Projects STRLCPY wirelesscomm Commits 51745868
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    readme.md
    skipped 49 lines
    50 50   * Lecture: [[PDF]](./lectures/Unit03_Fading.pdf) [[PPT]](./lectures/Unit03_Fading.pptx)
    51 51   * [Lecture videos](./unit03_fading/readme.md) and in-class exercises
    52 52   * Demo: Simulating fading [[PDF]](./unit03_fading/demo_fading.pdf) [[Matlab Live]](./fading/unit03_demo_fading.mlx)
     53 + * Problems: [[PDF]](./unit03_fading/prob/prob_fading.pdf) [[Latex]](./unit03_fading/prob/prob_fading.tex)
    53 54   * Lab: 5G channel sounding with Doppler [[PDF]](./unit03_fading/partial/lab_chan_sounder.pdf) [[Matlab]](./unit03_fading/partial/lab_chan_sounder.m)
    54  -* Unit 4. Diversity
    55  -* Unit 5. Capacity of Fading Channels and MAC-Layer Concepts
     55 +* Unit 4. Capaciy of Fadingin Channels
     56 +* Unit 5. MAC-Layer Concepts
    56 57  * Unit 6. Channel Estimation and Equalization
    57 58  * Unit 7. Multiple Antennas and Beamforming
    58 59   * Lecture: [[PDF]](./lectures/Unit06_Beamforming.pdf) [[PPT]](./lectures/Unit06_Beamforming.pptx)
    skipped 8 lines
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    unit03_fading/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'.
  • unit03_fading/prob/prob_fading.pdf
    Binary file.
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    unit03_fading/prob/prob_fading.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: Small-Scale Fading\\
     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{Doppler shift and frequency offset:}
     192 +\begin{enumerate}[label=(\alph*)]
     193 +\item Suppose that a vehicle is traveling at \SI{100}{km/h} directly towards a base station.
     194 +If the carrier frequency is \SI{2.5}{GHz}, what is the maximum Doppler?
     195 + 
     196 +\item Frequency shifts can also occur due to differences in the local oscillators
     197 +(LOs) at the TX and RX. Suppose that the LO error is 1 part per million (ppm)
     198 +meaning that the frequency error is 1 millionth of the carrier. What is the
     199 +frequency shift?
     200 +\end{enumerate}
     201 + 
     202 + 
     203 +\item \emph{Doppler and match filter}: Suppose that we receive a signal
     204 +\[
     205 + r(t) = u(t)e^{2\pi i ft} + w(t),
     206 +\]
     207 +where $u(t)$ is a transmitted signal, $f$ is a frequency shift
     208 +and $w(t)$ is WGN with PSD $N_0$. Assume $|u(t)|^2=P$ for all $t$
     209 +for some received power level $P > 0$.
     210 +We then compute a matched filter,
     211 +\[
     212 + z = \frac{1}{T} \int_0^T u(t)^*r(t)dt.
     213 +\]
     214 +\begin{enumerate}[label=(\alph*)]
     215 +\item Write the MF response as $z = x + v$ where $x$ is due to the signal
     216 +and $v$ is due to the noise. Your expression for $x$ should have a sinc
     217 +function in it.
     218 + 
     219 +\item Write an expression for the SNR defined as
     220 +\[
     221 + \SNR = \frac{|x|^2}{\Exp|v|^2}.
     222 +\]
     223 +The expression should be in terms of the integration time $T$,
     224 +SNR $P/N_0$ and frequency offset $f$.
     225 + 
     226 +\item When there is no frequency offset (i.e.\ $f=0$), the SNR increases
     227 +linearly with the integration time. However, when there is an uncompensated
     228 +frequency offset, there is an optimal integration time beyond which
     229 +the SNR begins to drop. Find the maximum SNR and optimal integration time.
     230 + 
     231 +The maximization will not have a closed-form answer. So, we will use MATLAB.
     232 +Specifically, plot the SNR as a function of $T$ when $P/N_0=1$ and $f=1$. Then,
     233 +write an expression to translate your answer to other values of $P/N_0$ and $f$.
     234 + 
     235 +\item Suppose that the frequency offset is $f=$ 100 Hz, the received power
     236 +is $P=$ -100 dBm and the noise power density is $N_0=$ -140 dBm/Hz.
     237 +What is the optimal integration time $T$ and maximum SNR?
     238 + 
     239 +\end{enumerate}
     240 + 
     241 +\item A received signal has two paths:
     242 +\begin{itemize}
     243 +\item Path 1: power -100 dBm and Doppler shift 100 Hz,
     244 +\item Path 2: power -103 dBm and Doppler shift -50 Hz.
     245 +\end{itemize}
     246 +\begin{enumerate}[label=(\alph*)]
     247 +\item Draw the signal power as a function of time.
     248 +\item What is the average receive power in dBm?
     249 +\item What is the time it takes to go from the maximum to minimum power?
     250 +\item What is the fraction of time the power is greater than -101 dBm?
     251 +\end{enumerate}
     252 + 
     253 +\item (*) Suppose that a narrowband complex channel $h(t)$ be modeled as
     254 +a wide-sense stationary random process.
     255 +\begin{enumerate}[label=(\alph*)]
     256 + 
     257 +\item Let $\rho(\tau)$ be the relative change
     258 +in $h(t)$ over a time $\tau$.
     259 +\[
     260 + \rho(\tau) = \frac{\Exp|h(t)-h(t+\tau)|^2}{\Exp|h(t)|^2}
     261 +\]
     262 +Write $\rho(\tau)$ in terms of the autocorrelation function
     263 +$R(\tau) = \Exp h(t)h^*(t-\tau)$.
