commit 8d86ce77eb74d4d55fcb5515385e70cbe34c77b2 Author: Peter Tanner Date: Tue Oct 29 04:59:21 2024 +0800 v1-rc diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..83384af --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +copyrighted_images \ No newline at end of file diff --git a/.markdownlint.json b/.markdownlint.json new file mode 100644 index 0000000..ff2e60d --- /dev/null +++ b/.markdownlint.json @@ -0,0 +1,4 @@ +{ + "MD013": false, + "no-duplicate-heading": false +} \ No newline at end of file diff --git a/COPYING.txt b/COPYING.txt new file mode 100644 index 0000000..f288702 --- /dev/null +++ b/COPYING.txt @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, your program's commands +might be different; for a GUI interface, you would use an "about box". + + You should also get your employer (if you work as a programmer) or school, +if any, to sign a "copyright disclaimer" for the program, if necessary. +For more information on this, and how to apply and follow the GNU GPL, see +. + + The GNU General Public License does not permit incorporating your program +into proprietary programs. If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/README.html b/README.html new file mode 100644 index 0000000..f419568 --- /dev/null +++ b/README.html @@ -0,0 +1,2300 @@ + + + + + Idiot's guide to ELEC4402 communication systems + + + + + + + + + + + + + +

Idiot's guide to ELEC4402 communication systems

+

License and information

+

Notes are open-source and licensed under the GNU GPL-3.0. You must include the full-text of the license 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

+
+GPL copyright information +Copyright (C) 2024 Peter Tanner +

This program is free software: you can redistribute it and/or modify +it under the terms of the GNU General Public License as published by +the Free Software Foundation, either version 3 of the License, or +(at your option) any later version.

+

This program is distributed in the hope that it will be useful, +but WITHOUT ANY WARRANTY; without even the implied warranty of +MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +GNU General Public License for more details.

+

You should have received a copy of the GNU General Public License +along with this program. If not, see http://www.gnu.org/licenses/.

+
+

Access a web-copy of the notes for printing

+

I accept pull requests or suggestions but the content must not be copyrighted under a non-GPL compatible license.

+

Fourier transform identities

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Time FunctionFourier Transform
rect(tT)Π(tT)\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)T sinc(fT)T \, \text{sinc}(fT)
sinc(2Wt)\text{sinc}(2Wt)12Wrect(f2W)12WΠ(f2W)\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\exp(-at)u(t), \, a>01a+j2πf\frac{1}{a + j2\pi f}
exp⁡(−a∣t∣), a>0\exp(-a\lvert t \rvert), \, a>02aa2+(2πf)2\frac{2a}{a^2 + (2\pi f)^2}
exp⁡(−πt2)\exp(-\pi t^2)exp⁡(−πf2)\exp(-\pi f^2)
{1−∣t∣T,∣t∣<T0,∣t∣≥T\begin{cases} 1 - \frac{\lvert t \rvert}{T}, & \lvert t \rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}T sinc2(fT)T \, \text{sinc}^2(fT)
δ(t)\delta(t)11
11δ(f)\delta(f)
δ(t−t0)\delta(t - t_0)exp⁡(−j2πft0)\exp(-j2\pi f t_0)
g(t−a)g(t-a)exp⁡(−j2πfa)G(f)shift property\exp(-j2\pi fa)G(f)\quad\text{shift property}
g(bt)g(bt)G(f/b)∣b∣scaling property\frac{G(f/b)}{|b|}\quad\text{scaling property}
g(bt−a)g(bt-a)1∣b∣exp⁡(−j2πa(f/b))⋅G(f/b)shift & scale\frac{1}{|b|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift \& scale}
ddtg(t)\frac{d}{dt}g(t)j2πfG(f)differentiation propertyj2\pi fG(f)\quad\text{differentiation property}
G(t)G(t)g(−f)duality propertyg(-f)\quad\text{duality property}
g(t)h(t)g(t)h(t)G(f)∗H(f)G(f)*H(f)
g(t)∗h(t)g(t)*h(t)G(f)H(f)G(f)H(f)
exp⁡(j2πfct)\exp(j2\pi f_c t)δ(f−fc)\delta(f - f_c)
cos⁡(2πfct)\cos(2\pi f_c t)12[δ(f−fc)+δ(f+fc)]\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]
sin⁡(2πfct)\sin(2\pi f_c t)12j[δ(f−fc)−δ(f+fc)]\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]
sgn(t)\text{sgn}(t)1jπf\frac{1}{j\pi f}
1πt\frac{1}{\pi t}−j sgn(f)-j \, \text{sgn}(f)
u(t)u(t)12δ(f)+1j2πf\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}
∑n=−∞∞δ(t−nT0)\sum_{n=-\infty}^{\infty} \delta(t - nT_0)1T0∑n=−∞∞δ(f−nT0)=f0∑n=−∞∞δ(f−nf0)\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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Function NameFormula
Unit Step Functionu(t)={1,t>012,t=00,t<0u(t) = \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}
Signum Functionsgn(t)={+1,t>00,t=0−1,t<0\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}
sinc Functionsinc(2Wt)=sin⁡(2πWt)2πWt\text{sinc}(2Wt) = \frac{\sin(2\pi W t)}{2\pi W t}
Rectangular Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5\text{rect}(t) = \Pi(t) = \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}
Convolutiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτg(t)*h(t)=(g*h)(t)=\int_\infty^\infty g(\tau)h(t-\tau)d\tau
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Fourier transform of continuous time periodic signal

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Required for some questions on sampling:

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Transform a continuous time-periodic signal xp(t)=∑n=−∞∞x(t−nTs)x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s) with period TsT_s:

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Xp(f)=∑n=−∞∞Cnδ(f−nfs)fs=1TsX_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} +

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Calculate CnC_n coefficient as follows from xp(t)x_p(t):

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Cn=1Ts∫Tsxp(t)exp⁡(−j2πfst)dt=1TsX(nfs)(TODO: Check)x(t−nTs) is contained in the interval Ts\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*} +

