# What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability distribution of $X+Y$. This is the only intuition I have for what convolution means.

Are there any other intuitive models for the process of convolution?

• See my previous question: mathoverflow.net/questions/1977/why-is-the-gradient-normal Nov 18, 2009 at 1:35
• There are some nice interpretations listed in the Wikipedia article: en.wikipedia.org/wiki/Convolution#Applications Nov 18, 2009 at 2:54
• Wikipedia Jan 17, 2014 at 16:20
• sites.google.com/site/butwhymath/m/convolution Feb 6, 2015 at 15:01
• On the space of functions from an abelian group to $\Bbb{C}$, a linear map $T$ is a convolution operator iff $(Tf)(x) = (Tf(.+x))(0)$. Shift, derivatives and $\int$ are convolution operators. Convolution by $1_{[0,1]}$ means $Tf(x)$ is the mean-value of $f$ on $x-[0,1]$, the minus sign is to make $f \ast g = g \ast f$ so that functions spaces are convolution algebras. Jun 22, 2019 at 19:52

I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad first approximation for this effect). I found this to be very helpful, not just for understanding convolution per se, but as a lesson that one should try to use physical intuition to model mathematical concepts whenever one can.

More generally, if one thinks of functions as fuzzy versions of points, then convolution is the fuzzy version of addition (or sometimes multiplication, depending on the context). The probabilistic interpretation is one example of this (where the fuzz is a a probability distribution), but one can also have signed, complex-valued, or vector-valued fuzz, of course.

• Also, Gaussian Blur is a convolution filter on some image manipulation programs that is often used to test computer speed. Nov 18, 2009 at 4:41
• Might one say generally one should use, not just physical intuition, but applications, which may come from physics or from statistics or even from something else in mathematics that may itself not have any known applications? $\qquad$ May 22, 2016 at 10:12
• The convolution theorem gives this "blurring" process a nice interpretation in Fourier space. The Fourier transform $\hat{f}(k)$ of a smooth function $f(x)$ falls off quickly (faster than any power-law) at large $k$, so convolution by a smooth function corresponds to suppressing the high-frequency components in Fourier space. In the context of renormalization in physics, this relates the Kadanoff real-space coarse-graining picture and the Wilson picture of truncating high-frequency modes in momentum space. Feb 5, 2018 at 20:42

I prefer sound to Terry Tao's light. Listen to my voice through a wall. At each moment in time, you hear not just what I am saying now, but also some reverberation from what I said moments ago. So if I make a sound given by $$f(t)$$ (density of air), you hear a linear combination $$h(0)f(t) + h(1)f(t-1) + h(2)f(t-2) + \dots$$, or a continuous version of that, i.e. $$h*f$$. The function $$h(\tau)$$ is how much you hear from $$\tau$$ seconds before the current time. If $$h(\tau)$$ decays slowly, my voice is muffled by reverb.

Fourier theory shows that recovering my voice $$f(t)$$ is difficult when $$\hat{h}(\xi)$$ is very small at some frequencies $$\xi$$: the wall doesn't vibrate at those frequencies.

If $$h(\tau) \ne 0$$ for some negative $$\tau$$, you can hear me before I speak!

• This is a really nice, concrete example. It definitely helped me think about what is going on in a convolution. Dec 11, 2015 at 18:20
• very nice example!!! Dec 12, 2015 at 17:05
• Sounds like the air between my wife and me has $h(\tau)\neq 0$ for negative $\tau$... Aug 31, 2021 at 22:41

What is the operator $C_f\colon g\mapsto f*g$? Consider the translation operator $T_y$ defined by $T_y(g)(x)=g(x-y)$, and look at $f*g(x)=\int_{\mathbb{R}}f(y)g(x-y) \, dy$. Rewriting this as an operator by taking out $g$, you end up with the operator equation $$C_f=\int_{\mathbb{R}}f(y)T_y \, dy.$$ This is only formally correct of course, but it roughly says that convolution with $f$ is a linear combination of translation operators, the integral being a sort of generalized sum.

Tying this in with Terry Tao's answer, which came in while I was writing the above, if $f$ is a bump function, say nonnegative, with integral equal to 1 and concentrated near the origin, then $f*g$ is a (generalized) linear combination of translates of $g$, each one translated just a short distance, hence the blurryness of the result.

Among things that it's good to know about convolution is that the identity element for convolution is the Dirac delta function $\delta$.

Another is that if you convolve a function $f$ with $\delta'$, the derivative of the delta function, you get $f'$. Since convolution is associative, that implies that $f'*g = f*g'$.

