Below is the output and image from the DFT R function, found in the Roots_of_unity.R file above.
It produces the identical complex number output from the standard fft() function in R, found in the f hat column.
x=seq(0,1,.01);y=sin(x)
DFT(y)
Real 1 Real 2 f hat Magnitude Amplitude
1 46.390121824 46.390121824 46.3901218+ 0.0000000i 92.7802 0.918616274
2 12.303667079 -15.634741445 -1.6655372-13.9692043i 28.1363 0.278577113
3 6.113549355 -7.569917230 -0.7281839- 6.8417333i 13.7608 0.136245058
4 3.978796427 -5.095993953 -0.5585988- 4.5373952i 9.1433 0.090527731
5 2.891525025 -3.890589791 -0.4995324- 3.3910574i 6.8553 0.067874311
6 2.231091904 -3.175579531 -0.4722438- 2.7033357i 5.4885 0.054342054
7 1.786547471 -2.701415809 -0.4574342- 2.2439816i 4.5803 0.045349127
8 1.466334577 -2.363353442 -0.4485094- 1.9148440i 3.9333 0.038943947
9 1.224266796 -2.109705320 -0.4427193- 1.6669861i 3.4495 0.034153926
10 1.034515440 -1.912017221 -0.4387509- 1.4732663i 3.0744 0.030439811
11 0.881504004 -1.753330577 -0.4359133- 1.3174173i 2.7753 0.027478476
12 0.755279537 -1.622908639 -0.4338146- 1.1890941i 2.5315 0.025064487
13 0.649184314 -1.513622269 -0.4322190- 1.0814033i 2.3292 0.023060988
14 0.558595961 -1.420551752 -0.4309779- 0.9895739i 2.1587 0.021373285
15 0.480204602 -1.340192135 -0.4299938- 0.9101984i 2.0133 0.019933776
16 0.411577647 -1.269978545 -0.4292004- 0.8407781i 1.8880 0.018692910
17 0.350886999 -1.207990610 -0.4285518- 0.7794388i 1.7790 0.017613541
18 0.296732080 -1.152761804 -0.4280149- 0.7247469i 1.6834 0.016667279
19 0.248021598 -1.103152681 -0.4275655- 0.6755871i 1.5990 0.015832055
20 0.203892479 -1.058264353 -0.4271859- 0.6310784i 1.5241 0.015090457
21 0.163653029 -1.017378037 -0.4268625- 0.5905155i 1.4573 0.014428568
22 0.126742264 -0.979911960 -0.4265848- 0.5533271i 1.3973 0.013835131
23 0.092700295 -0.945390059 -0.4263449- 0.5190452i 1.3434 0.013300945
24 0.061146402 -0.913418886 -0.4261362- 0.4872826i 1.2947 0.012818421
25 0.031762574 -0.883670301 -0.4259539- 0.4577164i 1.2505 0.012381243
26 0.004280983 -0.855868337 -0.4257937- 0.4300747i 1.2104 0.011984116
27 -0.021525666 -0.829779098 -0.4256524- 0.4041267i 1.1739 0.011622571
28 -0.045851644 -0.805202919 -0.4255273- 0.3796756i 1.1406 0.011292808
29 -0.068864110 -0.781968196 -0.4254162- 0.3565520i 1.1102 0.010991584
30 -0.090707806 -0.759926504 -0.4253172- 0.3346093i 1.0823 0.010716112
31 -0.111508813 -0.738948684 -0.4252287- 0.3137199i 1.0569 0.010463987
32 -0.131377592 -0.718921697 -0.4251496- 0.2937721i 1.0335 0.010233127
33 -0.150411455 -0.699746053 -0.4250788- 0.2746673i 1.0122 0.010021721
34 -0.168696591 -0.681333717 -0.4250152- 0.2563186i 0.9926 0.009828190
35 -0.186309746 -0.663606376 -0.4249581- 0.2386483i 0.9748 0.009651152
36 -0.203319606 -0.646494002 -0.4249068- 0.2215872i 0.9584 0.009489399
37 -0.219787969 -0.629933648 -0.4248608- 0.2050728i 0.9435 0.009341868
38 -0.235770723 -0.613868443 -0.4248196- 0.1890489i 0.9300 0.009207626
39 -0.251318675 -0.598246730 -0.4247827- 0.1734640i 0.9177 0.009085854
40 -0.266478264 -0.583021342 -0.4247498- 0.1582715i 0.9066 0.008975831
41 -0.281292165 -0.568148974 -0.4247206- 0.1434284i 0.8966 0.008876927
