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Projects STRLCPY wirelesscomm Commits 4363006c
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  • lectures/Unit07_ChanEst.pptx
    Binary file.
  • ■ ■ ■ ■ ■ ■
    readme.md
    skipped 64 lines
    65 65   * Demo: 802.11 MCS selection [[PDF]](./unit05_amc/demo_mcs.pdf) [[Matlab Live]](./unit05_amc/demo_mcs.mlx)
    66 66   * Demo: Channel Tracking with 5G NR CSI-RS [[PDF]](./unit05_amc/demo_csirs.pdf) [[Matlab Live]](./unit05_amc/demo_csirs.mlx)
    67 67   * Lab: 5G NR DL Throughput with Multi-Process HARQ [[PDF]](./unit05_amc/lab_partial/labHarq.pdf) [[Matlab Live]](./unit05_amc/lab_partial/labHarq.mlx)
    68  - * Problems: [[PDF]](./unit05_amc/prob/prob_amc.pdf) [[Latex]](./unit05_amc/prob/prob_amc .tex)
     68 + * Problems: [[PDF]](./unit05_amc/prob/prob_amc.pdf) [[Latex]](./unit05_amc/prob/prob_amc.tex)
    69 69  * Unit 6. Diversity
    70 70  * Unit 7. OFDM Channel Estimation and Equalization
    71 71   * Lecture: [[PDF]](./lectures/Unit07_ChanEst.pdf) [[Powerpoint]](../lectures/Unit07_ChanEst.pdf)
    skipped 1 lines
    73 73   * Demo: 5G NR DM-RS configuration [[PDF]](./unit07_chanest/demoDMRSConfig.pdf) [[Matlab Live]](./unit07_chanest/demoDMRSConfig.mlx)
    74 74   * Demo: Kernel regression channel estimation [[PDF]](./unit07_chanest/demoKernelEst.pdf) [[Matlab Live]](./unit07_chanest/demoKernelEst.mlx)
    75 75   * Lab: 5G NR DL Throughput with Channel Estimation [[PDF]](./unit07_chanest/lab_partial/labChanEst.pdf) [[Matlab Live]](./unit07_chanest/lab_partial/labChanEst.mlx)
     76 + * Problems: [[PDF]](./unit07_chanest/prob/prob_chanest.pdf) [[Latex]](./unit07_chanest/prob/prob_chanest.tex)
    76 77  * Unit 8. Multiple Antennas and Beamforming
    77 78   * Lecture: [[PDF]](./lectures/Unit06_Beamforming.pdf) [[PPT]](./lectures/Unit06_Beamforming.pptx)
    78 79   * Demo: Visualizing and simualting arrays [[PDF]](./beamforming/demo_bf.pdf) [[Matlab]](./beamforming/demo_bf.m)
    skipped 7 lines
  • ■ ■ ■ ■ ■ ■
    unit07_chanest/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'.
  • unit07_chanest/prob/prob_chan_est.pdf
    Binary file.
  • ■ ■ ■ ■ ■ ■
    unit07_chanest/prob/prob_chan_est.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 +\usepackage{mathtools}
     19 +%\usetikzlibrary{dsp,chains}
     20 + 
     21 +%\restylefloat{figure}
     22 +%\theoremstyle{plain} \newtheorem{theorem}{Theorem}
     23 +%\theoremstyle{definition} \newtheorem{definition}{Definition}
     24 + 
     25 +\def\del{\partial}
     26 +\def\ds{\displaystyle}
     27 +\def\ts{\textstyle}
     28 +\def\beq{\begin{equation}}
     29 +\def\eeq{\end{equation}}
     30 +\def\beqa{\begin{eqnarray}}
     31 +\def\eeqa{\end{eqnarray}}
     32 +\def\beqan{\begin{eqnarray*}}
     33 +\def\eeqan{\end{eqnarray*}}
     34 +\def\nn{\nonumber}
     35 +\def\binomial{\mathop{\mathrm{binomial}}}
     36 +\def\half{{\ts\frac{1}{2}}}
     37 +\def\Half{{\frac{1}{2}}}
     38 +\def\N{{\mathbb{N}}}
     39 +\def\Z{{\mathbb{Z}}}
     40 +\def\Q{{\mathbb{Q}}}
     41 +\def\R{{\mathbb{R}}}
     42 +\def\C{{\mathbb{C}}}
     43 +\def\argmin{\mathop{\mathrm{arg\,min}}}
