Stem style construct

f9612434-d67b-4d6a-a759-5cba4717c9a4 24 6 5 auto

&[DATE] &[TIME] - &[FILENAME]

Fourier discrete transform open style

data_1 1 0.132 - 2 0.11 - 3 0.045 4 0.0045 5 0.138 - 6 0.467 7 0.5 8 0.78 9 1.342 10 0.993 11 1.105 12 1.119 13 1.18 14 1.143 15 0.655 16 1.008 17 0.771 18 1.208 19 0.938 20 0.783 21 0.734 22 0.763 23 1.044 24 0.754 25 0.797 252 mat :

data_2 26 1.086 27 0.426 28 0.563 29 0.814 30 0.665 31 0.561 - 32 0.429 - 33 0.187 34 0.153 - 35 0.422 - 36 0.127 - 37 0.018 - 38 0.247 - 39 0.422 - 40 0.738 - 41 0.711 - 42 0.863 - 43 0.769 - 44 0.485 - 45 0.79 - 46 0.737 - 47 0.708 - 48 0.965 - 49 0.899 - 50 0.947 - 252 mat :

data_3 51 0.871 - 52 1.138 - 53 1.191 - 54 0.046 - 55 1.016 - 56 1.275 - 57 0.617 - 58 0.908 - 59 1.09 - 60 0.972 - 61 0.594 - 62 0.5 - 63 0.346 - 64 0.191 - 65 0.177 - 66 0.362 - 67 0.547 - 68 0.455 69 0.417 70 0.129 - 71 0.352 72 0.906 73 0.386 74 0.047 - 75 0.214 252 mat :

data_4 76 0.309 77 0.615 78 0.846 79 0.698 80 0.819 81 1.113 82 1.042 83 1.114 84 0.653 85 0.667 86 0.657 87 0.846 88 0.547 89 0.053 90 1.244 91 1.046 92 0.793 93 0.383 94 0.335 95 0.272 96 0.537 97 0.248 98 0.67 99 0.226 - 100 0.208 252 mat :

XY data_1 data_2 data_3 data_4 stack :

L H N xd U 0 : dx H L - N / : i 1 N range U i el L dx - (dx i * + : for U 21 line 31 line :

X XY 1 col :

Y XY 2 col :

Y Y 12511 submatrix :

t0 1 time :

Fourier complex  from sampled  Y vector Y fft N Y length : i 1 - sqrt : n 0 N range D n 1 + el 1 N sqrt / Y k 1 + el e i 2 π * n * N / (* k *^* (k 0 N 1 - sum * : for D 41 line :

Export complex for subuse F Y fft :

N F length :

H 5 :

L 5 - :

U L H N xd :

Vectorise 'F' to get spectrum New t U : Ω F mat2sys abs sys2mat : t Ω augment 31 line :

This Fourier "fft" will let you pair a user wild vector of "U" around [L,H]. It is not made stem style to keep things simple. This open style is compatible to even/odd length of 'Y'. It is more Edu. than practical: Smath chokes around N < 25

1 time t0 - 3533 sec *

Prepare Stem Style spec New 2 col :

View the crappy sampled data.

data_1

DC 0 spec 1 el 0022 mat :

<= Fake to view DC

N spec length :

N 26

Stem style construct

iii 1 N range q iii el 3 : q N el 2 : 21 line for

ii 1 N range v ii el ii : for

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

u q v replicate :

i 1 N range S 3 i * el spec i el : for

V u S augment :

V V 3 V rows 12 submatrix :

Unscaled spectrum ... random sampled data

V DC 2 1 sys

Raw spectrum located wrt [L, H]

New

In this "ifft(v)" demo, the Smath length is limited to 20. Also, we note that for stem style plot, it does not take 'C' from C:=ifft(Y) ... we must copy/paste 'C' in a vector.

