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Rearrange some stuff and resolve some lint errors

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Peter 2024-10-29 16:54:26 +08:00
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_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md
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@ -153,7 +153,7 @@ G_y(f)&=G(f)G_w(f)\\
| 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
@ -434,7 +434,7 @@ 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 `assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png`) ### 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`)
Cannot add directly due to copyright! Cannot add directly due to copyright!
<!-- ![sampling](copyrighted_assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png) --> <!-- ![sampling](copyrighted_assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png) -->
<!-- ![sampling](/assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png) --> <!-- ![sampling](/assets/img/2024-10-29-Idiots-guide-to-ELEC/sampling.png) -->
@ -593,22 +593,22 @@ Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as f
<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"]
@ -666,26 +666,6 @@ Bit error rate (BER) from matched filter outputs and filter output noise
## 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
| $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ |
@ -735,12 +715,34 @@ 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
@ -811,7 +813,7 @@ To solve this type of question:
### 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}
@ -830,8 +832,6 @@ P_r(kT)=\begin{cases}
\end{align*} \end{align*}
``` ```
<div style="page-break-after: always;"></div>
### Table of bandpass signalling and BER ### Table of bandpass signalling and BER
| **Binary Bandpass Signaling** | **$B_\text{null-null}$ (Hz)** | **$B_\text{abs-abs}\color{red}=2B_\text{abs}$ (Hz)** | **BER with Coherent Detection** | **BER with Noncoherent Detection** | | **Binary Bandpass Signaling** | **$B_\text{null-null}$ (Hz)** | **$B_\text{abs-abs}\color{red}=2B_\text{abs}$ (Hz)** | **BER with Coherent Detection** | **BER with Noncoherent Detection** |