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memo describing APERTIF Anti-Aliasing Filter (AAF)

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\documentclass[a4paper,10pt]{astron}
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\renewcommand{\theTitle}{APERTIF Anti-Aliasing Filter}
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E.~Kooistra\\
A.~Mika\\
R.~Oonk\\
T.~Oosterloo\\
S.~van~der~Tol\\
M.A.W.~Verheijen\\
S.J.~Wijnholds
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\section{Introduction}
\subsection{Purpose of this report}
In the APERTIF signal chains, Nyquist-sampled polyphase filter banks are used to split the baseband signal in 781-kHz wide subbands, which are then further split into 12-kHz wide frequency channels. Due to the finite length of the finite impulse response (FIR) filters used, the frequency response of these filters is not infinitely sharp. This causes aliasing effects at the edges of the subbands that become visible after channel separation. A similar frequency separation structure is used in LOFAR. As LOFAR observations mostly concern astronomical signals with a reasonably wide frequency response, the solution adopted in LOFAR is to simply flag the channels near the edges of the subband. This approach may not work for APERTIF, since APERTIF will mostly be used to conduct spectral line observations.
In this document, we
\begin{enumerate}
\item assess the impact of the aliasing effect at the subband edges on spectral line observations with the goal of determining the need for corrective processing steps to mitigate this effect;
\item discuss possible solutions and their impact the APERTIF signal chain and observing strategy;
\item select the most suitable solution and describe it in sufficient detail to provide a good starting point for implementation in the APERTIF system.
\end{enumerate}
\subsection{Outline of this document}
In the next section, we present a theoretical framework to describe the FIR filter response and use it to demonstrate the effect it has on the observed spectra. This section is concluded with an assessment on the potential impact on APERTIF observations. In Sec.~\ref{sec:corrective_measures}, we give an overview of possible corrective measures and assess their impact on the APERTIF signal chain and observing strategy. Based on that assessment, a preferred solution is proposed, which is described in more detail in Sec.~\ref{sec:AAF}. This detailed description includes an assessment of the level to which aliasing effects are reduced as well as an updated overview of the APERTIF signal chain that is sufficiently detailed to start work on the implementation of the solution. The code of the key routines used for the simulations presented in this report is provided in the Appendix.
\section{FIR filter response}
\subsection{Theoretical description}
\subsection{Impact on spectra}
\subsection{Impact on observations}
% source smearing? missed lindes? velocity field reconstruction issues?
\section{Corrective measures}
\label{sec:corrective_measures}
\subsection{Flag and ignore affected channels}
\subsection{Oversampled polyphase filter bank}
\subsection{Anti-aliasing filter}
\subsection{Re-observing with local oscillator tuning}
\subsection{Preferred solution}
\section{Anti-aliasing filter implementation}
\label{sec:AAF}
{\color{red} SJW: it is assumed that the outcome of the assessment of different options will be that the anti-aliasing filter is the preferred solution. We should, however, remain willing to accept a different solution if the assessment in the previous section dictates otherwise.}
\addtocontents{toc}{\contentsline{section}{References}{\pageref{literature}}}
\bibliographystyle{IEEEtran}
\bibliography{refs}
\label{literature}
\appendix
\section{Matlab code}
\label{sec:app_code}
\end{document}
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%% Author: G.W. Kant
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\addvspace{2.25em \@plus\p@}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
{\leavevmode
\large \bfseries #1\hfil \hb@xt@\@pnumwidth{\hss #2}}\par
\nobreak
\endgroup
\fi}
\newcommand*\l@section[2]{%
\ifnum \c@tocdepth >\z@
\addpenalty\@secpenalty
\addvspace{1.0em \@plus\p@}%
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries
\advance\leftskip\@tempdima
\hskip -\leftskip
#1\nobreak\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par
\endgroup
\fi}
\newcommand*\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\newcommand*\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\newcommand*\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\newcommand*\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\newcommand\listoffigures{%
\section*{\listfigurename
