Paper V: Polyphase Signal Calculus

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \draw[black!12] (0.65000,0.80000) -- (0.65000,4.60000);
3 \node[font=\sffamily\footnotesize] at (0.65000,0.53000) {$-1/2$};
4 \draw[black!12] (3.57500,0.80000) -- (3.57500,4.60000);
5 \node[font=\sffamily\footnotesize] at (3.57500,0.53000) {$-1/4$};
6 \draw[black!12] (6.50000,0.80000) -- (6.50000,4.60000);
7 \node[font=\sffamily\footnotesize] at (6.50000,0.53000) {$0$};
8 \draw[black!12] (9.42500,0.80000) -- (9.42500,4.60000);
9 \node[font=\sffamily\footnotesize] at (9.42500,0.53000) {$1/4$};
10 \draw[black!12] (12.35000,0.80000) -- (12.35000,4.60000);
11 \node[font=\sffamily\footnotesize] at (12.35000,0.53000) {$1/2$};
12 \draw[black!12] (0.65000,1.32414) -- (12.35000,1.32414);
13 \node[anchor=east,font=\sffamily\footnotesize] at (0.51000,1.32414) {0};
14 \draw[black!12] (0.65000,2.63448) -- (12.35000,2.63448);
15 \node[anchor=east,font=\sffamily\footnotesize] at (0.51000,2.63448) {$1/2$};
16 \draw[black!12] (0.65000,3.94483) -- (12.35000,3.94483);
17 \node[anchor=east,font=\sffamily\footnotesize] at (0.51000,3.94483) {1};
18 \draw[->] (0.65000,0.80000) -- (12.48000,0.80000);
19 \draw[->] (0.65000,0.80000) -- (0.65000,4.73000);
20 \draw[black!60,line width=1.35pt] plot coordinates {(0.65000,1.32414) (3.57500,1.32414) (3.57500,3.94483) (9.42500,3.94483) (9.42500,1.32414) (12.35000,1.32414)};
21 \draw[dashed,line width=.95pt] plot coordinates {(0.65000,1.18855) (0.67340,1.18895) (0.69680,1.19013) (0.72020,1.19209) (0.74360,1.19483) (0.76700,1.19832) (0.79040,1.20256) (0.81380,1.20751) (0.83720,1.21315) (0.86060,1.21945) (0.88400,1.22638) (0.90740,1.23390) (0.93080,1.24198) (0.95420,1.25056) (0.97760,1.25961) (1.00100,1.26908) (1.02440,1.27891) (1.04780,1.28905) (1.07120,1.29945) (1.09460,1.31005) (1.11800,1.32080) (1.14140,1.33164) (1.16480,1.34250) (1.18820,1.35333) (1.21160,1.36406) (1.23500,1.37464) (1.25840,1.38501) (1.28180,1.39510) (1.30520,1.40486) (1.32860,1.41423) (1.35200,1.42316) (1.37540,1.43159) (1.39880,1.43947) (1.42220,1.44675) (1.44560,1.45338) (1.46900,1.45932) (1.49240,1.46453) (1.51580,1.46897) (1.53920,1.47260) (1.56260,1.47540) (1.58600,1.47734) (1.60940,1.47839) (1.63280,1.47854) (1.65620,1.47778) (1.67960,1.47609) (1.70300,1.47346) (1.72640,1.46991) (1.74980,1.46543) (1.77320,1.46004) (1.79660,1.45374) (1.82000,1.44656) (1.84340,1.43851) (1.86680,1.42964) (1.89020,1.41997) (1.91360,1.40954) (1.93700,1.39840) (1.96040,1.38659) (1.98380,1.37416) (2.00720,1.36117) (2.03060,1.34767) (2.05400,1.33374) (2.07740,1.31945) (2.10080,1.30485) (2.12420,1.29003) (2.14760,1.27506) (2.17100,1.26002) (2.19440,1.24500) (2.21780,1.23008) (2.24120,1.21535) (2.26460,1.20089) (2.28800,1.18679) (2.31140,1.17315) (2.33480,1.16005) (2.35820,1.14758) (2.38160,1.13584) (2.40500,1.12492) (2.42840,1.11491) (2.45180,1.10589) (2.47520,1.09795) (2.49860,1.09118) (2.52200,1.08566) (2.54540,1.08148) (2.56880,1.07870) (2.59220,1.07741) (2.61560,1.07768) (2.63900,1.07957) (2.66240,1.08316) (2.68580,1.08849) (2.70920,1.09562) (2.73260,1.10462) (2.75600,1.11551) (2.77940,1.12835) (2.80280,1.14317) (2.82620,1.16000) (2.84960,1.17885) (2.87300,1.19977) (2.89640,1.22274) (2.91980,1.24779) (2.94320,1.27491) (2.96660,1.30409) (2.99000,1.33533) (3.01340,1.36861) (3.03680,1.40390) (3.06020,1.44118) (3.08360,1.48040) (3.10700,1.52153) (3.13040,1.56452) (3.15380,1.60932) (3.17720,1.65586) (3.20060,1.70408) (3.22400,1.75392) (3.24740,1.80530) (3.27080,1.85813) (3.29420,1.91234) (3.31760,1.96783) (3.34100,2.02452) (3.36440,2.08231) (3.38780,2.14110) (3.41120,2.20078) (3.43460,2.26125) (3.45800,2.32240) (3.48140,2.38413) (3.50480,2.44631) (3.52820,2.50884) (3.55160,2.57161) (3.57500,2.63448) (3.59840,2.69736) (3.62180,2.76012) (3.64520,2.82265) (3.66860,2.88484) (3.69200,2.94656) (3.71540,3.00771) (3.73880,3.06819) (3.76220,3.12787) (3.78560,3.18665) (3.80900,3.24444) (3.83240,3.30113) (3.85580,3.35663) (3.87920,3.41084) (3.90260,3.46367) (3.92600,3.51504) (3.94940,3.56488) (3.97280,3.61311) (3.99620,3.65965) (4.01960,3.70444) (4.04300,3.74743) (4.06640,3.78856) (4.08980,3.82779) (4.11320,3.86506) (4.13660,3.90036) (4.16000,3.93363) (4.18340,3.96488) (4.20680,3.99406) (4.23020,4.02118) (4.25360,4.04622) (4.27700,4.06920) (4.30040,4.09011) (4.32380,4.10897) (4.34720,4.12580) (4.37060,4.14061) (4.39400,4.15345) (4.41740,4.16435) (4.44080,4.17334) (4.46420,4.18048) (4.48760,4.18581) (4.51100,4.18939) (4.53440,4.19129) (4.55780,4.19155) (4.58120,4.19026) (4.60460,4.18749) (4.62800,4.18330) (4.65140,4.17778) (4.67480,4.17101) (4.69820,4.16308) (4.72160,4.15406) (4.74500,4.14404) (4.76840,4.13312) (4.79180,4.12138) (4.81520,4.10892) (4.83860,4.09582) (4.86200,4.08218) (4.88540,4.06808) (4.90880,4.05362) (4.93220,4.03888) (4.95560,4.02396) (4.97900,4.00894) (5.00240,3.99391) (5.02580,3.97894) (5.04920,3.96412) (5.07260,3.94952) (5.09600,3.93522) (5.11940,3.92129) (5.14280,3.90780) (5.16620,3.89481) (5.18960,3.88238) (5.21300,3.87057) (5.23640,3.85942) (5.25980,3.84899) (5.28320,3.83932) (5.30660,3.83045) (5.33000,3.82241) (5.35340,3.81523) (5.37680,3.80893) (5.40020,3.80353) (5.42360,3.79905) (5.44700,3.79550) (5.47040,3.79288) (5.49380,3.79119) (5.51720,3.79042) (5.54060,3.79057) (5.56400,3.79163) (5.58740,3.79357) (5.61080,3.79637) (5.63420,3.80000) (5.65760,3.80444) (5.68100,3.80965) (5.70440,3.81559) (5.72780,3.82222) (5.75120,3.82950) (5.77460,3.83738) (5.79800,3.84581) (5.82140,3.85473) (5.84480,3.86411) (5.86820,3.87387) (5.89160,3.88396) (5.91500,3.89433) (5.93840,3.90491) (5.96180,3.91564) (5.98520,3.92647) (6.00860,3.93733) (6.03200,3.94816) (6.05540,3.95891) (6.07880,3.96951) (6.10220,3.97992) (6.12560,3.99006) (6.14900,3.99989) (6.17240,4.00935) (6.19580,4.01840) (6.21920,4.02