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‹cra.sh›
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Projects STRLCPY wirelesscomm Commits 43940b75
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    unit08_bf/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'.
  • unit08_bf/prob/prob_bf.pdf
    Binary file.
  • ■ ■ ■ ■ ■ ■
    unit08_bf/prob/prob_bf.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 +\usepackage{physics}
     20 +%\usetikzlibrary{dsp,chains}
     21 + 
     22 +%\restylefloat{figure}
     23 +%\theoremstyle{plain} \newtheorem{theorem}{Theorem}
     24 +%\theoremstyle{definition} \newtheorem{definition}{Definition}
     25 + 
     26 +\def\del{\partial}
     27 +\def\ds{\displaystyle}
     28 +\def\ts{\textstyle}
     29 +\def\beq{\begin{equation}}
     30 +\def\eeq{\end{equation}}
     31 +\def\beqa{\begin{eqnarray}}
     32 +\def\eeqa{\end{eqnarray}}
     33 +\def\beqan{\begin{eqnarray*}}
     34 +\def\eeqan{\end{eqnarray*}}
     35 +\def\nn{\nonumber}
     36 +\def\binomial{\mathop{\mathrm{binomial}}}
     37 +\def\half{{\ts\frac{1}{2}}}
     38 +\def\Half{{\frac{1}{2}}}
     39 +\def\N{{\mathbb{N}}}
     40 +\def\Z{{\mathbb{Z}}}
     41 +\def\Q{{\mathbb{Q}}}
     42 +\def\R{{\mathbb{R}}}
     43 +\def\C{{\mathbb{C}}}
     44 +\def\argmin{\mathop{\mathrm{arg\,min}}}
     45 +\def\argmax{\mathop{\mathrm{arg\,max}}}
     46 +%\def\span{\mathop{\mathrm{span}}}
     47 +\def\diag{\mathop{\mathrm{diag}}}
     48 +\def\x{\times}
     49 +\def\limn{\lim_{n \rightarrow \infty}}
     50 +\def\liminfn{\liminf_{n \rightarrow \infty}}
     51 +\def\limsupn{\limsup_{n \rightarrow \infty}}
     52 +\def\MID{\,|\,}
     53 +\def\MIDD{\,;\,}
     54 + 
     55 +\newtheorem{proposition}{Proposition}
     56 +\newtheorem{definition}{Definition}
     57 +\newtheorem{theorem}{Theorem}
     58 +\newtheorem{lemma}{Lemma}
     59 +\newtheorem{corollary}{Corollary}
     60 +\newtheorem{assumption}{Assumption}
     61 +\newtheorem{claim}{Claim}
     62 +\def\qed{\mbox{} \hfill $\Box$}
     63 +\setlength{\unitlength}{1mm}
     64 + 
     65 +\def\bhat{\widehat{b}}
     66 +\def\ehat{\widehat{e}}
     67 +\def\phat{\widehat{p}}
     68 +\def\qhat{\widehat{q}}
     69 +\def\rhat{\widehat{r}}
     70 +\def\shat{\widehat{s}}
     71 +\def\uhat{\widehat{u}}
     72 +\def\ubar{\overline{u}}
     73 +\def\vhat{\widehat{v}}
     74 +\def\xhat{\widehat{x}}
     75 +\def\xbar{\overline{x}}
     76 +\def\zhat{\widehat{z}}
     77 +\def\zbar{\overline{z}}
     78 +\def\la{\leftarrow}
     79 +\def\ra{\rightarrow}
     80 +\def\MSE{\mbox{\small \sffamily MSE}}
     81 +\def\SNR{\mbox{\small \sffamily SNR}}
     82 +\def\SINR{\mbox{\small \sffamily SINR}}
     83 +\def\arr{\rightarrow}
     84 +\def\Exp{\mathbb{E}}
     85 +\def\var{\mbox{var}}
     86 +\def\Tr{\mbox{Tr}}
