data

c9e959db-4585-4988-a21b-637dadf6c943 18 15 12 decimal

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

t Φ t Heaviside : W x f 22 ln sqrt * W π * / 42 ln * x W / (2^* - exp * : A B y g A x Φ * B x * - exp * : 31 line

RealRange( a, b )

multiple assume 5 x < y −10 < # assume

3 appVersion 0.98.6179 ≡

Does not cut the mustard !

x Int W x y - f A B y g * y ∞ - ∞ int A ∞ - ∞ RealRange ≡ B 0 ∞ RealRange ≡ W 0 ∞ RealRange ≡ assume maple 11 line :

x Int 22 ln sqrt * W π * / 42 ln * x y - W / (2^* - exp * A 1 x 0 - sign + 2 / * B x * - exp * * y ∞ - ∞ int A ∞ - ∞ RealRange ≡ B 0 ∞ RealRange ≡ W 0 ∞ RealRange ≡ assume maple

height = A
decay = B
width = W h 1 : d 1 : w 1 : 31 sys

t Φ t Heaviside :

h d w xo x GaussExp 12 / h * 114 / 8 x xo - (* 2 ln * d w 2^* - (w 2 ln sqrt * / * erf + (* 1 - 16 / d * 16 x xo - (* 2 ln * d w 2^* - (2 ln / * exp * :

h d xo x g h x xo - (Φ * d x xo - (* - exp * :

x c 11 - x ≤ (x 5 ≤ (& cases :

play 'w' 0, 0.125, 0.25 w 0.125 : xo 0 : 12 mat

h d w xo 0 GaussExp 0.48

h d w xo x GaussExp x c * h d xo x g h d 0.25 xo x GaussExp x c * h d 0.5 xo x GaussExp x c * 0 0.48 + 50 blue 1 5 mat 5 1 sys

The blue function is the convolution integration. Model fit to "Laser". Further fitted wrt Laser experimental data ... not given here.

0.125 x f 0.25 x f 0.5 x f 3 1 sys

W x f 22 ln sqrt * W π sqrt * / 42 ln * x W / (2^* - exp * :

A 1 : B 1 : W 1 : 13 mat

t Φ t Heaviside :

x c 11 - x ≤ (x 1 ≤ (& cases :

A B x g A x Φ * B x * - exp * :

W x f x c * 2 ln sqrt W π sqrt * / W 2 / - 1 W 2 / - 0.25 - 2 2 mat W 2 / 1 W 2 / 0.25 - 2 2 mat 4 1 sys A B x g

W 0 f 0.9394

2 ln sqrt W π sqrt * / 0.4697

22 ln sqrt * W π sqrt * / 0.9394

A B W x GaussExp 12 / A * 114 / 8 x * 2 ln * B W 2^* - (W 2 ln sqrt * / * erf + (* 1 - 16 / B * 16 x * 2 ln * B W 2^* - (2 ln / * exp * :

height = A
decay = B
widthr = W h 1 : d 22 ln sqrt * W π sqrt * / : w 1.2085 : 31 sys

22 ln sqrt * W π sqrt * / 0.9394

w x f h d x g h d w x GaussExp 3 1 sys

Experimental trial/error [d,w] w 0.5 f h d 0 g h d 0.000001 g h d w 0.5 GaussExp 41 mat 0.48362 0.5 1 0.48362 41 mat

A B W xo x GaussExp 12 / A * 114 / 8 x xo - (* 2 ln * B W 2^* - (W 2 ln sqrt * / * erf + (* 1 - 16 / B * 16 x xo - (* 2 ln * B W 2^* - (2 ln / * exp * :

A B xo x g A x xo - (Φ * B x xo - (* - exp * :

x c 11 - x ≤ (x 5 ≤ (& cases :

Now, we have the model function to fit the data set. Should there be a need for, this document can be used to do other convolution(s) as well. The functions would be defined above and pieces needed added to the integral to allow the symbolic solver to come up with new function. The other thing to pay attention to is that the convolution we have done is a Gaussian and a single exponential. Below, you will see why that was done as it adds another degree of adjustability to this document.

play 'w' 0, 0.125, 0.25 w 0.125 : xo 0 : 12 mat

h d w xo x GaussExp x c * h d xo x g x c * 2 1 sys

observe blue ... w[0], w[0.125]

The final curve fitting . . .

