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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 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
@ -666,26 +666,6 @@ Bit error rate (BER) from matched filter outputs and filter output noise
## Value tables for $\text{erf}(x)$ and $Q(x)$
### $\text{erf}(x)$ 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
| $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ |
@ -735,12 +715,34 @@ Bit error rate (BER) from matched filter outputs and filter output noise
| $2.10$ | $0.017864$ | $4.40$ | $5.4125 \times 10^{-6}$ | $6.65$ | $1.4655 \times 10^{-11}$ | $8.90$ | $2.7923 \times 10^{-19}$ |
| $2.15$ | $0.015778$ | $4.45$ | $4.2935 \times 10^{-6}$ | $6.70$ | $1.0421 \times 10^{-11}$ | $8.95$ | $1.7774 \times 10^{-19}$ |
| $2.20$ | $0.013903$ | $4.50$ | $3.3977 \times 10^{-6}$ | $6.75$ | $7.3923 \times 10^{-12}$ | $9.00$ | $1.1286 \times 10^{-19}$ |
| $2.25$ | $0.012224$ | | | | |
| $2.25$ | $0.012224$ | | | | | | |
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.
<div style="page-break-after: always;"></div>
### Receiver output shit
```math
@ -811,7 +813,7 @@ To solve this type of question:
### Nyquist stuff
#### Condition for 0 ISI TODO:
#### TODO: Condition for 0 ISI
```math
P_r(kT)=\begin{cases}
@ -830,8 +832,6 @@ P_r(kT)=\begin{cases}
\end{align*}
```
<div style="page-break-after: always;"></div>
### 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** |