Solution for AX=B if A is not Square matrix!

> ....
>  r=y(147:151);
>  X2=Hankel(c,r);

If you set q = h - h1, A = X1 and B = X2h2. We are essentially solving
the following problem for q (a set of overdetermined system of
equations):

Aq = B

(since h1 is known, knowing q is equivalent to knowing h). If q' is
the LS solution to the above problem, obviously Aq' will NOT be
exactly equal to B, but will minimize the expression ||Aq - B||^2.
This is best you can go with a LS-type approach. As Fred pointed out,
this can be accomplished using various other norms (general p-norms,
weighted norms etc.,), but you will NOT get an exact answer to this
problem, in general.

The above problem is equivalent to fitting a straight line through a
set of ordered pairs of points. The straight line (found by any of the
solution methods), in general,  will not pass through all the points
(unless they form a straight line themselves).

Hope this helps,

-LL.


> 
> Try the above and see the general definition - and do not hide to solve the
> Eq. I have given - constructing like the above.
> Santosh
> 
> 
> > 
> > > The Eq. should work fine at high C/I (carrier to interference ratio)
> > > but perform very badly at low C/I.
> > 
> > What is your context ? You haven't mentioned what is "carrier" and
> > what is "interference" ? Please fill in the missing information and
> > I'll see if I can help ...
> > 
> > -LL.

There is an excellent coverage of this problem in 
the section entitled "Least Squares Fitting to Data"
of "Elementary Linear Algebra with Applications" by
Anton and Rorres (section 10.15 of my book, which is
the first edition, copyright 1987).

--Randy

santosh.nath@ntlworld.com (santosh nath) wrote in message news:<6afd943a.0309251339.19a0e93b@posting.google.com>...
> Hi all,
> 
> I am stuck with a bit difficult problem. Let us consider the follwing
> linear matrix Eq.
> AX=B
> If A is a square matrix then there are direct and indirect(LU/Cholesky
> etc
> depending on matrix property)  solutions by matrix
> inversion. But My problem involves the following matrix dimensions:
> A=147 x 5 (Hankel matrix )
> X=5 x 1(Number of taps - One dim vector)
> B=147 x 1 (Matched filter output.)
> 
> So inverse or any of the above solutions are not possible and in fact
> there
> is no unique solution for X. There might exist several solutions( More
> Eqs than unknowns) - we need to pick up optimum one.
> 
> Can SVD(singular value decomposition) or some kind of pseudo inverse
> help.
> 
> Please suggest me a solution which can be implemented in DSP processor
> i.e
> computation complexity should not grow high.
> 
> Thanks in advance.
> Santosh