17Calculus - Laplace Transform Convolution
The concept of convolution is pretty straight-forward but it takes some thinking to understand it clearly. You can think of it this way. Functions that are convolved are combined in a different way using an integral.
So what does that mean? Let's start backwards, with two functions multipled together in the s-domain. So we have some function \(h(t)\) whose Laplace Transform is the product of two functions.
\[ \mathcal{L}\{ h(t) \} = F(s)G(s)\]
So, your first inclination is to say that \(h(t)\) is just the product of \(f(t)\) and \(g(t)\) and, although that is a logical conclusion, it is wrong. Remember the product rule from basics derivatives, i.e. \((fg)' = f'g+g'f.\) We can't just take the derivative of each of them and multiply them together, i.e. \((fg)' \neq f'g'.\) Just like the product rule, we can't say that the functions are just multiplied together in the t-domain, i.e.
\[ \mathcal{L}\{ f(t)g(t) \} \neq F(s)G(s)\]
However, there is a relationship between \(f(t), g(t), F(s), G(s)\) expressed this way.
\[\displaystyle{ h(t) = \int_0^t{ f(t-x)g(x)~dx } }\]
The integral on the right of the equal sign is called the convolution of \(f\) and \(g\) and we write this as \(f \ast g\). So now we can write the Laplace Transform of a pair of convolved functions as
\[ \mathcal{L}\{ f(t) \ast g(t) \} = F(s)G(s)\]
where
\[\displaystyle{ f(t) \ast g(t) = \int_0^t{ f(t-x)g(x)~dx } }\]
Now, we have to be careful here, \(f(t) \ast g(t)\) is NOT \(f(t)\) TIMES \(g(t)\) and the \(\ast\) notation does NOT mean composite either. This is a unique new way of combining two functions. The reason it is so important is because of what happens in the s-domain, i.e. \(F(s)G(s)\) IS \(F(s)\) TIMES \(G(s)\) but only in the s-domain, not in the t-domain.
So if we are working in the s-domain and we end up with two functions multipled together, we can use the convolution integral to convert back to the t-domain. We may even be able to evaluate the integral to determine our answer.
Before we go on, let's put everything together that we have so far. The equations are summarized in the following table.
The Convolution Integral |
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\(\displaystyle{ f(t) \ast g(t) = \int_0^t{ f(t-x)g(x)~dx } }\) |
The Laplace Transform |
\( \mathcal{L}\{ f(t) \ast g(t) \} = F(s)G(s)\) |
Here is a video that explains this again. It is always good to hear the same concepts from different sources, multiple times.
MIT OCW - Laplace Transforms and Convolution [10mins-28secs]
video by MIT OCW |
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Okay, you should be convinced by now that \(f(t)g(t) \neq f(t) \ast g(t)\). However, this next video shows an example that will really help you cement that in your mind.
blackpenredpen - laplace transform of \(\sin(t)\cos(t)\) vs laplace transform \(\sin(t) \ast \cos(t)\) [10mins-28secs]
video by blackpenredpen |
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Convolution Properties |
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Convolution has the following properties, which work as you would expect.
Constant Property (c is a constant) |
\( cf \ast g = f \ast cg = c(f \ast g) \) |
Commutative Property |
\(f \ast g = g \ast f\) |
Associative Property |
\( (f \ast g) \ast h = f \ast (g \ast h) \) |
Distributive Property |
\( f \ast (g+h) = f \ast g + f \ast h \) |
Before we go on, let's do an example. Try it on your own before looking at the solution.
Find \(h(t)\) where \(\displaystyle{ \mathcal{L}\{ h(t) \} = \frac{3}{(s-1)(s-4)} }\). First, use partial fractions and then use the convolution integral, showing that the answers are the same.
