Convolution theorem — proof, applications: Foundations

4.6.32 · D1 · Maths › Ordinary Differential Equations › Convolution theorem — proof, applications

Is page par yeh assume kiya gaya hai ki tumne kuch nahi dekha. Isse pehle ki hum $L {f * g} = F (s) G (s)$ padh bhi sakein, humein iske har ek squiggle ko earn karna hoga: function kya hota hai, integral mein kya picture banti hai, woh bada $\int_{0}^{\infty}$ jo $e^{- s t}$ ke saath hai woh actually kya karta hai, dummy variable kya hoti hai, aur "arguments jo $t$ mein add hote hain" poora trick kyun hai. Hum inhe aise order mein build karenge jahan har ek cheez sirf usse pehle waali cheez par rely kare.

0. Jo hum build karne wale hain uska map

Isko upar se neeche padho: functions aur integrals sab kuch feed karte hain; exponential se Laplace transform banta hai; shifting/flipping se convolution banta hai; dono streams theorem par milte hain.

1. Ek function $f (t)$ — ek machine jo ek number ko ek number mein badal deti hai

Picture hai ek graph: horizontal axis input $t$ hai, vertical axis output $f (t)$ hai. Ek curve bas yeh batata hai ki "har $t$ ke liye, $f$ kitna tall hai".

Figure — Convolution theorem — proof, applications

2. Integral $\int$ — infinitely many thin slivers ko add karna

Stretched-S symbol $\int$ literally purana-fashioned "S" hai Sum ke liye. $d t$ decoration nahi hai — yeh batata hai ki tum kis variable ko sweep kar rahe ho aur remind karta hai ki har strip ki width " $d t$ " hai.

Recall Signed area, saare cases

$f (t) > 0$ on $[a, b]$ ::: integral positive hai (area axis ke upar).
$f (t) < 0$ kuch part par ::: woh part subtract hota hai (area axis ke neeche).
$a = b$ ::: integral $0$ hai (koi width nahi, kuch add nahi).
$f (t) = 0$ har jagah ::: integral $0$ hai.

3. Dummy variable $τ$ — ek naam jo sirf integral ke andar rehta hai

$τ$ ko ek pointer ki tarah picture karo jo lower limit se upper limit tak sweep karta hai, har strip ko ek baar tap karta hai.

4. Improper integral $\int_{0}^{\infty}$ — add karna kabhi band mat karo

5. Exponential $e^{- s t}$ — "shrinking weight"

Recall

$e^{- s t}$ ka behaviour har case mein ( $t \geq 0$ )

$t = 0$ ::: value hai $e^{0} = 1$ .
$s > 0$ , $t$ bada ::: value $\to 0$ (decay hoti hai — convergence ke liye achha).
$s = 0$ ::: value saare $t$ ke liye $1$ hai (koi shrinking nahi).
$s < 0$ ::: value grow karta hai, integral diverge ho sakta hai — Laplace ko $s$ kaafi bada chahiye.

6. Laplace transform $F (s) = L {f}$ — poori function ko ek $s$ -curve mein reweigh karo

7. Shift aur flip: $g (t - τ)$ — convolution ka dil

Convolution sense banana shuru kare, usse pehle humein yeh samajhna hoga ki kisi function mein $t - τ$ plug karne se uski picture par kya hota hai.

8. Convolution $f * g$ — saare earned pieces ko assemble karna

Is convolution se phir applications ko power milti hai: ODEs solve karna (dekho Solving Linear ODEs with Laplace Transforms) aur Volterra Integral Equations, aur yeh (alag limits ke saath) Fourier Transform Convolution Theorem aur Transfer Functions and Impulse Response mein dobara appear karta hai.

9. Sab kuch assembled hone ke baad ek chhota sanity computation

Equipment checklist

Test karo khud ko — har ::: ke right side ko cover karo aur dekho ki tum instantly answer de sakte ho ya nahi.

f (t)

ka ek phrase mein kya matlab hai?

Ek rule jo input number

t

ko exactly ek output number mein convert karta hai; iska graph height-vs-

t

hai.

\int_{a}^{b} f (t) d t

kya picture karta hai?

t = a

aur

t = b

ke beech curve ke neeche ki signed area = thin strips ka sum.

τ

jaisi dummy variable kya hoti hai?

Ek naam jo sirf integral ke andar rehta hai, current strip ko label karta hai; ise rename karne se kuch nahi badalta.

Laplace integral ko

e^{- s t}

kyun chahiye?

Yeh ek aisa weight hai jo

t = 0

par

1

hai,

\int_{0}^{\infty}

par convergence force karne ke liye shrink karta hai, aur exponentials ke products ko exponents ke sums mein convert karta hai.

F (s)

kya hai?

f

ka Laplace transform:

\int_{0}^{\infty} e^{- s t} f (t) d t

s

-world mein ek function (na ki

t

L {t}

aur

L {sin t}

kya hain?

1/ s^{2}

aur

1/ (s^{2} + 1)

g (t - τ)

g

ke graph ke saath kya karta hai?

Ise left-right flip karta hai, phir

t

se right slide karta hai.

Convolution arguments

t

mein kyun add hote hain?

τ + (t - τ) = t

; har woh pair of times jo

t

mein sum hote hain,

t

par value mein contribute karte hain.

Convolution ka upper limit

t

kyun hai?

Causality —

g (t - τ) = 0

jab

τ > t

hota hai, toh woh slivers zero add karte hain.

(1 * 1) (t)

kya hai?

t

Connections

Laplace Transform — definition and properties
Inverse Laplace Transform — partial fractions
Solving Linear ODEs with Laplace Transforms
Volterra Integral Equations
Fourier Transform Convolution Theorem
Transfer Functions and Impulse Response