Another is that often the convolution of two functions is as well-behaved as the better-behaved one of the two. If you convolve something with a smooth function, you get a smooth function; if you convolve something with a polynomial, you get a polynomial. In other words, many classes of "well-behaved" functions are ideals in a ring whose multiplication is convolution.

So if you convolve $f$ with a smooth approximation to Dirac's delta function, you get a smooth approximation to $f$. Thinking about why that works can probably shed a lot of intuitive light on the nature of convolution.

• As an illustration: The Dirac-Heaviside delta can be taken as the limiting case of a sinc function, and convolution of sinc functions gives the wider sinc function shifted horizontally by the narrower one. Fourier transforming the convolution accentuates this observation and makes it obvious. In this case in Fourier space, you are multiplying rectangle/window functions and the narrower one survives with a phase shift imposed by the broader one. Sep 1, 2021 at 0:24

I think one's standards of intuitiveness depend strongly on one's background. Even if a picture seems unintuitive at first, it can be helpful later.

1. If you're an algebraist, I'd suggest the multiplication operation in group rings or monoid rings (easiest example: polynomial rings).
2. If you like differential or integral operators, I'd suggest convolving with derivatives of delta and Heaviside functions to realize derivatives and integrals.
3. If you like multiplying large numbers (or power series) together, convolution in the places (possibly with a carry) is the process by which this is done.
• Convolution of probability distributions which are supported on the integers is a special case of multiplying power series together; it corresponds to multiplication of the probability generating functions. Nov 18, 2009 at 3:31
• Binomial convolution for the coefficients of an e.g.f. / formal Taylor series and the Cauchy product for an o.g.f. / formal power series / polynomial. Strang in Applied Mathematics and Bracewell in The Fourier Transform and its Applications give useful accounts of convolution. Any good book on operational calculus / fractional calc / generalized fcts. should give nice presentations of point 2 (with Mikusinski giving the axiomatic approach and Pincherle), involving convolution with $H(x-t) (x-t)^{\alpha} /\alpha!$ suitably regularized for frac calc (H(x) = Heaviside step function). Sep 1, 2021 at 0:42

The two things that first come to mind when I think 'convolution' are:

1. It's the thing that corresponds to multiplication on the other side of the Fourier transform. (This was already mentioned by John D. Cook) It works both ways, of course, $\mathcal F (f*g)=\mathcal F f\cdot \mathcal F g$ and $\mathcal F (f\cdot g)=\mathcal F f* \mathcal F g$. This fact is useful when used in combination with other simple facts about the Fourier transform (such as the fact that a rectangular function corresponds to sinc and, in the limit, a Dirac impulse corresponds to a constant function).

2. Imagine a black box that receives one number $x_n$ every second and must output a number $y_n$ every second. (DSP people call it a 'filter' and it's used, for example, to process audio signals in a mobile phone in real-time.) The simplest thing the box could do is to output some function of the current input. The natural next step is to remember the last k inputs and output some function of those k values. One of the simplest functions is a linear combination $$y_n=\sum_i c_i x_{n-i}$$ where $c_i$ is non-zero only for $0\le i<k$. That's a convolution! To generalize, you make the filter remember all previous values and even be clairvoyant. That is, you extend the support of $[c_n]$. Then, if you want, you replace digital circuits with analog ones. That is, you go from summing to integration.

As an example of combining these two points, if the filter always outputs the average of the last k inputs then that's a convolution with the rectangular function in the time domain so it must be a multiplication with a sinc in the frequency domain. Therefore, averaging the last k values attenuates high frequencies. (Hardly surprising, but at least you see immediately that the frequency response is not monotonic and there are only a few frequencies that are completely filtered out.)

I want to expand on a special case of Terry's answer which I think is particularly intuitive.

Suppose there is a function $f$ that you want to understand, but perhaps it is not smooth. Convolution gives you a way to construct new, possibly nicer functions which approximate $f$.

If you let $g$ be a bump function centered at the origin, then the convolution $f*g$ is a new function whose value at $x$ is given by averaging the values of $f$ around $x$. What do we mean exactly by "averaging"? Well, you use $g$ as your measure; translate it over so that it is centered at $x$, and then the integral $$f * g(x) = \int_{\mathbb{R}^n} f(y)g(x-y) \, dy$$ in the convolution corresponds to the $g$-weighted average of the values of $f$ around $x$ (i.e. in the small ball where $g$ doesn't vanish).

