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

Brian W,

Intriguing.

Thanks for the Smith link, that gives the mathematical explanation I was missing. I now see that multiplication of polynomials is indeed convolution of their coefficients. Part of my difficulty was because it sounded to me like you were referring to convolution as an operation on the set of polynomials themselves, or other functions, and that interpretation seemed possible from my background mathematical knowledge of Laplace and Fourier transforms as operators operating on functions -- but your examples were clearly simply polynomial multiplication, which is in fact not a convolution of the polynomials themselves (except in the sense that the polynomial can be taken as a representation of its coefficients and vice versa). Similarly I wouldn't have understood calling integer multiplication a convolution of the integers themselves, although I see now that it _is_ a convolution of the two sequences of digits. This also highlights my second difficulty, i.e., that I am more comfortable with the calculus than with discrete math. I had honestly never thought of a discrete analog of the convolution process, although it seems obvious enough once pointed out. (As do many things!)

I also thank you for taking the time to add additional explanations, although it's so hard to know what is needed, of course. In this case, the background of Laplace transformations is quite familiar to me (I've even taught it in math methods of physics), but the optical methods are new to me, and fascinating. Optics was always a hole in my background. Sigh!

Again, thanks for a very helpful response!

Ken

-----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 10:02 PM
To: Forum for Physics Educators
Subject: 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


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