# Idiot's guide to ELEC4402 communication systems

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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

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## Fourier transform identities

| **Time Function**                                                                                               | **Fourier Transform**                                                                                                                     |
| --------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$                                           | $T \, \text{sinc}(fT)$                                                                                                                    |
| $\text{sinc}(2Wt)$                                                                                              | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$                                           |
| $\exp(-at)u(t), \, a>0$                                                                                         | $\frac{1}{a + j2\pi f}$                                                                                                                   |
| $\exp(-a\lvert t \rvert), \, a>0$                                                                               | $\frac{2a}{a^2 + (2\pi f)^2}$                                                                                                             |
| $\exp(-\pi t^2)$                                                                                                | $\exp(-\pi f^2)$                                                                                                                          |
| $\begin{cases} 1 - \frac{\lvert t \rvert}{T}, & \lvert t \rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}$ | $T \, \text{sinc}^2(fT)$                                                                                                                  |
| $\delta(t)$                                                                                                     | $1$                                                                                                                                       |
| $1$                                                                                                             | $\delta(f)$                                                                                                                               |
| $\delta(t - t_0)$                                                                                               | $\exp(-j2\pi f t_0)$                                                                                                                      |
| $g(t-a)$                                                                                                        | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$                                                                                           |
| $g(bt)$                                                                                                         | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$                                                                                        |
| $g(bt-a)$                                                                                                       | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift \& scale}$                                                                |
| $\frac{d}{dt}g(t)$                                                                                              | $j2\pi fG(f)\quad\text{differentiation property}$                                                                                         |
| $G(t)$                                                                                                          | $g(-f)\quad\text{duality property}$                                                                                                       |
| $g(t)h(t)$                                                                                                      | $G(f)*H(f)$                                                                                                                               |
| $g(t)*h(t)$                                                                                                     | $G(f)H(f)$                                                                                                                                |
| $\exp(j2\pi f_c t)$                                                                                             | $\delta(f - f_c)$                                                                                                                         |
| $\cos(2\pi f_c t)$                                                                                              | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$                                                                                          |
| $\sin(2\pi f_c t)$                                                                                              | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$                                                                                        |
| $\text{sgn}(t)$                                                                                                 | $\frac{1}{j\pi f}$                                                                                                                        |
| $\frac{1}{\pi t}$                                                                                               | $-j \, \text{sgn}(f)$                                                                                                                     |
| $u(t)$                                                                                                          | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$                                                                                               |
| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$                                                                    | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ |

| **Function Name**    | **Formula**                                                                                             |
| -------------------- | ------------------------------------------------------------------------------------------------------- |
| Unit Step Function   | $u(t) = \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}$                     |
| Signum Function      | $\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}$                    |
| sinc Function        | $\text{sinc}(2Wt) = \frac{\sin(2\pi W t)}{2\pi W t}$                                                    |
| Rectangular Function | $\text{rect}(t) = \Pi(t) = \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}$ |
| Convolution          | $g(t)*h(t)=(g*h)(t)=\int_\infty^\infty g(\tau)h(t-\tau)d\tau$                                           |

### Fourier transform of continuous time periodic signal

Required for some questions on **sampling**:

<!-- Transform a continuous time-periodic signal $x(t)=\sum_{n=-\infty}^\infty x_p(t-nT_s)$ with period $T_s$: -->

Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$:

$$
X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
$$

Calculate $C_n$ coefficient as follows from $x_p(t)$:

<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->

$$
\begin{align*}
    % C_n&=X_p(nf_s)\\
    C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\
       &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
\end{align*}
$$

### $\text{rect}$ function

![rect](images/rect.drawio.svg)

### Bessel function

$$
\begin{align*}
    \sum_{n\in\Z}{J_n}^2(\beta)&=1\\
    J_n(\beta)&=(-1)^nJ_{-n}(\beta)
\end{align*}
$$

### White noise

$$
\begin{align*}
R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\
G_w(f)&=\frac{N_0}{2}\\
N_0&=kT\\
G_y(f)&=|H(f)|^2G_w(f)\\
G_y(f)&=G(f)G_w(f)\\
\end{align*}
$$

### WSS

$$
\begin{align*}
    \mu_X(t) &= \mu_X\text{ Constant}\\
    R_{XX}(t_1,t_2)&=R_X(t_1-t_2)=R_X(\tau)\\
    E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)]
\end{align*}
$$

### Ergodicity

$$
\begin{align*}
    \braket{X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t)dt\\
    \braket{X(t+\tau)X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t+\tau)x(t)dt\\
    E[\braket{X(t)}_T]&=\frac{1}{2T}\int_{-T}^{T}x(t)dt=\frac{1}{2T}\int_{-T}^{T}m_Xdt=m_X\\
\end{align*}
$$

| Type                                | Normal                                                  | Mean square sense                                           |
| ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- |
| 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**

### Other identities

$$
\begin{align*}
    f*(g*h) &=(f*g)*h\quad\text{Convolution associative}\\
    a(f*g) &= (af)*g \quad\text{Convolution associative}\\
    \sum_{x=-\infty}^\infty(f(x a)\delta(\omega-x b))&=f\left(\frac{\omega a}{b}\right)
\end{align*}
$$

### Other trig

$$
\begin{align*}
    \cos2\theta=2 \cos^2 \theta-1&\Leftrightarrow\frac{\cos2\theta+1}{2}=\cos^2\theta\\
    e^{-j\alpha}-e^{j\alpha}&=-2j \sin(\alpha)\\
    e^{-j\alpha}+e^{j\alpha}&=2 \cos(\alpha)\\
    \cos(-A)&=\cos(A)\\
    \sin(-A)&=-\sin(A)\\
    \sin(A+\pi/2)&=\cos(A)\\
    \sin(A-\pi/2)&=-\cos(A)\\
    \cos(A-\pi/2)&=\sin(A)\\
    \cos(A+\pi/2)&=-\sin(A)\\
    \int_{x\in\R}\text{sinc}(A x) &= \frac{1}{|A|}\\
\end{align*}
$$

$$
\begin{align*}
    \cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\
    \sin(A+B) &= \sin (A) \cos (B)+\cos (A) \sin (B) \\
    \cos(A)\cos(B) &= \frac{1}{2} (\cos (A-B)+\cos (A+B)) \\
    \cos(A)\sin(B) &= \frac{1}{2} (\sin (A+B)-\sin (A-B)) \\
    \sin(A)\sin(B) &= \frac{1}{2} (\cos (A-B)-\cos (A+B)) \\
\end{align*}
$$

$$
\begin{align*}
    \cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\
    \cos(A)-\cos(B) &= -2 \sin \left(\frac{A}{2}-\frac{B}{2}\right) \sin \left(\frac{A}{2}+\frac{B}{2}\right) \\
    \sin(A)+\sin(B) &= 2 \sin \left(\frac{A}{2}+\frac{B}{2}\right) \cos \left(\frac{A}{2}-\frac{B}{2}\right) \\
    \sin(A)-\sin(B) &= 2 \sin \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\
    \cos(A)+\sin(B)&= -2 \sin \left(\frac{A}{2}-\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}+\frac{B}{2}+\frac{\pi }{4}\right) \\
    \cos(A)-\sin(B)&= -2 \sin \left(\frac{A}{2}+\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}-\frac{B}{2}+\frac{\pi }{4}\right) \\
\end{align*}
$$

## IQ/Complex envelope

Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $e^{j\theta}$ form.