The convolution $f*g$ in this case has the advantage that it is much smoother than $f$. Intuitively, this should be not surprising since the value of $f*g(x)$ was gotten by averaging nearby $f$-values of $x$. Furthermore, you can approximate $f$ by smooth(er) things by considering a sequence of convolutions $f*(g_n)$ where $g_n$ is a sequence of bump functions which are more and more concentrated at the origin.

If you think of the second function $g$ in the convolution $f*g$ as a measure, then you can think of convolutions as $g$-weighted averages of $f$.

The fundamental theorem of calculus says that $\frac{d}{dx} \displaystyle \int_a^x f(t)dt = f(x)$. In other words, $f(x) \approx \displaystyle \int_{x-h}^{x+h} \frac{1}{2h} f(t)dt$. Letting $g_h(u) = \frac{1}{2h}$ on the interval $(-h,h)$ and 0 elsewhere, we see by pure algebraic manipulation that $f(x) \approx \displaystyle \int_{-\infty}^\infty g_h(x-t)f(t)dt$. So the fundamental theorem of calculus can very naturally be rephrased in terms of convolution with a bump function. Differentiation under the integral sign immediately gives the differentiation formula for convolutions, and thus that convolutions of two functions are at least as smooth as both factors. Thus finding good smooth approximations to the rectangular bump functions $g_h$ automatically gives us smooth approximations to any integrable function we like, just by convolving against these "smooth molifiers". Pretty cool stuff. As mentioned in other answers they really start to shine when you start thinking about fourier analysis, but that is a another story. If you google "low pass filter" you will find some pretty snazzy applications of the fact that the fourier transform turns convolution into multiplication.

I like the answer you gave when you asked the question. More generally, the convolution of two measures $\mu$ and $\nu$ is the pushforward of $\mu \times \nu$ by addition. In probability, that means you independently draw from $\mu$ and $\nu$ and add the resulting random vector. It's something that you can visualize to a certain extent if you do think of measures as fuzzy versions of points (like Terry Tao said).

One point of view of measures is that they are linear combinations of points (or limits of things you can get from linear combinations of points). If you take this point of view, then convolution is simply the extension of the addition law by linearity to the case of measures.

Since you can translate functions as well as measures, you can convolve, say, a probability measure with a function by randomly translating the function, giving the averaged out function $\int f(x-y) d\mu(y)$ which generally looks like a smoothed out version of your function $f$ -- $\mu$ tells you which translations you use and how to average. Again, you can view this as the extension of the operation of translating functions by linearity/continuity to the case of measures.

The Lebesgue measure allows you to identify functions with measures, $g \mapsto g(x) dx$, so you can also convolve functions with other functions, but you might think of this operation is a bit less basic.

Actually, the process of convolution extends by continuity to more than just measures but also to distributions. For example, you can approximate a tangent vector at $0$ (giving the distribution $u(x) = \sum_i c^i \partial_i \delta_0$) by differences of point masses, so convolution extends to distributions as well, but you can even get differential operators this way (in this example, $u \ast f$ is the derivative of $f$ in the $u$ direction). The technical difference here is that the approximation is only valid when integrated against $C^k$ functions (rather than $C^0$ functions in the case of measures). But the principle is the same -- it's the extension of the addition law by linearity and limits.

• "the convolution of two measure $\mu$ and $\nu$ is the pushforward of $\mu \times \nu$ by multiplication". Don't you mean "...by addition"? Sep 19, 2013 at 14:18
• Thanks. Yes, I meant addition. I guess I was thinking about convolution on a general group. I edited the answer. Sep 22, 2013 at 1:51

Maybe it would help your intuition to think about the discrete case first where the convolution is a sum rather than an integral. (f*g)(x) is the sum of f(i) g(j) over all (i, j) that sum to x.

Or maybe you could think of convolution as a kind of multiplication. Convolution makes certain function spaces into algebras.

Or you could think in terms of Fourier transforms: the Fourier transform of f*g is the product of the Fourier transforms of f and g.

This is really following on from John's answer, but is a bit long for a comment, so I thought I'd write it out here for extra clarity.

Say you have a semigroup S, and you take the free (complex) vector space this generates, call it $C^S$. By construction/definition this has a distinguished basis, indexed by elements of the semigroup; and since we have a semigroup structure, that means we can multiply basis elements to get other basis elements. But that now gives us a way to multiply two vectors $a$ and $b$ together: write $a$ and $b$ as linear combinations of basis elements, and then define their product as the obvious (bi)linear extension of the multiplication on basis elements. If you do all this starting with $S={\bf Z}$, the group of integers, then what we've done is defined multiplication of trigonometric polynomials, or convolution of finitely supported sequences.