     87 +\def\tm1{t\! - \! 1}
     88 +\def\tp1{t\! + \! 1}
     89 + 
     90 +\def\Xset{{\cal X}}
     91 + 
     92 +\newcommand{\one}{\mathbf{1}}
     93 +\newcommand{\abf}{\mathbf{a}}
     94 +\newcommand{\bbf}{\mathbf{b}}
     95 +\newcommand{\dbf}{\mathbf{d}}
     96 +\newcommand{\ebf}{\mathbf{e}}
     97 +\newcommand{\gbf}{\mathbf{g}}
     98 +\newcommand{\hbf}{\mathbf{h}}
     99 +\newcommand{\pbf}{\mathbf{p}}
     100 +\newcommand{\pbfhat}{\widehat{\mathbf{p}}}
     101 +\newcommand{\qbf}{\mathbf{q}}
     102 +\newcommand{\qbfhat}{\widehat{\mathbf{q}}}
     103 +\newcommand{\rbf}{\mathbf{r}}
     104 +\newcommand{\rbfhat}{\widehat{\mathbf{r}}}
     105 +\newcommand{\sbf}{\mathbf{s}}
     106 +\newcommand{\sbfhat}{\widehat{\mathbf{s}}}
     107 +\newcommand{\ubf}{\mathbf{u}}
     108 +\newcommand{\ubfhat}{\widehat{\mathbf{u}}}
     109 +\newcommand{\utildebf}{\tilde{\mathbf{u}}}
     110 +\newcommand{\vbf}{\mathbf{v}}
     111 +\newcommand{\vbfhat}{\widehat{\mathbf{v}}}
     112 +\newcommand{\wbf}{\mathbf{w}}
     113 +\newcommand{\wbfhat}{\widehat{\mathbf{w}}}
     114 +\newcommand{\xbf}{\mathbf{x}}
     115 +\newcommand{\xbfhat}{\widehat{\mathbf{x}}}
     116 +\newcommand{\xbfbar}{\overline{\mathbf{x}}}
     117 +\newcommand{\ybf}{\mathbf{y}}
     118 +\newcommand{\zbf}{\mathbf{z}}
     119 +\newcommand{\zbfbar}{\overline{\mathbf{z}}}
     120 +\newcommand{\zbfhat}{\widehat{\mathbf{z}}}
     121 +\newcommand{\Ahat}{\widehat{A}}
     122 +\newcommand{\Abf}{\mathbf{A}}
     123 +\newcommand{\Bbf}{\mathbf{B}}
     124 +\newcommand{\Cbf}{\mathbf{C}}
     125 +\newcommand{\Bbfhat}{\widehat{\mathbf{B}}}
     126 +\newcommand{\Dbf}{\mathbf{D}}
     127 +\newcommand{\Ebf}{\mathbf{E}}
     128 +\newcommand{\Gbf}{\mathbf{G}}
     129 +\newcommand{\Hbf}{\mathbf{H}}
     130 +\newcommand{\Ibf}{\mathbf{I}}
     131 +\newcommand{\Kbf}{\mathbf{K}}
     132 +\newcommand{\Pbf}{\mathbf{P}}
     133 +\newcommand{\Phat}{\widehat{P}}
     134 +\newcommand{\Qbf}{\mathbf{Q}}
     135 +\newcommand{\Rbf}{\mathbf{R}}
     136 +\newcommand{\Rhat}{\widehat{R}}
     137 +\newcommand{\Sbf}{\mathbf{S}}
     138 +\newcommand{\Ubf}{\mathbf{U}}
     139 +\newcommand{\Vbf}{\mathbf{V}}
     140 +\newcommand{\Wbf}{\mathbf{W}}
     141 +\newcommand{\Xhat}{\widehat{X}}
     142 +\newcommand{\Xbf}{\mathbf{X}}
     143 +\newcommand{\Ybf}{\mathbf{Y}}
     144 +\newcommand{\Zbf}{\mathbf{Z}}
     145 +\newcommand{\Zhat}{\widehat{Z}}
     146 +\newcommand{\Zbfhat}{\widehat{\mathbf{Z}}}
     147 +\def\alphabf{{\boldsymbol \alpha}}
     148 +\def\betabf{{\boldsymbol \beta}}
     149 +\def\mubf{{\boldsymbol \mu}}
     150 +\def\lambdabf{{\boldsymbol \lambda}}
     151 +\def\etabf{{\boldsymbol \eta}}
     152 +\def\xibf{{\boldsymbol \xi}}
     153 +\def\taubf{{\boldsymbol \tau}}
