<p> Stem(data) </p>

36ba6ce6-1538-48cd-a04f-c57e30aa58d4 7 15 12 decimal

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

Convolution Integral(s)

T 2 :

t x 0 t 0 ≤ 0 t 6 ≥ 1 cases :

t h 0 t 0 ≤ e t -^cases :

t f u x t u - h * u 0 t int eval :

Convolving Integral components

T x - h x x x x T x - h * 3 1 sys

System output [RC response]

x f x x 2 1 sys

http://lpsa.swarthmore.edu/Convolution/Convolution2.html

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

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

a 1 - : b 1 : 21 line

https://en.wikipedia.org/wiki/Convolution

slide 1.5 - :

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

t g 0 t α ≤ 0 t β ≥ 1 cases :

sliding window 
of integration α β 21 mat 1.5 - 0.5 21 mat

t f 0 t a ≤ 0 t b ≥ 1 cases :

t Int u f u g * u a t int eval :

t Σ t Int t 0 ≤ t Int - 2 + t β ≤ 0 cases :

Classical Triangular [pulses]

x Σ a 0 a 1 b 1 b 0 4 2 mat α 0 α 1 β 1 β 0 4 2 mat 3 1 sys

t Int u f t u - g * u 0 t int eval :

x Int a 0 a 1 b 1 b 0 4 2 mat α 0 α 1 β 1 β 0 4 2 mat 3 1 sys

t Int u f u t - g * u 0 t int eval :

x Int a 0 a 1 b 1 b 0 4 2 mat α 0 α 1 β 1 β 0 4 2 mat 3 1 sys <p> Stem(data) </p>

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 :

t U 2 t 21 mat min :

t L 2 - t 2 - 21 mat max :

t Φ t Heaviside : t X t 2 + Φ 2 t Φ * - t 2 - Φ + : t H t 0 t ≤ (t 2 ≤ (& 0 cases : 31 sys

m 1 :

x U x L 2 1 sys x X m x - H m 1 m 0.5 - 2 2 mat 3 1 sys

t Convol u X t u - H * u t L t U int eval t m ≤ (* :

Cvl t 2 - m 2 - 0.125 + 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 m 2 m 0.5 - 2 2 mat Cvl x Convol m 2 m 14 green 1 5 mat 7 1 sys

x Convol scalar canvas ≡

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t Φ t Heaviside : t X t 5 + Φ t 5 - Φ - : t H t 5 + Φ t 5 - Φ - : 31 sys

t U 5 t 21 mat min :

t L 5 - t 5 - 21 mat max :

m 7.5 :

The two undocumented functions: U(t), L(t) govern the convolution(s). Set adequately to avoid cigar convolution(s).

t Convol u X t u - H * u t L t U int eval t m ≤ (* :

Cvl t m - m m - 0.25 + range : i 1 t rows range y i el t i el Convol eval : for t 0.25 y * augment Stem 31 line :

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

x Convol scalar canvas ≡

Typical Convolution Integral

3 appVersion 0.98.6179 ≡ acc200 15 min ≡ 21 line

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 :

T δ t f t u - h u q * u 0 t int :

T t φ t u - q u q * u 0 t int :

T 5 : δ 0.5 : 12 mat

cumulative segment h(t)
under Φ(t,δ) ...
normal decay h(t)... T+δ ≤ Φ(t,δ)
free of Φ(t,δ) x cumul 1 δ x ≤ (x T δ + ≤ (& cases : x decay 1 T δ + x ≤ cases : 21 sys

T δ x f x cumul * T δ x f x decay * T δ + 1 T δ + 0.5 - 2 2 mat 3 1 sys T x φ T 1 T 0.5 - 2 2 mat 2 1 sys

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

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 :

set ∫(,)acc ...
typical: 200 T δ t f t u - h u q * u 0 t int :

T 5 : δ 0.5 : 12 mat

x user 3 π * x * sin :

t0 1 time :

System output [RC decay response]

T δ x f T δ x f x user * T δ + 1 T δ + 1 - 2 2 mat δ 0 + 50 blue 1 5 mat 4 1 sys

1 time t0 - min 2

ConvolIntegral u q t u - h * u 0 t int ≡

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

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 0 t int eval :

t0 1 time :

Symbolic wrt augment(,,) Convol t 05 0.025 range : i 1 t rows range y i el t i el L eval : for t y augment 31 line :

Convol Convol :

Convol δ 0 + 20 red 1 5 mat 2 1 sys

1 time t0 - min 2

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

Unary Convol exp(,) over Φ(t,δ) δ t Convol 1 t u - δ - sign + 2 e t u -^* / u 0 t int :

0 x Convol 1 x Convol 2 1 sys

Unary Convol exp(,) over Φ(t,δ) δ t Convol 1 t u - δ - sign + 2 e t^* / u 0 t int :

0 x Convol 1 x Convol 2 1 sys

x 05 0.1 range :

0 x Convol vectorize Max 0.368

Unary Convol exp(,) over Φ(t,δ) δ t Convol 1 t u - δ - sign + 2 e 0.1 t * -^* / u 0 t int :

0 x Convol 1 x Convol 2 1 sys