\@mkboth{\MakeUppercase\listfigurename}%
{\MakeUppercase\listfigurename}}%
\@starttoc{lof}%
}
\newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
\section*{\listtablename
\@mkboth{%
\MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
\@starttoc{lot}%
}
\let\l@table\l@figure
\newdimen\bibindent
\setlength\bibindent{1.5em}
\newenvironment{thebibliography}[1]
{\section*{\refname
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}}%
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\let\@openbib@code\@empty
\newenvironment{theindex}
{\if@twocolumn
\@restonecolfalse
\else
\@restonecoltrue
\fi
\columnseprule \z@
\columnsep 35\p@
\twocolumn[\section*{\indexname}]%
\@mkboth{\MakeUppercase\indexname}%
{\MakeUppercase\indexname}%
\thispagestyle{plain}\parindent\z@
\parskip\z@ \@plus .3\p@\relax
\let\item\@idxitem}
{\if@restonecol\onecolumn\else\clearpage\fi}
\newcommand\@idxitem{\par\hangindent 40\p@}
\newcommand\subitem{\@idxitem \hspace*{20\p@}}
\newcommand\subsubitem{\@idxitem \hspace*{30\p@}}
\newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax}
\renewcommand\footnoterule{%
\kern-3\p@
\hrule\@width.4\columnwidth
\kern2.6\p@}
\newcommand\@makefntext[1]{%
\parindent 1em%
\noindent
\hb@xt@1.8em{\hss\@makefnmark}#1}
\newcommand\contentsname{Contents}
\newcommand\listfigurename{List of Figures}
\newcommand\listtablename{List of Tables}
\newcommand\refname{References}
\newcommand\indexname{Index}
\newcommand\figurename{Figure}
\newcommand\tablename{Table}
\newcommand\partname{Part}
\newcommand\appendixname{Appendix}
\newcommand\abstractname{Abstract}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\day, \number\year}
\if@twoside
\else
\raggedbottom
\fi
\if@twocolumn
\twocolumn
\sloppy
\flushbottom
\else
\onecolumn
\fi
%% ENVIRONEMTS
\def\QED{\mbox{\rule[0pt]{1.5ex}{1.5ex}}}
\def\proof{\noindent\hspace{2em}{\it Proof: }}
\def\endproof{\hspace*{\fill}~\QED\par\endtrivlist\unskip}
\def\@begintheorem#1#2{\tmpitemindent\itemindent\topsep 0pt\rm\trivlist
\item[\hskip \labelsep{\indent\it #1\ #2:}]\itemindent\tmpitemindent}
\def\@opargbegintheorem#1#2#3{\tmpitemindent\itemindent\topsep 0pt\rm \trivlist
\item[\hskip\labelsep{\indent\it #1\ #2\ \rm(#3)}]\itemindent\tmpitemindent}
\def\@endtheorem{\endtrivlist\unskip}
%% set up default PAGESTYLE
\pagenumbering{arabic}
%%
\if@draftversion \def\baselinestretch{1.5}\fi
%%
%% Definition for Big letter at the beginning of a paragraph
%%
\def\PARstart#1#2{\begingroup\def\par{\endgraf\endgroup\lineskiplimit=0pt}
\setbox2=\hbox{\uppercase{#2} }\newdimen\tmpht \tmpht \ht2
\advance\tmpht by \baselineskip\font\hhuge=cmr10 at \tmpht
\setbox1=\hbox{{\hhuge #1}}
\count7=\tmpht \count8=\ht1\divide\count8 by 1000 \divide\count7 by\count8
\tmpht=.001\tmpht\multiply\tmpht by \count7\font\hhuge=cmr10 at \tmpht
\setbox1=\hbox{{\hhuge #1}} \noindent \hangindent1.05\wd1
\hangafter=-2 {\hskip-\hangindent \lower1\ht1\hbox{\raise1.0\ht2\copy1}%
\kern-0\wd1}\copy2\lineskiplimit=-1000pt}
%%
\if@technote\def\PARstart#1#2{#1#2}\fi % if technical note, disable it
\if@draftversion\def\PARstart#1#2{#1#2}\fi % if draft, disable it
%%
%% Allows to enter BIOGRAPHY leaving place for picture (adapts to font size)
%%
\newenvironment{biography}[1]{%
\section*{CV}
\vspace{-1cm}
\unitlength 1mm\bigskip\bigskip\bigskip\parskip=0pt\par%
\rule{0pt}{38mm}\vspace{-38mm}\par% garantees correct page breaking
\noindent\setbox0\hbox{#1}% box containing the frame
%\ht0=37mm\count10=\ht0\divide\count10 by\baselineskip% calculates lines
%\global\hangindent29mm\global\hangafter-\count10}%
\begin{picture}(0,0)(0,0)%
\put(130,-36){\box0}%
\end{picture}%
\global\hangindent-40mm\global\hangafter-\count50}%
{\par\rm\normalsize\global\hangindent0mm}
%% Allows to enter BIOGRAPHY leaving place for picture (adapts to font size)
%%
\newenvironment{biography2}[1]{%
\footnotesize\unitlength 1mm\bigskip\bigskip\bigskip\parskip=0pt\par%
\rule{0pt}{39mm}\vspace{-39mm}\par% garantees correct page breaking
\noindent\setbox0\hbox{\framebox(25,32){}}% box containing the frame
\ht0=37mm\count10=\ht0\divide\count10 by\baselineskip% calculates lines
\global\hangindent29mm\global\hangafter-\count10%
\hskip-28.5mm\setbox0\hbox to 28.5mm {\raise-30.5mm\box0\hss}%
\dp0=0mm\ht0=0mm\box0\noindent\bf#1\rm}{
\par\rm\normalsize\global\hangindent0mm}
% A Few PostScript definitions to use with \psboxit
% Call \PScommands once at the begining of your program, this will
% define : box roundedbox rectcartouche and cartouche. They are 4
% PostScript programs. Change the values before setlinewidth end
% setgray to customize your boxes
%
% Ex : \psboxit{25 cartouche}{blah blah}
% \psboxit{rectcartouche}{blah blah}
%
\long\def\PScommands{\special{! TeXDict begin
/box{% Processes the path of a rectangle.
