1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"> 2 3<!--Converted with LaTeX2HTML 2018.3 (Released July 19, 2018) --> 4<HTML lang="EN"> 5<HEAD> 6<TITLE>Pixel window functions</TITLE> 7<META NAME="description" CONTENT="Pixel window functions"> 8<META NAME="keywords" CONTENT="intro"> 9<META NAME="resource-type" CONTENT="document"> 10<META NAME="distribution" CONTENT="global"> 11 12<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=utf-8"> 13<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0"> 14<META NAME="Generator" CONTENT="LaTeX2HTML v2018.3"> 15 <link rel='apple-touch-icon' sizes='180x180' href='images/favicons/apple-touch-icon.png?v=2017'> 16 <link rel='icon' type='image/png' sizes='32x32' href='images/favicons/favicon-32x32.png?v=2017'> 17 <link rel='icon' type='image/png' sizes='16x16' href='images/favicons/favicon-16x16.png?v=2017'> 18 <link rel='manifest' href='images/favicons/manifest.json?v=2017'> 19 <link rel='mask-icon' href='images/favicons/safari-pinned-tab.svg?v=2017' color='#5bbad5'> 20 <link rel='shortcut icon' href='images/favicons/favicon.ico?v=2017'> 21 <meta name='apple-mobile-web-app-title' content='HEALPix'> 22 <meta name='application-name' content='HEALPix'> 23 <meta name='msapplication-config' content='images/favicons/browserconfig.xml?v=2017'> 24 <meta name='theme-color' content='#ffffff'> 25 26<LINK REL="STYLESHEET" HREF="intro.css"> 27 28<LINK REL="next" HREF="intro_Comment_on_Random_Number_Ge.htm"> 29<LINK REL="previous" HREF="intro_HEALPix_conventions.htm"> 30<LINK REL="next" HREF="intro_Comment_on_Random_Number_Ge.htm"> 31</HEAD> 32 33<body text="#000000" bgcolor="#FFFFFA"> 34 35<DIV CLASS="navigation"><!--Navigation Panel--> 36<A 37 HREF="intro_HEALPix_conventions.htm"> 38<IMG WIDTH="63" HEIGHT="24" ALT="previous" SRC="prev.png"></A> 39<A 40 HREF="intro_Introduction_HEALPix.htm"> 41<IMG WIDTH="26" HEIGHT="24" ALT="up" SRC="up.png"></A> 42<A 43 HREF="intro_Comment_on_Random_Number_Ge.htm"> 44<IMG WIDTH="37" HEIGHT="24" ALT="next" SRC="next.png"></A> 45<A ID="tex2html111" 46 HREF="intro_TABLE_CONTENTS.htm"> 47<IMG WIDTH="65" HEIGHT="24" ALT="contents" SRC="contents.png"></A> 48<BR> 49<B> Previous:</B> <A 50 HREF="intro_HEALPix_conventions.htm">HEALPix conventions</A> 51 52<B>Up:</B> <A 53 HREF="intro_Introduction_HEALPix.htm">Introduction to HEALPix</A> 54 55<B> Next:</B> <A 56 HREF="intro_Comment_on_Random_Number_Ge.htm">A Comment on the Random Number</A> 57<B> Top:</B> <a href="main.htm">Main Page</a></DIV> 58<!--End of Navigation Panel--> 59 60<H1><A ID="SECTION700"> 61Pixel window functions</A> 62</H1> 63 64<P> 65A pixelated signal <SPAN CLASS="MATH"><I>f</I>(<I>p</I>)</SPAN> is the average within each pixel <SPAN CLASS="MATH"><I>p</I></SPAN> (with surface 66area <!-- MATH 67 $\Omega_{\mathrm{pix}}$ 68 --> 69<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.75ex; " SRC="intro_img154.png" 70 ALT="$\Omega_{\mathrm{pix}}$"></SPAN>) of the underlying signal 71<BR> 72<DIV ALIGN="CENTER"> 73 74<!-- MATH 75 \begin{eqnarray} 76f(p) &\myequal & \int d\u w_p(\u )f(\u ) 77\end{eqnarray} 78 --> 79<SPAN CLASS="MATH"> 80<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 81<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>f</I>(<I>p</I>)</TD> 82<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 83 ALT="$\textstyle \myequal $"></TD> 84<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="intro_img155.png" 85 ALT="$\displaystyle \int d\u w_p(\u )f(\u )$"></TD> 86<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 87(<SPAN CLASS="arabic">29</SPAN>)</TD></TR> 88</TABLE> 89</SPAN></DIV> 90<BR CLEAR="ALL"><P></P> 91where <SPAN CLASS="MATH"><I>w</I><SUB><i>p</i></SUB></SPAN> is equal to <!