Y 3.5507 1.0822 - 0.4413 i * - 0.1808 - 0.7526 i * - 0.1185 0.2149 i * - 0.0714 - 0.1904 i * - 0.3499 - 110000 / i * - 75000 / 0.0334 i * - 0.1407 - 0.007 i * + 0.1444 - 0.1928 i * - 0.0077 - 0.1618 i * - 0.1656 0.0199 i * - 0.3104 - 0.0745 i * - 0.1033 - 0.0887 i * + 0.1033 - 0.0887 i * - 0.3104 - 0.0745 i * + 0.1656 0.0199 i * + 0.0077 - 0.1618 i * + 0.1444 - 0.1928 i * + 0.1407 - 0.007 i * - 75000 / 0.0334 i * + 0.3499 - 110000 / i * + 0.0714 - 0.1904 i * + 0.1185 0.2149 i * + 0.1808 - 0.7526 i * + 1.0822 - 0.4413 i * + 3.5507 261 mat :

Y Y 11511 submatrix :

Numeric vec ifft N vec length 1 - : i 1 - sqrt : n 0 N 1 - range ID n 1 + el 1 N sqrt / vec k 1 + el i - (2 π * n * N / (* k * exp * (k 0 N 1 - sum * : for ID 41 line :

Symbolic C Y ifft :

'C' Symbolic, Auto C 14.058 - 3.517 - 6.515 i * + 1.929 9.575 i * + 3.922 1.975 i * + 5.352 1.071 i * + 2.636 2.575 i * - 3.108 4.097 i * - 0.485 3.108 4.097 i * + 2.636 2.575 i * + 5.352 1.071 i * - 3.922 1.975 i * - 1.929 9.575 i * - 3.517 - 6.515 i * - 141 mat

Symbolic C Y ifft :

C 14.057968 - 165.876282 250.867766 55.261462 33.197186 64.303025 - 103.416916 - 0.485052 109.632838 69.574217 22.49396 - 47.418067 - 247.00893 - 172.910435 - 141 mat

C 14.057968 - 165.876282 250.867766 55.261462 33.197186 64.303025 - 103.416916 - 0.485052 109.632838 69.574217 22.49396 - 47.418067 - 247.00893 - 172.910435 - 141 mat :

N C rows :

N 14

Stem style ifft

iii 1 N range qq iii el 3 : qq N el 2 : 21 line for

ii 1 N range vv ii el ii : for

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

uu qq vv replicate :

i 1 N range SS 3 i * el C i el : for

uu rows 42

VV uu SS augment :

SS rows 42

VV VV 3 VV rows 12 submatrix :

This is NOT spectrum ... just ifft of sampled data for working in this domain.

VV 1 1 sys

Sanity check Mathematica 4.0

Y 1.3 2.4 1.7 4.1 2.9 1.7 5.1 2.7 18 mat transpose :

Fourier complex  from sampled  Y vector [Numeric] Y fft N Y length : i 1 - sqrt : n 0 N range D n 1 + el 1 N sqrt / Y k 1 + el e i 2 π * n * N / (* k *^* (k 0 N 1 - sum * : for D 41 line :

Symbolic F Y fft :

Fraction => Auto F 7.742819 0.740685 - 0.677082 i * - 0.919239 - 0.954594 i * - 0.390685 - 1.727082 i * + 0.035355 0.390685 - 1.727082 i * - 0.919239 - 0.954594 i * + 0.740685 - 0.677082 i * + 7.742819 91 mat

Y 1 - 1 - 1 - 1 - 111118 mat transpose :

F Y fft :

These two sanity check

F 0 0.707107 - 1.707107 i * - 0 0.707107 - 0.292893 i * - 0 0.707107 - 0.292893 i * + 0 0.707107 - 1.707107 i * + 091 mat

yy 0 0.707107 - 1.707107 i * - 0 0.707107 - 0.292893 i * - 0 0.707107 - 0.292893 i * + 0 0.707107 - 1.707107 i * + 091 mat :

k # :

=> copy/paste =>

Numeric vec ifft N vec length 1 - : i 1 - sqrt : n 0 N 1 - range ID n 1 + el 1 N sqrt / vec k 1 + el i - (2 π * n * N / (* k * exp * (k 0 N 1 - sum * : for ID 41 line :

Symbolic sanity yy ifft :

Fraction Auto sanity 1 - 14 i * 14.000004 i * 14 i * 114 i * - 14.000004 i * - 14 i * - 81 mat

There an unknown interaction that renders 'ifft' wrong to use in-situ here. Copy/paste in blank work sheet and it works fine. Visit: "Fourier discrete fft, ifft".

# Clear