The Mathworld explanation (with animation) is here: http://mathworld.wolfram.com/Convolution.html . For symmetric functions, it's a good explanation; for asymmetric time-series/functions/whatever, you should probably draw out a couple of data sets (maybe the boxcar for one, and a spike followed by a falling exponential for the other), and draw what you think the "overlap" is for different time-offsets.
You should rapidly be able to convince yourself, at a good visceral level, that the "reverse" operation is necessary; the "sliding average" should probably be more properly termed a "sliding product" or "sliding integral".
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As a species, we are forever sticking our fingers into the electric socket of the Universe to see what'll happen next. It's a trait that'll either save us or kill us, but by god it's what makes us human beings. I'd rather be in the company of people who look at Mars than people who contemplate humanity's navel -- other worlds are better than fluff. ~~Sir Terence David John Pratchett
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From: Stefan Jeglinski <jeglin@4pi.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Tue, April 6, 2010 2:56:27 PM
Subject: [Phys-l] whither convolution
Hopefully I won't get tied up in terminology here, which is not the
point of my question, namely, why is convolution the correct way to
describe the way one function "acts" on another?
When considering a "black box," one observes a given output, given a
known input. The transfer function is expressed mathematically as a
convolution:
output(t) = Integral[ h(t') x input(t-t'), dt' ]
Why is it that we don't naively claim
output(t) = h(t) x input(t)
and leave it at that? Why is it that a reversed sliding average, an
approach that seems highly non-intuitive, correct?
I know about things like the convolution providing us the framework
for the very useful notion of impulse response etc, but I don't think
my answer should be "that's just the way it works." However, is it
possible that the convolution exists merely to mathematically support
the concept of impulse response?
When I work with detector responses, analog electrical transfer
functions, time-domain signal processing, to name a few, I know that
the convolution approach is the fundamental framework, but I don't
know why it is correct on a fundamental level.
In reading about convolution on wikipedia, I'm lead to the more
general idea of integral transforms, and the desirability of
expressing a function as a sum of more simple basis functions. And of
course I'm familiar with limits and converting a sum into an
integral. OK, it's a connection, but seems uninspiring. A convolution
is a very specific form of integral transform, and most examples of
integral transforms (eg Fourier, Laplace) deal with transforming from
space to inverse space; the convolution transfers from space to the
same space (t-space, in my example above).