Stem style construct

f9612434-d67b-4d6a-a759-5cba4717c9a4 39 6 9 decimal

&[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 :

X XY 1 col :

Y XY 2 col :

L H N xd U 0 : dx H L - N / : i 1 N range U i el L dx - (dx i * + : for U 41 line :

Y Y 110011 submatrix :

t0 1 time :

Fourier complex  from sampled  Y vector Y fft N Y length : 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 eval * : for D 31 line :

Export complex for subuse F Y fft :

N F length : L 5 - : H 5 : U L H N xd : 41 line

Vectorise 'F' to get spectrum New Ω F mat2sys abs sys2mat : U Ω augment 21 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'. Smath support this 100 length vector.

1 time t0 - 7 sec *

Prepare Stem Style spec New 2 col :

View the crappy sampled data.

DC 0 spec 1 el 0022 mat :

<= Fake to view DC

N spec length :

N 101

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 :

iiii 1 N range S 3 iiii * el spec iiii 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 spec Min 2 1 sys

Numeric ... Smath takes this 100 length vector v ifft N v length 1 - : n 0 N 1 - range ID n 1 + el 1 N sqrt / v k 1 + el i - (2 π * n * N / (* k * exp * (k 0 N 1 - sum eval * : for ID 31 line :

Symbolic C Y ifft :

'C' Numeric, Auto C # ≡

C 14 - 166251553364 - 103 - 01107022 - 47 - 247 - 173 - 114 mat transpose :

N C rows : 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 :

iiii 1 N range SS 3 iiii * el C iiii 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

Recapitulate/Export 'fft(v)' ... 'ifft(v)'

fft ... ifft

Fourier 'fft' from sampled  'v' vector [Numeric] v fft N v length : n 0 N range D n 1 + el 1 N sqrt / v k 1 + el e i 2 π * n * N / (* k *^* (k 0 N 1 - sum eval * : for D 31 line :

Fourier 'ifft' from sampled 'v' vector[Numeric] v ifft N v length 1 - : n 0 N 1 - range ID n 1 + el 1 N sqrt / v k 1 + el i - (2 π * n * N / (* k * exp * (k 0 N 1 - sum eval * : for ID 31 line :

Symbolic vy 1 - 1 - 1 - 1 - 111118 mat transpose : FFT vy fft : IFFT FFT ifft : Paired paired for spectrum idx 1 vy rows 1 + range : FFT FFT mat2sys abs sys2mat : idx FFT augment 41 line : 41 line

Auto Paired transpose 1234567890 1.847759 0 0.765367 0 0.765367 0 1.847759 029 mat

Sanity check ..... Numeric/Auto .... FFT IFFT 12 mat 0 0.707 - 1.707 i * - 0 0.707 - 0.293 i * - 0 0.707 - 0.293 i * + 0 0.707 - 1.707 i * + 091 mat 1 - 1 - 1 - 1 - 111181 mat 12 mat

Bingo ! ... triangular wave

Paired 0 2 1 sys