     44 +\def\argmax{\mathop{\mathrm{arg\,max}}}
     45 +%\def\span{\mathop{\mathrm{span}}}
     46 +\def\diag{\mathop{\mathrm{diag}}}
     47 +\def\x{\times}
     48 +\def\limn{\lim_{n \rightarrow \infty}}
     49 +\def\liminfn{\liminf_{n \rightarrow \infty}}
     50 +\def\limsupn{\limsup_{n \rightarrow \infty}}
     51 +\def\MID{\,|\,}
     52 +\def\MIDD{\,;\,}
     53 + 
     54 +\newtheorem{proposition}{Proposition}
     55 +\newtheorem{definition}{Definition}
     56 +\newtheorem{theorem}{Theorem}
     57 +\newtheorem{lemma}{Lemma}
     58 +\newtheorem{corollary}{Corollary}
     59 +\newtheorem{assumption}{Assumption}
     60 +\newtheorem{claim}{Claim}
     61 +\def\qed{\mbox{} \hfill $\Box$}
     62 +\setlength{\unitlength}{1mm}
     63 + 
     64 +\def\bhat{\widehat{b}}
     65 +\def\ehat{\widehat{e}}
     66 +\def\phat{\widehat{p}}
     67 +\def\qhat{\widehat{q}}
     68 +\def\rhat{\widehat{r}}
     69 +\def\shat{\widehat{s}}
     70 +\def\uhat{\widehat{u}}
     71 +\def\ubar{\overline{u}}
     72 +\def\vhat{\widehat{v}}
     73 +\def\xhat{\widehat{x}}
     74 +\def\xbar{\overline{x}}
     75 +\def\zhat{\widehat{z}}
     76 +\def\zbar{\overline{z}}
     77 +\def\la{\leftarrow}
     78 +\def\ra{\rightarrow}
     79 +\def\MSE{\mbox{\small \sffamily MSE}}
     80 +\def\SNR{\mbox{\small \sffamily SNR}}
     81 +\def\SINR{\mbox{\small \sffamily SINR}}
     82 +\def\arr{\rightarrow}
     83 +\def\Exp{\mathbb{E}}
     84 +\def\var{\mbox{var}}
     85 +\def\Tr{\mbox{Tr}}
     86 +\def\tm1{t\! - \! 1}
     87 +\def\tp1{t\! + \! 1}
     88 + 
     89 +\def\Xset{{\cal X}}
     90 + 
     91 +\newcommand{\one}{\mathbf{1}}
     92 +\newcommand{\abf}{\mathbf{a}}
     93 +\newcommand{\bbf}{\mathbf{b}}
     94 +\newcommand{\dbf}{\mathbf{d}}
     95 +\newcommand{\ebf}{\mathbf{e}}
     96 +\newcommand{\gbf}{\mathbf{g}}
     97 +\newcommand{\hbf}{\mathbf{h}}
     98 +\newcommand{\pbf}{\mathbf{p}}
     99 +\newcommand{\pbfhat}{\widehat{\mathbf{p}}}
     100 +\newcommand{\qbf}{\mathbf{q}}
     101 +\newcommand{\qbfhat}{\widehat{\mathbf{q}}}
     102 +\newcommand{\rbf}{\mathbf{r}}
     103 +\newcommand{\rbfhat}{\widehat{\mathbf{r}}}
     104 +\newcommand{\sbf}{\mathbf{s}}
     105 +\newcommand{\sbfhat}{\widehat{\mathbf{s}}}
     106 +\newcommand{\ubf}{\mathbf{u}}
     107 +\newcommand{\ubfhat}{\widehat{\mathbf{u}}}
     108 +\newcommand{\utildebf}{\tilde{\mathbf{u}}}
     109 +\newcommand{\vbf}{\mathbf{v}}
     110 +\newcommand{\vbfhat}{\widehat{\mathbf{v}}}
     111 +\newcommand{\wbf}{\mathbf{w}}
     112 +\newcommand{\wbfhat}{\widehat{\mathbf{w}}}
     113 +\newcommand{\xbf}{\mathbf{x}}
     114 +\newcommand{\xbfhat}{\widehat{\mathbf{x}}}
     115 +\newcommand{\xbfbar}{\overline{\mathbf{x}}}
     116 +\newcommand{\ybf}{\mathbf{y}}
     117 +\newcommand{\zbf}{\mathbf{z}}
     118 +\newcommand{\zbfbar}{\overline{\mathbf{z}}}
     119 +\newcommand{\zbfhat}{\widehat{\mathbf{z}}}
     120 +\newcommand{\Ahat}{\widehat{A}}
     121 +\newcommand{\Abf}{\mathbf{A}}
     122 +\newcommand{\Bbf}{\mathbf{B}}
     123 +\newcommand{\Cbf}{\mathbf{C}}
     124 +\newcommand{\Bbfhat}{\widehat{\mathbf{B}}}
     125 +\newcommand{\Dbf}{\mathbf{D}}