699) (6.24260,4.03506) (6.26600,4.04259) (6.28940,4.04952) (6.31280,4.05582) (6.33620,4.06146) (6.35960,4.06641) (6.38300,4.07064) (6.40640,4.07414) (6.42980,4.07687) (6.45320,4.07884) (6.47660,4.08002) (6.50000,4.08041) (6.52340,4.08002) (6.54680,4.07884) (6.57020,4.07687) (6.59360,4.07414) (6.61700,4.07064) (6.64040,4.06641) (6.66380,4.06146) (6.68720,4.05582) (6.71060,4.04952) (6.73400,4.04259) (6.75740,4.03506) (6.78080,4.02699) (6.80420,4.01840) (6.82760,4.00935) (6.85100,3.99989) (6.87440,3.99006) (6.89780,3.97992) (6.92120,3.96951) (6.94460,3.95891) (6.96800,3.94816) (6.99140,3.93733) (7.01480,3.92647) (7.03820,3.91564) (7.06160,3.90491) (7.08500,3.89433) (7.10840,3.88396) (7.13180,3.87387) (7.15520,3.86411) (7.17860,3.85473) (7.20200,3.84581) (7.22540,3.83738) (7.24880,3.82950) (7.27220,3.82222) (7.29560,3.81559) (7.31900,3.80965) (7.34240,3.80444) (7.36580,3.80000) (7.38920,3.79637) (7.41260,3.79357) (7.43600,3.79163) (7.45940,3.79057) (7.48280,3.79042) (7.50620,3.79119) (7.52960,3.79288) (7.55300,3.79550) (7.57640,3.79905) (7.59980,3.80353) (7.62320,3.80893) (7.64660,3.81523) (7.67000,3.82241) (7.69340,3.83045) (7.71680,3.83932) (7.74020,3.84899) (7.76360,3.85942) (7.78700,3.87057) (7.81040,3.88238) (7.83380,3.89481) (7.85720,3.90780) (7.88060,3.92129) (7.90400,3.93522) (7.92740,3.94952) (7.95080,3.96412) (7.97420,3.97894) (7.99760,3.99391) (8.02100,4.00894) (8.04440,4.02396) (8.06780,4.03888) (8.09120,4.05362) (8.11460,4.06808) (8.13800,4.08218) (8.16140,4.09582) (8.18480,4.10892) (8.20820,4.12138) (8.23160,4.13312) (8.25500,4.14404) (8.27840,4.15406) (8.30180,4.16308) (8.32520,4.17101) (8.34860,4.17778) (8.37200,4.18330) (8.39540,4.18749) (8.41880,4.19026) (8.44220,4.19155) (8.46560,4.19129) (8.48900,4.18939) (8.51240,4.18581) (8.53580,4.18048) (8.55920,4.17334) (8.58260,4.16435) (8.60600,4.15345) (8.62940,4.14061) (8.65280,4.12580) (8.67620,4.10897) (8.69960,4.09011) (8.72300,4.06920) (8.74640,4.04622) (8.76980,4.02118) (8.79320,3.99406) (8.81660,3.96488) (8.84000,3.93363) (8.86340,3.90036) (8.88680,3.86506) (8.91020,3.82779) (8.93360,3.78856) (8.95700,3.74743) (8.98040,3.70444) (9.00380,3.65965) (9.02720,3.61311) (9.05060,3.56488) (9.07400,3.51504) (9.09740,3.46367) (9.12080,3.41084) (9.14420,3.35663) (9.16760,3.30113) (9.19100,3.24444) (9.21440,3.18665) (9.23780,3.12787) (9.26120,3.06819) (9.28460,3.00771) (9.30800,2.94656) (9.33140,2.88484) (9.35480,2.82265) (9.37820,2.76012) (9.40160,2.69736) (9.42500,2.63448) (9.44840,2.57161) (9.47180,2.50884) (9.49520,2.44631) (9.51860,2.38413) (9.54200,2.32240) (9.56540,2.26125) (9.58880,2.20078) (9.61220,2.14110) (9.63560,2.08231) (9.65900,2.02452) (9.68240,1.96783) (9.70580,1.91234) (9.72920,1.85813) (9.75260,1.80530) (9.77600,1.75392) (9.79940,1.70408) (9.82280,1.65586) (9.84620,1.60932) (9.86960,1.56452) (9.89300,1.52153) (9.91640,1.48040) (9.93980,1.44118) (9.96320