6.3 Convolution

Convolution

We said that the Laplace transformation of a product is not the product of the transforms. All hope is not lost however. We simply have to use a different type of a “product.” Take two functions $f(t)$ and $g(t)$ defined for $t \geq 0$ , and define the convolution1 of $f(t)$ and $g(t)$ as

$\label{eq:1} (f \ast g)(t) \overset{\rm{def}}{=} \int_0^t f(\tau)g(t- \tau)d\tau.$

(6.3.1)

As you can see, the convolution of two functions of $t$ is another function of $t$ .

A pale wave runs across the frame at fixed height. A second, dashed wave of the same height occupies only the region left of a moving vertical line, and shifts bodily as that line is dragged rightward. A heavy curve fills the same region, being the product of the two, swinging above and below the axis. A separate curve behind them oscillates with a height that grows steadily larger the further right it goes, staying inside a pair of dashed straight lines that widen away from the origin — so its swings start small near the left edge and become several times taller by the right. Adjustable parameters: t — how far g has slid (s) = 2.5, Frequency ω (w) = 1 rad/s. Viewing window: x from -2.07 to 8.07, y from -3.13 to 3.13. — Lebl's Example 6.3.2 worked as a picture: (sin ωt ∗ cos ωt)(t) = ½ t sin(ωt). The pale curve is f(τ) = sin(ωτ), fixed. The dashed curve is g(t − τ) = cos(ω(t − τ)) — g REVERSED and slid to the right by t, which is the whole content of the definition ∫₀ᵗ f(τ)g(t − τ)dτ. The heavy orange curve is their product, drawn only on 0 ≤ τ ≤ t, and its signed area is the answer. Drag t: the window widens, the reversed copy slides, and the bold blue curve is TRACED OUT as you go — that is the running result, and the pale blue curve ahead of it is where it is headed. Notice the blue curve does not just oscillate — it grows inside the dashed ±t/2 envelope. That factor of t is why §2.6's resonance solution grows without bound, and here you can watch it accumulate rather than take it from the algebra.

Example 1

Take $f(t)=e^t$ and $g(t) = t$ for $t \geq 0$ . Then

$(f \ast g)(t) = \int_0^t e^{\tau}(t- \tau)d\tau = e^t-t-1. \nonumber$

To solve the integral we did one integration by parts.

Example 2

Take $f(t)=\sin(\omega t)$ and $g(t)=\cos(\omega t)$ for $t \geq 0$ . Then

$(f \ast g)(t) = \int_0^t \sin(\omega \tau)\cos(\omega (t- \tau))d\tau. \nonumber$

We apply the identity

$\cos(\theta)\sin(\psi)= \dfrac{1}{2}(\sin(\theta + \psi)-\sin(\theta -\psi )) \nonumber$

Hence,

$\begin{align}\begin{aligned} (f \ast g)(t) & = \int_0^t \dfrac{1}{2}(\sin(\omega t) - \sin(\omega t-2 \omega \tau )) d\tau \\ &=\left[ \dfrac{1}{2} \tau \sin(\omega t)+\dfrac{1}{4\omega} \cos(2\omega \tau -\omega t) \right]_{\tau =0}^t \\ &= \dfrac{1}{2} t \sin(\omega t). \end{aligned} \end{align} \nonumber$

The formula holds only for $t \geq 0$ . We assumed that $f$ and $g$ are zero (or simply not defined) for negative $t$ .

The convolution has many properties that make it behave like a product. Let $c$ be a constant and $f$ , $g$ , and $h$ be functions then

$\begin{align}\begin{aligned} f*g &= g* f \\ (c\,f)*g &= f*(c\,g)=c\,(f*g) \\ (f*g)*h&=f*(g*h) \end{aligned}\end{align} \nonumber$

The most interesting property for us, and the main result of this section is the following theorem.

Note

Let $f(t)$ and $g(t)$ be of exponential type, then

$\mathcal{L}\{(f \ast g)(t)\}= \mathcal{L} \left\{ \int_0^t f(\tau)g(t- \tau)d\tau \right\}=\mathcal{L}\{f(t)\} \mathcal{L} \{g(t)\}. \nonumber$

In other words, the Laplace transform of a convolution is the product of the Laplace transforms. The simplest way to use this result is in reverse.

Example 3

Suppose we have the function of $s$ defined by

$\dfrac{1}{(s+1)s^2}=\dfrac{1}{s+1}\dfrac{1}{s^2}. \nonumber$

We recognize the two entries of Table 6.1.2. That is

$\mathcal{L}^{-1} \left\{ \dfrac{1}{s+1} \right\}=e^{-t}~~~~~~ {\rm{and}}~~~~~~ \mathcal{L}^{-1} \left\{ \dfrac{1}{s^2} \right\}=t. \nonumber$

Therefore,

$\mathcal{L}^{-1} \left\{ \dfrac{1}{s+1} \dfrac{1}{s^2} \right\}=\int_0^t \tau e^{-(t- \tau)}d\tau =e^{-t}+t-1. \nonumber$

The calculation of the integral involved an integration by parts.

