Stem(data)

45ae93f9-c19c-45ea-9ed4-23d4e20f6087 63 4 5 decimal

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click ■ scroll bar TOC 1. Kernelconvolution 2. Suite convolution 3. Two Gaussians convol 4. Parametric convol 5. Voltera 2nd kind 51 mat ≡

1. Create the convolution matrix

0 ln #

u v Kernel i 1 u rows range j 1 u rows v rows + (range H i j el 0 : for for i 1 u rows range jj 1 v rows range H i jj i + 1 - el v jj el : for for Kernel H : 31 line :

x 0.1 0.2 0.3 0.4 41 mat :

h 1.123456789 2.2 3.3 4.4 41 mat :

K x h Kernel eval :

K 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 00000 1.123457 2.2 3.3 4.4 048 mat

Isolate the demo ...

K 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 048 mat :

x 0.1 0.2 0.3 0.4 41 mat :

Convolve matrix over vector i 1 K cols range : Convol i x transpose K * (transpose augment : XY Convol : 31 line

i 1 Convol rows range p i el . : for

data XY 1 col XY 2 col p augment :

data XY 2 1 sys

Optimiz: numeric, Decimal Convol 1 0.1123 2 0.4447 3 1.107 4 2.2094 5 2.75 6 2.64 7 1.76 8082 mat

K x al_convr1d 0.1123 0.2247 0.337 0.4494 00071 mat

For this convoved vector, it is tempting to fit a function. A 5th degree polyfit is OK. The "user scale" is 0.5 in this polyfit demo. We can shift the fit, unshifted fir is better.

C 0.112346 0.444691 1.107037 2.209383 2.75 2.64 1.76 081 mat :

scale 0.5 :

shift 0 :

XY i 1 C rows range X i el i scale / : for X C augment 21 line :

x XY 1 col shift - :

XY 2 0.112346 4 0.444691 6 1.107037 8 2.209383 10 2.75 12 2.64 14 1.76 16082 mat

y XY 2 col :

degree 5 :

i 1 y length range j 1 degree range M i j el x i el (j^: for for 11 line

Cholesky solver
NOT commutative k M transpose M * (1 -^M transpose * y * :

k 0.088525 0.060087 - 0.024097 0.002169 - 5.474234 105 -^* 51 mat

u f k i el u i^* (i 1 k length sum 11 line :

i 1 XY rows range p i el . : for

x u 0 x ≤ (x 16 ≤ (& :

data XY 1 col shift - XY 2 col p augment :

User scaled polyfit. Note that some extrapolation is possible.

data x f x u * 2 1 sys

2. Suite convolution

0 ln #

u v Convol same as Mathcad built-in i 1 u rows range j 1 u rows v rows + (range H i j el 0 : for for i 1 u rows range jj 1 v rows range H i jj i + 1 - el v jj el : for for u transpose H * (transpose 41 line :

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

Simple convolution:CONVOL(x,h,n) is often confusing in CAS, with double convolution from above Convol(x,h). Let disambiguate by plotting both styles below.

x h n CONVOL simple convol prepare m n : x k 1 + el h m k - 1 + el * (k 0 m sum 31 line :

u v C simple convol construct N is wild i 0 N range C i 1 + el u v i CONVOL : for v C augment 41 line :

N 15 :

convol vectors n 0 N range x n 1 + el n 2 / : h n 1 + el n : 21 line for 21 line

Ω Double convolution Ω x h Convol : i 1 Ω length range u i el i 1 - : for u Ω augment 41 line :

C x h C :

C C : Ω Ω : 12 mat

C 00102 0.5 32455106 17.5 72884296010 82.5 11110121431318214 227.5 15280162 mat

x 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 161 mat

h 0123456789101112131415161 mat

Ω 00102 0.5 32455106 17.5 72884296010 82.5 11110121431318214 227.5 15280163401739218 435.5 19470204952151022 514.5 232 mat

a b c spec . 12 red 13 mat :

ptsΩ Ω a b c spec pts :

a b c spec o 7 black 13 mat :

ptsC C a b c spec pts :

Ω => Double convolution as smooth trace ptsΩ => Double convolution plotted points ptsC => Simple convolution plotted points

Ω ptsΩ ptsC 3 1 sys 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 :

3. Convolution of two Gaussians

0 ln #

t U 2 t 21 mat min :

t L 2 - t 2 - 21 mat max :

