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# Idiot's guide to ELEC4402 communication systems # Idiot's guide to ELEC4402 communication systems
<!-- PRINT NOTE: Use 0.20 margins all around, scale: fit to page width, and no headers or backgrounds -->
This unit allows you to bring infinite physical notes (except books borrowed from the UWA library) to all tests and the final exam. You can't rely on what material they provide in the test/exam, it is very _minimal_ to say the least. Hope this helps. This unit allows you to bring infinite physical notes (except books borrowed from the UWA library) to all tests and the final exam. You can't rely on what material they provide in the test/exam, it is very _minimal_ to say the least. Hope this helps.
If you have issues or suggestions, [raise them on GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/issues/new). I accept [pull requests](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/pulls) for fixes or suggestions but the content must not be copyrighted under a non-GPL compatible license. If you have issues or suggestions, [raise them on GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/issues/new). I accept [pull requests](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/pulls) for fixes or suggestions but the content must not be copyrighted under a non-GPL compatible license.
## [Download PDF 📄](/README.pdf) ## [Download PDF 📄](https://raw.githubusercontent.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/refs/heads/master/README.pdf)
It is recommended to refer to use [the PDF copy](/README.pdf) instead of whatever GitHub renders. It is recommended to refer to use [the PDF copy](https://raw.githubusercontent.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/refs/heads/master/README.pdf) instead of whatever GitHub renders.
## License and information ### License and information
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) 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)
@ -33,6 +35,41 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
</details> </details>
## Other advice for this unit
### Get more exam papers on OneSearch
- You can access up to **6** more papers with this method (You normally only get the previous year's paper on LMS in week 12).
- Either [search "Communications" and filter by type "Examination Papers"](https://onesearch.library.uwa.edu.au/discovery/search?query=any%2Ccontains%2Ccommunications&tab=Everything&search_scope=MyInst_and_CI&vid=61UWA_INST%3AUWA&facet=rtype%2Cinclude%2Cexampaper&lang=en&offset=0)
- Or search old unit codes
- ELEC4301 Digital Communications and Networking
- ENGT4301 Digital Communications and Networking
- ELEC3302 Communications Systems
- Note that ELEC5501 Advanced Communications is a different unit.
<details>
<summary>Listing of examination papers on OneSearch</summary>
<ul>
<li>Communications Systems ELEC3302 Examination paper [2008 Supplementary]</li>
<li>Communications Systems ELEC4402 Examination paper [2014 Semester 2]</li>
<li>Communications Systems ELEC3302 Examination paper [2014 Semester 2]</li>
<li>Communications Systems ELEC3302 Examination paper [2008 Semester 1]</li>
<li>Digital Communications and Networking ENGT4301 Examination paper [2005 Supplementary]</li>
<li>Digital Communications and Networking ELEC4301 Examination paper [2009 Supplementary]</li>
</ul>
</details>
### Tests
- A lot of the unit requires you to learn processes and apply them. This is quite time consuming to do during the semester and the marking of the tests will destroy your wam if you do not know the process (especially compared to signal processing and signals and systems), I do not recommend doing this unit during thesis year.
- This formula sheet will attempt to condense all processes/formulas you may need in this unit.
- **You do not get given a formula sheet**, so you are entirely dependent on your own notes (except for some exceptions, such as the $\text{erf}(x)$ table). So bring good notes.
- Doing this unit after signal processing is a good idea.
