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Aaÿ€°€‚€`€ffU0pð0ƒ@ ° àƒ@p@ 0€°à p HH $ ÃÂdÁ HHHH€ÌÌ€ÌÌ€ÌÌ€ff€@ÊøÊøÊøÿ€ÿÿÿ€ÿÿÿ€ÿÿÁÿÿö Êøÿÿÿÿÿÿÿÿ€™ d ÙFootnote TableFootnote* à* à. . / - Ð Ñ:;,.É!?-3 ^,þþþ8 i€{þþþTOCHeading seamlessly EquationVariables€€[6"oB G÷Ìõ¤ÅL SW[_dinqr}x€Á‚‹!$”(™+-¢/¼1À3ÃÄÙÚÜÝéâ5è7ëìþð9ô;ø=ü?ÿ 3AFH"J(L-N1P6R<$monthname> <$daynum>, <$year>"<$monthnum>/<$daynum>/<$shortyear>;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>"<$monthnum>/<$daynum>/<$shortyear><$monthname> <$daynum>, <$year>"<$monthnum>/<$daynum>/<$shortyear> <$fullfilename> <$filename><$paratext[Title]><$paratext[Heading]> <$curpagenum> <$marker1> <$marker2> (Continued)+ (Sheet <$tblsheetnum> of <$tblsheetcount>)Number in Parens(<$paranumonly>)Pagepage<$pagenum>Heading & Page Ò<$paratext>Ó on page<$pagenum>See Heading & Page%See Ò<$paratext>Ó on page<$pagenum>.Table & Page7Table<$paranumonly>, Ò<$paratext>,Ó on page<$pagenum>Number Only<$paranumonly>MATLAB2MATLAB kkllmmnnooAppAqqrrssAt‚bAvvHeadings€HTML? 8II. #I. s9*> (1)C(2)H`e(3)j(4)7 •ž(5)£Å'ÊÑÞ(6)ä(7)III. 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(10) (11)#$(12))2- (13)7 8(14)=>(15) ëC^_(16)de(17)jm(18)rs(19)x{*¥ €**¥ ‰**¥ *IV. ¡¨ª'«'°'¹'ô¾¿''Ä''Ø''Ù'Þß(20)á(21)ã''ä'(22)ù'þÿ''''(23)(24)(BG(25)L(26)ù!û!ý!ÿ%%%%& %! !"!$%&%(%*%,%.%I%K%M%O%Q%S%U%W%Y%[%]%_%a% c%e%g%i%k%Ÿ!¡!£!¥!§%©%«%%¯%±%³%µ%Î!Ð!Ò!Ô%Ö%Ø%Ú%Ü%Þ%÷!ù!û!ý!ÿ%%%%% %%%% %'%%)%%/%%2%4%6%8%:%<%[%%@%B%D%F%H%J%L%I%%P%R%T%V%X%Z%\%7%%`%b%d%f%h%9%%;%%y%%K%%M%%‹%%]%%_%%%%¦!¨!ª!¬%®%°%²%´%¶%Ç!É!Ë!Í!Ï!Ñ%Ó%Õ%×%Ù%Û%Ý%ß%á%ã%å%ç%é%ë%í%%%%% %"%$%&%(%*%,%.%0%2%4%6%8%:%<%>%@%B%D%F%H%J%L%N%P%R%T%V%X%Z%\%^%`%b%d%f%h%j%l%n%p%r%t%v%x%z%|%~%€%‚%„%†%ˆ%Š%Œ%Ž%%’%”%–%˜%š%œ%ž% %¢%¤%¦%¨%ª%¬%®%°%²%´%¶%¸%º%¼%¾%À%Â%Ä%Æ%È%Ê%Ì%Î%Ð%Ò%Ô%Ö%Ø%Ú%Ü%Þ%à%â%ä%æ%è%ê%ì%î%ð%ò%ô%ö%ø%ú%ü%þ%%%%%% %%%%%%%%%%% %"%$%&%(%*%,%.%0%2%4%6%?%%A%%c%%e%%·%%¹%%+%%-%%%%%®%°)²%´%¶(¸%I'‚hÂdÃ’kpÂdÃ“qs ŸƒÛmÁø| ŸŸƒÛmÁø||'graphicsdsp1FrameHeader.eps0001FRAMEPSFMACP0001¯üîHÂ$Àko»! € €Àvy,¨i@ Àvy,¨iÀvy,¨i¼<––” ÿ'equal[times[char[x],id[char[t]]],times[over[num[1.0000000000000000,"1"],times[num[2.0000000000000000,"2"],char[pi]]],int[(*n*)times[char[X],id[times[char[j],char[Omega]]],indexes[1,0,char[e],times[char[j],char[Omega],char[t]]]],minus[char[infty]],char[infty]],char[d],char[t]]]ÂdÃ?HHÁÔÂˆ@ HHÁÔÂˆÂ‚&ÄH•FÍ’†¢•FÍ h" Ž)q„ÀÀAöà (uNote that the complex exponential # in Eq. %3$ has the same role as the complex exponential ' in Eq. ˆ&>ÀVàQ HZ)1(. Now letÕs consider the expression + for the frequencies , and -: ‘œ–„ÀÀw£% h. ‰@„ÀÀ‘£% (oIn other words, the discrete-time frequency variable is periodic with period /. As a result, we only evaluÀ¡£% jate the IDTFT computation over a frequency region that is arbitrary but of extent 0: This periodicity ‡ƒS‚cÀ¯æx@@occurs because the time variable €nl is always integer. Žââ‘¡ôÀÈ,l h*1 ‡ƒS‚cÀíQ³ xwhere the single subscript on the integral sign indicates that integration can be performed over €anyl strip of ‰@„ÀÀúôÅ H`2 of extent 3. This occurs because the entire integrand of Eq. 554 is periodic. ‡ƒS‚cÁ8h^The validity of Eqs. 7(3)6 and 9(5)8 can be confirmed by simple substitution: •Ô“5 Á2þ h; •FÍ’†¢Áa˜Ô h:< ‰@„ÀÁ‰_v (€Because €nl and €ll are always integers, the integral in Eq. €>(7)=l is equal to @ when A and zero otherwise. Á™_v iHence the only nonzero term in the outer sum occurs when B, and it evaluates to C. This confirms ‡ƒS‚cÁ§¢ÉH?that Eqs. D(3)? and F(5)E are transform pairs. ‰UTÁÃ[/`Basic DTFT examples ‡ƒS‚cÁÛ‰,`EThe decaying exponential. lWe first consider the simple function !