W1 1 : W2 0.75 : m 0.5 : 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

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 5 1 sys

x X m x - H Cvl m 1 m 0.5 - 22 mat m 0.75 m 14 green 15 mat 51 sys

4. Parametric Convolution ... suggestive example

0 ln #

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 g(A,B,δ,t)
B ... pulse shape g(A,B,δ,t)
W ... shape width f(W,t) 
δ ... shift ≥ 0 A 1 : B 0.5 : W 2 : δ 0.5 : 41 sys

acc 100 t L W t u - f A B δ u g * u δ t int :

This demo example suggests best fitting some data set otherwise difficult from lack of models. Mathcad/Mathsoft [8.Pro, 11] would best Minerr [A, B, W, δ]. Not attempted Smath.

Convol t δ 20 δ 0.1 + (range : i 1 t rows range y i el t i el L eval : for t y augment 31 line :

t0 1 time :

store/isolate Convol Convol eval :

W x f Convol δ 0 + 40 black 1 5 mat δ 0 δ 12 black 1 5 mat 4 1 sys

1 time t0 - min 1.3

X Convol 1 col : Y Convol 2 col : 12 mat

x convol X Y x cinterp :

3 convol 0.16

5.Voltera 2nd kind

0 ln #

u Φ 1 u mwbesselj u 1015 -^+ / :

x c 10 x ≤ (x 12 ≤ (& cases :

x abs Φ 1 1 sys

Voltera 2nd kind x λ f 1 λ 2^- sqrt x t - (* sin t Φ * t 0 x int :

u 9 :

in-situ explorer u 0.5 f 0.3697

Scalar plot on canvas ... 3 min

x 0.9 f x c * x 0.5 f x c * u u 0.5 f + 20 black 1 5 mat 3 1 sys

x 0121 range :

XY x x 0.5 f vectorize eval augment :

isolate XY XY :

linfit modules

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 :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

The linfitCheby Procedure

order 10 :

t0 1 time :

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin XY . 12 black plot : 41 line

4 appVersion 0.98.6179.21440 ≡

minXd maxXd 21 mat 01221 mat

1. Convert Original XY => Cheby domain [±1] v i 1 XY rows range v i el i : for v 2 v * v Min - v Max - v Max v Min - / : 21 line : XY v XY 2 col augment : vx XY 1 col : vy XY 2 col : data XY o 7 red plot : 51 line

2. Create ChebyShev Optimiz: Symbolic x β p n 0 order range F n 1 + el β n 1 + el n x acos * cos * : for F 21 line :

0. Create the original reserve 1. The original XY is converted in ChebyShev domain. 2. The ChebyShev vector function is created. 3. Collect the model f(x,β). 4. Cheby coefficients are spooled to file. 5. Collocate the "linfitCheby" Colln(x) 6. Reverse back the fit & the data set

3. Collect  the model function x β f x β p n el n 1 order 1 + sum 11 line :

Prepare the 'Unary β' i 1 order 1 + range idx i el 1 : for

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

idx transpose 11111111111111 mat

set the 'Unary β' & length 'n' of the vector gradient

β idx : n idx rows : 21 line

<= vector gradient x β φ x β f n φ :

The "linfit" coefficients Cheby XY f φ β βlinfit :

The "linfit" coefficients Cheby XY f φ β βlinfit :

b Cheby 6 round vectorize :

Smath 6179 may create extra imaginary residual of low value, that must be elimitated for export. module has that virtue.

c b eval :

c 0.229274 0.120372 0.211426 0.186005 0.534365 - 0.309602 - 0.190889 0.086014 0.030006 - 0.012421 - 0.003152 111 mat

# # round

minXd maxXd 21 mat 01221 mat

x p x β p :

5. Collocate the "linfitCheby" x Colln c k el x p k el * (k 1 order 1 + sum 11 line :

6. reverse scale 'x domain' ω X 2 ω * minXd - maxXd - maxXd minXd - / :

5. Data set & the fit in ChebyShev domain [-1..1]

x Colln data 2 1 sys

6. linfitCheby in the original domain

Origin x X Colln 2 1 sys

◄ sanity  0.3697 9 X Colln 0.3699

Export fitted data  ► spline in-situ data ω 012 0.1 range : n 1 ω rows range Ex n el ω n el X Colln eval : for ω Ex augment 31 line :

Origin data 2 1 sys

x X Colln x minXd maxXd int 2.273

1 time t0 - 27 s *