int_{x\in\mathbb{R}}\text{sinc}(A x) &= \frac{1}{|A|}\\
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\int_{x\in\R}\text{sinc}(A x) &= \frac{1}{|A|}\\
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\end{align*}
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```
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$$
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|
||||
```math
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$$
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\begin{align*}
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\cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\
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\sin(A+B) &= \sin (A) \cos (B)+\cos (A) \sin (B) \\
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|
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@ -180,9 +176,9 @@ G_y(f)&=G(f)G_w(f)\\
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\cos(A)\sin(B) &= \frac{1}{2} (\sin (A+B)-\sin (A-B)) \\
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\sin(A)\sin(B) &= \frac{1}{2} (\cos (A-B)-\cos (A+B)) \\
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\end{align*}
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```
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$$
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```math
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$$
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\begin{align*}
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\cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\
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\cos(A)-\cos(B) &= -2 \sin \left(\frac{A}{2}-\frac{B}{2}\right) \sin \left(\frac{A}{2}+\frac{B}{2}\right) \\
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|
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@ -191,7 +187,7 @@ G_y(f)&=G(f)G_w(f)\\
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\cos(A)+\sin(B)&= -2 \sin \left(\frac{A}{2}-\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}+\frac{B}{2}+\frac{\pi }{4}\right) \\
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\cos(A)-\sin(B)&= -2 \sin \left(\frac{A}{2}+\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}-\frac{B}{2}+\frac{\pi }{4}\right) \\
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\end{align*}
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```
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$$
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## IQ/Complex envelope
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@ -199,7 +195,7 @@ Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $
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|
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### Convert complex envelope representation to time-domain representation of signal
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```math
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$$
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\begin{align*}
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g(t)&=g_I(t)\cos(2\pi f_c t)-g_Q(t)\sin(2\pi f_c t)\\
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&=\text{Re}[\tilde{g}(t)\exp{(j2\pi f_c t)}]\\
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@ -209,23 +205,23 @@ A(t)&=|g(t)|=\sqrt{g_I^2(t)+g_Q^2(t)}\quad\text{Amplitude}\\
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g_I(t)&=A(t)\cos(\phi(t))\quad\text{In-phase component}\\
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g_Q(t)&=A(t)\sin(\phi(t))\quad\text{Quadrature-phase component}\\
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\end{align*}
|
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```
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$$
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|
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### For transfer function
|
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|
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```math
|
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$$
|
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\begin{align*}
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h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
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&=2\text{Re}[\tilde{h}(t)\exp{(j2\pi f_c t)}]\\
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\Rightarrow\tilde{h}(t)&=h_I(t)/2+jh_Q(t)/2=A(t)/2\exp{(j\phi(t))}
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\end{align*}
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```
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$$
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## AM
|
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|
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### CAM
|
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|
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```math
|
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$$
|
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\begin{align*}
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m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
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m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\
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|
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@ -237,7 +233,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
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\eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\
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B_T&=2f_m=2B
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\end{align*}
|
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```
|
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$$
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|
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$B_T$: Signal bandwidth
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$B$: Bandwidth of modulating wave
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|
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@ -246,16 +242,16 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
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|
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### DSB-SC
|
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|
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```math
|
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$$
|
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\begin{align*}
|
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x_\text{DSB}(t) &= A_c \cos{(2\pi f_c t)} m(t)\\
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B_T&=2f_m=2B
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\end{align*}
|
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```
|
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$$
|
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|
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## FM/PM
|
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|
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```math
|
||||
$$
|
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\begin{align*}
|
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s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\
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s(t) &= A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_0^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\
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|
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@ -264,30 +260,34 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
|
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\Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\
|
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D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
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|
||||
### Bessel form and magnitude spectrum (single tone)
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t]
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
<!-- ### Bessel repr.