,1.40390) (9.98660,1.36861) (10.01000,1.33533) (10.03340,1.30409) (10.05680,1.27491) (10.08020,1.24779) (10.10360,1.22274) (10.12700,1.19977) (10.15040,1.17885) (10.17380,1.16000) (10.19720,1.14317) (10.22060,1.12835) (10.24400,1.11551) (10.26740,1.10462) (10.29080,1.09562) (10.31420,1.08849) (10.33760,1.08316) (10.36100,1.07957) (10.38440,1.07768) (10.40780,1.07741) (10.43120,1.07870) (10.45460,1.08148) (10.47800,1.08566) (10.50140,1.09118) (10.52480,1.09795) (10.54820,1.10589) (10.57160,1.11491) (10.59500,1.12492) (10.61840,1.13584) (10.64180,1.14758) (10.66520,1.16005) (10.68860,1.17315) (10.71200,1.18679) (10.73540,1.20089) (10.75880,1.21535) (10.78220,1.23008) (10.80560,1.24500) (10.82900,1.26002) (10.85240,1.27506) (10.87580,1.29003) (10.89920,1.30485) (10.92260,1.31945) (10.94600,1.33374) (10.96940,1.34767) (10.99280,1.36117) (11.01620,1.37416) (11.03960,1.38659) (11.06300,1.39840) (11.08640,1.40954) (11.10980,1.41997) (11.13320,1.42964) (11.15660,1.43851) (11.18000,1.44656) (11.20340,1.45374) (11.22680,1.46004) (11.25020,1.46543) (11.27360,1.46991) (11.29700,1.47346) (11.32040,1.47609) (11.34380,1.47778) (11.36720,1.47854) (11.39060,1.47839) (11.41400,1.47734) (11.43740,1.47540) (11.46080,1.47260) (11.48420,1.46897) (11.50760,1.46453) (11.53100,1.45932) (11.55440,1.45338) (11.57780,1.44675) (11.60120,1.43947) (11.62460,1.43159) (11.64800,1.42316) (11.67140,1.41423) (11.69480,1.40486) (11.71820,1.39510) (11.74160,1.38501) (11.76500,1.37464) (11.78840,1.36406) (11.81180,1.35333) (11.83520,1.34250) (11.85860,1.33164) (11.88200,1.32080) (11.90540,1.31005) (11.92880,1.29945) (11.95220,1.28905) (11.97560,1.27891) (11.99900,1.26908) (12.02240,1.25961) (12.04580,1.25056) (12.06920,1.24198) (12.09260,1.23390) (12.11600,1.22638) (12.13940,1.21945) (12.16280,1.21315) (12.18620,1.20751) (12.20960,1.20256) (12.23300,1.19832) (12.25640,1.19483) (12.27980,1.19209) (12.30320,1.19013) (12.32660,1.18895) (12.35000,1.18855)};
22 \draw[line width=.8pt] plot coordinates {(0.65000,1.37608) (0.67340,1.37502) (0.69680,1.37192) (0.72020,1.36687) (0.74360,1.36009) (0.76700,1.35185) (0.79040,1.34247) (0.81380,1.33234) (0.83720,1.32185) (0.86060,1.31142) (0.88400,1.30148) (0.90740,1.29243) (0.93080,1.28463) (0.95420,1.27840) (0.97760,1.27400) (1.00100,1.27162) (1.02440,1.27135) (1.04780,1.27322) (1.07120,1.27717) (1.09460,1.28305) (1.11800,1.29063) (1.14140,1.29962) (1.16480,1.30966) (1.18820,1.32037) (1.21160,1.33131) (1.23500,1.34204) (1.25840,1.35213) (1.28180,1.36117) (1.30520,1.36878) (1.32860,1.37465) (1.35200,1.37852) (1.37540,1.38021) (1.39880,1.37964) (1.42220,1.37680) (1.44560,1.37179) (1.46900,1.36478) (1.49240,1.35603) (1.51580,1.34587) (1.53920,1.33470) (1.56260,1.32295) (1.58600,1.31110) (1.60940,1.29961) (1.63280,1.28896) (1.65620,1.27960) (1.67960,1.27191) (1.70300,1.26624) (1.72640,1.26284) (1.74980,1.26190) (1.77320,1.26349) (