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Projects STRLCPY wirelesscomm Commits 0a60dbce
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  • ■ ■ ■ ■ ■ ■
    unit03_fading/prob/prob_fading.tex
    skipped 249 lines
    250 250  \item What is the fraction of time the power is greater than -101 dBm?
    251 251  \end{enumerate}
    252 252   
    253  -\item \emph{Two path channel:} Suppose that a narrowband complex channel $h(t)$ be modeled as
     253 +\item \emph{Auto-correlation with time.}
     254 +Suppose that a narrowband complex channel $h(t)$ be modeled as
    254 255  a wide-sense stationary random process.
    255 256  \begin{enumerate}[label=(\alph*)]
    256 257   
    skipped 18 lines
    275 276  what is the time it takes the channel to change by 10\%?
    276 277  \end{enumerate}
    277 278   
    278  -\item \emph{Auto-correlation.} Consider a multipath fading channel of the form
     279 +\item \emph{Auto-correlation with delay.} Consider a multipath fading channel of the form
    279 280  \[
    280 281   y(t) = \frac{1}{\sqrt{L}}\sum_{\ell = 1}^L g_\ell
    281 282   e^{2\pi if_\ell} x(t-\tau_\ell), \quad f_\ell = f_{max}\cos(\theta_\ell),
    skipped 63 lines
  • unit04_coding/lab_partial/labPdsch.mlx
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  • unit04_coding/lab_partial/labPdsch.pdf
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  • ■ ■ ■ ■ ■ ■
    unit04_coding/prob/mcode.sty
     1 +%%
     2 +%% This is file `mcode.sty'
     3 +%%
     4 +%% It is supposed to help you easily include MATLAB source code
     5 +%% into LaTeX document, but have it nicely highlighted, using
     6 +%% the great listings package.
     7 +%%
     8 +%% PLEASE NOTE that this package does nothing but save you from
     9 +%% figuring out some configurations in setting up the LISTINGS
     10 +%% package. ALL the work is done by that package! Thus, please
     11 +%% refer your questions to the listings package documentation.
     12 +%%
     13 +%% Usage: You have three ways of including your MATLAB code. As
     14 +%% environment, as inline object and directly from an external
     15 +%% file.
     16 +%%
     17 +%% 1) Environment:
     18 +%%
     19 +%% \begin{lstlisting}
     20 +%% YOUR CODE HERE
     21 +%% \end{lstlisting}
     22 +%%
     23 +%%
     24 +%% 2) Inline object:
     25 +%%
     26 +%% Bla bla \mcode{CODEFRAGMENT} bla bla.
     27 +%%
     28 +%%
     29 +%% 3) Include external file (in environment form)
     30 +%%
     31 +%% \lstinputlisting{YOUR-FILE.m}
     32 +%%
     33 +%%
     34 +%% For your convenience this package has the following options:
     35 +%%
     36 +%% - bw if you intend to print the document (highlighting done
     37 +%% via text formatting (bold, italic) and shades of gray)
     38 +%%
     39 +%% - numbered if you want line numbers
     40 +%%
     41 +%% - autolinebreaks if you want the package to automatically
     42 +%% wrap your code. This is buggy as it may well break
     43 +%% break syntax and it doesn't work well with comments.
     44 +%% You REALLY should wrap your code manually.
     45 +%%
     46 +%% - useliterate if you want some characters / relations in
     47 +%% your code to be replace with something more readable.
     48 +%% Example: ~= becomes $\neq$, >= becomes $\geq$, delta
     49 +%% becomes $\delta$ and so on.
     50 +%%
     51 +%% - framed if you want a frame around the source code blocks
     52 +%%
     53 +%% - final if you have ``gloablly'' set the draft option, the
     54 +%% listings package will not output the code at all. to
     55 +%% force it to do so anyway, load this package with the
     56 +%% final option (passes the ``final'' on to listings).
     57 +%%
     58 +%% For example, you may use \usepackage[numbered,framed]{mcode}
     59 +%% in your document preamble.