Ž)q„ÀÁó¯ hH "‡ƒS‚cÂY`Substituting directly produces #•FÍ’†¢Â*á hI ‡ƒS‚cÂPÖ`bAs shown in class, the real and imaginary parts can be obtained by rationalizing the denominator: ]$•Ô“5Ânñ¼ hJ HHÁÔÂˆBHHÁÔÂˆtE Žÿðl~ÁÝçsÀxy,ªiA €“Ôt€ÀmdÊ¨iEÀmdÊ¨iÀmdÊ¨i·²e–” ÿ'€@equal[times[char[X],id[times[char[j],char[Omega]]]],int[times[char[x],id[char[t]],indexes[1,0,char[e],minus[times[char[j],char[Omega],char[t]]]],diff[char[t]]],minus[char[infty]],char[infty]]]~ÂP‡ÀodÊªiF € “Ôt €‰ràJ ‰rà‰rà…¹p‰@ÿ'char[Omega]ÀûìpÂE¹›‹ràK €„À€’yQ’y’yŠ<ˆ‰@ÿ'times[char[x],id[char[t]]]À¦0Âa ”yR € „À€ “FéqU“Féq“FéqŠ£Ž)qÿ'7indexes[1,0,char[e],times[char[j],char[Omega],char[t]]]HÂq •F’éqV €„À€h Yh h ´‰@ÿ'-times[char[X],id[times[char[j],char[Omega]]]]Á¹[Âv‰qŸh Z €„À€‰rà]‰rà‰rà…¹p‰@ÿ'char[Omega]À}q@Â†‰q‹rà^ €„À€ŸÂ/’‡ÖbŸÂ/’‡ÖŸÂ/’‡ÖáŽ)qÿ'Btimes[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]ÀzÂ ‰q¡Â/”‡Öc €†^e€ÀolÚ¥ÍogÀolÚ¥ÍoÀolÚ¥Ío¸¶m•FÍÿ'€Vequal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega],char[n]]]]],sum[times[char[x],id[(*i1i*)char[n]],indexes[1,0,char[e],minus[times[char[j],char[omega],char[n]]]]],minus[char[infty]],char[infty]]]~HÀqlÚ§Íoh ’†¢”ˆþéql”ˆþéq”ˆþéq‹DŽ)qÿ'7indexes[1,0,char[e],times[char[j],char[omega],char[n]]]Àá9ƒÀ{Ío–ˆþ’éqm „À“Féqv“Féq“FéqŠ£Ž)qÿ'7indexes[1,0,char[e],times[char[j],char[Omega],char[t]]]Áç^ÕÀ{Ío•F’éqw „ÀÀéÆPÀ¶à–ˆþ’éqƒ !„À”ˆþéq„”ˆþéq”ˆþéq‹DŽ)qÿ'7indexes[1,0,char[e],times[char[j],char[omega],char[n]]]vÀf>‰!vÀf>vÀf>‡»`‰@ÿ'4indexes[0,1,char[omega],num[0.0000000000000000,"0"]]ÁXDNÀ• QvÀ‘f>Š $ˆ&> ÂdÃõ'* ¥Xf>$¥Xf>¥Xf>“¬H‰@ÿ'fplus[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],times[num[2.0000000000000000,"2"],char[pi]]]Á}îÀ• Q§X‘f>Ž !(ˆ&>##ÀÌJ”\–’(ÀÌJ”\–ÀÌJ”\–ÀOæ%‘œ–ÿ'‚7equal[indexes[1,0,char[e],times[char[j],id[plus[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],times[num[2.0000000000000000,"2"],char[pi]]]],char[n]]],times[indexes[1,0,char[e],times[char[j],indexes[0,1,char[omega],num[0.0000000000000000,"0"]],char[n]]],indexes[1,0,char[e],times[char[j],num[2.0000000000000000,"2"],char[pi],char[n]]]],indexes[1,0,char[e],times[char[j],indexes[0,1,char[omega],num[0.0000000000000000,"0"]],char[n]]]]HHÁÔÂˆö "HHÁÔÂˆÂwf®i‚E'8ÊùŽ~-Êù h\ 9‡ƒS‚c²Ìy dThe term on the left contributes a linear phase shift to the DTFT. The quantity inside the parentheŽââŒ–ÀFm fses,`, is sometimes referred to as the Òdiscrete-time sinc functionÓ. It has the following proper‡ƒS‚cÀ\+Ö@ties: :-L2€Àz¼4 hBy LÕHopitalÕs rule, ] ;ŽââŒ–À¢Ñ– hM^ has regularly-occurring zero crossings at _ for all integer €k <ÀÊJŽ hQlThe envelope of a tapers downward as b increases from zero to c =‰UTÀð5ø`1Some properties and additional examples of DTFTs >‡ƒS‚cÁcõ bIn this section we summarize some of the properties of DTFTs along with some additional DTFT exam0ÁJZjples. In some cases, brief proofs were provided in the lecture ... for the most part these proofs are not @6included in these notes in the interests of brevity. ?