     126 +\newcommand{\Ebf}{\mathbf{E}}
     127 +\newcommand{\Gbf}{\mathbf{G}}
     128 +\newcommand{\Hbf}{\mathbf{H}}
     129 +\newcommand{\Ibf}{\mathbf{I}}
     130 +\newcommand{\Kbf}{\mathbf{K}}
     131 +\newcommand{\Pbf}{\mathbf{P}}
     132 +\newcommand{\Phat}{\widehat{P}}
     133 +\newcommand{\Qbf}{\mathbf{Q}}
     134 +\newcommand{\Rbf}{\mathbf{R}}
     135 +\newcommand{\Rhat}{\widehat{R}}
     136 +\newcommand{\Sbf}{\mathbf{S}}
     137 +\newcommand{\Ubf}{\mathbf{U}}
     138 +\newcommand{\Vbf}{\mathbf{V}}
     139 +\newcommand{\Wbf}{\mathbf{W}}
     140 +\newcommand{\Xhat}{\widehat{X}}
     141 +\newcommand{\Xbf}{\mathbf{X}}
     142 +\newcommand{\Ybf}{\mathbf{Y}}
     143 +\newcommand{\Zbf}{\mathbf{Z}}
     144 +\newcommand{\Zhat}{\widehat{Z}}
     145 +\newcommand{\Zbfhat}{\widehat{\mathbf{Z}}}
     146 +\def\alphabf{{\boldsymbol \alpha}}
     147 +\def\betabf{{\boldsymbol \beta}}
     148 +\def\mubf{{\boldsymbol \mu}}
     149 +\def\lambdabf{{\boldsymbol \lambda}}
     150 +\def\etabf{{\boldsymbol \eta}}
     151 +\def\xibf{{\boldsymbol \xi}}
     152 +\def\taubf{{\boldsymbol \tau}}
     153 +\def\sigmahat{{\widehat{\sigma}}}
     154 +\def\thetabf{{\bm{\theta}}}
     155 +\def\thetabfhat{{\widehat{\bm{\theta}}}}
     156 +\def\thetahat{{\widehat{\theta}}}
     157 +\def\mubar{\overline{\mu}}
     158 +\def\muavg{\mu}
     159 +\def\sigbf{\bm{\sigma}}
     160 +\def\etal{\emph{et al.}}
     161 +\def\Ggothic{\mathfrak{G}}
     162 +\def\Pset{{\mathcal P}}
     163 +\newcommand{\bigCond}[2]{\bigl({#1} \!\bigm\vert\! {#2} \bigr)}
     164 +\newcommand{\BigCond}[2]{\Bigl({#1} \!\Bigm\vert\! {#2} \Bigr)}
     165 + 
     166 +\def\Rect{\mathop{Rect}}
     167 +\def\sinc{\mathop{sinc}}
     168 +\def\NF{\mathrm{NF}}
     169 +\def\Real{\mathrm{Re}}
     170 +\def\Imag{\mathrm{Im}}
     171 +\newcommand{\tran}{^{\text{\sf T}}}
     172 +\newcommand{\herm}{^{\text{\sf H}}}
     173 + 
     174 + 
     175 +% Solution environment
     176 +\definecolor{lightgray}{gray}{0.95}
     177 +\newmdenv[linecolor=white,backgroundcolor=lightgray,frametitle=Solution:]{solution}
     178 + 
     179 + 
     180 + 
     181 +\begin{document}
     182 + 
     183 +\title{Problems: Channel Estimation and Equalization\\
     184 +ECE-GY 6023. Wireless Communications}
     185 +\author{Prof.\ Sundeep Rangan}
     186 +\date{}
     187 + 
     188 +\maketitle
     189 + 
     190 + 
     191 +\begin{enumerate}
     192 + 
     193 +\item \emph{Kernel regression:}
     194 +Consider a kernel regression estimate of the form
     195 +\begin{equation} \label{eq:kernel}
     196 + \widehat{h}[n] = \frac{\sum_{\ell \in I} w_\ell \widehat{h}_0[n+\ell]}
     197 + {\sum_{\ell \in I} w_\ell},
     198 +\end{equation}
     199 +where $I$ is a set of reference symbol locations, $\widehat{h}_0[n]$ are raw estimates of the
     200 +channel on the reference symbols and $w_\ell$ is a kernel.
     201 +Suppose you use a triangular kernel,
     202 +\[
     203 + w_\ell = \max\{ 1-\ell/L, 0 \},
     204 +\]
     205 +for some width $L$ and the reference symbols are spaced every $d$ positions,
     206 +\[
     207 + I = \{0, d, 2d, \quad, Md\}.