\( h(t) = e^t(e^{3t}-1) \)
1. Using partial fractions, we get |
\(\displaystyle{ \frac{3}{(s-1)(s-4)} = \frac{-1}{s-1} + \frac{1}{s-4} }\) |
From the table of Laplace Transforms, we know \(\displaystyle{\mathcal{L}\left\{\frac{1}{s-a}\right\} = e^{at}}\) |
So \(\displaystyle{ \mathcal{L}\left\{\frac{-1}{s-1}\right\} = -e^t }\) and \(\displaystyle{ \mathcal{L}\left\{\frac{1}{s-4}\right\} = e^{4t} }\) |
Putting the results together, we have \(h(t)=e^{4t}-e^{t}=e^t(e^{3t}-1)\) |
Final Answer from partial fractions: \(h(t)=e^t(e^{3t}-1)\) |
2. To use the convolution integral, we need to assign \(F(s)\) and \(G(s)\). |
Let \(\displaystyle{ F(s)=\frac{3}{s-1} }\) and \(\displaystyle{ G(s)=\frac{1}{s-4} }\) |
Using the table of Laplace Transforms, we get |
\(f(t)=3e^t\) and \(g(t)=e^{4t}\) |
Our convolution integral is |
\(\displaystyle{ h(t)=f(t)\ast g(t) = \int_0^t{ f(t-x)g(x)~dx } }\) |
With our \(f(t)\) and \(g(t)\) we have |
\(\displaystyle{ h(t)=3\int_0^t{ e^{t-x}e^{4x}~dx } }\) |
\(\displaystyle{ h(t)=3\int_0^t{ e^t e^{-x} e^{4x}~dx } }\) |
\(\displaystyle{ h(t) = 3e^t \int_0^t{ e^{3x}~dx } }\) |
\(\displaystyle{ h(t) = 3e^t \left[ \frac{e^{3x}}{3} \right]_0^t }\) |
\( h(t) = e^t [ e^{3t} - e^0 ] = e^t(e^{3t}-1) \) |
Final Answer from convolution: \( h(t) = e^t(e^{3t}-1) \) |
The answers are equal. |
Final Answer |
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\( h(t) = e^t(e^{3t}-1) \) |
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close solution |
Proof |
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Okay, you should be asking yourself, why does this work? Here is a video with a proof.
blackpenredpen - Proof of the Convolution Theorem [18mins-9secs]
video by blackpenredpen |
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Graphical Concept of Convolution |
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To understand what is going on with the convolution integral in a graphical sense, here is a set of videos that explains it and shows several examples.
Graphical Concept of Convolution Videos
videos by Michel vanBiezen |
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Convolution of Periodic Functions |
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Here is a video containing a summary of the Laplace Transforms of some periodic functions.
Michel vanBiezen - Laplace Transform of Periodic Functions Summary [3mins-38secs]
video by Michel vanBiezen |
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Here are some practice problems to help you get familiar with the convolution integral. After that, you can use the convolution integral to solve differential equations.
Practice
Instructions - Unless otherwise instructed, solve these problems using the convolution integral.
Note: Although we use \(x\) on this page as our variable of integration, some of these practice problem solutions use the greek letter tau \(\tau\) for the same purpose.
Basic Problems |
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Problem Statement |
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Evaluate \( \mathcal{L}\{ \int_0^t{ 3(t-\tau)^2 \sin(2\tau)d\tau }\} \) using the convolution integral.
Final Answer |
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Problem Statement |
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Evaluate \( \mathcal{L}\{ \int_0^t{ 3(t-\tau)^2 \sin(2\tau)d\tau }\} \) using the convolution integral.
Solution |
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2236 video
video by Gregg Waterman |
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Final Answer |
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\(\displaystyle{ \frac{12}{s^3(s^2+4)} }\) |
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Problem Statement |
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Evaluate \( \mathcal{L}\{ \int_0^t{ \tau e^{-2(t-\tau)}~d\tau } \} \) using the convolution integral.
Final Answer |
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Problem Statement |
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Evaluate \( \mathcal{L}\{ \int_0^t{ \tau e^{-2(t-\tau)}~d\tau } \} \) using the convolution integral.
Solution |
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2237 video
video by Gregg Waterman |
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Final Answer |
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\(\displaystyle{ \frac{1}{s^2(s+2)} }\) |
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{5}{s^3(s+4)} \right\} }\) as a convolution integral.
Final Answer |
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{5}{s^3(s+4)} \right\} }\) as a convolution integral.
Solution |
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2238 video
video by Gregg Waterman |
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Final Answer |
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\(\displaystyle{ \frac{5}{2}\int_0^t{ (t-\tau)^2 e^{-4\tau} ~ d\tau } }\) |
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{3s}{(s^2+16)^2} \right\} }\) as a convolution integral.