     60 +%%
     61 +%% Note: Inside code blocks you can escape to LaTeX text mode
     62 +%% using §...§. For ex. §text and some math: $x^2$§, which is
     63 +%% especially useful in comments for putting nicely typeset
     64 +%% equations etc. To get the same colour/style as in the rest
     65 +%% of the comment use \mcommentfont, i.e. §\mcommentfont $x^2$§
     66 +%%
     67 +%% To change the font used, edit the first line in the "custo-
     68 +%% mise below" section. And feel free to edit other things as
     69 +%% well. Refer to the documentation of the listings package to
     70 +%% see what else you could do. If an extra small font is re-
     71 +%% quired, use {\fontfamily{pcr}\fontsize{3}{4.6}\selectfont}
     72 +%% in the definition of \lstbasicfont.
     73 +%%
     74 +%% Author:
     75 +%% Florian Knorn | [email protected] | www.florian-knorn.com
     76 +%%
     77 +%% Version history:
     78 +%% 2.3 -- More keywords (thanks Dominik Wild!)
     79 +%% 2.2 -- Bugfix (thanks Willi Gerbig!)
     80 +%% 2.1 -- Finally automatic detection between end and end
     81 +%% 2.0 -- New options for line breaking and literate prog.
     82 +%% 1.8 -- Fixed typo in documentation regarding §...§
     83 +%% 1.7 -- Added MATLAB block comment syntax %{ ...... %}
     84 +%% 1.6 -- Added some infos, dealing with keyword ``end''
     85 +%% 1.5 -- Tweaked check to see wether textcomp is loaded
     86 +%% 1.4 -- Fixed misconfig (mathescape now set to false)
     87 +%% 1.3 -- Purely cosmetic (tabs replaced by spaces)
     88 +%% 1.2 -- Added \lstset{showstringspaces=false}
     89 +%% 1.1 -- Added \mcode command and [final] option
     90 +%% 1.0 -- Release
     91 + 
     92 + 
     93 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     94 +% D O N ' T T O U C H T H I S %
     95 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     96 +\def\fileversion{2.3}
     97 +\def\filedate{2012/08/31}
     98 + 
     99 +\typeout{-- Package: `mcode' \fileversion\space <\filedate> --}
     100 +\NeedsTeXFormat{LaTeX2e}
     101 +\ProvidesPackage{mcode}[\filedate\space\fileversion]
     102 + 
     103 +% for bw-option
     104 +\newif\ifbw
     105 +\DeclareOption{bw}{\bwtrue}
     106 + 
     107 +% numbered option
     108 +\newif\ifnumbered
     109 +\DeclareOption{numbered}{\numberedtrue}
     110 + 
     111 +% final option
     112 +\newif\iffinal
     113 +\DeclareOption{final}{\finaltrue}
     114 + 
     115 +% autolinebreaks option
     116 +\newif\ifautolinebreaks
     117 +\DeclareOption{autolinebreaks}{\autolinebreakstrue}
     118 + 
     119 +% literate programming (replace certain characters/relations
     120 +\newif\ifuseliterate
     121 +\DeclareOption{useliterate}{\useliteratetrue}
     122 + 
     123 +% framed option
     124 +\newif\ifframed
     125 +\DeclareOption{framed}{\framedtrue}
     126 + 
     127 +\DeclareOption*{% default
     128 + \PackageWarning{mcode}{Unknown option `\CurrentOption' !}%
     129 +}
     130 +\ProcessOptions
     131 + 
     132 +\ifbw\typeout{ - settings optimized for printing (bw formating)}
     133 +\else\typeout{ - settings optimized for display (colour formating)}\fi
     134 +\ifnumbered\typeout{ - line numbering enabled}\else\fi
     135 +\ifuseliterate\typeout{ - literate programming (character replacements) enabled}\else\fi
     136 +\ifautolinebreaks\typeout{ - automatic line breaking enabled (careful, buggy!)}\else\fi
     137 +\ifframed\typeout{ - framed listings}\else\fi
     138 + 
     139 +% This command allows you to typeset syntax highlighted Matlab
     140 +% code ``inline''. The font size \small seems to look best...