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rect\text{rect} function

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rect

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Bessel function

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∑n∈ZJn2(β)=1Jn(β)=(−1)nJ−n(β)\begin{align*} + \sum_{n\in\Z}{J_n}^2(\beta)&=1\\ + J_n(\beta)&=(-1)^nJ_{-n}(\beta) +\end{align*} +

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White noise

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RW(τ)=N02δ(τ)=kT2δ(τ)=σ2δ(τ)Gw(f)=N02N0=kTGy(f)=∣H(f)∣2Gw(f)Gy(f)=G(f)Gw(f)\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*} +

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WSS

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μX(t)=μX ConstantRXX(t1,t2)=RX(t1−t2)=RX(τ)E[X(t1)X(t2)]=E[X(t)X(t+τ)]\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*} +

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Ergodicity

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⟨X(t)⟩T=12T∫−TTx(t)dt⟨X(t+τ)X(t)⟩T=12T∫−TTx(t+τ)x(t)dtE[⟨X(t)⟩T]=12T∫−TTx(t)dt=12T∫−TTmXdt=mX\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*} +

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TypeNormalMean square sense
ergodic in mean

lim⁡T→∞⟨X(t)⟩T=mX(t)=mX\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X

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lim⁡T→∞VAR[⟨X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0

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ergodic in autocorrelation function

lim⁡T→∞⟨X(t+τ)X(t)⟩T=RX(τ)\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)

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lim⁡T→∞VAR[⟨X(t+τ)X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0

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A WSS random process needs to be both ergodic in mean and autocorrelation to be considered an ergodic process

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Other identities

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f∗(g∗h)=(f∗g)∗hConvolution associativea(f∗g)=(af)∗gConvolution associative∑x=−∞∞(f(xa)δ(ω−xb))=f(ωab)\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*} +

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Other trig

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cos⁡2θ=2cos⁡2θ−1⇔cos⁡2θ+12=cos⁡2θe−jα−ejα=−2jsin⁡(α)e−jα+ejα=2cos⁡(α)cos⁡(−A)=cos⁡(A)sin⁡(−A)=−sin⁡(A)sin⁡(A+π/2)=cos⁡(A)sin⁡(A−π/2)=−cos⁡(A)cos⁡(A−π/2)=sin⁡(A)cos⁡(A+π/2)=−sin⁡(A)∫x∈Rsinc(Ax)=1∣A∣\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*} +

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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)=12(cos⁡(A−B)+cos⁡(A+B))cos⁡(A)sin⁡(B)=12(sin⁡(A+B)−sin⁡(A−B))sin⁡(A)sin⁡(B)=12(cos⁡(A−B)−cos⁡(A+B))\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*} +

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cos⁡(A)+cos⁡(B)=2cos⁡(A2−B2)cos⁡(A2+B2)cos⁡(A)−cos⁡(B)=−2sin⁡(A2−B2)sin⁡(A2+B2)sin⁡(A)+sin⁡(B)=2sin⁡(A2+B2)cos⁡(A2−B2)sin⁡(A)−sin⁡(B)=2sin⁡(A2−B2)cos⁡(A2+B2)cos⁡(A)+sin⁡(B)=−2sin⁡(A2−B2−π4)sin⁡(A2+B2+π4)cos⁡(A)−sin⁡(B)=−2sin⁡(A2+B2−π4)sin⁡(A2−B2+π4)\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*} +

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IQ/Complex envelope

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

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Convert complex envelope representation to time-domain representation of signal

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g(t)=gI(t)cos⁡(2πfct)−gQ(t)sin⁡(2πfct)=Re[g~(t)exp⁡(j2πfct)]=A(t)cos⁡(2πfct+ϕ(t))A(t)=∣g(t)∣=gI2(t)+gQ2(t)Amplitudeϕ(t)PhasegI(t)=A(t)cos⁡(ϕ(t))In-phase componentgQ(t)=A(t)sin⁡(ϕ(t))Quadrature-phase component\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*} +

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For transfer function

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h(t)=hI(t)cos⁡(2πfct)−hQ(t)sin⁡(2πfct)=2Re[h~(t)exp⁡(j2πfct)]⇒h~(t)=hI(t)/2+jhQ(t)/2=A(t)/2exp⁡(jϕ(t))\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*} +

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AM

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CAM

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ma=min⁡t∣kam(t)∣Acka is the amplitude sensitivity (volt−1), ma is the modulation index.ma=Amax−AminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))x(t)=Accos⁡(2πfct)[1+kam(t)]=Accos⁡(2πfct)[1+mam(t)/Ac],where m(t)=Amm^(t) and m^(t) is the normalized modulating signalPc=Ac22Carrier powerPx=14ma2Ac2η=Signal PowerTotal Power=PxPx+PcBT=2fm=2B\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*} +

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BTB_T: Signal bandwidth +BB: Bandwidth of modulating wave

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

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DSB-SC

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xDSB(t)=Accos⁡(2πfct)m(t)BT=2fm=2B\begin{align*} + x_\text{DSB}(t) &= A_c \cos{(2\pi f_c t)} m(t)\\ + B_T&=2f_m=2B +\end{align*} +

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FM/PM

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s(t)=Accos⁡[2πfct+kpm(t)]Phase modulated (PM)s(t)=Accos⁡[2πfct+2πkf∫0tm(τ)dτ]Frequency modulated (FM)s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]FM single toneβ=Δffm=kfAmModulation indexΔf=βfm=kfAmfm=max⁡t(kfm(t))−min⁡t(kfm(t))Maximum frequency deviationD=ΔfWmDeviation ratio, where Wm is bandwidth of m(t) (Use FT)\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*} +

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Bessel form and magnitude spectrum (single tone)

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s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]⇔s(t)=Ac∑n=−∞∞Jn(β)cos⁡[2π(fc+nfm)t]\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*} +

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FM signal power

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Pav=Ac22Pband_index=Ac2Jband_index2(β)2band_index=0  ⟹  fc+0fmband_index=1  ⟹  fc+1fm,…\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*} +

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Carson's rule to find BB (98% power bandwidth rule)

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B=2Mfm=2(β+1)fm=2(Δf+fm)=2(kfAm+fm)=2(D+1)WmB={2(Δf+fm)FM, sinusoidal message2(Δϕ+1)fmPM, sinusoidal message\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*} +

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Δf\Delta f of arbitrary modulating signal

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Find instantaneous frequency fFMf_\text{FM}.

