Visual walkthrough — Impulse response and transfer function (GNC connection)

4.6.33 · D2 · Maths › Ordinary Differential Equations › Impulse response and transfer function (GNC connection)

Yeh page parent topic ka picture-version hai. Agar koi word naya lage, toh hum use pehle define karte hain, phir use karte hain.

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Step 1 — "Ek single kick" dikhti kaisi hai?

KYA HAI. Hum woh input draw karte hain jisse poori theory shuru hoti hai: Dirac delta $\delta (t)$. Yeh ek idealised push hai jo exactly ek instant mein hoti hai.

YEH OBJECT KYUN. Real pushes (jaise thruster burst) thodi der tak chalti hain. System ka pure "fingerprint" pane ke liye hum uss push ko shrink karte hain: total shove ko $1$ par fixed rakhte hain lekin use thinner aur taller banate hain. Limit mein hume ek infinitely thin, infinitely tall spike milta hai jiska area exactly 1 hi rehta hai. Area = total shove woh cheez hai jo physically matter karti hai, isliye hum ise constant rakhte hain.

PICTURE. Dekho kaise rectangle taller hota jaata hai jaise jaise thinner hota hai — shaded area (andar likha number) kabhi nahi badlta.

Ek property jo hum actually use karenge woh hai sifting property: kisi bhi smooth function $f$ ko $t=a$ par spike se multiply karke integrate karo toh sirf $f(a)$ milta hai. ${\int }_{-\infty }^{\infty }f(t)\delta (t-a)dt=f(a).$ Poori kahani ke liye Dirac delta function dekho.

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Step 2 — Uss ek kick ki response: $h(t)$

KYA HAI. Spike $\delta (t)$ ko ek aise system mein daalo jo at rest hai (kuch nahi chal raha, output $t=0$ se pehle flat zero hai). Jo output nikalti hai use impulse response $h(t)$ kehte hain.

YAHAN SE KYUN SHURU KAREN. Time-invariant + linear systems mein ek magical property hoti hai (Steps 4–5): agar tum ek standard input ka jawab jaante ho, toh har input ka jawab jaante ho. Delta sabse simple possible standard input hai — ek single instant of forcing — isliye iska response natural "unit" hai measure karne ka.

PICTURE. Ek vertical spike $t=0$ par andar jaati hai; curve $h(t)$ baad mein ring karta hai aur (stable system ke liye) die away ho jaata hai. Note karo ki kick se pehle yeh flat aur zero hai — yeh hai causality: cause se pehle koi output nahi.

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Step 3 — Kisi bhi input ko chhoti kicks ki row mein kaato

KYA HAI. Ek arbitrary input $u(t)$ lo — ek wiggly thruster command. Hum ise bahut saari chhoti spikes ke sum ke roop mein likhte hain, har instant $\tau$ par ek, har ek $u(\tau )$ height par scaled.

KYUN. Hum sirf jaante hain ki system spike ka kya jawab deta hai. Toh hum $u$ ko spikes ki language mein express karte hain. Sifting property hume exactly yeh karne deti hai: $u(t)={\int }_{-\infty }^{\infty }u(\tau )\delta (t-\tau )d\tau .$

PICTURE. Smooth grey curve $u$ ko picket-fence of thin bars se approximate kiya gaya hai. Har bar ek scaled, shifted spike hai; uski height uss jagah $u$ ki value hai.

Term by term:

• $\tau$ ::: woh instant jahan ek particular tiny kick baithti hai.
• $\delta (t-\tau )$ ::: ek spike jo time $\tau$ par move ho gayi hai ($-\tau$ ise right slide karta hai).
• $u(\tau )$ ::: hum uss kick ko kitna strong banate hain — signal ki local height.
• $\int d\tau$ ::: har instant mein saari kicks ko add karo.

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Step 4 — Do rules kicks ko outputs mein convert karti hain

KYA HAI. Hum LTI system ki do defining properties use karte hain har input-spike ko output mein convert karne ke liye.