     141 +\newcommand{\mcode}[1]{\lstinline[basicstyle=\lstbasicfont\small]|#1|}
     142 + 
     143 +% check if color command exists
     144 +\ifx\color\undefined%
     145 + \RequirePackage{xcolor}%
     146 +\fi
     147 + 
     148 +% check if listings has been loaded
     149 +\ifx\lstset\undefined%
     150 + \iffinal
     151 + \RequirePackage[final]{listings}
     152 + \else
     153 + \RequirePackage{listings}
     154 + \fi
     155 +\fi
     156 + 
     157 +% Check if textcomp has been loaded (this package is needed for
     158 +% upright quotes '' (instead of typographic ones `´)...
     159 +\ifx\textquotesingle\undefined%
     160 + \RequirePackage{textcomp}%
     161 +\fi
     162 + 
     163 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     164 +% C U S T O M I S E B E L O W %
     165 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     166 + 
     167 +% ---------------------------------------------------------------------------------
     168 +% default font
     169 +\def\lstbasicfont{\fontfamily{pcr}\selectfont\footnotesize}
     170 + 
     171 +% ---------------------------------------------------------------------------------
     172 +% matlat languate definition
     173 +\lstdefinelanguage{matlabfloz}{%
     174 + alsoletter={...},%
     175 + morekeywords={% % keywords
     176 + break,case,catch,continue,elseif,else,end,%
     177 + for,function,global,if,otherwise,persistent,%
     178 + return,switch,try,while,methods,properties,%
     179 + events,classdef,...},%
     180 + comment=[l]\%, % comments
     181 + morecomment=[l]..., % comments
     182 + morecomment=[s]{\%\{}{\%\}}, % block comments
     183 + morestring=[m]' % strings
     184 +}[keywords,comments,strings]%
     185 + 
     186 +% ---------------------------------------------------------------------------------
     187 +% general definitions
     188 +\lstset{%
     189 + basicstyle={\lstbasicfont}, % set font
     190 + showstringspaces=false, % do not emphasize spaces in strings
     191 + tabsize=4, % number of spaces of a TAB
     192 + mathescape=false,escapechar=§, % escape to latex with §...§
     193 + upquote=true, % upright quotes
     194 + aboveskip={1.5\baselineskip}, % a bit of space above listings
     195 + columns=fixed % nice spacing
     196 +}
     197 + 
     198 +% ---------------------------------------------------------------------------------
     199 +% define colours and styles
     200 +\ifbw % use font formating and gray 'colors'
     201 + \def\mcommentfont{\color[gray]{.75}\itshape} %comments light gray and italic
     202 + \lstset{language=matlabfloz, % use our version of highlighting
     203 + keywordstyle=\bfseries, % keywords in bold
     204 + commentstyle=\mcommentfont, % comments
     205 + stringstyle=\color[gray]{0.5} % strings darker gray
     206 + }
     207 +\else% notbw => use colors : )
     208 + \def\mcommentfont{\color[rgb]{.133,.545,.133}} %comments in green
     209 + \lstset{language=matlabfloz, % use our version of highlighting
     210 + keywordstyle=\color[rgb]{0,0,1}, % keywords in blue
     211 + commentstyle=\mcommentfont, % comments
     212 + stringstyle=\color[rgb]{.627,.126,.941} % strings in purple
     213 + }
     214 +\fi%bw
     215 + 
     216 +% ---------------------------------------------------------------------------------
     217 +% automatic line breaking --- warning, this is buggy and
     218 +% doesn't break comments correctly!
     219 +\ifautolinebreaks
     220 + \newsavebox{\lbreakdots}\sbox{\lbreakdots}{\lstbasicfont\mcommentfont...}
     221 + \lstset{breaklines=true,breakatwhitespace=true,prebreak=\usebox{\lbreakdots}}
     222 +\fi
     223 + 
     224 +% ---------------------------------------------------------------------------------
     225 +% literate replacements
     226 +% the following is for replacing some matlab relations like >= or ~=
     227 +% by the corresponding LaTeX symbols, which are much easier to read ...