### Convert complex envelope representation to time-domain representation of signal

$$
\begin{align*}
g(t)&=g_I(t)\cos(2\pi f_c t)-g_Q(t)\sin(2\pi f_c t)\\
&=\text{Re}[\tilde{g}(t)\exp{(j2\pi f_c t)}]\\
&=A(t)\cos(2\pi f_c t+\phi(t))\\
A(t)&=|g(t)|=\sqrt{g_I^2(t)+g_Q^2(t)}\quad\text{Amplitude}\\
\phi(t)&\quad\text{Phase}\\
g_I(t)&=A(t)\cos(\phi(t))\quad\text{In-phase component}\\
g_Q(t)&=A(t)\sin(\phi(t))\quad\text{Quadrature-phase component}\\
\end{align*}
$$

### For transfer function

$$
\begin{align*}
h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
&=2\text{Re}[\tilde{h}(t)\exp{(j2\pi f_c t)}]\\
\Rightarrow\tilde{h}(t)&=h_I(t)/2+jh_Q(t)/2=A(t)/2\exp{(j\phi(t))}
\end{align*}
$$

## AM

### CAM

$$
\begin{align*}
    m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
    m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\
    m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\
    x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\
    &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\
    P_c &=\frac{{A_c}^2}{2}\quad\text{Carrier power}\\
    P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\
    \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\
    B_T&=2f_m=2B
\end{align*}
$$

$B_T$: Signal bandwidth
$B$: Bandwidth of modulating wave

Overmodulation (resulting in phase reversals at crossing points): $m_a>1$

### DSB-SC

$$
\begin{align*}
    x_\text{DSB}(t) &= A_c \cos{(2\pi f_c t)} m(t)\\
    B_T&=2f_m=2B
\end{align*}
$$

## FM/PM

$$
\begin{align*}
    s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\
    s(t) &= A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_0^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\
    s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\quad\text{FM single tone}\\
    \beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\
    \Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\
    D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}
\end{align*}
$$

### Bessel form and magnitude spectrum (single tone)

$$
\begin{align*}
    s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t]
\end{align*}
$$

<!-- ### Bessel repr.

$$s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m) t)$$ -->

### FM signal power

$$
\begin{align*}
    P_\text{av}&=\frac{{A_c}^2}{2}\\
    P_\text{band\_index}&=\frac{{A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\
    \text{band\_index}&=0\implies f_c+0f_m\\
    \text{band\_index}&=1\implies f_c+1f_m,\dots\\
\end{align*}
$$

### Carson's rule to find $B$ (98% power bandwidth rule)

$$
\begin{align*}
B &= 2Mf_m = 2(\beta + 1)f_m\\
    &= 2(\Delta f+f_m)\\
    &= 2(k_f A_m+f_m)\\
    &= 2(D+1)W_m\\
B &= \begin{cases}
    2(\Delta f+f_m) & \text{FM, sinusoidal message}\\
    2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message}
\end{cases}\\
\end{align*}
$$

#### $\Delta f$ of arbitrary modulating signal

Find instantaneous frequency $f_\text{FM}$.

$M$: Number of **pairs** of significant sidebands

$$
\begin{align*}
s(t)&=A_c\cos(\theta_\text{FM}(t))\\
f_\text{FM}(t) &= \frac{1}{2\pi}\frac{d\theta_\text{FM}(t)}{dt}\\
A_m &= \max_t|m(t)|\\
\Delta f &= \max_t(f_\text{FM}(t)) - f_c\\
W_m &= \text{max}(\text{frequencies in $\theta_\text{FM}(t)$...}) \\
\text{Example: }&\text{sinc}(At+t)+2\cos(2\pi t)=\frac{\sin(2\pi((At+t)/2))}{\pi(At+t)}+2\cos(2\pi t)\to W_m=\max\left(\frac{A+1}{2},1\right)\\
D &= \frac{\Delta f}{W_m}\\
B_T &= 2(D+1)W_m
\end{align*}
$$

### Complex envelope

$$
\begin{align*}
    s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t)) \Leftrightarrow \tilde{s}(t) = A_c\exp(j\beta\sin(2\pi f_m t))\\
    s(t)&=\text{Re}[\tilde{s}(t)\exp{(j2\pi f_c t)}]\\
    \tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t)
\end{align*}
$$

### Band

| Narrowband    | Wideband      |
| ------------- | ------------- |
| $D<1,\beta<1$ | $D>1,\beta>1$ |

## Power, energy and autocorrelation

$$
\begin{align*}
    G_\text{WGN}(f)&=\frac{N_0}{2}\\
    G_x(f)&=|H(f)|^2G_w(f)\text{ (PSD)}\\
    G_x(f)&=G(f)G_w(f)\text{ (PSD)}\\
    G_x(f)&=\lim_{T\to\infty}\frac{|X_T(f)|^2}{T}\text{ (PSD)}\\
    G_x(f)&=\mathfrak{F}[R_x(\tau)]\text{ (WSS)}\\
    P&=\sigma^2=\int_\mathbb{R}G_x(f)df\\
    P&=\sigma^2=\lim_{t\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\\
    P[A\cos(2\pi f t+\phi)]&=\frac{A^2}{2}\quad\text{Power of sinusoid }\\
    E&=\int_{-\infty}^{\infty}|x(t)|^2dt=|X(f)|^2\\
    R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}
\end{align*}
$$

##

## Noise performance

$$
\begin{align*}
    \text{CNR}_\text{in} &= \frac{P_\text{in}}{P_\text{noise}}\\
    \text{CNR}_\text{in,FM} &= \frac{A^2}{2WN_0}\\
    \text{SNR}_\text{FM} &= \frac{3A^2k_f^2P}{2N_0W^3}\\
    \text{SNR(dB)} &= 10\log_{10}(\text{SNR}) \quad\text{Decibels from ratio}
\end{align*}
$$

## Sampling

$$
\begin{align*}
    t&=nT_s\\
    T_s&=\frac{1}{f_s}\\
    x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\
    X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\
    B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\
\end{align*}
$$

### Procedure to reconstruct sampled signal

Analog signal $x'(t)$ which can be reconstructed from a sampled signal $x_s(t)$: Put $x_s(t)$ through LPF with maximum frequency of $f_s/2$ and minimum frequency of $-f_s/2$. Anything outside of the BPF will be attenuated, therefore $n$ which results in frequencies outside the BPF will evaluate to $0$ and can be ignored.