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MM: Number of pairs of significant sidebands

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s(t)=Accos⁡(θFM(t))fFM(t)=12πdθFM(t)dtAm=max⁡t∣m(t)∣Δf=max⁡t(fFM(t))−fcWm=max(frequencies in θFM(t)...)Example: sinc(At+t)+2cos⁡(2πt)=sin⁡(2π((At+t)/2))π(At+t)+2cos⁡(2πt)→Wm=max⁡(A+12,1)D=ΔfWmBT=2(D+1)Wm\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*} +

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Complex envelope

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s(t)=Accos⁡(2πfct+βsin⁡(2πfmt))⇔s~(t)=Acexp⁡(jβsin⁡(2πfmt))s(t)=Re[s~(t)exp⁡(j2πfct)]s~(t)=Ac∑n=−∞∞Jn(β)exp⁡(j2πfmt)\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*} +

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Band

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NarrowbandWideband
D<1,β<1D<1,\beta<1D>1,β>1D>1,\beta>1
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Power, energy and autocorrelation

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GWGN(f)=N02Gx(f)=∣H(f)∣2Gw(f) (PSD)Gx(f)=G(f)Gw(f) (PSD)Gx(f)=lim⁡T→∞∣XT(f)∣2T (PSD)Gx(f)=F[Rx(τ)] (WSS)P=σ2=∫RGx(f)dfP=σ2=lim⁡t→∞1T∫−T/2T/2∣x(t)∣2dtP[Acos⁡(2πft+ϕ)]=A22Power of sinusoid E=∫−∞∞∣x(t)∣2dt=∣X(f)∣2Rx(τ)=F(Gx(f))PSD to 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*} +

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Noise performance

+

CNRin=PinPnoiseCNRin,FM=A22WN0SNRFM=3A2kf2P2N0W3SNR(dB)=10log⁡10(SNR)Decibels from ratio\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

+

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)∑n∈Zδ(t−nTs)=∑n∈Zx(nTs)δ(t−nTs)Xs(f)=X(f)∗∑n∈Zδ(f−nTs)=X(f)∗∑n∈Zδ(f−nfs)B>12fs,2B>fs→Aliasing\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)x'(t) which can be reconstructed from a sampled signal xs(t)x_s(t): Put xs(t)x_s(t) through LPF with maximum frequency of fs/2f_s/2 and minimum frequency of −fs/2-f_s/2. Anything outside of the BPF will be attenuated, therefore nn which results in frequencies outside the BPF will evaluate to 00 and can be ignored.

+

Example: fs=5000  ⟹  LPF∈[−2500,2500]f_s=5000\implies \text{LPF}\in[-2500,2500]

+

Then iterate for n=0,1,−1,2,−2,…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 Xˉs(f)\bar X_s(f) back to time domain to get xs(t)x_s(t)

+

Fourier transform of continuous time periodic signal (1)

+

Required for some questions on sampling:

+ +

Transform a continuous time-periodic signal xp(t)=∑n=−∞∞x(t−nTs)x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s) with period TsT_s:

+

Xp(f)=∑n=−∞∞Cnδ(f−nfs)fs=1TsX_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} +

+

Calculate CnC_n coefficient as follows from xp(t)x_p(t):

+ +

Cn=1Ts∫Tsxp(t)exp⁡(−j2πfst)dt=1TsX(nfs)(TODO: Check)x(t−nTs) is contained in the interval Ts\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*} +

+ +

Nyquist criterion for zero-ISI

+

Do not transmit more than 2B2B samples per second over a channel of BB bandwidth.

+

By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142

+

Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to images/sampling.png)

+

sampling +sampling

+

Quantizer

+

Δ=xMax−xMin2kfor k-bit quantizer (V/lsb)\begin{align*} + \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\ +\end{align*} +

+

Quantization noise

+

e:=y−xQuantization errorμE=E[E]=0Zero meanσE2=E[E2]−02=∫−Δ/2Δ/2e2×(1Δ)deWhere E∼1/Δ uniform over (−Δ/2,Δ/2)SQNR=Signal powerQuantization noiseSQNR(dB)=10log⁡10(SQNR)\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 +quantizer

+

Line codes

+

binary_codes

+

Rb→Bit rateD→Symbol rate | Rd | 1/TbA→maV(f)→Pulse shapeVrectangle(f)=Tsinc(fT×DutyCycle)GMunipolarNRZ(f)=(M2−1)A2D12∣V(f)∣2+(M−1)24(DA)2∑l=−∞∞∣V(lD)∣2δ(f−lD)GMpolarNRZ(f)=(M2−1)A2D3∣V(f)∣2GunipolarNRZ(f)=A24Rb(sinc2(fRb)+Rbδ(f)),NB0=RbGpolarNRZ(f)=A2Rbsinc2(fRb)GunipolarNRZ(f)=A24Rb(sinc2(fRb)+Rbδ(f))GunipolarRZ(f)=A216(∑l=−∞∞δ(f−lTb)∣sinc(duty×l)∣2+Tb∣sinc(duty×fTb)∣2),NB0=2Rb\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