     228 +\ifuseliterate
     229 + \lstset{%
     230 + literate=%
     231 + {~}{{$\neg$}}1 % \neg
     232 + {<=}{{\tiny$\leq$}}1 % \leq
     233 + {>=}{{\tiny$\geq$}}1 % \geq
     234 + {~=}{{\tiny$\neq$}}1 % \neq
     235 + {delta}{{\tiny$\Delta$}}1 % \Delta
     236 + {(end)}{\lstbasicfont (end)}{5} % black ``end'' when indexing last vector element
     237 + {({ }end)}{\lstbasicfont ({ }end)}{6}
     238 + {(end{ })}{\lstbasicfont (end{ })}{6}
     239 + {({ }end{ })}{\lstbasicfont ({ }end{ })}{7}
     240 + {:end}{\lstbasicfont :end}{4}
     241 + {:{ }end}{\lstbasicfont :{ }end}{5}
     242 + {end:}{\lstbasicfont end:}{4}
     243 + {end{ }:}{\lstbasicfont end{ }:}{5}
     244 + {,end}{\lstbasicfont ,end}{4}
     245 + {,{ }end}{\lstbasicfont ,{ }end}{5}
     246 + }
     247 +\else
     248 + \lstset{%
     249 + literate=%
     250 + {(end)}{\lstbasicfont (end)}{5} % black ``end'' when indexing last vector element
     251 + {({ }end)}{\lstbasicfont ({ }end)}{6}
     252 + {(end{ })}{\lstbasicfont (end{ })}{6}
     253 + {({ }end{ })}{\lstbasicfont ({ }end{ })}{7}
     254 + {:end}{\lstbasicfont :end}{4}
     255 + {:{ }end}{\lstbasicfont :{ }end}{5}
     256 + {end:}{\lstbasicfont end:}{4}
     257 + {end{ }:}{\lstbasicfont end{ }:}{5}
     258 + {,end}{\lstbasicfont ,end}{4}
     259 + {,{ }end}{\lstbasicfont ,{ }end}{5}
     260 + }
     261 +\fi%literates
     262 + 
     263 +% ---------------------------------------------------------------------------------
     264 +% line numbering
     265 +\ifnumbered% numbered option
     266 + \lstset{%
     267 + numbersep=3mm, numbers=left, numberstyle=\tiny, % number style
     268 + }
     269 +\fi
     270 + 
     271 +\ifframed% framed option
     272 + \lstset{%
     273 + frame=single, % frame
     274 + }
     275 + \ifnumbered%
     276 + \lstset{%
     277 + framexleftmargin=6mm, xleftmargin=6mm % tweak margins
     278 + }
     279 + \fi
     280 +\fi
     281 + 
     282 +\endinput
     283 +%% End of file `mcode.sty'.
  • unit04_coding/prob/prob_coding.pdf
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    unit04_coding/prob/prob_coding.tex
     1 +\documentclass[11pt]{article}
     2 + 
     3 +\usepackage{fullpage}
     4 +\usepackage{amsmath, amssymb, bm, cite, epsfig, psfrag}
     5 +\usepackage{graphicx}
     6 +\usepackage{float}
     7 +\usepackage{amsthm}
     8 +\usepackage{amsfonts}
     9 +\usepackage{listings}
     10 +\usepackage{cite}
     11 +\usepackage{hyperref}
     12 +\usepackage{tikz}
     13 +\usepackage{enumitem}
     14 +\usetikzlibrary{shapes,arrows}
     15 +\usepackage{mdframed}
     16 +\usepackage{mcode}
     17 +\usepackage{siunitx}
     18 +%\usetikzlibrary{dsp,chains}
     19 + 
     20 +%\restylefloat{figure}
     21 +%\theoremstyle{plain} \newtheorem{theorem}{Theorem}
     22 +%\theoremstyle{definition} \newtheorem{definition}{Definition}
     23 + 
     24 +\def\del{\partial}
     25 +\def\ds{\displaystyle}
     26 +\def\ts{\textstyle}
     27 +\def\beq{\begin{equation}}
     28 +\def\eeq{\end{equation}}