|
||||
|
||||
$$s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m) t)$$ -->
|
||||
|
||||
### FM signal power
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
P_\text{av}&=\frac{{A_c}^2}{2}\\
|
||||
P_\text{band\_index}&=\frac{{A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\
|
||||
\text{band\_index}&=0\implies f_c+0f_m\\
|
||||
\text{band\_index}&=1\implies f_c+1f_m,\dots\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Carson's rule to find $B$ (98% power bandwidth rule)
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
B &= 2Mf_m = 2(\beta + 1)f_m\\
|
||||
&= 2(\Delta f+f_m)\\
|
||||
|
|
@ -298,7 +298,7 @@ B &= \begin{cases}
|
|||
2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message}
|
||||
\end{cases}\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### $\Delta f$ of arbitrary modulating signal
|
||||
|
||||
|
|
@ -306,7 +306,7 @@ Find instantaneous frequency $f_\text{FM}$.
|
|||
|
||||
$M$: Number of **pairs** of significant sidebands
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s(t)&=A_c\cos(\theta_\text{FM}(t))\\
|
||||
f_\text{FM}(t) &= \frac{1}{2\pi}\frac{d\theta_\text{FM}(t)}{dt}\\
|
||||
|
|
@ -317,17 +317,17 @@ W_m &= \text{max}(\text{frequencies in $\theta_\text{FM}(t)$...}) \\
|
|||
D &= \frac{\Delta f}{W_m}\\
|
||||
B_T &= 2(D+1)W_m
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Complex envelope
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t)) \Leftrightarrow \tilde{s}(t) = A_c\exp(j\beta\sin(2\pi f_m t))\\
|
||||
s(t)&=\text{Re}[\tilde{s}(t)\exp{(j2\pi f_c t)}]\\
|
||||
\tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t)
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Band
|
||||
|
||||
|
|
@ -337,7 +337,7 @@ B_T &= 2(D+1)W_m
|
|||
|
||||
## Power, energy and autocorrelation
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
G_\text{WGN}(f)&=\frac{N_0}{2}\\
|
||||
G_x(f)&=|H(f)|^2G_w(f)\text{ (PSD)}\\
|
||||
|
|
@ -350,24 +350,24 @@ B_T &= 2(D+1)W_m
|
|||
E&=\int_{-\infty}^{\infty}|x(t)|^2dt=|X(f)|^2\\
|
||||
R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
##
|
||||
|
||||
## Noise performance
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{CNR}_\text{in} &= \frac{P_\text{in}}{P_\text{noise}}\\
|
||||
\text{CNR}_\text{in,FM} &= \frac{A^2}{2WN_0}\\
|
||||
\text{SNR}_\text{FM} &= \frac{3A^2k_f^2P}{2N_0W^3}\\
|
||||
\text{SNR(dB)} &= 10\log_{10}(\text{SNR}) \quad\text{Decibels from ratio}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
## Sampling
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
t&=nT_s\\
|
||||
T_s&=\frac{1}{f_s}\\
|
||||
|
|
@ -375,7 +375,7 @@ B_T &= 2(D+1)W_m
|
|||
X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\
|
||||
B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Procedure to reconstruct sampled signal
|
||||
|
||||
|
|
@ -397,21 +397,21 @@ Required for some questions on **sampling**:
|
|||
|
||||
Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$:
|
||||
|
||||
```math
|
||||
$$
|
||||
X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
|
||||
```
|
||||
$$
|
||||
|
||||
Calculate $C_n$ coefficient as follows from $x_p(t)$:
|
||||
|
||||
<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
% C_n&=X_p(nf_s)\\
|
||||
C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\
|
||||
&=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
<!-- Reconstruct from $\bar{X_s}(f)$ within the range $[-f_s/2,f_s/2]$ -->
|
||||
|
||||
|
|
@ -428,15 +428,15 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
|
|||
|
||||
## Quantizer
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Quantization noise
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