## Printable notes begins on next page (in PDF)
<div style="page-break-after: always;"></div>
## Fourier transform identities ## Fourier transform identities
| **Time Function** | **Fourier Transform** | | **Time Function** | **Fourier Transform** |
@ -71,6 +108,8 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Fourier transform of continuous time periodic signal ### Fourier transform of continuous time periodic signal
Required for some questions on **sampling**: Required for some questions on **sampling**:
@ -83,18 +122,21 @@ Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-
X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
``` ```
<!-- MATH END -->
Calculate $C_n$ coefficient as follows from $x_p(t)$: Calculate $C_n$ coefficient as follows from $x_p(t)$:
<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ --> <!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->
```math ```math
\begin{align*} \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\\ 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 $T_s$} &=\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*} \end{align*}
``` ```
<!-- MATH END -->
### $\text{rect}$ function ### $\text{rect}$ function
![rect](images/rect.drawio.svg) ![rect](images/rect.drawio.svg)
@ -108,6 +150,8 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$:
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### White noise ### White noise
```math ```math
@ -120,6 +164,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### WSS ### WSS
```math ```math
@ -130,6 +176,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Ergodicity ### Ergodicity
```math ```math
@ -140,12 +188,14 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
| Type | Normal | Mean square sense | | Type | Normal | Mean square sense |
| ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- | | ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- |
| ergodic in mean | $$\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0$$ | | ergodic in mean | $$\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0$$ |
| ergodic in autocorrelation function | $$\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0$$ | | ergodic in autocorrelation function | $$\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0$$ |
**A WSS random process needs to be both ergodic in mean and autocorrelation to be considered an ergodic process** Note: **A WSS random process needs to be both ergodic in mean and autocorrelation to be considered an ergodic process**
### Other identities ### Other identities
@ -157,6 +207,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Other trig ### Other trig
```math ```math
@ -174,6 +226,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
```math ```math
\begin{align*} \begin{align*}
\cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\ \cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\
@ -184,6 +238,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
```math ```math
\begin{align*} \begin{align*}
\cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\ \cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\
@ -195,6 +251,8 @@ G_y(f)&=G(f)G_w(f)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## IQ/Complex envelope ## IQ/Complex envelope
Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $e^{j\theta}$ form. Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $e^{j\theta}$ form.
@ -213,6 +271,8 @@ g_Q(t)&=A(t)\sin(\phi(t))\quad\text{Quadrature-phase component}\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### For transfer function ### For transfer function
```math ```math
@ -223,6 +283,8 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## AM ## AM
### CAM ### CAM
@ -234,13 +296,15 @@ 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
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
$B_T$: Signal bandwidth $B_T$: Signal bandwidth
$B$: Bandwidth of modulating wave $B$: Bandwidth of modulating wave
@ -255,6 +319,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## FM/PM ## FM/PM
```math ```math
@ -268,6 +334,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Bessel form and magnitude spectrum (single tone) ### Bessel form and magnitude spectrum (single tone)
```math ```math
@ -276,17 +344,21 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### FM signal power ### FM signal power
```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*}
``` ```
<!-- MATH END -->
### Carson's rule to find $B$ (98% power bandwidth rule) ### Carson's rule to find $B$ (98% power bandwidth rule)
```math ```math
@ -302,6 +374,8 @@ B &= \begin{cases}
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### $\Delta f$ of arbitrary modulating signal #### $\Delta f$ of arbitrary modulating signal
Find instantaneous frequency $f_\text{FM}$. Find instantaneous frequency $f_\text{FM}$.
@ -321,6 +395,8 @@ B_T &= 2(D+1)W_m
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Complex envelope ### Complex envelope
```math ```math
@ -331,6 +407,8 @@ B_T &= 2(D+1)W_m
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Band ### Band
| Narrowband | Wideband | | Narrowband | Wideband |
@ -354,6 +432,8 @@ B_T &= 2(D+1)W_m
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## ##
## Noise performance ## Noise performance
@ -367,6 +447,8 @@ B_T &= 2(D+1)W_m
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## Sampling ## Sampling
```math ```math
@ -379,6 +461,8 @@ B_T &= 2(D+1)W_m
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Procedure to reconstruct sampled signal ### Procedure to reconstruct sampled signal
Analog signal $x'(t)$ which can be reconstructed from a sampled signal $x_s(t)$: Put $x_s(t)$ through LPF with maximum frequency of $f_s/2$ and minimum frequency of $-f_s/2$. Anything outside of the BPF will be attenuated, therefore $n$ which results in frequencies outside the BPF will evaluate to $0$ and can be ignored. Analog signal $x'(t)$ which can be reconstructed from a sampled signal $x_s(t)$: Put $x_s(t)$ through LPF with maximum frequency of $f_s/2$ and minimum frequency of $-f_s/2$. Anything outside of the BPF will be attenuated, therefore $n$ which results in frequencies outside the BPF will evaluate to $0$ and can be ignored.