Á6$`LP1. Linearity. lAs noted in class, the DTFT operation itself is linear. FŽ)q†^eÁR£§ hP2. Time shift. d G“Š‹¿ ÁxŒ% h3P3. Multiplication by a complex exponential. e J•FÍ’†¢Á¥‘ÿ h3E3. DTFT of an impulse. f lfor all €n. K‡ƒS‚cÁË›ô`WE4. DTFT of a constant. lIn similar fashion, working from the right side we obtain O•FÍ’†¢ÁêEÓ hg N‡ƒS‚cÂOÈ mWhile this notation is cumbersome, it merely expresses the fact that a constant in time has a DTFT that is a ‰@„ÀÂòÚ kdelta function in frequency at h and that this function repreats periodically in frequency with period Â-òÚ Hi. L‡ƒS‚cÂF6-`TE5. DTFT of a complex exponential. lUsing Property P3, we can now easily obtain _P•FÍ’†¢Âdà hj. HHÁÔÂˆø"*HHÁÔÂˆE‚?&& Žÿðl~À®ÀŸÌJ–\–“ $+„À%%/p—+/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]Á.ÀÁƒÀ†€ý"'Á.ÀÁƒÀ†€Á.ÀÁƒÀ†€Á2ÁŒÿ'prompt[]Á“4°ÀÐc%/p˜ (-„À))/p›-/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]Á®'ÀÀàc%/pœ +/„À,, Àœž„Ö /Àœž„ÖÀœž„ÖÀ@ŠÎŽââÿ';equal[times[char[x],id[(*i1i*)char[n]]],times[over[num[1.0000000000000000,"1"],times[num[2.0000000000000000,"2"],char[pi]]],int[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]],indexes[1,0,char[e],times[char[j],char[omega],char[n]]],diff[char[omega]]],times[num[2.0000000000000000,"2"],char[pi]]]]]~ÁIŠÀœ „Ö¡ -1‘¡ô..!ˆ‹àº1ˆ‹àˆ‹à…Eð‰@ÿ'char[omega]HÁ9´ÅŠ‹à» /3„À00"/p¾3/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]À~«€Á9´Å/p¿ 15„À22#À´d9¥Íoæ7À´d9¥ÍoÀ´d9¥ÍoÀ[2•FÍÿ'‚Uequal[prompt[],sum[(*n*)times[char[x],id[(*i1i*)char[l]],times[(*n*)over[num[1.0000000000000000,"1"],times[num[2.0000000000000000,"2"],char[pi]]],int[(*n*)times[string[" "],sum[(*n*)indexes[1,0,times[indexes[1,0,char[e],minus[times[char[j],char[omega],char[l]]]],char[e]],times[char[j],char[omega],char[n]]],equal[char[l],minus[char[infty]]],char[infty]],diff[char[omega]]],times[num[2.0000000000000000,"2"],char[pi]]]]],equal[char[l],minus[char[infty]]],char[infty]]]~Ád›*Áµ¨¶Ýá 37“5 66+†p;ÁÕ¥Íoã5†p;ÁÕ¥Ío†p;ÁÕ¥ÍoÀ‡Ú•Ôÿ'„Dequal[times[char[x],id[(*i1i*)char[n]]],times[over[num[1.0000000000000000,"1"],times[num[2.0000000000000000,"2"],char[pi]]],int[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]],indexes[1,0,char[e],times[char[j],char[omega],char[n]]],diff[char[omega]]],times[num[2.0000000000000000,"2"],char[pi]]]],times[over[num[1.0000000000000000,"1"],times[num[2.0000000000000000,"2"],char[pi]]],int[(*n*)times[string[" "],sum[(*n*)times[char[x],id[(*i1i*)char[l]]],equal[char[l],minus[char[infty]]],char[infty]],indexes[1,0,times[indexes[1,0,char[e],minus[times[char[j],char[omega],char[l]]]],char[e]],times[char[j],char[omega],char[n]]],diff[char[omega]]],times[num[2.0000000000000000,"2"],char[pi]]]]]~Á”RÀ¶d9§Íoç 59’†¢44,/pî9/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]ÁxFÐÁÈv/pï 7;„À880š¿ò;š¿š¿Ž_€‰@ÿ'equal[char[n],char[l]]Á¤€`ÁÈvœ¿ó 9=„À::1š¿ö=š¿š¿Ž_€‰@ÿ'equal[char[l],char[n]]ÁM 4ÁØvœ¿÷ ;?„À<<2”ê°ú?”ê°”ê°‹uX‰@ÿ'!times[char[x],id[(*i1i*)char[n]]]ÁÀ—ÀÁØv–ê°û =A„À>>3Àt4éqAÀt4éqÀt4éq»Ž)qÿ'€Àv4’éq ?F„À@@8ÁQ.õ¥ÍoFÁQ.õ¥ÍoÁQ.