e &:= y-x\quad\text{Quantization error}\\
|
||||
\mu_E &= E[E] = 0\quad\text{Zero mean}\\
|
||||
|
|
@ -444,7 +444,7 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
|
|||
\text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise}}\\
|
||||
\text{SQNR(dB)}&=10\log_{10}(\text{SQNR})
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `images/quantizer.png`)
|
||||
|
||||
|
|
@ -455,7 +455,7 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
|
|||
|
||||
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
R_b&\rightarrow\text{Bit rate}\\
|
||||
D&\rightarrow\text{Symbol rate | }R_d\text{ | }1/T_b\\
|
||||
|
|
@ -469,7 +469,7 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
|
|||
G_\text{unipolarNRZ}(f)&=\frac{A^2}{4R_b}\left(\text{sinc}^2\left(\frac{f}{R_b}\right)+R_b\delta(f)\right)\\
|
||||
G_\text{unipolarRZ}(f)&=\frac{A^2}{16} \left(\sum _{l=-\infty }^{\infty } \delta \left(f-\frac{l}{T_b}\right) \left| \text{sinc}(\text{duty} \times l) \right| {}^2+T_b \left| \text{sinc}\left(\text{duty} \times f T_b\right) \right| {}^2\right), \text{NB}_0=2R_b
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
## Modulation and basis functions
|
||||
|
||||
|
|
@ -479,27 +479,25 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
|
|||
|
||||
#### Basis functions
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Symbol mapping
|
||||
|
||||
```math
|
||||
b_n:\{1,0\}\to a_n:\{1,0\}
|
||||
```
|
||||
$$b_n:\{1,0\}\to a_n:\{1,0\}$$
|
||||
|
||||
#### 2 possible waveforms
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\
|
||||
s_1(t)&=0\\
|
||||
&\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + 0)=\frac{{A_c}^2}{4}T_b$}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
Distance is $d=\sqrt{2E_b}$
|
||||
|
||||
|
|
@ -507,27 +505,25 @@ Distance is $d=\sqrt{2E_b}$
|
|||
|
||||
#### Basis functions
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Symbol mapping
|
||||
|
||||
```math
|
||||
b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\}
|
||||
```
|
||||
$$b_n:\{1,0\}\to a_n:\{1,\color{lime}-1\color{white}\}$$
|
||||
|
||||
#### 2 possible waveforms
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{E_b}\varphi_1(t)\\
|
||||
s_1(t)&=-A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=-\sqrt{E_b}\varphi_2(t)\\
|
||||
&\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + \frac{{A_c}^2}{2}\times T_b)=\frac{{A_c}^2}{2}T_b$}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
Distance is $d=2\sqrt{E_b}$
|
||||
|
||||
|
|
@ -535,45 +531,45 @@ Distance is $d=2\sqrt{E_b}$
|
|||
|
||||
#### Basis functions
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
T &= 2 T_b\quad\text{Time per symbol for two bits $T_b$}\\
|
||||
\varphi_1(t) &= \sqrt{\frac{2}{T}}\cos(2\pi f_c t)\quad0\leq t\leq T\\
|
||||
\varphi_2(t) &= \sqrt{\frac{2}{T}}\sin(2\pi f_c t)\quad0\leq t\leq T\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### 4 possible waveforms
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s_1(t)&=\sqrt{E_s/2}\left[\varphi_1(t)+\varphi_2(t)\right]\\
|
||||
s_2(t)&=\sqrt{E_s/2}\left[\varphi_1(t)-\varphi_2(t)\right]\\
|
||||
s_3(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)+\varphi_2(t)\right]\\
|
||||
s_4(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)-\varphi_2(t)\right]\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows:
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
s_i(t)&=A_c\sqrt{T/2}/\sqrt{2}\times\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
|
||||
&=\sqrt{T{A_c}^2/4}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
|
||||
&=\sqrt{E_s/2}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Signal
