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Fix sampling function (scale $f_s$)

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Peter 2024-11-02 20:30:59 +08:00
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commit 81935d491c
1 changed files with 9 additions and 2 deletions
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_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md
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@ -438,8 +438,9 @@ $$
t&=nT_s\\ t&=nT_s\\
T_s&=\frac{1}{f_s}\\ T_s&=\frac{1}{f_s}\\
x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\ x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\
X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ X_s(f)&=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\
B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ \implies X_s(f)&=\sum_{n\in\mathbb{Z}}f_s X\left(f-n f_s\right)\quad\text{Sampling (FT)}\\
B&>\frac{1}{2}f_s\implies 2B>f_s\rightarrow\text{Aliasing}\\
\end{align*} \end{align*}
$$ $$
@ -484,6 +485,12 @@ $$
Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth. Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth.
```math
\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B}
```
<!-- MATH END -->
![By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142](/assets/img/2024-10-29-Idiots-guide-to-ELEC/Nyquist_frequency_&_rate.svg) ![By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142](/assets/img/2024-10-29-Idiots-guide-to-ELEC/Nyquist_frequency_&_rate.svg)
### 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`)