@ -403,18 +487,21 @@ Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-
X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
``` ```
<!-- MATH END -->
Calculate $C_n$ coefficient as follows from $x_p(t)$: Calculate $C_n$ coefficient as follows from $x_p(t)$:
<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ --> <!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->
```math ```math
\begin{align*} \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\\ 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 $T_s$} &=\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*} \end{align*}
``` ```
<!-- MATH END -->
<!-- Reconstruct from $\bar{X_s}(f)$ within the range $[-f_s/2,f_s/2]$ --> <!-- Reconstruct from $\bar{X_s}(f)$ within the range $[-f_s/2,f_s/2]$ -->
### Nyquist criterion for zero-ISI ### Nyquist criterion for zero-ISI
@ -425,6 +512,8 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `images/sampling.png`) ### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `images/sampling.png`)
Cannot add directly due to copyright!
![sampling](copyrighted_images/sampling.png) ![sampling](copyrighted_images/sampling.png)
![sampling](images/sampling.png) ![sampling](images/sampling.png)
@ -436,6 +525,8 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Quantization noise ### Quantization noise
```math ```math
@ -448,8 +539,12 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `images/quantizer.png`) ### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `images/quantizer.png`)
Cannot add directly due to copyright!
![quantizer](copyrighted_images/quantizer.png) ![quantizer](copyrighted_images/quantizer.png)
![quantizer](images/quantizer.png) ![quantizer](images/quantizer.png)
@ -473,6 +568,8 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## Modulation and basis functions ## Modulation and basis functions
![Constellation diagrams](./images/Constellation.drawio.svg) ![Constellation diagrams](./images/Constellation.drawio.svg)
@ -487,22 +584,28 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Symbol mapping #### Symbol mapping
```math ```math
b_n:\{1,0\}\to a_n:\{1,0\} b_n:\{1,0\}\to a_n:\{1,0\}
``` ```
<!-- MATH END -->
#### 2 possible waveforms #### 2 possible waveforms
```math ```math
\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*}
``` ```
<!-- MATH END -->
Distance is $d=\sqrt{2E_b}$ Distance is $d=\sqrt{2E_b}$
### BPSK ### BPSK
@ -515,22 +618,28 @@ Distance is $d=\sqrt{2E_b}$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Symbol mapping #### Symbol mapping
```math ```math
b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\} b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\}
``` ```
<!-- MATH END -->
#### 2 possible waveforms #### 2 possible waveforms
```math ```math
\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*}
``` ```
<!-- MATH END -->
Distance is $d=2\sqrt{E_b}$ Distance is $d=2\sqrt{E_b}$
### QPSK ($M=4$ PSK) ### QPSK ($M=4$ PSK)
@ -545,6 +654,8 @@ Distance is $d=2\sqrt{E_b}$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### 4 possible waveforms ### 4 possible waveforms
```math ```math
@ -556,6 +667,8 @@ Distance is $d=2\sqrt{E_b}$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows: Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows:
```math ```math
@ -566,6 +679,8 @@ Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as f
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Signal #### Signal
```math ```math
@ -577,27 +692,29 @@ Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as f
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Example of waveform ### Example of waveform
<details> <details>
<summary>Code</summary> <summary>Code</summary>