% Needs : x0 y0 x1 y1.
newpath 2 copy moveto 3 copy pop exch lineto 4 copy pop pop
lineto 4 copy exch pop exch pop lineto closepath } bind def
%
%
/min{ 2 copy gt { exch } if pop } bind def%
/max{ 2 copy lt { exch } if pop } bind def%
/roundedbox{% Processes the path of a rounded rectangle.
% Needs : x0 y0 x1 y1 radius.
% The bounding box is augmented by +/- radius to allow easily to
% frame several rounded boxes around the same Texture box. Ex:
% \psboxit{4 copy 15 roundedbox 25 roundedbox} {\spbox{Some Text}}
% draws two scaled boxes arond the same word. Delete the `radius
% sub' and `radius add' commands to suppress that enlargement.
%
/radius exch store
3 2 roll % x0 x1 y1 y0
2 copy min radius sub /miny exch store
max radius add /maxy exch store
2 copy min radius sub /minx exch store
max radius add /maxx exch store
newpath
minx radius add miny moveto
maxx miny maxx maxy radius arcto
maxx maxy minx maxy radius arcto
minx maxy minx miny radius arcto
minx miny maxx miny radius arcto 16 {pop} repeat
closepath
}bind def
%
%
/rectcartouche{% Draws a filled and framed box
% Needs : x0 y0 x1 y1
box gsave .95 setgray fill grestore 1 setlinewidth stroke }bind def
%
%
/cartouche{% Draws a filled and framed rounded box
% Needs : x0 y0 x1 y1 radius
roundedbox gsave .95 setgray fill grestore 1 setlinewidth stroke }bind def
%
end }% Closes dictionnary
}\PScommands%
%%
%% PSBOXIT
%%
%% \psboxit{PS program}{TeX stuff}
%%
%% The bounding box of the TeX stuff is pushed on the PostScript stack
%% and then the program in the first argument is called
%%
%% EXAMPLE: set some text on a gray background, Use the SPBOX macro to
%% give some space around the text.
%%
%% \psboxit{box 0.5 setgray fill}{\spbox{Some Text}}
%%
%% See \PScommands for the \box definition
%%
\chardef \atcode = \the \catcode `\@
\catcode `\@ = 11
\long\def\psboxit#1#2{%
\begingroup\setbox0=\hbox{#2}%
\dimen0=\ht0 \advance\dimen0 by \dp0%
% Write out the PS code to set the current path using HEIGHT,
% WIDTH , DEPTH of box0.
\hbox{%
\special{ps: gsave currentpoint translate
0
\number\dp0 \space 15800 300 VResolution div mul DVImag div div % han
\number\wd0 \space 15800 300 Resolution div mul DVImag div div % han
\number\ht0 \space -15800 300 VResolution div mul DVImag div div % hand
% \number\dp0 \space 16384 div
% \number\wd0 \space 16384 div
% \number\ht0 \space -16384 div % Bounding box
% \number\dp0 \space 65536 div
% \number\wd0 \space 65536 div
% \number\ht0 \space -65536 div % Bounding box
#1 grestore}%
\copy0%
}%HBOX
\endgroup%
}%
% SPACEBOX
%
% This macro simply takes some TeX stuff, and puts FOUR sides on it
% so that the box is the same size as the thing you'd get with
% an \fbox{} command. (All I did was modify the code for \fbox{}
% so that all rules were replaced with struts).
