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Fix $\Delta f_max$ formulas

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Peter 2024-11-02 22:21:40 +08:00
parent 81935d491c
commit 96c6185640
1 changed files with 8 additions and 9 deletions
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_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md
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@ -321,11 +321,11 @@ $$
s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\
s(t) &= A_c\cos(\theta_i(t))=A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_{-\infty}^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\
s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\quad\text{FM single tone}\\
f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)\quad\text{Instantaneous frequency from instantaneous phase}\\
\Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\
\Delta f&=\max_t(f_i(t))- \min_t(f_i(t))\quad\text{Maximum frequency deviation}\\
\beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\
D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\
f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)=f_c+k_f m(t)=f_c+\Delta f_\text{max}\hat m(t)\quad\text{Instantaneous frequency}\\
\Delta f_\text{max}&=\max_t|f_i(t)-f_c|=k_f \max_t |m(t)|\quad\text{Maximum frequency deviation}\\
\Delta f_\text{max}&=k_f A_m\quad\text{Maximum frequency deviation (sinusoidal)}\\
\beta&=\frac{\Delta f_\text{max}}{f_m}\quad\text{Modulation index}\\
D&=\frac{\Delta f_\text{max}}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\
\end{align*}
$$
@ -353,12 +353,11 @@ $$
$$
\begin{align*}
B &= 2Mf_m = 2(\beta + 1)f_m\\
&= 2(\Delta f+f_m)\\
&= 2(k_f A_m+f_m)\\
&= 2(\Delta f_\text{max}+f_m)\\
&= 2(D+1)W_m\\
B &= \begin{cases}
2(\Delta f+f_m) & \text{FM, sinusoidal message}\\
2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message}
2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\
2(\Delta\phi_\text{max} + 1)f_m=2(\Delta \phi_\text{max}+1)W_m & \text{PM, sinusoidal message}
\end{cases}\\
\end{align*}
$$