Stem(data)

2086b40a-29ac-4b68-bc80-0a74cdf126cd 28 4 5 decimal

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

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 :

Part_1: Integral Convolution

W1 1 : W2 0.75 : m 0.75 : α 2 : 14 mat

t U α t 21 mat min :

t L α - t α - 21 mat max :

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 ≤ (* : acc 50 for this symmetric Gaussian 41 sys

x U x L 2 1 sys

t0 1 time :

0.225 1 min t Convol m t - H - 0 ≡ t 0 0.25 solve

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

0.225 Convol 0.241408

1 time t0 - min 0.9

Part_2: Convolution Laser 560 TOPO/HDA NC's

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 * :

A B x g W x f 2 1 sys

W 0 f 0.9394

22 ln sqrt * W π sqrt * / 0.9394

Borrowed from
MCD Symbolic A B W x Convol 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 1.25 : 31 sys

22 ln sqrt * W π sqrt * / 0.9394

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

A B W xo x Convol 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 :

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

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

Now, we have the blue 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.

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x c 14000 - x ≤ (x 40000 ≤ (& cases :

Credit and thanks to Paul W. for those optimized Mathcad 11 Minerr [A,B,W,Xo]

c 22 ln sqrt / :

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

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 incomplete from original immense

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 rows 200

t0 1 time :

Laser x 0120002 range : i 1 x rows range y i el x i el Laser eval : for x y augment 31 line :

pts data . 9 black plot :

pts Laser 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 1.6

Symbolic Convolution Mathcad

Part_3: Sliding pulse

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 :

Optimiz: NONE a b t pulse 1 s / a s * - exp b s * - exp - (* t 0 > assume s t invlaplace maple :

a b t pulse t a - sign t b - sign - 2 /

a 1 - : b 1 : 21 line

a b x f x a - sign x b - sign - 2 / : convolving function ≡

α 2 - slide + : β 0 slide + : 21 line

α β x g x α - sign x β - sign - 2 / : convolved function ≡

Mesh U, V  within the bounds L, H L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx i * dx - (+ : for U 31 line :

slide 0.5 - :

sliding window 
of integration α β 21 mat 2.5 - 0.5 - 21 mat

a 1 - : b 1 : 21 line

α 0 slide + : β b a - (slide + : 21 line

a b 21 mat 1 - 121 mat

Convolution integration α β x Σ a b x f α β x g * x α β int :

α β x Σ 1.5

The sliding pulse is of constant width α β - 2 -

α β u F a b x f α β x g * :

pulse a 0 a 1 b 1 b 042 mat :

convol α 0 α 1 β 1 β 042 mat :

α β z Int α β u F u 1 - x int :

s slide 2 slide 1 - 22 mat :

recast canvas α β x Int α β z Int :

cnv N 51 : x α b N xd : 12 mat i 1 x rows range α x i el ≤ (x i el β ≤ (& Σ i el α β x i el α - (Int eval : 0 if 11 line for x Σ augment Stem 31 line :

Yo α β α Int : 0.25

p pulse convol cnv s 61 sys :

Differential

b slide - b α β x Σ 2 2 mat 2 1 sys α β x Int 1 1 sys p

Part_4: Convolution observer.

Exploded symbolic convolution experimental wrt parameters ... A B W xo x Convol 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 * :

t Φ t Heaviside :

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

Convolution Observer ... can be used to fit data sets that have convolution shape. Does NOT exclude PDF's fit.

[W d]  companions
slide up/dn decay W 1 : d 22 ln sqrt * W π sqrt * / : h 1 : w 0.25 : xo 0 : 13 mat 31 sys 1 0.9394 1 0.25 013 mat 31 sys

x f h d w xo x Convol :

Golden-Ratio search [min/max]

TOL 1012 -^:

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

X f 0 0.5 y1 y2 ≥ (TOL Extrema 1 el :

X X f 12 mat 0.1903 0.8024 12 mat

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

A 1 : B 1 : W 1 : 13 mat

Unary convolution observer ...

h d w xo x Convol h d xo x g X X f + 20 black 1 5 mat 3 1 sys

t Φ t Heaviside :

A B x g A x Φ * B x * - exp * :

A B x g W x f 2 1 sys

W 0 f 0.9394

Borrowed from
MCD Symbolic U V Ω x Convol 12 / U * 114 / 8 x * 2 ln * V Ω 2^* - (Ω 2 ln sqrt * / * erf + (* 1 - 16 / V * 16 x * 2 ln * V Ω 2^* - (2 ln / * exp * :

system h 1 : d 22 ln sqrt * W π sqrt * / : w 1.5 : 31 sys

22 ln sqrt * W π sqrt * / 0.9394

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

x Cvl h d w x Convol :

x q w x f :

x Φ x q x Cvl - eval eval :

a 0.25 : b 0.75 : ε 109 -^: 13 mat

"RootSecant" is viable  => refine from "sanity check" Φ a b ε X Root Secant u 1 el 1015 -^: u 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : u i el u i 1 - el - abs ε ≥ u i 1 + el u i el u i el Φ u i 1 - el u i el - (* u i 1 - el Φ u i el Φ - / - : i i 1 + : 21 line while u 71 line :

sol Φ a b ε X :

sol 11015 -^* 0.561142453914 0.548858993177 0.550209675159 0.550205461003 0.550205459433 0.550205459433 71 mat

sanity check
adjust [a b] sol sol rows el Φ 7.7 1016 -^* -