+

BASK

+

Basis functions

+

φ1(t)=2Tbcos⁡(2πfct)0≤t≤Tb\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

+

bn:{1,0}→an:{1,0}b_n:\{1,0\}\to a_n:\{1,0\} +

+

2 possible waveforms

+

s1(t)=AcTb2φ1(t)=2Ebφ1(t)s1(t)=0Since Eb=Eaverage=12(Ac22×Tb+0)=Ac24Tb\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=2Ebd=\sqrt{2E_b}

+

BPSK

+

Basis functions

+

φ1(t)=2Tbcos⁡(2πfct)0≤t≤Tb\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

+

bn:{1,0}→an:{1,−1}b_n:\{1,0\}\to a_n:\{1,\color{lime}-1\color{white}\} +

+

2 possible waveforms

+

s1(t)=AcTb2φ1(t)=Ebφ1(t)s1(t)=−AcTb2φ1(t)=−Ebφ2(t)Since Eb=Eaverage=12(Ac22×Tb+Ac22×Tb)=Ac22Tb\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=2Ebd=2\sqrt{E_b}

+

QPSK (M=4M=4 PSK)

+

Basis functions

+

T=2TbTime per symbol for two bits Tbφ1(t)=2Tcos⁡(2πfct)0≤t≤Tφ2(t)=2Tsin⁡(2πfct)0≤t≤T\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

+

s1(t)=Es/2[φ1(t)+φ2(t)]s2(t)=Es/2[φ1(t)−φ2(t)]s3(t)=Es/2[−φ1(t)+φ2(t)]s4(t)=Es/2[−φ1(t)−φ2(t)]\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 ∣si(t)∣=Ac|s_i(t)|=A_c, have to normalize distance as follows:

+

si(t)=AcT/2/2×[α1iφ1(t)+α2iφ2(t)]=TAc2/4[α1iφ1(t)+α2iφ2(t)]=Es/2[α1iφ1(t)+α2iφ2(t)]\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

+

Symbol mapping: {1,0}→{1,−1}I(t)=b2nφ1(t)Even bitsQ(t)=b2n+1φ2(t)Odd bitsx(t)=Ac[I(t)cos⁡(2πfct)−Q(t)sin⁡(2πfct)]\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

+
+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)"]
+

+
+

Remember that T=2TbT=2T_b

+ + + + + + + + + + + + + + + + + + + + + +
bnb_nQPSK bits
I(t)I(t) (Odd, 1st bits)QPSK bits
Q(t)Q(t) (Even, 2nd bits)QPSK bits
+

Matched filter

+

1. Filter function

+

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

+

h(t)=s1(T−t)−s2(T−t)h(t)=s∗(T−t)((.)* is the conjugate)son(t)=h(t)∗sn(t)=∫∞∞h(τ)sn(t−τ)dτFilter outputno(t)=h(t)∗n(t)Noise at filter output\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

+

Q(x)=12−12erf(x2)⇔erf(x2)=1−2Q(x)Eb=d2=∫−∞∞∣s1(t)−s2(t)∣2dtEnergy per bit/DistanceT=1/RbRb: BitrateEb=PT=Pav/RbEnergy per bitP(W)=10P(dB)10PRX(W)=PTX(W)⋅10Ploss(dB)10Ploss is expressed with negative sign e.g. "-130 dB"BERMatchedFilter=Q(d22N0)=Q(Eb2N0)BERunipolarNRZ|BASK=Q(d2N0)=Q(EbN0)BERpolarNRZ|BPSK=Q(2d2N0)=Q(2EbN0)\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*} +

+
+

Value tables for erf(x)\text{erf}(x) and Q(x)Q(x)

+

erf(x)\text{erf}(x) function

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
xxerf(x)\text{erf}(x)xxerf(x)\text{erf}(x)xxerf(x)\text{erf}(x)
0.000.000.000000.000000.750.750.711160.711161.501.500.966110.96611
0.050.050.056370.056370.800.800.742100.742101.551.550.971620.97162
0.100.100.112460.112460.850.850.770670.770671.601.600.976350.97635
0.150.150.168000.168000.900.900.796910.796911.651.650.980380.98038
0.200.200.222700.222700.950.950.820890.820891.701.700.983790.98379
0.250.250.276330.276331.001.000.842700.842701.751.750.986670.98667
0.300.300.328630.328631.051.050.862440.862441.801.800.989090.98909
0.350.350.379380.379381.101.100.880210.880211.851.850.991110.99111
0.400.400.428390.428391.151.150.896120.896121.901.900.992790.99279
0.450.450.475480.475481.201.200.910310.910311.951.950.994180.99418
0.500.500.520500.520501.251.250.922900.922902.002.000.995320.99532
0.550.550.563320.563321.301.300.934010.934012.502.500.999590.99959
0.600.600.603860.603861.351.350.943760.943763.003.000.999980.99998
0.650.650.642030.642031.401.400.952290.952293.303.300.9999980.999998**
0.700.700.677800.677801.451.450.959700.95970
+