     154 +\def\sigmahat{{\widehat{\sigma}}}
     155 +\def\thetabf{{\bm{\theta}}}
     156 +\def\thetabfhat{{\widehat{\bm{\theta}}}}
     157 +\def\thetahat{{\widehat{\theta}}}
     158 +\def\mubar{\overline{\mu}}
     159 +\def\muavg{\mu}
     160 +\def\sigbf{\bm{\sigma}}
     161 +\def\etal{\emph{et al.}}
     162 +\def\Ggothic{\mathfrak{G}}
     163 +\def\Pset{{\mathcal P}}
     164 +\newcommand{\bigCond}[2]{\bigl({#1} \!\bigm\vert\! {#2} \bigr)}
     165 +\newcommand{\BigCond}[2]{\Bigl({#1} \!\Bigm\vert\! {#2} \Bigr)}
     166 + 
     167 +\def\Rect{\mathop{Rect}}
     168 +\def\sinc{\mathop{sinc}}
     169 +\def\NF{\mathrm{NF}}
     170 +\def\Real{\mathrm{Re}}
     171 +\def\Imag{\mathrm{Im}}
     172 +\newcommand{\tran}{^{\text{\sf T}}}
     173 +\newcommand{\herm}{^{\text{\sf H}}}
     174 + 
     175 + 
     176 +% Solution environment
     177 +\definecolor{lightgray}{gray}{0.95}
     178 +\newmdenv[linecolor=white,backgroundcolor=lightgray,frametitle=Solution:]{solution}
     179 + 
     180 + 
     181 + 
     182 +\begin{document}
     183 + 
     184 +\title{Problems: Antenna Arrays and Beamforming\\
     185 +ECE-GY 6023. Wireless Communications}
     186 +\author{Prof.\ Sundeep Rangan}
     187 +\date{}
     188 + 
     189 +\maketitle
     190 + 
     191 + 
     192 +\begin{enumerate}
     193 + 
     194 +\item \emph{ULA.} Consider a ULA with spatial signature,
     195 +\[
     196 + \ubf(\phi) = \left[ 1, e^{i \beta \sin(\phi)}, \cdots,
     197 + e^{i \beta(N-1) \sin(\phi)} \right]\tran, \quad
     198 + \beta = \frac{2\pi d}{\lambda},
     199 +\]
     200 +where $\phi$ is the azimuth angle relative to boresight.
     201 +\begin{enumerate}[label=(\alph*)]
     202 +\item Find the optimal beamforming vector for an angle $\phi= 30^\circ$
     203 +with $N=8$ antennas and $d=\lambda/2$.
     204 +\item Now suppose a path arrives at an angle $\phi = 60^\circ$.
     205 +What is the beamforming gain on that path using the BF vector in part (a)?
     206 +\end{enumerate}
     207 + 
     208 +\item \emph{Constrained beamforming.} Consider a MISO channel,
     209 +\[
     210 + \rbf = \hbf x + \vbf, \quad \vbf \sim C{\mathcal N}(0,N_0 \Ibf), \quad |x|^2 = E_x.
     211 +\]
     212 +\begin{enumerate}[label=(\alph*)]
     213 +\item What is the SNR after beamforming with a vector $\wbf$?
     214 +\item What is the maximum SNR after beamforming if we are allowed any vector $\wbf$?
     215 +\item What is the maximum SNR after beamforming if the components of $\wbf$
     216 +must be constant magnitude, $|w_n| = 1$. That is, you can only change the phase of $w_n$.
     217 +This commonly occurs in analog beamforming systems.
     218 +\item What is the maximum SNR after beamforming if
     219 +$w_n \neq 0$ for only one antenna $n$. This is called antenna selection beamforming.