õ¥ÍoÀ©—z•FÍÿ'„>equal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]],sum[times[char[x],id[(*i1i*)char[n]],indexes[1,0,char[e],minus[times[char[j],char[omega],char[n]]]]],equal[char[n],minus[char[infty]]],char[infty]],sum[times[indexes[1,0,char[alpha],char[n]],indexes[1,0,char[e],minus[times[char[j],char[omega],char[n]]]]],equal[char[n],num[0.0000000000000000,"0"]],char[infty]],sum[indexes[1,0,id[times[char[alpha],indexes[1,0,char[e],minus[times[char[j],char[omega]]]]]],char[n]],equal[char[n],num[0.0000000000000000,"0"]],char[infty]],over[num[1.0000000000000000,"1"],plus[num[1.0000000000000000,"1"],minus[indexes[1,0,times[char[alpha],char[e]],times[minus[(*n*)char[(*n*)j]],char[omega]]]]]]]ÂdÃÉEƒ/HHÁÔÂˆÊ CHHÁÔÂˆÂc¼˜JgE%‡ƒS‚c‡ƒS`%By comparison of the terms we obtain &ŽÁ‘¡RŸ§| hK '‡ƒS‚cÀDÌ!`and (ŽÁ‘¡RÀ\ðJ hL )Ú©Ž~-À‹lE h9Because M and N for any complex variable €Z, *“ªˆ•uÀ´”ú hO +‡ƒS‚cÀÝÝ`Hence ,ŽÁ”RŠÀõ² hP -‡ƒS‚cÁ‡ã`and .ÊùŽ~-Á7µî hS 0Ž)q†^eÁ`]Œ (ZWe note that T is an even function of U, V is odd, W is even and X is ‡ƒS‚cÁp?D@1odd, at least for this particular time function. 1Á†%©`XE2. The finite-duration pulse. lWe next consider the finite duration causal pulse, 2•1e’ä™Á¤º hY 3‡ƒS‚cÁË"`"This sum is also easily obtained: 4•FÍ’†¢ÁéËë hZ 5‡ƒS‚cÂÕà mPlease note that we used the relation for the finite sum of exponentials discussed in recition in developing Â¼E@fthe last result. This expression can be further simplified by balancing the terms in the parentheses: 6•Ô“5Â:¡+ h[ ]7‡ƒS‚cÂaY†`or HHÁÔÂˆÌCƒ/HHÁÔÂˆ'DD Žÿðl~Â\¼ÁS.õ§Ío AH’†¢BB9‚Á»m(¦¶ÜHÁ½©(¦¶Ü‚Á»m(¦¶ÜÀßÔ”•Ôÿ'ˆSequal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]],over[num[1.0000000000000000,"1"],plus[num[1.0000000000000000,"1"],minus[times[char[alpha],indexes[1,0,char[e],minus[times[char[j],char[omega]]]]]]]],times[id[over[(*n*)num[1.0000000000000000,"1"],plus[num[1.0000000000000000,"1"],minus[times[char[alpha],indexes[1,0,char[e],minus[times[char[j],char[omega]]]]]]]]],id[over[plus[num[1.0000000000000000,"1"],minus[times[char[alpha],indexes[1,0,char[e],times[char[j],char[omega]]]]]],plus[num[1.0000000000000000,"1"],minus[times[char[alpha],indexes[1,0,char[e],times[char[j],char[omega]]]]]]]]],over[plus[(*n*)num[1.0000000000000000,"1"],minus[times[char[alpha],indexes[1,0,char[e],times[char[j],char[omega]]]]]],plus[num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]],over[plus[(*n*)num[1.0000000000000000,"1"],minus[times[char[alpha],cos[id[char[omega]]]]],minus[times[char[j],char[alpha],sin[id[char[omega]]]]]],plus[num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]]ÀN$„Â¡oèÁ¿©(¨¶Ü F“5GG:ÀÌžbi JÀÌžbiÀÌžbiÀOæŽÁÿ'tequal[times[char[R],char[e],id[(*i1i*)times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]],over[plus[num[1.0000000000000000,"1"],minus[times[char[alpha],cos[id[char[omega]]]]]],plus[num[2.0000000000000000,"2"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]]~ÀXæeÀŸÌ bi! CL‘¡RIID;ÀÍžbi&LÀÍžbiÀÍžbiÀOæ¿ŽÁÿ'Zequal[times[char[I],char[m],id[(*i1i*)times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]],over[minus[times[char[j],char[alpha],sin[id[char[omega]]]]],plus[num[2.0000000000000000,"2"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]]~À–/3ÀŸÍ bi' CJN‘¡RKKD<À‰iÐ“š©+NÀ‰iÐ“š©À‰iÐ“š©ÀE´èÚ©ÿ'€vequal[indexes[1,0,abs[char[Z]],num[2.0000000000000000,"2"]],sqrt[plus[indexes[1,0,id[times[char[R],char[e],id[(*i1i*)char[Z]]]],num[2.0000000000000000,"2"]],indexes[1,0,id[times[char[I],char[m],id[(*i1i*)char[Z]]]],num[2.0000000000000000,"2"]]]]]Àoe