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{Symbol mapping: }& \left\{1,0\right\}\to\left\{1,-1\right\}\\
|
||||
I(t) &= b_{2n}\varphi_1(t)\quad\text{Even bits}\\
|
||||
Q(t) &= b_{2n+1}\varphi_2(t)\quad\text{Odd bits}\\
|
||||
x(t) &= A_c[I(t)\cos(2\pi f_c t)-Q(t)\sin(2\pi f_c t)]
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Example of waveform
|
||||
|
||||
|
|
@ -619,20 +615,20 @@ Remember that $T=2T_b$
|
|||
|
||||
Find transfer function $h(t)$ of matched filter and apply to an input:
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
h(t)&=s_1(T-t)-s_2(T-t)\\
|
||||
h(t)&=s^*(T-t) \qquad\text{((.)* is the conjugate)}\\
|
||||
s_{on}(t)&=h(t)*s_n(t)=\int_\infty^\infty h(\tau)s_n(t-\tau)d\tau\quad\text{Filter output}\\
|
||||
n_o(t)&=h(t)*n(t)\quad\text{Noise at filter output}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### 2. Bit error rate
|
||||
|
||||
Bit error rate (BER) from matched filter outputs and filter output noise
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
% H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right)
|
||||
|
||||
|
|
@ -647,7 +643,7 @@ Bit error rate (BER) from matched filter outputs and filter output noise
|
|||
\text{BER}_\text{unipolarNRZ|BASK}&=Q\left(\sqrt{\frac{d^2}{N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)\\
|
||||
\text{BER}_\text{polarNRZ|BPSK}&=Q\left(\sqrt{\frac{2d^2}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
<div style="page-break-after: always;"></div>
|
||||
|
||||
|
|
@ -677,7 +673,7 @@ Bit error rate (BER) from matched filter outputs and filter output noise
|
|||
|
||||
| $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ |
|
||||
| ------ | ---------- | ------ | ----------------------- | ------ | ------------------------ | ------ | ------------------------ |
|
||||
| $0.00$ | $0.5$ | $2.30$ | $0.010724$ | $4.55$ | $2.6823 \times 10^{-6}$ | $6.80$ | $5.231 \times 10^{-12}$ |
|
||||
| $0.00$ | $0.5 | $2.30$ | $0.010724$ | $4.55$ | $2.6823 \times 10^{-6}$ | $6.80$ | $5.231 \times 10^{-12}$ |
|
||||
| $0.05$ | $0.48006$ | $2.35$ | $0.0093867$ | $4.60$ | $2.1125 \times 10^{-6}$ | $6.85$ | $3.6925 \times 10^{-12}$ |
|
||||
| $0.10$ | $0.46017$ | $2.40$ | $0.0081975$ | $4.65$ | $1.6597 \times 10^{-6}$ | $6.90$ | $2.6001 \times 10^{-12}$ |
|
||||
| $0.15$ | $0.44038$ | $2.45$ | $0.0071428$ | $4.70$ | $1.3008 \times 10^{-6}$ | $6.95$ | $1.8264 \times 10^{-12}$ |
|
||||
|
|
@ -730,7 +726,7 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
|
|||
|
||||
### Receiver output shit
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
r_o(t)&=\begin{cases}
|
||||
s_{o1}(t)+n_o(t) & \text{code 1}\\
|
||||
|
|
@ -738,14 +734,14 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
|
|||
\end{cases}\\
|
||||
n&: \text{AWGN with }\sigma_o^2\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
<!--
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
G_x(f)
|
||||
\end{align*}
|
||||
``` -->
|
||||
$$ -->
|
||||
|
||||
## ISI, channel model
|
||||
|
||||
|
|
@ -755,22 +751,20 @@ TODO:
|
|||
|
||||
### Nomenclature
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
D&\rightarrow\text{Symbol Rate, Max. Signalling Rate}\\
|
||||
T&\rightarrow\text{Symbol Duration}\\
|
||||
M&\rightarrow\text{Symbol set size}\\
|
||||
W&\rightarrow\text{Bandwidth}\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Raised cosine (RC) pulse
|
||||
|
||||
|
||||
|
||||
```math
|
||||
0\leq\alpha\leq1
|
||||
```
|
||||
$$0\leq\alpha\leq1$$
|
||||
|
||||
⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method.
|
||||
|
||||
|
|
@ -779,43 +773,50 @@ To solve this type of question:
|
|||
1. Use the formula for $D$ below
|
||||
2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$.