     220 +\item Suppose $\hbf = [4,2+i,-1,i]\tran$ and $E_x/N_0=$\, \si{5}{dB}. What is the SNR
     221 +after beamforming in parts (b), (c) and (d)?
     222 +\end{enumerate}
     223 + 
     224 + 
     225 +\item \label{prob:freq_resp}
     226 +\emph{Vector frequency response}. Consider a MISO channel where
     227 +\[
     228 + \rbf = \hbf(f)\xbf + \vbf, \quad \vbf \sim C{\mathcal N}(0,N_0 \Ibf), \quad |x|^2 = E_x,
     229 +\]
     230 +where $\hbf(f)$ is the frequency-dependent channel
     231 +\[
     232 + \hbf = \sum_{\ell=1}^L a_\ell e^{2\pi if \tau_\ell} \ubf(\theta_\ell),
     233 +\]
     234 +and $L$ is the number of paths. Suppose the
     235 +array is $N=8$ ULA with antenna spacing $d=\lambda/2$ and the channel
     236 +has three paths with parameters shown in Table \ref{tab:freq_resp}.
     237 +In the table, the column ``SNR" is $E_x|a_\ell|^2/N_0$, and the column ``path phase" is the angle of $a_\ell$.
     238 + 
     239 +\begin{table}
     240 +\centering
     241 +\footnotesize
     242 +\begin{tabular}{|l|l|l|l|l|}
     243 +\hline
     244 +Path & SNR [dB] & Path phase [deg] & AoA $\phi_\ell$ [deg] & Delay $\tau_\ell$ [ns] \\ \hline
     245 +1 & 4 & 0 & 30 & 0 \\ \hline
     246 +2 & 1 & 180 & -30 & 100 \\ \hline
     247 +3 & -2 & 65 & 80 & 130 \\ \hline
     248 +\end{tabular}
     249 +\caption{Problem \ref{prob:freq_resp}: Path parameters.}
     250 +\label{tab:freq_resp}
     251 +\end{table}
     252 + 
     253 +Write a short MATLAB program to plot the following SNR values for 100
     254 +frequency points in the range $f \in [-10,10]$\, \si{MHz}:
     255 +\begin{enumerate}[label=(\alph*)]
     256 +\item The SNR only using the signal from antenna 1.
     257 +\item The SNR after beamforming where the beamforming vector is optimally selected at each
     258 +frequency.
     259 +\item The SNR after beamforming where the beamforming vector is optimized for $f=0$.
     260 +\end{enumerate}
     261 + 
     262 +\item \emph{Beamforming with SVDs}. Below are a set of channel matrices $\Hbf$
     263 +described by TX and RX spatial signatures $\ubf_{\rm tx}(\Omega^{\rm tx}_\ell)$
     264 +and $\ubf_{\rm rx}(\Omega^{\rm rx}_\ell)$ and complex gains
     265 +$g_\ell$.
     266 +For each of the channel matrices,
     267 +find the rank $r$, the maximum singular value and the optimal TX and RX
     268 +beamforming vectors. Assume all matrices have dimensions
     269 + $\Hbf \in \C^{N_r \times N_t}$,
     270 +and the spatial signatures are normalized so that
     271 +\[
     272 + \|\ubf_{\rm tx}(\Omega^{\rm tx})\|^2=N_t, \quad
     273 + \|\ubf_{\rm rx}(\Omega^{\rm rx})\|^2=N_r,
     274 +\]
     275 +for all angles $\Omega^{\rm tx}$ and $\Omega^{\rm rx}$.
     276 +\begin{enumerate}[label=(\alph*)]
     277 +\item Single path channel:
     278 +\[
     279 + \Hbf = g_1
     280 + \ubf_{\rm rx}(\Omega_1^{\rm rx}) \ubf_{\rm tx}(\Omega_1^{\rm tx}).
     281 +\]
     282 +\item Two channel paths with same RX angle $\Omega_1^{\rm rx}$
     283 +and two different TX angles, $\Omega_1^{\rm tx}$ and $\Omega_2^{\rm tx}$:
     284 +\[
     285 + \Hbf = g_1
     286 + \ubf_{\rm rx}(\Omega_1^{\rm rx}) \ubf_{\rm tx}(\Omega_1^{\rm tx}) +
     287 + g_2 \ubf_{\rm rx}(\Omega_1^{\rm rx}) \ubf_{\rm tx}(\Omega_2^{\rm tx}).