Example: $f_s=5000\implies \text{LPF}\in[-2500,2500]$

Then iterate for $n=0,1,-1,2,-2,\dots$ until the first iteration where the result is 0 since all terms are eliminated by the LPF.

TODO: Add example

Then add all terms and transform $\bar X_s(f)$ back to time domain to get $x_s(t)$

### Fourier transform of continuous time periodic signal (1)

Required for some questions on **sampling**:

<!-- Transform a continuous time-periodic signal $x(t)=\sum_{n=-\infty}^\infty x_p(t-nT_s)$ with period $T_s$: -->

Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$:

$$
X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}
$$

Calculate $C_n$ coefficient as follows from $x_p(t)$:

<!-- Remember $X_p(f)\leftrightarrow x_p(t)$ and **NOT** $\color{red}X_p(f)\leftrightarrow x_p(t-nT_s)$ -->

$$
\begin{align*}
    % C_n&=X_p(nf_s)\\
    C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\
       &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
\end{align*}
$$

<!-- Reconstruct from $\bar{X_s}(f)$ within the range $[-f_s/2,f_s/2]$ -->

### Nyquist criterion for zero-ISI

Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth.

![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`)

![sampling](copyrighted_images/sampling.png)
![sampling](images/sampling.png)

## Quantizer

$$
\begin{align*}
    \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\
\end{align*}
$$

### Quantization noise

$$
\begin{align*}
    e &:= y-x\quad\text{Quantization error}\\
    \mu_E &= E[E] = 0\quad\text{Zero mean}\\
    {\sigma_E}^2&=E[E^2]-0^2=\int_{-\Delta/2}^{\Delta/2}e^2\times\left(\frac{1}{\Delta}\right) de\quad\text{Where $E\thicksim 1/\Delta$ uniform over $(-\Delta/2,\Delta/2)$}\\
    \text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise}}\\
    \text{SQNR(dB)}&=10\log_{10}(\text{SQNR})
\end{align*}
$$

### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `images/quantizer.png`)

![quantizer](copyrighted_images/quantizer.png)
![quantizer](images/quantizer.png)

## Line codes

![binary_codes](images/Line_Codes.drawio.svg)

$$
\begin{align*}
    R_b&\rightarrow\text{Bit rate}\\
    D&\rightarrow\text{Symbol rate | }R_d\text{ | }1/T_b\\
    A&\rightarrow m_a\\
    V(f)&\rightarrow\text{Pulse shape}\\
    V_\text{rectangle}(f)&=T\text{sinc}(fT\times\text{DutyCycle})\\
    G_\text{MunipolarNRZ}(f)&=\frac{(M^2-1)A^2D}{12}|V(f)|^2+\frac{(M-1)^2}{4}(DA)^2\sum_{l=-\infty}^{\infty}|V(lD)|^2\delta(f-lD)\\
    G_\text{MpolarNRZ}(f)&=\frac{(M^2-1)A^2D}{3}|V(f)|^2\\
    G_\text{unipolarNRZ}(f)&=\frac{A^2}{4R_b}\left(\text{sinc}^2\left(\frac{f}{R_b}\right)+R_b\delta(f)\right), \text{NB}_0=R_b\\
    G_\text{polarNRZ}(f)&=\frac{A^2}{R_b}\text{sinc}^2\left(\frac{f}{R_b}\right)\\
    G_\text{unipolarNRZ}(f)&=\frac{A^2}{4R_b}\left(\text{sinc}^2\left(\frac{f}{R_b}\right)+R_b\delta(f)\right)\\
    G_\text{unipolarRZ}(f)&=\frac{A^2}{16} \left(\sum _{l=-\infty }^{\infty } \delta \left(f-\frac{l}{T_b}\right) \left| \text{sinc}(\text{duty} \times l) \right| {}^2+T_b \left| \text{sinc}\left(\text{duty} \times f T_b\right) \right| {}^2\right), \text{NB}_0=2R_b
\end{align*}
$$

## Modulation and basis functions

![Constellation diagrams](./images/Constellation.drawio.svg)

### BASK

#### Basis functions

$$
\begin{align*}
    \varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\
\end{align*}
$$

#### Symbol mapping

$$b_n:\{1,0\}\to a_n:\{1,0\}$$

#### 2 possible waveforms

$$
\begin{align*}
    s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\
    s_1(t)&=0\\
    &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + 0)=\frac{{A_c}^2}{4}T_b$}
\end{align*}
$$

Distance is $d=\sqrt{2E_b}$

### BPSK

#### Basis functions

$$
\begin{align*}
    \varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\
\end{align*}
$$

#### Symbol mapping

$$b_n:\{1,0\}\to a_n:\{1,\color{lime}-1\color{white}\}$$

#### 2 possible waveforms

$$
\begin{align*}
    s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{E_b}\varphi_1(t)\\
    s_1(t)&=-A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=-\sqrt{E_b}\varphi_2(t)\\
    &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + \frac{{A_c}^2}{2}\times T_b)=\frac{{A_c}^2}{2}T_b$}
\end{align*}
$$

Distance is $d=2\sqrt{E_b}$

### QPSK ($M=4$ PSK)

#### Basis functions

$$
\begin{align*}
    T &= 2 T_b\quad\text{Time per symbol for two bits $T_b$}\\
    \varphi_1(t) &= \sqrt{\frac{2}{T}}\cos(2\pi f_c t)\quad0\leq t\leq T\\
    \varphi_2(t) &= \sqrt{\frac{2}{T}}\sin(2\pi f_c t)\quad0\leq t\leq T\\
\end{align*}
$$

### 4 possible waveforms

$$
\begin{align*}
    s_1(t)&=\sqrt{E_s/2}\left[\varphi_1(t)+\varphi_2(t)\right]\\
    s_2(t)&=\sqrt{E_s/2}\left[\varphi_1(t)-\varphi_2(t)\right]\\
    s_3(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)+\varphi_2(t)\right]\\
    s_4(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)-\varphi_2(t)\right]\\
\end{align*}
$$

Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows:

$$
\begin{align*}
    s_i(t)&=A_c\sqrt{T/2}/\sqrt{2}\times\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
          &=\sqrt{T{A_c}^2/4}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
          &=\sqrt{E_s/2}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\
\end{align*}
$$

#### Signal

$$
\begin{align*}
    \text{Symbol mapping: }& \left\{1,0\right\}\to\left\{1,-1\right\}\\
    I(t) &= b_{2n}\varphi_1(t)\quad\text{Even bits}\\
    Q(t) &= b_{2n+1}\varphi_2(t)\quad\text{Odd bits}\\
    x(t) &= A_c[I(t)\cos(2\pi f_c t)-Q(t)\sin(2\pi f_c t)]
\end{align*}
$$

### Example of waveform

<details>
<summary>Code</summary>
<pre><code>
tBitstream[bitstream_, Tb_, title_] := 
  Module[{timeSteps, gridLines, plot},
   timeSteps = 
    Flatten[Table[{(n - 1)     Tb, bitstream[[n]]}, {n, 1, 
        Length[bitstream]}] /. {t_, v_} :> {{t, v}, {t + Tb, v}}, 1]; 
   gridLines = {Join[
      Table[{n  Tb, Dashed}, {n, 1, 2  Length[bitstream], 2}], 
      Table[{n  Tb, Thin}, {n, 0, 2  Length[bitstream], 2}]], None};
   plot = 
    Labeled[ListLinePlot[timeSteps, InterpolationOrder -> 0, 
      PlotRange -> Full, GridLines -> gridLines, PlotStyle -> Thick, 
      Ticks -> {Table[{n     Tb, 
          Row[{n, "\!\(\*SubscriptBox[\(T\), \(b\)]\)"}]}, {n, 0, 
          Length[bitstream]}], {-1, 0, 1}}, 
      LabelStyle -> Directive[Bold, 12], 
      PlotRangePadding -> {Scaled[.05]}, AspectRatio -> 0.1, 
      ImageSize -> Large], {Style[title, "Text", 16]}, {Right}]];

tBitstream[{0, 1, 0, 0, 1, 0, 1, 1, 1, 0}, 1, "Bitstream Step Plot"]
tBitstream[{-1, -1, -1, -1, 1, 1, 1, 1, 1, 1}, 1, "I(t)"]
tBitstream[{1, 1, -1, -1, -1, -1, 1, 1, -1, -1}, 1, "Q(t)"]
</code></pre>

</details>

Remember that $T=2T_b$

|                         |                                     |
| ----------------------- | ----------------------------------- |
| $b_n$                   | ![QPSK bits](/images/qpsk-bits.svg) |
| $I(t)$ (Odd, 1st bits)  | ![QPSK bits](/images/qpsk-it.svg)   |
| $Q(t)$ (Even, 2nd bits) | ![QPSK bits](/images/qpsk-qt.svg)   |

## Matched filter

### 1. Filter function

Find transfer function $h(t)$ of matched filter and apply to an input:

$$
\begin{align*}
    h(t)&=s_1(T-t)-s_2(T-t)\\
    h(t)&=s^*(T-t) \qquad\text{((.)* is the conjugate)}\\
    s_{on}(t)&=h(t)*s_n(t)=\int_\infty^\infty h(\tau)s_n(t-\tau)d\tau\quad\text{Filter output}\\
    n_o(t)&=h(t)*n(t)\quad\text{Noise at filter output}
\end{align*}
$$

### 2. Bit error rate

Bit error rate (BER) from matched filter outputs and filter output noise

$$
\begin{align*}
    % H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right)

    % \text{BER}_\text{bin}&=p Q\left(\frac{s_{o1}-V_T}{\sigma_o}\right)+(1-p)Q\left(\frac{V_T-s_{o2}}{\sigma_o}\right)\text{, $p\rightarrow$Probability $s_1(t)$ sent, $V_T\rightarrow$Threshold voltage}
    Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\
    E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\
    T&=1/R_b\quad\text{$R_b$: Bitrate}\\
    E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\
    P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\
    P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\
    \text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\
    \text{BER}_\text{unipolarNRZ|BASK}&=Q\left(\sqrt{\frac{d^2}{N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)\\
    \text{BER}_\text{polarNRZ|BPSK}&=Q\left(\sqrt{\frac{2d^2}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\\
\end{align*}
$$

