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

This page is the picture-version of the parent topic. If a word here feels new, we define it before we use it.

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Step 1 — What does "a single kick" even look like?

WHAT. We draw the input that starts the whole theory: the Dirac delta $\delta (t)$. It is an idealised push that happens at exactly one instant.

WHY this object. Real pushes (a thruster burst) last a short time. To find the system's pure "fingerprint" we shrink that push: keep the total shove fixed at $1$ but squeeze it thinner and taller. In the limit we get an infinitely thin, infinitely tall spike whose area is still exactly 1. Area = total shove is the thing that matters physically, so we hold it constant.

PICTURE. Look at how the rectangle grows taller as it grows thinner — the shaded area (the number written inside) never changes.

The one property we will actually use is the sifting property: multiplying any smooth function $f$ by a spike at $t=a$ and integrating just reads off $f(a)$. ${\int }_{-\infty }^{\infty }f(t)\delta (t-a)dt=f(a).$ See Dirac delta function for the full story.

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Step 2 — The response to that one kick: $h(t)$

WHAT. Feed the spike $\delta (t)$ into a system that is at rest (nothing moving, output flat at zero before $t=0$). The output that comes out is called the impulse response $h(t)$.

WHY start here. Time-invariant + linear systems have a magic property (Steps 4–5): if you know the answer to one standard input, you know the answer to every input. The delta is the simplest possible standard input — a single instant of forcing — so its response is the natural "unit" to measure.

PICTURE. A vertical spike goes in at $t=0$; the curve $h(t)$ rings out afterward and (for a stable system) dies away. Note it is flat and zero before the kick — this is causality: no output before the cause.

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Step 3 — Chop any input into a row of tiny kicks

WHAT. Take an arbitrary input $u(t)$ — a wiggly thruster command. We rewrite it as a sum of many little spikes, one at each instant $\tau$, each scaled to the height $u(\tau )$.

WHY. We only know how the system answers a spike. So we express $u$ purely in the language of spikes. The sifting property lets us do exactly that: $u(t)={\int }_{-\infty }^{\infty }u(\tau )\delta (t-\tau )d\tau .$

PICTURE. The smooth grey curve $u$ is approximated by a picket-fence of thin bars. Each bar is a scaled, shifted spike; its height is $u$ at that spot.

Term by term:

• $\tau$ ::: the instant where a particular tiny kick sits.
• $\delta (t-\tau )$ ::: a spike moved to time $\tau$ (the $-\tau$ slides it right).
• $u(\tau )$ ::: how strong we make that kick — the local height of the signal.
• $\int d\tau$ ::: add up all the kicks across every instant.

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Step 4 — Two rules turn kicks into outputs

WHAT. We use the two defining properties of an LTI system to convert each input-spike into an output.

WHY these two, and no others. They are the only assumptions we made about the system, so the whole result must follow from just them:

1. Time-invariance — kicking later gives the same shape, shifted. So input $\delta (t-\tau )$ produces output $h(t-\tau )$.
2. Linearity — scaling the input scales the output, and inputs add. So the scaled kick $u(\tau )\delta (t-\tau )$ produces the scaled response $u(\tau )h(t-\tau )$.

PICTURE. Left column: three shifted/scaled input spikes. Right column: each maps to a shifted, scaled copy of the same fingerprint $h$. Same shape, moved and stretched.

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Step 5 — Add the copies: convolution appears

WHAT. Sum (integrate) all the shifted, scaled copies of $h$. That superposition is the output.

WHY. The input was a sum of kicks (Step 3); each kick gives a copy of $h$ (Step 4); linearity says the total output is the total of the copies. Integrating over $\tau$: $\boxed{y(t)={\int }_{-\infty }^{\infty }u(\tau )h(t-\tau )d\tau =(u\ast h)(t)}$

PICTURE. Slide a flipped copy of $h$ across the input; the overlap area at each position $t$ traces out $y(t)$. This sliding-overlap is exactly what Convolution means.

Term by term:

• $h(t-\tau )$ ::: the fingerprint, flipped ($-\tau$) and shifted to line up under time $t$.
• $u(\tau )h(t-\tau )$ ::: how much the kick at $\tau$ still contributes at the later time $t$.
• $u\ast h$ ::: shorthand for "slide, multiply, and add up the overlap."