42 -0.295799820 -0.553589648 -0.4246947- 0.1288949i 0.8876 0.008788589
43 -0.310037889 -0.539306238 -0.4246721- 0.1146342i 0.8797 0.008710336
44 -0.324040656 -0.525264064 -0.4246524- 0.1006117i 0.8728 0.008641752
45 -0.337840385 -0.511430530 -0.4246355- 0.0867951i 0.8668 0.008582477
46 -0.351467631 -0.497774797 -0.4246212- 0.0731536i 0.8618 0.008532209
47 -0.364951531 -0.484267496 -0.4246095- 0.0596580i 0.8576 0.008490694
48 -0.378320061 -0.470880463 -0.4246003- 0.0462802i 0.8542 0.008457723
49 -0.391600277 -0.457586495 -0.4245934- 0.0329931i 0.8517 0.008433135
50 -0.404818538 -0.444359126 -0.4245888- 0.0197703i 0.8501 0.008416809
51 -0.418000717 -0.431172409 -0.4245866- 0.0065858i 0.0000 0.000000000
52 -0.431172409 -0.418000717 -0.4245866+ 0.0065858i 0.0000 0.000000000
53 -0.444359126 -0.404818538 -0.4245888+ 0.0197703i 0.0000 0.000000000
54 -0.457586495 -0.391600277 -0.4245934+ 0.0329931i 0.0000 0.000000000
55 -0.470880463 -0.378320061 -0.4246003+ 0.0462802i 0.0000 0.000000000
56 -0.484267496 -0.364951531 -0.4246095+ 0.0596580i 0.0000 0.000000000
57 -0.497774797 -0.351467631 -0.4246212+ 0.0731536i 0.0000 0.000000000
58 -0.511430530 -0.337840385 -0.4246355+ 0.0867951i 0.0000 0.000000000
59 -0.525264064 -0.324040656 -0.4246524+ 0.1006117i 0.0000 0.000000000
60 -0.539306238 -0.310037889 -0.4246721+ 0.1146342i 0.0000 0.000000000
61 -0.553589648 -0.295799820 -0.4246947+ 0.1288949i 0.0000 0.000000000
62 -0.568148974 -0.281292165 -0.4247206+ 0.1434284i 0.0000 0.000000000
63 -0.583021342 -0.266478264 -0.4247498+ 0.1582715i 0.0000 0.000000000
64 -0.598246730 -0.251318675 -0.4247827+ 0.1734640i 0.0000 0.000000000
65 -0.613868443 -0.235770723 -0.4248196+ 0.1890489i 0.0000 0.000000000
66 -0.629933648 -0.219787969 -0.4248608+ 0.2050728i 0.0000 0.000000000
67 -0.646494002 -0.203319606 -0.4249068+ 0.2215872i 0.0000 0.000000000
68 -0.663606376 -0.186309746 -0.4249581+ 0.2386483i 0.0000 0.000000000
69 -0.681333717 -0.168696591 -0.4250152+ 0.2563186i 0.0000 0.000000000
70 -0.699746053 -0.150411455 -0.4250788+ 0.2746673i 0.0000 0.000000000
71 -0.718921697 -0.131377592 -0.4251496+ 0.2937721i 0.0000 0.000000000
72 -0.738948684 -0.111508813 -0.4252287+ 0.3137199i 0.0000 0.000000000
73 -0.759926504 -0.090707806 -0.4253172+ 0.3346093i 0.0000 0.000000000
74 -0.781968196 -0.068864110 -0.4254162+ 0.3565520i 0.0000 0.000000000
75 -0.805202919 -0.045851644 -0.4255273+ 0.3796756i 0.0000 0.000000000
76 -0.829779098 -0.021525666 -0.4256524+ 0.4041267i 0.0000 0.000000000
77 -0.855868337 0.004280983 -0.4257937+ 0.4300747i 0.0000 0.000000000
78 -0.883670301 0.031762574 -0.4259539+ 0.4577164i 0.0000 0.000000000
79 -0.913418886 0.061146402 -0.4261362+ 0.4872826i 0.0000 0.000000000
80 -0.945390059 0.092700295 -0.4263449+ 0.5190452i 0.0000 0.000000000
81 -0.979911960 0.126742264 -0.4265848+ 0.5533271i 0.0000 0.000000000
82 -1.017378037 0.163653029 -0.4268625+ 0.5905155i 0.0000 0.000000000
83 -1.058264353 0.203892479 -0.4271859+ 0.6310784i 0.0000 0.000000000
84 -1.103152681 0.248021598 -0.4275655+ 0.6755871i 0.0000 0.000000000
85 -1.152761804 0.296732080 -0.4280149+ 0.7247469i 0.0000 0.000000000
86 -1.207990610 0.350886999 -0.4285518+ 0.7794388i 0.0000 0.000000000