<pre><code> <pre><code>
tBitstream[bitstream_, Tb_, title_] := 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};
plot = plot =
Labeled[ListLinePlot[timeSteps, InterpolationOrder -> 0, Labeled[ListLinePlot[timeSteps, InterpolationOrder -> 0,
PlotRange -> Full, GridLines -> gridLines, PlotStyle -> Thick, PlotRange -> Full, GridLines -> gridLines, PlotStyle -> Thick,
Ticks -> {Table[{n Tb, Ticks -> {Table[{n Tb,
Row[{n, "\!\(\*SubscriptBox[\(T\), \(b\)]\)"}]}, {n, 0, Row[{n, "\!\(\*SubscriptBox[\(T\), \(b\)]\)"}]}, {n, 0,
Length[bitstream]}], {-1, 0, 1}}, Length[bitstream]}], {-1, 0, 1}},
LabelStyle -> Directive[Bold, 12], LabelStyle -> Directive[Bold, 12],
PlotRangePadding -> {Scaled[.05]}, AspectRatio -> 0.1, PlotRangePadding -> {Scaled[.05]}, AspectRatio -> 0.1,
ImageSize -> Large], {Style[title, "Text", 16]}, {Right}]]; ImageSize -> Large], {Style[title, "Text", 16]}, {Right}]];
tBitstream[{0, 1, 0, 0, 1, 0, 1, 1, 1, 0}, 1, "Bitstream Step Plot"] tBitstream[{0, 1, 0, 0, 1, 0, 1, 1, 1, 0}, 1, "Bitstream Step Plot"]
@ -630,19 +747,19 @@ Find transfer function $h(t)$ of matched filter and apply to an input:
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### 2. Bit error rate ### 2. Bit error rate
Bit error rate (BER) from matched filter outputs and filter output noise Bit error rate (BER) from matched filter outputs and filter output noise
```math ```math
\begin{align*} \begin{align*}
% H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right)
% \text{BER}_\text{bin}&=p Q\left(\frac{s_{o1}-V_T}{\sigma_o}\right)+(1-p)Q\left(\frac{V_T-s_{o2}}{\sigma_o}\right)\text{, $p\rightarrow$Probability $s_1(t)$ sent, $V_T\rightarrow$Threshold voltage}
Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\ Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\
E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\ E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\
T&=1/R_b\quad\text{$R_b$: Bitrate}\\ T&=1/R_b\quad\text{$R_b$: Bitrate}\\
E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ E_b&=P_\text{av}T=P_\text{av}/R_b\quad\text{Energy per bit}\\
P_\text{av}&=E_b/T=E_bR_b\quad\text{Average power}\\
P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\ P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\
P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\ P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\
\text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\ \text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\
@ -651,30 +768,12 @@ Bit error rate (BER) from matched filter outputs and filter output noise
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
<div style="page-break-after: always;"></div> <div style="page-break-after: always;"></div>
## Value tables for $\text{erf}(x)$ and $Q(x)$ ## Value tables for $\text{erf}(x)$ and $Q(x)$
### $\text{erf}(x)$ function
| $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ |
| ------ | --------------- | ------ | --------------- | ------ | --------------- |
| $0.00$ | $0.00000$ | $0.75$ | $0.71116$ | $1.50$ | $0.96611$ |
| $0.05$ | $0.05637$ | $0.80$ | $0.74210$ | $1.55$ | $0.97162$ |
| $0.10$ | $0.11246$ | $0.85$ | $0.77067$ | $1.60$ | $0.97635$ |
| $0.15$ | $0.16800$ | $0.90$ | $0.79691$ | $1.65$ | $0.98038$ |
| $0.20$ | $0.22270$ | $0.95$ | $0.82089$ | $1.70$ | $0.98379$ |
| $0.25$ | $0.27633$ | $1.00$ | $0.84270$ | $1.75$ | $0.98667$ |
| $0.30$ | $0.32863$ | $1.05$ | $0.86244$ | $1.80$ | $0.98909$ |
| $0.35$ | $0.37938$ | $1.10$ | $0.88021$ | $1.85$ | $0.99111$ |
| $0.40$ | $0.42839$ | $1.15$ | $0.89612$ | $1.90$ | $0.99279$ |
| $0.45$ | $0.47548$ | $1.20$ | $0.91031$ | $1.95$ | $0.99418$ |
| $0.50$ | $0.52050$ | $1.25$ | $0.92290$ | $2.00$ | $0.99532$ |
