Re: [Phys-l] whither convolution

[John Denker <jsd@av8n.com>]

On 04/06/2010 01:56 PM, Stefan Jeglinski wrote:

> 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' ]    [1]

That's not the most general "action" a black box
could have.

> Why is it that we don't naively claim
>
> 	output(t) = h(t) x input(t)             [2]

That's possible, but even less general than [1].  Ohm's law
is of form [2].  When we have capacitors as well as resistors,
form [1] is needed.  When we have any kind of nonlinearity,
e.g. light bulbs, diodes, varactors, etc., neither [1] nor
[2] will do the job.
 
> and leave it at that? Why is it that a reversed sliding average, an 
> approach that seems highly non-intuitive, correct?

One person's "intuition" is another person's "magic".
There's a proverb that defines education as the process
of cultivating your intuition.

> 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?

Impulse response is a sufficient explanation for the 
structure of equation [1], for a time-invariant linear
system.

There exist things such as correlation functions and
dot products that do not require a minus sign in
front of the dt'.  And you can disguise said minus
sign by time-reversing the kernel h(t) ... but then
h(t) is not a picture of the impulse response but
rather the reverse thereof.

If you want h(t) to be the non-reversed impulse
response and you want the overall system to be
linear and invariant w.r.t to time-shifts, then
equation [1] is pretty much the only game in town.

It just expresses the idea that the output is a
superposition of impulse responses.

You can work up to this idea by considering an input
that is a single impulse, then the sum of two impulses,
then three, and so forth.