-- MATH 92 $1/\Omega_{\mathrm{pix}}$ 93 --> 94<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img156.png" 95 ALT="$1/\Omega_{\mathrm{pix}}$"></SPAN> within the pixel, and equal to 0 outside, so 96that <!-- MATH 97 $\int d\u w_p(\u ) = 1$ 98 --> 99<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img157.png" 100 ALT="$\int d\u w_p(\u ) = 1$"></SPAN>. 101Eq. (<A HREF="intro_HEALPix_conventions.htm#eq:alms">4</A>) then becomes 102<BR> 103<DIV ALIGN="CENTER"> 104 105<!-- MATH 106 \begin{eqnarray} 107f(p)&\myequal &\sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}w_{\ell m}(p), 108\end{eqnarray} 109 --> 110<SPAN CLASS="MATH"> 111<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 112<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>f</I>(<I>p</I>)</TD> 113<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 114 ALT="$\textstyle \myequal $"></TD> 115<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 6.06ex; vertical-align: -2.66ex; " SRC="intro_img158.png" 116 ALT="$\displaystyle \sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}w_{\ell m}(p),$"></TD> 117<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 118(<SPAN CLASS="arabic">30</SPAN>)</TD></TR> 119</TABLE> 120</SPAN></DIV> 121<BR CLEAR="ALL"><P></P> 122where 123<BR> 124<DIV ALIGN="CENTER"><A ID="eq:pixel_lmp"></A> 125<!-- MATH 126 \begin{eqnarray} 127w_{\ell m}(p) &\myequal & \int d\u w_p(\u ) Y_{\ell m}(\u ), 128\end{eqnarray} 129 --> 130<SPAN CLASS="MATH"> 131<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 132<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img159.png" 133 ALT="$\displaystyle w_{\ell m}(p)$"></TD> 134<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 135 ALT="$\textstyle \myequal $"></TD> 136<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="intro_img160.png" 137 ALT="$\displaystyle \int d\u w_p(\u ) Y_{\ell m}(\u ),$"></TD> 138<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 139(<SPAN CLASS="arabic">31</SPAN>)</TD></TR> 140</TABLE> 141</SPAN></DIV> 142<BR CLEAR="ALL"><P></P> 143is the Spherical Harmonic Transform of the pixel <SPAN CLASS="MATH"><I>p</I></SPAN>. 144 145<P> 146However, complete analysis of a pixelated map with the exact <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img161.png" 147 ALT="$w_{\ell m}(p)$"></SPAN> 148defined above would be computationally intractable (because of azimutal 149variation of pixel shape over the polar caps of the <b>HEALPix</b> grid), 150and some simplifying asumptions have to be 151made. If the pixel is small compared to the signal correlation length 152(determined by the beam size), the exact structure of the pixel can be ignored 153in the subsequent analysis and we can <EM>assume</EM> 154<BR> 155<DIV ALIGN="CENTER"> 156 157<!-- MATH 158 \begin{eqnarray} 159w_{\ell m}(p) &\myequal & w_\ell(p) Y_{\ell m}(p) 160\end{eqnarray} 161 --> 162<SPAN CLASS="MATH"> 163<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 164<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img159.png" 165 ALT="$\displaystyle w_{\ell m}(p)$"></TD> 166<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 167 ALT="$\textstyle \myequal $"></TD> 168<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img162.png" 169 ALT="$\displaystyle w_\ell(p) Y_{\ell m}(p)$"></TD> 170<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 171(<SPAN CLASS="arabic">32</SPAN>)</TD></TR> 172</TABLE> 173</SPAN></DIV> 174<BR CLEAR="ALL"><P></P> 175where we introduced the <SPAN CLASS="MATH"><I>m</I></SPAN>-averaged window function 176<BR> 177<DIV ALIGN="CENTER"><A ID="eq:pixel_lp"></A> 178<!