     264 + 
     265 +\item If $h(t)$ follows a Jakes' spectrum with maximum Doppler $f_{max} = $ 200 Hz,
     266 +what is the time it takes the channel to change by 10\%?
     267 +You will need MATLAB to evaluate the Bessel function.
     268 +\end{enumerate}
     269 + 
     270 +\item (*) Consider a multipath fading channel of the form
     271 +\[
     272 + y(t) = \frac{1}{\sqrt{L}}\sum_{\ell = 1}^L g_\ell
     273 + e^{2\pi if_\ell} x(t-\tau_\ell), \quad f_\ell = f_{max}\cos(\theta_\ell),
     274 +\]
     275 +where $L$ is the number of paths, and for each path $\ell$,
     276 +$g_\ell$ is its complex gain, $\theta_\ell$ the AoA, $f_\ell$ the Doppler shift
     277 +and $\tau_\ell$ the delay. This channel has a time-varying frequency response
     278 +\beq \label{eq:Htf}
     279 + H(t,f) = \frac{1}{\sqrt{L}}
     280 + \sum_{\ell =1}^L g_\ell e^{2\pi i(f_\ell t - \tau_\ell f)},
     281 + \quad f_\ell = f_{max}\cos\theta_\ell.
     282 +\eeq
     283 +Suppose that we model the path parameters statistically. That is,
     284 +we model $g_\ell,\theta_\ell,\tau_\ell$ as random variables that
     285 +are all independent
     286 +from one another and across different values of $\ell$. Assume that the
     287 +distributions of the random variables are identical for different
     288 +path indices $\ell$.
     289 +In addition, assume that
     290 +$g_\ell$ are zero mean, with average power gain
     291 +$\Exp|g_\ell|^2 = G$ for some value $G > 0$.
     292 + 
     293 +\begin{enumerate}[label=(\alph*)]
     294 +\item Suppose that the delays are exponentially distributed with pdf,
     295 +\[
     296 + p(\tau_\ell) = \frac{1}{\lambda}e^{-\tau_\ell/\lambda},
     297 +\]
     298 +for some $\lambda$ representing the delay spread.
     299 +What is the autocorrelation function in frequency
     300 +\[
     301 + R(\delta f) = \Exp\left[ H(t,f)H^*(t,f-\Delta f) \right].
     302 +\]
     303 + 
     304 +\item Suppose we define the 3~dB coherence bandwidth as the frequency $W$
     305 +where the frequency response changes by less than 3~dB. That is,
     306 +\[
     307 + \Exp|H(t,f)-H(t,f+W)|^2 \leq \beta \Exp|H(t,f)|^2, \quad \beta = 0.5.
     308 +\]
     309 +Under the above exponential delay model,
     310 +if the delay spread is $\lambda =$ 100 ns, what is the 3~dB bandwidth?
     311 +\end{enumerate}
     312 + 
     313 +\item
     314 +\begin{enumerate}[label=(\alph*)]
     315 +\item (*) Consider the time-varying frequency response \eqref{eq:Htf}.
     316 +Suppose that the AoAs are uniformly distributed in $\theta_\ell \in [-\Delta/2,\Delta/2]$.
     317 +Write an expression for the autocorrelation over time,
     318 +\[
     319 + R(\tau) := \Exp H(t,f)H^*(t-\tau,f).
     320 +\]
     321 +Your expression should have an integral. Do not evaluate this integral.
     322 + 
     323 +\item In 3D channel models, the paths arrive with an \emph{elevation} (vertical)
     324 +angle $\omega_\ell$ to the $z$-axis
     325 +and \emph{azimuth} (horizontal) angle $\varphi_\ell$ to the $x$-axis on the
     326 +$xy$-plane.
     327 +Suppose that the receiver moves along the positive $x$-axis. What is the Doppler
     328 +shift of a path arriving at angles $(\omega_\ell,\varphi_\ell)$.
     329 + 
     330 +\end{enumerate}
     331 + 
     332 +\end{enumerate}
     333 + 
     334 +\end{document}
     335 + 
     336 + 
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    unit03_fading/readme.md
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    7 7  * Introduction and Learning Objectives: [[YouTube]](https://youtu.be/JQ5IuK1NgHc)
    8 8  * Review of Up and Down-Conversion: [[YouTube]](https://youtu.be/WOLkM6GSFEY)
    9 9   * Problems 1 and 2 in in-class exercises: [[MATLAB live]](./fading_inclass.mlx) [[PDF]](./fading_inclass.pdf)
    10  -* Review of Sampling: [[YouTube](https://youtu.be/hLcqiPjbfJM)
     10 +* Review of Sampling: [[YouTube]](https://youtu.be/hLcqiPjbfJM)
    11 11   * Problem 3 in in-class exercises: [[MATLAB live]](./fading_inclass.mlx) [[PDF]](./fading_inclass.pdf)
    12  -* Doppler and Multi-Path Fading: [[YouTube](https://youtu.be/86xTpIRAqvY)
     12 +* Doppler and Multi-Path Fading: [[YouTube]](https://youtu.be/86xTpIRAqvY)
    13 13   * Problem 4 in in-class exercises: [[MATLAB live]](./fading_inclass.mlx) [[PDF]](./fading_inclass.pdf)
    14  -* Statistical Models for Fading: [[YouTube](https://youtu.be/Vkf1gJWz48s)
     14 +* Statistical Models for Fading: [[YouTube]](https://youtu.be/Vkf1gJWz48s)
    15 15   
    16 16  Solutions: [[MATLAB Live]](./fading_inclass_soln.mlx) [[PDF]](./fading_inclass_soln.pdf)
    17 17   
     18 + 
     19 +Mom check init conditions
     20 +Explain Hk
     21 +Explain matrix inv lemma
     22 + 
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