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Fix some bugs and add more shannon sampling/zero isi stuff
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README-copyright.pdf
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README.md
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README.md
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# Idiot's guide to ELEC4402 communication systems
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<style>
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@media print{
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.copyrighted{
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display: none !important;
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}
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}
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</style>
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<!-- PRINT NOTE: Use 0.20 margins all around, scale: fit to page width, and no headers or backgrounds -->
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This unit allows you to bring infinite physical notes (except books borrowed from the UWA library) to all tests and the final exam. You can't rely on what material they provide in the test/exam, it is very _minimal_ to say the least. Hope this helps.
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@ -70,7 +78,7 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
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<div style="page-break-after: always;"></div>
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## Fourier transform identities
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## Fourier transform identities and properties
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| Time domain $x(t)$ | Frequency domain $X(f)$ |
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| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
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@ -85,25 +93,30 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
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| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ |
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| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ |
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| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ |
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| $\cos(2\pi f_c t+\theta)$ | $\frac{1}{2}[\delta(f - f_c)\exp(j\theta) + \delta(f + f_c)\exp(-j\theta)]\quad\text{Use for coherent recv.}$ |
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| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ |
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| $\sin(2\pi f_c t+\theta)$ | $\frac{1}{2j} [\delta(f - f_c)\exp(j\theta) - \delta(f + f_c)\exp(-j\theta)]$ |
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| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ |
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| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ |
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| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ |
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| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ |
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| Time domain $x(t)$ | Frequency domain $X(f)$ | Property |
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| ------------------------------- | ------------------------------------------------ | ------------------------- |
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| ---------------------------------------- | ------------------------------------------------ | ----------------------------- |
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| $g(t-a)$ | $\exp(-j2\pi fa)G(f)$ | Time shifting |
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| $\exp(-j2\pi f_c t)g(t)$ | $G(f-f_c)$ | Frequency shifting |
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| $g(bt)$ | $\frac{G(f/b)}{\|b\|}$ | Time scaling |
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| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)$ | Time scaling and shifting |
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| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad$ | Differentiation wrt time |
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| $tg(t)$ | $\frac{1}{2\pi}\frac{d}{df}G(f)\quad$ | Differentiation wrt frequency |
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| $g^*(t)$ | $G^*(-f)$ | Conjugate functions |
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| $G(t)$ | $g(-f)$ | Duality |
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| $\int_{-\infty}^t g(\tau)d\tau$ | $\frac{1}{j2\pi f}G(f)+\frac{G(0)}{2}\delta(f)$ | Integration wrt time |
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| $g(t)h(t)$ | $G(f)*H(f)$ | Time multiplication |
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| $g(t)*h(t)$ | $G(f)H(f)$ | Time convolution |
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| $ag(t)+bh(t)$ | $aG(f)+bH(f)$ | Linearity $a,b$ constants |
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| $\int_{-\infty}^\infty x(t)y^*(t)dt$ | $\int_{-\infty}^\infty X(f)Y^*(f)df$ | Parseval's theorem |
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| $E_x=\int_{-\infty}^\infty \|x(t)\|^2dt$ | $E_x=\int_{-\infty}^\infty \|X(f)\|^2df$ | Parseval's theorem |
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| Description | Property |
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| ----------------------------------- | ----------------- |
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@ -338,11 +351,13 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$
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```math
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\begin{align*}
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s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\
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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)}\\
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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)}\\
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s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\quad\text{FM single tone}\\
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\beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\
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f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)\quad\text{Instantaneous frequency from instantaneous phase}\\
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\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}\\
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D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}
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\Delta f&=\max_t(f_i(t))- \min_t(f_i(t))\quad\text{Maximum frequency deviation}\\
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\beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\
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D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\
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\end{align*}
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```
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@ -388,27 +403,6 @@ B &= \begin{cases}
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<!-- MATH END -->
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#### $\Delta f$ of arbitrary modulating signal
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Find instantaneous frequency $f_\text{FM}$.
