Fix $\Delta f_max$ formulas

_posts/2024-10-29-Idiots-guide-to-ELEC4402-Communications-Systems.md

@@ -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*}
 $$
@@ -438,8 +437,9 @@ $$
     t&=nT_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(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)\\
-    B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\
+    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)\\
+    \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*}
 $$
 
@@ -484,6 +484,12 @@ $$
 
 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)
 
 ### 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`)