Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources that easily explain it. Note: I don't understand math notation very well, so a watered-down explanation would do just fine. $$(f * g)(t) = \int_0^t f(\tau)g(t-\tau)\ \mathrm{d}\tau$$

This is what my textbook has written. What do those lowercase t-like symbols represent (I haven't seen them before).

• That's a greek letter called tau. It's just a variable here. – Pedro Dec 11 '12 at 1:53
• Lowercase t-like symbol is a greek letter "tau". Here it represents an integration (dummy) variable, which "runs" from lower integration limit, "0", to upper integration limit, "t". So, the convolution is a function, which value for any value of argument (independent variable) "t" is expressed as an integral over dummy variable "tau". – mbaitoff Dec 11 '12 at 1:55
• ... and it's common to choose related names for related variables. Some other examples of letters that are often paired are $(z, \zeta)$, $(x, \xi)$, $(r, \rho)$, $(s, \sigma)$, $(w, \omega)$, $(a, \alpha)$, $(b, \beta)$, $(d, \delta)$, $(e, \epsilon)$. Also, the letters $\iota, \kappa, \lambda, \mu, \nu$ are often used in places where one might also tend to use $i, k, l, m, n$ respectively. All of the "strange" letters I've used here are from the Greek alphabet. – Hurkyl Dec 11 '12 at 1:57
• Maybe the animated gifs at Wikipedia will give you some ideas. – dtldarek Dec 11 '12 at 2:05
• I'm late to the party, I know, but I recently stumbled across Terry Tao's explanation on mathoverflow. – user170231 Nov 10 '15 at 21:46

Intuitively speaking when you are given two signals/or functions $f$ and $g$. You time reverese one of the signals, it doesnt matter which one, and shift it by a value of $t$ then you simply multiply and then sum the area under the intersection.

If you consider a function say function of $x$, then time reversal means inserting $-x$ wherever you see $x$ in this function.

Example:

Question1: Assume you have a function $f(x)$ that is $1$ if $x\in[0,1]$, and $0$ elsewhere then how should you plot $f(-x)$?

Question2: Assume you have $g(x)=f(x)$ is there any intersecting area between $f(x)$ and $g(x)$?

Question3: Now shift $g(-x)$ by $0.5$, that is to find $g(-x+0.5)$. How does it look like when you plot it?

Question4: Where does the intersecting region lie in this case $x\in?$ what is the are of the intersecting region? Answer: below white area$=0.5$ at $x=0.5$ shift.

Question5: If you select the shifting parameter not $0.5$ but all reals in $[0,2]$ what function should you get at the output? check $x=0.5$ and see $f*g(x)$ is $0.5$ as found at step 4

EDIT: You define convolution integral in $[0,t]$ for bounded signals. The integral limits depend on where your signal is non-zero.

If you have two signals as you suggested $f(t)=e^{at}$ and $g(t)=e^{bt}$ then the first question: what is the relation between $a$ and $b$? are they positive? where is the function defined? For example when $a$ and $b$ are some positive terms then we have the following integral

$$h(t)=\int f(\tau)g(t-\tau)d\tau= \int e^{a\tau}e^{b(t-\tau)}d\tau=e^{bt}\int e^{(a-b)\tau}d\tau=\Bigg]_{\tau\in\Omega}\frac{e^{(a-b)\tau}}{a-b}$$

clearly $\Omega=\mathbb{R}$ is not possible because the integral does not converge.

• Can you use a more concrete example? For example, how can we find the convolution of $f(t) = \sin2t$ and $g(t) = e^t$ or $f(t) = e^{at}$ and $g(t) = e^{bt}$ with $a \neq b$ in the second example? Or perhaps using the same theorem to find an inverse Laplace of $F(s) = \frac{4}{s(s^2+4)}$? – Bailor Tow Dec 11 '12 at 2:36
• please see the edit – Seyhmus Güngören Dec 11 '12 at 3:10

I believe the convolution functions makes the most sense when you see it applied in probability theory.

Let X and Y be two random variables and f(X) and g(Y) be the probability distributions of the random variables.

Then the distribution of the sum of two random variables:

$$(f*g)(t) = \int^t_0 f(-\tau)g(\tau - t )d\tau$$

Why is this? Let us visualize the simple case of rolling dices. and X be the outcome of the first roll and Y be the outcome of the second roll. What is the distribution of the sum?

Since our distributions are discrete, $$(f*g)(t) = \sum^t_{i=0} f(t)g(i-t) \quad t\in[2,12]$$ This basically translates to, sum up all probabilities such that it has this probability.

i.e. $$(f*g)(4)= \sum^4_{i=0} f(t)g(i-t) =f(1)g(3) + f(2)g(2)+ f(3)g(1) = 1/12$$

Which is the answer we expect. We can also look at the question from a more physics point of view where it is a time reversed signals but I find this much more intuitive.

• Shouldn't that be $g(t-i)$? And why isn't the convolution integrand $f(\tau)g(t-\tau)$? – Mateen Ulhaq Aug 2 '15 at 7:15
• This result is proved in a "The Fourier Transform and its applications" lecture on YouTube: youtube.com/watch?v=LA4Uv6PMRTM – Pramod Dec 6 '15 at 23:01

Consider the sequences $x_0, x_1, \dotsc$ and $y_0, y_1, \dotsc$.

Now, $$\left( \sum_{j=0}^n x_j \right) \left( \sum_{j=0}^n y_j \right) = \sum_{z=0}^n z_j,$$ where $$z_j = \sum_{k=0}^j x_k y_{j-k}.$$

• The $j$ subindex on the RHS seems not be bound by anything. – siamii Sep 26 '16 at 16:10
• @siamii: The $j$ is fixed on the LHS. For each $j$ you have one $z_j$. The RHS is how to calculate $z_j$. – André Caldas Oct 2 '16 at 3:51

If you want a broad overview, the convolution "blends" two functions together & is the expression of the amount of overlap of one function as it is shifted over another. The convolution takes two functions (& one of them may be a kernel). Writing one of them as a translation, multiply them together & they give you a new function that takes the best properties of both functions. If you take a kernel (as I mentioned above) the new function may have properties from that kernel. A good example would be from something I am interested in: Littlewood-Paley theory. When embarking on various LP constructions, we end up with LP operators such as

\begin{equation*} \Delta_j(f)=\Delta^{\Psi}_{j}=\Psi_{2^{-j}}\ast f. \end{equation*}

These have been defined by constructing a partition of unity (where we are constructing results locally & extending them globally) where $\Psi$ is a radial Schwartz function on $\mathbb{R}^n$ with certain support properties.

Looking at the convolution, $\Delta_j(f)$ has properties that $f$ had as well as support properties passed onto it from $\Psi.$