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$M$: Number of **pairs** of significant sidebands
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```math
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\begin{align*}
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s(t)&=A_c\cos(\theta_\text{FM}(t))\\
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f_\text{FM}(t) &= \frac{1}{2\pi}\frac{d\theta_\text{FM}(t)}{dt}\\
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A_m &= \max_t|m(t)|\\
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\Delta f &= \max_t(f_\text{FM}(t)) - f_c\\
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W_m &= \text{max}(\text{frequencies in $\theta_\text{FM}(t)$...}) \\
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\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)\\
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D &= \frac{\Delta f}{W_m}\\
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B_T &= 2(D+1)W_m
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\end{align*}
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```
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<!-- MATH END -->
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### Complex envelope
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```math
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@ -436,20 +430,44 @@ B_T &= 2(D+1)W_m
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G_x(f)&=G(f)G_w(f)\text{ (PSD)}\\
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G_x(f)&=\lim_{T\to\infty}\frac{|X_T(f)|^2}{T}\text{ (PSD)}\\
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G_x(f)&=\mathfrak{F}[R_x(\tau)]\text{ (WSS)}\\
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P&=\sigma^2=\int_\mathbb{R}G_x(f)df\\
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P&=\sigma^2=\lim_{t\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\\
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P_x&={\sigma_x}^2=\int_\mathbb{R}G_x(f)df\quad\text{For zero mean}\\
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P_x&={\sigma_x}^2=\lim_{t\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\quad\text{For zero mean}\\
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P[A\cos(2\pi f t+\phi)]&=\frac{A^2}{2}\quad\text{Power of sinusoid }\\
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E&=\int_{-\infty}^{\infty}|x(t)|^2dt=|X(f)|^2\\
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E_x&=\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{-\infty}^{\infty}|X(f)|^2df\quad\text{Parseval's theorem}\\
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R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}
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\end{align*}
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```
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<!-- MATH END -->
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##
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## Noise performance
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Coherent detection system.
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```math
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\begin{align*}
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y(t) &= m(t)\cos(2\pi f_c t+\theta) = m(t)\cos^2(2\pi f_c t+\theta)\text{ After IF mixing.}\\
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&= m(t)\frac{1}{2}(1+\cos(4 f_c t+2\theta))\\
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&= \frac{1}{2}m(t)+\frac{1}{2}\cos(4 f_c t+2\theta)\\
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\implies S_u(f) &= \left(\frac{1}{2}\right)^2S_u(f)\quad\text{After LPF.}
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\end{align*}
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```
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<!-- MATH END -->
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Use formualas from previous section, [Power, energy and autocorrelation](#power-energy-and-autocorrelation).\
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Use these formulas in particular:
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```math
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\begin{align*}
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G_\text{WGN}(f)&=\frac{N_0}{2}\\
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G_x(f)&=|H(f)|^2G_w(f)&\text{Note the square in $|H(f)|^2$}\\
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P_x&={\sigma_x}^2=\int_\mathbb{R}G_x(f)df&\text{Often perform graphical integration}\\
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\end{align*}
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```
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<!-- MATH END -->
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```math
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\begin{align*}
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\text{CNR}_\text{in} &= \frac{P_\text{in}}{P_\text{noise}}\\
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@ -526,7 +544,8 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt
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Cannot add directly due to copyright!
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<img src="copyrighted_images/sampling.png" alt="sampling" class="copyrighted">
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## Quantizer
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@ -557,7 +576,8 @@ Cannot add directly due to copyright!
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Cannot add directly due to copyright!
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<img src="copyrighted_images/quantizer.png" alt="quantizer" class="copyrighted">
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## Line codes
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@ -748,7 +768,8 @@ Remember that $T=2T_b$
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### 1. Filter function
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Find transfer function $h(t)$ of matched filter and apply to an input:
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Find transfer function $h(t)$ of matched filter and apply to an input:\
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Note that $x(T-t)$ is equivalent to horizontally flipping $x(t)$ around $x=T/2$.
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```math
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\begin{align*}
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@ -761,7 +782,7 @@ Find transfer function $h(t)$ of matched filter and apply to an input:
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<!-- MATH END -->
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### 2. Bit error rate
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### 2. Bit error rate of matched filter
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Bit error rate (BER) from matched filter outputs and filter output noise
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@ -886,9 +907,34 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems
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## ISI, channel model
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### Raised cosine (RC) pulse
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```math
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0\leq\alpha\leq1
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```
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<!-- MATH END -->
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⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method.