Credits and thanks to Mathsoft Collabs:

Richard, Tom, Paul, Jean [suggested convolution] .

Three advanced art fitting techniques have been applied.

1. Data Reduction

2. Careful initialization

3. PW Minerr Null style.

]]>

Laser model fit

c 22 ln sqrt / :

x B p x Xo - (2 ln * B W 2^* - eval :

x c 14000 - x ≤ (x 40000 ≤ (& cases :

A 166.937 436.795 126.295 31 mat :

B 0.00013405 0.000010719 0.000355983 31 mat :

W 443.03334 : Xo 1994.32623 : offset 107.5887 : 31 line

x Laser A i el 1 c x B i el p * W / erf + (* B i el - (x B i el p 2 ln / * exp * i 13 sum offset + :

data

data 3192 - 873119 - 1143045 - 1302972 - 1002999 - 1032825 - 1042752 - 1152679 - 1112605 - 972532 - 1142458 - 1212385 - 1152312 - 932238 - 1102165 - 1062091 - 1052018 - 971945 - 971871 - 1031798 - 1091725 - 1201651 - 1211578 - 1101504 - 1081431 - 1161358 - 1081284 - 851211 - 981137 - 921064 - 124991 - 117917 - 112844 - 116771 - 91697 - 129624 - 126550 - 112477 - 94404 - 115330 - 79257 - 117183 - 122110 - 12737 - 1083711611010818385257983309240412447711755011262411669712677110084410591711199113310641111137147121199128410513581141431101150410415781161651135172518117982811871440194562220188472091102721651206223813352312143223851593245814732532150026051470267914592752144628251505289914542972144630451457311914173192139132661417333914273706136837791365385313723926137739991373407313134146136342201315429312784366135444401319451312904587126846601292473312904807128848801255495313145027131451001304517412145247121753201213539411955467128455401177561412355687131857611169583412325907127459811168605412016128123562011174627412056348119564211185649411566568114266411198671511426788116368611115693511837008118270821137715511787228116573021122737511387448108875221125759511117669110877421096781511567889107079621116803610938109111681821115825611388329108284021066847610288549107486231096869610838769104588431057891610538990109290631049913610319210108092839929356102794301077950311029577109496501099972310309797102798701033994410171001711081009010771001711081009010771016410581023799010310101910384104410457107010531103710604101710677967107519761082410641089810061097110291104496211118968111919671126498111338100611411104611485999115589402002 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

data rows 200

t0 1 time :

L x 0120001 range : i 1 x rows range y i el x i el Laser eval : for x y augment 31 line :

pts data . 8 black plot :

pts L 1024 - offset 3072 offset 2 2 mat Xo Xo Laser + 50 blue 1 5 mat 2485 1500 + 50 blue 1 5 mat 5 1 sys

sanity MCD 2485 Laser 1500

1 time t0 - min 3.9

Stem(data)

Stem construction over a data set data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Convolution of two Gaussian

α 2 :

t U α t 21 mat min :

t L α - t α - 21 mat max :

W1 1 : W2 0.75 : m 0.75 : 13 mat

<= 'm' =>

x U x L 2 1 sys

t X 1.5 2 ln sqrt * W1 π sqrt * / 42 ln * t W1 / (2^* - exp * : t H 1.5 2 ln sqrt * W2 π sqrt * / 42 ln * t W2 / (2^* - exp * : t Convol u X t u - H * u t L t U int t m ≤ (* : 31 sys

t0 1 time :

Cvl t 2 - m 1.95 - range : i 1 t rows range y i el t i el Convol eval : for t y augment Stem 31 line :

x X m x - H Cvl m 1 m 0.5 - 2 2 mat m 0.75 m 14 green 1 5 mat x Convol 6 1 sys

x Convol w/o 0.6 min scalar 2.6 min 21 sys ≡

0.25 Convol 0.241414

1 time t0 - min 2.5