ÀÂ‘œÀ‹iÐ•š©, CLP„ÀMMD=ÀZI&/PÀ[,I&ÀZI&®–HÊùÿ'yequal[angle[char[Z]],atan[id[over[times[char[I],char[m],id[(*i1i*)char[Z]]],times[char[R],char[e],id[(*i1i*)char[Z]]]]]]]Á1PÀÂ¡LÀ],ŸI&0 CNRŽ~-OOD>‚Á†¾§ 4RÁˆú§ ‚Á†¾§ ÀÅ}“ªˆÿ'‡requal[indexes[1,0,abs[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]],num[2.0000000000000000,"2"]],plus[indexes[1,0,id[over[plus[(*n*)num[1.0000000000000000,"1"],minus[times[char[alpha],cos[id[char[omega]]]]]],plus[num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]],num[2.0000000000000000,"2"]],indexes[1,0,id[over[minus[times[char[alpha],sin[id[char[omega]]]]],plus[num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]],num[2.0000000000000000,"2"]]],over[(*n*)plus[(*n*)num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]],indexes[1,0,id[plus[(*n*)num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]],num[2.0000000000000000,"2"]]]]ÀlÛiÀèêrÁŠú© 5 CPT•uQQD?À•¡¡:TÀ•¡¡À•¡¡ÀK‹GŽÁÿ'-equal[abs[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]],over[num[1.0000000000000000,"1"],sqrt[plus[(*n*)num[1.0000000000000000,"1"],minus[times[num[2.0000000000000000,"2"],char[alpha],cos[plus[(*n*)id[(*n*)char[omega]],indexes[1,0,char[alpha],num[2.0000000000000000,"2"]]]]]]]]]]~Á.ðïÀ—£¡; CRW”RŠSSD@ÂdÃòvvÀ¶¿I&@WÀÔ¿I&À¶¿I&ÀIj_Êùÿ'€eequal[angle[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]],atan[id[over[minus[times[char[alpha],sin[id[char[omega]]]]],plus[(*n*)num[1.0000000000000000,"1"],minus[times[char[alpha],cos[id[char[omega]]]]]]]]]]~ÁnêõÀ’Ô¿ŸI&A CTYŽ~-VVDCµŠï’‡ÖEYµŠï’‡ÖµŠï’‡Ö›ÅwŽ)qÿ'dtimes[char[R],char[e],id[(*i1i*)times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]]ÀãcÁš4·Šï”‡ÖF CW[†^eXXDDˆ‹àI[ˆ‹àˆ‹à…Eð‰@ÿ'char[omega]ÁohÁŸŒŠ‹àJ CY\„ÀZZDEÁ-l Áš4·ŒO”‡ÖN C[^†^e]]DFµŒO’‡ÖO\µŒO’‡ÖµŒO’‡Ö›Æ'Ž)qÿ'dtimes[char[I],char[m],id[(*i1i*)times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]]Á‡ ÛÁš4§$Ÿ”‡ÖU C\a†^e__DG¥$Ÿ’‡ÖV^¥$Ÿ’‡Ö¥$Ÿ’‡Ö“’OŽ)qÿ'Gabs[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]¨5’‡Ö[a¨5’‡Ö¨5’‡Ö•‡Ž)qÿ'Iangle[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]ÁåîÁš4ª5”‡Ö\ C^c†^e``DHðÀzŸp¦þacÀzÇP¦þðÀzŸp¦þ¾c¨•1eÿ'equal[times[char[x],id[(*i1i*)char[n]]],lparen[(*i2i*)matrix[(*i1in*)2,1,comma[num[1.0000000000000000,"1"],leq[num[0.0000000000000000,"0"],char[n],plus[char[N],minus[num[1.0000000000000000,"1"]]]]],comma[num[0.0000000000000000,"0"],string[" otherwise"]]]]]~Á×ˆ»À|ÇP¨þb Cae’ä™bbDIÁ< D¥ÍogeÁ< D¥ÍoÁ< D¥ÍoÀŸ¢•FÍÿ'…equal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]],sum[times[char[x],id[(*i1i*)char[n]],indexes[1,0,char[e],minus[times[char[j],char[omega],char[n]]]]],equal[char[n],minus[char[infty]]],char[infty]],sum[indexes[1,0,char[e],minus[times[char[j],char[omega],char[n]]]],equal[char[n],num[0.0000000000000000,"0"]],plus[char[N],minus[num[1.0000000000000000,"1"]]]],sum[indexes[1,0,id[indexes[1,0,char[e],minus[times[char[j],char[omega]]]]],char[n]],equal[char[n],num[0.0000000000000000,"0"]],plus[char[N],minus[num[1.0000000000000000,"1"]]]],over[plus[num[1.0000000000000000,"1"],minus[indexes[1,0,char[e],minus[times[char[j],char[omega],char[N]]]]]],plus[num[1.0000000000000000,"1"],minus[indexes[1,0,char[e],times[minus[(*n*)char[(*n*)j]],char[omega]]]]]]]~Â…Á> D§Íoh