     208 +\]
     209 +Assume $L=12$, $d=4$ and $M=10$. For each value $n$, below write $\widehat{h}[n]$ as a linear
     210 +combination of the values $\widehat{h}_0[k]$.
     211 +\begin{enumerate}[label=(\alph*)]
     212 +\item The symbol on the edge, $n=0$.
     213 +\item A symbol in the middle located on a reference symbol location, $n=20$.
     214 +\item A symbol in the middle located between two reference symbol locations, $n=22$.
     215 +\end{enumerate}
     216 + 
     217 +\item \emph{Channel estimation error for a stochastic model:} Suppose that
     218 +we form an estimate,
     219 +\[
     220 + \widehat{h}[n] = \frac{1}{2} \left[ \widehat{h}_0[n-L] + \widehat{h}_0[n+L] \right].
     221 +\]
     222 +with measurements
     223 +\[
     224 + \widehat{h}_0[k] = h[k] + v[k], \quad v[k] \sim C{\mathcal N}(0,N_v).
     225 +\]
     226 +Suppose we can model $h[k]$ as a wide sense stationary Gaussian random process
     227 +with auto-correlation $R[k] = \Exp( h[n]h[n+k]^* )$.
     228 +\begin{enumerate}[label=(\alph*)]
     229 +\item Find the mean squared error,
     230 +\[
     231 + \epsilon = \Exp | \widehat{h}[n] - h[n]|^2
     232 +\]
     233 +in terms of the auto-correlation $R[n]$ and the error in the raw channel estimate $N_v$.
     234 + 
     235 +\item If $h[n]$ is a narrowband fading process with Jake's spectrum then its auto-correlation is
     236 +\[
     237 + R[k] = E_s J_0(2\pi f_{\rm max} k T)
     238 +\]
     239 +where $E_s$ is the energy per sample,
     240 +$f_{\rm max}$ is the maximum Doppler spread and $T$ is the sample period.
     241 +Write and plot the normalized MSE,
     242 +$\epsilon/E_s$ as a function of $f_{\rm max}LT$ for the case when $N_v=0$.
     243 +Note that when there is zero noise, $\mathrm{MSE}$, represents the bias squared.
     244 + 
     245 +\item In the above model, there is one reference symbol every $2L$ samples,
     246 +so the overhead is $1/(2L)$.
     247 +Suppose that $f_{\rm max} = $\, \SI{100}{Hz}, and the sample period is $T=$\, \SI{1}{\micro\second}. Using the model in part (b),
     248 +what is the minimum overhead if we need the MSE, $\epsilon$,
     249 +to be less than \SI{20}{dB} below $E_s$?
     250 + 
     251 +\end{enumerate}
     252 + 
     253 + 
     254 +\item \emph{Joint Likelihoods}:
     255 +Suppose that we have two BPSK symbols, $i=0,1$:
     256 +\[
     257 + r_i = hx_i + w_i, \quad w_i \sim C{\mathcal N}(0,N_0),
     258 + \quad x_i = \pm 1,
     259 +\]
     260 +where the noise $w_i$ is i.i.d. The channel gain $h$ is unknown and can
     261 +be modeled as a complex Gaussian $h \sim C{\mathcal N}(0,E_s)$.
     262 +We use the first symbol, $x_0=1$, as a reference symbol. So, there are
     263 +two possibilities:
     264 +\[
     265 + \xbf = \xbf^{(1)}=(1,1) \mbox{ or } \xbf^{(2)}=(1,-1).
     266 +\]
     267 +We will compute the LLR for these two possibilties.
     268 +\begin{enumerate}[label=(\alph*)]
     269 +\item What is the mean and covariance matrix of the vector $\rbf=(r_0,r_1)$
     270 +for the two values of $\xbf$?
     271 + 
     272 +\item Using the fact that $\rbf$ is a Gaussian random vector,
     273 +what is the log likelihood ratio
     274 +\[
     275 + \mathrm{LLR} = \log \left[ \frac{p(\rbf|\xbf=(1,1))}{p(\rbf|\xbf=(1,-1))} \right].
     276 +\]
     277 +You can simplify the expression with the matrix identity,
     278 +\[
     279 + (\Ibf + \gamma \ubf \ubf^*)^{-1} = \Ibf - \frac{\gamma}{1 + \gamma\|\ubf\|^2} \ubf\ubf^*
     280 +\]
     281 +for any vector $\ubf$.
     282 +\end{enumerate}
     283 + 
     284 + 
     285 +\end{enumerate}
     286 + 
     287 + 
     288 + 
     289 + 
     290 + 
     291 + 
     292 +\end{document}
     293 + 
     294 + 
     295 + 
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