87 -1.269978545 0.411577647 -0.4292004+ 0.8407781i 0.0000 0.000000000
88 -1.340192135 0.480204602 -0.4299938+ 0.9101984i 0.0000 0.000000000
89 -1.420551752 0.558595961 -0.4309779+ 0.9895739i 0.0000 0.000000000
90 -1.513622269 0.649184314 -0.4322190+ 1.0814033i 0.0000 0.000000000
91 -1.622908639 0.755279537 -0.4338146+ 1.1890941i 0.0000 0.000000000
92 -1.753330577 0.881504004 -0.4359133+ 1.3174173i 0.0000 0.000000000
93 -1.912017221 1.034515440 -0.4387509+ 1.4732663i 0.0000 0.000000000
94 -2.109705320 1.224266796 -0.4427193+ 1.6669861i 0.0000 0.000000000
95 -2.363353442 1.466334577 -0.4485094+ 1.9148440i 0.0000 0.000000000
96 -2.701415809 1.786547471 -0.4574342+ 2.2439816i 0.0000 0.000000000
97 -3.175579531 2.231091904 -0.4722438+ 2.7033357i 0.0000 0.000000000
98 -3.890589791 2.891525025 -0.4995324+ 3.3910574i 0.0000 0.000000000
99 -5.095993953 3.978796427 -0.5585988+ 4.5373952i 0.0000 0.000000000
100 -7.569917230 6.113549355 -0.7281839+ 6.8417333i 0.0000 0.000000000
101 -15.634741445 12.303667079 -1.6655372+13.9692043i 0.0000 0.000000000The output from the Roots.of.unity R function will consist of the following image and complex numbers along with their simultaneous reals.
Roots.of.unity(39)
$`Simultaneous Reals`
Real 1 Real 2
1 1.00000000 1.00000000
2 0.82663898 1.14746154
3 0.63186845 1.26520444
4 0.42073285 1.35017920
5 0.19870050 1.40018503
6 -0.02847809 1.41392680
7 -0.25491912 1.39104861
8 -0.47475787 1.33214300
9 -0.68230065 1.23873558
10 -0.87217219 1.11324555
11 -1.03945494 0.95892306
12 -1.17981635 0.77976496
13 -1.28962113 0.58041136
14 -1.36602540 0.36602540
15 -1.40705034 0.14215959
16 -1.41163341 -0.08538809
17 -1.37965591 -0.31072426
18 -1.31194605 -0.52801283
19 -1.21025748 -0.73162615
20 -1.07722388 -0.91629074
21 -0.91629074 -1.07722388
22 -0.73162615 -1.21025748
23 -0.52801283 -1.31194605
24 -0.31072426 -1.37965591
25 -0.08538809 -1.41163341
26 0.14215959 -1.40705034
27 0.36602540 -1.36602540
28 0.58041136 -1.28962113
29 0.77976496 -1.17981635
30 0.95892306 -1.03945494
31 1.11324555 -0.87217219
32 1.23873558 -0.68230065
33 1.33214300 -0.47475787
34 1.39104861 -0.25491912
35 1.41392680 -0.02847809
36 1.40018503 0.19870050
37 1.35017920 0.42073285
38 1.26520444 0.63186845
39 1.14746154 0.82663898
$`Complex Points`
[1] 1.0000000+0.0000000i 0.9870503+0.1604113i 0.9485364+0.3166680i 0.8854560+0.4647232i
[5] 0.7994428+0.6007423i 0.6927244+0.7212024i 0.5680647+0.8229839i 0.4286926+0.9034504i
[9] 0.2782175+0.9605181i 0.1205367+0.9927089i -0.0402659+0.9991890i -0.2000257+0.9797907i
[13] -0.3546049+0.9350162i -0.5000000+0.8660254i -0.6324454+0.7746050i -0.7485107+0.6631227i
[17] -0.8451901+0.5344658i -0.9199794+0.3919666i -0.9709418+0.2393157i -0.9967573+0.0804666i
[21] -0.9967573-0.0804666i -0.9709418-0.2393157i -0.9199794-0.3919666i -0.8451901-0.5344658i
[25] -0.7485107-0.6631227i -0.6324454-0.7746050i -0.5000000-0.8660254i -0.3546049-0.9350162i
[29] -0.2000257-0.9797907i -0.0402659-0.9991890i 0.1205367-0.9927089i 0.2782175-0.9605181i
[33] 0.4286926-0.9034504i 0.5680647-0.8229839i 0.6927244-0.7212024i 0.7994428-0.6007423i
[37] 0.8854560-0.4647232i 0.9485364-0.3166680i 0.9870503-0.1604113i