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Re: [Phys-l] Convolution (was Sun's image ...)

Ken,

I am not motivated, and almost certainly not capable of providing a satisfying insight for you
into the convolution of polynomials. Perhaps this note from Julius Smith who is professor of music (!) at Stanford will commend itself to you instead:
<http://www.dsprelated.com/dspbooks/mdft/Polynomial_Multiplication.html>

You are familiar I expect with the concept originated by that hero of engineers, Laplace
who provided an avenue to ditch systems of differential or integral equations describing dynamic objects by showing a mapping of functions of time onto functions of complex frequency which could be handled as systems of algebraic equations - a much more sympathetic pursuit!

It is this s transform that makes a matrix algebra package attractive to some engineers.
There is a connection between the discrete fourier transform and a polynomial which can be used to carry the DFT's values at particular data points, so this is another avenue where vector algebra is used to massage DFT information.

But I should mention the great ease with which the whole edifice of numerical computation can be set aside, by using optical methods of applying convolutions and fourier transforms.
The materials are relatively simple:
A laser with a beam expander. A converging lens whose aperture can be filled with the beam. A blank screen placed at the far focal plane of the lens.
The effect at this plane of placing a 35 mm slide in the beam with defined spatial features - like a cosine grid image for example - shows an image at the far focal plane which can be associated with a spectrum of the spatial frequencies available in the slide image. Placing two such slides in the beam can be associated with a convolution of the images, providing a spatial frequency spectrum in two dimensions at the far focal plane.

If the blank screen is removed, it may be replaced with an optical filter which can stop the lower optical frequencies near the optical axis ( a central blanking spot) so that the remaining image may be again transformed by a second lens which then provides an optically processed image, in this case of two convolved functions representing the original slide materials, which have been optically filtered to remove the low spatial frequencies, with the effect of intensifying or selecting higher optical frequencies like edges..... An aperture placed at the focal plane will filter out the higher frequencies represented further from the optical axis.

I am sad to realise that what I have written is so far from the elegant clarity which I would like to bring to bear on this topic, but it is all that I can presently summon.

Brian W

Ken Caviness wrote:
Brian or anyone who wants to chip in,

Am I misunderstanding something here? It seems to me that you are not talking about convolution at all, just ordinary function multiplication, showcasing integer multiplication as a special case of polynomial multiplication (with x=10) and then putting in the carried digits (the sense of this also escapes me).

Can you tie this in for me with the only way I've used the idea of convolution, namely in regard to Fourier and Laplace transforms? For example, http://mathworld.wolfram.com/ConvolutionTheorem.html reminds us that F(f*g) = F(f) F(g), where ordinary multiplication of the Fourier transforms of the functions leads to the same result as the transform of the convolution of the original functions: '*' represents function convolution, defined as

f * g = Integral[g(t')f(t-t')dt',{t,-Infinity,Infinity}].

Feeling lost,

Ken

Ken Caviness
Physics
Southern Adventist University

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of Brian Whatcott
Sent: Friday, May 01, 2009 1:18 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Convolution (was Sun's image ...)

Now hang on a darn' minute!
Every single one of your students has been convolving three and four
term
polynomials since third grade, and then some!

Look at this polynomial a(x) = x^3 + 2x^2 + 3x + 4
and b(x) = x^3 + 3x^2 + x + 3

The convolution is 1 5 10 18 21 13 12


Now if I mention that your students typically use x = 10
And they express two polynomials such as....
a(x) = 10^3 + 2X 10^2 + 3X 10 + 4 as 1234
and
b(x) as 10^3 + 3 X 10^2 + 10 + 3 X 10^0 as 1313


They convolve these polynomials.

In other words, they not only derive the convolution of these two
polynomials, but they THEN
carry values greater than the base to the next column:
so (from the lowest power)
12 = 2 carry 1
13 plus 1 = 14 = 4 carry 1
21 plus 1 = 22 = 2 carry 2
18 plus 2 = 20 = 0 carry 2
10 plus 2 = 12 = 2 carry 1
5 plus 1 = 6
1

They then write
1 6 2 0 2 4 2 = 1234 X 1313

:-)

Who knew?

Brian W