%
% USAGE: \spbox{text} is just like \fbox{text} but makes no rules
%
% REASON: so that if using \pspath{...}{\fbox{stuff}}
% there is a way to get another box the same size:
% \pspath{...}{\spbox{stuff}}
%
\long\def\spbox#1{\leavevmode\setbox1\hbox{#1}%
\dimen0\fboxrule \advance\dimen0 \fboxsep%
\advance\dimen0 \dp1%
\hbox{\lower \dimen0\hbox%
{\vbox{\hrule height \fboxrule width 0pt%
\hbox{\vrule width \fboxrule height 0pt \hskip\fboxsep%
\vbox{\vskip\fboxsep \box1\vskip\fboxsep}\hskip%
\fboxsep\vrule width \fboxrule height 0pt}%
\hrule height \fboxrule width 0pt}}}}%
\def\Gbox#1{\psboxit{box 0.9 setgray fill}{#1}}
%%
\newsavebox{\gray@box}%
\newdimen\gray@space
%
\long\def\Beginboxitpara#1{\edef\gray@tmp{#1}%
\gray@space=\fboxsep \advance\gray@space by -\fboxrule
\par\prevdepth=-1000pt\vskip 0.5\baselineskip\noindent
\setbox\gray@box=%
\hbox\bgroup \vrule width \fboxrule
\vtop\bgroup \hrule
\vbox\bgroup
\kern\gray@space
\hbox\bgroup
\kern\gray@space
\vtop\bgroup
\hsize=\linewidth\advance\hsize-2\fboxsep\noindent
\ignorespaces}
\def\Endboxitpara{%
\egroup
\kern\gray@space
\egroup
\kern\gray@space
\egroup \hrule
\egroup\vrule width \fboxrule
\egroup
\psboxit{\gray@tmp}{\box\gray@box}%
\vskip 0.5\baselineskip
\par}
\newenvironment{boxitpara}[1]{\Beginboxitpara{#1}}{\Endboxitpara}
\newenvironment{boxitpara*}[1]{\Beginboxitpara{#1}\hbox to\hsize{}}%
{\Endboxitpara}
\catcode `\@ = \the \atcode
\def\@normalsize{\@setsize\normalsize{15pt}\xpt\@xpt
\abovedisplayskip 1em plus2pt minus5pt\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus3pt\belowdisplayshortskip .6em plus3pt minus3pt
\topsep \belowdisplayshortskip%!PN
}
\pagestyle{fancy}
\fancyhead[RE,RO]{\epsfig{file=astronlogo_text,width=8cm}}
\fancyfoot[CE,CO]{}
\fancyfoot[LE,LO]{
\begin{tabular}{p{6.0cm}p{4.5cm}p{1cm}p{4.0cm}}
\hline
& & Doc.nr.: & \theFootDocNr \\
& \centering {\bf \Large \theDepartment} & Rev.: & \theFootRev \\
& \centering {\bf \Large \theProject} & Date: & \theFootDate \\
& & Class.: & \theFootClass \\
{\thepage /\pageref{LastPage}} & & &
\end{tabular}
}
\newcommand{\theDistributionList} {%
%\cleardoublepage
%\addcontentsline{toc}{section}{Distribution List}
{\bf \Large Distribution list:}
\renewcommand{\arraystretch}{1.15}%
\vspace*{-1.5\baselineskip}
\rule{\linewidth}{0.5mm}
\begin{tabularx}{\linewidth}[t]{|X|X|}\hline
{\bf Group:} & {\bf Others:} \\ \hline
\begin{tabular}[t]{@{}l}
\dGList
\end{tabular}\begin{tabular}[t]{@{\hspace*{1cm}}l}
\ddGList
\end{tabular} & \begin{tabular}[t]{@{}l}
\dOList
\end{tabular}\\ \hline
\end{tabularx}
%\vspace{\fill}
%\mbox{}
}
\newcommand{\theDocumentRevision}{%
%\cleardoublepage
%\addcontentsline{toc}{section}{Document Revision}
{\bf \Large Document history:}
%\renewcommand{\arraystretch}{1.15}%
\vspace*{-1.5\baselineskip}
\rule{\linewidth}{0.5mm}
\begin{tabular}{|c|l|c|l|}\hline
\makebox[2.0cm]{{\bf \hfill Revision\hfill}} &
\makebox[2.0cm]{{\bf Date\hfill}} &
\makebox[4.0cm]{{\bf \hfill Chapter/Page\hfill}} &
\makebox[7.0cm]{{\bf Modification/Change\hfill}}\\ \hline
\begin{tabular}[t]{@{}c}
\revList
\end{tabular} &
\begin{tabular}[t]{@{}l}
\dateList
\end{tabular} &
\begin{tabular}[t]{@{}c}
\pageList
\end{tabular} &
\begin{tabular}[t]{@{}l}
\changeList
\end{tabular} \\ \hline
\end{tabular}}%
\endinput
%%
%% End of file `astron.cls'.
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