The thing I like about this point of view is that it immediately generalises to $l^1(S)$, and makes $l^1(S)$ into a Banach algebra. If $S$ is a topological group with a Haar measure on it -- such as the real line with Lebesgue measure -- then the same idea gives us the usual Banach algebra structure on $L^1(S)$, which in the case $S={\bf R}$ is precisely the convolution of integrable functions in the usual sense.

(At this point someone -- often me -- usually wants to mention forgetful functors from algebras to vector spaces and from semigroups to sets, but that's probably getting a bit OTT for the question at hand.)

Have a look here - its rather helpful!

A heuristic that I like is that convolution is like measuring temperature with a large thermometer.

In one dimension, imagine a trough filled with some sort of warm goo, not necessarily at a uniform temperature. If $x$ is the position along the trough (say, in cm), let $f(x)$ denote the temperature of the goo at position $x$.

Now imagine you will insert a thermometer to measure the temperature of the goo. Your thermometer has a center point, and at a displacement $y$ from the center point, it has a sensitivity $g(y)$. So when you insert the thermometer, the reading on your thermometer is an average of the actual temperature at points around the center, weighted by the thermometer's sensitivity at each point.

Then $(f \ast g)(x)$ tells you the reading on the thermometer if you insert it in the goo with its center point at position $x$.

If $g = \delta$ is a Dirac delta, this corresponds to a "point" thermometer that measures temperature exactly at the center point, and so we see that $f \ast \delta = f$.

This analogy also makes sense in higher dimensions, of course.

I'll give two answers. Put together I think they build a decent intuition.

1. The first is algorithmic. I think about convolution of two lists $$\vec{a} \ast \vec{b}$$ as contrasted to the dot product $$\vec{a} \cdot \vec{b}$$. In fact convolution is how I and other students intuitively wanted to describe multiplication of vectors in my first attempt at multivariable calc. The dot product maps $$K^N \times K^N \to K^1$$ but $$\ast: K^N \times K^N \to K^N$$. Otherwise they are similar in that they multiply pairs $$a_i \cdot b_i$$. But where the dot product $$\tt{reduce}$$s dimensionality by summing $$\displaystyle \sum_i$$ over the entries, convolution leaves the products in place $$\left( a_1 \cdot b_1, \ \ldots\ , \ a_N \cdot b_N \right)$$. In other words convolution is the entrywise multiplication your linear algebra students wanted to do when you first asked them how they should multiply two matrices (except on vectors).

2. The second is visual. Picture a choppy curve, e.g. (source: fedprimerate.com)
. To smooth that you would convolve the long data series against a much shorter list. To get the weekly rolling average you would convolve it against $${1 \over 5} \cdot \left[1 \ 1 \ 1 \ 1 \ 1 \right]$$ (implied zeroes to the left and right—this is convolving against a $$\mathrm{Rect}$$ function). To get the monthly rolling average that would be 25 ones. If you want to convolve against something smoother than $$\mathrm{Rect}$$ you could convolve against a Gaussian. Gaussian blur in 2-D looks like this: . I'm now getting out of my knowledge area but I think box blur is the image-processing word for convolving against a 2D $$\mathrm{Rect}$$ function, perhaps only meaning with width=three ones: (source: codecave.org)
. Perhaps with some computer knowledge you can code up a smoother with data of your choice.

As a post-script to this answer, I think Gaussian vs rectangular convolution is actually a good example to explain to non-mathematicians how mathematicians think about ugliness''. There's something intuitively stupid, even to a non-mathematician, about integrating against something like $$\begin{bmatrix}0&0&0&0&0\\ 0&0 & 1 & 0&0 \\ 0&1 & 1 & 1&0 \\ 0&0 & 1 & 0&0 \\ 0&0&0&0&0 \end{bmatrix}$$ when a circle or ellipse would be a more logical shape. There's also something stupid or ugly or pointless or strange about integrating against $$\left[ 0 \ \cdots\ 0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 0 \ \cdots\ 0 \right]$$ , but what's the right way to "smoothly" or "logically" go down on both sides? Cue discussion on $$\mathcal{F} \left( \exp(-x^2/2) \right) = \exp( -x^2/2)$$....

There is a nice relationship between convolution of probability measures and random walks which is very clear on finite groups.