     288 +\]
     289 +Assume $|g_1| > |g_2|$ and $\ubf_{\rm tx}(\Omega_1^{\rm tx}) \perp \ubf_{\rm tx}(\Omega_2^{\rm tx})$
     290 + 
     291 +\item Two channel paths with two different RX angles
     292 +and two different TX angles:
     293 +\[
     294 + \Hbf = g_1
     295 + \ubf_{\rm rx}(\Omega_1^{\rm rx}) \ubf_{\rm tx}(\Omega_1^{\rm tx}) +
     296 + g_2 \ubf_{\rm rx}(\Omega_2^{\rm rx}) \ubf_{\rm tx}(\Omega_2^{\rm tx}).
     297 +\]
     298 +Assume $|g_1| > |g_2|$, $\ubf_{\rm rx}(\Omega_1^{\rm rx}) \perp \ubf_{\rm rx}(\Omega_2^{\rm rx})$ and $\ubf_{\rm tx}(\Omega_1^{\rm tx}) \perp \ubf_{\rm tx}(\Omega_2^{\rm tx})$.
     299 + 
     300 +\item Same as part (b) except the two TX spatial signatures are not orthongonal:
     301 +\[
     302 + \ubf_{\rm tx}(\Omega_1^{\rm tx})^*\ubf_{\rm tx}(\Omega_2^{\rm tx}) =\rho N_t,
     303 +\]
     304 +for some $\rho$ with $|\rho|\leq 1$.
     305 + 
     306 +\end{enumerate}
     307 + 
     308 +\item \emph{Array normalization.} When there is no mutual coupling,
     309 +an array with $N$ elements can generally obtain a gain of $N$ (in linear scale).
     310 +However, in this exercise, we will show that the gain is more limited when the
     311 +antennas are closely spaced. To this end, consider a ULA with $N$ elements
     312 +with total length $L$, so the antennas are spaced by $L/N$.
     313 +Given a beamforming vector $\wbf \in \C^N$,
     314 +suppose the power intensity at angle $\phi$ is,
     315 +\[
     316 + U(\phi) = c|\wbf\tran \ubf(\phi)|^2,
     317 +\]
     318 +where $\ubf(\phi)$ is the spatial signature and $c > 0$ is some constant.
     319 +Suppose we use the all ones beamforming vector $w_n = \frac{1}{N}$ so that the
     320 +energy is maximized at $\phi = 0$.
     321 +\begin{enumerate}[label=(\alph*)]
     322 +\item Write the spatial signature $\ubf(\phi)$ as a function of the total array length
     323 +$L$, wavelength $\lambda$ and number of elements $N$.
     324 + 
     325 +\item Find the limit
     326 +\[
     327 + \lim_{N \rightarrow \infty} U(\phi).
     328 +\]
     329 + 
     330 +\item Recall the antenna directivity is
     331 +\[
     332 + D(\phi) = \frac{4\pi U(\phi)}{P_{\rm rad}},
     333 +\]
     334 +where $P_{\rm rad}$ is the total radiated power,
     335 +\[
     336 + P_{\rm rad} = \int_{-\pi/2}^{\pi/2} \int_{-\pi}^{\pi} U(\phi)\cos \theta \,
     337 + \dd \phi \,\dd \theta.
     338 +\]
     339 +Use MATLAB to plot $D(\phi)$ using the limit of large numbers of antennas for $L=2\lambda$,
     340 +$4\lambda$ and $8\lambda$. What is the peak gain in each case.
     341 + 
     342 + 
     343 +\end{enumerate}
     344 + 
     345 + 
     346 + 
     347 + 
     348 +\end{enumerate}
     349 + 
     350 + 
     351 + 
     352 + 
     353 + 
     354 +\end{document}
     355 + 
     356 + 
     357 + 
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