|
||||
|
||||
<!-- $$
|
||||
V_\text{RC}(f)=\begin{cases}
|
||||
T & 0\le|f|\le(1-\alpha)/2T\\
|
||||
\frac{T}{2}\left(1+\cos\left[\right]\right)
|
||||
\end{cases}
|
||||
$$ -->
|
||||
|
||||
| Linear modulation ($M$-PSK, $M$-QAM) | NRZ unipolar encoding |
|
||||
| --------------------------------------------------- | -------------------------------------------------- |
|
||||
| $W=B_\text{\color{green}abs-abs}$ | $W=B_\text{\color{green}abs}$ |
|
||||
| $W=B_\text{\color{lime}abs-abs}$ | $W=B_\text{\color{lime}abs}$ |
|
||||
| $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ |
|
||||
| $D=\frac{W\text{ symbol/s}}{1+\alpha}$ | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$ |
|
||||
|
||||
#### Symbol set size $M$
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
D\text{ symbol/s}&=\frac{2W\text{ Hz}}{1+\alpha}\\
|
||||
R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\
|
||||
M\text{ symbol/set}&=2^k\\
|
||||
E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Nyquist stuff
|
||||
|
||||
#### Condition for 0 ISI TODO:
|
||||
#### Condition for 0 ISI
|
||||
|
||||
```math
|
||||
$$
|
||||
P_r(kT)=\begin{cases}
|
||||
1 & k=0\\
|
||||
0 & k\neq0
|
||||
\end{cases}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Other
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{Excess BW}&=B_\text{abs}-B_\text{Nyquist}=\frac{1+\alpha}{2T}-\frac{1}{2T}=\frac{\alpha}{2T}\quad\text{FOR NRZ (Use correct $B_\text{abs}$)}\\
|
||||
\alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\
|
||||
T&=1/D
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
<div style="page-break-after: always;"></div>
|
||||
|
||||
|
|
@ -853,7 +854,7 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems
|
|||
|
||||
### Entropy for discrete random variables
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
H(x) &\geq 0\\
|
||||
H(x) &= -\sum_{x_i\in A_x} p_X(x_i) \log_2(p_X(x_i))\\
|
||||
|
|
@ -865,7 +866,7 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems
|
|||
H(x|y) &= H(x,y)-H(y)\\
|
||||
H(x,y) &= H(x) + H(y|x) = H(y) + H(x|y)\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
Entropy is **maximized** when all have an equal probability.
|
||||
|
||||
|
|
@ -873,63 +874,64 @@ Entropy is **maximized** when all have an equal probability.
|
|||
|
||||
TODO: Cut out if not required
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
h(x) &= -\int_\mathbb{R}f_X(x)\log_2(f_X(x))dx
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Mutual information
|
||||
|
||||
Amount of entropy decrease of $x$ after observation by $y$.
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
I(x;y) &= H(x)-H(x|y)=H(y)-H(y|x)\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Channel model
|
||||
|
||||
Vertical, $x$: input\
|
||||
Horizontal, $y$: output
|
||||
|
||||
```math
|
||||
$$
|
||||
\mathbf{P}=\left[\begin{matrix}
|
||||
p_{11} & p_{12} &\dots & p_{1N}\\
|
||||
p_{21} & p_{22} &\dots & p_{2N}\\
|
||||
\vdots & \vdots &\ddots & \vdots\\
|
||||
p_{M1} & p_{M2} &\dots & p_{MN}\\
|
||||
\end{matrix}\right]
|
||||
```
|
||||
$$
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{array}{c|cccc}
|
||||
P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\\hline
|
||||
P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\
|
||||
\hline
|
||||
x_1 & p_{11} & p_{12} & \dots & p_{1N} \\
|
||||
x_2 & p_{21} & p_{22} & \dots & p_{2N} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots \\
|
||||
x_M & p_{M1} & p_{M2} & \dots & p_{MN} \\
|
||||
\end{array}
|
||||
```
|
||||
$$
|
||||
|