Q(x)Q(x) function

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
xxQ(x)Q(x)xxQ(x)Q(x)xxQ(x)Q(x)xxQ(x)Q(x)
0.000.00$0.52.302.300.0107240.0107244.554.552.6823×10−62.6823 \times 10^{-6}6.806.805.231×10−125.231 \times 10^{-12}
0.050.050.480060.480062.352.350.00938670.00938674.604.602.1125×10−62.1125 \times 10^{-6}6.856.853.6925×10−123.6925 \times 10^{-12}
0.100.100.460170.460172.402.400.00819750.00819754.654.651.6597×10−61.6597 \times 10^{-6}6.906.902.6001×10−122.6001 \times 10^{-12}
0.150.150.440380.440382.452.450.00714280.00714284.704.701.3008×10−61.3008 \times 10^{-6}6.956.951.8264×10−121.8264 \times 10^{-12}
0.200.200.420740.420742.502.500.00620970.00620974.754.751.0171×10−61.0171 \times 10^{-6}7.007.001.2798×10−121.2798 \times 10^{-12}
0.250.250.401290.401292.552.550.00538610.00538614.804.807.9333×10−77.9333 \times 10^{-7}7.057.058.9459×10−138.9459 \times 10^{-13}
0.300.300.382090.382092.602.600.00466120.00466124.854.856.1731×10−76.1731 \times 10^{-7}7.107.106.2378×10−136.2378 \times 10^{-13}
0.350.350.363170.363172.652.650.00402460.00402464.904.904.7918×10−74.7918 \times 10^{-7}7.157.154.3389×10−134.3389 \times 10^{-13}
0.400.400.344580.344582.702.700.0034670.0034674.954.953.7107×10−73.7107 \times 10^{-7}7.207.203.0106×10−133.0106 \times 10^{-13}
0.450.450.326360.326362.752.750.00297980.00297985.005.002.8665×10−72.8665 \times 10^{-7}7.257.252.0839×10−132.0839 \times 10^{-13}
0.500.500.308540.308542.802.800.00255510.00255515.055.052.2091×10−72.2091 \times 10^{-7}7.307.301.4388×10−131.4388 \times 10^{-13}
0.550.550.291160.291162.852.850.0021860.0021865.105.101.6983×10−71.6983 \times 10^{-7}7.357.359.9103×10−149.9103 \times 10^{-14}
0.600.600.274250.274252.902.900.00186580.00186585.155.151.3024×10−71.3024 \times 10^{-7}7.407.406.8092×10−146.8092 \times 10^{-14}
0.650.650.257850.257852.952.950.00158890.00158895.205.209.9644×10−89.9644 \times 10^{-8}7.457.454.667×10−144.667 \times 10^{-14}
0.700.700.241960.241963.003.000.00134990.00134995.255.257.605×10−87.605 \times 10^{-8}7.507.503.1909×10−143.1909 \times 10^{-14}
0.750.750.226630.226633.053.050.00114420.00114425.305.305.7901×10−85.7901 \times 10^{-8}7.557.552.1763×10−142.1763 \times 10^{-14}
0.800.800.211860.211863.103.100.00096760.00096765.355.354.3977×10−84.3977 \times 10^{-8}7.607.601.4807×10−141.4807 \times 10^{-14}
0.850.850.197660.197663.153.150.000816350.000816355.405.403.332×10−83.332 \times 10^{-8}7.657.651.0049×10−141.0049 \times 10^{-14}
0.900.900.184060.184063.203.200.000687140.000687145.455.452.5185×10−82.5185 \times 10^{-8}7.707.706.8033×10−156.8033 \times 10^{-15}
0.950.950.171060.171063.253.250.000577030.000577035.505.501.899×10−81.899 \times 10^{-8}7.757.754.5946×10−154.5946 \times 10^{-15}
1.001.000.158660.158663.303.300.000483420.000483425.555.551.4283×10−81.4283 \times 10^{-8}7.807.803.0954×10−153.0954 \times 10^{-15}
1.051.050.146860.146863.353.350.000404060.000404065.605.601.0718×10−81.0718 \times 10^{-8}7.857.852.0802×10−152.0802 \times 10^{-15}
1.101.100.135670.135673.403.400.000336930.000336935.655.658.0224×10−98.0224 \times 10^{-9}7.907.901.3945×10−151.3945 \times 10^{-15}
1.151.150.125070.125073.453.450.000280290.000280295.705.705.9904×10−35.9904 \times 10^{-3}7.957.959.3256×10−169.3256 \times 10^{-16}
1.201.200.115070.115073.503.500.000232630.000232635.755.754.4622×10−94.4622 \times 10^{-9}8.008.006.221×10−166.221 \times 10^{-16}
1.251.250.105650.105653.553.550.000192620.000192625.805.803.3157×10−93.3157 \times 10^{-9}8.058.054.1397×10−164.1397 \times 10^{-16}
1.301.300.09680.09683.603.600.000159110.000159115.855.852.4579×10−92.4579 \times 10^{-9}8.108.102.748×10−162.748 \times 10^{-16}
1.351.350.0885080.0885083.653.650.000131120.000131125.905.901.8175×10−91.8175 \times 10^{-9}8.158.151.8196×10−161.8196 \times 10^{-16}
1.401.400.0807570.0807573.703.700.00010780.00010785.955.951.3407×10−91.3407 \times 10^{-9}8.208.201.2019×10−161.2019 \times 10^{-16}
1.451.450.0735290.0735293.753.758.8417×10−58.8417 \times 10^{-5}6.006.009.8659×10−109.8659 \times 10^{-10}8.258.257.9197×10−177.9197 \times 10^{-17}
1.501.500.0668070.0668073.803.807.2348×10−57.2348 \times 10^{-5}6.056.057.2423×10−107.2423 \times 10^{-10}8.308.305.2056×10−175.2056 \times 10^{-17}
1.551.550.0605710.0605713.853.855.9059×10−55.9059 \times 10^{-5}6.106.105.3034×10−105.3034 \times 10^{-10}8.358.353.4131×10−173.4131 \times 10^{-17}
1.601.600.0547990.0547993.903.904.8096×10−54.8096 \times 10^{-5}6.156.153.8741×10−103.8741 \times 10^{-10}8.408.402.2324×10−172.2324 \times 10^{-17}
1.651.650.0494710.0494713.953.953.9076×10−53.9076 \times 10^{-5}6.206.202.8232×10−102.8232 \times 10^{-10}8.458.451.4565×10−171.4565 \times 10^{-17}
1.701.700.0445650.0445654.004.003.1671×10−53.1671 \times 10^{-5}6.256.252.0523×10−102.0523 \times 10^{-10}8.508.509.4795×10−189.4795 \times 10^{-18}
1.751.750.0400590.0400594.054.052.5609×10−52.5609 \times 10^{-5}6.306.301.4882×10−101.4882 \times 10^{-10}8.558.556.1544×10−186.1544 \times 10^{-18}
1.801.800.035930.035934.104.102.0658×10−52.0658 \times 10^{-5}6.356.351.0766×10−101.0766 \times 10^{-10}8.608.603.9858×10−183.9858 \times 10^{-18}
1.851.850.0321570.0321574.154.151.6624×10−51.6624 \times 10^{-5}6.406.407.7688×10−117.7688 \times 10^{-11}8.658.652.575×10−182.575 \times 10^{-18}
1.901.900.0287170.0287174.204.201.3346×10−51.3346 \times 10^{-5}6.456.455.5925×10−115.5925 \times 10^{-11}8.708.701.6594×10−181.6594 \times 10^{-18}
1.951.950.0255880.0255884.254.251.0689×10−51.0689 \times 10^{-5}6.506.504.016×10−114.016 \times 10^{-11}8.758.751.0668×10−181.0668 \times 10^{-18}
2.002.000.022750.022754.304.308.5399×10−68.5399 \times 10^{-6}6.556.552.8769×10−112.8769 \times 10^{-11}8.808.806.8408×10−196.8408 \times 10^{-19}
2.052.050.0201820.0201824.354.356.8069×10−66.8069 \times 10^{-6}6.606.602.0558×10−112.0558 \times 10^{-11}8.858.854.376×10−194.376 \times 10^{-19}
2.102.100.0178640.0178644.404.405.4125×10−65.4125 \times 10^{-6}6.656.651.4655×10−111.4655 \times 10^{-11}8.908.902.7923×10−192.7923 \times 10^{-19}
2.152.150.0157780.0157784.454.454.2935×10−64.2935 \times 10^{-6}6.706.701.0421×10−111.0421 \times 10^{-11}8.958.951.7774×10−191.7774 \times 10^{-19}
2.202.200.0139030.0139034.504.503.3977×10−63.3977 \times 10^{-6}6.756.757.3923×10−127.3923 \times 10^{-12}9.009.001.1286×10−191.1286 \times 10^{-19}
2.252.250.0122240.012224
+

Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems

+

**The value of erf(3.30)\text{erf}(3.30) should be ≈0.999997\approx0.999997 instead, but this value is quoted in the formula table.

+

Receiver output shit

+

ro(t)={so1(t)+no(t)code 1so2(t)+no(t)code 0n:AWGN with σo2\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*} +

+ +

ISI, channel model

+

Nyquist criterion for zero ISI

+

TODO:

+

Nomenclature

+

D→Symbol Rate, Max. Signalling RateT→Symbol DurationM→Symbol set sizeW→Bandwidth\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

+

0≤α≤10\leq\alpha\leq1 +

+

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

+

To solve this type of question:

+
    +
  1. Use the formula for DD below
    2. +
  2. Consult the BER table below to get the BER which relates the noise of the channel N0N_0 to EbE_b and to RbR_b.
    3. +
+ + + + + + + + + + + + + + + + + + + + + + +
Linear modulation (MM-PSK, MM-QAM)NRZ unipolar encoding
W=Babs-absW=B_\text{\color{lime}abs-abs}W=BabsW=B_\text{\color{lime}abs}
W=Babs-abs=1+αT=(1+α)DW=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)DW=Babs=1+α2T=(1+α)D/2W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2
D=W symbol/s1+αD=\frac{W\text{ symbol/s}}{1+\alpha}D=2W symbol/s1+αD=\frac{2W\text{ symbol/s}}{1+\alpha}
+

Symbol set size MM

+

D symbol/s=2W Hz1+αRb bit/s=(D symbol/s)×(k bit/symbol)M symbol/set=2kEb=PT=Pav/RbEnergy per bit\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

+

Pr(kT)={1k=00k≠0P_r(kT)=\begin{cases} + 1 & k=0\\ + 0 & k\neq0 +\end{cases} +

+

Other

+

Excess BW=Babs−BNyquist=1+α2T−12T=α2TFOR NRZ (Use correct Babs)α=Excess BWBNyquist=Babs−BNyquistBNyquistT=1/D\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*} +

+
+

Table of bandpass signalling and BER

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Binary Bandpass SignalingBnull-nullB_\text{null-null} (Hz)Babs-abs=2BabsB_\text{abs-abs}\color{red}=2B_\text{abs} (Hz)BER with Coherent DetectionBER with Noncoherent Detection
ASK, unipolar NRZ2Rb2R_bRb(1+α)R_b (1 + \alpha)Q(Eb/N0)Q\left( \sqrt{E_b / N_0} \right)0.5exp⁡(−Eb/(2N0))0.5\exp(-E_b / (2N_0))
BPSK2Rb2R_bRb(1+α)R_b (1 + \alpha)Q(2Eb/N0)Q\left( \sqrt{2E_b / N_0} \right)Requires coherent detection
Sunde's FSK3Rb3R_bQ(Eb/N0)Q\left( \sqrt{E_b / N_0} \right)0.5exp⁡(−Eb/(2N0))0.5\exp(-E_b / (2N_0))
DBPSK, MM-ary Bandpass Signaling2Rb2R_bRb(1+α)R_b (1 + \alpha)0.5exp⁡(−Eb/N0)0.5\exp(-E_b / N_0)
QPSK/OQPSK (M=4M=4, PSK)RbR_bRb(1+α)2\frac{R_b (1 + \alpha)}{2}Q(2Eb/N0)Q\left( \sqrt{2E_b / N_0} \right)Requires coherent detection
MSK1.5Rb1.5R_b3Rb(1+α)4\frac{3R_b (1 + \alpha)}{4}Q(2Eb/N0)Q\left( \sqrt{2E_b / N_0} \right)Requires coherent detection
MM-PSK (M>4M > 4)2Rb/log⁡2M2R_b / \log_2 MRb(1+α)log⁡2M\frac{R_b (1 + \alpha)}{\log_2 M}2log⁡2MQ(2log⁡2Msin⁡2(π/M)Eb/N0)\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
MM-DPSK (M>4M > 4)2Rb/log⁡2M2R_b / \log_2 MRb(1+α)2log⁡2M\frac{R_b (1 + \alpha)}{2 \log_2 M}2log⁡2MQ(4log⁡2Msin⁡2(π/(2M))Eb/N0)\frac{2}{\log_2 M} Q\left( \sqrt{4 \log_2 M \sin^2 \left( \pi / (2M) \right) E_b / N_0} \right)
MM-QAM (Square constellation)2Rb/log⁡2M2R_b / \log_2 MRb(1+α)log⁡2M\frac{R_b (1 + \alpha)}{\log_2 M}4log⁡2M(1−1M)Q(3log⁡2MM−1Eb/N0)\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
MM-FSK Coherent(M+3)Rb2log⁡2M\frac{(M + 3) R_b}{2 \log_2 M}M−1log⁡2MQ((log⁡2M)Eb/N0)\frac{M - 1}{\log_2 M} Q\left( \sqrt{(\log_2 M) E_b / N_0} \right)
Noncoherent2MRb/log⁡2M2M R_b / \log_2 MM−12log⁡2M0.5exp⁡(−(log⁡2M)Eb/2N0)\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