Ccg’†¢ddDJ‚Á…„Ç¦¶ÜogÁˆÞÇ¦¶Ü‚Á…„Ç¦¶ÜÀÅoc•Ôÿ'Šequal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]],over[plus[num[1.0000000000000000,"1"],minus[indexes[1,0,char[e],minus[times[char[j],char[omega],char[N]]]]]],plus[num[1.0000000000000000,"1"],minus[indexes[1,0,char[e],times[minus[(*n*)char[(*n*)j]],char[omega]]]]]],times[id[over[indexes[1,0,char[e],minus[times[char[j],char[omega],fract[(*n*)char[(*n*)N],num[(*n*)2.0000000000000000,"2"]]]]],indexes[1,0,char[e],minus[times[char[j],fract[(*n*)char[(*n*)omega],num[(*n*)2.0000000000000000,"2"]]]]]]],id[over[plus[indexes[1,0,char[e],times[char[j],char[omega],fract[(*n*)char[(*n*)N],num[(*n*)2.0000000000000000,"2"]]]],minus[indexes[1,0,char[e],minus[times[char[j],char[omega],fract[(*n*)char[(*n*)N],num[(*n*)2.0000000000000000,"2"]]]]]]],plus[indexes[1,0,char[e],times[char[j],fract[(*n*)char[(*n*)omega],num[(*n*)2.0000000000000000,"2"]]]],minus[indexes[1,0,char[e],minus[times[char[j],fract[(*n*)char[(*n*)omega],num[(*n*)2.0000000000000000,"2"]]]]]]]]]],times[indexes[1,0,char[e],minus[times[char[j],char[omega],fract[(*n*)id[plus[(*n*)char[(*n*)N],minus[(*n*)num[(*n*)1.0000000000000000,"1"]]]],num[(*n*)2.0000000000000000,"2"]]]]],id[over[times[num[2.0000000000000000,"2"],char[j],sin[id[fract[(*n*)times[(*n*)char[(*n*)N],char[(*n*)omega]],num[(*n*)2.0000000000000000,"2"]]]]],times[num[2.0000000000000000,"2"],char[j],sin[id[fract[char[omega],num[2.0000000000000000,"2"]]]]]]]]]Àe¶ÕÂmWÁŠÞÇ¨¶Üp Ce“5ffDKÀ§äéI&uiÀ©éI&À§äéI&ÀUtÊùÿ'‚equal[times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]],times[indexes[1,0,char[e],minus[times[char[j],char[omega],fract[(*n*)id[plus[(*n*)char[(*n*)N],minus[(*n*)num[(*n*)1.0000000000000000,"1"]]]],num[(*n*)2.0000000000000000,"2"]]]]],id[over[sin[id[fract[(*n*)times[(*n*)char[(*n*)N],char[(*n*)omega]],num[(*n*)2.0000000000000000,"2"]]]],sin[id[fract[char[omega],num[2.0000000000000000,"2"]]]]]]]]~HÀ«éŸI&v "‚-Ž~-hh&LÀeØä›_Ì}xÀeØä›_ÌÀeØä›_Ì³ìr-Lÿ'€eequal[lim[over[sin[id[fract[(*n*)times[(*n*)char[(*n*)N],char[(*n*)omega]],num[(*n*)2.0000000000000000,"2"]]]],sin[id[fract[char[omega],num[2.0000000000000000,"2"]]]]],rightarrow[char[omega],num[0.0000000000000000,"0"]]],char[N]]H$ÁÔŠlH$ÁÔŠ€€ŽÿðlHÂìwÁÔŠkmHÂìwÁÔŠ€€ŽÿðlH$ÁÔŠlnH$ÁÔŠ€€ŽÿðlHÂìwÁÔŠmoHÂìwÁÔŠ€€ŽÿðlHHÁÔÂˆnpHHÁÔÂˆ€€ŽÿðlHHÁÔÂˆoHHÁÔÂˆ€€ŽÿðlH$ÁÔŠrH$ÁÔŠ€€ŽÿðlHÂìwÁÔŠqsHÂìwÁÔŠ€ € ŽÿðlHHÁÔÂˆrHHÁÔÂˆ€ € ŽÿðlHHÁÔÂˆ€HHÁÔÂˆ€€ Žÿðl$$ÂÂÐó U$$ÂÂÐÀaæcw‚%v_H‡ƒS‚c‡ƒSBm$$ÂÂÐõ U$$ÂÂÐuuŽÿðl€}$³æeÂø U{$³æeÂuW}eHeadings TableÀ»=°À²ŽèÀgØä_Ì~ "‚-‚*2€jj&M€}À³æeÂú UÀ³æeÂuW~e€}Á ³æeÂü UÁ ³æeÂuWe€}$ÀCæelþ Uw|$ÀCæelu!W€ eHeading Level€}ÀÀCæeÀ U{}ÀÀCæeÀu!W€ eParagraph Format€}Á ÀCæeÀ U|~Á ÀCæeÀu!W€ e Comments€}$ÀSæel U}$ÀSæelu"W€ e3€}ÀÀSæeÀ U~€ÀÀSæeÀŽÿþu"W€ ‡UT‰UTeCellBodyNum€}Á ÀSæeÀ U‚&Á ÀSæeÀu"W€ eÂdÃ€€H$ÁÔŠùH$ÁÔŠ‰ÿÿkW†ªª†ªªdHÂìwÁÔŠúHÂìwÁÔŠ‰ÿÿlW†ªª†ªªdH$ÁÔŠûH$ÁÔŠ‰ÿÿmW†ªª†ªªdHÂìwÁÔŠüHÂìwÁÔŠ‰ÿÿnW†ªª†ªªdHHÁÔÂˆýHHÁÔÂˆˆÿÿo_†Ôz‚+…†ÔzdHHÁÔÂˆþHHÁÔÂˆˆÿÿp_†Ôz‚+…†ÔzdH$ÁÔŠÿH$ÁÔŠ‰ÿÿqW†ªª†ªªh018-791 Lecture 3 Notes-Q6R-Fall, 2005HÂìwÁÔŠHÂìwÁÔŠ‰ÿÿrW†ªª†ªª`HHÁÔÂˆHHÁÔÂˆˆÿÿs_†Ôz‚+…†Ôz`ÂdÃ€€H—µ£³Rò‹ €€ H—µ£³Rò‹ H RH R!FootnoteHÀr3À@‹ €€€HÀr3À@‹ HÀz·¯HÀz·¯!Single LineH§À´Žÿð€€ €€€Footnote ‚ À€ ‚ À ‚ À™‚ HÀ¨3ÀD‹ €€€HÀ¨3ÀD‹ HÀ°·¯HÀ°·¯!Double LineHÀ¹ÿðÁÔŒ€€€€€Double LineÁÔ €€ÁÔÁÔ„ÁÔ €€„ÁÔ„ÁÔ„HÀ…ÿúÁÔŠ€€€€€Single