1.79660,1.26758) (1.82000,1.27407) (1.84340,1.28273) (1.86680,1.29324) (1.89020,1.30523) (1.91360,1.31822) (1.93700,1.33173) (1.96040,1.34520) (1.98380,1.35809) (2.00720,1.36987) (2.03060,1.38003) (2.05400,1.38812) (2.07740,1.39377) (2.10080,1.39669) (2.12420,1.39669) (2.14760,1.39369) (2.17100,1.38774) (2.19440,1.37901) (2.21780,1.36777) (2.24120,1.35440) (2.26460,1.33940) (2.28800,1.32332) (2.31140,1.30679) (2.33480,1.29046) (2.35820,1.27502) (2.38160,1.26111) (2.40500,1.24935) (2.42840,1.24031) (2.45180,1.23445) (2.47520,1.23212) (2.49860,1.23356) (2.52200,1.23884) (2.54540,1.24791) (2.56880,1.26056) (2.59220,1.27640) (2.61560,1.29494) (2.63900,1.31553) (2.66240,1.33742) (2.68580,1.35978) (2.70920,1.38170) (2.73260,1.40226) (2.75600,1.42055) (2.77940,1.43569) (2.80280,1.44687) (2.82620,1.45341) (2.84960,1.45476) (2.87300,1.45055) (2.89640,1.44057) (2.91980,1.42488) (2.94320,1.40372) (2.96660,1.37757) (2.99000,1.34715) (3.01340,1.31339) (3.03680,1.27743) (3.06020,1.24058) (3.08360,1.20431) (3.10700,1.17021) (3.13040,1.13992) (3.15380,1.11512) (3.17720,1.09750) (3.20060,1.08865) (3.22400,1.09007) (3.24740,1.10309) (3.27080,1.12886) (3.29420,1.16827) (3.31760,1.22197) (3.34100,1.29030) (3.36440,1.37329) (3.38780,1.47065) (3.41120,1.58178) (3.43460,1.70576) (3.45800,1.84138) (3.48140,1.98714) (3.50480,2.14133) (3.52820,2.30202) (3.55160,2.46713) (3.57500,2.63448) (3.59840,2.80183) (3.62180,2.96694) (3.64520,3.12763) (3.66860,3.28182) (3.69200,3.42759) (3.71540,3.56320) (3.73880,3.68718) (3.76220,3.79832) (3.78560,3.89568) (3.80900,3.97867) (3.83240,4.04700) (3.85580,4.10070) (3.87920,4.14011) (3.90260,4.16588) (3.92600,4.17890) (3.94940,4.18031) (3.97280,4.17146) (3.99620,4.15384) (4.01960,4.12905) (4.04300,4.09876) (4.06640,4.06465) (4.08980,4.02839) (4.11320,3.99154) (4.13660,3.95558) (4.16000,3.92182) (4.18340,3.89140) (4.20680,3.86525) (4.23020,3.84409) (4.25360,3.82839) (4.27700,3.81842) (4.30040,3.81420) (4.32380,3.81555) (4.34720,3.82209) (4.37060,3.83328) (4.39400,3.84841) (4.41740,3.86670) (4.44080,3.88727) (4.46420,3.90919) (4.48760,3.93154) (4.51100,3.95343) (4.53440,3.97402) (4.55780,3.99256) (4.58120,4.00841) (4.60460,4.02105) (4.62800,4.03012) (4.65140,4.03541) (4.67480,4.03684) (4.69820,4.03452) (4.72160,4.02865) (4.74500,4.01961) (4.76840,4.00786) (4.79180,3.99395) (4.81520,3.97850) (4.83860,3.96217) (4.86200,3.94564) (4.88540,3.92956) (4.90880,3.91456) (4.93220,3.90120) (4.95560,3.88996) (4.97900,3.88122) (5.00240,3.87527) (5.02580,3.87228) (5.04920,3.87227) (5.07260,3.87519) (5.09600,3.88084) (5.11940,3.88893) (5.14280,3.89910) (5.16620,3.91087) (5.18960,3.92377) (5.21300,3.93724) (5.23640,3.95074) (5.25980,3.96374) (5.28320,3.97572) (5.30660,3.98624) (5.33000,3.99490) (5.35340,4.00138) (5.37680,4.00548) (5.40020,4.00707) (5.42360,4.00612) (5.44700,4.00273) (5.47040,3.99706) (5.49380,3.98937) (5.51720,3.98000