-- MATH 179 \begin{eqnarray} 180w_{\ell}(p) &\myequal & \left(\frac{4 \pi}{2\ell+1}\sum_{m=-\ell}^{\ell} \left|w_{\ell m}(p)\right|^2\right)^{1/2}, 181\end{eqnarray} 182 --> 183<SPAN CLASS="MATH"> 184<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 185<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img163.png" 186 ALT="$\displaystyle w_{\ell}(p)$"></TD> 187<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 188 ALT="$\textstyle \myequal $"></TD> 189<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 7.58ex; vertical-align: -3.00ex; " SRC="intro_img164.png" 190 ALT="$\displaystyle \left(\frac{4 \pi}{2\ell+1}\sum_{m=-\ell}^{\ell} \left\vert w_{\ell m}(p)\right\vert^2\right)^{1/2},$"></TD> 191<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 192(<SPAN CLASS="arabic">33</SPAN>)</TD></TR> 193</TABLE> 194</SPAN></DIV> 195<BR CLEAR="ALL"><P></P> 196which is independent of the pixel location on the sky. 197 198<P> 199If we assume all the pixels to be identical, the power spectrum of the 200pixelated map, <!-- MATH 201 $C_{\ell}^{\mathrm{pix}}$ 202 --> 203<SPAN CLASS="MATH"><IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="intro_img165.png" 204 ALT="$C_{\ell}^{\mathrm{pix}}$"></SPAN>, is related to the hypothetical unpixelated 205one, <!-- MATH 206 $C_{\ell}^{\mathrm{unpix}}$ 207 --> 208<SPAN CLASS="MATH"><IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="intro_img166.png" 209 ALT="$C_{\ell}^{\mathrm{unpix}}$"></SPAN>, by 210<BR> 211<DIV ALIGN="CENTER"><A ID="eq:cl_pixel"></A> 212<!-- MATH 213 \begin{eqnarray} 214C_{\ell}^\mathrm{pix} &\myequal & w^2_{\ell} C_{\ell}^\mathrm{unpix} 215\end{eqnarray} 216 --> 217<SPAN CLASS="MATH"> 218<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 219<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="intro_img167.png" 220 ALT="$\displaystyle C_{\ell}^\mathrm{pix}$"></TD> 221<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 222 ALT="$\textstyle \myequal $"></TD> 223<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="intro_img168.png" 224 ALT="$\displaystyle w^2_{\ell} C_{\ell}^\mathrm{unpix}$"></TD> 225<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 226(<SPAN CLASS="arabic">34</SPAN>)</TD></TR> 227</TABLE> 228</SPAN></DIV> 229<BR CLEAR="ALL"><P></P> 230where the effective pixel window function <SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img169.png" 231 ALT="$w_{\ell}$"></SPAN> is defined as 232<BR> 233<DIV ALIGN="CENTER"><A ID="eq:pixel_l"></A> 234<!-- MATH 235 \begin{eqnarray} 236w_{\ell} &\myequal & \left(\frac{1}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} w^2_{\ell}(p)\right)^{1/2}. 237\end{eqnarray} 238 --> 239<SPAN CLASS="MATH"> 240<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 241<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img170.png" 242 ALT="$\displaystyle w_{\ell}$"></TD> 243<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 244 ALT="$\textstyle \myequal $"></TD> 245<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 7.58ex; vertical-align: -3.00ex; " SRC="intro_img171.png" 246 ALT="$\displaystyle \left(\frac{1}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} w^2_{\ell}(p)\right)^{1/2}.$"></TD> 247<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 248(<SPAN CLASS="arabic">35</SPAN>)</TD></TR> 249</TABLE> 250</SPAN></DIV> 251<BR CLEAR="ALL"><P></P> 252This function is provided with the <b>HEALPix</b> package for <!-- MATH 253 $\ell\le 4N_{\mathrm{side}}$ 254 --> 255<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img172.png" 256 ALT="$\ell\le 4N_{\mathrm{side}}$"></SPAN> for each 257resolution parameter <!-- MATH 258 $N_{\mathrm{side}}$ 259 --> 260<SPAN CLASS="MATH"><I>N</I><SUB>side</SUB></SPAN>. 