LineÁÔ€ÁÔÁÔHZÀ´…ÿð €€€ TableFootnoteÀE¸ÏÀGyûÀR«‹ €€ÀE¸ÏÀGyûÀR«‹ ÀE¸ÏÀPwÀE¸ÏÀPw! TableFootnoteÂdÃttHHÁÔÂˆ€HHÁÔÂˆÂmGt/‰vÈ‚Þ¸À€æƒ`# NOTES FOR 18-791 LECTURES 3 and 4 ‚|À™< h>Introduction to Discrete-Time Fourier Transforms (DTFTs) ‡ƒS‚cÀ¯ž`$Distributed: lSeptember 8, 2005 ÀÅ„s pNotes: lThis handout contains in brief outline form the lecture notes used for 18-791 lectures Number 3 and 2ÀÑjØg4, presented by videotape on September 6 and 8. Because the notes were transcribed some time after the jlecture was taped, there may be some minor differences between these notes and what is seen on the video. eAlso, topics have been merged across the two lectures, so the order of presentation may vary somwhat @from what is in the lecture. ‰UTÁÖm` Introduction ‡ƒS‚cÁ)j gLast week we talked about the time domain behavior of discrete-time signals and systems. Today we will 0Á4êÏibegin (or more accurately we will begin to review) how we characterize discrete-time signals and systems @Cin the frequency domain using the discrete-time Fourier transform. ‰UTÁ\‰š`The DTFT and its inverse ‡ƒS‚cÁt·— hProbably the easiest way to introduce the discrete-time Fourier transform (DTFT) is through its counter0Á€ühpart, the continuous-time Fourier transform (CTFT). As you will recall, the CTFT and its inverse can be @expressed as: –” “ÔtÁ¬| h& ÁÝå' h ‰@„ÀÂù› (mNote that we use the upper case variable to indicate continuous-time angular frequency in this course. ‡ƒS‚cÂ<înAs we have discused, the first equation (which is actually the inverse CTFT) expresses the fact that a finite-‰@„ÀÂ"à genergy time function can be represented as a weighted linear combination of complex exponentials Ž)q„ÀÂ7Éq g. The second equation (the actual CTFT) tells you how to compute the weights . Note that the ‰@„ÀÂGÉq H,frequencies include all real numbers. ]Ž)q†^eÂf²â h=The DTFT ! is computed very similar fashion to the CTFT: $$ÂÂÐ €$$ÂÂÐÂ©æe€}€€t€w€z‚ €L€P€T€p€l€h€d€`€\€X€C€F€I€7€;€?€€€"€4€1€.€+€(€%_€‡ƒS‚c‡ƒSBm$$ÂÂÐ €$$ÂÂÐ€€ Žÿðl€}$Â2Â €B€$Â2Â#W€eCharacter Macros€}lÂ2Â lÂ2Â#W€e€}ÀüÂ2Â! 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Qi¸`SE6. DTFT of a cosine. lFrom EulerÕs representation of trig functions we obtain R—?x’†¢¾B hk I‰@ˆ&>ÀeÒä (mThis is simply a pair of delta functions of area l occuring at m, repeating periodically with period „ÀÀy9" Hn. DŽ)q†^eÀ˜"“ hP4. Time reversal. o UÀ¸ªi h9We note that if a time function is €evenl, q V h6Similarly, if a time function is €odd, lr W‡ƒS‚cÀó÷ mIn other words, if a time function is even, its DTFT is also even, and if a time function is odd its DTFT is Àþú\@also odd. W SŽ)q†^eÁ†ß hP5. Complex conjugation. p TÁ<µ h:We note that if a time function is €reall, s. X (wThis latter property is referred to as €Hermitian symmetry. lIf t is Hermetian symmetric, its real part is ‡ƒS‚cÁlxCneven, its imaginary part is odd, its magnitude is real, and its phase angle is odd. We note further that if a 0Áx^¨jreal time function is even or odd, both the Hermitian symmetry and the time reversal properties constrain qthe transform. Specifically, if a time function is both real and even, its DTFT is also real and even. If a time @kfunction is real and odd, its DTFT is both imaginary and odd. For