<p> data </p>

5362d258-8828-49db-af21-91b22c8bd124 34 4 5 decimal

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

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

t Φ t Heaviside :

A B x g A x Φ * B x * - exp * :

Unsolved

W x t - f A B t g * t int assume A ∞ - ∞ RealRange ≡ assume B 0 ∞ RealRange ≡ assume W 0 ∞ RealRange ≡ 61 line maple 22 ln sqrt * W π sqrt * / 42 ln * x t - W / (2^* - exp * A 1 t 0 - sign + 2 / * B t * - exp * * t int W 0 ∞ RealRange ≡ maple

Two numerical convolutions

T 5 :

δ 1 :

t δ Φ t δ Heaviside :

t h 0 t 0 ≤ t δ Φ e t -^(* cases :

t q 0 t 0 ≤ 0 t T ≥ 1 cases :

sensible to acc. t f u q t u - h * u 0 t int eval :

t0 1 time :

System output [RC response]

x q x f 2 1 sys

1 time t0 - min 1.8

x δ Φ e x - ^ ( *

t f 0 u 0 ≤ 0 u 5 ≥ 1 cases 0 t u - 0 ≤ 11 t - u + (- sign + 2 e t u -^* / cases * u 0 t int eval

===============================================================

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

t δ Φ t δ Heaviside :

A B δ t g A t δ Φ * B t * - exp * :

A ... magnitude
B ... shape
W ... shape
δ ... shift A 1 : B 0.75 : W 0.25 : δ 0.25 : 41 sys

t L W t u - f A B δ u g * u δ t int eval :

t0 1 time :

Convol t δ 5 δ 0.025 + (range : i 1 t rows range y i el t i el L eval : for t y augment 31 line :

Convol δ 0 + 20 red 1 5 mat 2 1 sys

1 time t0 - min 3.8

About fitting a convolution . . .

For attempting to fit a data set with an integral convolution,

both : Conjugate Gradient [CG], Levenberg-Marquardt [LM]

will fail. Neither one recognizes the UnitStep function.

Thus, you will have to fit manually from trial/error.

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This Solved Convolution

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 * /

22 ln sqrt * W π sqrt * / 0.9394

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
width = W h 0.9394 : d 1 : w 1 : 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

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

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

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.

The final curve fitting . . .

Credits and thanks to Mathsoft Collabs:

Richard, Tom, Paul, Jean.

Three advanced art fitting techniques have been applied.

1. Data Reduction

2. Careful initialization

3. PW Minerr Null style.

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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 + :

L x 100040001001 range : i 1 x rows range y i el x i el Laser eval : for x y augment 31 line :

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 data 45 data rows range 12 range el :

pts data . 8 black plot :

Laser [560 TOPO/HDA NC's] => original data set [600].

x Laser x c * Xo 750 + 30 red 1 5 mat 2485 1500 + 30 red 1 5 mat 37 116 . 8 black 110 108 . 8 black 183 85 . 8 black 257 98 . 8 black 330 92 . 8 black 404 124 . 8 black 477 117 . 8 black 550 112 . 8 black 624 116 . 8 black 697 126 . 8 black 771 100 . 8 black 844 105 . 8 black 917 111 . 8 black 991 133 . 8 black 1064 111 . 8 black 1137 147 . 8 black 1211 99 . 8 black 1284 105 . 8 black 1358 114 . 8 black 1431 101 . 8 black 1504 104 . 8 black 1578 116 . 8 black 1651 135 . 8 black 1725 181 . 8 black 1798 281 . 8 black 1871 440 . 8 black 1945 622 . 8 black 2018 847 . 8 black 2091 1027 . 8 black 2165 1206 . 8 black 2238 1335 . 8 black 2312 1432 . 8 black 2385 1593 . 8 black 2458 1473 . 8 black 2532 1500 . 8 black 2605 1470 . 8 black 2679 1459 . 8 black 2752 1446 . 8 black 2825 1505 . 8 black 2899 1454 . 8 black 2972 1446 . 8 black 3045 1457 . 8 black 3119 1417 . 8 black 3192 1391 . 8 black 3266 1417 . 8 black 3339 1427 . 8 black 3706 1368 . 8 black 3779 1365 . 8 black 3853 1372 . 8 black 3926 1377 . 8 black 3999 1373 . 8 black 4073 1313 . 8 black 4146 1363 . 8 black 4220 1315 . 8 black 4293 1278 . 8 black 4366 1354 . 8 black 4440 1319 . 8 black 4513 1290 . 8 black 4587 1268 . 8 black 4660 1292 . 8 black 4733 1290 . 8 black 4807 1288 . 8 black 4880 1255 . 8 black 4953 1314 . 8 black 5027 1314 . 8 black 5100 1304 . 8 black 5174 1214 . 8 black 5247 1217 . 8 black 5320 1213 . 8 black 5394 1195 . 8 black 5467 1284 . 8 black 5540 1177 . 8 black 5614 1235 . 8 black 5687 1318 . 8 black 5761 1169 . 8 black 5834 1232 . 8 black 5907 1274 . 8 black 5981 1168 . 8 black 6054 1201 . 8 black 6128 1235 . 8 black 6201 1174 . 8 black 6274 1205 . 8 black 6348 1195 . 8 black 6421 1185 . 8 black 6494 1156 . 8 black 6568 1142 . 8 black 6641 1198 . 8 black 6715 1142 . 8 black 6788 1163 . 8 black 6861 1115 . 8 black 6935 1183 . 8 black 7008 1182 . 8 black 7082 1137 . 8 black 7155 1178 . 8 black 7228 1165 . 8 black 7302 1122 . 8 black 7375 1138 . 8 black 7448 1088 . 8 black 7522 1125 . 8 black 7595 1111 . 8 black 7669 1108 . 8 black 7742 1096 . 8 black 7815 1156 . 8 black 7889 1070 . 8 black 7962 1116 . 8 black 8036 1093 . 8 black 8109 1116 . 8 black 8182 1115 . 8 black 8256 1138 . 8 black 8329 1082 . 8 black 8402 1066 . 8 black 8476 1028 . 8 black 8549 1074 . 8 black 8623 1096 . 8 black 8696 1083 . 8 black 8769 1045 . 8 black 8843 1057 . 8 black 8916 1053 . 8 black 8990 1092 . 8 black 9063 1049 . 8 black 9136 1031 . 8 black 9210 1080 . 8 black 9283 992 . 8 black 9356 1027 . 8 black 9430 1077 . 8 black 9503 1102 . 8 black 9577 1094 . 8 black 9650 1099 . 8 black 9723 1030 . 8 black 9797 1027 . 8 black 9870 1033 . 8 black 9944 1017 . 8 black 10017 1108 . 8 black 10090 1077 . 8 black 10017 1108 . 8 black 10090 1077 . 8 black 10164 1058 . 8 black 10237 990 . 8 black 10310 1019 . 8 black 10384 1044 . 8 black 10457 1070 . 8 black 10531 1037 . 8 black 10604 1017 . 8 black 10677 967 . 8 black 10751 976 . 8 black 10824 1064 . 8 black 10898 1006 . 8 black 10971 1029 . 8 black 11044 962 . 8 black 11118 968 . 8 black 11191 967 . 8 black 11264 981 . 8 black 11338 1006 . 8 black 11411 1046 . 8 black 11485 999 . 8 black 11558 940 . 8 black 156 5 mat 4 1 sys pts L Xo Xo Laser + 20 blue 1 5 mat 3 1 sys

sanity MCD 2485 Laser 1500