For a particularly concrete example, suppose you are shuffling a deck of cards. You can model this as picking elements of $S_{52}$ according to some probability measure $P$ on $S_{52}$. This generates a Markov chain with transition matrix whose $(s,t)$-th entry is given by $P(ts^{-1})$ --- if I am permitted to abuse notation somewhat, the element $ts^{-1}$ is the shuffle that takes the deck from ordering $s$ to ordering $t$. If one wants to know the transition matrix for two shuffles, this corresponds to the square of the original matrix. One can then check that this new matrix corresponds to constructing the transition matrix generated by $P*P$, that is the matrix whose $(s,t)$-th entry is given by $(P*P)(ts^{-1})$, and in fact $k$ shuffles corresponds to the the $k$-fold convolution of $P$ with itself.

Convolving two different probability measures then corresponds to shuffling your deck according to one technique and then a different technique.

To complement/supplement other answers, it may be worthwhile to note that the question itself blurs two substantially different mechanisms. Namely, there is, first, for any group representation of a topological group $G$ on a topological vector space $V$, an action of compactly-supported continuous functions on $G$ on $V$, by (e.g., Gelfand-Pettis/weak) integrals $f\cdot v=\int_G f(g)\cdot gv\,dg$. It is of some moment to note that this does not depend on $v$ being in any sort of natural function-space. The second point is that $f\cdot (g\cdot v)=(f*g)\cdot v$, where $*$ denotes the convolution. That is, the notion of convolution is externally determined by being what it has to be for (for example) compactly-supported continuous functions to act (associatively) on any representation space.

Depending on one's outlook, this may reduce some element of seeming whimsy in "defining" convolution, since, in a larger context, _there_is_no_choice_.

Arthur Mattuck gives an interesting lecture on the convolution, its construction, and its applications:

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-21-convolution-formula/

The way you have described is the best way to think about convolution. More generally, if you have a group and a class of square-integrable functions (really I should say "half-densities") on it, then the convolution product precisely extends the group product.

If you convolve an image with a discrete matrix of values - so like a function that is zero outside a few pixles then you can create almost an unlimited number of filtering effects around each point. Fof example you can do some kind of averaging or weighted integration - which looks like blurring as Professor Tao mentions if you use a matrix whose values drop off smoothly, radially from the centre - a bump. You can also compute directional derivatives, look for edges, circles, blobs, steps - basically anything you like.

The interpretation in terms of multiplication of Fourier coeficients is interesting and makes applications of the above in reality fast, especially if the filter is fixed, because you can use the Fast Fourier Transform on both images but you only need to update one of them. However I a not sure how intuitive it is!

I'm not sure I have added much additional information but I hope this helps anyway,

Ivan

Here is my take on this. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Discretize the real axis and thing of it as the collection of point $\Lambda_\hbar:=\hbar \bZ$, where $\hbar>0$ is a small number. Then a function $f:\Lambda_\hbar\to \bR$ is determined by its generating function, i.e., the formal power series $\newcommand{\ii}{\boldsymbol{i}}$

$$G^\hbar_f(t)=\sum_{n\in\bZ}f(n\hbar)t^n.$$

Then

$$G^\hbar_{f_0\ast f_1}(t)= G^\hbar_{f_0}(t)\cdot G^\hbar_{f_1}(t).\tag{1}$$

Observe that if we set $t=e^{-\ii\xi \hbar}$, then

$$G^\hbar_f(t)=\sum_{x\in\Lambda_\hbar} f(x) e^{-\ii \xi x}.$$

Moreover

$$\hbar G^\hbar_f(e^{-\ii\xi \hbar})=\sum _{n\in \bZ} \hbar f(n\hbar) e^{-\ii\xi(n\hbar)}, \tag{2}$$

and the expression in the right hand sum is a "Riemann sum" approximating

$$\int_{\bR} f(x)^{-\ii\xi x} dx.$$

Above we recognize the Fourier transform of $f$. If we let $\hbar\to 0$ in (\ref{2}) and we use (\ref{1}) we obtain the wellknown fact that the Fourier transform maps the convolution to the usual pointwise product of functions. (The fact that this rather careless passing to the limit can be rigorous is what the Poisson formula is all about.)

The above argument shows that we can regard $\hbar G_f^\hbar(1)$ as an approximation for $\int_{\bR} f(x) dx$.