Final Answer |
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{3s}{(s^2+16)^2} \right\} }\) as a convolution integral.
Solution |
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2239 video
video by Gregg Waterman |
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Final Answer |
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\(\displaystyle{ \int_0^t{ \cos[4(t-\tau)]\sin(4\tau)~d\tau } }\) |
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close solution
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{H(s)}{s+2} \right\} }\) as a convolution integral.
Final Answer |
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Problem Statement |
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Write \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{H(s)}{s+2} \right\} }\) as a convolution integral.
Solution |
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2240 video
video by Gregg Waterman |
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Final Answer |
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\(\displaystyle{ \int_0^t{ h(t-\tau)e^{-2\tau}~d\tau } }\) |
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Problem Statement |
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For a certain system, the impulse response is \(h(t)=2e^{-2t}\) and the source voltage is \(v_s(t)=10e^{-t}\). Determine the output response \(v_o(t)=h(t)\ast v_s(t)\).
Final Answer |
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Problem Statement |
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For a certain system, the impulse response is \(h(t)=2e^{-2t}\) and the source voltage is \(v_s(t)=10e^{-t}\). Determine the output response \(v_o(t)=h(t)\ast v_s(t)\).
Solution |
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2260 video
video by Michel vanBiezen |
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Final Answer |
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\( v_o(t) = 20[e^{-t}-e^{-2t}] \) |
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Intermediate Problems |
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Problem Statement |
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Evaluate \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{s}{(s^2+1)^2} \right\} }\) using the convolution integral.
Final Answer |
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Problem Statement |
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Evaluate \(\displaystyle{ \mathcal{L}^{-1}\left\{ \frac{s}{(s^2+1)^2} \right\} }\) using the convolution integral.
Solution |
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2241 video
video by blackpenredpen |
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Final Answer |
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\(\displaystyle{ \frac{t\sin(t)}{2} }\) |
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Problem Statement |
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Calculate the Laplace Transform of the periodic function which is a constant pulse of height \(k\) and width \(a\). The period is \(2a\).
Hint |
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The Laplace Transform of a general periodic function \(f(t)\) of period P is \(\displaystyle{ \mathcal{L}[f(t)] = \frac{1}{1-e^{-sP}}\int_0^P{e^{-s\tau}f(\tau)~d\tau} }\), which is derived in practice problem 2257.
Problem Statement |
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Calculate the Laplace Transform of the periodic function which is a constant pulse of height \(k\) and width \(a\). The period is \(2a\).
Final Answer |
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Problem Statement |
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Calculate the Laplace Transform of the periodic function which is a constant pulse of height \(k\) and width \(a\). The period is \(2a\).
Hint |
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The Laplace Transform of a general periodic function \(f(t)\) of period P is \(\displaystyle{ \mathcal{L}[f(t)] = \frac{1}{1-e^{-sP}}\int_0^P{e^{-s\tau}f(\tau)~d\tau} }\), which is derived in practice problem 2257.
Solution |
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2258 video
video by Michel vanBiezen |
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Final Answer |
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\(\displaystyle{ \mathcal{L}[f(t)] = \frac{k(1-e^{-as})}{s(1-e^{-2as})} }\) |
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Advanced Problems |
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Problem Statement |
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Find the Laplace Transform of a periodic function \(f(t)\) with period \(P\), i.e. \(f(t+P)=f(t)\).
Hint |
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Your answer will contain a definite integral since you are not given a specific function \(f(t)\).
Problem Statement |
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Find the Laplace Transform of a periodic function \(f(t)\) with period \(P\), i.e. \(f(t+P)=f(t)\).
Final Answer |
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Problem Statement |
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Find the Laplace Transform of a periodic function \(f(t)\) with period \(P\), i.e. \(f(t+P)=f(t)\).
Hint |
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Your answer will contain a definite integral since you are not given a specific function \(f(t)\).
Solution |
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2257 video
video by Michel vanBiezen |
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Final Answer |
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\(\displaystyle{ \mathcal{L}[f(t)] = \frac{1}{1-e^{-sP}}\int_0^P{ e^{-s\tau} f(\tau) ~d\tau } }\) |
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