<div style="page-break-after: always;"></div>

## 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

| $x$    | $Q(x)$     | $x$    | $Q(x)$                  | $x$    | $Q(x)$                   | $x$    | $Q(x)$                   |
| ------ | ---------- | ------ | ----------------------- | ------ | ------------------------ | ------ | ------------------------ |
| $0.00$ | $0.5       | $2.30$ | $0.010724$              | $4.55$ | $2.6823 \times 10^{-6}$  | $6.80$ | $5.231 \times 10^{-12}$  |
| $0.05$ | $0.48006$  | $2.35$ | $0.0093867$             | $4.60$ | $2.1125 \times 10^{-6}$  | $6.85$ | $3.6925 \times 10^{-12}$ |
| $0.10$ | $0.46017$  | $2.40$ | $0.0081975$             | $4.65$ | $1.6597 \times 10^{-6}$  | $6.90$ | $2.6001 \times 10^{-12}$ |
| $0.15$ | $0.44038$  | $2.45$ | $0.0071428$             | $4.70$ | $1.3008 \times 10^{-6}$  | $6.95$ | $1.8264 \times 10^{-12}$ |
| $0.20$ | $0.42074$  | $2.50$ | $0.0062097$             | $4.75$ | $1.0171 \times 10^{-6}$  | $7.00$ | $1.2798 \times 10^{-12}$ |
| $0.25$ | $0.40129$  | $2.55$ | $0.0053861$             | $4.80$ | $7.9333 \times 10^{-7}$  | $7.05$ | $8.9459 \times 10^{-13}$ |
| $0.30$ | $0.38209$  | $2.60$ | $0.0046612$             | $4.85$ | $6.1731 \times 10^{-7}$  | $7.10$ | $6.2378 \times 10^{-13}$ |
| $0.35$ | $0.36317$  | $2.65$ | $0.0040246$             | $4.90$ | $4.7918 \times 10^{-7}$  | $7.15$ | $4.3389 \times 10^{-13}$ |
| $0.40$ | $0.34458$  | $2.70$ | $0.003467$              | $4.95$ | $3.7107 \times 10^{-7}$  | $7.20$ | $3.0106 \times 10^{-13}$ |
| $0.45$ | $0.32636$  | $2.75$ | $0.0029798$             | $5.00$ | $2.8665 \times 10^{-7}$  | $7.25$ | $2.0839 \times 10^{-13}$ |
| $0.50$ | $0.30854$  | $2.80$ | $0.0025551$             | $5.05$ | $2.2091 \times 10^{-7}$  | $7.30$ | $1.4388 \times 10^{-13}$ |
| $0.55$ | $0.29116$  | $2.85$ | $0.002186$              | $5.10$ | $1.6983 \times 10^{-7}$  | $7.35$ | $9.9103 \times 10^{-14}$ |
| $0.60$ | $0.27425$  | $2.90$ | $0.0018658$             | $5.15$ | $1.3024 \times 10^{-7}$  | $7.40$ | $6.8092 \times 10^{-14}$ |
| $0.65$ | $0.25785$  | $2.95$ | $0.0015889$             | $5.20$ | $9.9644 \times 10^{-8}$  | $7.45$ | $4.667 \times 10^{-14}$  |
| $0.70$ | $0.24196$  | $3.00$ | $0.0013499$             | $5.25$ | $7.605 \times 10^{-8}$   | $7.50$ | $3.1909 \times 10^{-14}$ |
| $0.75$ | $0.22663$  | $3.05$ | $0.0011442$             | $5.30$ | $5.7901 \times 10^{-8}$  | $7.55$ | $2.1763 \times 10^{-14}$ |
| $0.80$ | $0.21186$  | $3.10$ | $0.0009676$             | $5.35$ | $4.3977 \times 10^{-8}$  | $7.60$ | $1.4807 \times 10^{-14}$ |
| $0.85$ | $0.19766$  | $3.15$ | $0.00081635$            | $5.40$ | $3.332 \times 10^{-8}$   | $7.65$ | $1.0049 \times 10^{-14}$ |
| $0.90$ | $0.18406$  | $3.20$ | $0.00068714$            | $5.45$ | $2.5185 \times 10^{-8}$  | $7.70$ | $6.8033 \times 10^{-15}$ |
| $0.95$ | $0.17106$  | $3.25$ | $0.00057703$            | $5.50$ | $1.899 \times 10^{-8}$   | $7.75$ | $4.5946 \times 10^{-15}$ |
| $1.00$ | $0.15866$  | $3.30$ | $0.00048342$            | $5.55$ | $1.4283 \times 10^{-8}$  | $7.80$ | $3.0954 \times 10^{-15}$ |
| $1.05$ | $0.14686$  | $3.35$ | $0.00040406$            | $5.60$ | $1.0718 \times 10^{-8}$  | $7.85$ | $2.0802 \times 10^{-15}$ |
| $1.10$ | $0.13567$  | $3.40$ | $0.00033693$            | $5.65$ | $8.0224 \times 10^{-9}$  | $7.90$ | $1.3945 \times 10^{-15}$ |
| $1.15$ | $0.12507$  | $3.45$ | $0.00028029$            | $5.70$ | $5.9904 \times 10^{-3}$  | $7.95$ | $9.3256 \times 10^{-16}$ |
| $1.20$ | $0.11507$  | $3.50$ | $0.00023263$            | $5.75$ | $4.4622 \times 10^{-9}$  | $8.00$ | $6.221 \times 10^{-16}$  |
| $1.25$ | $0.10565$  | $3.55$ | $0.00019262$            | $5.80$ | $3.3157 \times 10^{-9}$  | $8.05$ | $4.1397 \times 10^{-16}$ |
| $1.30$ | $0.0968$   | $3.60$ | $0.00015911$            | $5.85$ | $2.4579 \times 10^{-9}$  | $8.10$ | $2.748 \times 10^{-16}$  |
| $1.35$ | $0.088508$ | $3.65$ | $0.00013112$            | $5.90$ | $1.8175 \times 10^{-9}$  | $8.15$ | $1.8196 \times 10^{-16}$ |
| $1.40$ | $0.080757$ | $3.70$ | $0.0001078$             | $5.95$ | $1.3407 \times 10^{-9}$  | $8.20$ | $1.2019 \times 10^{-16}$ |
| $1.45$ | $0.073529$ | $3.75$ | $8.8417 \times 10^{-5}$ | $6.00$ | $9.8659 \times 10^{-10}$ | $8.25$ | $7.9197 \times 10^{-17}$ |
| $1.50$ | $0.066807$ | $3.80$ | $7.2348 \times 10^{-5}$ | $6.05$ | $7.2423 \times 10^{-10}$ | $8.30$ | $5.2056 \times 10^{-17}$ |
| $1.55$ | $0.060571$ | $3.85$ | $5.9059 \times 10^{-5}$ | $6.10$ | $5.3034 \times 10^{-10}$ | $8.35$ | $3.4131 \times 10^{-17}$ |
| $1.60$ | $0.054799$ | $3.90$ | $4.8096 \times 10^{-5}$ | $6.15$ | $3.8741 \times 10^{-10}$ | $8.40$ | $2.2324 \times 10^{-17}$ |
| $1.65$ | $0.049471$ | $3.95$ | $3.9076 \times 10^{-5}$ | $6.20$ | $2.8232 \times 10^{-10}$ | $8.45$ | $1.4565 \times 10^{-17}$ |
| $1.70$ | $0.044565$ | $4.00$ | $3.1671 \times 10^{-5}$ | $6.25$ | $2.0523 \times 10^{-10}$ | $8.50$ | $9.4795 \times 10^{-18}$ |
| $1.75$ | $0.040059$ | $4.05$ | $2.5609 \times 10^{-5}$ | $6.30$ | $1.4882 \times 10^{-10}$ | $8.55$ | $6.1544 \times 10^{-18}$ |
| $1.80$ | $0.03593$  | $4.10$ | $2.0658 \times 10^{-5}$ | $6.35$ | $1.0766 \times 10^{-10}$ | $8.60$ | $3.9858 \times 10^{-18}$ |
| $1.85$ | $0.032157$ | $4.15$ | $1.6624 \times 10^{-5}$ | $6.40$ | $7.7688 \times 10^{-11}$ | $8.65$ | $2.575 \times 10^{-18}$  |
| $1.90$ | $0.028717$ | $4.20$ | $1.3346 \times 10^{-5}$ | $6.45$ | $5.5925 \times 10^{-11}$ | $8.70$ | $1.6594 \times 10^{-18}$ |
| $1.95$ | $0.025588$ | $4.25$ | $1.0689 \times 10^{-5}$ | $6.50$ | $4.016 \times 10^{-11}$  | $8.75$ | $1.0668 \times 10^{-18}$ |
| $2.00$ | $0.02275$  | $4.30$ | $8.5399 \times 10^{-6}$ | $6.55$ | $2.8769 \times 10^{-11}$ | $8.80$ | $6.8408 \times 10^{-19}$ |
| $2.05$ | $0.020182$ | $4.35$ | $6.8069 \times 10^{-6}$ | $6.60$ | $2.0558 \times 10^{-11}$ | $8.85$ | $4.376 \times 10^{-19}$  |
| $2.10$ | $0.017864$ | $4.40$ | $5.4125 \times 10^{-6}$ | $6.65$ | $1.4655 \times 10^{-11}$ | $8.90$ | $2.7923 \times 10^{-19}$ |
| $2.15$ | $0.015778$ | $4.45$ | $4.2935 \times 10^{-6}$ | $6.70$ | $1.0421 \times 10^{-11}$ | $8.95$ | $1.7774 \times 10^{-19}$ |
| $2.20$ | $0.013903$ | $4.50$ | $3.3977 \times 10^{-6}$ | $6.75$ | $7.3923 \times 10^{-12}$ | $9.00$ | $1.1286 \times 10^{-19}$ |
| $2.25$ | $0.012224$ |        |                         |        |                          |

Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems

\*\*The value of $\text{erf}(3.30)$ should be $\approx0.999997$ instead, but this value is quoted in the formula table.

### Receiver output shit

$$
\begin{align*}
    r_o(t)&=\begin{cases}
        s_{o1}(t)+n_o(t) & \text{code 1}\\
        s_{o2}(t)+n_o(t) & \text{code 0}\\
    \end{cases}\\
    n&: \text{AWGN with }\sigma_o^2\\
\end{align*}
$$

<!--
$$
\begin{align*}
    G_x(f)
\end{align*}
$$ -->

## ISI, channel model

### Nyquist criterion for zero ISI

TODO:

### Nomenclature

$$
\begin{align*}
    D&\rightarrow\text{Symbol Rate, Max. Signalling Rate}\\
    T&\rightarrow\text{Symbol Duration}\\
    M&\rightarrow\text{Symbol set size}\\
    W&\rightarrow\text{Bandwidth}\\
\end{align*}
$$

### Raised cosine (RC) pulse

![Raised cosine pulse](images/RC.drawio.svg)

$$0\leq\alpha\leq1$$

⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method.

To solve this type of question:

1. Use the formula for $D$ below
2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$.

<!-- $$
V_\text{RC}(f)=\begin{cases}
    T & 0\le|f|\le(1-\alpha)/2T\\
    \frac{T}{2}\left(1+\cos\left[\right]\right)
\end{cases}
$$ -->

| Linear modulation ($M$-PSK, $M$-QAM)                | NRZ unipolar encoding                              |
| --------------------------------------------------- | -------------------------------------------------- |
| $W=B_\text{\color{lime}abs-abs}$                    | $W=B_\text{\color{lime}abs}$                       |
| $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ |
| $D=\frac{W\text{ symbol/s}}{1+\alpha}$              | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$            |

#### Symbol set size $M$

$$
\begin{align*}
    D\text{ symbol/s}&=\frac{2W\text{ Hz}}{1+\alpha}\\
    R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\
    M\text{ symbol/set}&=2^k\\
    E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\
\end{align*}
$$

### Nyquist stuff

#### Condition for 0 ISI

$$
P_r(kT)=\begin{cases}
    1 & k=0\\
    0 & k\neq0
\end{cases}
$$

#### Other

$$
\begin{align*}
    \text{Excess BW}&=B_\text{abs}-B_\text{Nyquist}=\frac{1+\alpha}{2T}-\frac{1}{2T}=\frac{\alpha}{2T}\quad\text{FOR NRZ (Use correct $B_\text{abs}$)}\\
    \alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\
    T&=1/D
\end{align*}
$$

<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**                                                                |
| --------------------------------- | -------------------------------- | ---------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------- |
| ASK, unipolar NRZ                 | $2R_b$                           | $R_b (1 + \alpha)$                                   | $Q\left( \sqrt{E_b / N_0} \right)$                                                                                   | $0.5\exp(-E_b / (2N_0))$                                                                          |
| BPSK                              | $2R_b$                           | $R_b (1 + \alpha)$                                   | $Q\left( \sqrt{2E_b / N_0} \right)$                                                                                  | Requires coherent detection                                                                       |
| Sunde's FSK                       | $3R_b$                           |                                                      | $Q\left( \sqrt{E_b / N_0} \right)$                                                                                   | $0.5\exp(-E_b / (2N_0))$                                                                          |
| DBPSK, $M$-ary Bandpass Signaling | $2R_b$                           | $R_b (1 + \alpha)$                                   |                                                                                                                      | $0.5\exp(-E_b / N_0)$                                                                             |
| QPSK/OQPSK (**$M=4$, PSK**)       | $R_b$                            | $\frac{R_b (1 + \alpha)}{2}$                         | $Q\left( \sqrt{2E_b / N_0} \right)$                                                                                  | Requires coherent detection                                                                       |
| MSK                               | $1.5R_b$                         | $\frac{3R_b (1 + \alpha)}{4}$                        | $Q\left( \sqrt{2E_b / N_0} \right)$                                                                                  | Requires coherent detection                                                                       |
| $M$-PSK ($M > 4$)                 | $2R_b / \log_2 M$                | $\frac{R_b (1 + \alpha)}{\log_2 M}$                  | $\frac{2}{\log_2 M} Q\left( \sqrt{2 \log_2 M \sin^2 \left( \pi / M \right) E_b / N_0} \right)$                       | Requires coherent detection                                                                       |
| $M$-DPSK ($M > 4$)                | $2R_b / \log_2 M$                | $\frac{R_b (1 + \alpha)}{2 \log_2 M}$                |                                                                                                                      | $\frac{2}{\log_2 M} Q\left( \sqrt{4 \log_2 M \sin^2 \left( \pi / (2M) \right) E_b / N_0} \right)$ |
| $M$-QAM (Square constellation)    | $2R_b / \log_2 M$                | $\frac{R_b (1 + \alpha)}{\log_2 M}$                  | $\frac{4}{\log_2 M} \left( 1 - \frac{1}{\sqrt{M}} \right) Q\left( \sqrt{\frac{3 \log_2 M}{M - 1} E_b / N_0} \right)$ | Requires coherent detection                                                                       |
| $M$-FSK Coherent                  | $\frac{(M + 3) R_b}{2 \log_2 M}$ |                                                      | $\frac{M - 1}{\log_2 M} Q\left( \sqrt{(\log_2 M) E_b / N_0} \right)$                                                 |                                                                                                   |
| Noncoherent                       | $2M R_b / \log_2 M$              |                                                      |                                                                                                                      | $\frac{M - 1}{2 \log_2 M} 0.5\exp({-(\log_2 M) E_b / 2N_0})$                                      |

Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems

### PSD of modulated signals

| Modulation | $G_x(f)$                                                                                          |
| ---------- | ------------------------------------------------------------------------------------------------- |
| Quadrature | $\color{red}\frac{{A_c}^2}{4}[G_I(f-f_c)+G_I(f+f_c)+G_Q(f-f_c)+G_Q(f+f_c)]$                       |
| Linear     | $\color{red}\frac{\|V(f)\|^2}{2}\sum_{l=-\infty}^\infty R(l)\exp(-j2\pi l f T)\quad\text{What??}$ |

### Symbol error probability

- Minimum distance between any two point
- Different from bit error since a symbol can contain multiple bits

## Information theory

### Entropy for discrete random variables

$$
\begin{align*}
    H(x) &\geq 0\\
    H(x) &= -\sum_{x_i\in A_x} p_X(x_i) \log_2(p_X(x_i))\\
    H(x,y) &= -\sum_{x_i\in A_x}\sum_{y_i\in A_y} p_{XY}(x_i,y_i)\log_2(p_{XY}(x_i,y_i)) \quad\text{Joint entropy}\\
    H(x,y) &= H(x)+H(y) \quad\text{Joint entropy if $x$ and $y$ independent}\\
    H(x|y=y_j) &= -\sum_{x_i\in A_x} p_X(x_i|y=y_j) \log_2(p_X(x_i|y=y_j)) \quad\text{Conditional entropy}\\
    H(x|y) &= -\sum_{y_j\in A_y} p_Y(y_j) H(x|y=y_j) \quad\text{Average conditional entropy, equivocation}\\
    H(x|y) &= -\sum_{x_i\in A_x}\sum_{y_i\in A_y} p_X(x_i,y_j) \log_2(p_X(x_i|y=y_j))\\
    H(x|y) &= H(x,y)-H(y)\\
    H(x,y) &= H(x) + H(y|x) = H(y) + H(x|y)\\
\end{align*}
$$

Entropy is **maximized** when all have an equal probability.

### Differential entropy for continuous random variables

TODO: Cut out if not required

$$
\begin{align*}
    h(x) &= -\int_\mathbb{R}f_X(x)\log_2(f_X(x))dx
\end{align*}
$$

### Mutual information

Amount of entropy decrease of $x$ after observation by $y$.

$$
\begin{align*}
    I(x;y) &= H(x)-H(x|y)=H(y)-H(y|x)\\
\end{align*}
$$

### Channel model

Vertical, $x$: input\
Horizontal, $y$: output

$$
\mathbf{P}=\left[\begin{matrix}
    p_{11} & p_{12} &\dots & p_{1N}\\
    p_{21} & p_{22} &\dots & p_{2N}\\
    \vdots & \vdots &\ddots & \vdots\\
    p_{M1} & p_{M2} &\dots & p_{MN}\\
\end{matrix}\right]
$$

$$
\begin{array}{c|cccc}
    P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\
    \hline
    x_1 & p_{11} & p_{12} & \dots & p_{1N} \\
    x_2 & p_{21} & p_{22} & \dots & p_{2N} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_M & p_{M1} & p_{M2} & \dots & p_{MN} \\
\end{array}
$$

Input has probability distribution $p_X(a_i)=P(X=a_i)$

Channel maps alphabet $\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}$

Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$

$$
\begin{align*}
    p_Y(b_j) &= \sum_{i=1}^{M}P[x=a_i,y=b_j]\quad 1\leq j\leq N \\
        &= \sum_{i=1}^{M}P[X=a_i]P[Y=b_j|X=a_i]\\
       [\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] &= [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}
\end{align*}
$$

#### Fast procedure to calculate $I(y;x)$

$$
\begin{align*}
    &\text{1. Find }H(x)\\
    &\text{2. Find }[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] = [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}\\
    &\text{3. Multiply each row in $\textbf{P}$ by $p_X(a_i)$ since $p_{XY}(x_i,y_i)=P(y_i|x_i)P(x_i)$}\\
    &\text{4. Find $H(x,y)$ using each element from (3.)}\\
    &\text{5. Find }H(x|y)=H(x,y)-H(y)\\
    &\text{6. Find }I(y;x)=H(x)-H(x|y)\\
\end{align*}
$$

### Channel types

| Type              | Definition                                                                                                                                            |
| ----------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------- |
| Symmetric channel | Every row is a permutation of every other row, Every column is a permutation of every other column. $\text{Symmetric}\implies\text{Weakly symmetric}$ |
| Weakly symmetric  | Every row is a permutation of every other row, Every column has the same sum                                                                          |

#### Channel capacity of weakly symmetric channel

$$
\begin{align*}
    C &\to\text{Channel capacity (bits/channels used)}\\
    N &\to\text{Output alphabet size}\\
    \mathbf{p} &\to\text{Probability vector, any row of the transition matrix}\\
    C &= \log_2(N)-H(\mathbf{p})\quad\text{Capacity for weakly symmetric and symmetric channels}\\
    R &< C \text{ for error-free transmission}
\end{align*}
$$

#### Channel capacity of an AWGN channel

$$
y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2)
$$

$$
C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)
$$

#### Channel capacity of a bandwidth AWGN channel

Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.

$$
\begin{align*}
    P_s&\to\text{Bandwidth limited average power}\\
    y_i&=\text{bandpass}_W(x_i)+n_i\quad n_i\thicksim N(0,N_0/2)\\
    C&=W\log_2\left(1+\frac{P_s}{N_0 W}\right)\\
    C&=W\log_2(1+\text{SNR})\quad\text{SNR}=P_s/(N_0 W)
\end{align*}
$$

## Channel code

|                  |                                   |                                                                                                                            |
| ---------------- | --------------------------------- | -------------------------------------------------------------------------------------------------------------------------- |
| Hamming weight   | $w_H(x)$                          | Number of `'1'` in codeword $x$                                                                                            |
| Hamming distance | $d_H(x_1,x_2)=w_H(x_1\oplus x_2)$ | Number of different bits between codewords $x_1$ and $x_2$ which is the hamming weight of the XOR of the two codes.        |
| Minimum distance | $d_\text{min}$                    | **IMPORTANT**: $x\neq\bold{0}$, excludes weight of all-zero codeword. For a linear block code, $d_\text{min}=w_\text{min}$ |

### Linear block code

Code is $(n,k)$

$n$ is the width of a codeword

$2^k$ codewords

A linear block code must be a subspace and satisfy both:

1. Zero vector must be present at least once
2. The XOR of any codeword pair in the code must result in a codeword that is already present in the code table.