YEH DON, AUR KOI NAHI, KYUN. Yeh sirf wahi assumptions hain jo humne system ke baare mein ki hain, isliye poora result sirf inhi se follow karna chahiye:

1. Time-invariance — baad mein kick karo toh same shape milti hai, shifted. Toh input $\delta (t-\tau )$ se output $h(t-\tau )$ milta hai.
2. Linearity — input scale karo toh output scale hoti hai, aur inputs add hote hain. Toh scaled kick $u(\tau )\delta (t-\tau )$ se scaled response $u(\tau )h(t-\tau )$ milta hai.

PICTURE. Left column: teen shifted/scaled input spikes. Right column: har ek map hota hai usi fingerprint $h$ ki shifted, scaled copy mein. Same shape, moved aur stretched.

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Step 5 — Copies add karo: convolution appear hoti hai

KYA HAI. $h$ ki saari shifted, scaled copies ko sum (integrate) karo. Woh superposition hi output hai.

KYUN. Input kicks ka sum tha (Step 3); har kick $h$ ki ek copy deti hai (Step 4); linearity kehti hai total output total copies ka total hai. $\tau$ par integrate karte hue: $\boxed{y(t)={\int }_{-\infty }^{\infty }u(\tau )h(t-\tau )d\tau =(u\ast h)(t)}$

PICTURE. $h$ ki ek flipped copy ko input par slide karo; har position $t$ par overlap area $y(t)$ trace karta hai. Yeh sliding-overlap exactly wahi hai jo Convolution ka matlab hai.

Term by term:

• $h(t-\tau )$ ::: fingerprint, flipped ($-\tau$) aur shifted taaki time $t$ ke neeche line up ho.
• $u(\tau )h(t-\tau )$ ::: $\tau$ par ki kick baad ke time $t$ par kitna contribute karti hai.
• $u\ast h$ ::: "slide, multiply, aur overlap add karo" ka shorthand.

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Step 6 — Convolution annoying hai; Laplace se language switch karo

KYA HAI. Hum Laplace transform $\mathcal{L}$ par move karte hain, jo sliding integral ko plain multiplication mein trade karta hai.

YEH TOOL KYUN. Convolution (slide-multiply-add) directly compute karna painful hai. Laplace transform mein convolution theorem hai: yeh $u\ast h$ ko simple product $U(s)H(s)$ mein convert kar deta hai. Hum Laplace choose karte hain (Fourier nahi) kyunki yeh $t\ge 0$, at-rest setup aur derivatives ko bhi cleanly handle karta hai.

$\mathcal{L}\{f\}(s)=F(s)={\int }_0^{\infty }{e}^{-st}f(t)dt.$

PICTURE. Ek two-world map: "time world" (hard convolution) left mein bridge $\mathcal{L}$ se connect hota hai "$s$ world" (easy multiplication) se right mein.

Do facts jo chahiye, aur kyun yeh hold karte hain:

• $\mathcal{L}\{\delta (t)\}={\int }_0^{\infty }{e}^{-st}\delta (t)dt=e^0=1$ — sifting $e^{-st}$ ko $t=0$ par read karta hai.
• $\mathcal{L}\{f^{\prime }\}=sF(s)-f(0)$ — integration by parts se. Jab sab kuch at rest ho, $f(0)=0$, toh har derivative sirf multiply-by-$s$ ban jaata hai.

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Step 7 — Transfer function nikal aati hai

KYA HAI. Poori ODE $L[y]=u$ par $\mathcal{L}$ apply karo, at rest. Har $y^{(k)}$ ban jaata hai $s^{k}Y(s)$.

KYUN. Kyunki derivatives powers of $s$ ban jaate hain, differential equation ek algebraic equation ban jaati hai — koi calculus nahi bacha, bas divide karo.