Denote by $\delta(x)$ the Delta function 9concentrated at $0$. The Delta function concentrated at $x_0$ is then $\delta(x-x_0)$. What could be the generating function of $\delta(x)$, $G\delta^\hbar$? First, we know that $\delta(x)=0$, $\forall x\neq 0$ so that

$$G_\delta^\hbar(t) =ct^0=c.$$

The constant $c$ can be determined from the equality

$$1= \int_{\bR} \delta(x) dx=\hbar G_\delta^\hbar(1)=\hbar c$$

Hence $\hbar G_\delta^\hbar(1)=1$. Similarly

$$G^\hbar_{\delta(\cdot-n\hbar)} =\frac{1}{\hbar} t^n.$$

Putting together all of the above we obtain an equivalemt description for the generating functon af a function $f:\Lambda_\hbar\to\bR$. More precisely

$$G_f(t)=\hbar\sum_{\lambda\in\Lambda_\hbar}f(\lambda) G_{\delta(\cdot-\lambda)}(t).$$

The last equality suggests an interpretation for the generating function as an algebraic encoding of the fact that $f:\Lambda_\hbar\to\bR$ is a superposition of $\delta$ functions concentrated along the points of the lattice $\Lambda_\hbar$.

It looks I am a bit late to the party; but hopefully not too late. I can contribute one instance where the application of convolution is very enlightening. Look at this in the light of Harald Hanche-Olsen's answer.

Let us consider the operation of a signal processing system, or an electrical network, or a control system.

Let $i(t)$ be the response of the system to the unit impulse function, ie the Dirac delta function, $\delta(t)$. Now we give an arbitrary signal $f(t)$ as input to the system. Then, the response of the system to $f(t)$ is

$(f * i) (t)$,

i.e. the convolution of $i$ and $f$.

Next, in the spirit of rgrig's answer, I add that convolution becomes multiplication in the frequency domain. In that domain, it is like you multiply individually at each frequency component and add up again(ie integrate).

• You may want to add that you talk about a linear system. Mar 21, 2010 at 13:45
• Thanks. The systems considered in electrical engineering and signal processing are usually linear. So it escaped my mind to mention it explicitly. Mar 21, 2010 at 13:50
• More specifically, linear and time-invariant. Nov 19, 2010 at 18:10

I think convoluton gets more intuitive when looking at the measure algebra instead of the special case of “densities”: Consider two measures $\mu,\nu$, for a test-function $f$ we get:

$\int f\mathrm{d}(\mu*\nu)=\int\int f(x+y)\mathrm{d}\mu(x)\mathrm{d}\nu(y)$

This definition is very “symmetric”, we do not have to think about the minus-sign, and especially in the case of non-commutative groups it is much more obvious than getting the “right” formula for the convolution of functions. We directly recover the intuition of the sum of two random variables (and it is even more general, because we can have arbitrary distributions (for example $\delta$-distributions) for our random variables): “We account the ‘probability’ for the pair $(x,y)$ at the point $x+y$”.

As far as I understand, in simple words, considering the simple moving average algorithm, when you convolve F with G, then G defines how you are going to do the weightings to get the average. G can be seen as a component for defining the weighting policy.

I asked myself this same question recently and had difficulty finding an answer which I found satisfying. Eventually I satisfied myself with a version of the `bialgebra' characterization of convolution (I was helped very much by this MO answer by Theo Johnson-Freyd). I'd like to record it here in case anyone else finds it useful.

Our setting is a space $\mathrm{Hom}(A,B)$ of maps from a set $A$ to a set $B$ (both with extra structure). Let's say I furthermore have an operation $-\cdot-$ ("multiplication") which combines each pair of elements $b_1$ and $b_2$ in $B$ (or $b_1\otimes b_2$ in $B\otimes B$) to an element $b_1\cdot b_2\in B$, and another operation $\Delta$ ("comultiplication") which splits each element $a$ in $A$ to a collection (a formal sum) of pairs $a_1^i\otimes a_2^i$. A convolution algebra structure on $\mathrm{Hom}(A,B)$ is induced as follows:

Think of a function from $A$ to $B$ as a black box that accepts inputs $a$ in $A$ and responds by outputting corresponding elements $f(a)$ in $B$. I would like to fuse together a pair of functions $f,g\colon\, A\longrightarrow B$ into a single function $f\ast g\colon\, A\longrightarrow B$. To do so, I must specify which output $f\ast g$ generates when it is fed $a$ for each $a\in A$. I have in my hand just the structure to do so.

