FPGA Central

For the impatient, my question comes first, motivation for the
question afterward.

I started looking into the fractional Fourier transform. Since a
Fourier transform converts a convolution operation into a pointwise-
multiplication, what can we say about a /fractional/ Fourier
transform? If I transform "half way" between time and frequency, do I
convert the convolution operation into something which is somehow
"half way" between convolution and point-wise multiplication? If such
an operation exists, is there any comprehensible way to describe it?

Here's why I'm wondering:

I've been experimenting with the use of FFTs in the numerical solution
of partial differential equations. I know that I'm not the first to
think of it, but I suddenly recognized that the discrete Laplace
operator can be described as a convolution, thus, could be implemented
as pointwise multiplication in the frequency domain.

The problem I encounter is the treatment of the velocity field. Since
my solver does not assume a constant wave velocity at every grid
point, I need to convolve the frequency-domain representation of the
velocity field with the frequency-domain representation of the
displacement field. Convolution with the Laplace kernel is inexpensive
(there are only 5 points in the stencil) but for an arbitrary velocity
field the spectrum will be extremely complex.

So I started looking into the fractional Fourier transform and had a
thought. If I operated, not in time or frequency, but in a domain
halfway between the two, perhaps the simple pointwise multiplication
of the Laplace operator, and the much more involved convolution with
the velocity field, could be expressed as two operations which are in
some sense "opposite" to each other but equal in complexity. Although
it probably would not improve the efficiency of the method (in fact,
it would probably be awful), it's something that I'd like to find an
answer to if I can.

Thanks,
Scott L

Scott L wrote:
> For the impatient, my question comes first, motivation for the
> question afterward.
>
> I started looking into the fractional Fourier transform. Since a
> Fourier transform converts a convolution operation into a pointwise-
> multiplication, what can we say about a /fractional/ Fourier
> transform? If I transform "half way" between time and frequency, do I
> convert the convolution operation into something which is somehow
> "half way" between convolution and point-wise multiplication? If such
> an operation exists, is there any comprehensible way to describe it?

Hi Scott

If the question is: windowing (pointwise multiplication) in fractional
Fourier space corresponds to what in time domain?, then from the
wikipedia article it looks like fractional Fourier transforms can be
viewed as time-varying convolution. This is also suggested in the
short summary here:

http://www.ee.bilkent.edu.tr/~haldun/wileybook.html

It just occured to me that windowing in fractional Fourier domain
could be used to "linearize" swept-sine measuremnts. So many ideas, so
little time ...

Regards,
Andor

On Aug 26, 5:33*am, Scott L <smxl...@gmail.com> wrote:
> The problem I encounter is the treatment of the velocity field. Since
> my solver does not assume a constant wave velocity at every grid
> point, I need to convolve the frequency-domain representation of the
> velocity field with the frequency-domain representation of the
> displacement field.

Convolution in the frequency domain is pointwise multiplication in the
spatial domain. Or am I missing something?

illywhacker;

On Aug 26, 3:29 am, illywhacker <illywac...@gmail.com> wrote:
> Convolution in the frequency domain is pointwise multiplication in the
> spatial domain. Or am I missing something?
>
> illywhacker;

It is, but I do not want to transform back and forth between space and
frequency at every timestep.

On Aug 26, 6:07*pm, Scott L <smxl...@gmail.com> wrote:
> On Aug 26, 3:29 am, illywhacker <illywac...@gmail.com> wrote:
>
> > Convolution in the frequency domain is pointwise multiplication in the
> > spatial domain. Or am I missing something?
>
> > illywhacker;
>
> It is, but I do not want to transform back and forth between space and
> frequency at every timestep.

It is probably more efficient than performing the convolution. To be
honest, I am not really sure it is worth going to the frequency domain
if the only spatial convolution you need to perform is the Laplacian.

illywhacker;