Solving ODEs

The next example demonstrates the full power of the convolution and the Laplace transform. We can give the solution to the forced oscillation problem for any forcing function as a definite integral.

Example 4

Find the solution to

$x''+\omega_0^2x=f(t),~~~~~x(0)=0,~~~~~x'(0)=0, \nonumber$

for an arbitrary function $f(t)$ .

We first apply the Laplace transform to the equation. Denote the transform of $x(t)$ by $X(s)$ and the transform of $f(t)$ by $F(s)$ as usual.

$s^2X(s)+ \omega_0^2X(s)=F(s), \nonumber$

or in other words

$X(s)=F(s) \dfrac{1}{s^2+ \omega_0^2}. \nonumber$

We know

$\mathcal{L}^{-1} \left\{ \dfrac{1}{s^2+ \omega_0^2} \right\}= \dfrac{\sin(\omega_0 t)}{\omega_0}. \nonumber$

Therefore,

$x(t)= \int_0^tf(\tau)\dfrac{\sin(\omega_0( t-\tau))}{\omega_0}d\tau, \nonumber$

or if we reverse the order

$x(t)= \int_0^t \dfrac{\sin(\omega_0 \tau)}{\omega_0}f(t-\tau)d\tau. \nonumber$

Let us notice one more feature of this example. We can now see how Laplace transform handles resonance. Suppose that $f(t)=\cos(\omega_0t)$ . Then

$x(t)= \int_0^t \dfrac{\sin(\omega_0 \tau)}{\omega_0}\cos(\omega_0(t-\tau))d\tau = \dfrac{1}{\omega_0}\int_0^t \sin(\omega_0 \tau)\cos(\omega_0(t-\tau))d\tau. \nonumber$

We have computed the convolution of sine and cosine in Example 6.3.2. Hence

$x(t)=\left( \dfrac{1}{\omega_0}\right) \left( \dfrac{1}{2}t\sin(\omega_0t)\right) = \dfrac{1}{2\omega_0} \sin(\omega_0t). \nonumber$

Note the $t$ in front of the sine. The solution, therefore, grows without bound as $t$ gets large, meaning we get resonance.

Similarly, we can solve any constant coefficient equation with an arbitrary forcing function $f(t)$ as a definite integral using convolution. A definite integral, rather than a closed form solution, is usually enough for most practical purposes. It is not hard to numerically evaluate a definite integral.

Volterra Integral Equation

A common integral equation is the Volterra integral equation2

$x(t) = f(t) + \int_0^t g(t-\tau)x(\tau)\, d\tau \nonumber$

where $f(t)$ and $g(t)$ are known functions and $x(t)$ is an unknown we wish to solve for. To find $x(t)$ , we apply the Laplace transform to the equation to obtain

$X(s)=F(s)+G(s)X(s), \nonumber$

where $X(s)$ , $F(s)$ , and $G(s)$ are the Laplace transforms of $x(t)$ , $f(t)$ , and $g(t)$ , respectively. We find

$X(s) = \dfrac{F(s)}{1-G(s)}. \nonumber$

To find $x(t)$ we now need to find the inverse Laplace transform of $X(s)$ .

Example 5

Solve

$x(t) = e^{-t} + \int _0^t \sinh(t-\tau)x(\tau)\, d\tau \nonumber$

We apply Laplace transform to obtain

$X(s) = \dfrac{1}{s+1} + \dfrac{1}{s^2-1}X(s), \nonumber$

$X(s) = \dfrac{\dfrac{1}{s+1}}{1-\dfrac{1}{s^2-1}} = \dfrac{s-1}{s^2-2}= \dfrac{s}{s^2-2}-\dfrac{1}{s^2-2}. \nonumber$

It is not hard to apply Table 6.1.1 to find

$x(t) = \cosh \left( \sqrt{2}\, t \right) -\dfrac{1}{\sqrt{2}} \sinh \left(\sqrt{2}\,t \right). \nonumber$

Footnotes

[1] For those that have seen convolution defined before, you may have seen it defined as $f\ast g)(t)=\int_{-\infty}^{\infty}f(\tau )g(t-\tau )d\tau$ . This definition agrees with (6.3.1) if you define $f(t)$ and $g(t)$ to be zero for $t<0$ . When discussing the Laplace transform the definition we gave is sufficient. Convolution does occur in many other applications, however, where you may have to use the more general definition with infinities.

[2] Named for the Italian mathematician Vito Volterra (1860–1940).

Adapted from Differential Equations for Engineers by Jiří Lebl (https://www.jirka.org/diffyqs/), © Jiří Lebl, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.