Fix $\Delta f_max$ formulas

README.md

@@ -1,12 +1,6 @@
 # Idiot's guide to ELEC4402 communication systems
 
-<style>
+<div class="info-text">
-@media print{
-   .copyrighted{
-       display: none !important;
-   }
-}
-</style>
 
 <!-- PRINT NOTE: Use 0.20 margins all around, scale: fit to page width, and no headers or backgrounds -->
 
@@ -20,7 +14,7 @@ It is recommended to refer to use [the PDF copy](https://raw.githubusercontent.c
 
 ### License and information
 
-Notes are open-source and licensed under the GNU GPL-3.0. **You must include the [full-text of the license](/COPYING.txt) and follow its terms when using these notes or any diagrams in derivative works** (but not when printing as notes)
+Notes are open-source and licensed under the GNU GPL-3.0. **You must include the [full-text of the license](COPYING.txt) and follow its terms when using these notes or any diagrams in derivative works** (but not when printing as notes)
 
 Copyright (C) 2024 Peter Tanner
 
@@ -74,9 +68,16 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 - **You do not get given a formula sheet**, so you are entirely dependent on your own notes (except for some exceptions, such as the $\text{erf}(x)$ table). So bring good notes.
 - Doing this unit after signal processing is a good idea.
 
-## Printable notes begins on next page (in PDF)
-
 <div style="page-break-after: always;"></div>
+</div>
+
+<div class="short-info">
+
+**[https://www.petertanner.dev/posts/Idiots-guide-to-ELEC4402-Communications-Systems/](https://www.petertanner.dev/posts/Idiots-guide-to-ELEC4402-Communications-Systems/)**
+
+**Notes are open-source and licensed under the [GNU GPL-3.0](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/blob/master/COPYING.txt). Suggest any corrections or changes on [GitHub](https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems).**
+
+</div>
 
 ## Fourier transform identities and properties
 
@@ -316,11 +317,11 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
 
 ```math
 \begin{align*}
+    x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\
+    &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\
     m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
     m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\
     m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\
-    x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\
-    &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\
     P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\
     P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\
     \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\
@@ -353,11 +354,11 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
     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}\\
+    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&=\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_\text{max}&=\max_t|f_i(t)-f_c|=k_f \max_t |m(t)|\quad\text{Maximum frequency deviation}\\
-    \Delta f&=\max_t(f_i(t))- \min_t(f_i(t))\quad\text{Maximum frequency deviation}\\
+    \Delta f_\text{max}&=k_f A_m\quad\text{Maximum frequency deviation (sinusoidal)}\\
-    \beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\
+    \beta&=\frac{\Delta f_\text{max}}{f_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)}\\
+    D&=\frac{\Delta f_\text{max}}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\
 \end{align*}
 ```
 
@@ -391,12 +392,11 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
 ```math
 \begin{align*}
 B &= 2Mf_m = 2(\beta + 1)f_m\\
-    &= 2(\Delta f+f_m)\\
+    &= 2(\Delta f_\text{max}+f_m)\\
-    &= 2(k_f A_m+f_m)\\
     &= 2(D+1)W_m\\
 B &= \begin{cases}
-    2(\Delta f+f_m) & \text{FM, sinusoidal message}\\
+    2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\
-    2(\Delta\phi + 1)f_m & \text{PM, 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*}
 ```
@@ -440,6 +440,9 @@ B &= \begin{cases}
 
 <!-- MATH END -->
 
+<!-- ADJUST ACCORDING TO PDF OUTPUT -->
+<div style="page-break-after: always;"></div>
+
 ## Noise performance
 
 Coherent detection system.
@@ -486,8 +489,9 @@ Use these formulas in particular:
     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)\\
+    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*}
 ```
 
@@ -538,6 +542,12 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$:
 
 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](images/Nyquist_frequency_&_rate.svg)
 
 ### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `images/sampling.png`)
@@ -758,11 +768,14 @@ tBitstream[{1, 1, -1, -1, -1, -1, 1, 1, -1, -1}, 1, "Q(t)"]
 
 Remember that $T=2T_b$
 
-|                         |                                     |
+|                         |                                    |
-| ----------------------- | ----------------------------------- |
+| ----------------------- | ---------------------------------- |
-| $b_n$                   | ![QPSK bits](/images/qpsk-bits.svg) |
+| $b_n$                   | ![QPSK bits](images/qpsk-bits.svg) |
-| $I(t)$ (Odd, 1st bits)  | ![QPSK bits](/images/qpsk-it.svg)   |
+| $I(t)$ (Odd, 1st bits)  | ![QPSK bits](images/qpsk-it.svg)   |
-| $Q(t)$ (Even, 2nd bits) | ![QPSK bits](/images/qpsk-qt.svg)   |
+| $Q(t)$ (Even, 2nd bits) | ![QPSK bits](images/qpsk-qt.svg)   |
+
+<!-- ADJUST ACCORDING TO PDF OUTPUT -->
+<div style="page-break-after: always;"></div>
 
 ## Matched filter
 
@@ -1005,6 +1018,9 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems
 - Minimum distance between any two point
 - Different from bit error since a symbol can contain multiple bits
 
+<!-- ADJUST ACCORDING TO PDF OUTPUT -->
+<div style="page-break-after: always;"></div>
+
 ## Information theory
 
 ### Entropy for discrete random variables
@@ -1176,6 +1192,9 @@ C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)
 
 <!-- MATH END -->
 
+<!-- ADJUST ACCORDING TO PDF OUTPUT -->
+<div style="page-break-after: always;"></div>
+
 ## Channel code
 
 Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
@@ -1372,3 +1391,25 @@ Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.
 - Transfer function in complex envelope form $\tilde{h}(t)$ should be divided by two.
 - Convolutions: do not forget width when using graphical method
 - todo: add more items to check
+
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+    h2::after, h3::after, h4::after, h5::after {
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