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### Nyquist criterion for zero ISI
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TODO:
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<!-- P_r(kT)=\begin{cases}
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1 & k=0\\
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0 & k\neq0
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\end{cases} -->
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```math
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\begin{align*}
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D &> 2W\quad\text{Use $W$ from table below depending on modulation scheme.}\\
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B_\text{Nyquist} &= \frac{W}{1+\alpha}\\
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\alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\
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\end{align*}
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```
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<!-- MATH END -->
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### Nomenclature
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@ -903,25 +949,17 @@ TODO:
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<!-- MATH END -->
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### Raised cosine (RC) pulse
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```math
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0\leq\alpha\leq1
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```
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<!-- MATH END -->
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⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method.
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### Bandwidth $W$ and bit error rate of modulation schemes
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To solve this type of question:
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1. Use the formula for $D$ below
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2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$.
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| Linear modulation ($M$-PSK, BPSK, $M$-QAM) | NRZ unipolar encoding, BASK |
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| Linear modulation | Half |
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| --------------------------------------------------- | -------------------------------------------------- |
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| BPSK, QPSK, $M$-PSK, $M$-QAM, ASK, FSK | $M$-PAM, PAM |
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| RZ unipolar, Manchester | NRZ Unipolar, NRZ Polar, Bipolar RZ |
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| $W=B_\text{\color{green}abs-abs}$ | $W=B_\text{\color{green}abs}$ |
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| $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ |
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| $D=\frac{W\text{ symbol/s}}{1+\alpha}$ | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$ |
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\begin{align*}
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R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\
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M\text{ symbol/set}&=2^k\\
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T\text{ s/symbol}&=1/(D\text{ symbol/s})\\
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E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\
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\end{align*}
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```
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<!-- MATH END -->
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### Nyquist stuff
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#### TODO: Condition for 0 ISI
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```math
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P_r(kT)=\begin{cases}
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1 & k=0\\
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0 & k\neq0
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\end{cases}
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```
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<!-- MATH END -->
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#### Other
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```math
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\begin{align*}
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\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 $B_\text{abs}$)}\\
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\alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\
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T&=1/D
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\end{align*}
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```
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<!-- MATH END -->
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### Table of bandpass signalling and BER
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| **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** |
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N &\to\text{Output alphabet size}\\
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\mathbf{p} &\to\text{Probability vector, any row of the transition matrix}\\
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C &= \log_2(N)-H(\mathbf{p})\quad\text{Capacity for weakly symmetric and symmetric channels}\\
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R &< C \text{ for error-free transmission}
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R_b &< C \text{ for error-free transmission}
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\end{align*}
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```
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@ -1133,16 +1147,30 @@ C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)
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<!-- MATH END -->
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#### Channel capacity of a bandwidth AWGN channel
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Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
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#### Channel capacity of a bandwidth limited AWGN channel
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```math
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\begin{align*}
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P_s&\to\text{Bandwidth limited average power}\\
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y_i&=\text{bandpass}_W(x_i)+n_i\quad n_i\thicksim N(0,N_0/2)\\
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C&=W\log_2\left(1+\frac{P_s}{N_0 W}\right)\\
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C&=W\log_2(1+\text{SNR})\quad\text{SNR}=P_s/(N_0 W)
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C&=W\log_2(1+\text{SNR})\\
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\text{SNR}&=P_s/(N_0 W)
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\end{align*}
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```
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<!-- MATH END -->
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#### Shannon limit
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```math
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\begin{align*}
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R_b &< C\\
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\implies R_b &< W\log_2\left(1+\frac{P_s}{N_0 W}\right)\quad\text{For bandwidth limited AWGN channel}\\
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\frac{E_b}{N_0} &> \frac{2^\eta-1}{\eta}\quad\text{SNR per bit required for error-free transmission}\\
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\eta &= \frac{R_b}{W}\quad\text{Spectral efficiency (bit/(s-Hz))}\\
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\eta &\gg 1\quad\text{Bandwidth limited}\\
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\eta &\ll 1\quad\text{Power limited}
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\end{align*}
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```
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## Channel code
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Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition.
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| | | |
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| ---------------- | --------------------------------- | ---------------------------------------------------------------------------------------------------------------------------- |
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| Hamming weight | $w_H(x)$ | Number of `'1'` in codeword $x$ |
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1. Zero vector must be present at least once
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2. The XOR of any codeword pair in the code must result in a codeword that is already present in the code table.
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For a linear block code, $d_\text{min}=w_\text{min}$
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3. $d_\text{min}=w_\text{min}$ (Implied by (1) and (2).)
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### Code generation
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