) (5.54060,3.96935) (5.56400,3.95787) (5.58740,3.94601) (5.61080,3.93426) (5.63420,3.92309) (5.65760,3.91294) (5.68100,3.90419) (5.70440,3.89717) (5.72780,3.89216) (5.75120,3.88932) (5.77460,3.88875) (5.79800,3.89045) (5.82140,3.89432) (5.84480,3.90018) (5.86820,3.90780) (5.89160,3.91684) (5.91500,3.92693) (5.93840,3.93766) (5.96180,3.94860) (5.98520,3.95930) (6.00860,3.96935) (6.03200,3.97834) (6.05540,3.98592) (6.07880,3.99180) (6.10220,3.99574) (6.12560,3.99762) (6.14900,3.99735) (6.17240,3.99496) (6.19580,3.99056) (6.21920,3.98433) (6.24260,3.97654) (6.26600,3.96748) (6.28940,3.95754) (6.31280,3.94712) (6.33620,3.93663) (6.35960,3.92649) (6.38300,3.91711) (6.40640,3.90887) (6.42980,3.90209) (6.45320,3.89705) (6.47660,3.89394) (6.50000,3.89289) (6.52340,3.89394) (6.54680,3.89705) (6.57020,3.90209) (6.59360,3.90887) (6.61700,3.91711) (6.64040,3.92649) (6.66380,3.93663) (6.68720,3.94712) (6.71060,3.95754) (6.73400,3.96748) (6.75740,3.97654) (6.78080,3.98433) (6.80420,3.99056) (6.82760,3.99496) (6.85100,3.99735) (6.87440,3.99762) (6.89780,3.99574) (6.92120,3.99180) (6.94460,3.98592) (6.96800,3.97834) (6.99140,3.96935) (7.01480,3.95930) (7.03820,3.94860) (7.06160,3.93766) (7.08500,3.92693) (7.10840,3.91684) (7.13180,3.90780) (7.15520,3.90018) (7.17860,3.89432) (7.20200,3.89045) (7.22540,3.88875) (7.24880,3.88932) (7.27220,3.89216) (7.29560,3.89717) (7.31900,3.90419) (7.34240,3.91294) (7.36580,3.92309) (7.38920,3.93426) (7.41260,3.94601) (7.43600,3.95787) (7.45940,3.96935) (7.48280,3.98000) (7.50620,3.98937) (7.52960,3.99706) (7.55300,4.00273) (7.57640,4.00612) (7.59980,4.00707) (7.62320,4.00548) (7.64660,4.00138) (7.67000,3.99490) (7.69340,3.98624) (7.71680,3.97572) (7.74020,3.96374) (7.76360,3.95074) (7.78700,3.93724) (7.81040,3.92377) (7.83380,3.91087) (7.85720,3.89910) (7.88060,3.88893) (7.90400,3.88084) (7.92740,3.87519) (7.95080,3.87227) (7.97420,3.87228) (7.99760,3.87527) (8.02100,3.88122) (8.04440,3.88996) (8.06780,3.90120) (8.09120,3.91456) (8.11460,3.92956) (8.13800,3.94564) (8.16140,3.96217) (8.18480,3.97850) (8.20820,3.99395) (8.23160,4.00786) (8.25500,4.01961) (8.27840,4.02865) (8.30180,4.03452) (8.32520,4.03684) (8.34860,4.03541) (8.37200,4.03012) (8.39540,4.02105) (8.41880,4.00841) (8.44220,3.99256) (8.46560,3.97402) (8.48900,3.95343) (8.51240,3.93154) (8.53580,3.90919) (8.55920,3.88727) (8.58260,3.86670) (8.60600,3.84841) (8.62940,3.83328) (8.65280,3.82209) (8.67620,3.81555) (8.69960,3.81420) (8.72300,3.81842) (8.74640,3.82839) (8.76980,3.84409) (8.79320,3.86525) (8.81660,3.89140) (8.84000,3.92182) (8.86340,3.95558) (8.88680,3.99154) (8.91020,4.02839) (8.93360,4.06465) (8.95700,4.09876) (8.98040,4.12905) (9.00380,4.15384) (9.02720,4.17146) (9.05060,4.18031) (9.07400,4.17890) (9.09740,4.16588) (9.12080,4.14011) (9.14420,4.10070) (9.16760,4.04700) (9.19100,3.97867) (9.21440,3.89568) (9.23780,3.79832) (9.26120,3