For a linear block code, $d_\text{min}=w_\text{min}$

### Code generation

Each generator vector is a binary string of size $n$. There are $k$ generator vectors in $\mathbf{G}$.

$$
\begin{align*}
\mathbf{g}_i&=[\begin{matrix}
    g_{i,0}& \dots & g_{i,n-2} & g_{i,n-1}
\end{matrix}]\\
\color{darkgray}\mathbf{g}_0&\color{darkgray}=[1010]\quad\text{Example for $n=4$}\\
\mathbf{G}&=\left[\begin{matrix}
    \mathbf{g}_0\\
    \mathbf{g}_1\\
    \vdots\\
    \mathbf{g}_{k-1}\\
\end{matrix}\right]=\left[\begin{matrix}
    g_{0,0}& \dots & g_{0,n-2} & g_{0,n-1}\\
    g_{1,0}& \dots & g_{1,n-2} & g_{1,n-1}\\
    \vdots & \ddots & \vdots & \vdots\\
    g_{k-1,0}& \dots & g_{k-1,n-2} & g_{k-1,n-1}\\
\end{matrix}\right]
\end{align*}
$$

A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$:

$$
\begin{align*}
\mathbf{m}&=[\begin{matrix}
    m_{0}& \dots & m_{n-2} & m_{k-1}
\end{matrix}]\\
\color{darkgray}\mathbf{m}&\color{darkgray}=[101001]\quad\text{Example for $k=6$}\\
\mathbf{x} &= \mathbf{m}\mathbf{G}=m_0\mathbf{g}_0+m_1\mathbf{g}_1+\dots+m_{k-1}\mathbf{g}_{k-1}
\end{align*}
$$

### Systemic linear block code

Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits.

$$
\begin{align*}
\mathbf{G}&=\begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\left[
    \begin{array}{c|c}
    \begin{matrix}
    1 & 0 & \dots & 0\\
    0 & 1 & \dots & 0\\
    \vdots & \vdots & \ddots & \vdots\\
    0& 0 & \dots & 1\\
\end{matrix}
&
\begin{matrix}
    p_{0,0}& \dots & p_{0,n-2} & p_{0,n-1}\\
    p_{1,0}& \dots & p_{1,n-2} & p_{1,n-1}\\
    \vdots & \ddots & \vdots & \vdots\\
    p_{k-1,0}& \dots & p_{k-1,n-2} & p_{k-1,n-1}\\
\end{matrix}\end{array}\right]\\
\mathbf{m}&=[\begin{matrix}
    m_{0}& \dots & m_{n-2} & m_{k-1}
\end{matrix}]\\
\mathbf{x} &= \mathbf{m}\mathbf{G}= \mathbf{m} \begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\begin{array}{c|c}[\mathbf{mI}_k & \mathbf{mP}]\end{array}=\begin{array}{c|c}[\mathbf{m} & \mathbf{b}]\end{array}\\
\mathbf{b} &= \mathbf{m}\mathbf{P}\quad\text{Parity bits of $\mathbf{x}$}
\end{align*}
$$

#### Parity check matrix $\mathbf{H}$

Transpose $\mathbf{P}$ for the parity check matrix

$$
\begin{align*}
\mathbf{H}&=\begin{array}{c|c}[\mathbf{P}^\text{T} & \mathbf{I}_{n-k}]\end{array}\\
&=\left[
    \begin{array}{c|c}
    \begin{matrix}
        {\textbf{p}_0}^\text{T} & {\textbf{p}_1}^\text{T} & \dots & {\textbf{p}_{k-1}}^\text{T}
    \end{matrix}
    &
    \mathbf{I}_{n-k}\end{array}\right]\\
&=\left[
    \begin{array}{c|c}
    \begin{matrix}
        p_{0,0}& \dots & p_{0,k-2} & p_{0,k-1}\\
        p_{1,0}& \dots & p_{1,k-2} & p_{1,k-1}\\
        \vdots & \ddots & \vdots & \vdots\\
        p_{n-1,0}& \dots & p_{n-1,k-2} & p_{n-1,k-1}\\
    \end{matrix}
    &
    \begin{matrix}
    1 & 0 & \dots & 0\\
    0 & 1 & \dots & 0\\
    \vdots & \vdots & \ddots & \vdots\\
    0& 0 & \dots & 1\\
\end{matrix}\end{array}\right]\\
\mathbf{xH}^\text{T}&=\mathbf{0}\implies\text{Codeword is valid}
\end{align*}
$$

#### Procedure to find parity check matrix from list of codewords

1. From the number of codewords, find $k=\log_2(N)$
2. Partition codewords into $k$ information bits and remaining bits into $n-k$ parity bits. The information bits should be a simple counter (?).
3. Express parity bits as a linear combination of information bits
4. Put coefficients into $\textbf{P}$ matrix and find $\textbf{H}$

Example:

$$
\begin{array}{cccc}
    x_1 & x_2 & x_3 & x_4 & x_5 \\
    \hline
    \color{magenta}1&\color{magenta}0&1&1&0\\
    \color{magenta}0&\color{magenta}1&1&1&1\\
    \color{magenta}0&\color{magenta}0&0&0&0\\
    \color{magenta}1&\color{magenta}1&0&0&1\\
\end{array}
$$

Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$.

$$
\begin{align*}
\begin{align*}
    x_3 &= x_1\oplus x_2\\
    x_4 &= x_1\oplus x_2\\
    x_5 &= x_2\\
\end{align*}
\implies\textbf{P}&=
\begin{array}{c|ccc}
    & x_1 & x_2 \\
    \hline
    x_3&1&1&\\
    x_4&1&1&\\
    x_5&0&1&\\
\end{array}\\
\textbf{H}&=\left[
    \begin{array}{c|c}
    \begin{matrix}
    1&1\\
    1&1\\
    0&1\\
    \end{matrix}
    &
    \begin{matrix}
    1 & 0 & 0\\
    0 & 1 & 0\\
    0 & 0 & 1\\
\end{matrix}\end{array}\right]
\end{align*}
$$

#### Error detection and correction

**Detection** of $s$ errors: $d_\text{min}\geq s+1$

**Correction** of $u$ errors: $d_\text{min}\geq 2u+1$

## CHECKLIST

- 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