||||
Input has probability distribution $p_X(a_i)=P(X=a_i)$
|
||||
|
||||
Channel maps alphabet $`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`$
|
||||
Channel maps alphabet $\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}$
|
||||
|
||||
Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
p_Y(b_j) &= \sum_{i=1}^{M}P[x=a_i,y=b_j]\quad 1\leq j\leq N \\
|
||||
&= \sum_{i=1}^{M}P[X=a_i]P[Y=b_j|X=a_i]\\
|
||||
[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] &= [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Fast procedure to calculate $I(y;x)$
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
&\text{1. Find }H(x)\\
|
||||
&\text{2. Find }[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] = [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}\\
|
||||
|
|
@ -938,7 +940,7 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
|
|||
&\text{5. Find }H(x|y)=H(x,y)-H(y)\\
|
||||
&\text{6. Find }I(y;x)=H(x)-H(x|y)\\
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Channel types
|
||||
|
||||
|
|
@ -949,7 +951,7 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
|
|||
|
||||
#### Channel capacity of weakly symmetric channel
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
C &\to\text{Channel capacity (bits/channels used)}\\
|
||||
N &\to\text{Output alphabet size}\\
|
||||
|
|
@ -957,38 +959,38 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
|
|||
C &= \log_2(N)-H(\mathbf{p})\quad\text{Capacity for weakly symmetric and symmetric channels}\\
|
||||
R &< C \text{ for error-free transmission}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Channel capacity of an AWGN channel
|
||||
|
||||
```math
|
||||
$$
|
||||
y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2)
|
||||
```
|
||||
$$
|
||||
|
||||
```math
|
||||
$$
|
||||
C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)
|
||||
```
|
||||
$$
|
||||
|
||||
#### Channel capacity of a bandwidth AWGN channel
|
||||
|
||||
Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
P_s&\to\text{Bandwidth limited average power}\\
|
||||
y_i&=\text{bandpass}_W(x_i)+n_i\quad n_i\thicksim N(0,N_0/2)\\
|
||||
C&=W\log_2\left(1+\frac{P_s}{N_0 W}\right)\\
|
||||
C&=W\log_2(1+\text{SNR})\quad\text{SNR}=P_s/(N_0 W)
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
## Channel code
|
||||
|
||||
| | | |
|
||||
| ---------------- | --------------------------------- | ---------------------------------------------------------------------------------------------------------------------------- |
|
||||
| ---------------- | --------------------------------- | -------------------------------------------------------------------------------------------------------------------------- |
|
||||
| Hamming weight | $w_H(x)$ | Number of `'1'` in codeword $x$ |
|
||||
| Hamming distance | $d_H(x_1,x_2)=w_H(x_1\oplus x_2)$ | Number of different bits between codewords $x_1$ and $x_2$ which is the hamming weight of the XOR of the two codes. |
|
||||
| Minimum distance | $d_\text{min}$ | **IMPORTANT**: $x\neq\textbf{0}$, excludes weight of all-zero codeword. For a linear block code, $d_\text{min}=w_\text{min}$ |
|
||||
| Minimum distance | $d_\text{min}$ | **IMPORTANT**: $x\neq\bold{0}$, excludes weight of all-zero codeword. For a linear block code, $d_\text{min}=w_\text{min}$ |
|
||||
|
||||
### Linear block code
|
||||
|
||||
|
|
@ -1009,7 +1011,7 @@ For a linear block code, $d_\text{min}=w_\text{min}$
|
|||
|
||||
Each generator vector is a binary string of size $n$. There are $k$ generator vectors in $\mathbf{G}$.