example, the sine wave has the transform Y—?x’†¢Á°Íü hu E‡o‚AGÁÖÄ;`P6. ParsevalÕs theorem. Z•FÍ’†¢ÁõLO hv [‡ƒS‚cÂVD qThis relationship is important because the quantity on the left side of the equation is clearly the total energy Â'<©gof the time function. Hence the energy in a frequency band can be obtained by integrating the function _J†^eÂ;ÿ hw over that frequency band. (Be sure to take the negative frequencies into account!) Because of its ÂT¼´ Hephsycial meanining, the function x is sometimes referred to as the €energy density spectrum. \‡ƒS‚cÂnžl mP7. Initial value theorems. lThese trivial-to-prove theorems are sometimes confused with ParsevalÕs theoÂz„Ñ@+rem. They sometimes help with computation. ]]•FÍ’†¢Â™.° hy HHÁÔÂ²$ Ì‚=HHÁÔÂ²$ '‚b‚>‚> ŽÿðlŸ<€Ñ‚AŸ<€Ÿ<€ž@‰@ÿ'.equal[char[omega],num[0.0000000000000000,"0"]]ÀÌÂ\²Ú¡<€Ò "‚<‚C„À‚@‚@&X/pÕ‚C/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]HÂl²Ú/pÖ "‚A‚E„À‚B‚B&YÀ—=¥ÍoÛ‚EÀ—=¥ÍoÀ—=¥ÍoÀLž†•FÍÿ'‚LRarrow[indexes[1,0,char[e],times[char[j],indexes[0,1,char[omega],num[0.0000000000000000,"0"]],char[n]]],times[num[2.0000000000000000,"2"],char[pi],sum[times[char[delta],id[plus[char[omega],minus[times[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],minus[(*n*)times[(*n*)num[(*n*)2.0000000000000000,"2"],char[(*n*)pi],char[(*n*)r]]]]]]]],equal[char[r],minus[char[infty]]],char[infty]]]]~Â—™?À™=§ÍoÜ "‚C’†¢‚D‚D&ZÁLó×§Ææ‚GÁLó×§ÆÁLó×§ÆÀ§yë—?xÿ'…equal[cos[id[times[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],char[n]]]],LRarrow[over[plus[indexes[1,0,char[e],times[char[j],indexes[0,1,char[omega],num[0.0000000000000000,"0"]]]],indexes[1,0,char[e],minus[times[char[j],indexes[0,1,char[omega],num[0.0000000000000000,"0"]],char[n]]]]],num[2.0000000000000000,"2"]],times[char[pi],sum[id[plus[times[char[delta],id[plus[char[omega],minus[times[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],minus[(*n*)times[(*n*)num[(*n*)2.0000000000000000,"2"],char[(*n*)pi],char[(*n*)r]]]]]]]],times[char[delta],id[plus[char[omega],times[indexes[0,1,char[omega],num[0.0000000000000000,"0"]],minus[(*n*)times[(*n*)num[(*n*)2.0000000000000000,"2"],char[(*n*)pi],char[(*n*)r]]]]]]]]],equal[char[r],minus[char[infty]]],char[infty]]]]]~ÀnÌÊÁNó×©Æç ‚=‚I’†¢‚F‚F‚>[‡ €ê‚I‡ €‡ €„„À‰@ÿ'char[pi]Á[xÀ¤’ä‰ €ë ‚=‚G‚K„À‚H‚H‚>\¬§ f>î‚K¬§ f>¬§ f>—SÐ‰@ÿ'Kequal[char[omega],pm[indexes[0,1,char[omega],num[0.0000000000000000,"0"]]]]ÁQ0ÌÀ¤’ä®§ ‘f>ï ‚=‚I‚Mˆ&>‚J‚J‚>]/pò‚M/p/p‡—¸‰@ÿ'+times[num[2.0000000000000000,"2"],char[pi]]HÀ·ù"/pó ‚=‚K‚O„À‚L‚L‚>^ÀOc’‡Öö‚OÀOc’‡ÖÀOc’‡Ö¨±ÖŽ)qÿ'€LRarrow[times[char[x],id[(*i1i*)minus[char[n]]]],times[char[X],id[indexes[1,0,char[e],times[minus[(*n*)char[(*n*)j]],char[omega]]]]]]ÀŸEÀÑù"ÀQc”‡Ö÷ ‚=‚M‚S†^e‚N‚N‚>_ÀO¡_’‡Öû‚QÀO¡_’‡ÖÀO¡_’‡Ö¨Ð¯Ž)qÿ'wLRarrow[times[ast[char[x]],id[(*i1i*)char[n]]],times[ast[char[X]],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]]ÀÃÀÁU]nÀQ¡_”‡Öü ‚=‚T‚W†^e‚P‚P‚>`À¤qÌ’‡Ö‚SÀ¤qÌ’‡ÖÀ¤qÌ’‡ÖÀS8æŽ)qÿ'€gequal[times[char[x],id[(*i1i*)char[n]]],LRarrow[times[char[x],id[(*i1i*)minus[char[n]]]],times[char[X],id[indexes[1,0,char[e],times[char[j],char[omega]]]]]],times[char[X],id[indexes[1,0,char[e],minus[times[char[j],char[omega]]]]]]]ÀôqÀÀò€øÀ¦qÌ”‡Ö ‚=‚O‚T†^e‚R‚R‚>aÀäŸ°ÁÎÀ´Bì”‡Ö 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