     29 +\def\beqa{\begin{eqnarray}}
     30 +\def\eeqa{\end{eqnarray}}
     31 +\def\beqan{\begin{eqnarray*}}
     32 +\def\eeqan{\end{eqnarray*}}
     33 +\def\nn{\nonumber}
     34 +\def\binomial{\mathop{\mathrm{binomial}}}
     35 +\def\half{{\ts\frac{1}{2}}}
     36 +\def\Half{{\frac{1}{2}}}
     37 +\def\N{{\mathbb{N}}}
     38 +\def\Z{{\mathbb{Z}}}
     39 +\def\Q{{\mathbb{Q}}}
     40 +\def\R{{\mathbb{R}}}
     41 +\def\C{{\mathbb{C}}}
     42 +\def\argmin{\mathop{\mathrm{arg\,min}}}
     43 +\def\argmax{\mathop{\mathrm{arg\,max}}}
     44 +%\def\span{\mathop{\mathrm{span}}}
     45 +\def\diag{\mathop{\mathrm{diag}}}
     46 +\def\x{\times}
     47 +\def\limn{\lim_{n \rightarrow \infty}}
     48 +\def\liminfn{\liminf_{n \rightarrow \infty}}
     49 +\def\limsupn{\limsup_{n \rightarrow \infty}}
     50 +\def\MID{\,|\,}
     51 +\def\MIDD{\,;\,}
     52 + 
     53 +\newtheorem{proposition}{Proposition}
     54 +\newtheorem{definition}{Definition}
     55 +\newtheorem{theorem}{Theorem}
     56 +\newtheorem{lemma}{Lemma}
     57 +\newtheorem{corollary}{Corollary}
     58 +\newtheorem{assumption}{Assumption}
     59 +\newtheorem{claim}{Claim}
     60 +\def\qed{\mbox{} \hfill $\Box$}
     61 +\setlength{\unitlength}{1mm}
     62 + 
     63 +\def\bhat{\widehat{b}}
     64 +\def\ehat{\widehat{e}}
     65 +\def\phat{\widehat{p}}
     66 +\def\qhat{\widehat{q}}
     67 +\def\rhat{\widehat{r}}
     68 +\def\shat{\widehat{s}}
     69 +\def\uhat{\widehat{u}}
     70 +\def\ubar{\overline{u}}
     71 +\def\vhat{\widehat{v}}
     72 +\def\xhat{\widehat{x}}
     73 +\def\xbar{\overline{x}}
     74 +\def\zhat{\widehat{z}}
     75 +\def\zbar{\overline{z}}
     76 +\def\la{\leftarrow}
     77 +\def\ra{\rightarrow}
     78 +\def\MSE{\mbox{\small \sffamily MSE}}
     79 +\def\SNR{\mbox{\small \sffamily SNR}}
     80 +\def\SINR{\mbox{\small \sffamily SINR}}
     81 +\def\arr{\rightarrow}
     82 +\def\Exp{\mathbb{E}}
     83 +\def\var{\mbox{var}}
     84 +\def\Tr{\mbox{Tr}}
     85 +\def\tm1{t\! - \! 1}
     86 +\def\tp1{t\! + \! 1}
     87 + 
     88 +\def\Xset{{\cal X}}
     89 + 
     90 +\newcommand{\one}{\mathbf{1}}
     91 +\newcommand{\abf}{\mathbf{a}}
     92 +\newcommand{\bbf}{\mathbf{b}}
     93 +\newcommand{\dbf}{\mathbf{d}}
     94 +\newcommand{\ebf}{\mathbf{e}}
     95 +\newcommand{\gbf}{\mathbf{g}}
     96 +\newcommand{\hbf}{\mathbf{h}}
     97 +\newcommand{\pbf}{\mathbf{p}}
     98 +\newcommand{\pbfhat}{\widehat{\mathbf{p}}}
     99 +\newcommand{\qbf}{\mathbf{q}}
     100 +\newcommand{\qbfhat}{\widehat{\mathbf{q}}}
     101 +\newcommand{\rbf}{\mathbf{r}}
     102 +\newcommand{\rbfhat}{\widehat{\mathbf{r}}}
     103 +\newcommand{\sbf}{\mathbf{s}}
     104 +\newcommand{\sbfhat}{\widehat{\mathbf{s}}}
     105 +\newcommand{\ubf}{\mathbf{u}}
     106 +\newcommand{\ubfhat}{\widehat{\mathbf{u}}}
     107 +\newcommand{\utildebf}{\tilde{\mathbf{u}}}
     108 +\newcommand{\vbf}{\mathbf{v}}
     109 +\newcommand{\vbfhat}{\widehat{\mathbf{v}}}
     110 +\newcommand{\wbf}{\mathbf{w}}
     111 +\newcommand{\wbfhat}{\widehat{\mathbf{w}}}