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Step 6 — Convolution is annoying; switch languages with Laplace

WHAT. We move to the Laplace transform $\mathcal{L}$, which trades the sliding integral for plain multiplication.

WHY this tool. Convolution (slide-multiply-add) is painful to compute directly. The Laplace transform has the convolution theorem: it turns $u\ast h$ into a simple product $U(s)H(s)$. We pick Laplace (not Fourier) because it also handles the $t\ge 0$, at-rest setup and derivatives cleanly.

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

PICTURE. A two-world map: the "time world" (hard convolution) on the left connects by a bridge $\mathcal{L}$ to the "$s$ world" (easy multiplication) on the right.

Two facts we need, and why they hold:

• $\mathcal{L}\{\delta (t)\}={\int }_0^{\infty }{e}^{-st}\delta (t)dt=e^0=1$ — sifting reads $e^{-st}$ at $t=0$.
• $\mathcal{L}\{f^{\prime }\}=sF(s)-f(0)$ — from integration by parts. With everything at rest, $f(0)=0$, so each derivative just becomes multiply-by-$s$.

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Step 7 — The transfer function drops out

WHAT. Apply $\mathcal{L}$ to the whole ODE $L[y]=u$, at rest. Every $y^{(k)}$ becomes $s^{k}Y(s)$.

WHY. Since derivatives become powers of $s$, the differential equation becomes an algebraic equation — no calculus left, just divide.

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

Term by term:

• $s^k$ ::: what the $k$-th derivative turned into.
• $a_n{s}^n+\cdots +a_0$ ::: the characteristic polynomial wearing an $s$ instead of a $\lambda$.
• $H(s)$ ::: the transfer function — output-over-input in the $s$ world.

PICTURE. The ODE (with derivative arrows) morphs into a polynomial box; $Y=H\cdot U$ is a single multiply.

Because the impulse has $U(s)=\mathcal{L}\{\delta \}=1$, its output is $Y=H\cdot 1=H$. Undoing the transform: $H(s)=\mathcal{L}\{h(t)\}.$ The transfer function is literally the fingerprint, re-written.

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Step 8 — Every case of the poles (where stability lives)

WHAT. The denominator's roots are the poles $p$. Inverting $H(s)$ gives time-terms $e^{pt}$, so each pole's location in the plane fixes the shape of $h$. We must cover every case.

WHY. A pole is a complex number $p=\sigma +j\omega$: $\sigma$ (real part) controls grow/decay, $\omega$ (imaginary part) controls wobble. Missing a case means missing a behaviour of the spacecraft.

PICTURE. The complex plane split by the imaginary axis. Each region carries the matching time-plot of $h(t)$.

Pole location	$e^{pt}$ behaves as	$h(t)$ shape	Verdict
$\sigma <0$, real	shrinking exponential	smooth decay	stable
$\sigma <0$, complex pair	decaying $\times$ sine	ringing that dies	stable
$\sigma =0$, simple pair	pure sine	forever oscillates	marginal
$\sigma =0$, repeated (e.g. $1/s^2$)	$t$	ramp, unbounded	marginal / unstable
$\sigma >0$	growing exponential	blows up	unstable

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The one-picture summary

Everything above, on one canvas: kick $\delta$ → fingerprint $h$ → convolve for any output → cross the Laplace bridge → multiply by $H$ → read stability from the poles.

Recall Feynman: the whole walk in plain words

Give the system one super-quick tap and watch how it rings and settles — that ring is its personality, called $h$. Any complicated push is really just a long line of tiny taps, one after another, each a little stronger or weaker. Since the system answers every tap with the same personality (just moved and resized), the full answer is a pile of shifted, resized copies of $h$ added together — that pile-up is convolution. Adding piles is slow, so we walk across a bridge called the Laplace transform into a world where "pile up" becomes plain "multiply." Over there the personality is a fraction, $H(s)$, one over the characteristic polynomial. The bottom of that fraction has special numbers called poles; if any pole sits on the right side of the map, the ring grows and the spacecraft tumbles, so engineers push every pole to the left, where rings die down. That single idea — tap, ring, pile up, multiply, watch the poles — is the entire chapter.

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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)