261 262<P> 263The pixel window functions are now available for both temperature and 264polarization. 265 266<P> 267For <!-- MATH 268 $N_{\mathrm{side}}\le 128$ 269 --> 270<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img173.png" 271 ALT="$N_{\mathrm{side}}\le 128$"></SPAN>, those window functions are computed exactly using 272Eqs. (<A HREF="#eq:pixel_lp">33</A>) and (<A HREF="#eq:pixel_l">35</A>). For <!-- MATH 273 $N_{\mathrm{side}}> 128$ 274 --> 275<SPAN CLASS="MATH"><I>N</I><SUB>side</SUB>> 128</SPAN> the 276calculations are too costly to be done exactly at all <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 277 ALT="$\ell$"></SPAN>. The temperature 278windows are 279extrapolated from the case <!-- MATH 280 $N_{\mathrm{side}}= 128$ 281 --> 282<SPAN CLASS="MATH"><I>N</I><SUB>side</SUB>= 128</SPAN> assuming a scaling in <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 283 ALT="$\ell$"></SPAN> similar 284to the one exhibited by the window of a tophat pixel. The polarization 285windows are assumed to be proportional to those for temperature, with a 286proportionality factor given by the exact calculation of <SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img169.png" 287 ALT="$w_{\ell}$"></SPAN> at low 288<SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 289 ALT="$\ell$"></SPAN>. 290 291<P> 292Because of a change of the extrapolation scheme used, the temperature window 293functions provided with <b>HEALPix</b> 1.2 and higher for <!-- MATH 294 $N_{\mathrm{side}}> 128$ 295 --> 296<SPAN CLASS="MATH"><I>N</I><SUB>side</SUB>> 128</SPAN> are slighty different from those 297provided with <b>HEALPix</b> 1.1. For a given <!-- MATH 298 $N_{\mathrm{side}}$ 299 --> 300<SPAN CLASS="MATH"><I>N</I><SUB>side</SUB></SPAN>, the relative difference 301increases almost linearly with <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 302 ALT="$\ell$"></SPAN>, and is of the order of <!-- MATH 303 $\Delta w/w < 7\ 10^{-4}$ 304 --> 305<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="intro_img174.png" 306 ALT="$\Delta w/w < 7\ 10^{-4}$"></SPAN> at 307<!-- MATH 308 $\ell=2N_{\mathrm{side}}$ 309 --> 310<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img175.png" 311 ALT="$\ell=2N_{\mathrm{side}}$"></SPAN> and <!-- MATH 312 $\Delta w/w < 1.7\ 10^{-3}$ 313 --> 314<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="intro_img176.png" 315 ALT="$\Delta w/w < 1.7\ 10^{-3}$"></SPAN> at <!-- MATH 316 $\ell=4N_{\mathrm{side}}$ 317 --> 318<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img177.png" 319 ALT="$\ell=4N_{\mathrm{side}}$"></SPAN>. 320 321<P> 322 323<DIV CLASS="navigation"><HR> 324<!--Navigation Panel--> 325<A 326 HREF="intro_HEALPix_conventions.htm"> 327<IMG WIDTH="63" HEIGHT="24" ALT="previous" SRC="prev.png"></A> 328<A 329 HREF="intro_Introduction_HEALPix.htm"> 330<IMG WIDTH="26" HEIGHT="24" ALT="up" SRC="up.png"></A> 331<A 332 HREF="intro_Comment_on_Random_Number_Ge.htm"> 333<IMG WIDTH="37" HEIGHT="24" ALT="next" SRC="next.png"></A> 334<A ID="tex2html111" 335 HREF="intro_TABLE_CONTENTS.htm"> 336<IMG WIDTH="65" HEIGHT="24" ALT="contents" SRC="contents.png"></A> 337<BR> 338<B> Previous:</B> <A 339 HREF="intro_HEALPix_conventions.htm">HEALPix conventions</A> 340 341<B>Up:</B> <A 342 HREF="intro_Introduction_HEALPix.htm">Introduction to HEALPix</A> 343 344<B> Next:</B> <A 345 HREF="intro_Comment_on_Random_Number_Ge.htm">A Comment on the Random Number</A> 346<B> Top:</B> <a href="main.htm">Main Page</a></DIV> 347<!--End of Navigation Panel--> 348<ADDRESS> 349Version 3.50, 2018-12-10 350</ADDRESS> 351</BODY> 352</HTML> 353