     112 +\newcommand{\xbf}{\mathbf{x}}
     113 +\newcommand{\xbfhat}{\widehat{\mathbf{x}}}
     114 +\newcommand{\xbfbar}{\overline{\mathbf{x}}}
     115 +\newcommand{\ybf}{\mathbf{y}}
     116 +\newcommand{\zbf}{\mathbf{z}}
     117 +\newcommand{\zbfbar}{\overline{\mathbf{z}}}
     118 +\newcommand{\zbfhat}{\widehat{\mathbf{z}}}
     119 +\newcommand{\Ahat}{\widehat{A}}
     120 +\newcommand{\Abf}{\mathbf{A}}
     121 +\newcommand{\Bbf}{\mathbf{B}}
     122 +\newcommand{\Cbf}{\mathbf{C}}
     123 +\newcommand{\Bbfhat}{\widehat{\mathbf{B}}}
     124 +\newcommand{\Dbf}{\mathbf{D}}
     125 +\newcommand{\Ebf}{\mathbf{E}}
     126 +\newcommand{\Gbf}{\mathbf{G}}
     127 +\newcommand{\Hbf}{\mathbf{H}}
     128 +\newcommand{\Kbf}{\mathbf{K}}
     129 +\newcommand{\Pbf}{\mathbf{P}}
     130 +\newcommand{\Phat}{\widehat{P}}
     131 +\newcommand{\Qbf}{\mathbf{Q}}
     132 +\newcommand{\Rbf}{\mathbf{R}}
     133 +\newcommand{\Rhat}{\widehat{R}}
     134 +\newcommand{\Sbf}{\mathbf{S}}
     135 +\newcommand{\Ubf}{\mathbf{U}}
     136 +\newcommand{\Vbf}{\mathbf{V}}
     137 +\newcommand{\Wbf}{\mathbf{W}}
     138 +\newcommand{\Xhat}{\widehat{X}}
     139 +\newcommand{\Xbf}{\mathbf{X}}
     140 +\newcommand{\Ybf}{\mathbf{Y}}
     141 +\newcommand{\Zbf}{\mathbf{Z}}
     142 +\newcommand{\Zhat}{\widehat{Z}}
     143 +\newcommand{\Zbfhat}{\widehat{\mathbf{Z}}}
     144 +\def\alphabf{{\boldsymbol \alpha}}
     145 +\def\betabf{{\boldsymbol \beta}}
     146 +\def\mubf{{\boldsymbol \mu}}
     147 +\def\lambdabf{{\boldsymbol \lambda}}
     148 +\def\etabf{{\boldsymbol \eta}}
     149 +\def\xibf{{\boldsymbol \xi}}
     150 +\def\taubf{{\boldsymbol \tau}}
     151 +\def\sigmahat{{\widehat{\sigma}}}
     152 +\def\thetabf{{\bm{\theta}}}
     153 +\def\thetabfhat{{\widehat{\bm{\theta}}}}
     154 +\def\thetahat{{\widehat{\theta}}}
     155 +\def\mubar{\overline{\mu}}
     156 +\def\muavg{\mu}
     157 +\def\sigbf{\bm{\sigma}}
     158 +\def\etal{\emph{et al.}}
     159 +\def\Ggothic{\mathfrak{G}}
     160 +\def\Pset{{\mathcal P}}
     161 +\newcommand{\bigCond}[2]{\bigl({#1} \!\bigm\vert\! {#2} \bigr)}
     162 +\newcommand{\BigCond}[2]{\Bigl({#1} \!\Bigm\vert\! {#2} \Bigr)}
     163 + 
     164 +\def\Rect{\mathop{Rect}}
     165 +\def\sinc{\mathop{sinc}}
     166 +\def\NF{\mathrm{NF}}
     167 +\def\Real{\mathrm{Re}}
     168 +\def\Imag{\mathrm{Im}}
     169 +\newcommand{\tran}{^{\text{\sf T}}}
     170 +\newcommand{\herm}{^{\text{\sf H}}}
     171 + 
     172 + 
     173 +% Solution environment
     174 +\definecolor{lightgray}{gray}{0.95}
     175 +\newmdenv[linecolor=white,backgroundcolor=lightgray,frametitle=Solution:]{solution}
     176 + 
     177 + 
     178 + 
     179 +\begin{document}
     180 + 
     181 +\title{Problems: Coding and Capacity on Fading Channels\\
     182 +ECE-GY 6023. Wireless Communications}
     183 +\author{Prof.\ Sundeep Rangan}
     184 +\date{}
     185 + 
     186 +\maketitle
     187 + 
     188 + 
     189 +\begin{enumerate}
     190 + 
     191 +\item \emph{Slow vs.\ fast fading:}.
