<p> Data set </p>

7f4b16d3-d644-4e76-a763-7239283963c5 73 6 9 decimal

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

Fourier DCT-II & IDCT-II

f dct2 DCT-II -> Discrete Cosine Transform n f length : g 1 el 1 n / sqrt f j 1 + el j 0 n 1 - sum * : k 1 n 1 - range g k 1 + el 2 n / sqrt f j 1 + el 2 j * 1 + (k * π * 2 n * / cos * (j 0 n 1 - sum * eval : for g n el 1 n / sqrt f j 1 + el 2 j * 1 + (n 1 - (* π * 2 n * / cos * (j 0 n 1 - sum * eval : g 61 line :

g idct2 IDCT-II Inverse Discrete Cosine IF k 0 ≡ 12 sqrt / 1 if : n g length : m g length 1 - : j n n m + range f j m - el 2 n / sqrt g k 1 + el IF (* 2 j * 1 + (k * π * 2 n * / cos * (k 0 n 1 - sum * eval : for f reverse 61 line :

Matlab DCT-II algo not same.

α V W the window weight vector i 1 V rows range i i α ≤ (i α ≤ (& (W i el 1 : α 1 + (i ≤ (i V rows ≤ (& W i el 0 : NOT window if if : for W 31 line :

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 :

t0 1 time :

m XY rows : 100

Low 10 : High 0 : 13 mat

1. Set the range of experiment

j 1 m range j Low ≤ (m High - j ≤ (¤ W j el 1 : 0 if for W 21 line

Project note: Smath smooths this data set in reasonable time. No limit can be predicted if not tested. The window width 10 results in complete smoothing. Experiment smaller window for variable smoothing.

Y Y 1 m High - 11 submatrix :

X X 1 m High - 11 submatrix :

numeric Fx Y dct2 eval :

2. Enter the Fourier domain

3. Weight the dct2 ...... 4. Recover from idct2

U window weight Fx i 1 Fx rows range F i el W i el Fx i el * eval : for F 31 line :

Xf U idct2 eval :

dctII X Xf augment :

1 time t0 - 1232 sec *

Fourier DCT-II window smoothing

XY dctII 2 1 sys

Variable smoothing from Fourier Budapest 39

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

XY data_1 data_2 data_3 data_4 stack :

X XY 1 col :

Y XY 2 col :

N XY rows : 100

Δx X N el X 1 el - (N 1 - / : 1

ω 2 π N / * : 0.062832

u λ u X 1 el - (ω Δx / * :

n 0 N range A n 1 + el 2 N / Y k 1 + el n k * ω * cos * k 0 N 1 - sum * eval : B n 1 + el 2 N / Y k 1 + el n k * ω * sin * k 0 N 1 - sum * eval : 21 line for

u B39 A 1 el 2 / A n 1 + el u λ n * cos * B n 1 + el u λ n * sin * + (n 110 sum + :

fit x 1100 1.25 range : i 1 x length range v i el x i el B39 eval : 11 line for x v augment 31 line :

a b c spec . 10 black 13 mat :

pts XY a b c spec pts :

Fourier polynomial fit to periodic data set

pts fit 2 1 sys

This type of fit is generally successful to reconstruct trains of "chirps". Not exemplified in this WS.