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\mathbf{g}_i&=[\begin{matrix}
|
||||
g_{i,0}& \dots & g_{i,n-2} & g_{i,n-1}
|
||||
|
|
@ -1027,11 +1029,11 @@ Each generator vector is a binary string of size $n$. There are $k$ generator ve
|
|||
g_{k-1,0}& \dots & g_{k-1,n-2} & g_{k-1,n-1}\\
|
||||
\end{matrix}\right]
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$:
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\mathbf{m}&=[\begin{matrix}
|
||||
m_{0}& \dots & m_{n-2} & m_{k-1}
|
||||
|
|
@ -1039,13 +1041,13 @@ A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codew
|
|||
\color{darkgray}\mathbf{m}&\color{darkgray}=[101001]\quad\text{Example for $k=6$}\\
|
||||
\mathbf{x} &= \mathbf{m}\mathbf{G}=m_0\mathbf{g}_0+m_1\mathbf{g}_1+\dots+m_{k-1}\mathbf{g}_{k-1}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
### Systemic linear block code
|
||||
|
||||
Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits.
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\mathbf{G}&=\begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\left[
|
||||
\begin{array}{c|c}
|
||||
|
|
@ -1068,13 +1070,13 @@ Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits afte
|
|||
\mathbf{x} &= \mathbf{m}\mathbf{G}= \mathbf{m} \begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\begin{array}{c|c}[\mathbf{mI}_k & \mathbf{mP}]\end{array}=\begin{array}{c|c}[\mathbf{m} & \mathbf{b}]\end{array}\\
|
||||
\mathbf{b} &= \mathbf{m}\mathbf{P}\quad\text{Parity bits of $\mathbf{x}$}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Parity check matrix $\mathbf{H}$
|
||||
|
||||
Transpose $\mathbf{P}$ for the parity check matrix
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\mathbf{H}&=\begin{array}{c|c}[\mathbf{P}^\text{T} & \mathbf{I}_{n-k}]\end{array}\\
|
||||
&=\left[
|
||||
|
|
@ -1101,7 +1103,7 @@ Transpose $\mathbf{P}$ for the parity check matrix
|
|||
\end{matrix}\end{array}\right]\\
|
||||
\mathbf{xH}^\text{T}&=\mathbf{0}\implies\text{Codeword is valid}
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Procedure to find parity check matrix from list of codewords
|
||||
|
||||
|
|
@ -1112,28 +1114,30 @@ Transpose $\mathbf{P}$ for the parity check matrix
|
|||
|
||||
Example:
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{array}{cccc}
|
||||
x_1 & x_2 & x_3 & x_4 & x_5 \\\hline
|
||||
x_1 & x_2 & x_3 & x_4 & x_5 \\
|
||||
\hline
|
||||
\color{magenta}1&\color{magenta}0&1&1&0\\
|
||||
\color{magenta}0&\color{magenta}1&1&1&1\\
|
||||
\color{magenta}0&\color{magenta}0&0&0&0\\
|
||||
\color{magenta}1&\color{magenta}1&0&0&1\\
|
||||
\end{array}
|
||||
```
|
||||
$$
|
||||
|
||||
Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
|
||||
|
||||
```math
|
||||
$$
|
||||
\begin{align*}
|
||||
\begin{align*}
|
||||
\begin{aligned}
|
||||
x_3 &= x_1\oplus x_2\\
|
||||
x_4 &= x_1\oplus x_2\\
|
||||
x_5 &= x_2\\
|
||||
\end{aligned}
|
||||
\end{align*}
|
||||
\implies\textbf{P}&=
|
||||
\begin{array}{c|cc}
|
||||
& x_1 & x_2 \\\hline
|
||||
\begin{array}{c|ccc}
|
||||
& x_1 & x_2 \\
|
||||
\hline
|
||||
x_3&1&1&\\
|
||||
x_4&1&1&\\
|
||||
x_5&0&1&\\
|
||||
|
|
@ -1152,7 +1156,7 @@ Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
|
|||
0 & 0 & 1\\
|
||||
\end{matrix}\end{array}\right]
|
||||
\end{align*}
|
||||
```
|
||||
$$
|
||||
|
||||
#### Error detection and correction
|
||||
|
||||
|
|
@ -1162,6 +1166,6 @@ Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
|
|||
|
||||
## CHECKLIST
|
||||
|
||||
- Transfer function in complex envelope form $\tilde{h}(t)$ should be divided by two.
|
||||
- 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
|
||||
|
|
|
|||
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Reference in New Issue
Block a user