     192 +For each scenario below state whether the variations would likely be
     193 +slow or fast fading relative to the coding block.
     194 +Use reasonable assumptions and explain your reasoning. There is no single correct answer.
     195 +\begin{enumerate}[label=(\alph*)]
     196 +\item A 5G NR base stations transmits over a channel with a \SI{100}{ns} delay spread,
     197 +to a UE moving at $v=$ \SI{30}{m/s} with a $180^\circ$ angular spread.
     198 +The carrier frequency is $f_c=$ \SI{28}{GHz}.
     199 +The transmission is over a \SI{100}{MHz} bandwidth in \SI{125}{\micro\second} slots.
     200 +\item A UAV is connected to a ground base station via point-to-point link with a line-of-sight.
     201 +So, there is no multipath fading. But the UAV rotates 360$^\circ$ about once a second.
     202 +The beamwidth of the UAV
     203 +antenna element is 60$^\circ$ and packets are transmitted once every \SI{1}{ms}.
     204 +\end{enumerate}
     205 + 
     206 +\item \emph{Error rate on uncoded modulation}:
     207 +\begin{enumerate}[label=(\alph*)]
     208 +\item Use any reference to find the symbol error rate (SER) of 16-QAM as
     209 +a function of the SNR $\gamma_s = E_s/N_0$. Your expression will have a $Q$-function.
     210 +\item Find the SNR $\gamma_s$ requred for a SER of $(10)^{-3}$ assuming a constant channel.
     211 +You can use MATLAB to invert the $Q$-function.
     212 +\item Suppose that the channel is Rayleigh fading, so $\gamma_s$ is exponentially distributed.
     213 +Find the average SNR, $\Exp(\gamma_s)$ so that the average SER is $(10)^{-3}$.
     214 +\end{enumerate}
     215 + 
     216 +\item \emph{Slow fading and outage probability:} An access point is installed in an office
     217 +area with four rooms. The path loss from the access point to each room and the percentage of
     218 +users in each room are as follows:
     219 +\begin{center}
     220 +\begin{tabular}{|c|c|c|}
     221 + \hline
     222 + % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
     223 + Room & Path loss [dB] & Fraction users \\ \hline
     224 + 1 & 60 & 0.6 \\ \hline
     225 + 2 & 80 & 0.3 \\ \hline
     226 + 3 & 90 & 0.06 \\ \hline
     227 + 4 & 100 & 0.04 \\ \hline
     228 +\end{tabular}
     229 +\end{center}
     230 +The AP has a transmit power of \SI{15}{dBm} and bandwidth of \SI{18}{MHz}.
     231 +The thermal noise at the receivers, including noise figure is \SI{-165}{dBm/Hz}.
     232 +\begin{enumerate}[label=(\alph*)]
     233 +\item If there is no fading, what SNR can be guaranteed to at least 95\% of the users?
     234 +\item Now suppose that, at each location, there is Rayleigh fading that can be modeled as flat
     235 +over the transmissions.
     236 +Write an expression for the CDF of the SNR including variation in both location and fading.
     237 +\item What is the SNR that can be guaranteed to at least 95\% of the users if we need
     238 +to account for slow fading? You can use MATLAB to invert the expression in part (b).
     239 +\end{enumerate}
     240 + 
     241 +\item \emph{Ergodic capacity:} A channel has two paths. One path would be received
     242 +at power, $P_1$ and delay $\tau_1$, and the second path would be received at power
     243 +$P_2$ and delay $\tau_2$ where $\tau_2 > \tau_1$.