.68718) (9.28460,3.56320) (9.30800,3.42759) (9.33140,3.28182) (9.35480,3.12763) (9.37820,2.96694) (9.40160,2.80183) (9.42500,2.63448) (9.44840,2.46713) (9.47180,2.30202) (9.49520,2.14133) (9.51860,1.98714) (9.54200,1.84138) (9.56540,1.70576) (9.58880,1.58178) (9.61220,1.47065) (9.63560,1.37329) (9.65900,1.29030) (9.68240,1.22197) (9.70580,1.16827) (9.72920,1.12886) (9.75260,1.10309) (9.77600,1.09007) (9.79940,1.08865) (9.82280,1.09750) (9.84620,1.11512) (9.86960,1.13992) (9.89300,1.17021) (9.91640,1.20431) (9.93980,1.24058) (9.96320,1.27743) (9.98660,1.31339) (10.01000,1.34715) (10.03340,1.37757) (10.05680,1.40372) (10.08020,1.42488) (10.10360,1.44057) (10.12700,1.45055) (10.15040,1.45476) (10.17380,1.45341) (10.19720,1.44687) (10.22060,1.43569) (10.24400,1.42055) (10.26740,1.40226) (10.29080,1.38170) (10.31420,1.35978) (10.33760,1.33742) (10.36100,1.31553) (10.38440,1.29494) (10.40780,1.27640) (10.43120,1.26056) (10.45460,1.24791) (10.47800,1.23884) (10.50140,1.23356) (10.52480,1.23212) (10.54820,1.23445) (10.57160,1.24031) (10.59500,1.24935) (10.61840,1.26111) (10.64180,1.27502) (10.66520,1.29046) (10.68860,1.30679) (10.71200,1.32332) (10.73540,1.33940) (10.75880,1.35440) (10.78220,1.36777) (10.80560,1.37901) (10.82900,1.38774) (10.85240,1.39369) (10.87580,1.39669) (10.89920,1.39669) (10.92260,1.39377) (10.94600,1.38812) (10.96940,1.38003) (10.99280,1.36987) (11.01620,1.35809) (11.03960,1.34520) (11.06300,1.33173) (11.08640,1.31822) (11.10980,1.30523) (11.13320,1.29324) (11.15660,1.28273) (11.18000,1.27407) (11.20340,1.26758) (11.22680,1.26349) (11.25020,1.26190) (11.27360,1.26284) (11.29700,1.26624) (11.32040,1.27191) (11.34380,1.27960) (11.36720,1.28896) (11.39060,1.29961) (11.41400,1.31110) (11.43740,1.32295) (11.46080,1.33470) (11.48420,1.34587) (11.50760,1.35603) (11.53100,1.36478) (11.55440,1.37179) (11.57780,1.37680) (11.60120,1.37964) (11.62460,1.38021) (11.64800,1.37852) (11.67140,1.37465) (11.69480,1.36878) (11.71820,1.36117) (11.74160,1.35213) (11.76500,1.34204) (11.78840,1.33131) (11.81180,1.32037) (11.83520,1.30966) (11.85860,1.29962) (11.88200,1.29063) (11.90540,1.28305) (11.92880,1.27717) (11.95220,1.27322) (11.97560,1.27135) (11.99900,1.27162) (12.02240,1.27400) (12.04580,1.27840) (12.06920,1.28463) (12.09260,1.29243) (12.11600,1.30148) (12.13940,1.31142) (12.16280,1.32185) (12.18620,1.33234) (12.20960,1.34247) (12.23300,1.35185) (12.25640,1.36009) (12.27980,1.36687) (12.30320,1.37192) (12.32660,1.37502) (12.35000,1.37608)};
23 \draw[fill=black] (3.57500,2.63448) circle (0.065);
24 \draw[fill=black] (9.42500,2.63448) circle (0.065);
25 \node[] at (6.50000,5.03000) {pulse, cutoff $m=5$ (dashed), cutoff $m=15$ (thin solid)};
26 \node[] at (6.50000,0.05000) {$2m+1$ signed frequency bins; marked values are exactly $1/2$};
27 \end{tikzpicture}

Back to source controls

Back to Paper V: Polyphase Signal Calculus