<p> Stem style construct </p>

795e5dbd-d7c4-4aa4-a2aa-a35370bd18e9 59 4 5 decimal

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

Fourier trig fit to periodic data set

The Budapest algorithm 1939 is Courtesy from Theodore to Jean. Works great in Smath after repair by Dave Carpi [2016_01_03]. This tool is most precious for fitting periodic data sets.

data 0 100.639 0.0032 100.45 0.0064 99.88 0.0096 99.29 0.013 99.126 0.016 99.533 0.019 100.253 0.022 100.844 0.026 100.965 0.029 100.515 0.032 99.681 0.035 98.915 0.038 98.772 0.042 99.511 0.045 100.739 0.048 101.55 0.051 101.218 0.054 99.912 0.058 98.655 0.061 98.468 0.064 99.455 0.067 100.736 0.07 101.291 0.074 100.843 0.077 99.96 0.08 99.377 0.083 99.342 0.086 99.601 0.09 99.861 0.093 100.094 0.096 100.382 0.099 100.619 322 mat :

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 :

X data 1 col :

Y data 2 col :

N data rows : 32

ω 2 π N / * : 0.19635

Δx X N el X 1 el - (N 1 - / : 0.003194

a b c spec . 14 black 13 mat :

u λ u X 1 el - (ω Δx / * :

pts data a b c spec pts :

Δx 0.003194

0.05 λ 3.074159

Δx 0.003194

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 1 N 2 / sum + :

fit x 0 0.1 0.00025 range : i 1 x length range v i el x i el B39 eval : 11 line for x v augment 31 line :

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.

fit #

0.5 B39 #

Extract & plot "Unscaled" spectrum

Q n 2 N range c n el A n el i (B n 1 - el * - : for c 21 line :

spectral t 1 N 2 range : i 2 N range v Q mat2sys abs sys2mat : v 1 el A 1 el (: 21 line for t v augment 31 line :

Isolated for stem style spec spectral 2 col :

DC 0 A 1 el 0022 mat :

<= Fake to view DC

<p> Stem style construct </p>

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. Remember: the 1rst component of a spectrum is the DC component, very large in this application [2*Mean(Y)].

V DC 2 1 sys

Pairwise coefficients CosSin A B augment :

cos_sin idx 1 CosSin rows range : Cos idx CosSin 1 col augment : Sin idx CosSin 2 col augment : 21 line Cos Sin 21 mat 31 line :

Cos cos_sin 1 el :

Sin cos_sin 2 el :

"spectral" is pure convention. The collection of the respective coefficients is the true representation of the composition of the Y vector. Sampled data don't come with X vector, in such case we just fake index X for Y. The data set we have fitted is an Engineering data set not revealed but in the public domain. From recollection it was a test set for quantisation.

CosSin rows 33

data rows 32