Factoring a filter (s^2) * h into (s*h1) . (s*h2)

Hello all,

Thanks for your replies. Indeed, there are some stringent conditions.
The number of equations scales as ~n^2 but the number of unknowns
(size of h1 and h2) scales as ~n.

h1[n]h2[n] = h[n]
h1[n]h2[p] + h1[p]h2[n] = 0 (for n != p)

I was wondering if this problem has been studied before, otherwise,
I'll try to create a solution from scratch.

Going into Fourier does not seem to help.

On Jan 22, 5:02 am, "sarw...@YouEyeYouSee.edu" <dvsarw...@gmail.com>
wrote:
> On Jan 22, 1:16 am, Laurent Schmalen <lo...@gmx.de> wrote:
>
>
>
> > sarw...@YouEyeYouSee.edu wrote:
> > > On Jan 18, 2:41 pm, Antoine  Bruguier <tony.brugu...@gmail.com> wrote:
>
> > >> I was looking for conditions on h such that we can find h1 and h2 such
> > >> that for every s:
> > >> (s^2) * h = (s*h1) . (s*h2)
>
> > > If the condition is required to hold for *every* input s, then
> > > it must hold when the input is the unit pulse function given
> > > by p[n] = 1 for n = 0 and p[n] = 0 otherwise.  Since p^2 = p,
> > > and p is the identity for convolution, it must be that
> > > (p^2) * h = p * h = h, and p*h1 = h1, p*h2 = h2.  Hence,
> > > h = h1.h2, that is,
>
> > >          h[n] = h1[n]h2[n] for all n.
>
> > > Since multiplication in the time domain corresponds to convolution
> > > inn the frequency domain, perhaps you need to look at spectral
> > > convolution H = (H1) * (H2) rather than spectral factorization
> > > H = (H1)(H2).
>
> > Wrong.
> > Squaring is a non-linear operation and thus your condition does NOT
> > hold. Take this little example:
>
> > h1 = [1 2 3 4];
> > h2 = [-1 -2 -3 -4];
> > s = [0 -0.1 0.2 -0.3 0.4];
> > h = h1.*h2;
> > conv(s.^2,h)
> > conv(s,h1) .* conv(s,h2)
>
> > and compare the outputs.
>
> > Regards,
> > Laurent
>
> The condition h[n] = h1[n]h2[n] for all n is NECESSARY
> if it is desired that (s^2) * h = (s*h1) . (s*h2) hold for ALL
> signals s; it is by no means SUFFICIENT.  I very much
> doubt that the decomposition sought by the OP is possible
> for all h; indeed the OP was looking for conditions on h
> that would allow for the decomposition that he wanted