$(a_n{s}^n+\cdots +a_{1}s+a_0)Y(s)=U(s)$ $\boxed{H(s):=\frac{Y(s)}{U(s)}=\frac{1}{a_n{s}^n+\cdots +a_0}}$

Term by term:

• $s^k$ ::: $k$-th derivative kya ban gaya.
• $a_n{s}^n+\cdots +a_0$ ::: characteristic polynomial jisme $\lambda$ ki jagah $s$ hai.
• $H(s)$ ::: transfer function — $s$ world mein output-over-input.

PICTURE. ODE (derivative arrows ke saath) ek polynomial box mein morph ho jaati hai; $Y=H\cdot U$ ek single multiply hai.

Kyunki impulse ka $U(s)=\mathcal{L}\{\delta \}=1$ hai, uski output hai $Y=H\cdot 1=H$. Transform undo karo: $H(s)=\mathcal{L}\{h(t)\}.$ Transfer function literally fingerprint hai, re-written.

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Step 8 — Poles ke har case (jahan stability rehti hai)

KYA HAI. Denominator ke roots poles $p$ hain. $H(s)$ invert karne se time-terms $e^{pt}$ milte hain, isliye har pole ki plane mein location $h$ ki shape fix karti hai. Hume har case cover karna hai.

KYUN. Ek pole ek complex number $p=\sigma +j\omega$ hai: $\sigma$ (real part) grow/decay control karta hai, $\omega$ (imaginary part) wobble control karta hai. Ek case miss karna matlab spacecraft ka ek behaviour miss karna.

PICTURE. Complex plane imaginary axis se split. Har region mein $h(t)$ ka matching time-plot hai.

Pole location	$e^{pt}$ behave karta hai	$h(t)$ ki shape	Verdict
$\sigma <0$, real	shrinking exponential	smooth decay	stable
$\sigma <0$, complex pair	decaying $\times$ sine	ringing jo die ho jaaye	stable
$\sigma =0$, simple pair	pure sine	hamesha oscillate	marginal
$\sigma =0$, repeated (e.g. $1/s^2$)	$t$	ramp, unbounded	marginal / unstable
$\sigma >0$	growing exponential	blow up ho jaata hai	unstable

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Ek-picture summary

Upar sab kuch, ek canvas par: kick $\delta$ → fingerprint $h$ → kisi bhi output ke liye convolve → Laplace bridge cross karo → $H$ se multiply karo → poles se stability padho.

Recall Feynman: poora walk plain words mein

System ko ek super-quick tap do aur dekho woh kaise ring karta hai aur settle hota hai — woh ring uski personality hai, jise $h$ kehte hain. Koi bhi complicated push asli mein sirf lambi line of tiny taps hai, ek ke baad ek, har ek thodi strong ya weak. Kyunki system har tap ka same personality se jawab deta hai (bas moved aur resized), poora jawab $h$ ki shifted, resized copies ka dhera hai ek saath jodke — yeh dhera lagna hi convolution hai. Dhera jodan slow hai, isliye hum Laplace transform naam ke ek bridge par chalke ek aisi duniya mein jaate hain jahan "dhera jodan" plain "multiply" ban jaata hai. Wahan personality ek fraction hai, $H(s)$, characteristic polynomial ke upar ek. Us fraction ke neeche special numbers hote hain jinhe poles kehte hain; agar koi bhi pole map ke right side par baitha hai, toh ring badhta hai aur spacecraft tumble karta hai, isliye engineers har pole ko left push karte hain, jahan rings die down ho jaate hain. Yeh ek idea — tap, ring, dhera jodo, multiply karo, poles dekho — poora chapter hai.

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Connections

• Parent topic
• Dirac delta function
• Laplace transform
• Convolution
• Characteristic polynomial and roots
• Linear constant-coefficient ODEs
• Stability and poles (left-half plane)
• Feedback control systems
• Guidance Navigation and Control (GNC)