Convolution theorem — proof, applications: Foundations

This page assumes you have seen nothing. Before we can even read $L {f * g} = F (s) G (s)$ , we must earn every squiggle in it: what a function is, what an integral pictures, what that big $\int_{0}^{\infty}$ with $e^{- s t}$ actually does, what a dummy variable is, and why "arguments that add to $t$ " is the whole trick. We build them in an order where each one leans only on the ones before it.

0. The map of what we are about to build

Read it top to bottom: functions and integrals feed everything; the exponential builds the Laplace transform; shifting/flipping builds convolution; the two streams meet at the theorem.

1. A function $f (t)$ — a machine that turns a number into a number

The picture is a graph: the horizontal axis is the input $t$ , the vertical axis is the output $f (t)$ . A curve is just "for each $t$ , how tall is $f$ ".

Figure — Convolution theorem — proof, applications

2. The integral $\int$ — adding up infinitely many thin slivers

The stretched-S symbol $\int$ is literally an old-fashioned "S" for Sum. The $d t$ is not decoration — it tells you which variable you are sweeping and reminds you each strip has width " $d t$ ".

Recall Signed area, all cases

$f (t) > 0$ on $[a, b]$ ::: integral is positive (area above axis).
$f (t) < 0$ on part ::: that part subtracts (area below axis).
$a = b$ ::: integral is $0$ (no width, nothing to add).
$f (t) = 0$ everywhere ::: integral is $0$ .

3. The dummy variable $τ$ — a name that lives only inside the integral

Picture $τ$ as a pointer sweeping from the lower limit to the upper limit, tapping each strip once.

4. The improper integral $\int_{0}^{\infty}$ — never stop adding

5. The exponential $e^{- s t}$ — the "shrinking weight"

Recall Behaviour of

$e^{- s t}$ in every case ( $t \geq 0$ )

$t = 0$ ::: value is $e^{0} = 1$ .
$s > 0$ , $t$ large ::: value $\to 0$ (decays — good for convergence).
$s = 0$ ::: value is $1$ for all $t$ (no shrinking).
$s < 0$ ::: value grows, integral may diverge — Laplace needs $s$ large enough.

6. The Laplace transform $F (s) = L {f}$ — reweigh a whole function into one $s$ -curve

7. Shift and flip: $g (t - τ)$ — the heart of convolution

Before convolution can make sense, we must understand what plugging $t - τ$ into a function does to its picture.

8. Convolution $f * g$ — assembling all the earned pieces

This convolution then powers the applications: solving ODEs (see Solving Linear ODEs with Laplace Transforms) and Volterra Integral Equations, and it reappears (with different limits) in the Fourier Transform Convolution Theorem and in Transfer Functions and Impulse Response.

9. A tiny sanity computation with everything assembled

Equipment checklist

Test yourself — cover the right side of each ::: and see if you can answer instantly.

What does

f (t)

mean in one phrase?

A rule turning input number

t

into exactly one output number; its graph is height-vs-

t

What does

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

picture?

The signed area under the curve between

t = a

and

t = b

= sum of thin strips.

What is a dummy variable like

τ

A name that lives only inside the integral, labelling the current strip; renaming it changes nothing.

Why does the Laplace integral need

e^{- s t}

It's a weight that is

1

t = 0

, shrinks to force convergence over

\int_{0}^{\infty}

, and turns products of exponentials into sums of exponents.

What is

F (s)

The Laplace transform of

f

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

, a function in the

s

-world (not

t

What are

L {t}

and

L {sin t}

1/ s^{2}

and

1/ (s^{2} + 1)

What does

g (t - τ)

do to the graph of

g

Flips it left-right, then slides it right by

t

Why do the convolution arguments add to

t

τ + (t - τ) = t

; each pair of times summing to

t

contributes to the value at

t

Why is the convolution upper limit

t

Causality —

g (t - τ) = 0

for

τ > t

, so those slivers add zero.

What is

(1 * 1) (t)

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