     244 +Suppose you signal over a bandwidth $W \gg 1/(\tau_2 - \tau_1)$ and noise power spectral density
     245 +is $N_0$.
     246 +\begin{enumerate}[label=(\alph*)]
     247 +\item What is the average SNR over the band?
     248 +\item What is the ergodic capacity over the band?
     249 +\item Evaluate the expressions in (a) and (b) with $P_1/(W N_0)=$ \SI{8}{dB}
     250 +and $P_2/(W N_0)=$ \SI{5}{dB}.
     251 +\end{enumerate}
     252 + 
     253 + 
     254 +\item \emph{LLRs:} For each of the following channels, find the log likelihood ratio (LLR):
     255 +\[
     256 + L(r) = \log \frac{ p(r|c=1) }{ p(r|c=0) }
     257 +\]
     258 +for the following channels:
     259 +\begin{enumerate}[label=(\alph*)]
     260 +\item Real-valued binary channel with fading:
     261 +\[
     262 + r= Ax + w, \quad w \sim {\mathcal N}(0,N_0/2), \quad
     263 + x = \begin{cases}
     264 + \sqrt{E_x/2} & \mbox{if } c = 1, \\
     265 + -\sqrt{E_x/2} & \mbox{if } c = 0.
     266 + \end{cases}
     267 +\]
     268 +The LLR $L$ should depend on $A$ and $N_0$.
     269 + 
     270 +\item Binary symmetric channel:
     271 +\[
     272 + r = c + w ~(\mbox{mod } 2), \quad w =
     273 + \begin{cases}
     274 + 1 & \mbox{with probability } p \\
     275 + 0 & \mbox{with probability } 1-p
     276 + \end{cases}
     277 +\]
     278 +Thus, $r \in \{0,1\}$ where there is a bit error with probability $p$.
     279 + 
     280 +\item Non-coherent channel:
     281 +\[
     282 + r = \begin{cases}
     283 + h + n & \mbox{when } c = 1\\
     284 + n & \mbox{when } c = 0,
     285 + \end{cases}
     286 + \quad h \sim {\mathcal CN}(0,E_s), ~n\sim {\mathcal CN}(0,N_0).
     287 +\]
     288 + 
     289 +\end{enumerate}
     290 + 
     291 +\item \emph{Bitwise likelihood}: Suppose that two bits $(c_0,c_1)$ are modulated to
     292 +a $4$-PAM constellation
     293 +(the real or imaginary component of a 16-QAM constellation):
     294 +\[
     295 + r = x + n, \quad n \sim {\mathcal N}(0,N_0/2),
     296 +\]
     297 +where the transmitted symbol
     298 +\[
     299 + x = \begin{cases}
     300 + -3A & \mbox{if } (c_0,c_1) = (00) \\
     301 + -A & \mbox{if } (c_0,c_1) = (01) \\
     302 + A & \mbox{if } (c_0,c_1) = (11) \\
     303 + 3A & \mbox{if } (c_0,c_1) = (10)
     304 + \end{cases}
     305 +\]
     306 +Assume all the transmitted bits are equally likely.
     307 +\begin{enumerate}[label=(\alph*)]
     308 +\item Given a symbol energy, $E_s$, find $A$ such that $\Exp|x|^2 = E_s/2$.
     309 +\item Find the bitwise LLR for $c_0$:
     310 +\[
     311 + L_0(r) = \log \frac{p(r|c_0=1)}{p(r|c_0=1)}.
     312 +\]
     313 +Use total probability
     314 +\[
     315 + p(r|c_0) = \frac{1}{2}\left[ p(r|c_0,c_1=1) + p(r|c_0,c_1=0) \right].
     316 +\]
     317 +Find the bitwise LLR for $c_1$ as well.
     318 +\end{enumerate}
     319 + 
     320 +\item \emph{Row-column interleavers:} One simple way of doing interleaving is as follows.
     321 +The input is a sequence of bits of length $MN$ for some parameters $M$ and $N$.
     322 +We read the bits into an $M \times N$ array, one row at a time. Then, we read out
     323 +the bits one column at a time. If two bits are adjacent on the input what is the minimum separation on the output?
     324 +\end{enumerate}
     325 + 
     326 +\end{document}
     327 + 
     328 + 
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