+ + + + + + + + + + + + + + + + + +
ModulationGx(f)G_x(f)
QuadratureAc24[GI(f−fc)+GI(f+fc)+GQ(f−fc)+GQ(f+fc)]\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∣V(f)∣22∑l=−∞∞R(l)exp⁡(−j2πlfT)What??\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

+

H(x)≥0H(x)=−∑xi∈AxpX(xi)log⁡2(pX(xi))H(x,y)=−∑xi∈Ax∑yi∈AypXY(xi,yi)log⁡2(pXY(xi,yi))Joint entropyH(x,y)=H(x)+H(y)Joint entropy if x and y independentH(x∣y=yj)=−∑xi∈AxpX(xi∣y=yj)log⁡2(pX(xi∣y=yj))Conditional entropyH(x∣y)=−∑yj∈AypY(yj)H(x∣y=yj)Average conditional entropy, equivocationH(x∣y)=−∑xi∈Ax∑yi∈AypX(xi,yj)log⁡2(pX(xi∣y=yj))H(x∣y)=H(x,y)−H(y)H(x,y)=H(x)+H(y∣x)=H(y)+H(x∣y)\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

+

h(x)=−∫RfX(x)log⁡2(fX(x))dx\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 xx after observation by yy.

+

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

+

Channel model

+

Vertical, xx: input
+Horizontal, yy: output

+

P=[p11p12…p1Np21p22…p2N⋮⋮⋱⋮pM1pM2…pMN]\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] +

+

P(yj∣xi)y1y2…yNx1p11p12…p1Nx2p21p22…p2N⋮⋮⋮⋱⋮xMpM1pM2…pMN\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 pX(ai)=P(X=ai)p_X(a_i)=P(X=a_i)

+

Channel maps alphabet {a1,…,aM}→{b1,…,bN}\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}

+

Output has probabiltiy distribution pY(bj)=P(y=bj)p_Y(b_j)=P(y=b_j)

+

pY(bj)=∑i=1MP[x=ai,y=bj]1≤j≤N=∑i=1MP[X=ai]P[Y=bj∣X=ai][pY(b0)pY(b1)…pY(bj)]=[pX(a0)pX(a1)…pX(ai)]×P\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)I(y;x)

+

1. Find H(x)2. Find [pY(b0)pY(b1)…pY(bj)]=[pX(a0)pX(a1)…pX(ai)]×P3. Multiply each row in P by pX(ai) since pXY(xi,yi)=P(yi∣xi)P(xi)4. Find H(x,y) using each element from (3.)5. Find H(x∣y)=H(x,y)−H(y)6. Find I(y;x)=H(x)−H(x∣y)\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

+ + + + + + + + + + + + + + + + + +
TypeDefinition
Symmetric channelEvery row is a permutation of every other row, Every column is a permutation of every other column. Symmetric  ⟹  Weakly symmetric\text{Symmetric}\implies\text{Weakly symmetric}
Weakly symmetricEvery row is a permutation of every other row, Every column has the same sum
+

Channel capacity of weakly symmetric channel

+

C→Channel capacity (bits/channels used)N→Output alphabet sizep→Probability vector, any row of the transition matrixC=log⁡2(N)−H(p)Capacity for weakly symmetric and symmetric channelsR<C for error-free transmission\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

+

yi=xi+nini∼N(0,N0/2)y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2) +

+

C=12log⁡2(1+PavN0/2)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.

+

Ps→Bandwidth limited average poweryi=bandpassW(xi)+nini∼N(0,N0/2)C=Wlog⁡2(1+PsN0W)C=Wlog⁡2(1+SNR)SNR=Ps/(N0W)\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 weightwH(x)w_H(x)Number of '1' in codeword xx
Hamming distancedH(x1,x2)=wH(x1⊕x2)d_H(x_1,x_2)=w_H(x_1\oplus x_2)Number of different bits between codewords x1x_1 and x2x_2 which is the hamming weight of the XOR of the two codes.
Minimum distancedmind_\text{min}IMPORTANT: x≠0x\neq\bold{0}, excludes weight of all-zero codeword. For a linear block code, dmin=wmind_\text{min}=w_\text{min}
+

Linear block code

+

Code is (n,k)(n,k)

+

nn is the width of a codeword

+

2k2^k codewords

+

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

+
    +
  1. Zero vector must be present at least once
    2. +
  2. The XOR of any codeword pair in the code must result in a codeword that is already present in the code table.
    3. +
+

For a linear block code, dmin=wmind_\text{min}=w_\text{min}

+

Code generation

+

Each generator vector is a binary string of size nn. There are kk generator vectors in G\mathbf{G}.