How do $a\mapsto f(a)$ and $a\mapsto g(a)$ combine to induce a black box to produce an output from an input? Well, I first split $a$ into pairs $\sum\limits_i a_1^i\otimes a_2^i$ using $\Delta$, then I apply $f$ to the first elements in the pair and $g$ to the second to obtain $\sum\limits_i f(a_1^i)\otimes g(a_2^i)$, and finally I multiply each pair using $-\cdot-$ to conclude with $(f\ast g)(a)=\sum\limits_i f(a_1^i)\cdot g(a_2^i)$. The function $f\ast g$ is called the convolution of $f$ with $g$.

If I've convinced myself that both $-\cdot-$ and $\Delta$ are fundamental and canonical, then convolution is inevitable as a binary operation on functions $f,g\in \mathrm{Hom}(A,B)$ because $f$ and $g$ together act on pairs (as $f\otimes g\colon\, A\otimes A \longrightarrow B\otimes B$) and I have fundamental and canonical ways both to split elements of $A$ into pairs and also to combine pairs in $B$.

The cleanest example I know, which is also quite general, is when $A$ is a group and $B$ is a ring. Now, the coproduct $\Delta$ is the map $a\mapsto \sum\limits_{a_1a_2=a}a_1\otimes a_2$, and the product $-\cdot-$ is $b_1\otimes b_2 \mapsto b_1b_2$. Convolution is now given by the expression $(f\ast g)(a)= \sum\limits_{a_1a_2=a}f(a_1)\cdot g(a_2)$. In the special case that $A$ is the additive group $(\mathbb{R},+)$ and $B$ is the ring $(\mathbb{R},+,\cdot\,)$, this reduces to the familiar $(f\ast g)(x)= \int\limits_{t\in \mathbb{R}}f(t)\cdot g(x-t)\, dt$.

In signal processing, convolution $$f(t)*g(t)$$ can be seen as a weighted mean so $$f(t)*g(t)=\int{f(t-u)g(u)du}$$ is the mean of $$f(t)$$ through the window $$g$$ weighted by the value at the center of this window, $$g(u)$$. In this way we obtain a varying mean of $$f$$ depending on the parameter $$u$$.‌

For a physicist (or signal processor), perhaps the most intuitive way to think of convolution is in terms of translation invariant (in time and/or space) Green's functions (also called influence functions) which encode how a disturbance at one point affects other points, either in a deterministic or probabilistic manner. In signal processing, one considers the impulse response function for a linear translation invariant system or step response function.

See Sections 3.5 through 3.9 of Cartier's "Mathemagics" for an additional, interesting presentation of these ideas, particularly the Bargmann-Segal transform.

Edit 8/31/2021:

The idea of blurring as a fundamental intuitive perspective on convolution in general is seriously flawed. Look at the images in Figure 3 on pg. 11 of "A Tutorial on Synthetic Aperture Radar" of the effects of convolution on a raw SAR return and tell me 'convolution is a blurring process'. Analogously, it's as misleading as saying for the reals that multiplication of a vector by a scalar 'stretches it out'--depends on whether the scalar has magnitude greater than one, right?

More precisely: The convolutional processing of the data in SAR is in the complex domain. The depiction of the azimuth reference functions in the figure is of the azimuthal Doppler frequency of point reflectors at different ranges whereas the range reference function has a single, centered frequency modulation. The raw return after range processing (complex data) is convolved in the azimuthal direction with a complex reference function in such a way that the Doppler FM return from a strong point reflector constructively adds over the synthetic aperture, i.e., over the time the radar can 'see' it. The concepts of superposition and constructive/destructive interference lie at the heart of the convolutional processing. Similar ideas pertain to sine, cosine, and other integral transforms (and the discrete Walsh transform) that don't have complex values but do have negative and positive values. The Gaussian transform, albeit an important, useful transform, is an exception though certainly the idea of superposition is important in understanding solutions to the heat/diffusion equation with a given initial boundary function--the Gaussian convolution smoothing out the initial, possibly highly irregular, function over time.

It is true that any physical measurements involve instruments with limited resolution/bandwidth so a blurring of the 'true' underlying physical parameters occurs in any attempt to measure them and this can be modeled mathematically by a convolution, but this is only one use of the method and not necessarily the most important/productive one. For example, a SAR image has azimuthal resolution (proportional to $$\lambda/\theta$$) limited by the wavelength of the radar and the width of the synthetic aperture (a.k.a. the radar's azimuthal beamwidth), giving at best essentially a sinc function for the resolution for an isolated point reflector, but this is far better than the beamwidth resolution ($$\theta$$) of the point reflector without the convolutional processing.