| $0.55$ | $0.56332$ | $1.30$ | $0.93401$ | $2.50$ | $0.99959$ |
| $0.60$ | $0.60386$ | $1.35$ | $0.94376$ | $3.00$ | $0.99998$ |
| $0.65$ | $0.64203$ | $1.40$ | $0.95229$ | $3.30$ | $0.999998$\*\* |
| $0.70$ | $0.67780$ | $1.45$ | $0.95970$ | | |
### $Q(x)$ function ### $Q(x)$ function
| $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ |
@ -724,12 +823,34 @@ Bit error rate (BER) from matched filter outputs and filter output noise
| $2.10$ | $0.017864$ | $4.40$ | $5.4125 \times 10^{-6}$ | $6.65$ | $1.4655 \times 10^{-11}$ | $8.90$ | $2.7923 \times 10^{-19}$ | | $2.10$ | $0.017864$ | $4.40$ | $5.4125 \times 10^{-6}$ | $6.65$ | $1.4655 \times 10^{-11}$ | $8.90$ | $2.7923 \times 10^{-19}$ |
| $2.15$ | $0.015778$ | $4.45$ | $4.2935 \times 10^{-6}$ | $6.70$ | $1.0421 \times 10^{-11}$ | $8.95$ | $1.7774 \times 10^{-19}$ | | $2.15$ | $0.015778$ | $4.45$ | $4.2935 \times 10^{-6}$ | $6.70$ | $1.0421 \times 10^{-11}$ | $8.95$ | $1.7774 \times 10^{-19}$ |
| $2.20$ | $0.013903$ | $4.50$ | $3.3977 \times 10^{-6}$ | $6.75$ | $7.3923 \times 10^{-12}$ | $9.00$ | $1.1286 \times 10^{-19}$ | | $2.20$ | $0.013903$ | $4.50$ | $3.3977 \times 10^{-6}$ | $6.75$ | $7.3923 \times 10^{-12}$ | $9.00$ | $1.1286 \times 10^{-19}$ |
| $2.25$ | $0.012224$ | | | | | | $2.25$ | $0.012224$ | | | | | | |
Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
### $\text{erf}(x)$ function
| $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ | $x$ | $\text{erf}(x)$ |
| ------ | --------------- | ------ | --------------- | ------ | --------------- |
| $0.00$ | $0.00000$ | $0.75$ | $0.71116$ | $1.50$ | $0.96611$ |
| $0.05$ | $0.05637$ | $0.80$ | $0.74210$ | $1.55$ | $0.97162$ |
| $0.10$ | $0.11246$ | $0.85$ | $0.77067$ | $1.60$ | $0.97635$ |
| $0.15$ | $0.16800$ | $0.90$ | $0.79691$ | $1.65$ | $0.98038$ |
| $0.20$ | $0.22270$ | $0.95$ | $0.82089$ | $1.70$ | $0.98379$ |
| $0.25$ | $0.27633$ | $1.00$ | $0.84270$ | $1.75$ | $0.98667$ |
| $0.30$ | $0.32863$ | $1.05$ | $0.86244$ | $1.80$ | $0.98909$ |
| $0.35$ | $0.37938$ | $1.10$ | $0.88021$ | $1.85$ | $0.99111$ |
| $0.40$ | $0.42839$ | $1.15$ | $0.89612$ | $1.90$ | $0.99279$ |
| $0.45$ | $0.47548$ | $1.20$ | $0.91031$ | $1.95$ | $0.99418$ |
| $0.50$ | $0.52050$ | $1.25$ | $0.92290$ | $2.00$ | $0.99532$ |
| $0.55$ | $0.56332$ | $1.30$ | $0.93401$ | $2.50$ | $0.99959$ |
| $0.60$ | $0.60386$ | $1.35$ | $0.94376$ | $3.00$ | $0.99998$ |
| $0.65$ | $0.64203$ | $1.40$ | $0.95229$ | $3.30$ | $0.999998$\*\* |
| $0.70$ | $0.67780$ | $1.45$ | $0.95970$ | | |
\*\*The value of $\text{erf}(3.30)$ should be $\approx0.999997$ instead, but this value is quoted in the formula table. \*\*The value of $\text{erf}(3.30)$ should be $\approx0.999997$ instead, but this value is quoted in the formula table.
<div style="page-break-after: always;"></div>
### Receiver output shit ### Receiver output shit
```math ```math
@ -742,6 +863,8 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
<!-- <!--
```math ```math
\begin{align*} \begin{align*}
@ -766,6 +889,8 @@ TODO:
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Raised cosine (RC) pulse ### Raised cosine (RC) pulse
![Raised cosine pulse](images/RC.drawio.svg) ![Raised cosine pulse](images/RC.drawio.svg)
@ -774,6 +899,8 @@ TODO:
0\leq\alpha\leq1 0\leq\alpha\leq1
``` ```
<!-- MATH END -->
⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method. ⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method.
To solve this type of question: To solve this type of question:
@ -781,26 +908,25 @@ To solve this type of question:
1. Use the formula for $D$ below 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 $R_b$. 2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$.