+

gi=[gi,0…gi,n−2gi,n−1]g0=[1010]Example for n=4G=[g0g1⋮gk−1]=[g0,0…g0,n−2g0,n−1g1,0…g1,n−2g1,n−1⋮⋱⋮⋮gk−1,0…gk−1,n−2gk−1,n−1]\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 m\mathbf{m} is coded as x\mathbf{x} using the generation codewords in G\mathbf{G}:

+

m=[m0…mn−2mk−1]m=[101001]Example for k=6x=mG=m0g0+m1g1+⋯+mk−1gk−1\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 kk message bits (Copy m\mathbf{m} as-is) and (n−k)(n-k) parity bits after the message bits.

+

G=[IkP]=[10…001…0⋮⋮⋱⋮00…1p0,0…p0,n−2p0,n−1p1,0…p1,n−2p1,n−1⋮⋱⋮⋮pk−1,0…pk−1,n−2pk−1,n−1]m=[m0…mn−2mk−1]x=mG=m[IkP]=[mIkmP]=[mb]b=mPParity bits of x\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 H\mathbf{H}

+

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

+

H=[PTIn−k]=[p0Tp1T…pk−1TIn−k]=[p0,0…p0,k−2p0,k−1p1,0…p1,k−2p1,k−1⋮⋱⋮⋮pn−1,0…pn−1,k−2pn−1,k−110…001…0⋮⋮⋱⋮00…1]xHT=0  ⟹  Codeword is valid\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)k=\log_2(N)
    2. +
  2. Partition codewords into kk information bits and remaining bits into n−kn-k parity bits. The information bits should be a simple counter (?).
    3. +
  3. Express parity bits as a linear combination of information bits
    4. +
  4. Put coefficients into P\textbf{P} matrix and find H\textbf{H}
    5. +
+

Example:

+

x1x2x3x4x510110011110000011001\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 x1,x2x_1,x_2 as information bits. Express x3,x4,x5x_3,x_4,x_5 in terms of x1,x2x_1,x_2.

+

x3=x1⊕x2x4=x1⊕x2x5=x2  ⟹  P=x1x2x311x411x501H=[111101100010001]\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 ss errors: dmin≥s+1d_\text{min}\geq s+1

+

Correction of uu errors: dmin≥2u+1d_\text{min}\geq 2u+1

+

CHECKLIST

+
    +
  • Transfer function in complex envelope form (h~(t)\tilde{h}(t)) should be divided by two.
    • +
  • Convolutions: do not forget width when using graphical method
    • +
+ + + + + \ No newline at end of file diff --git a/README.md b/README.md new file mode 100644 index 0000000..7e77b17 --- /dev/null +++ b/README.md @@ -0,0 +1,1171 @@ +# Idiot's guide to ELEC4402 communication systems + +## 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) + +Copyright (C) 2024 Peter Tanner + +
+GPL copyright information +Copyright (C) 2024 Peter Tanner + +This program is free software: you can redistribute it and/or modify +it under the terms of the GNU General Public License as published by +the Free Software Foundation, either version 3 of the License, or +(at your option) any later version. + +This program is distributed in the hope that it will be useful, +but WITHOUT ANY WARRANTY; without even the implied warranty of +MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +GNU General Public License for more details. + +You should have received a copy of the GNU General Public License +along with this program. If not, see . + +
+ +[Access a web-copy of the notes for printing](/README.html) + +I accept pull requests or suggestions but the content must not be copyrighted under a non-GPL compatible license. + +## 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 $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ 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 $x_p(t)$: + + + +$$ +\begin{align*} + % C_n&=X_p(nf_s)\\ + C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ + &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} +\end{align*} +$$ + +### $\text{rect}$ 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) + +$$ +\begin{align*} + s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t] +\end{align*} +$$ + + + +### 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 $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ 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 $x_p(t)$: + + + +$$ +\begin{align*} + % C_n&=X_p(nf_s)\\ + C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ + &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} +\end{align*} +$$ + + + +### 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 + +
+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)"]
+
+ +
+ +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 + +$$ +\begin{align*} + % H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right) + + % \text{BER}_\text{bin}&=p Q\left(\frac{s_{o1}-V_T}{\sigma_o}\right)+(1-p)Q\left(\frac{V_T-s_{o2}}{\sigma_o}\right)\text{, $p\rightarrow$Probability $s_1(t)$ sent, $V_T\rightarrow$Threshold voltage} + Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\ + E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\ + T&=1/R_b\quad\text{$R_b$: Bitrate}\\ + E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ + P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\ + P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\ + \text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\ + \text{BER}_\text{unipolarNRZ|BASK}&=Q\left(\sqrt{\frac{d^2}{N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)\\ + \text{BER}_\text{polarNRZ|BPSK}&=Q\left(\sqrt{\frac{2d^2}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\\ +\end{align*} +$$ + +
+ +## 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 + +$$ +\begin{align*} + r_o(t)&=\begin{cases} + s_{o1}(t)+n_o(t) & \text{code 1}\\ + s_{o2}(t)+n_o(t) & \text{code 0}\\ + \end{cases}\\ + n&: \text{AWGN with }\sigma_o^2\\ +\end{align*} +$$ + + + +## 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 $R_b$. + + + +| 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 + +$$ +\begin{align*} + \text{Excess BW}&=B_\text{abs}-B_\text{Nyquist}=\frac{1+\alpha}{2T}-\frac{1}{2T}=\frac{\alpha}{2T}\quad\text{FOR NRZ (Use correct $B_\text{abs}$)}\\ + \alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\ + T&=1/D +\end{align*} +$$ + +
+ +### 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 diff --git a/images/Constellation.drawio.svg b/images/Constellation.drawio.svg new file mode 100644 index 0000000..590c981 --- /dev/null +++ b/images/Constellation.drawio.svg @@ -0,0 +1,4 @@ + + + +
BASK constellation
QPSK constellation
BPSK constellation
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Linear modulation RC pulse
Linear modulation RC pulse
NRZ unipolar RC pulse
NRZ unipolar RC pulse
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$$W=B_\text{abs}$$
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