• A 2-D intensity plot of the coherent raw radar return for synthetic aperture radar appears as a white noise image. Convolving (or deconvolving) the raw return with the correct complex function produces an image of the reflecting surface with essentially sinc fct resolution (with a phase factor) of isolated point reflectors. This is an example of de-blurring through convolution. May 16, 2017 at 18:57
• No need to invoke 'bump' or Gaussian functions for an understanding of the simplistic 'blurring' perspective either--a simple running average/mean will do or even a running median. Using the SAR radar without azimuthal convolution is tantamount to a running average. Aug 31, 2021 at 22:49
• A dated but good starter for really getting a practical feel for the subject and its use in digital signal processing is Digital Filters by Hamming. Fourier Transforms and their Applications by Bracewell and Applied Mathematics by Strang give useful material to master before you can really say you understand convolution. Sep 1, 2021 at 1:04
• On the history of the interplay of combinatorics, probability theory, harmonic analysis, and convolution, see mathoverflow.net/questions/111970/… Sep 1, 2021 at 5:11

In signals and systems, "Weighted sum of past inputs". That's how I would put it. This is because in order to find out the current output of a system, you need to consider past inputs as well as current input because past inputs also leave certain amount of energy in the system. Convolution gives you a way of adding them respect to time. I made a video on it because no one would give me an intuitive insight then. Hope this helps! https://www.youtube.com/watch?v=1Y8wHa3fCKs&t=40s

As Terry Tao noted, the convolution with a (positive, somewhat smooth) bump function is "blurring". One might add that this does not have to be the case when the convolution is taken with a function taking values of both signs, especially with a sharply varying function of this kind. As a somewhat extreme example of this sort, note that the convolution of a smooth function $f$ with the $n$th derivative $\delta^{(n)}$ of the Dirac delta function $\delta$ is $f^{(n)}$, which will usually be much less "blurred" than $f$; for instance, think of $f(x)=e^{-x^2}$.

The way I explain convolution $$f*g(x)=\int_{-\infty}^{+\infty} f(x-y)g(y)dy$$ to my (mostly engineering) students is this:

-keep the function $$g(y)$$ (to be "analyzed" or "filtered") fixed;

-invert the graph of $$f$$ to make it $$f_{inv}$$ and translate it forward by $$x$$ along the $$y$$-axis, as if on rails (in my picture $$f$$ is a bump centered at the origin, or the derivative of such a bump, or a delayed bump...);

-multiply and integrate (i.e. take an "average" of $$g(y)$$ "around" $$x$$).

This is just picturing the formula, but it contains the "blurring" interpretation (I always make an example of a "black-and-white" $$g$$ where the edges become grey spots); the translation invariance of $$g\mapsto f*g$$ (moving $$g$$ backward by $$a$$ means that $$f_{inv}$$ has to move $$a$$ meters less to reproduce the same effect); the fact that $$\delta_0*g=g$$ (at least on an intuitive level). It also makes it easier to "design" a function $$f$$ so that $$g\mapsto f*g$$ has desired properties.

The illustration can be easily translated from $$\mathbb R$$ to $$\mathbb T$$, $$\mathbb Z$$, etcetera, and most students get some intuition of what goes on. When they learn that using only past information ("causality") brings about holomorphic functions on the Fourier side, some of them are sincerely gratified.

PS If $$g$$ and $$h$$ represent "linear systems" $$G:f\mapsto g*f,H:f\mapsto h*f$$ (or, as engineers put it, unit-impulse responses to such systems), then $$g*h$$ represents the systems "in series", $$G\circ H$$; and this provides further intuition of what's going on and of what's lying ahead.

Although I like @TerryTao's answer, I would phrase a little differently. If you view $$f$$ as just a function and $$g$$ as a weight function with integral $$1$$ centered at $$0$$, then the convolution $$f*g$$ at $$x$$ is the weighted average of $$f$$ centered at $$x$$. This gives a "softened" approximation to $$f$$. If you rescale $$g$$ to maintain its integral at $$1$$ but concentrate it more and more at the origin, then it's clear the convolution will converge in some sene to $$f$$ itself. That's why it is often called an approximation of the identity.

If $$f$$ is a function of time, then the convolution of $$f$$ by $$g$$ can be viewed as a continuous analogue of a moving average of a sequence of measurements over time (except that the moving average usually has a time lag because it is using only values from the past). Both the moving average and convolution are used to soften the spikes that occur in the original sequence or function.