| Linear modulation ($M$-PSK, $M$-QAM) | NRZ unipolar encoding | | Linear modulation ($M$-PSK, BPSK, $M$-QAM) | NRZ unipolar encoding, BASK |
| --------------------------------------------------- | -------------------------------------------------- | | --------------------------------------------------- | -------------------------------------------------- |
| $W=B_\text{\color{green}abs-abs}$ | $W=B_\text{\color{green}abs}$ | | $W=B_\text{\color{green}abs-abs}$ | $W=B_\text{\color{green}abs}$ |
| $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ | | $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ |
| $D=\frac{W\text{ symbol/s}}{1+\alpha}$ | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$ | | $D=\frac{W\text{ symbol/s}}{1+\alpha}$ | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$ |
#### Symbol set size $M$
```math ```math
\begin{align*} \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})\\ R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\
M\text{ symbol/set}&=2^k\\ M\text{ symbol/set}&=2^k\\
E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Nyquist stuff ### Nyquist stuff
#### Condition for 0 ISI TODO: #### TODO: Condition for 0 ISI
```math ```math
P_r(kT)=\begin{cases} P_r(kT)=\begin{cases}
@ -809,6 +935,8 @@ P_r(kT)=\begin{cases}
\end{cases} \end{cases}
``` ```
<!-- MATH END -->
#### Other #### Other
```math ```math
@ -819,7 +947,7 @@ P_r(kT)=\begin{cases}
\end{align*} \end{align*}
``` ```
<div style="page-break-after: always;"></div> <!-- MATH END -->
### Table of bandpass signalling and BER ### Table of bandpass signalling and BER
@ -843,7 +971,7 @@ 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
@ -869,6 +997,8 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
Entropy is **maximized** when all have an equal probability. Entropy is **maximized** when all have an equal probability.
### Differential entropy for continuous random variables ### Differential entropy for continuous random variables
@ -881,6 +1011,8 @@ TODO: Cut out if not required
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Mutual information ### Mutual information
Amount of entropy decrease of $x$ after observation by $y$. Amount of entropy decrease of $x$ after observation by $y$.
@ -891,6 +1023,8 @@ Amount of entropy decrease of $x$ after observation by $y$.
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Channel model ### Channel model
Vertical, $x$: input\ Vertical, $x$: input\
@ -905,6 +1039,8 @@ Horizontal, $y$: output
\end{matrix}\right] \end{matrix}\right]
``` ```
<!-- MATH END -->
```math ```math
\begin{array}{c|cccc} \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
@ -915,6 +1051,8 @@ Horizontal, $y$: output
\end{array} \end{array}
``` ```
<!-- MATH END -->
Input has probability distribution $p_X(a_i)=P(X=a_i)$ 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 $`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`$
@ -929,6 +1067,8 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Fast procedure to calculate $I(y;x)$ #### Fast procedure to calculate $I(y;x)$
```math ```math
@ -942,6 +1082,8 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Channel types ### Channel types
| Type | Definition | | Type | Definition |
@ -961,16 +1103,22 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Channel capacity of an AWGN channel #### Channel capacity of an AWGN channel
```math ```math
y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2) y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2)
``` ```
<!-- MATH END -->
```math ```math
C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right) C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)
``` ```
<!-- MATH END -->
#### Channel capacity of a bandwidth AWGN channel #### Channel capacity of a bandwidth AWGN channel
Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition. Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
@ -984,6 +1132,8 @@ Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
## Channel code ## Channel code
| | | | | | | |
@ -1031,6 +1181,8 @@ Each generator vector is a binary string of size $n$. There are $k$ generator ve
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$: A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$:
```math ```math
@ -1043,6 +1195,8 @@ A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codew
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
### Systemic linear block code ### Systemic linear block code
Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits. Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits.
@ -1072,6 +1226,8 @@ Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits afte
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Parity check matrix $\mathbf{H}$ #### Parity check matrix $\mathbf{H}$
Transpose $\mathbf{P}$ for the parity check matrix Transpose $\mathbf{P}$ for the parity check matrix
@ -1105,6 +1261,8 @@ Transpose $\mathbf{P}$ for the parity check matrix
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Procedure to find parity check matrix from list of codewords #### Procedure to find parity check matrix from list of codewords
1. From the number of codewords, find $k=\log_2(N)$ 1. From the number of codewords, find $k=\log_2(N)$
@ -1124,6 +1282,8 @@ Example:
\end{array} \end{array}
``` ```
<!-- MATH END -->
Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$. Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
```math ```math
@ -1156,6 +1316,8 @@ Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
\end{align*} \end{align*}
``` ```
<!-- MATH END -->
#### Error detection and correction #### Error detection and correction
**Detection** of $s$ errors: $d_\text{min}\geq s+1$ **Detection** of $s$ errors: $d_\text{min}\geq s+1$

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